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Full text of "Naval research logistics quarterly"

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NAVAL RESEARCH 

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MARCH 1980 
VOL. 27, NO. 1 




OFFICE OF NAVAL RESEARCH 



NAVSO P-1278 



<9h7~3 



NAVAL RESEARCH LOGISTICS QUARTERLY 

EDITORIAL BOARD 

Marvin Denicoff, Office of Naval Research, Chairman Ex Officio Members 

Murray A. Geisler, Logistics Management Institute Thomas C Varley, Office of Naval Research 

Program Director 

W. H. Marlow, The George Washington University 

Seymour M. Selig, Office of Naval Research 

Bruce J. McDonald, Office of Naval Research Tokyo Managing Editor 

MANAGING EDITOR 

Seymour M. Selig 

Office of Naval Research 

Arlington, Virginia 22217 

ASSOCIATE EDITORS 

Frank M. Bass, Purdue University Kenneth O. Kortanek, Carnegie-Mellon University 

Jack Borsting, Naval Postgraduate School Charles Kriebel, Carnegie-Mellon University 

Leon Cooper, Southern Methodist University Jack Laderman, Bronx, New York 

Eric Denardo, Yale University Gerald J. Lieberman, Stanford University 

Marco Fiorello, Logistics Management Institute Clifford Marshall, Polytechnic Institute of New York 

Saul I. Gass, University of Maryland John A. Muckstadt, Cornell University 

Neal D. Glassman, Office of Naval Research William P. Pierskalla, Northwestern University 

Paul Gray, University of Southern California Thomas L. Saaty, University of Pennsylvania 

Carl M. Harris, Mathematica, Inc. Henry Solomon, The George Washington University 

Arnoldo Hax, Massachusetts Institute of Technology Wlodzimierz Szwarc, University of Wisconsin, Milwauket 

Alan J. Hoffman, IBM Corporation James G. Taylor, Naval Postgraduate School 

Uday S. Karmarkar, University of Chicago Harvey M. Wagner, The University of North Carolina 

Paul R. Kleindorfer, University of Pennsylvania John W. Wingate, Naval Surface Weapons Center, White I 

Darwin Klingman, University of Texas, Austin Shelemyahu Zacks, Case Western Reserve University 

The Naval Research Logistics Quarterly is devoted to the dissemination of scientific information in logistics 
will publish research and expository papers, including those in certain areas of mathematics, statistics, and econom 
relevant to the over-all effort to improve the efficiency and effectiveness of logistics operations. 

Information for Contributors is indicated on inside back cover. 

The Naval Research Logistics Quarterly is published by the Office of Naval Research in the months of March, . 
September, and December and can be purchased from the Superintendent of Documents, U.S. Government Prin 
Office, Washington, D.C. 20402. Subscription Price: $11.15 a year in the U.S. and Canada, $13.95 elsewhere. Co 
individual issues may be obtained from the Superintendent of Documents. 

The views and opinions expressed in this Journal are those of the authors and not necessarily those of the O 

of Naval Research. 

Issuance of this periodical approved in accordance with Department of the Navy Publications and Printing Regulati 

P-35 (Revised 1-74). 



ANALYZING AVAILABILITY USING TRANSFER FUNCTION 
MODELS AND CROSS SPECTRAL ANALYSIS* 

Nozer D. Singpurwalla 

The George Washington University 

School of Engineering and Applied Science 

Institute for Management Science and Engineering 

Washington, D.C. 

ABSTRACT 

If we look at the literature of reliability and life testing we do not see much 
on the use of the powerful methods of time series analysis. In this paper we 
show how the methods of multivariate time series analysis can be used in a 
novel way to investigate the interrelationships between a series of operating 
(running) times and a series of maintenance (down) times of a complex sys- 
tem. Specifically, we apply the techniques of cross spectral analysis to help us 
obtain a Box-Jenkins type transfer function model for the running times and 
the down times of a nuclear reactor. A knowledge' of the interrelationships 
between the running times and the down times is useful for an evaluation of 
maintenance policies, for replacement policy decisions, and for evaluating the 
availability and the readiness of complex systems. 

1. INTRODUCTION AND SUMMARY 

The investigation reported here was undertaken to determine if a stochastic interrelation- 
ship exists between the runnng times and the down times of the Robinson Nuclear Power 
Plant. The data was provided to us by the Probabilistic Analysis Staff of the Nuclear Regulatory 
Commission (NRC). Our goal was to understand what the data were telling us about the rela- 
tionship between the series of running times and the series of down times. One way of achiev- 
ing this goal is to obtain a Box- Jenkins [1] type of "transfer function model" between the run- 
ning times and the down times. The interpretation and uses of a transfer function model for 
the situation considered here are discussed in Section 1.1. 

A first step in the analysis of the data was its careful screening. This was done in order to 
eliminate those observations that were judged to be questionable or that had arisen under 
unusual circumstances. Such observations introduce spurious autocorrelations and cross corre- 
lations, and thus tend to obscure the identification of a simple relationship that may exist 
between the running times and the down times. 

We would like to emphasize that, for an analysis of data of the type discussed here (often 
referred to as "messy data"), an examination and screening of the data prior to model building 
are important preliminary operations. If one neglects to perform these operations, one may 
face the frustrating task of attempting to fit several transfer function models, none of which 
may be satisfactory. 



'Jointly Sponsored by Contract AT(49-24)-021 1 Nuclear Regulatory Commission and Program in Logistics Contract 
N00014-75-C-0729 Project NR 347 020 Office of Naval Research. 



2 N.D. SINGPURWALLA 

In Figures 1.1 and 1.2, we display a time sequence plot of the screened down times X, and 
the corresponding screened running times Y t , t = 1, 2, ... , 28. Note that X x represents the 
first down time, X 2 the second down time, and so on, whereas Y\ denotes the first running 
time, Y 2 the second running time, and so on. Note also that the two plots are not drawn to the 
same scale. In Figure 1.3 we indicate the relative positions of the X^s and the Y,'s, t = 1, 2, 



TIME IN 
DAYS 




SERIAL NUMBER 
OF DOWNTIMES 



Figure 1.1. Time series plot of screened down times 



In transfer function model building observations must be considered in pairs. In our case, 
the paired observations will be 

U 1( YJ, (X 2 , Y 2 ), ... , (X 2i , Y u ). 

In considering the above pairs, we will have to bear in mind that X, precedes Y, in chronologi- 
cal time. 

In Table 1.1 we present the actual values of the screened down times X, and the 
corresponding screened running times Y t , t = 1, 2, .. . , 28. 



If changes in a series of observations Y,, t = 1, 2, . . . , tend to be anticipated by changes 
in another series of observations, say X„ t — 1, 2, . . . , then X, is said to be a leading indicator 
of Y,. In our case it is reasonable to assume that the down times X, are a leading indicator of 
the running times Y,. Based upon this consideration, and together with an analysis of the avail- 
able data, albeit an insufficient amount, the best transfer function model we have identified and 
fitted is given as 



AVAILABILITY USING TRANSFER FUNCTION 




SERIAL NUMBER 
OF RUNNING TIMES 



Figure 1.2. Time series plot of screened running times 



SYSTEM 
STATE 



UP -- 



DOWN P 



Y, 



Y 2 





-A. TIME 



Figure 1.3. State of the system versus time 



N.D. SINGPURWALLA 

TABLE 1.1 — Screened Down Times and 

Running Times of the Robinson Nuclear 

Power Plant 



Down Times 


Prewhitened 


Runnng Times 


Prewhitened 


(days): X, 


Down Times 
(days): a, 


(days): Y, 


Running Times 
(days): /3, 


0.87 


0.07 


22.83 


6.40 


0.26 


-0.54 


15.75 


-1.43 


0.42 


-0.30 


3.50 


-11.55 


0.95 


-0.09 


0.00 


-14.50 


1.13 


0.19 


21.00 


3.88 


0.06 


-0.78 


8.00 


-13.17 


0.19 


-0.47 


11.00 


-5.09 


0.02 


-0.84 


1.00 


-16.82 


0.08 


-0.81 


16.00 


-1.18 


0.06 


-0.86 


101.00 


80.84 


0.40 


... -0.55 


0.58 


27.76 


0.01 


-0.99 


1.00 


2.97 


0.15 


-0.75 


0.83 


-10.46 


0.08 


-0.89 


6.90 


-11.44 


0.17 


-0.80 


0.83 


15.85 


5.48 


4.51 


27.00 


8.86 


2.81 


1.21 


44.00 


22.01 


0.06 


-0.13 


8.00 


11.43 


0.08 


0.13 


17.00 


7.25 


0.98 


0.30 


25.00 


8.48 


0.87 


0.04 


17.00 


-0.57 


0.70 


-0.03 


10.30 


-3.64 


0.63 


-0.11 


11.36 


-3.64 


0.19 


-0.59 


2.81 


-13.93 


4.60 


3.85 


63.00 


47.04 


0.19 


-1.19 


8.00 


-17.05 


0.14 


0.13 


3.90 


-3.48 


0.41 


-0.12 


12.50 


-0.50 



(1.1) 



Y, - 10.630 = 6.6Z, - 0.55^,_!. 



Interpretation and uses of the Transfer Function Model 



Transfer function models are generally used to forecast the future values of a time series 
Y, (in our case the running times) given the previous values of the leading indicator series X, 
(the down times in our case). However, forecasts of the running times based upon the previ- 
ous and present values of the down times, via Equation (1.1), must be used with great caution 
for the following two reasons. First, the occurrence of unforeseen but rare circumstances may 
cause the future running times to be abnormally small (or even large). An example of this is a 
reactor shutdown due to an unforeseen operator error. Second, since Equation (1.1) is based 
on only 28 observations, it may not be too reliable as a model for forecasting. We can, how- 
ever, make several observations of practical interest based upon Equation (1.1). 



AVAILABILITY USING TRANSFER FUNCTION 5 

We first note that previous values of Y t , such as K,_,, K,_ 2 , •• , etc., do not appear in 
Equation (1.1). This implies that the running time history gives us little information about the 
individual future running times; that is, the next running time may be unpredictable from a 
knowledge of the previous running times. However, future running times on the average may 
be estimated from prevous running times. 

An important consequence of Equation (1.1) is that the running times Y, appear to be 
strongly influenced by the immediately preceding down times X,. Since the down times gen- 
erally correspond to maintenance actions, we can make the following conjecture: 

Conjecture: Barring unforeseeable circumstances, and confining ourselves to the limits of the 
observed data, the operating times are, on the average, increased by a factor of about six per 
unit increase in the maintenance times. 

An explanation to support the above conjecture is that the more thorough a job of repair 
that is performed, the longer the next running time becomes. This is perhaps one of the most 
important conclusions that can be reached from our analyses. 

Since the coefficient of —0.55 associated with X,^ is small compared to the coefficient of 
6.6 associated with X„ we will ignore the effect of X,^ on Y,. Even though the transfer func- 
tion model is obtained after an involved analysis, as discussed in the remainder of this report, 
the simplicity of Equation (1.1) suggests that a plot of Y, versus X„ t = 1, 2, . . . , 28, should 
be approximately linear. The actual plot confirms the reasonableness of Equation (1.1), includ- 
ing the values of its coefficients. 

In conclusion, for the situation considered here the transfer function model is more 
efficient as a tool that gives us some insight into the manner in which the system operates, 
rather than as a tool that can give us reliable forecasts of future running times. 

The remainder of this paper is devoted to a discussion of the pertinent details that lead us 
to our model. In Section 2, by way of presenting some aspects of transfer function model 
building, we also introduce some terminology and notation. In Section 3, we present an 
analysis of our data. 

In what follows, we require the reader to have some familiarity with the material in Box 
and Jenkins [1] and with that in Jenkins and Watts [3]. 

2. TRANSFER FUNCTION MODELS AND THEIR ESTIMATION 

Univariate transfer function models as described by Box and Jenkins [1] are models that 
specify the stochastic interrelationships between two time series. They are more general than 
regression models with lag structures on predetermined variables, in that the dependent vari- 
able can also have a lag structure. In addition, the transfer function models can have a super- 
imposed error structure which may be of a very general nature. Engineers often refer to error 
with the term "noise," and "white noise" refers to errors that are independent and identically 
distributed. 

There are two equivalent representations of a univariate transfer function model. One is 
the infinite or reduced form, and the other is the finite form. In the infinite form, the output 
series, say Y, (in our case the running time), is explicitly represented as a function of the input 
series X, (in our case the down time) and its lagged (previous) values; that is, 



6 N.D. SINGPURWALLA 

(2.1) Y, = v + v, AT f _, + v 2 X,_ 2 + . . . + N, , 

where the constants v , Vj, ... are called the impulse response weights. In cases where there is 
no immediate response, one or more of the initial v's, say v , v b ... , v 6 _ b is equal to zero. 
The process N, represents noise, which is assumed to be independent of the level of the input 
series, but is additive with respect to the influence of the input; N, can have any general struc- 
ture. 

It can be shown that an equivalent representation of the model given by Equation (2.1) is 
the following finite form: 

(2.2) Y, - 8, y,_, - ... - 8 r Y,_ r = oi X,_ b - ^i X,_ b _ x - ... - (o s X,_ h _ s + N, , 

where the 8's, the w's, and b are unknown constants. The constant b associated with the lead- 
ing indicator series X, indicates which of the previous values of X, affect the present Y,. In our 
application, the value of b represents the number of previous maintenance times affecting the 
present running time. 

A first step towards estimating the transfer function model is a tentative identification of 
the values of r, b, and s. This can be accomplished by an examination of the estimated impulse 
response weights v k , k = 0, I, 2, ... . A plot of \ k versus k is known as the impulse response 
function. 

There are two general approaches for obtaining the impulse response function. The first 
one, outlined by [1, p. 379], is based on a "prewhitening" of the input series. Prewhitening the 
input series means fitting a time series model to the X, series such that the residuals from the 
model, say a,, are independent and identically distributed random variables with mean zero and 
a constant variance. When the v k are estimated using the prewhitening of the input series pro- 
cedure, their neighboring values tend to be correlated. Thus the graph of the impulse response 
function tends to be misleading. This ultimately affects our ability to obtain a realistic transfer 
function model. We are therefore interested in considering an alternate approach for estimating 
the impulse response weights. 

The second approach for estimating the impulse response weights involves the use of 
"cross spectral analysis". Such an approach removes the difficulties associated with the problem 
of the correlated estimates of v^, and also provides us with some additional insight into the 
nature of the dependencies between the input and output series. These are illustrated at the 
end of Section 3. 

Once the impulse response function is obtained, we can isolate the noise series, N„ by 
using Equation (2.1). Specifically, we estimate the noise series by 

(2-3) A> = ^-vo-v,*,.,-^*,^-.... 

A knowledge of N,, plus a knowledge of the tentative values of r, b, and s, helps us to 
estimate the parameters of the transfer function model, Equation (2.2). One way of accom- 
plishing this is by using the TIMES program package described by Willie [7]. 

The adequacy of the proposed model can be checked by an analysis of the residues from 
the model. The details of such an analysis are given in [1, p. 392]. 



AVAILABILITY USING TRANSFER FUNCTION 7 

3. ANALYSIS OF RUNNING TIME AND DOWN TIME DATA 

Data on the operating history of nuclear reactors are generally available showing dates on 
which the reactors ceased operation and the duration of the stoppage. Among other facts, the 
reasons for the stoppage are also given. Stoppages are categorized according to whether they 
were scheduled or forced. In addition to this, there is a further breakdown indicating whether 
the stoppage was due to equipment failure, testing, refueling, regulatory reasons, operator train- 
ing, administrative reasons, operational error, or other causes. 

3.1 Screening the Data 

The data that were given to us described the Robinson Power Plant's operating history 
from June 1974 through April 1976. These data did not contain any stoppages due to adminis- 
trative reasons, operational error, or other causes; they contained one stoppage for regulatory 
reasons and one stoppage for refueling. In one instance, the data contained an unrealistic 
combination for the cause of stoppage— a scheduled failure. In this case we used our discretion 
to alter it to a forced failure. Whenever there were stoppages due to operator training, these 
were treated as running times rather than as down times. This was done for two reasons. First, 
the duration of each stoppage was very short (on the average about 1/2 hour); second, we 
would like to concentrate on those down times that pertain to the physical operation of the sys- 
tem rather than on those external to it. 

The single stoppage due to a forced regulatory restriction was for a period of 3.67 hours, 
and since it immediately followed a forced equipment failure of 15.52 hours, it was combined 
with the equipment failure stoppage. 

Refueling the reactor takes place annually and is generally of a very long duration. In our 
data, we had only one stoppage for scheduled refueling, and it was of 960 hours duration. 
Since the duration of this stoppage is out of line with the duration of the other stoppages (see 
Table 1.1), it was excluded from consideration. 

We remark here that any time a stoppage (running time), say X,(Y,), was excluded from 
consideration, its corresponding running time (stoppage) Y, (X,) was also excluded. This is to 
ensure that no bias is introduced into the relationship between the two variables of interest 
because of the elimination of observations of either one. 

Another convention followed in our analysis arises from the fact that the raw data show 
the dates and the duration (in hours) of each stoppage, rather than the actual time of stoppage. 
We assume that each down time commenced at 0000 hours (unless in some rare instances there 
is a second stoppage occurring during the same day). Whenever two or more breakdowns 
occurred during the same day, they were combined into One down time period and the inter- 
vening operating period was ignored. 

Because of the paucity of data, we chose not to distinguish between stoppages due to 
equipment failure and those due to testing. This is reasonable because whenever there is a 
forced equipment faiure, maintenance and test actions on other (nonfailed) components are 
routine. Thus, in practice it is difficult to differentiate clearly between the consequences of 
equipment failure and those of testing. This strategy was suggested by some staff members of 
the Probabilistic Analysis Staff at NRC. 

The preliminary screening and examination described above gives us a series of values of 
the down times X, (in days) and a series of values of the corresponding running times Y, (in 
days). However, as we shall soon see, some further screening is necessary. 



8 N.D. SINGPURWALLA 

The next step in our analysis involved prewhitening the X, series. This turned out to be 
quite a frustrating endeavor, since no simple univariate time series model of the Box-Jenkins 
type seemed to provide a reasonable fit. The difficulty turned out to have been caused by two 
unusually large down times due to scheduled testing and forced failure of 18.08 and 25.43 days, 
respectively. These were incompatible with the other down times (see Table 1.1), and thus 
defied the use of a simple model as a prewhitening transformation. Perhaps a time series 
model with an indicator variable (such as those used in the "intervention analysis" of Box and 
Tiao [2]) might have been adequate for these and for the refueling stoppage, but this was not 
attempted. In the interest of expediency, it was preferable to eliminate the two large A^'s and 
their corresponding K/s. Thus, in effect, some data screening was done during the prewhiten- 
ing phase. Table 1.1 presents the 28 screened values of the down times X, and the correspond- 
ing running times Y t . We remind the reader that the subscript / is a sequential index rather 
than an index representing time. That is, X, and Y, are not simultaneously observed in time; 
X, precedes Y,. 

3.2 Transfer Function Model Development 

In Figures 3.1 and 3.2 we show plots of the autocorrelation functions of the (screened) 
down times X, and the (screened) running times Y„ respectively. Based upon these plots we 
are able to conclude that the two time series can be treated as stationary [1, p. 174]. 



ESTIMATED 
AUTOCORRELATION 



12 3 4 5 6| 7 



^A 



Figure 3.1. Estimated autocorrelations 
of down times X, 



In Figure 3.3 we show a plot of the logarithm of the smoothed estimator of the power 
spectrum of the screened down times X„ as well as the approximate 95% confidence limits. 
The smoothing was performed using a rectangular window of band width .098. As a matter of 
fact, all the smoothing that is discussed in this paper was performed using a rectangular window 
of band width .098. The power spectrum curve shows us how the variance of the X, time series 
is distributed with frequency. For a detailed understanding of the power spectrum, its smooth- 
ing, and the band width of a smoothing window, we refer the reader to Chapter 6 of Jenkins 
and Watts [3]. 



AVAILABILITY USING TRANSFER FUNCTION 



ESTIMATED 
AUTOCORRELATION 



1.0] 
































0.1 




























4 




1 


2 


3 


4 


1 


1 


' 8 


•0.1 


; 














0.2 
















J 


' 


: r 










-i n- 

















Figure 3.2. Estimated autocorrelations of running 
times Y, 



-1.200 



-3.200 



95% UPPER 
CONFIDENCE 

LIMIT 




95% LOWER 

CONFIDENCE 

LIMIT 



FREQUENCY 



0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 



Figure 3.3. Log 10 of the smoothed estimator of the spectrum of down limes versus frequency, using 

rectangular window of band width 0.98 



10 



N.D. SINGPURWALLA 



Our next step involves the determination of a suitable prewhitening transformation for 
the down times X,. Based upon the several models that we attempted, we conclude that a mov- 
ing average process of order 3 best describes the X, series. Specifically, we find that 



(3.1) 



X, - 0.7963 = a, + 0.117a, _, - 0.189a, _ 2 - 0.133a, _ 3> 



where .7963 is the mean of the X, series. The a, represent the residuals when a moving aver- 
age process of order 3 is fitted to the X, series. If the prewhitening transformation given by 
Equation (3.1) is correct, then the a, will be independently and identically distributed with a 
constant mean and variance. 

In Table 1.1 we give the values of the a,'s. In order to verify the appropriateness of the 
model given by Equation (3.1), we plot the estimated autocorrelation function and the 
estimated power spectrum of the a, series. These plots are given in Figures 3.4 and 3.5, respec- 
tively. We remark that the plot of the estimated power spectrum of the a, series given in Fig- 
ure 3.5 is relatively constant as compared to the plot of the estimated power spectrum of the X, 
series given in Figure 3.3. This is because the effect of prewhitening is to remove the depen- 
dencies among the X^s and give us a set of independent a,. Figures 3.4 and 3.5 confirm the 
appropriateness of the prewhitening transformation given by Equation (3.1). 



ESTIMATED 
AUTOCORRELATION 



. STANDARO DEVIATION 



V 



Figure 3.4. Estimated autocorrelations of residuals 
from the model used for prewhitening X, 



Following Box and Jenkins [1, p. 380], we next apply the same prewhitening transforma- 
tion (i.e., the one given by Equation (3.1)) to the running times Y, and obtain the j8,'s as resi- 
duals. In Table 1.1 we give the values of /3, under the heading "Prewhitened Running Times." 
In Figure 3.6 we show a plot of the smoothed estimator of the power spectrum of (3 t . We 
remark that except at the very low frequencies, the plot of the estimated power spectrum of the 
0,'s ' s fairly constant. Thus it appears that the prewhitening transformation given by Equation 
(3.1), when applied to the running times Y,, also yields a sequence of independent and identi- 
cally distributed random variables /3,. 



AVAILABILITY USING TRANSFER FUNCTION 



11 



-1 600 
-1.650 

- 1 800 



-2.400 
-2.550 



95% UPPER 

CONFIDENCE 

LIMIT 





95% LOWER 

CONFIDENCE 

LIMIT 



0.050 0.100 0.150 0.200 0.250 0.300 350 400 450 0.500 FREQUENCY 



Figure 3.5. Log| of ihe smoothed eslimator of the spectrum of the prewhitened down times versus frequency, 

using rectangular window of band width .098 



0.900 




95% LOWER 

CONFIDENCE 

LIMIT 



FREQUENCY 



0050 0.100 150 200 0.250 300 0.350 400 450 0.500 



Figure 3.6. Log| of the smoothed eslimator of the spectrum of B, versus frequency, using a 
rectangular window of band width .098 



12 



N.D. SINGPURWALLA 



Our next step is to obtain the cross correlation between the a, and 0, at lags k, k — 0, 1, 
2, . . . . If 5„ and s^ denote the estimated standard deviations of the a, and the /3, series, 
respectively, and if r a(i (k) denotes the estimated cross correlation between the a, and the j8, at 
lag A:, then v*., an estimate of v^, is 



v t = r 



afi 



(k) 






k =0, 1, 2, 



(see Box and Jenkins [1, p. 380]). 



In Figure 3.7 we show a plot of the impulse response function; that is, a plot of v k versus 
k, k = 0, 1, 2, .. . . We remark that in Figure 3.7, the value v is significantly larger than the 
other values of v k , k — 1, 2, . . . . 



ESTIMATED 
IMPULSE 
RESPONSE 
WEIGHT 



2.0 
0.0 
-2.0 
-4.0 



J— T 



Figure 3.7. The impulse response funeiion using 
i he prewhitening of the input series 

Because the neighboring values of the v k in Figure 3.7 tend to be correlated, we also 
obtained the impulse response function using the "cross spectrum" between the X, and the Y, 
series (see Jenkins and Watts [3, p. 424]). In Figure 3.8 we show a plot of the impulse 
response function using the cross spectrum. Note that this plot is quite similar to the one given 
in Figure 3.7; that is, v is significantly larger than the other values of v k . Based upon Figures 
3.7 and 3.8, we can conclude that the greatest influence on the running time is exerted by the down 
time immediately preceding it. This is, of course, a major point of our conclusions. 

We now estimate the noise series N, using Equation (2.3). An inspection of the 
estimated autocorrelation function of the estimated noise series TV,, and a "portmanteau lack of 
fit test" (1.69 with 8 degrees of freedom) [1, p. 290], lead us to conclude that the N, are 
independent and identically distributed. In addition to the above, we show in Figure 3.9 a plot 
of the smoothed estimator of the power spectrum of the N,. Here again, except at the very low 
frequencies, the estimated power spectrum of the N, series is fairly constant. Thus the 
estimated noise series N, can be described reasonably well by a white noise process. 



A final step in the analysis involves the fitting of a transfer function model to the running 
times Y,. This was accomplished by using the TIMES program package. Of the several models 
that were attempted, the model 



AVAILABILITY USING TRANSFER FUNCTION 



13 





















6.0 




































3.0 


















1.5 


I 






! 


I 













5 


1 




10 


1 


1 


1 1 


1.6 


















30 



















Figure 3.8. The impulse response function using cross speclral analysis 



95% UPPER 

CONFIDENCE 

LIMIT 




0050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 



FREQUENCY 



Figure 3.9. Log !0 of ihe smoothed estimator of the spectrum of the estimated noise series, using a 

rectangular window of band width .098 



14 



N.D. SINGPURWALLA 



(3.2) Y, - 10.630 = 6.6*, - 0.55*,_, 

appears to be the best; 10.630 is the mean of the Y, series. 

In order to verify the reasonableness of the model, two diagnostic checks were suggested 
in [1]. One depends on the autocorrelation of the N, and the other depends on the estimated 
cross correlation between the N, and the a,. For both cases a portmanteau lack of fit test was 
used. In the former case, the test statistic is 1.69 with 8 degrees of freedom, whereas in the 
latter case the test statistic is .698 with 6 degrees of freedom. These values support the reason- 
ableness of the proposed transfer function model. 

As stated earlier, a cross spectral analysis of the X, and the Y, series can give us further 
insight into the nature of the dependencies between the two series. For example, G(f), the 
"gain" at frequency /, behaves like the regression coefficient in a linear regression model 
between the output and the input at frequency /(see Jenkins and Watts [3, p. 352]. In Figure 
3.10 we plot the gain of the running times on the down times at various frequencies. Another 
important function in cross spectral analysis is the "squared coherence" between the input and 
the output at frequency / This quantity measures the correlation between the sinusoidal com- 
ponent of Y, and that of X, at frequency /. The square coherence is also in some sense a meas- 
ure of the proportion of information in the Y, series that is attributable to the X, series. For 
more information on the coherence and the coherence spectrum, we refer the reader to Jenkins 
and Watts [3, p. 352]. In Figure 3.11 we show a plot of the coherence for the running time and 
down time data. Figures 3.10 and 3.11 give us some additional assurance on the dependence of 
the running times on the down times. 



95% UPPER 

CONFIDENCE 

LIMIT 




95% LOWER 

CONFIDENCE 

LIMIT 



FREQUENCY 



0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 

Figure 3.10. The gain versus frequency of running limes on down times 



AVAILABILITY USING TRANSFER FUNCTION 



15 




4 FREQUENCY 



0.060 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 

Figure 3.11. The coherence versus frequency of running limes and down times 



4. SUMMARY AND CONCLUSIONS 

In the foregoing analysis we have demonstrated the use of time series analysis methodol- 
ogy for studying the interrelationships between the maintenance times and the running times of 
a complex system. Our analysis enables a decision maker to assess the impact of his mainte- 
nance policies on running times, or to influence the operating times by managing the mainte- 
nance times. In addition, given a down time a decision maker can, to a limited extent, forecast 
the next running time. This type of information may be very valuable, especially for large and 
complex systems. 

Our analysis can be criticized on the grounds that it is based on an insufficient amount of 
data. We hope that this criticism can be overlooked in the light of the fact that our approach is 
to be viewed as a prototype for the analysis of reliability data involving two interrelated sources 
of data. For other uses of the time series analysis methodology for reliability and life data, we 
refer the reader to Singpurwalla [4], [5] and [6]. 

ACKNOWLEDGMENTS 



The written comments by Drs. Ray Waller and Gary Tietjens, and the helpful conversa- 
tions with Professor Robert Shumway and Mr. Randall Willie, are much appreciated. Mr. 
Mahesh Chandra's efforts on behalf of the work reported here is also acknowledged. Finally, 
the several comments by Professor Frank Proschan with respect to the direction and the 
emphasis of this report are gratefully acknowledged. 



16 N.D. SINGPURWALLA 

REFERENCES 

[1] Box, G.E.P. and G.M. Jenkins, Time Series Analysis, Forecasting, and Control, revised edition 
(Holden-Day, San Francisco 1976). 

[2] Box, G.E.P. and G.C. Tiao, "Intervention Analysis with Applications to Economic and 
Environmental Problems," J. American Statistical Association 70, 70-79 (1975). 

[3] Jenkins, G.M. and D.G. Watts, Spectral Analysis and its Applications (Holden-Day, San Fran- 
cisco 1968). 

[4] Singpurwalla, N.D. "Time Series Analysis and Forecasting of Failure Rate Processes," Relia- 
bility and Fault Tree Analysis (R.E. Barlow, J.B. Fussell, and N.D. Singpurwalla, Eds.) 
Society for Industrial and Applied Mathematics (1975). 

[5] Singpurwalla, N.D. "Time Series Analysis of Failure Data. Proceedings," Annual Symposium 
on Reliability and Maintainability, pp. 107-112 (1978). 

[6] Singpurwalla, N.D. "Estimating Reliability Growth Using Time Series Analysis," The Naval 
Research Logistics Quarterly, Vol. 25, No. 1, pp. 1-14 (1978). 

[7] Willie, R.R. "Everyman's guide to TIMS," ORC 77-2, Operations Research Center, Univer- 
sity of California at Berkeley (1977). 



OPTIMAL MAINTENANCE-REPAIR POLICIES 
FOR THE MACHINE REPAIR PROBLEM 

S. Christian Albright 

Department of Quantitative Business Analysis 

Graduate School of Business 

Indiana University 

Bloomington, Indiana 

ABSTRACT 

We consider a model willi M + N idenlical machines. As many as N of 
these can be working ai any given time and the others act as standby spares. 
Working machines fail at exponential rate \, spares fail at exponential rale y, 
and failed machines are repaired at exponential rale /x The control variables 
are \. m. and the number of removable repairman, S, to be operated at any 
given lime. Using ihe criterion of total expected discounted cost, we show that 
A..V. and tx are monotonic functions of the number of failed machines M, V, 
ihe discount factor, and for the finite lime horizon model, the amount of lime 
remaining. 

1. INTRODUCTION 

In this paper we examine the structure of the optimal maintenance and repair policies for 
an exponential repair model. We assume that there are M + /V identical machines, /Vof which 
can be working at any given time. The others function as standby spares. The working 
machines fail at rate A, and the failed machines are repaired at rate /a. We also include the pos- 
sibility of spare machines failing at rate y. Our control variables are A, which can be thought of 
as a maintenance control variable, ti, and the number of removable repairmen, S. Our princi- 
ple results are that A decreases and S and ti increase as more machines are in the failed state. 
(Throughout this paper, we use the terms increasing and decreasing to mean nondecreasing and 
nonincreasing, respectively.) We also show how these control variables behave as functions of 
M, N, the discount factor, and the amount of time remaining. 

Our basic model has been studied by several researchers. Some of these have described 
the model and given its operating characteristics under exponential and nonexponential assump- 
tions. See, for example, Barlow [3], Gnedenko, Belyayev, and Solovyev [8], and Iglehart and 
Lemoine [11], [12]. There has been less literature on the control of these models. Goheen [9] 
and Cinlar [4] study the problem of where to send failed machines when there are several 
repair facilities. Cinlar assumes exponential times and reduces the problem to a linear program- 
ming problem, whereas Goheen assumes Erlang times and reduces to the problem to a 
mathematical programming problem. 

More in line with the model presented in the current paper are the papers by Crabill [5] 
and Winston [18]. Each considers a single-repairman system (and Winston assumes no spares) 

17 



18 S.C.ALBRIGHT 

which can be operated at a finite number of service rates, and each shows that the optimal ser- 
vice rate is an increasing function of the number of failed machines. Crabill does this with a 
long run average cost criterion by means of tedious manipulations of policy improvement equa- 
tions. Winston uses a discounted cost criterion, and he builds up his infinite horizon model as 
a sequence of discrete time ^-period problems. He then analyzes these quite easily be means of 
induction on n. This type of approach is also used in the present paper although our ^-period 
model is defined differently than Winston's. This approach allows us to obtain more results, 
and to do so by simpler means, than was evidently possible for Crabill. 

2. THE BASIC MODEL 

In this section we present the basic model and main results. The system we will study is 
composed of /V + M identical machines. As many as A 7 of these can be working simultaneously 
in parallel, while the rest function as warm standby spares. Each of the machines which is 
currently working fails, independently of the others, with exponential rate \. Whenever one of 
these machines fails, it is immediately replaced by a spare if any is available. We also assume 
each of the available spares fails, independently of the other machines, with exponential rate y. 
(This is the rationale behind the term "warm standby".) There is a repair facility with as many 
as N + M removable servers in parallel. Whenever a working machine or a spare fails, it is 
sent to this facility, where it is served on a first-come-first-served basis with exponential rate fx. 

The costs and rewards are as follows. Whenever j machines, ^ ;4 A 7 , are working, 
revenue is earned at a rate /;(./ ), where // is an increasing function with li(0) = 0. The service 
cost for having S servers each working at rate fx is Sf(fx), where /is a nonnegative increasing 
function of /x. Also, because we will wish to control A, we assume the cost of having j 
machines working, each subject to failure rate A, is ,y#(A). Here # is a decreasing function of A. 
and can be thought of as a maintenance cost. Finally, we will continuously discount all reve- 
nues and costs by a discount rate a > 0. Our objective is to maximize expected discounted 
rewards over an infinite horizon. The decision variables at any decision point are the number 
of servers S, the service rate per service fx, and the failure rate per working machine A. These 
must satisfy 0<S^/V+ /V/, 0< A mm ^ X < \ max < oo, and < /x min < fx < fi mSiX < ». 
We assume the failure rate for spares, y, is a fixed constant. 

The problem is formulated most naturally as a continuous time Markov decision model. 
The state space is {0, 1, 2, . . . , N + M }, where state / means that / machines are currently in 
repair or waiting for repair. The actions, transition rates, and reward rates are easily obtained 
from the above description of the model. For example, in state /', let N, and M, be, respec- 
tively, the number of working machines and the number of available spares. Then N, = 
(N + M — i) A A 7 and M, = V (M - /), where "A" and "V" stand for "min" and "max". This 
implies that if we are in state /and use actions S, A, and /x, the transition rates q,j(S, A, fx) are 
q,, + \(S, A, (x) = A 7 , A. + M,y, q,,_\(S, \, /x) = (S A /)/u, and ^(S, k, /x) = 
-A^A. - M,y - (5 A i)fi. The reward rate in state / is r,(S, X, fx) = h(N,) - N,g(\) — 
Sf(fx). Because of the exponential assumptions and infinite time horizon, we may assume that 
decisions are made only at failure times and at service completion times. 

Let V(i) be the maximal expected discounted reward which can be obtained when starting 
in state /. Then it follows from well-known results (see, for example, Howard [10] or Ross 
[16]) that V(i) satisfies a certain functional equation and that the solution to this equation 
yields the optimal actions to use in state /. In this paper, however, we are interested in struc- 
tural properties of the optimal actions and, in particular, how they behave as functions of /. To 
discover these properties, the above functional equation is unfortunately not very useful. 



OPTIMAL MAINTENANCE/REPAIR POLICIES 19 

Instead, it is better to build up our infinite horizon problem as a sequence of /7-period discrete 
time problems. This enables us to use induction arguments (on n) to prove the desired struc- 
tural properties of the optimal actions. 

To this end, we define a finite horizon discrete time Markov decision model which in the 
limit (on the number of periods) is equivalent to our original problem in the sense that each 
gives the same optimal expected rewards and each prescribes the same optimal actions. This 
method of using a discrete time problem to prove structural properties of a continuous time 
problem was first utilized by Lippman [14], and has since been discussed and used by several 
other authors [1], [7], [13], [17]. An especially lucid description of the relationship between 
the two problems may be found in Anderson's thesis [2]. Because the method is by now quite 
well-known, we will only sketch the procedure. The reader is referred to the above references 
for further details. 

Let A = A/X max + My + (N + M)/x max , so that A ^ —Qa(S, k, //.) for all states i and 
actions S, X , /x. The possible states and actions of the discrete time process are the same as for 
the continuous time process, but the transition probabilities, rewards, and discount factor are 
modified as follows. The transition probabilities are p iJ+ \ (S, k, (x) = q ii+ \(S, k, fx)/A = 
(Njk + A/,y)/A, p u -\(S, k, /x) = qjj-\(S, X, (x)/A = (SAO/i/A, and p u (S, X, fi) = 
(A + q tl (S, k, fx))/A = (A - Njk — M,y — (S A /')//, )/A. The one-period rewards are 
J,(S, k, fx) = r,(S, k, fx)/(A +a) = (h(Nj) - N/giX) - Sf(jx))/(A +a), and the discount 
factor is/8 = A/ (A + a). 

Let V„(i) be the maximal expected discounted reward over n periods for the above 
discrete time process. Then it follows from the previously referenced results that V„(i) con- 
verges to V(i) and the optimal actions for the w-period problem converge to the optimal actions 
for the continuous time infinite horizon problem. Furthermore, it is easy to see that V„(i) 
satisfies the following recursive equation: 

V„(i) = max {/„(/, S, k, /x)}/(A + a), 

S, K. fJL 

where K (/) = 0, and 

/„(/, S, k, fx) = r,(S, k, fi) + (N,k + M,y) V n _ x {i + 1) 

+ (S A ftfi V n _ x (i - 1) + (A - N^ - M,y - (S A i)fi) K„_,(/). 

By letting A„(/) = K„(/) — y„(i — 1), we may rewrite J„ in the more useful form: 

/„(/, 5, k, tx) = h(N,) + Nji-gik) + XA„_,(/ + 1)) -I- M,yA n _ x {i + 1) 

- Sf(fi) -(S A /)^A H _,(i) + A ^_i(/). 

Since f(fx) ^ 0, it makes no sense to have S > i. Therefore, S A i = S and the possible 
action space in state / is ^ S ^ /, \ min < k < X max , Mmin ■< n < ^ ma x- 

We will assume sufficient continuity conditions on /and g to insure that the above max- 
imum is obtained. With this in mind, let S n (i) be optimal in state /, let X„(/) be the largest 
optimizing action in state /, and let fx n (i) be the smallest optimizing action in state /, all when n 
periods remain. More concisely, we will say these actions are optimal for (n, ft. Our main 
result below is that S n (i) and /x n (i) are increasing functions of /and X„(/) is a decreasing func- 
tion of /. We also show how these optimal actions behave as functions of n, a, M, and N. Our 
first observation follows easily (and we immediately see the benefit of this particular discrete 
time formulation). From the above form of J n it is clear that 

(i) k„(i) maximizes -g(k) + XA„_,(/ -I- 1), 



20 ' S.C. ALBRIGHT 



(ii) /*„(/) minimizes /(/u.) +/liA„_ 1 (/), 

if /(/*„(/)) +/*„(/) A.-iO") > 
if f(fi„(i)) +M„(')A„_ 1 (/) < 0. 



(iii) S„(/) = 



We are now ready for the results of this section. 

PROPOSITION 1: If h(N,) - h(N l+x ) > g(\ mm ) for / ^ M, then A„(/) ^ for n ^ 0, 
1 " < / < M + N. 

PROOF: We use induction on n. For n = 0, A (/) = 0, so the result is trivial. Assum- 
ing A„_i(/) ^ 0, let (S\, \\, /u.]) be the optimal actions for («, /) and (S 2> X 2 , ^2) be the 
optimal actions for (n, i — 1), and we have 

(A + «)A„(/) = /„(/, S,, A.,, /x\) - J„(i - 1, S 2 , X 2> M2) 

< /„(/, 5,, X,, /a,) - J„(i - 1, S 2 , X,, ^i 2 ) 

= /*(#,) - //(#,_,) - (ty - ^,_,)^(Xj) + UV,X, + A/,y)A / ,_ 1 (/ + 1) 
- S,/W + S 2 (f(fi 2 ) + /x 2 A„_,(/ - 1)) 
+ (A - ty_,X, - M,_,y - S , 1 /*i)A B _i(/) 

< Ji(Ni) - //(/V,_,) - (A/, - ^,_,)g(X,) + S 2 (f(fi 2 ) + m 2 A„^(/ - D), 

by the induction hypothesis and definition of A. Also, if / ^ M, //(A 7 ,) — h(N,_\) — 
(N, - N,_ x )g(X : ) = 0, and if 1 > M + 1, //(A,) - /;(#,_,) - (AT, - Af,_,)g(X,) = //(A,) - 
hiN,-^ + g(\\) ^ 0, by the assumption of the proposition. Finally, either S 2 = or f(fx. 2 ) + 
/i 2 A„_i(/' — 1) < 0, so that S 2 (/(/i 2 ) + /Lt 2 A„_|(/ — 1)) ^ 0. This completes the proof. 

THEOREM 1: If &(ty) - h(N i+l ) > #(X max ) for / > M, then A„ +1 (/) < A„(/) for n > 
0, 1 < / < M + N. 

PROOF: For n = 0, we need A,(/) ^ 0. We have 

(A +«) K,(i) = max {//(A,) - N,g{k) - Sf(fi)) = h(N,) - N,g(k m J, 

S.K.fJL 

so that A,(/) = for / ^ Mand A,(/) = h(N,) - //(A,_,) + £(X max ) ^ for 1 > M. Now 
assume A„(/) < A„_](/) for each /, and let (S\, Xi, /jl\) and (S 2 , X 2 , /a 2 ) be optimal for 
(« + 1, /) and (n, / — 1), respectively. Now we have 

(A + a)(A„+,(/) - A„(/)) ^ /„+,(/, S lf X,, fi x ) - j„{i, S v X,, m) 

- J n + X {i - 1, S 2 , X 2> fi 2 ) + J„0 ~ 1. S 2 , X 2 , M 2 ) 

= (/V,X, + A/,y)(A„(/ + 1) - A„_,(/ + 1)) 

+ 5 2 /x 2 (A„(/ - 1) - A„_,(/ - 1)) 

+ (A - W,_,X 2 - M,_,y - SvixH&.il) - A„_,(/)) < 0, 

by the induction hypothesis. 

COROLLARY 1: If h(N,) - h(N l+l ) ^ #(X max ) for / ^ M, then X„(/) is decreasing in n 
for ^ / < M + N — I, and ix„{i) and S„(i) are increasing in n for 1 < / < M + A 7 . 

PROOF: If X f is the largest value of X which maximizes the function k{\\ c) = — g(k) + 
Xc, then it is easy to show that X ( is increasing in c. If we let c = A„(/ + 1), the monotonicity 



OPTIMAL MAINTENANCE/REPAIR POLICIES 21 

of X„(/) in n follows from Theorem 1. The proof that fi„(i) increases in n is similar. To show 
that S„(i) increases with n, it suffices to show that f((x„(i)) + /u„(/)A„_|(/') decreases with n. 
We have 

Afi n+l (i)) +fi n+l (i)b„(» < /WO) +/*„'(/) A„(/) 

< An„(i)) +m„(/)A„_ 1 (/), 

from Theorem 1 and the definition of /li„ +1 (/). 

Corollary 1 states that when there is more time remaining, the system works harder, in 
the sense of more maintenance (lower X) and more repair (higher 5 and fx). The next theorem 
and its corollary show that for a given /?, the system works harder when more machines are bro- 
ken down. From now on (except for Theorem 3 and Corollary 3), we will need to assume that 
h is a linear function, so that h{j) = jr, < j ^ N. 

THEOREM 2: Assume g is convex, differentiable, and satisfies — r + g(X min ) — 
s'UmmUmin < for i > M. Then A„(/ + 1) < A„(0 for n > 0, 1 < / < M+ N- 1. 

PROOF: Before proceeding, we note that the above conditions imply h(N,) — h(N l + i ) ^ 
g(\ min ) for / ^ M, so that the results of Proposition 1 and Corollary 1 are in effect. This fol- 
lows since g' < 0. 

For n = 0, the result is trivial. Now assume inductively that A„_i(/ + 1) < A„_,(/), 1 
< i < M + N- 1 and that r - g(\„(i)) + \„(/)A„_,(/ + 1) > for / ^ M + 1. 

First we show that A„(/' + 1) ^ A„(/). The difference (A + a) (A„(/ + 1) - A„(/)) is 
the sum of four expressions: the terms with rand X, the terms with y, the terms with Sand /x, 
and the terms with A. We look at each of these separately. 

(i) (r,X terms) = A - N i+l (r - g(\„(i + D) + \„(i + i)A /; _,(/' + 2)) 

- 2N,(r - g(X„(0) + X„(/)A„_,(/ -I- 1)) 

+ tf,_,(i- - g(\„0 - 1)) + \„(i - 1)A„_,(0). 

If / < M - 1 , 

/I < N\„0 + 1)(A„_,(/ + 2) - A„_,(/ + 1)) 

- N\„(i - 1)(A„_,(/ + 1) - A„_,(/)) 

< -N\„(i - 1)(A„_,(/ + 1) -A„_,(0). 

If / ^ M + 1 , 

/I ^ W, + 1 X„(/ + 1)(A„_,(/ + 2) - A„_,(/ + 1)) 

- N,_ x k n (i - 1)(A„_,(/ + 1) -A„_i(/)) 

< - N^O - 1)(A„_,(/ + 1) - A„_,(/)), 
since 2N, = A/ /+1 + #,_,. 
If / = M, 
A =(N - 1)0- - s(X„(M + D) + X„(M + l)A fl _!(M + 2)) 

- 2N(r - g(\„(M)) + X„(A/)A„_,(A/ + 1)) 



22 



S.C. ALBRIGHT 



+ N(r - g{k„{M - 1)) + \„(M - 1)A„_,(M)) 
< WA„(A/ + l)(A /f _,(Af + 2) - A„_,(M + 1)) 
- N\ n (M - 1)(A„_,(A/ + 1) - A„_,(A/)) 
^ -Nk„(M - 1)(A„_,(A/ + 1) - A„_,(M)), 
where the first inequality follows from the induction assumption. 

(ii) (y terms) = A/ /+1 yA„_,(/ + 2) - 2A/,yA„_,(/ + 1) + M,_,yA„_,(/) 

if / > M + 1 

yA„_,(/) < if / - M 

(M, - 1)7(A„_,(/ + 2) - A„_,(/ + D) 

- (M, + l)y(A„_,(/ + 1) - A„_,(/)) 
< - (Mi + DyCA.-iO + 1) - A.-id')) if / < M - 1. 

(iii) (S, ix terms) = B = -S„(/ + 1)(./V„0 + 0) + fi n (i + 1)A B _,(/ + 1)) 
+ 25 fl (/)(/0* fl (0) +^„(/)A„_ 1 (/)) 

- s„(/ - 1)(/W' - 0) + /*„(/ - l)A„_,(/ - D). 

From the induction assumption, it follows as in Corollary 1 that S„(i) is increasing in /. There- 
fore, we examine expression B by looking at the following four cases. 

CASE 1. S„{i + 1) = 0. Then B = 0. 

CASE 2. S„(i + 1) = / + 1, S„(/) = 0. Then 
B = -(/ + 1)(./V„0 + D) + (i a (i + 1)A„_,0 + D) 
= -(/ + l)ji„(/ + 1)(A„_,(/ + 1) - A„_,(/)) 
- (/ + l)C/(/*„(/ + D) + fi n (i + l)A fl _j(/)) 
^ -(/ + i)At„0 + 1)(A„_,(/ + 1) - A„_i(/)) 
-(/ + 1)(/W'» +/i H (/)A,_ 1 (/)) 

< -(/ + 1) M „(/ + 1)(A„_,(/ + 1) - A.^C/)). 
since 5„(/) = implies f(fi„(i)) + (i„(i)b„-i(i) > 0. 

CASE 3. S„(/) = /, S„(/ - 1) = 0. Then 
B ^ -(/ + !)/*„(/ + 1)(A„_,(/ + 1) - A„_,(/)) 
+ (/ - Dili,* ,,(')) + /* I1 (/)A /I _,(/)) 

< -(/ + l)/i fl (/ + l)(A fl _!(/ + 1) - A„_,(/)), 
since £„(/) = / implies /(/*„(/')) + /*.„(/) A„_,(/) ^ 0. 

CASE 4. S„0 - 1) = / - 1. In this case 
5 < -(/ + l)/i„(/ + 1)(A„_,(/ + 1) - A„_,(/)) 
+ (/- 1) M „(, -1)(A„_,(/) -A„_,(/ -1)) 

< -(/ + !)/*„(/ + l)(A„_i(i + 1) - A„_,(/)). 






OPTIMAL MAINTENANCE/REPAIR POLICIES 23 

(iv) (A terms) = A(A„_|(/ + 1) — A„_](/)). The only thing left to do now is to combine the 
above "leftover" terms, all negative multiples of b„-\(i + 1) — A„_,(/), with this A term 
to establish the result. 

We finally show that r - g(k, l+] (i)) + A, I+ , (/) A„(/ + 1) > for / > Af + 1 to establish 
the induction. We have, for / ^ M + 1, 

A„ + 1 (/)A„(/ +1) = U„ + 1 (/)/(A + «)) [N i+[ (r - g(k„(i + 1)) 

+ \„(i + l)A„_i(/ +2)) -S„(i + 1) (/(/*„(/ + D) +/*„(/ + 1) A„_!(/ + 1)) 

+ S„(i)W(fjL„(i)) + fi n U)A„-i(i)) - N,(r - g(k n (i)) 

+ \„(/)A„_i(/ + 1)) + AA„_,(/ + 1)]. 

By induction, r — g(k„(i + D) 4- A.,,0' + l)A„_i (/ + 2) ^ 0. By an argument similar to 
above, it is easy to show that the combined terms with S and /x are non-negative. Therefore, 
we have 

\, ; + 1 (/)A„(/ + 1) > (A II+1 (/)/(A + a)) [-N,(r - g(k„(i)) 

+ (A- N,\„(i)) b,,^/ + 1)] 

> (X„ +1 (/)/(A + a))[-W,(r - g(\„(/)) 
+ (A - fyA„(/))(-r + g(\ w (/)))/A„(/)] 

= (A/(A + «))\„ +1 (/)[(-a- + g(k n (i)))Jk„(i)] 

For fixed /, consider the graph of the function k(k) = —r + g(k). By the condition of 
the theorem, the linear extension of this graph from A. min back to A. = stays nonpositive. If 
we recognize k(k)/k as the slope of the line from (0, 0) to (k, k(k)), then the convexity of g 
and the fact that k„ +] (i) ^ X n (i) imply that /c(\„ + ,(/))/\„ +1 (/) < k(k n (i))/k„(i). There- 
fore, 

A„ +1 (/)A„(/ + 1) > (A/ (A +«))(-/• + g(k ll+l (i))) 

> -r + g(k„ +x (i)), 
and the proof is complete. 

The following corollary follows from Theorem 2 exactly as Corollary 1 follows from 
Theorem 1. 

COROLLARY 2: Given the same conditions as in Theorem 2, A.,,0) is decreasing in i 
and S„(i) and /&„(/) are increasing in /. 

Before proceeding, we remark that the differentiability of g assumed in Theorem 2 is 
obviously not necessary. All we need is that a linear (or convex) extension of g from \ min back 
to k = remains sufficiently small, and this condition is most easily stated in terms of g' when 
g'exists. 

We now examine how the optimal actions behave as functions of «, A/, and N. The addi- 
tional subscripts will show the dependence of the various quantities upon these parameters. 

THEOREM 3: If h(N,) - h(N l+l ) > g(k mn ) for / ^ A/, then A„ a| (/) ^ *„,«,(/) for 

discount factors «, ^ a 2 , and all /? $s 0, 1 < / < A/ + N - 1. (In this theorem and the fol- 
lowing corollary, // need not be linear.) 



24 S.C. ALBRIGHT 

PROOF: For n = 0, the result is trivial. Assuming it is true for n — 1, let V na (i) = 
J,M $,.«(/) ._X B .«(/), /*„.«(/)) and_write A,,, M (/) = K„ „(/) - V„Ji-\). Then_A„,' (M (/) - 
A,,„ 2 (/) = A„ ai (/)/(A + «,) - A,,„ 2 (/)/(A +a 2 ). If we can show that > *«.«,(/) ^ 
A„ „ (/) for each /, then the result will follow since 1/(A + «,) ^ 1/(A + a 2 ). But A„ „(/') ^ 
from Proposition 1, and A„ i(¥ (/) ^ A„ ( , (/) by a proof almost identical to the proof of 
Theorem 1. 

COROLLARY 3: If li(N,) - h(N l+1 ) > g(k m J for / ^ A/, then \„ „(/) is increasing in 
a and S„ ,„(/) and (i„ a (i) are decreasing in a. That is, the more the future is discounted, the 
less the system works. 

The following theorem and its corollary say that if we compare two systems in states 
which have the same number of working machines, the only difference being that one system 
has an additional spare, then the system with less spares will work harder. 

THEOREM 4: Consider two systems which are identical in every respect except that one 
has M spares while the other has M + 1 spares. Assume that the conditions of Theorem 2 
hold for each system. Then A„ M (i) ^ A„ M+ \{i + 1) for n ^ 0, 1 ^ / < M + N, if A is 
redefined as A = N\ max + {M + \)y + (N + M + l)/u max for each system. 

PROOF: We only sketch the proof, since the details are similar to those in Theorem 2. 
First, notice that N lM — N l+]M+l and M lM = M ( + , w+1 , since the only difference between 
state / in the first system and state / + 1 in the second system is the one extra machine in 
repair. This observation allows us to group the \ and y terms of A„ w (/) — A nM+] (i + 1) in 
the obvious manner, while the r terms cancel. Finally, for the S, //. terms, the observation that 
5„.a/(/' - 1) = / - 1 or S„ , M+ \(i + 1) = / + 1 implies S„ M (i) = i and that S„ M+] (i) = / 
implies S„ w+1 (/ + 1) = / + 1 defines the possible cases we need to consider. From this point, 
the proof is entirely analogous to the proof of Theorem 2. 

COROLLARY 4: Given the same conditions as in Theorem 4, k llM+ \d + 1) ^ ^„.w('), 
At/,.A/(') ^ Pn.M+ib + 1)' ancl 'f Sn.M+ib + 1) = ' + 1, then S„ M (i) = i. 

We next consider two systems with the same number of spares, but where one system has 
one more working machine than the other. If we compare two states where each system has 
the same number of machines in repair, then we find the possibly surprising result that the sys- 
tem with more working machines works harder. 

THEOREM 5: Consider two systems which are identical in every respect except that the 
first has N working machines while the second has N + 1 working machines. Assume that the 
conditions of Theorem 2 hold for each system. Then A„ v+ i(') ^ A„ v (/) for n ^ 0, 1 < / < 

M + N, if A is redefined as A = (N + 1) \ max + My + (M + N + 1) ^, max for each system. 

PROOF: Again we only sketch the proof. First we notice that N iN+] = A^, v + 1 and 
M, N+] = M //v , since the extra machine in the second system will be working for any state /. 
This allows us to group the A. and y terms in A„ N+] (/') — A„ N (i) in an obvious way and the r 
terms cancel. Finally we deal with the S, /x terms in the standard way by using the inequalities 
S„,n (' - 1) < S„ N (i) < S„ N+] (i) and S„ N (i - 1) ^ S„ N+ \(i - 1) < S„ N +\(i) to establish 
the possible cases. 

COROLLARY 5: Given the same conditions as in Theorem 5, \„ v (') is decreasing in N 
and /i,„ v (/) and S„ N (i) are increasing in N. 



OPTIMAL MAINTENANCE/REPAIR POLICIES 25 

In our final model of this section, we examine the case where there are no spares, that is, 
M = 0. Then N, = N - i and M, = 0. Here we find that if // is linear, then X„(/) and /*„(/) 
are independent of /, and for a given n we either never turn on servers, or we always turn on 
one server for each broken down machine. 

THEOREM 6: Assume h(j) = jr, < j < /V, and M = 0. Then A„(/) = A„ (/' + 1) 
for n ^ and 1 < / < N — 1. Therefore, X„(/) = X„ (/' + 1) and (i„(i) = /u.„ (/' 4- 1) for all 

/, and either S„(i) = /for all /or S„(i) = for all /. 

PROOF: For n = 0, the result is trivial. Assume A„_|(/) = A„_i(/' + 1) for each /. This 
obviously implies that X„(/) and fx„(i) are constant functions of /and that S„(i) = /for all /or 
S„(i) = for all /. Letting X = X„(/), y. = fi„(i), and A = A„_i(/), this yields 

(A + a)A„(/) = (N - i) (r - g(\) + XA) - S„(i) (gi/x) + /nA) 

- (N - i + 1) (r - g(X) + XA) + S„ (/ - 1) (g(fi) + utA) + AA 

= -(/•- ^(X) + XA) - (S„(/) - S„ (/ - 1)) (g(fi) + /it A) + AA. 

Since S„(/') — S n (i — 1) = 1 for all / or for all /, we see that A„(/) is constant in /, and the 
result is proved. 

As a corollary to this result, consider the two systems which are identical except that their 
failure rates, X-! and X 2 , no longer considered control variables, are different. That is, suppose 
each working machine in system /fails at rate X,, where X, < X 2 . Also assume M = and that 
S and p. are the only control variables. Intuitively, we might expect that the system with the 
larger failure rate would work harder to repair to its machines. This result turns out to be false. 
In fact, its exact opposite is true, as exhibited in the following corollary. 

COROLLARY 6: Given the above model with g(\) = and h(j) = jr, A„ Ai (/) < 
A„ „ 2 (/) for each n > 0, 1 < / ^ N, and X mm ^ X, < X 2 < X max . Hence ti„. K] (i) ^ ai„.a 2 (/) 
and S„ X] (i) ^ S„.x 2 0') for each n > 1, 1 < / < N. 

PROOF: By Theorem 6, we can write A„ K (/) = A /;/ and /a„ A (/) = fi nj for each /. 
Therefore, 

(A + «)(A,, X| (/) - A„, 2 (/)) = (A r - /)(X,A„_ U - X 2 A„_,. 2 ) 

- (N - i + DCX.A^,., - X 2 A„_,. 2 ) - (S ILK] (i) - S„, Ki (i - 1)) 
• (/(m,,,,) +^ / ,.,A„-i,i) + (S,,, K2 (i) - 5,, 2 (/ - 1)) 
' (/(A*n.2) + ^«.2A n -i, 2 ) + A(A„_ L1 - A„_, 2 ). 

By induction, (S„. X] (/) - S„ , X] (/ - D) - (S„ h (i) - S„ ,^(/ - 1)) = or 1. Both cases may 

be handled as in Theorem 2 to dispose of the S, fx terms (after possibly combining part of 
them with the A term). What is left is 

= (A - X 1 )(A„_ 1 ,, - A„_ 12 ) + (X 2 - X,)A„_,. 2 < 0. 

In general, if M > 0, the result of the above corollary seems to be partly true and partly 
false. For most small states (/' ^ M), it seems that ix nK] (i) ^ n„ k (/), but that for most large 
states (/ > A/), /t*„^(/) ^ P-„,k 2 0)- The following numerical example, whose results are for 
the original continuous time, infinite horizon problem, is typical. In this example, N = 2, M 
- 1, r - 200, a = 25, and /(/n) = .05/u 2 . 



26 S.C. ALBRIGHT 







X = 1 








X = 5 






X 


= 20 




/ 


SO) 


/*(/) 


K(i) 


/ 


so) 


/*(/) 


V(i) 


/ 


so) 


M/) 


V(i) 











15.98 











15.67 











13.44 


1 


1 


2.39 


15.74 


1 


1 


8.35 


14.83 


1 


1 


16.00 


11.84 


2 


2 


33.65 


12.38 


2 


2 


32.73 


11.56 


2 


2 


29.20 


8.92 


3 


3 


37.84 


8.59 


3 


3 


36.34 


7.92 


3 


3 


31.12 


5.81 



We end this section by noting that all of the previous results are true for the infinite hor- 
izon continuous time problem. This follows by letting n — - °° and using the results from Lipp- 
man [14]. Furthermore, because of Theorem 1 and the results of Lippman [15], these mono- 
tonicity properties also hold for the finite horizon continuous time problem. To state these 
results, we will drop the subscript n to denote the infinite horizon problem, and we will substi- 
tute t for n to denote that there is t time left in the finite horizon problem. 

THEOREM 7: Suppose h(j) = jr, j ^ 0, and the assumptions of Theorem 2 hold. Then 
for the infinite horizon problem, V(i) is concave and decreasing in /, ix{i) and SO) are increas- 
ing in /and A' and are decreasing in «, and XO) is decreasing in /and Wand is increasing in «. 
Also, X w+1 + 1) > X w 0), fixf(i) > Pm+\0 + 1), and if S M+1 + 1) = / + 1, then S M (i) 
= i. For the finite horizon problem with t time remaining, these same properties hold for 
V,(i), X,0), S,0), and /*,(/). Finally, /u.,0) and S,0) are increasing in /, and X,0) is decreas- 
ing in t. 

3. SUMMARY AND POSSIBLE EXTENSIONS 

In this paper we have been able to extend previous results concerning the structure of 
optimal policies for the machine repair problem. Namely, we have let X and S, as well as /a, be 
control variables and we have allowed spares to fail before they are put into use. There are 
still, however, many other complications of this model which are worthy of further study. 
Unfortunately, the ones we have in mind will probably present many more mathematical 
difficulties than the model we have examined here. 

One obvious feature which would be good to include is the presence of fixed costs for 
turning on or turning off servers. Since there has been only limited success with this type of 
queuing control in other, probably less complicated, queuing models, we are not optimistic 
about success here. Another possible extension, or set of extensions, is to complicate the prob- 
lem in such a way that there are two or more state variables. For example, we could assume 
that the time to install spares is an exponential random variable, or that failed spares go to their 
own repair facility, or that working machines may fail in more than one way and that each type 
of failure requires a different repair facility (see [11], [12]). In each of these models, the state 
space is multidimensional, which serves to complicate the mathematics immensely. In fact, we 
have investigated these models quite extensively. Unfortunately, aside from an analogue of 
Theorem 6, not any of even the most intuitive structural results could be proved by the above 
induction arguments. At the same time, no counterexamples to our conjectures were found. 
Hopefully, researchers in the future will discover how (or if) Lippman's "new device for 
exponential queuing systems" is able to solve these models with multidimensional state spaces. 



OPTIMAL MAINTENANCE/REPAIR POLICIES 27 

REFERENCES 

Albright, S.C., and W. Winston, "A Birth-Death Model of Advertising and Pricing," 

Advances in Applied Probability, 77, 134-152 (1979). 
Anderson, M., "Monotone Optimal Maintenance Policies for Equipment Subject to Mar- 

kovian Deterioration," Doctoral Dissertation, Indiana University, Bloomington, Ind. 

(1977). 
Barlow, R.E., "Repairman Problems," Chapter 2 in Studies in Applied Probability and 

Management Science, ed. Arrow, Karlin, and Scarf (Stanford University Press, Stanford, 

Ca. 1962). 
Cinlar, E., "Optimal Operating Policy for the Machine Repair Problem with Two Service 

Stations," Technical Report No. 266-3, Control Analysis Corp, Palo Alto, Ca. (1972). 
Crabill, T., "Optimal Control of a Maintenance System with Variable Service Rates," 

Operations Research, 22, 736-745 (1975). 
Denardo, E., "Contraction Mappings in the Theory Underlying Dynamic Programming," 

SIAM Review, 9, 165-177 (1967). 
Deshmukh, S.D., and W. Winston, "A Controlled Birth and Death Process Model of 

Optimal Product Pricing Under Stochastically Changing Demand," Journal of Applied 

Probability, 14, 328-339 (1977). 
Gnedenko, B.V., Y.K. Belyayev, and A.D. Solovyev, Mathematical Methods of Reliability 

Theory (Academic Press, New York 1969). 
Goheen, L., "On the Optimal Operating Policy for the Machine Repair Problem when 

Failure and Repair Times Have Erlang Distribution," Operations Research, 25, 484-492 

(1977). 
Howard, R.A., Dynamic Programming and Markov Processes (M.I.T. Press, Cambridge, 

Mass 1960). 
Iglehart, D.L., and A. Lemoine, "Approximations for the Repairman Problem with Two 

Repair Facilities, I: No Spares," Advances in Applied Probability, 5, 595-613 (1973). 
Iglehart, D.L., and A. Lemoine, "Approximations for the Repairman Problem with Two 

Repair Facilities, II: Spares," Advances in Applied Probability, 6, 147-158 (1974). 
Kakumanu, P., "Relation Between Continuous and Discrete Time Markovian Decision 

Problems," Naval Research Logistics Quarterly, 24, 431-439 (1977). 
Lippman, S.A., "Applying a New Device in the Optimization of Exponential Queuing Sys- 
tems," Operations Research, 23, 687-710 (1975). 
Lippman, S.A., "Countable-State, Continuous-Time Dynamic Programming with Struc- 
ture," Operations Research, 24, 477-490 (1976). 
Ross, S.M., Applied Probability Models with Optimization Applications (Holden-Day, San 

Francisco 1970). 
Serfozo, R., "An Equivalence Between Continuous and Discrete Time Markov Decision 

Processes," Tech. Report, Dept. of I.E. and O.R., Syracuse University (1976). 
Winston, W., "Optimal Control of Discrete and Continuous Time Maintenance Systems 

with Variable Service Rates," Operations Research, 25, 259-268 (1977). 



BENDERS' PARTITIONING SCHEME APPLIED TO A NEW 
FORMULATION OF THE QUADRATIC ASSIGNMENT PROBLEM* 

Mokhtar S. Bazaraa 

School of Industrial and Systems Engineering 

Georgia Institute of Technology 

Atlanta, Georgia 

Hanif D. Sherali 

School of Industrial Engineering and Operations Research 

Virginia Polytechnic Institute and State University 

Blacksburg, Virginia 

ABSTRACT 

In this paper we present a new formulation of the quadratic assignment 
problem. This is done by transforming the quadratic objective function into a 
linear objective function by introducing a number of new variables and con- 
straints. The resulting problem is a 0-1 linear integer program with a highly 
specialized structure. This permits the use of the partitioning scheme of 
Benders where only the original variables need be considered. The algorithm 
described thus iterates between two problems. The master problem is a pure 
0-1 integer program, and the subproblem is a transportation problem whose op- 
timal solution is shown to be readily available from the master problem in 
closed form. Computational experience on problems available in the literature 
is provided. 



1. INTRODUCTION 

In this paper, we revisit the quadratic assignment problem which was first formulated by 
Koopmans and Beckmann [19] for assigning m indivisible entities, called facilities, to m mutu- 
ally exclusive locations. More specifically, we consider the following most general formulation 
of this problem as introduced by Graves and Whinston [14]. 



mm m m m m p 

QAP 1: minimize £ £ a u ^ + £ £ J J £ b$ k , x tj x k , 

i=\ j=\ i=\ j=\ /c=l /=1 «=1 

m m m m p 

i=\ j=\ k=\ /=! n=\ 



subject to 

x€ X A - \ 



(x n , ... , x mm ): £ Xy =1,7=1, ... m, £ Xjj = 1, 



O.D /= 1, ... , m, xi- 0, 1, ij = 1, 



m 



Here there are p products that flow among the m facilities. Particularly, f" k is the amount of 
flow of product n from facility /' to facility k and d", is a distance measure from location j to 



•This study is supported under NSF Grant # ENG77-07468 

29 



30 M.S. BAZARAA AND H.D. SHERALI 

location / when transporting product n. Further, a n is the fixed cost of locating facility / at loca- 
tion j, and b" Jk i is a fixed cost for product n dependent on a pair of assignments, viz, facility / to 
location y and facility k to location /. Note, that without loss in generality, we can take b" m = 
if / = k or j = / and also, /// = dj) = for /, j = 1 , .... m, n = 1 , .... p. 

Using a simple transformation introduced by Lawler [22], and extended by Pierce and 
Crowston [25], the above problem may be written as: 



QAP 2: minimize 



m m m m 



X 1 Sx — 2^ ZL X X S ijkl x ij x kl 

1=1 7=1 k=\ l=\ 



where, 
(1.2) 



Vjkl 



£ m m + £ m df, if / *-k ox j* i 



n=\ n=\ 

Qjj otherwise 



and where a superscript /, will throughout this paper, denote the transpose operation. However, 
we will find it more convenient to use the following transformation: 



■ijkl 



a '! + f" + t ibSu + bgnj) + £ if',1 d>) + fl rf/p, 






n=\ n=\ 



(1.3) / — 1, . . . , m - 1, k = / + 1 m, I, j = 1 , ... , m, I ^ j. 

It may be easily seen that this transformation leads to the equivalent problem 

m—\ m m m 

QAP 3: minimize Z Z Z L c m x u x ki 

The quadratic assignment problem has enjoyed great popularity mostly because of its ver- 
satile applicability, but partly because of its insurmountable resistance to efficient solution tech- 
niques. For a review on quadratic assignment problems, we refer the reader to Francis and 
White [6], Gaschiitz and Ahrens [8], Gavett and Plyter [9], Gilmore [11], Graves and Whins- 
ton [14], Hanan and Kurtzberg [15], Heider [16], Hillier and Connors [17], Koopmans and 
Beckmann [19], Land [20], Lawler [22], and Pierce and Crowston [25]. For computational stu- 
dies on this problem, we refer the reader to Nugent, Vollmann and Ruml [23] and, more 
recently, to an excellent study by Burkard and Stratmann [2]. 

In the following sections, we first employ a transformation which converts problem QAP 3 
into a mixed integer linear program and thus makes it most amenable to the partitioning 
scheme of Benders [1]. This latter scheme decomposes the problem into a linear integer mas- 
ter program and several linear subproblems which are essentially transportation problems. The 
solution to the subproblems is shown to be readily available in closed form. We then develop 
an approach to the master problem. Finally, computational results and certain relevant conclud- 
ing remarks are presented. 

2. REFORMULATING THE QUADRATIC ASSIGNMENT PROBLEM 

For the purpose of this development, let us introduce m 2 (m — l) 2 /2 new variables, 

(2.1) y ijk i — Xjj x kl for / = 1 m — 1, k = / + 1, .... m, j, I — I, .... m, j ^ / 

In addition, introducing 2m{m — 1) new linear constraints, we formulate the following linear 
mixed integer problem which is then shown to be equivalent to problem QAP 3. 



BENDERS' PARTITIONING FOR QUADRATIC ASSIGNMENT 31 

in — 1 in m in 

QAP 4: minimize Z Z Z Z C W y>M 
subject to 

m m 

2.2) Z Z yijM - (« - x u ■ ° for ' = ! w - 1 , y = i /w 

/^ 

fc-l m 

2.3) Z Z yifki - (k-l) x kl = for k = 2 m, / - 1 /w 

/=i /-i 

/*/ 

2.4) Z x '/ = ! for / = 1 m 



2.5) Z^/ = 1 for y-1 



/?? 



/ = ! 



2.6) Xy binary for / J = 1 /w 

2.7) -y m ^ - 1 for / = 1 m - \, k = i + \, . . . , m,j,l = 1, .... m, j * I 

2.8) y m ^0 for / = 1, ... , m - 1, k - / + 1, ... , mj,/ = 1, . . . , mj ^ I 

*iote that problem QAP 4 has m 2 integer and /w 2 (m - l) 2 /2 continuous variables and 2m 2 
inear constraints. As opposed to this, Lawler's [22] linear integer formulation involves 
m 2 + m 4 integer variables and w 4 + 2m + 1 constraints. A more recent linearization technique 
lue to Kaufman and Broeckx [18] results in a mixed integer program with m 1 zero-one and m 2 
:ontinuous variables and m 2 + 2m constraints. In any case, we will be demonstrating how 
> roblem QAP 4 can be solved by exploiting its structure and handling explicitly only its m 2 
:ero-one variables and its 2m linear assignment constraints (2.4), (2.5). 

Before proceeding to propose a solution technique for problem QAP 4 we first establish 
hat every feasible solution to Problem QAP 4 must satisfy Equation (2.1). As a consequence 
)f this result, Theorem 1 stating that Problems QAP 3 and QAP 4 are equivalent, follows. 

LEMMA 1: Let (x,y) be any feasible solution to Problem QAP 4. Then, y likl = x,jX k i for 
= 1 , . . . , m — 1 , k — i + 1 , . . . , m, j,l = 1 , . . . , m, j ^ I. 

PROOF: First of all, note that for any p,q,r,s, p < r,q ^ s, x pq x rs = implies from 
equation (2.2), (2.3) that y pqrs = 0. Now consider p,q,r,s,p < r,q ^ s such that x pq = x n = 1. 
t suffices to show that this implies y pqn = 1. Since x n = 1, then by Equation (2.3), we have, 

r — \ in 

[2.9) . Z Z yurs - r - 1 

But since x satisfies the assignment constraints (2.4) through (2.6), and since x rs = 1, there are 
irecisely r — 1 nonzero variables x tj for / < r, j ^ s. Since x tj = implies y ljrs = from 



32 M.S. BAZARAA AND H.D. SHERALI 

above, this in turn means that there are at most r — 1 nonzero variables y ljrs for / < r, j ^ s. 
However, in view of Equations (2.7), (2.8), (2.9), we must have precisely r — 1 unit variables 
y iJrs , each corresponding to Xy — 1 for / < r, j ^ s. In particular, since x pq = 1, 
p < r, q ^ 5, then y pqrs = 1. This completes the proof. 

THEOREM 1: Problems QAP 3 and QAP 4 are equivalent in the following sense. For 
every feasible solution to Problem QAP 3 there corresponds a feasible solution to QAP 4 with 
the same objective function value and vice versa. 

PROOF: Let x be a feasible solution to Problem QAP 3 and let us define the vector y 
through Equation (2.1). We will show that (x,y) qualifies for the corresponding solution we are 
seeking to Problem QAP 4. Now, Equation (2.1) implies that 

in in m m 

(2.10) £ Z y lik i - Z Z x u x M = for / = 1 m - 1. J = 1 m 

k = , + \ l = \ A- = / + l / = 1 

If for any such ij, we have x n = 0, then from Equation (2.10), Equation (2.2) must hold. On 

in 

the other hand, if x n = 1, then since we must have x k/ = for each k > /, we get £ x kl = 

k = i + \ 

0. Thus, using Equation (2.4), we have, 

in in in m m in m 

(2.11) £ Z *</*«- Z Z *ki + Z x k, = Z Z x A/=(m-/) 

Thus, from Equations (2.10), (2.11), again Equation (2.2) holds. Similarly, one may show that 
Equations (2.3) holds. Also, from the definition of y through Equation (2.1), we have that 
Equations (2.7), (2.8) hold and that the objective function values of Problems QAP 3 and QAP 
4 are equal. 

Conversely, let (x,y) be a feasible solution to QAP 4. We will show that x qualifies for 
the corresponding solution we are seeking to Problem QAP 3. Clearly, x is feasible to QAP 3. 
That the objective function values are equal follows from Lemma 1 which asserts that Equation 
(2.1) holds. This completes the proof. 

We now proceed to develop a solution procedure for Problem QAP 4. This technique 
exploits the following special structure of the problem. The variables x are restricted to be 
extreme points of the assignment polytope, or points in the set X A . For each such point, the 
resulting problem in j is a transportation problem whose solution, as seen above, is given 
through Equation (2.1). This structure is most efficiently exploited by Benders' Decomposition 
scheme [1]. 

3. APPLICATION OF THE PARTITIONING SCHEME OF BENDERS. 

In this section, we will decompose Problem QAP 4 into a linear integer master problem in 
m 2 zero-one variables and a linear subproblem, using the method due to Benders (see Benders 
[1] and Lasdon [21]). At each iteration, the master problem generates a point of X A , based on 
which a suitable subproblem is solved to generate a cut-constraint. This cut is then appended 
to the other constraints in the master problem and the solution to the latter is updated. The 
procedure hence iterates between the master problem and the subproblem until a suitable ter- 
mination criterion is met in a finite number of steps. 



BENDERS' PARTITIONING FOR QUADRATIC ASSIGNMENT 



33 



To conduct such a decomposition, observe that for a fixed x€ X A , problem QAP 4 is a 
transportation problem in the y-variables over a set, say, Y(x) defined by constraints (2.2), 
(2.3), (2.7) and (2.8). Thus, we may write QAP 4 as 



QAP 4: minimize 



X e x A 



m — \ m m m 

minimum XIII c nki y,/ki 



lȣ Y(x) 



i-\ /-] k>i l*j 



For the inner minimization problem above, letting u lh / = 1, . . . , m — \,j = \, .... m be the 
dual variables associated with constraints (2.2), \ kh k = 2, ... , m, I = 1, .... m those with 



(2.3) and mw / = 1, 



m 



1, A: - / + 1, 



m, l,j = \, . . . , m, I ^ j those with 



(2.7), we may use the linear programming dual to rewrite the above problem as 

QAP 4: minimize [S/>U)] 
where, for a fixed x€X A , we have, 

m — I m m m 

SP(x): maximize I I (w - /) u u x u + £ £ (A: - 1) v kl x k , 

m — 1 w /« in 

~ I I I I »tyw 

i-I A=/+l /=1 /=l 



subject to 



for i — 1, ... , m — 1 , A: = /' + 1 , . . . , m, /', / = 1 , 



m 



Now, using the standard transformation of replacing unrestricted variables by the difference of 
two nonnegative variables, one may show that SP(x) attains an optimal solution at an extreme 

point of the resulting constraint set. Thus, letting (u p , v p , w n ) , p£ E = {1 P) be the 

finite number of points corresponding to the extreme points of this set, then SP(x), and hence, 
QAP 4 may be rewritten with obvious notation as 



QAP 4: minimize 



a€ X a 



maximum (a'x — «„) 

/>€£-{! P) P P 



Finally, the above problem may be represented as the master problem 

MP(E): minimize z 

subject to z ^ a' p x — a p for each p € E 

x<iX A 



The scheme of Benders' asserts that one need not generate the entire set E, but rather, 
generate suitable elements of E as and when needed. Thus, at any particular stage r, letting 
E r C E, 1 < |£,| = r < P, and denoting the corresponding relaxed master program by 
MP(E r ), the following scheme is validated by Benders. 



Initialization: 



Let z 1 = -oo, choose an arbitrary, preferably good, 
solution x l €.X A , set r = 1 and E = 0, and go to Step 2. 



Step 1: Solve the master problem MP{E r ). Let (x' +1 , z r+l ) be an optimal solution. 
Increment r by one and go to Step 2. 



34 



M.S. BAZARAA AND H.D. SHERALI 



Step 2: 



Solve the subproblem SPix 1 ). Let (u r , \ r , w 1 ) be an optimal extreme point 
solution with objective function z. If z = z r , terminate with x r as an optimal 
solution to the quadratic assignment problem. Otherwise, z > z r , and add 
the cut 



> 



alx 



to the current master problem. Letting E r = E r -\ U {/-}, go to Step 1. 

The cardinality of E being finite, the above procedure is finitely convergent since if any element 
of E\s re-generated then the termination criterion of Step 2 is necessarily met (see [1] or [21]). 
Over the next two subsections, we will now discuss the solution strategies for the master prob- 
lem and the subproblem. 

3.1 On Solving the Subproblem 

Recall that for the sake of finiteness of Benders' scheme, we need to generate an optimal 
extreme point solution to problem SP(x') where all unrestricted variables are transformed into 
nonnegative variables. Writing the Kuhn Tucker conditions for problem SP(x r ) and using a 
rather elaborate proof, one can show (see [27]) that the following solution qualifies, as such an 
extreme point solution: 











maximum [c /:/ - w ] if xfr = 1 




u r+x = 


k>! fori- 1 

minimum [c rM — v^ +1 ] if x[. = 

(k,l):k>i 


r+\ _ 
kl — 


ifxfc- 

minimum 


1 

minimum {c ijkj — «,y +1 }> minimum {c ijk i) 








Uj):xlj=\ (i.j):xf r 








i<k i<k 




if xjjxu = ° 


u/ +1 — 
w ijkl - 


for /' = 1, . . . , m — 1,/c = / + 1, 






« r+1 - 




c ijkl ^ X ij x kl ~ 1 



m — 1 J — 1, 



m 



for k = 2, ... , m,l = 1, .. . , m 



We briefly note at this point that one may write out an alternative optimal extreme point solu- 
tion by simply interchanging the roles of the variables u and v above. Through computational 
testing we found that, depending on the data structure, one of these solutions yields cuts that 
perform significantly better than those obtained from the other solution. 

3.2 On Solving the Master Problem 



First of all, note that the master problem need not be solved exactly at each iteration. 
Recall that in using Benders' decomposition, we are merely interested in generating elements of 
E = [1, .... P} until such time as the value of the master problem equals the current best 
value of Problem QAP 4. Moreover, while accomplishing this, we wish to ensure for the sake 
of finiteness that we do not regenerate any point of E until the termination criterion is met. 
Now suppose we find a possibly suboptimal solution (x r+ \ z r+x ) to the master problem at stage 
r. If necessary, we then update the current best solution using x' +1 . 



Let the current best value be z and suppose that z r+] 
tion value z r+l of the master problem satisfies 



< z. Then clearly, the exact solu- 



,r + l 



^ 



sr+1 



< Z 



BENDERS' PARTITIONING FOR QUADRATIC ASSIGNMENT 35 

Moreover, we can also assert that x r+x has not been previously generated. For, by contradic- 
tion, if it has been generated, at the q' h stage, q < r, then let the corresponding Benders' cut 
be z ^ a' q x — a q . Noting through Lemma 1 that the subproblem yields the same value as the 
quadratic assignment problem for any given solution, we get z ^ a' q x r+l — a q < z r+1 which 
contradicts z < z. 

Thus, so long as an approximate solution yields a value for the master problem which is 
strictly less than that of the current best solution to the quadratic assignment problem, includ- 
ing the value of the new point generated, we may use this approximate solution with the 
assurance that it has not previously been generated. It is only when this condition does not 
hold that we need to solve the master problem exactly. Before discussing the proposed approxi- 
mate and exact solution techniques below, we emphasize that this modification still maintains 
the scheme as an exact, finitely convergent procedure. 

3.2.1 Scheme for Generating an Approximate Solution to the Master Problem 

In this section, we will develop a heuristic to obtain a good quality approximate solution 
to the master problem MP(E r ). Towards this end, suppose we compute 

P p = maximum {1, | minimum {a' p x — a p )\\ for each p 6 E, 
and formulate the problem 

MP(E r ): minimize{A,'x: x€ X A ) 
where, 

1 







(3.2) A 

A r = Z 
p=\ 

Consider the solution obtained through the linear assignment problem MP(E r ). Since A/x is 
essentially the sum of the terms a' p x, p€ E r , each normalized by the magnitude (3 P , Problem 
MP(E r ) tends not to minimize any cut expression too much at the expense of increasing the 
value of others. In other words, the solution to MP(E r ) tends to achieve the criterion of Prob- 
lem MP(E r ), viz, to minimize the maximum of several cut expressions. This is further sub- 
stantiated by the fact that the quantities minimum [a p x — a p : x€ X A ) are almost always nega- 
tive, with the result that (3.2) gives more weight to cut expressions which tend to have algebra- 
ically larger values. Thereby, cuts which are likely to be binding in an optimal solution to 
MP(E r ) are given more weightage. 

As an additional improvement routine for this scheme, we adopted the following strategy 
which enabled us to recover good quality solutions. Consider the linear bounding form g'x of 
Cabot and Francis [3] satisfying g'x < x' Sx for each x€ X A , with coefficients given by 



gn = minimum 



2- 2- s ijkl x kl' x € %a> x u ~ 1 
fc-i /=i 



Note that g u is obtained by solving a linear assignment problem in m-1 variables after fixing 
the assignment x u — 1. Now, let . 

^min = minimum{g'x: x€ X A ) and i^ max = maximum{g'x: x€ X A ). 

Working with some test problems in the literature, we found that optimal or good quality solu- 
tions often had values of g'x lying in the initial 10-15% of the range [v mm , ^ ma J. Hence, we 
found it advantageous to replace MP(E r ) by the problem 



36 M.S. BAZARAA AND H.D. SHERALI 

(3.3) 



minimize 



A/x + 



w 



r'x: x€ X A 



where w is a suitable weightage parameter. In our experience, the solution x /+ to problem 
(3.3) almost always yielded a corresponding value of z' +l satisfying z < z, except towards 
the end of the procedure when an exact solution of the master problem was frequently 
required. Moreover, the solution x' +1 tended to be of a good quality. 

Finally, we note that each time an approximate solution was obtained through (3.3), we 
attempted to improve on this solution through pairwise interchanges, and elected to use the 
resulting solution in case it also qualified as an approximate solution. Thus, only when neither 
the solution to (3.3), nor the solution obtained through pairwise interchange improvements on 
it, yielded a value of z less than the incumbent value, did we need to resort to an exact solution 
technique for the master problem. 

3.2.2 Exact Solution Method for the Master Problem 

We now consider the exact solution of Problem MP(E r ). For this purpose, note that if 
we consider the linear relaxation A' of the set X A , then integral solutions feasible to X are in a 
one-to-one correspondence with the points in X A . In other words, all integral solutions in A" are 
zero-one assignment solutions. Thus, we first attempted the use of Gomory's [13] dual all- 
integer cuts to obtain an optimal solution to MP(E,). However, in spite of incorporating all the 
rules for finite convergence (see [13] and [7]), we experienced the same problems as Trauth 
and Woolsey [29] and as formally predicted by Finkelstein [5]. Namely the procedure works 
very well in some instances, but is quite unpredictable in most instances. In some cases, 
several thousand dual-integer cuts were unable to find an integer primal feasible solution. 
Further, in this process, the updated coefficients in the simplex tableau tended to blow-up in 
magnitude. 

We thus resorted to Glover's [12] pseudo-primal-dual procedure which iterates between a 
lexicographic dual feasible stage related to Gomory's dual all-integer algorithm [13] and a dual 
infeasible stage related to Young's primal all-integer algorithm [30]. As reported by Glover, 
this method results in a lexicographic decrease in the updated solution column of the simplex 
tableau on two successive visits to the dual feasible stage by an amount at least as much as the 
decrease which would be obtained through two successive iterations of the dual simplex 
method. Of course, if we have dual feasibility and the condition z = z' of Step 2 holds, then 
we may terminate without regard to primal feasibility. The revised-simplex method was special- 
ized and rules for finite convergence (see [12, 13 and 7]) were also incorporated. Although 
this technique contained the magnitudes of the tableau coefficients, it also experienced conver- 
gence problems. 

Finally, we adopted an implicit enumeration scheme which was initialized just once at the 
first visit of the procedure to the exact solution routine for the master problem, and was simply 
updated at each subsequent visit. The updating feature permitted us to delete Benders' cuts for 
larger sized problems since it ensured that at least fathomed solutions would not be regenerated 
during the exact solution of the master problem. This scheme either resulted in a solution with 
a value of z less than the incumbent value, in which case this solution was adopted as our 
approximate solution, or else it verified optimality by solving the problem to termination. We 
avoid giving details here of this procedure in view of our experience reported in the following 
section. The interested reader may refer to Sherali [27] , however. 



BENDERS' PARTITIONING FOR QUADRATIC ASSIGNMENT 



37 



4. COMPUTATIONAL EXPERIENCE 

In this section, we report computational experience using test problems available in the 
literature [4, 23, 29]. We found that the procedure performed satisfactorily in detecting 
optimal solutions early on in the search process. Even in cases where storage limitations forced 
a premature termination, solutions of quality often better than any other known in the litera- 
ture were obtained during the early stages of the search. In fact, recently, Burkard and Strat- 
mann [2] have described excellent heuristics that significantly improve upon all the best known 
solution values of problems in the literature We were able to still improve further upon the 
values for Nugent et al's [23] problems of sizes m = 20 and m = 30 and perform at least as 
good on the other problems attempted by Burkard and Stratmann. We also obtained a 
significant improvement over the best known solution to Elshafei's [4] hospital layout problem. 

We note that unlike the experience of Geoffrion and Graves [10] on using Benders' parti- 
tioning on a multicommodity distribution flow problem, our problem required a large number 
of Benders' cuts. The reason for this, as conjectured later in Section 5, may be that a solution 
to the linear relaxation of Problem QAP 4 is not "close" to an integral solution. In fact, when 
working with Nugent, Vollmann and Ruml's [23] test problems, we found that the procedure 
required to generate close to m ! cuts in order to verify optimality, even when the starting 
solution was optimal. Table 1 below reports this experience. We remark that the objective 
function value of the master problem increased very slowly except for the late stages of the 
procedure when a rapid increase was obtained. As a result, when the problems of sizes m > 8 
were prematurely terminated, the master problem value was as yet negative. 

TABLE 1. Exact Implementation of Benders' Scheme 



Problem 


m 


w! 


# of cuts 
generated 


Optimality verified? 


cpu seconds 
execution time (1) 


Nugent, Vollmann 
and Ruml's 
Problems [23] 


5 
6 
7 
8 


120 

720 

5040 

40320 


113 

690 

4711 (2) 
3722(2) 


YES 
YES 
NO 
NO 


6.9 
154.4 
770 
770 



(1) On a CDC Cyber 70, Model 74-28/CDC 6400 machine, with coding in Forlran IV 

(2) Forced termination as time limit is reached 



In view of these results, we chose to operate the procedure as a heuristic by terminating it 
prematurely. It may be noted that the performance of the procedure as a heuristic with regard 
to the quality of the best solution obtained is sensitive to both the starting solution employed, 
as well as the value of w in Equation (3.3). In most cases, we found it appropriate to start with 
a value of w = 1 and increment it by one every three to ten cuts. For constructing a starting 
solution, we adopted the following strategy. For each facility, we computed the sum of flows to 
all the other facilities and for each location, we computed the sum of distances to all the other 
locations. We then arranged the facilities in nonincreasing order of their flow sums and the 
locations in nondecreasing order of their distance sums. The desired solution was constructed 
by matching the arrangement of facilities and locations element by element, and then improv- 
ing upon the resulting solution through pairwise exchanges. 

Table 2 reports our computational experience with some problems available in the litera- 
ture [4, 23, 28]. Column 'a' gives the best values obtained on using the above starting solution 
and terminating the process after 25 cuts. Column 'b' gives the cut index at which this best 
solution was found. Through a few subsequent runs using better quality starting solutions as 
obtained over previous runs, we were able to improve on the initial run solutions. Column 'c' 
gives the best objective values we were able to obtain in this manner. The quality of these 



38 



M.S. BAZARAA AND H.D. SHERALI 



TABLE 2. Implementation of Benders' Scheme as a Heuristic 

















Best Locations of Facilities 1,2, . . . , m 




Problem 


m 


a 


b 


c 


d 


Respectively for Solutions of Values in 


e 
















Column c 




Nugent, Vollmann 


5 


25 


7 


25 


25 


3,4,5,1,2 


0.69 


and Ruml 


s 


6 


43 


1 


43 


43 


3,2,1,6,5,4 


1.30 


Problems [23] 


7 


74 


3 


74 


74 


1,2,5,3,4,7,6 


2.25 






8 


107 


4 


107 


107 


3,4,8,2,1,5,6,7 


3.56 






12 


289 


14 


289 


289 


5,1,9,8,4,3,11,7,10,2,6,12 


13.69 






15 


575 


9 


575 


575 


5,4,9,10,12,13,7,2,1,15,6,11,3,8,14 


33.93 






20 


1285 


3 


1285 


1287 


4,12,19,11,2,5,3,9,20,18,14,13,10,17,7,15,1, 
16,6,8 


100.61 






30 


3095 


10 


3077 


3079 


20,29,11,5,30,28,15,9,23,21,2,27,22,6,1,10, 
19,8,16,12,24,14,7,26,13,25,3,18,17,4 


393.96 




Rectilinear 


















Distance 


36 


4975 


10 


4802 


4802 


14,27,35,25,34,33,15,26,7,16,32,23,24,31,6,8, 


829.48 


Stein- 














18,17,12,22,20,29,21,28,19,10,30,13,5,4,3,11, 




berg's 














1,2,9,36 




Problems 
[28] 


















Squared 












31,10,2,12,3,4,22,11,29,21,5,14,13,15,30,28, 






Euclidean 


36 


8232 


16 


7926 


7926 


19,20,24,23,7,8,16,9,17,18,6,32,33,34,35,25, 


921.02 




Distance 












27,26,36,1 




Euclidean 


















Distance 


36 




1 ( » 


Ci 


Ci 


Same as for the squared euclidean distance 

solution, but with facilities 1 and 15 

interchanged in location. 


958.6 


Elshafei's 


Problem 
















[4] 




19 


Cl 


12 


C\ 


Cl 


17,18,19,11,12,9,3,14,1,2,10,13,7,5,16,15,8,4,6 


96.15 



(1) This resulted through the use of the best recorded squared euclidean solution as a starting solution. 

a, b, c, d, e — See the text of Section 4 for connotation. 

c, = 8, 606, 274 
c 2 = 11,281,888 
c 3 = 4125.168 



solutions may be compared with the previously best known solutions reported in the literature 
[2] as given in column 'd'. Note that one may use such trial runs to select between the two 
subproblem solutions given in Section 3.1, preferring the one which yields larger values for the 
master problem, and one may also derive information on the manner in which w may be incre- 
mented in Equation (3.3). Finally, column 'e' gives the execution time in cpu seconds for a 
run which generates 25 cuts on a CDC Cyber 70 Model 74-28/CDC 6400 computer, with cod- 
ing in FORTRAN IV. These times do not include the effort for generating either the linear 
bounding form g'x of Section 3.2.1 or the starting solution. 



Before concluding, we note that further improvements of the procedure as a heuristic may 
be possible through the implementation of more sophisticated exchange schemes [24] in lieu of 
the simple pairwise exchange operations we have employed at each iteration. Also, one may 
choose to attempt a few short trial runs with the aim of gaining the type of information alluded 
to above, and then execute a final run using the best solution found in the trial runs as a start- 
ing solution. 



BENDERS' PARTITIONING FOR QUADRATIC ASSIGNMENT 39 

5. CONCLUDING REMARKS 

It is our purpose in this section to discuss, at least for our problem, the relationship 
between the performance of Benders' partitioning scheme in regard to the number of cuts 
required and the closeness of the solution of the original problem to that of the linearly relaxed 
mixed integer program. In fact, we show below that if the linear relaxation of Problem QAP 4 
has an integral optimal solution, then Benders' procedure terminates as soon as this solution is 
detected. 

Suppose that an optimal integer solution (x, y) to the linear relaxation of Problem QAP 4 
exists so that x solves the quadratic assignment problem. The Kuhn-Tucker conditions can be 
shown to assert the following, where u u , v kh w ljk , are the dual variables associated with con- 
straints (2.2), (2.3), and (2.7) respectively: 

Juki = — ' w<jki = ° and U U + v « < c m 
(5-1) y llk , = 1 — u u +v kt - w ijk , = c iJkh w, Jk , ^ 

for / = 1 , . . . , m—\,j = \, . . . , m, k — i + 1, . . . , m, / = 1 , . . . , m, I ^ j 

Suppose that x is generated by Benders' scheme, say as a solution to the Master Problem 
MP(E r _\). Since x solves the quadratic assignment problem, and since by Lemma 1, 
Vijki = XjjX k i, then the current incumbent value z is given by: 

(c -j\ m — \mmm 

z= I I Z I c m y„k\- 

i = \ /-l k = i + \ l=\ 

Note, however, that (5.1) implies that («, v,w) is indeed an optimal solution to problem SP(x). 
By examining the dual of problem SP(x), it immediately follows that the right hand side of 

w m 

(5.2) is equal to £ £ a ti x„ — «, where, 

i- 1 /-i 

Ojj = (m — i) u u + (i — 1) v /:/j and 

m—\ m m m 

«=II Z Z w uki 

/=1 7=1 k=i+\ l=\ 

yielding 

(5.3) - v v - 



The next Benders' cut generated is of the form z ^ £ £ a n x n - a, and hence an 



'-1 7 = 1 

optimal objective value z r+] to problem MP(E r ) must satisfy the following inequality 



z r+X > £ I «(/ *u ~ « 

/=i /-l 

In view of (5.3) and (5.4), we have z' +1 > z, which is the termination criterion for Benders' 
scheme. 

To summarize, if a solution (x, y) to the linear relaxation of problem QAP 4 is integer, 
and if x is generated at some iteration r — 1, then Benders' scheme will terminate at iteration r 
with the conclusion that x is optimal. In particular, if xis used as a starting solution, then only 
a single cut is sufficient to verify optimality. We emphasize, however, according to our compu- 
tational testing, that if the solution to the relaxed problem is not integer, even if we start with 



40 M.S. BAZARAA AND H.D. SHERAL1 

the optimal solution to the quadratic assignment problem, many Benders' cuts may be needed 
to verify optimality. We hence conjecture that a fewer number of Benders' cuts may be 
required if the solution of the linear relaxation of QAP 4 is close to an integral solution. For 
additional insight into Benders' partitioning scheme and in particular to the strength of bounds 
derived therefrom, we refer the reader to Rardin and Unger [26] . 



[1 

[2 

[3 

[4 

[5 

[6 

[7 
[8 

[9 

[10 

[11 

[12 

[13 

[14 

[15 

[16 

[17 

[18 

[19 
[20 

[21 



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BENDERS' PARTITIONING FOR QUADRATIC ASSIGNMENT 41 

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[27] Sherali, H.D., "The Quadratic Assignment Problem: Exact and Heuristic Methods," 
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[29] Trauth, C.A. and R.E. Woolsey, "MESA: A Heuristic Integer Linear Programming Tech- 
nique," Res. Rep. SC-RR-68-299, Sandia Labs, Albuquerque, N.M. (1968). 

[30] Young, R.D., "A Primal (All-Integer) Integer Programming Algorithm," Journal of 
Research of the National Bureau of Standards, 69B, (Math and Math Phys.) Vol. 213, 
No. 3 (1965). 



TEST SELECTION FOR A MASS SCREEENING PROGRAM* 

John A. Voelker 

Argonne National Laboratory 
Argonne, Illinois 

William P. Pierskalla 

University of Pennsylvania 
Philadelphia, Pennsylvania 

ABSTRACT 

Periodic mass screening is the scheduled application of a test to all members 
of a population to provide early detection of a randomly occurring defect or 
disease. This paper considers periodic mass screening with particular reference 
to the imperfect capacity of the test to detect an existing defect and the associ- 
ated problem of selecting the kind of test to use. Alternative kinds of tests 
differ with respect to their reliability characteristics and their cost per applica- 
tion. 

Two kinds of imperfect lest reliability are considered. In the first case, the 
probability that the test will detect an existing defect is constant over all values 
of elapsed time since the incidence of the defect. In the second case, the test 
will delect the defect if, and only if, the lapsed time since incidence exceeds a 
critical threshold T which characterizes the test. 

The cost of delayed detection is an arbitrary increasing function (the "dis- 
utility function") of the duration of the delay. Expressions for the long-run ex- 
pected disutility per unit time are derived for the above two cases along with 
results concerning the best choice of type of test (where the decision rules 
make reference to characteristics of the disutility function). 



INTRODUCTION 

Mass screening is the process of inspecting all members of a large population for defects. 
If the early detection of a defect provides benefits, it may be advantageous to employ a test 
capable of revealing the defect's existence in its earlier stages. (Throughout this paper, the 
words "defect" and "unit" or "individual" will refer to defect, disorder, or disease and to a 
member of the population, respectively). 

Defects may arrive in a seemingly random fashion such as many types of machine failure, 
the incidence of certain types of cancer, diabetes, glaucoma, heart disease, etc.; or they may 
arrive as the result of some contagion such as smallpox, polio, etc. It is the former type of 
arrival process, random and independent arrivals, which is studied in this paper. 



•This research was supported by the Office of Naval Research under Contracts N00014-67-0356-0030 and N00014-75- 
C-0797. 

43 



44 J.A. VOELKER AND W.P. PIERSKALLA 

Continuous monitoring would provide the most immediate detection of a defect, but con- 
siderations of expense and practicality will frequently rule out continuous monitoring so that a 
schedule of periodic testing— a periodic screening program — may be the most practical means of 
achieving early detection of the defect. In general terms, the question then becomes one of 
how best to trade off the expense of testing which increases both with the frequency of test 
applications and with the cost of the type of test used against the benefits to be achieved from 
detecting the defect in an earlier state of development. 

The benefits of early detection depend upon the application considered. For example, in a 
human population being screened for some chronic disease, the benefits of early detection 
might include an improved probability of ultimate cure, diminished time period of disability, 
discomfort, and loss of earnings; and reduced treatment costs. If the population being screened 
consists of machines engaged in some kind of production, the benefits of early detection might 
include a less costly ultimate repair and a reduction in the time period during which a faulty 
product is being unknowingly produced. If the population being screened consists of machines 
held in readiness to meet some emergency situation, an early detection of a defect would 
reduce the time the machine was not serving its protective function. 

The expense of testing includes easily quantifiable economic costs such as those of the 
labor and materials needed to administer the testing. However, there can also be other impor- 
tant cost components which are more difficult to quantify. For example, in the case of a human 
population subject to medical screening, the cost of testing includes the inconvenience and pos- 
sible discomfort necessitated by the test, the cost of false positives which entails both emotional 
distress and the need to do unnecessary follow-up testing, and even the risk of physical harm to 
the testee; e.g., from the cumulative effect of x-ray exposure. 

The design of a mass screening program must address two important questions: How fre- 
quently to test and what kind of test to use. Optimal testing frequency has been investigated as 
a function of the defect incidence rate and other factors by Derman [3], Roeloffs [8,9], Barlow, 
et al. [1], Keller [4], Kirch and Klein [5], and Lincoln and Weiss [6]. 

The second question follows from the fact that more than one kind of test may be avail- 
able for use in a mass screening situation. The alternate tests can be entirely different pro- 
cedures; or they can be the same procedure with different criteria for what constitutes a positive 
outcome, e.g., alternate levels at which a recording of systolic blood pressure would induce 
follow-up testing. 

Alternative tests will generally differ both in their reliability characteristics and in their 
cost of application. How to select which test to use is a question which, to our knowledge, has 
not previously been examined in the context of a general model. This paper will examine this 
question. For two different ways of modeling test reliability, we develop a framework for test 
selection and present certain specific decision rules: 



incidence. Define 8 s (t) = 



Test reliability is assumed to be a function, pit), of elapsed time, f, since defect 

1 if t € S 
otherwise' Usually S will be an interval, e.g., [T,oo). The two 

cases pit) = p and pit) = 8[ Too) it) are considered, pit) = p indicates that test reliability is 
independent of defect age. For pit) = 8 [7 - oo) (/), the test will detect a defect if, and only if, 
the defect has existed for at least T units of time. In a sense, these two classes of pit) 
represent polar extremes in the responsiveness of test reliability to defect maturity. The test 
choice decision is posed within each of these two reliability classes. 



TEST SELECTION FOR MASS SCREENING 45 

A crucial feature in any optimization model of mass screening is the characterization of 
the cost due to detection delay. Detection delay is the gap between the time of detection and 
the time of defect incidence (or the time the defect becomes potentially detectable by a screen- 
ing test). The mapping between detection delay and the resultant cost we call the disutility 
function, £>(•). Obviously, the shape of the disutility depends upon the particular application 
considered. Section 1 gives examples. 

Pierskalla and Voelker [7] demonstrated that the shape of D() impacts the optimal alloca- 
tion of a screening budget among segments of the client population characterized by differing 
defect incidence rates. Results in Sections 3 and 4 below demonstrate the role £>(•) plays in 
test choice decision rules. 

Previous research in the area of optimal mass screening which utilized closed form 
expressions for expected cost (or disutility) placed restrictive assumptions upon the shape of 
£>(■). Early work assumed D(t) = ct (Barlow, et al. [1]). In Kirch and Klein [5], 
D(t) = min (t, T) where t is detection delay and T is the (possibly random) delay between 
defect incidence and the time when the defect would be discovered in the absence of a screen- 
ing program. Keller [4] restricts the generality of D{) by requiring that D() and the test fre- 

quency (density) r{t) be such that r(t) I D(s)dsbe well approximated by £>(l/2r(/)). (A 

•'0 

density is employed to represent the schedule of test times so that the calculus of variations 
could serve as the optimization tool.) 

Lincoln and Weiss [6] derive two kinds of optimal testing schedules. Both schedules 
maximize the time between successive tests subject to, respectively, a bound on the mean 
detection delay and a bound on the probability that detection delay will exceed a fixed thres- 
hold. Neither version is equivalent to using a general disutility function. 

A few authors, Schwartz and Galliher [10], Thompson and Disney [11], and Voelker [12] 
let both the reliability of the test and the disutility (or utility) of detection be a function of the 
defect's state rather than of time since the defect's incidence. Although such models are more 
general and do utilize a general concept of disutility, they have not been amenable to closed 
form evaluation of expected disutility. 

To incorporate random defect arrivals into their models, previous researchers focus upon 
an individual who will incur the defect. They use the density function for the age when that 
individual incurs the defect as a fundamental element of their model. Since the density func- 
tion reflects age-specific incidence rates, a "life time" testing schedule can, thereby, be 
developed to tailor testing frequency at each age to the probability that the defect will occur at 
that age. 

Our way of modeling the randomness of defect arrivals reflects a somewhat different per- 
spective on the mass screening problem. We look through the eyes of a decision maker 
charged with intelligently allocating a fixed budget. The time frame over which the allocation 
must be made is often short compared to a typical life time of a member of the client popula- 
tion. Therefore, the decision maker does not plan lifetime screening schedules for particular 
individuals. Instead, he tries to maximize the benefit that can be derived from his available 
budget over a much shorter planning horizon. (For modeling purposes the objective of minim- 
izing expected long run cost per unit time is not unreasonable with the problem viewed in this 
way since steady state conditions should approximately obtain after the initial screening. This is 
especially the case when an existing and ongoing screening program is being optimized by the 
decision maker. Also, lacking information to the contrary, the decision maker has no reason to 
anticipate abrupt changes in the screening policy at the end of the planning horizon.) 



46 J.A. VOELKER AND W.P. PIERSKALLA 

With the problem viewed in this perspective, the random nature of defect arrivals is most 
naturally modeled as a Poisson process with its parameter determined by the incidence rate of 
the defect and the size of the population. This approach has proved particularly useful in the 
following context: If different segments of the client population exhibit different incidence 
rates, subpopulations can be defined with defect incidence within each modeled as a Poisson 
process with its respective parameter. Then the budget can be so allocated among the subpopu- 
lation to permit appropriate relative testing frequencies (cf. Pierskalla and Voelker [7]). In this 
way, age-specific incidence rates can be incorporated into the notion of Poisson defect arrivals. 
Moreover, factors other than age which affect defect incidence rates (family history, smoking 
habits, work environment, etc.) can also be incorporated into the model. 

Although this paper does not follow Pierskalla and Voelker [7] in studying the case of 
heterogeneous populations, we use the same Poisson model of random defect arrivals. Section 
1 presents examples of disutility functions. Section 2 sketches the basic structure of the model 
and represents the expected long run disutility per unit time for general reliability and disutility 
functions. Section 3 considers the class of tests with pit) = p. Section 4 does so for pit) = 
h[ Too) it). In both Sections 3 and 4, results regarding and test choice criteria are presented. 
Proofs are deferred to the APPENDIX. 

SECTION 1: SOME EXAMPLES OF DISUTILITY FUNCTIONS 

Suppose a production process is subject to a randomly occurring defect. Although produc- 
tion appears to proceed normally after the incidence of the defect, the product produced is, 
thereafter, defective to an extent which remains constant until the production process is 
returned to its proper mode of operation. The only way to learn if the production process is in 
this degraded state is to perform a costly test. Now, if a test detects the existence of the 
degraded mode of production t units of time after its incidence, the harm done will be propor- 
tional to the amount of defective product (unknowingly) produced which, in turn, is propor- 
tional to /. Hence, Dit) = at for some a > 0. 

Another example where a linear £)(■) function may be appropriate would be for the 
periodic inspection of an inactive device (such as a missile) stored for possible use in an emer- 
gency. If t is the time between the incidence of the disorder and its detection, the disutility 
incurred is proportional to the probability that the device would be needed in that time interval. 
If such "emergencies" arise according to a Poisson process with rate /*, then the probability of 
an emergency in a time interval of length / is 1 — e _M ', which, for /x small, is approximately fxt. 
Hence, if b is the cost incurred should there be an emergency while the device is defective, and 
if /a is the (small) arrival rate of emergencies, then Dit) = bjxt. 

A quadratic disutility could arise in the following situation. Suppose the magnitude of a 
randomly occurring defect increases linearly with time since the occurrence of the defect. For 
example, the magnitude of the defect might be the size of a small leak in a storage container 
for a fluid, and as fluid escapes, the leak gets larger. Further suppose that the harm done accu- 
mulates at a rate proportional to the magnitude of the defect. Hence, the quantity of fluid lost 
(at least initially) increases the longer the defect exists, and the rate of fluid loss is proportional 
to the size of the leak. 

Let the size of the leak (as measured by rate of fluid loss), at time 5 since the leak's 
incidence, be cs. Then, if the defect is detected at time t since incidence, the disutility incurred 



(fluid lost) is Dit) = f'csds = l/2(c/) 2 . 

•'0 



TEST SELECTION FOR MASS SCREENING 47 

SECTION 2: EXPECTED DISUTILITY WITH A GENERAL RELIABILITY FUNCTION 

We assume that the times of defect arrivals in the screened population form a stationary 
Poisson process. Since there is a certain intuitive appeal to considering the defect arrival rate 
proportional to both the size of the population, N, and the intrinsic incidence rate, A.; let N\ 
designate the parameter of the above arrival process. It is not necessary to know the value of 
NX. in order to apply the decision rules of test selection developed in this paper. 

Let p/(t) be the probability that a test^)f type /will detect a defect which has been present 
t units of time. p/U) = for / < 0. Let Sfj be a random variable denoting the time at which 
the k th defect is detected. Sfj depends upon the arrival time of the defect, 8^; the type of test 
used, / ; and the testing frequency, r. 

Given the application of test type / at the times {l/r,2/r, . . .}, the disutility incurred by 
the /c th defect is D(Sfj — S k ). The total disutility incurred due to defects which occur before 
time A is 

iD(S^j-S k )8 l0 , A) (S k ). 

k = \ 

In Pierskalla and Voelker [7], the long run expected disutility per unit time under the 
above screening program r,p( )) was shown to be 

°° pnlr n-\ 

, n rNX £ J„_ 1/r ^(") PiM II 11 " Pi(» ~ mlM du - 

*■ ' n=l m — 1 

This result will serve as our starting piont for the technical results of this paper. Lincoln 
and Weiss established essentially the same result based upon a different probabilistic model of 
defect arrival. 

SECTION 3: CONSTANT TEST RELIABILITY 

The mass screening model yields interesting results for a test which has a fixed probability 
p of detecting the disorder if it is present in an individual. Such a model would arise if the 
unreliability of the test is entirely intrinsic to the test procedure rather than partially dependent 
upon the state or age of the defect. An example of this is the administration of a Mantoux test 
for tuberculosis in, say, a population of grade school children. The test has a small but rela- 
tively constant level of false negatives. There are other medical tests with similar characteris- 
tics. 

A Quality Control Example 

To see how another type of situation with constant test reliability could arise, consider a 
production system which is subject to a randomly occurring defect which degrades the system's 
performance. Once the defect occurs, the level of degradation of the process remains constant 
until the defect is discovered. Suppose the defect is such that each item produced has probabil- 
ity 8 of being defective and that the system without the defect never produces defective items. 



48 



J.A. VOELKER AND W.P. PIERSKALLA 



The only way to discover the existence of the defect in the production system is to exam- 
ine an item produced which is itself defective. Now, if the examination of an item is expensive 
(e.g., the item is destroyed as a result of the inspection) and if the capacity to examine a 
sequence of items involves a set-up cost (say a), the following strategy might be called for: At 
specified times \/r, 2/r, .... set up the capacity to examine a sequence of items and examine, 

say, / items at each of those times. The times \/r, 2/r are then the times of testing and 

sample size /specifies the test type. 

Assume that if a defective item is examined, the defect is always observed and the pro- 
duction process is, thereby, discovered to be in the degraded state. Hence, a degraded state of 
the production process will go undetected at the testing occasion k/r if, and only if, each of the 
/ items sampled at time k/r is, by chance, not defective. But the probability of that event is 
(1 — 8)'. Note that the elapsed time / between the entry of the production process into the 
degraded state and the test time k/r does not affect this probability. Hence, p, = p,{t) = 1 
(1 — 8)' which represents the probability that a test (the inspection of / items) will detect a 
degraded production process. Note that the choice of / affects both the test's reliability and 
cost. 

Expected Disutility 



Designate the expected long run disutility per unit time per member of a population 
screened with frequency r using a test of constant reliability {pit) = p, t > 0) by C(r,p). 
From Eq. (2.1), 



C(r.p) = Nkrp £ (1 

n=0 



n/r 



(n + D/r 



D{u)du. 



The question examined next is how do changes in rand p affect C(r,p). After that, expli- 
cit solutions are given for C(r,p) when £>(■) takes certain simple forms. And lastly, some gen- 
eral rules are indicated for selecting between a particular kind of test and a more expensive but 
more reliable alternative test when £)(•) takes certain forms. 

PROPOSITION 1: If D() is a strictly increasing function, [dC(r,p)]/dp < and 
[BC(r,p)]/Br < 0. 

Note that for £>(•) nondecreasing, the above inequalities still hold, but not strictly. 

From this proposition, as anticipated, when p increases, the expected disutility decreases. 
Similarly as r increases, the interval \/r between tests decreases and the expected disutility 
decreases. Consequently, as better test types are used or the tests are more frequently applied, 
the value of such changes in terms of reduced disutility versus the costs of the changes can, in 
principle, be assessed and the tradeoffs evaluated. 



It is easy to compute the Hessian for C(r,p) when D{) is differentiate: 



B 2 C _ 
Br 2 

B 2 C 

Br 2 



NXp £ qT 3 



n=0 



(n + l) 2 D' 



(n + 1) 



- n 2 D' 



= N\r £ n(n - 1) q"~ 2 [W{n,r) - W{n - \,r)}. 



«=2 



B 2 C 
BpBr 



NX £ [1 - (n + Dp] q"~ ] [W(n,r) - B(n + \,r) + B(n,r)], 



TEST SELECTION FOR MASS SCREENING 49 

where 

q = 1 - p, W(n,r) - f " ' D{s)ds, and B{n,r) = - D(n/r). 

J n/r r 

Note that if £>'(•) is increasing, then (d 2 C/dr 2 ) ^ and (d 2 C/dp 2 ) > 0. Hence, along 
coordinate directions C(r,p) is convex. 

Simple expressions for C(r,p) can be given when £>(•) is specialized to a polynomial or an 
exponential function. Since these two types of functions are reasonably general, they can be 
quite useful as realistic approximations in applications. 

/ 
PROPOSITION 2: If DO) = £ a, t ', then 

„ n C(/V^) = JV\/> £ k/r m '(m, + 1)] £ « m ' + V-'- 

If the ntj; i ' = 1, . . . , Ware positive integers, the inner summation of Eq. (3.1) is simply 
the (m, + 1) moment of a geometric random variable. Hence, using Laplace transforms, Eq. 
(3.1) becomes 

(3 2) C(r,p) = NXp £ [ajr^im, + 1)] ^ m ' + ' (0:p) 

where tyif.p) = /»'/(l - qe') and i// (m) (0:/j) = </ M ty(t:p)ldt m |, =0 . For example, when w = 1, 
C(r,/j) = (aN\/2r)[(2 - p)/p] and for m = 2, C(r,p) = [(flyVA/3r 2 )] [1 + (6 - 6p)/p 2 ]. 

PROPOSITION 3: If D(t) = fie" 1 for a. > 0, then C(r,p) = BNkpr(e a,r - I)/ 
ail- qe alr ), for r > -a/log(^). 

Test Selection 

Propositions 2 and 3 can provide a means to select between two alternative kinds of tests 
which differ with respect to reliability of detection and cost per application. Let test No. 1 have 
cost per application c x and reliability p x . The corresponding parameters for test No. 2 are c 2 
and p 2 . If test No. 1 is administered with frequency rci/c^ and test No. 2 administered with 
frequency r, both testing regimes will consume equal quantities of the budgeted resource; vis., 
Nrc 2 per unit time. If C{rc 2 lcx,px) ^ C(r,p 2 ) for all r ^ 0, then the expected disutility per 
unit time will be less with test No. 1 at all levels of budget Nrc 2 . That is, if test No. 2 is being 
used with frequency r, the expected disutility can be decreased without any additional allotment 
of budget, simply by switching to test No. 1 and testing as frequently as the budget permits. 

Suppose, for example, that D{t) = at m for m a positive integer. Then C{rc 2 /c x ,p x ) ^ 
C(r,p 2 ) is equivalent by Eq. (3.2) to 

<3 ' 3> (C|/C2> * „ ♦<-> (0:„> ' 

Therefore, test No. 1 is preferred over test No. 2 if, and only if, Eq. (3.3) obtains. 



50 JA. VOELKER AND W.P. PIERSKALLA 

SECTION 4: THRESHOLD TEST RELIABILITY 

In the previous section^ the reliability of the test depended only on factors intrinsic to the 
test itself and did not depend at all on the elapsed time since incidence at the time of the test. 
In this section, a special form for pit) is considered which is very different from the case of 
constant test reliability. Here the test reliability is zero if the elapsed time since the defect's 
incidence is less than T; otherwise, the reliability is one. That is, pit) = 8[ roo) it) where the 
number T\s a characteristic of the type of test chosen. 

For a screening program in which a test with the above reliability characteristics is applied 
with, frequency r, let A ir,T) represent the expected long run disutility per unit time. 

Of course, the "blind period" of the test for to Tdoes not, in a mass screening situation, 
delay detection of each arriving defect exactly T units of time. The amount of delay depends 
on the interplay among the time of arrival of the defect, the testing schedules {\/r,2/r, ...}, 
and the magnitude of T, 

The primary results in this section are a simple characterization of Air,T) and rules 
which, in some cases, will permit selection betwseen two tests which differ in their reliability 
(i.e., in their detection threshold T) and in their cost per application. 

Long Run Disutility 

PROPOSITION 4: If pi-) = 8 [Too) i-) for some T > 0, then 

Air,T) = Nkr j DiT + u) du. 

Suppose, for example, that Dit) = cxpiat) for a > 0. Then, Air.T) = 
Nkr e\piaT)[expia/ r) — \]/a. 

Test Selection 

Suppose the decision maker has two kinds of tests available and he must choose one of 
them for implementation in a mass screening program. Suppose the first kind of test— call it 
test No. 1— has sensitivity characterized by the "time-until-detectability" threshold T\. Let C] 
> be the cost per application (to an individual unit) of this kind of test. For the second kind 
of test under consideration, test No. 2, let T 2 and c 2 be the corresponding parameters. 

Assume test No. 1 is better in the sense that T x < T 2 . To avoid triviality, assume 
c, > c 2 . 

If the exact shape of the function Di) is known, Proposition 4 can be used to decide 
which test to use for each possible level of budget. Let b be the budget per unit time per indi- 
vidual in the population. Then the use of test No. 1 will permit a testing frequency of b/c f and 
the use of test No. 2 permits frequency b/c 2 . To decide which test to use, compare the 
expected disutilities per unit time assuming a fully allocated budget, i.e., compare Aib/t^.T]) 
and A ib/c 2 ,T 2 ). With Di-) known, these quantities can be evaluated explicity by Proposition 4 
and compared. 

It is clear that the entire budget should be allocated because, when Di-) is an increasing 
function, 



TEST SELECTION FOR MASS SCREENING 5 1 



4~ A(r,T) = Nk 
dr 



f ' D(T + u) - (1/r) D(T + -) du 



< 0. 



When the exact form of the disutility function is not known, Proposition 4 does not 
suffice to select between tests No. 1 and 2. However, the two following theorems will permit 
such a determination at least for certain relative configurations of budget, relative test sensi- 
tivity T 2 — T u and cost differential c 2 — c, of the tests. 

Specifically, Proposition 5 will show that for any (increasing) disutility function, test No. 1 
is indicated if the budget (per unit population) exceeds (c^ - c 2 )/(T 2 - T\). On the other 
hand, Proposition 6 shows that for a convex increasing disutility function, test No. 2 is better if 
the budget is less than (c, - c 2 )/[2(T 2 - T,)]. 

A decision rule for the case where the budget falls between (c] — c 2 )/(T 2 — T x ) and 
(<?! — c 2 )/[2(T 2 — T,)] has not been found for general disutility functions. 

Just how the statement of Propositions 5 and 6 are translated into the above decision 
rules is explained after the statements of the respective theorems. 

PROPOSITION 5: Given £>(•) a strictly increasing function, F, < T 2 and c, > c 2 , then 
T 2 — T x > (c x — c 2 )/rc 2 implies 

(4.1) A(rc 2 /c x ,T x ) < A(r,T 2 ), 

making test No. 1 preferable at the per unit population budget level of rc 2 . 

To apply test No. 1 with frequency rc-Jcy versus test No. 2 applied with frequency r 
(actions reflected, respectively, in the left- and right-hand sides of Eq. (4.1)) would require the 
same budget, b = rc 2 , per unit population. The hypothesis of theorem 2 implies b = rc 2 > 
(c\ — c 2 )/(T 2 — T{). With the hypothesis in this form, the theorem provides a lower bound on 
the budget which is a sufficient condition for test No. 1 to entail lower expected disutility per 
unit time vis-a-vis test No. 2, were the two tests scheduled at their maximal (subject to budget) 
frequencies rc-J C\ and /\ respectively. 

The following lemma is needed for the proof of Proposition 6 and is recorded here for 
general interest. 

LEMMA 1: If /is a convex function, the (\/y)f " f(s)ds < (1/2) [fU) + fit + v)]. 

PROPOSITION 6: If £>(•) is convex and increasing, T x < T 2 and c, > c 2 , then T 2 - T x 
< (c { - c 2 )/2rc 2 implies A(rc 2 /c u T l ) ^ A(r,T 2 ), making test No. 2 preferable at the per unit 
population budget level of b = rc 2 . 

The hypothesis of this proposition implies b = rc 2 < (c x - c 2 )/2(T 2 - T { ). Hence, Pro- 
position 6 indicates the superiority of test No. 2 when the budget per unit population is less 
than (c, -c 2 )/2(r 2 - T,). 

REFERENCES 

[1] Barlow, R., L. Hunter, and F. Proschan, "Optimum Checking Procedures," Journal SIAM, 
Vol. /7, No. 4, pp. 1078-95 (1963). 



52 J A. VOELKER AND W.P. PIERSKALLA 

[2] Butler, David A., "A Hazardous Inspection Model," Technical Report No. 187, Depart- 
ment of Operations Research and Department of Statistics, Stanford University (1977). 

[3] Derman, C, "On Minimax Surveillance Schedules," Naval Research Logistics Quarterly, 
Vol. 8, pp. 415-9 (1961). 

[4] Keller, J., "Optimum Checking Schedules for Systems Subject to Random Failure," 
Management Science, Vol. 21, pp. 256-fcO (1974). 

[5] Kirch, R. and M. Klein, "Surveillance Schedules for Medical Examinations," Management 
Science, Vol. 20, pp. 1403-9 (1974). 

[6] Lincoln, T. and G. H. Weiss, "A Statistical Evaluation of Recurrent Medical Examina- 
tions," Operations Research, Vol. 12, pp. 187,205 (1964). 

[7] Pierskalla, W.P. and J. A. Voelker, "A Model for Optimal Mass Screening and the Case of 
Perfect Test Reliability," Technical Report #3, Department of Industrial Engineering 
and Management Science, Northwestern University (1976). 

[8] Roeloffs, R., "Minimax Surveillance Schedules with Partial Information," Naval Research 
Logistics Quarterly, Vol. 10, pp. 307-22 (1963). 

[9] Roeloffs, R., "Minimax Surveillance Schedules for Replacement Units," Naval Research 

Logistics Quarterly, Vol. 14, pp. 461-71 (1967). 

[10] Shwartz, M., and H. Galliher, "Analysis of Serial Screening in an Asymptotic Individual to 

Detect Breast Cancer," Tech. Report, Dept. of Ind. and Operations Engr., College of 

Engineering, University of Michigan (1975). 

[11] Thompson, D., and R. Disney, "A Mathematical Model of Progressive Diseases and 

Screening," presented at the November 1976 ORSA TIMS Meeting. 
[12] Voelker, J. A., "Contributions to the Theory of Mass Screening," Ph.D. Dissertation, 
Northwestern University (1976). 



APPENDIX 



PROPOSITION 1: With q = 1 - p 

a 



C(n,p) = Nkr T q"W{n) - Nkrp Y n q"~ x W{n) 
9P „- 



n=0 



where 



(n + \)lr 



W{n) = f D(s)ds. 

J nl r 



Now let V = Wq, V„ = W n - W„_ u n — \, 2, Note that £>(•) non-negative increasing 

implies Vj > for J - 0, 1, 2, . . . . Then W n = £ V, and 

y-o 

-f- C(r,p) = N\r £ <?"£ V, - Nkrp £ n q"~ l £ V, 
QP n =o /-o 



n=0 



7=0 



CO CO 



7=0 n=j n=j 

oo 

= -Nkr £7 Q J ~ l Vj < 0. 

7=0 

-f- C(r,p) = Nkp £ q" [ , D(s)ds 

or *T n J "/r 



«=0 



- A^Xp X q" 

«=0 



n + 1 



D 



n + 1 


-±D 


n 


r 


r 


r 



TEST SELECTION FOR MASS SCREENING 

(n + \)lr 



53 



= NXp £ q" 
»=0 



f D(s)ds-- 

J njr /■ 





n + 1 




n 


D 




- D 






r 




r 



D 



n + 1 



< 0. 



This inequality follows from £>(•) increasing through the relations D[{n + \)/r] — D(n/r) > 
and 



(n + \)lr 



f , D(s)ds - — D 

J njr /• 



n + 1 



(n+l)/r 



< 



L D 



n + 1 





1 




n + 1 


ds - 




D 






r 




r 



= 0. 



Q.E.D. 



(n+\)lr , 

PROOF OF PROPOSITION 2: C(r,/>) = NXrp T <?" I Y a, /'"' 

n = J " /f ^i 



<// 



/ 
= NXrp£ £ fl/ ^/(m,+ 1) 



= NXp £ a,/r m '(w, + 1) 



- 1) 


A7 + 1 


w,+ l 




m,+l' 


oo oo 


n=l 




n=l 







m +1 



= JV\p £ <i,./r '(«, + 1) £ «' " r * /> q'" 

/-I n=\ 



Q.E.D. 



oo 

PROOF OF PROPOSITION 3: C(r,p) = NXp T q k — e aklr (e a/r - 1) 

* to a 

= H±PL {e a/r_ l) ^ {q e alr )k 



k=0 



The geometric series £ (q e alr ) k converges if, and only if, r > — a /log q. Therefore, for r > 

A-=0 

-a /log q, 



C (r,p) = ^^ (e a/r -l) 



1 



1 - q e alr 



Q.E.D. 



m 



PROOF OF PROPOSITION 4: Note that 

n-\ 

n i - 8(7-00) 

m = l 

ByEq. (2.1), 

A(r,T)-N\r J,] D(u)b lT ^ (u) ]J 

, •'(« — l)/r -*■-*■ 



= 1-8, 



r.oo) 



n-\ 



n = \ 



1 -8, 



,«//■ 



= "'i:J (fl _ n/ ,£<">8[r.~)(«) 

«=i 



1 ~~ 8 ir. 



oo) 



r.oo) 



w — 



w — 



m 



du 



du. 



54 



J.A. VOELKER AND W.P. P1ERSKALLA 



Now u < T + (1/r) «*£> u - (1/r) < T «*■> 1 - 8 [Too) [u - (Mr)] = 1. Therefore, 

J.oo 
D(w)S (roo) (u) 8 [ _ 00 ,r+i/r)(«) du 

r T+\lr 

= WXr J D(u)du Q.E.D. 



PROOF OF PROPOSITION 5: By Proposition 4, Eq. (4.1) is equivalent to 
(rc 2 /c,) J'' " 2 Z)(r, + 5) ds < r J" 'z)(r 2 + 5) eft. 

By hypothesis, T 2 — T\ > (c, - c 2 )/(rc 2 ) which implies T 2 -T x > u(c x -c 2 )/c 2 for 
u € [0,1/r] or T 2 - T x + u > uc x /c 2 for u € [0,1/r]. Since £>(•) is increasing, this inequality 
implies 

rCi r c./{rc 7 ) ~\/r 

— D(T, +s) * = rl D(T X + uc x /c 2 ) du 

C, •'0 •'0 

< r J* r Z)(7, + (r 2 - 7, + «)) </m = /■ J ' D{T 2 + u) du. 

Q.E.D. 



PROOF OF LEMMA 1: Let h{t + s) = fit) + (s/y)[f(t + y) - f(t)], s € [0,v]. 

1 r ,+j ' 1 r' +v ' 

Since /is convex, /(/ + 5) ^ h(t + s), s 6 [0j>]. Hence, — f(s)ds ^ — h(s)ds 

2 

Q.E.D. 



= - r />(? + 5)* = \[/U+y) + /(*)]. 

y J ° 



PROOF OF PROPOSITION 6: By Proposition 4, it suffices to show 

(rcJci) f ' 2 D(T X + u) du ^ r f D(T 2 + u) du. 

•/o •'0 

Letting /(/) = D(T X + t) and T = T 2 — T u this becomes 



(rc 2 /c x ) f ]n2 f(s)ds > r f ' f(T + s)ds 

*J •'0 



(A.l) 

'0 ' ^0 

Note that £)(•) being an arbitrary convex increasing function implies /(•) is also convex 

increasing. 



Let x = c x /rc 2 and let a = c 2 /c x < 1. Then Eq. (A.l) becomes 



T+ax 



J 1 / -t-ax t» x 

t f(s)d S < (1/x) J o /(*) 



rfs 



which will follow from 



1 /• / + ax 

ox ^ ' 



1 



X, 



2 7!+ax 



/(s)<fc < 



1 



f* As)ds 

♦'0 



(A. 2) ax j t * ' ^ 2T + ax J o x 

Note that the above expressions are all average values of /(s) over their respective intervals of 
integration. The right-hand inequality follows from /(■) increasing and IT -V ax ^ x. To 
prove the theorem, it only remains to estabilsih the left-hand inequality in Eq. (A. 2). 



(A.3) 



T+ax 



1 /• 2 / +ax 1 /• / +ax 

l ~— J As)ds-^- J As) 

-4- /tv •'0 nv v I 



2 T + ax J o 



C T _d_ 1 r 

~ Jo W8 ?s 4- ^r Jr 



ox •'J 

T+ax +8 



rfs 



28 -I- ax 



/(s)<fc 



^8 



TEST SELECTION FOR MASS SCREENING 



55 



d 
db 



1 /. T+ax+h 

4- TX Jt 



ax+28 J T-i 



ax + 28 



y (/(F-8) + /(T + ax +8)) 

-^T I f(s)ds\ 

+ 28 j t-s J J 



ax + 16 "z-8 
which is non-negative for 8 6 [0,71 by Lemma 1. Therefore, (A. 3) is non-negative and 

1 f 2 T+ax i y» T+ax 

-7 J /(*)* > — L fU)ds. 

+ ax **0 ax ** T 



IT + ax 



Q.E.D. 



OPTIMAL ADMISSION PRICING POLICIES 
FOR M/E k /1 QUEUES 

Michael Q. Anderson 

The Robert O. Anderson Schools of Management 

The University of New Mexico 

Albuquerque, New Mexico 

ABSTRACT 

This paper extends the Low-Lippman M/M/l model to the case of Gamma 
service times. Specifically, we have a queue in which arrivals are Poisson, ser- 
vice time is Gamma-distributed, and the arrival rate to the system is subject to 
setting an admission fee p. The arrival rate \(p) is non-increasing in p. We 
prove that the optimal admission fee p* is a non-decreasing function of the cus- 
tomer work load on the server. The proof is for an infinite capacity queue and 
holds for the infinite horizon continuous time Markov decision process. In the 
special case of exponential service time, we extend the Low-Lippman model to 
include a state-dependent service rate and service cost structure (for finite or 
infinite time horizon and queue capacity). Relatively recent dynamic program- 
ming techniques are employed throughout the paper. Due to the large class of 
functions represented by the Gamma family, the extension is of interest and 
utility. 

In this paper, we consider an M/E k /\ queuing system in which the arrival rate to the sys- 
tem is subject to control by the service facility. Low [10] considered a similiar control model 
for M/M/C systems and Lippman [6] has approached the Low model using new techniques. In 
this paper we extend Lippman's treatment (see pp 707-708 of [6]) to the case of gamma service 
time distribution. 

The model we treat is the following. Customers arrive to a single server infinite capacity 
queue according to a Poisson process having rate \(p), where p is the decision variable 
representing the price charged for admission to the system. We take p € P and assume X is a 
nonincreasing function of p. The service time is gamma with parameters (k,/x) (defined such 
that the mean service time is kf/x). We think in terms of the customer going through k serial 
phases of service, each having exponential service time with rate /x > 0. Furthermore, we 
assume that these phases are observable by the decision maker (See "Applications" below, and 
the comment following Theorem 2). We define the state of the system to be the number of 
phases of service left to complete. Thus, if there are c > 1 customers in the system, and the 
customer being served has / phases of service remaining, the state of the system would be 
(c — \)k + f. Denote the state by / and the state space by S = {0, 1, 2, . . .}. 

The cost structure has two components. First the admission price p is treated as a cost 
-p. Secondly, there is a holding cost h(i) defined as a function of the state variable. The 
optimality criterion is the minimization of expected discounted cost. We treat the infinite time 

57 



58 M.Q. ANDERSON 

horizon. This defines our model as a continuous time Markov decision process (see Ross [11], 
Bertsekas [2]). 

Our aim is to prove that the optimal admission price is a nondecreasing function of the 
state variable. In addition, we extend the results of the Low-Lippman model [6] to include 
state-dependent service rates and service cost rate in the special case of exponential service dis- 
tributions (k = 1). 

In concluding this summary we remark that the techniques employed in this paper 
represent an effort to employ more general dynamic programming tools which have recently 
appeared in the literature. Lippman [6] notes the desirability of a more general class of tech- 
niques applicable to queuing optimization problems. In this paper we have formulated our 
problem in the general setting of Topkis' framework for analyzing monotone policies [13], 
Lippman's approximating techniques for finite and infinite horizon processes [6,8], and 
Lippman's recent result on unbounded 1 -period cost functions [7]. 

APPLICATIONS 

Since an arrival to the system by a single customer results in the state variable increasing 
by k units, we may view the model as applying to batch arrivals of jobs to a single facility server 
who services each job in series (at an exponential service rate), completing the entire batch 
before starting work on the next batch. This would apply to batch arrivals by identical jobs to a 
machine for processing. Or, taking the other view that arrivals to the system represent single 
jobs, the gamma service time model would apply to situations where the server must perform a 
series of k well-defined (observable) processing operations on each job, where each of the k 
subtasks are (essentially) identical in service time requirements. 

The entrance price introduces a revenue component to the models. Holding cost might 
reflect costs of in-process inventories or higher machine (or service facility) maintenance cost. 

Finally, the extension of the Low-Lippman model to accommodate state-dependent arrival 
rates is applicable to two general classes of situations: (1) Where a single service facility adjusts 
its service rate in response to increased work load by speeding up or slowing down, and (2) 
where such adjustment is made by adding or relieving additional service facility support (for 
example, when the service facility is a team, by adjusting the number of team members-at pos- 
sibly a higher service cost rate). 

1. EQUIVALENCE BETWEEN CTMDP AND DTMDP 

In this section we summarize a new technique for transforming a continuous time Markov 
decision process (CTMDP) to an equivalent discrete time Markov decision process (DTMDP). 
Lippman appears to be the first to employ the method for queuing optimization problems [6]. 
See also [8], Serfozo [12], Kakumanu [4], Winston [14], and Anderson [1]. 

For the following discussion refer to Ross [11] or Bertsekas [2]. A DTMDP is defined by 
specifying four objects: a state space S, a collection of action spaces {A,:i € S), transition proba- 
bilities {Pjj{a)\i, j € s, a € yl,}, and cost functions [C(i,a);i € S, a € A/}. For a CTMDP, in 
addition to these four objects we must specify transition time probability distributions 
{Fjj(a):i, j € S, a € A,} where F tJ (a) is an exponential distribution having parameter \(i,a). 
Also costs C(i,a) are realized per unit time. For either process we specify a discount factor 
a, < a < 1 and define V to be the optimal return function: V(i) equals the minimum 



OPTIMAL POLICIES FOR M/E*/l QUEUES 59 

ected discounted cost realized over an infinite time horizon starting in state i € S. We may 
1 define the following functional equations of dynamic programming. 

For the DTMDP, < /i < + «>, and / 6 S, 

V o (i) = 

V n+{ (i) = mm{CU,a) + £ P^a) V H (J))/(A + a) 
a<iA i y-o 

; define V x = V.) 
the CTMDP, 



V(i) = min{C(/,a) + tf(/,a)£ P u {a) V(j)} 



a * A i j*i 

:re 

q(i,a) = k(i,a)/(a + X(/,a)) 



C(i,a)= C(i,a)/(a +\(i,a)). 

iolicy is a rule for choosing actions, i.e., is a function 8:7" x S —> A where 7 is the time axis 
he process (T = {0, 1,2, . . .} for DTMDP and T = [0,L], L ^ + oo for the CTMDP), and 
/) € 4 for each /' € 5. Define V b to be the return function when using policy 8. An 
imal policy 8 * is one satisfying V h * = K 8 * is stationary if it is constant on the factor T. 
iditions which insure the existence of an optimal stationary policy are specified by the well- 
iwn contraction and monotonicity properties of Denardo [3]. When an optimal stationary 
icy exists, V then satisfies equation (2) (in the case of the DTMDP) or (3) (in the case of 
CTMDP). 

Given a CTMDP for which an optimal stationary policy exists, assume that 
= sup {X (/,«):/ €5, a € A] < + °o. 



ine 



Pv'ia) = \ 



(A-X(/,o))/A i = j 
\(i,a)P u (a)/A i*j 



an then be shown (see [6,8], [4], [12], [5, p. 121], or [1]) that the CTMDP is computation- 
equivalent to a DTMDP having the same state and action spaces, cost function 
,a)/(A + a), transition probabilities Pi/ia), and discount factor A/(A + a). Thus, the 
mizing actions for the CTMDP can be computed from the following functional equations, 
i€S, 

V(i) = min{C(/,fl)/(A + a) + (A/ (A + a))£P y '(a) V(j)} 

a£Ai 

(3), \(i,a)Pij(a) is typically referred to as the rate of the transition i — j). For a treatment 
CTMDP having lump sum costs see [1] or [12]. 

10NOTONE OPTIMAL POLICIES 

In a somewhat technical paper, Topkis [13] presents a general framework for analyzing 
lotone optimal policies in optimization problems. For our purposes here, we specialize his 



60 M.Q. ANDERSON 

results to the problem we will be treating. Let J be a function defined on A x 5, where A , 
B are lattices. A function J is said to be submodular on A x B if for a x , a 2 € A, b x , \ 
€ B, a 2 ^ a\, b 2 ^ b\, J(a h b\) + J(a 2 , b 2 ) < J(a h b 2 ) + J(a 2 , b\). Suppose we h 
the following optimization problem, f(a) = minJia, b). Let b*(a) be the infimum of the 

bZB 

of points in B which minimizes J (a, b). Then under certain conditions on A, B, and J, Toi 
shows that to establish that b* is non-decreasing in a, it is sufficient that J be submodular 
A x B. (See Theorem 6.2 of [13] for details.) 

3. AN MIEJX QUEUE WITH CONTROLLED ARRIVAL RATE: 
THE INFINITE TIME HORIZON PROBLEM 

We make the following assumptions: 

(1) p G P, where Pis a compact subspace of [0, + °°]. 

(2) A. is a nonincreasing continuous function of p € P. Also for p € P, ^ X (p) 4 
oo. 

(3) // is nondecreasing in / 6 S = [0,1,2 .. .} and satisfies h(i + k)-h(i + k ■ 
> //(/) -//(/ -1), /> 1. 

Notation: Let /be a real-valued function on (0, 1,2, . . .}. Define A/(/) = /(/') - /(/' - 1), 
1. Define A = sup k(p) + /a. 

From Section 1, the equivalent discrete time recursions for our CTMDP are easily seen tn 
the following: 

For / <E S, 

(6) M') = o 

(7) y„d) = min J(n,i,p)/ (A + a), n > 

p 

where J is defined as follows: 
For i = 0, and ^ n ^ + °°, 

(8) J(n,0,p) = -pk(p) + k(p)V n _ ] (k) + V„-i(0)(\-\(p)), 
and for / > 0, 

J(n,i,p) = h(i) - pk{p) +k(p)V n _ l (i + k) +nV„_ l (i- 1) 

(9) + V^iOiA- kip) -fi) 

Notation: In (7) define the infimum of the minimizing actions by p* (/>) for n < + °°n 
p*(i) when n = + <*>. Also write V x = V and when n = + oo W e suppress n in the li] 
arguments for J, i.e., we will write J{i,p). 

THEOREM 1. There exists an optimal stationary policy to the CTMDP defined above 

PROOF. The primary consideration here is the presence of the unbounded 1 -period) 
function h. Although Denardo's results [3] do not directly apply, Lippman [7] has establi* 
conditions under which Denardo's contraction and monotonicity results do apply. In our cl 
is trivial to verify that Assumptions 1,2, and 3 of [7, p. 1227] are satisfied, and the theori 
established. 



OPTIMAL POLICIES FOR M/E*/l QUEUES 61 

We now establish two supporting lemmas. 

LEMMA 1. For < » < + oo and / ^ 1, A V„(i) > 0. 

PROOF. We prove the lemma for finite n (by finite induction) ; the case for infinite n then 
follows by taking limits. From (6) the result holds trivially for n = 0. First assume / > 1 and 
define p = p*(n,i). Then from (7) and (9), 

K0) ~ V n (i ~ 1) > Jin,i,p) - Jin J - ,p) 

= A/;(/') +AK, H (/ + k)k(p) + AK„_,(/)(A -kip) - fi) + AK„_,(/ - Dm 

^ 0, where the inequality follows from Assumption (3), the definition of A, and the inductive 
hypothesis AV n _ x ~^0. When / = 1, the same computation holds except that the term 
A V„-\(i — D/x, is not present. ■ 

The next result establishes a condition on V n which is somewhat weaker than the convex- 
ity required in Lippman's treatment [6, p. 708, Theorem 15]. 

LEMMA 2. For < n < + °o and / ^ 1, A V n (i + k) ^ A V n {i) 

PROOF. For n = 0, the result holds trivially. Assume it true for n — 1 < + °o. Define 
p x = p*(n,i-\) and p 2 = p*(n,i + k). Then for / > 1, from (7), (8) and (9), 
A V„U + k) - A V„U) > U(n,i + k,p 2 ) - J(n,i + k - \,p 2 )] - [J{n,i,p x ) - J{n,i - \, Pl )] 

= AM/ + k) - Ah(i) 

+ AK„_ 1 (/ + 2A:)\(p 2 ) 

+ AF„_,(/ + k)(A-ti - k(p 2 ) - \(pi)) 

+ AK„_ 1 (/ + k- l)fi 

+ AV„_ 1 (i)(-A+ f jL+k(pO) 

> A V n _ x (i + k - \)/x > 0, where the next to last inequality follows from Assumption (3), the 
inductive hypothesis and the definition of A , and the last inequality follows from Lemma 1 . 

For / = 1, a similar argument combined with the result for / > 1 yields the desired result. For 
n = + oo^ the result obtains by taking the limit as n — - +°o on the finite case. ■ 

We now establish our first main result, 

THEOREM 2. The optimal entrance price is a nondecreasing function of / € S, i.e., 
p*(i + 1) > p*(i). 

PROOF. From Section 2, it suffices to show that J is submodular on S x P. Let p x ^ p 2 . 
Define A ip by 

A = Jii.Px) + JU + \,p 2 ) - J(i,p 2 ) - J(i + \,p x ). 

Then from (9), for / > 0. 

A ip = (AV(i+ l)-AV(i + k + l))(k( Pl )-k(p 2 )) 

< 0, where the inequality follows from Assumption (1) and Lemma 2. when / = we must 
use (8) and (9) and the same computation results. To establish the general case 



62 M.Q. ANDERSON 

JU,P\) + J(i + j.Pi) — Jii.Pi) — J(i + j,Pj) ^ 0. Note that this expression can be written as 

^ A j+Lp each term of which is nonpositive by the case above. ■ 

/=o 

Thus, the greater the "load" on the server (as measured by / € 5), the greater is the price 
charged for admission to the system and thus, the lower the arrival rate. From the definition of 
the state variable we conclude that the optimal entrance fee p* is nondecreasing in both the 
number of customers in the system and the number of phases of service remaining on the cus- 
tomer currently being served. Along these lines, as a special case, we can take // to be a func- 
tion only of the number of customers in the system by defining h to be constant on each seg- 
ment. [Ok), [k, 2k), ... and requiring that the restriction of /; to the set {0,k,2k, . . .} be con- 
vex. It is easily verified that all proofs go through. 

It would be of interest to extend Theorem 2 to the truly finite horizon CTMDP. However 
we have thus far been unable to do this. Using results of Lippman ([8]— see the connectedness 
condition on page 483 and also Theorem 4 of that paper) the monotonicity of p* for the finite 
time horizon would follow if we could prove that J is submodular in n and /?, and this in turn 
would follow from AV„(i) > AV„-\(i). We have been unable to establish this last inequality 
(or its reverse which would also be sufficient). 

4. THE LOW-LIPPMAN MODEL WITH STATE-DEPENDENT SERVICE RATES 

Once again we consider Lippman's model [6, p. 707], this time incorporating a state- 
dependent service rate and a state-dependent service cost. Specifically we have an M/M/\ 
queue with queue capacity Q, where Q ^ + °°. An entrance price p is charged and the arrival 
rate A. is a function of p. The state of the system is the number of customers present and is 
denoted by / € S = {0, 1, ... , Q). Given the state is /, a holding cost //(/) per unit time is 
incurred. Furthermore, we shall assume that the server works at rate p., and incurs a service 
cost C((jl,) per unit time when the state is /'. 

The functional equations corresponding to (6), (7), (8), and (9) are, for Q = + oo, 

M') = 

V„{i) = min J(n,i,p)/(A +a), n > 

For / = 0, < /? ^ + oo 

J(n,Q,p) = -pk(p) + \(p)V n _ ] (\) 

+ V„-i(0) (A -k(p)) 

and for / > 0, 

J(n,i,p) = /;(/) + C(fii) - p\(p) + \{p)V„_ x (i + 1) 

+ (it K fl _,(/ - 1) + K„ _,(/)( A -ii,- \(p)) 

when Q < + °°, and / = Q, the transition / — » / + 1 has rate zero. In addition to Assumptions 
(1) and (2) of Section 3 above, we make the following: 



(2b) When Q < + oo we make the convention that p*(Q) = sup [p £ P] and 
X(p*(Q)) =0. 



OPTIMAL POLICIES FOR M/E*/l QUEUES 63 

(3) // is nondecreasing and convex in / € S = {0,1,2, ... ,Q) 

(4) C(/JL,) is convex increasing in / € S — {0, 1,2, ... ,Q}. (As a special case we may 
take CifjLi) =0). 

(5) ix i is concave in i and satisfies 2/x, — /x 2 ^ 0. When Q < + °° we require addi- 
tionally that 2/ip — ixq ^ sup {\ (/?)}. When Q = + °o 5 in order that A be finite 
we assume thatyu., is non-decreasing in i. 

When fij is taken to be non-decreasing in /, this may be interpreted in terms of 
the server choosing to increase the service rate (at higher costs in case 
C(jjlj) > 0). The case where /a, is non-increasing in i would represent a system 
where the effectiveness of the service facility decreases as the work load increases. 

We may now prove the analogs of all the previous results. The proofs obtain by employing the 
same techniques as before. We remark that A is defined to be the sup [k(p) + //,,} < + °° (by 

Assumptions (2) and (5)). Lemma 1 is straightforward. Lemma 2 is restated: 
A^„(/) ^ AF„(; — 1) i.e., V n is convex in i. To establish this requires Assumptions (4) and 
(5). In Theorem 2, we establish submodularity as before; when Q < + °° we observe 
Assumption (2b). Furthermore, for this model we may prove that p* is non-decreasing in i for 
the finite horizon continuous time problem as follows. We first show that p* is non-increasing 
as a function of n by establishing that J is submodular in n and p. This last condition will fol- 
low from AK„(/) ^ AK„_,(/) which is easily proved. We then invoke Lippman's connected- 
ness condition to establish Theorem 4 of [8] (see page 483). In every instance we observe the 
various special cases (/ = 0, /' = Q < + <»). We summarize these remarks in the following 
Theorem. 

THEOREM 3. In the Low-Lippman Model [6, p. 707] with state-dependent service rates 
and service costs, under Assumptions (1) through (5), and for finite or infinite queue capacity, 
in both the finite and infinite time horizon CTMDP, the optimal entrance price is a non- 
decreasing function of the number of customers in the system. 

We note that the special case C = and h linear was treated by Lippman and Stidham 
[91. 

5. TOPICS FOR FURTHER RESEARCH 

We have thus far been unable to entend Theorem 2 to either the finite queue capacity 
case or to the state-dependent service rate model. A few numerical computations have been 
made but no counter-examples yet found. This might be a research topic of interest to other 
authors working in the area. 

In general the technique of transforming a CTMDP to a DTMDP appears to be a very 
effective tool for analyzing birth-death processes. However, when more general state transi- 
tions are allowed, complications seem to appear. The author in [1] has had some success in 
this respect in treating various machine maintenance models. Another difficult class of control 
problems (using the transformation technique CTMDP — DTMDP) are Markov decision 
processes involving two dimensional state spaces (such as one would have in treating tandem 
queues, or the operation of two machines, for example). This might be a rich field of research 
for authors interested in applying the new methods employed in this paper for analyzing Mar- 
kov processes with structure. 



64 M.Q. ANDERSON 

REFERENCES 

[1] Anderson, M.Q., "Monotone Optimal Maintenance Policies For Equipment Subject to Mar- 
kovian Deterioration," Doctoral Dissertation, Indiana University (1977). 

[2] Bertsekas, D.P., Dynamic Programming and Stochastic Control (Academic Press, New York, 
1976). 

[3] Denardo, E., "Contraction Mappings in the Theory Underlying Dynamic Programming," 
SIAM Review, 9, 165-177 (1967). 

[4] Kakumanu, P., "Relation Between Continuous and Discrete Time Markovian Decision 
Problems," Naval Research Logistics Quarterly, 24, 431-441 (1977). 

[5] Howard, R., Dynamic Programming and Markov Processes (John Wiley, New York, 1970). 

[6] Lippman, S., "Applying a New Device in the Optimization of Exponential Queuing Sys- 
tems," Operations Research, Vol. 23, 687-710 (1975). 

[7] Lippman, S., "On Dynamic Programming With Unbounded Rewards," Management Sci- 
ence, Vol. 21, 1225-1233 (1975). 

[8] Lippman, S., "Countable-State, Continuous-Time-Dynamic Programming With Structure," 
Operations Research, Vol. 24, 477-490 (1976). 

[9] Lippman, S., and S. Stidham, "Individual Versus Social Optimization in Exponential 

Congestion Systems," Operations Research, Vol. 25, 233-247 (1977). 

[10] Low, D., "Optimal Dynamic Pricing Policies for an M/M/S Queue," Operations Research, 

Vol. 22, 545-561 (1974). 
[11] Ross, S., Applied Probability Models Helden-Day, San Francisco, CA (1970). 
[12] Serfozo, R.F., "An Equivalence Between Continuous and Discrete Time Markov Decision 

Processes," Operations Research, Vol 27, 616-620 (May-June 1976). 
[13] Topkis, D.M., "Minimizing a Submodular Function on a Lattice," Operations Research, 

Vol. 28, 305-321 (1978). 
[14] Winston, W.L., Optimal Operation of Congestion Systems with Heterogeneous Arrivals 
and Servers, Ph.d. Dissertation, Yale University (1975). 



THE DYNAMIC TRANSPORTATION PROBLEM: A SURVEY 

James H. Bookbinder and Suresh P. Sethi 

Faculty of Management Studies 

University of Toronto 

Toronto, Ontario 

ABSTRACT 

The dynamic transportation problem is a transportation problem over time. 
That is, a problem of selecting at each instant of time /, the optimal flow of 
commodities from various sources to various sinks in a given network so as to 
minimize the total cost of transportation subject to some supply and demand 
constraints. While the earliest formulation of the problem dates back to 1958 
as a problem of finding the maximal flow through a dynamic network in a 
given time, the problem has received wider attention only in the last ten years. 
During these years, the problem has been tackled by network techniques, linear 
programming, dynamic programming, combinational methods, nonlinear pro- 
gramming and finally, the optimal control theory. This paper is an up-to-date 
survey of the various analyses of the problem along with a critical discussion, 
comparison, and extensions of various formulations and techniques used. The 
survey concludes with a number of important suggestions for future work. 



1. INTRODUCTION 

The classical transportation problem refers to the shipment of goods from a set of sources 
to a collection of sinks at a minimum cost. Analysis of these problems dates back to the works 
of Kantorovich [28], Hitchcock [26], Koopmans [31], and Dantzig [9]. Subsequently, much 
work has been done. (See e.g., Holladay [27], Potts and Oliver [35], Bradley [5], Christofides 
[7], and Kennington [29].) 

The classical problem is static in nature in the sense that shipments are instantaneous, and 
costs as well as supply and demand requirements are independent of time. A notational formu- 
lation of the static transportation problem appears in Section 2. 

That the dynamics of the transportation problem are important was first recognized by 
Ford [15] in his formulation of the maximal dynamic flow problem. (See also Gale [19] and 
Ford and Fulkerson [16].) Since then, a number of other dynamic extensions have been stu- 
died. These include the minimum-time transportation problem (Szwarc [44,45,46], Hammer 
[23], Tapiero and Soliman [51], Tapiero [49], and Srinivasan and Thompson [41]); the 
minimum-cost transportation problem (Bellmore, Eklof, and Nemhauser [2], Szwarc [47], Tapi- 
ero [48], Tapiero and Soliman [51], and Srinivasan and Thompson [41,42]); and the maximal 
dynamic flow problem, both for a single commodity (Ford [15], Ford and Fulkerson [16]) and 
in the multi-commodity case (Bellmore and Vemuganti [3]). 

65 



66 



J.H. BOOKBINDER AND S.P. SETHI 



It is the purpose of the present paper to survey the above work as well as some applica- 
tions of the dynamic transportation problem. A schematic representation of the various treat- 
ments of this problem is given in Figure 1. 



TRANSPORTATION PROBLEM 



DETERMINISTIC 



STOCHASTIC 



KANTOROVICH 11939] 
HITCHCOCK (1941] 
KOOPMANS 11947] 
DANTZIG 11951] 



COMBINATORIAL METHODS 



- HAMMER 11969] 

- SZWARC |1"71a| 

GARFINKEL AND 
RAO 119711 

- SRINIVASAN AND 
THOMPSON 119761 



DETERMINISTIC 



DANTZIG [1955] 
- WILLIAMS 119631 
SZWARC 11964] 



STOCHASTIC 



MIDLER 11969] 
ELLIS AND 
RISHEL (1974] 

SEGALL AND MOSS 11976] 



NON LINEAR PROGRAMMING 



FORD (1958] 

GALE [1959] 

FORD AND FULKERSON (19621 

SZWARC [1966] 

BELLMORE, EKLOF AND NEMHAUSER [1969] 

SZWARC (1970, 1971b] 

SRINIVASAN AND THOMPSON (19721 

BELLMORE AND VEMUGANTI [1973] 

FONG AND SRINIVASAN (19761 

SRINIVASAN AND THOMPSON (1977] 

Figure 1 



CONTROL THEORY 



- HAUSMAN 

AND GILMOUR [1967] 



FRANK [1967] 



FRANK AND 
ELBARDAI (1969] 



MAXIMUM PRINCIPLE 



TAPIERO (19711 

TAPIERO AND 
SOLIMAN 11972] 

TAPIERO (19751 



The organization of our survey is as follows. Section 2 establishes the basic notation and 
points out some of the possible ways in which the static problem can be generalized to the 
time-dependent case. Sections 3, 4, and 5 deal with the case of discrete time, and continuous 
time is treated in Section 6. 

Section 3 is concerned with the deterministic dynamic transportation problems, i.e., the 
minimum-time, minimum-cost, and the maximal dynamic flow problems. We consider in Sec- 
tion 4 a stochastic multiperiod, multimode model (Midler [34]), a problem of discrete-time sto- 
chastic optimal control which is solved via a dynamic programming algorithm. 

Section 5 deals with two illustrative applications of the discrete time dynamic transporta- 
tion problem. One involves a multiperiod truck delivery problem (Hausman and Gilmour 
[25]), and the other studies the optimal rescheduling of air traffic in response to stochastic 
influences (Ellis and Rishel [11]). 



Section 6 is concerned with the dynamic transportation problem over continuous time, 
and moreover, when explicit account is taken of the time delay between dispatch of a shipment 
at the sink and its receipt at the source. The appropriate framework is that of optimal control 



DYNAMIC TRANSPORTATION PROBLEM 67 

theory, and papers surveyed include those of Frank [17], Frank and El-Bardai [18], Tapiero and 
Soliman [51], and Tapiero [49]. 

Section 7 concludes the paper with extensive outlines and suggestions for future research. 
Portions of this work are already under way (Sethi and Bookbinder [38]). 

2. STATIC TRANSPORTATION PROBLEM 

By the static transportation problem, we shall mean the well-known linear programming 
problem 



Min i=£ £ % u v> 



i=i y= 



subject to 



£ Ujj = A, (Supply constraints) 
/-) 
in 
£ Ujj = Bj (Demand constraints) 

< u jt ^ c,j (Capacity restrictions). 

In this notation (which has been chosen to conform with that of the control-theoretic treatment 
of transportation problems), u u is the shipment from origin i to destination j, at a cost of q {j . 
Each source node / has a supply A h the sink nodes j require an amount Bj, and so a transporta- 
tion schedule is to be found that minimizes the total shipping costs while not exceeding the 
capacity c y of any arc (ij). It should be noted that the above problem refers to the shipment 
of a single commodity (or at least only a standard "product mix") and only a single mode of tran- 
sportation is considered for each link (i,j). 

There are a number of possible ways in which this problem can be made time-dependent: 

2.1. A, and/or B, Functions of Time 

If the supply capabilities or the demand requirements are time-dependent, then the production 
schedule can be smoothed by incurring the costs of holding inventory which was delivered 
early. Inventory variables could be allowed at sources as well as sinks, and there may be 
included upperbounds, possibly time-dependent, on those inventories. 

2.2. Qjj Functions of Time 

The remarks of the preceding paragraph are applicable here as well. 

2.3. Associate a Time t, 7 with Each u f j 

The notation t, 7 refers to the time required to ship the u l} units over the route (i,j), so that the 
relevant problem is the minimization of the time needed to satisfy all the demand require- 
ments. It should be noted that the costs q {J would not normally enter into this problem 
(though they could of course be taken into account via subsidiary constraints; see Glickman 
and Berger [21], and also Srinivasan and Thompson [42]). 



68 J.H. BOOKBINDER AND S.P. SETHI 

A convenient division of the papers in this survey is according to whether the analytical 
techniques involve discrete time or continuous time. We shall begin with the former, and in fact 
with a problem of type 3 above. 

3. DETERMINISTIC DYNAMIC TRANSPORTATION PROBLEM: DISCRETE TIME 

3.1. The Minimum-Time Transportation Problem 

One is interested in minimization of the total time required to transport all the goods 
from the origins to their destinations, i.e., 

Min t* = {max(T /y )|w„ > Ol . 

Possible applications include the transport of perishable goods; the movements of military units 
from bases to fronts; or the shipment of customers 1 orders which are of a priority or "rush" 
nature. 

The minimum-time transportation problem has been considered by Szwarc [44,45,46] and 
Hammer [23,24]. (See also Garfinkel and Rao [20] and Srinivasan and Thompson [41].) 
Szwarc [45] modified his original [44] algorithm to prevent it from cycling. Szwarc [46] in a 
similar way refined Hammer's [23] algorithm and revised a proof concerning the equivalence of 
local and global optima. Hammer [24] also furnished some amendment of his own work. 

Szwarc [46] has shown that his algorithm [45] is essentially equivalent to that of Hammer 
[23]. By this, we mean they both produce the same sequence of basic feasible solutions, given 
that they begin with the same initial solution. The Szwarc-Hammer algorithm is a primal algo- 
rithm, but Garfinkel and Rao [20] employ a "threshold" algorithm that yields a primal-feasible 
solution only upon termination. Srinivasan and Thompson [41] have studied the Bottleneck 
Time and Bottleneck Shipment transportation problems, and have furnished an algorithm for 
the former (i.e., for the minimum-time transportation problem) that is virtually identical to the 
Szwarc-Hammer algorithm. Srinivasan and Thompson [41] have also shown that their algo- 
rithm, and hence the Szwarc-Hammer algorithm, is computationally more efficient than that of 
Garfinkel and Rao [20]. 

The main points of the Szwarc-Hammer algorithm involve the following steps: 

1. Find an initial basic feasible solution (BFS) to the problem. This is, of course, a basis 
of order (m + n — 1). Let t denote the current value of the bottleneck time. 

2. Find an adjacent (i.e., a change of only 1 basis element) BFS which is better (i.e., 
either t is lower, or if t is unchanged, there is a smaller quantity shipped on the bottleneck arc). 
Hammer gives a 4-step procedure by which this can be done: 

(i) Among these basic arcs (i,j) for which t l} = t, identify the maximum quantity 
shipped. Call the quantity u hk . 

(ii) Determine the set S hk of all non-basic elements (p,q), such that if u pq were intro- 
duced into the basis, the shipment u hk would thereby be reduced. 

(iii) Choose among the elements of S hk the one, say (po.qo), for which t pq is the 
smallest. 



DYNAMIC TRANSPORTATION PROBLEM 69 

(iv) Enter u p q into the basis, as in the usual transportation problem. Update l. 
(Szwarc, at this stage, eliminates from further consideration any non-basic u u for 
which t u ^ t.) 

3. Check for optimality, i.e., for the existence of a still better BFS. Szwarc's procedure 
follows from his Theorem 2, which as above, indicates optimality when the only vacant "cells" 
Uy available for entry into the basis turn out to have r y > t. 

4. Return to step 2 or terminate at optimal solution. 

We remark that the preceding treatments have emphasized the time objectives with little 
mention of the concomitant costs. Hammer [23] did give an extension of his algorithm to 
determine, from among the alternative optimal solutions with the given r* that solution with 
minimum cost. An interesting extension would be the imposition of an additional constraint 
involving the upper bound of total cost. Alternatively, Glickman and Berger [21] have 
analyzed the trade-off between the cost and time of transport, by solving the usual minimum 
cost problem, subject to an upper bound on the time of transportation. Srinivasan and Thomp- 
son [42] have determined cost/time efficient frontiers for the multimodal problem. For every 
Pareto-optimal point, their algorithm furnishes the routes, modes, and quantities shipped. 

3.2. Multiperiod Transportation Problem 

Attention is now addressed to the problem of minimization of total costs when the n sinks 
have demands for each of T periods. This problem has been studied by Bellmore, Eklof, and 
Nemhauser [2] (B.E.N.) , and by Szwarc [47]. (The latter reference first appeared in 1967 as a 
C.O.R.E. Discussion Paper, No. 6704.) 

B.E.N, treat the general case in which A h B n and q,j are all functions of time, which is 
denoted by the presence of a superscript t. Shipments to the sinks from the m sources can 
either take place in the same period as the demand requirement, corresponding to a shipping 
cost of 

qjj ujj, 

or earlier, in which case a linear cost for holding inventory is then also incurred. If the notation 
yj denotes the inventory carried from period (t — 1) to period t at sink j, and dj the 
corresponding inventory holding cost, the problem of B.E.N, can then be formulated as: 

Min/=2; i qbufr + t £ djyj 

U t=\ y=1 1=7 



subject to: 



£ ulj < A(, V i, t 

m 

I,4 + yj-yj +l >B}, v;, t 



^ > 

< yj < Nj, 

where Nj is the maximum inventory which can be stored at sink j between periods {t — 1) and 



70 JH. BOOKBINDER AND S.P. SETHI 

Were it not for the inclusion of the upperbounds A/j, the above problem could be solved 
by creation of a large single-period problem, with (mT) sources and (nT) sinks (Bowman [4] 
and Kreibel [32]). This approach, as B.E.N, point out, yields computer storage requirements 
that are proportional to T 2 . 

B.E.N, approach the problem through a minimum cost flow or transshipment model, i.e., 
Tim — source, // — sink) problems linked by inventory variables. (The "nodes" are the A' and 
B\ for all values of /.) The computer storage requirements are then proportional to T. B.E.N. 
also point out a decomposition of the transshipment problem that requires rapid access computer 
storage of data for only // more nodes and n more arcs than does the standard transportation 
problem. That is, the storage requirements for the decomposition are then independent of T 
(and, in fact, are at anytime comprised of the data for a single period in core). 

The B.E.N, transshipment algorithm is effectively composed of the following steps: 

1. Add an artificial source node s* and an artificial sink note t*. 

2. Introduce arc "distances", which are essentially qfj or d], suitably modified by the 
existing flow in the network. (Initial flow = 0.) 

3. Find a shortest path from 5* to f* and ship the maximum amount possible, con- 
sistent with the upperbounds on inventories. 

4. Increase flows, revise "distances", and re-calculate shortest path. 

5. Algorithm terminates when all demands have been satisfied. 

The paper of Szwarc [47] is addressed to the same problem as that of B.E.N. , but differs 
in that Szwarc does not include an inventory holding cost. B.E.N, include this cost, as well as 
upperbounds on inventories at each sink node. Neither Szwarc nor B.E.N, include upper 
bounds on the shipments u' n although as B.E.N, point out, the modification to the transship- 
ment algorithm in this case is clear. 

It may be argued that the B.E.N.'s desire to reduce the core requirements was motivated 
by desires more academic than practical, in that few real problems would assume as regular or 
straightforward a form. However, the work of B.E.N, should be quite practical for a more com- 
plicated problem with additional constraints, since the B.E.N, solution could be used for the ini- 
tial iteration to the larger problem. 

B.E.N, treated Aj, B' n and Nj as given, and determined optimal shipments u'j and inven- 
tories y'j to minimize the sum of the latter two costs over the horizon t = 1,2, ..., T. For 
intermediate values of T, related questions are how the prduction at each plant should be 
increased over time and how the shipping patterns should change, when market demands have 
given growth rates and there is thus an increase in the total volume handled by the system. 
Such optimal growth paths in logistics operations have been studied by Srinivasan and Thomp- 
son [40] when supply costs are linear or piecewise linear and convex. For still larger T, Fong 
and Srinivasan [14] have considered the multiperiod capacity expansion problem when market 
demands are nondecreasing over time and must be met exactly during each period. In the case 
of linear costs, they furnished an efficient algorithm to schedule capacity expansions and ship- 
ments to markets to minimize the discounted capacity expansion costs plus the costs of produc- 
tion and transportation. 



DYNAMIC TRANSPORTATION PROBLEM 71 

3.3. Maximal Dynamic Flows 

3.3.1. Single Commodity 

The problem of maximal dynamic flow requires the determination of the largest commo- 
dity flow v(D which can occur between source and sink within a specified time horizon T. A 
capacity c u and a minimum transit time t u are associated with each arc (ij). Let u'j be the 
flow leaving node / at time t, enroute to node j ; for the deterministic case, arrival at node j will 
occur at time (/ + t, ;/ ). Transshipment is allowed, with u' n denoting the commodity inventory 
at node i held over from time t to (t + 1). (It turns out, however, that there always exists a 
maximal dynamic flow in which there are no holdovers at intermediate nodes.) 

Letting P, Q denote the source and sink nodes, respectively, the problem is then: 

Maximize \(T) 
subject to 

1=0 j 

"£ [uj J -Uj~ T »] = Q- t*P,Q\ f=0, 1 T 

j 

£ r[«fc-«5 TjB ]--v(n 

'=0 J 

< ujj < Cjj. 

This problem has been studied in detail by Ford [15] and by Ford and Fulkerson [16]. 
Their analysis is based upon a Time Expanded (T.E.) network, in which a new node is intro- 
duced for each (discrete) time period. The maximal dynamic flow is obtained by: 

1. Solving a static transshipment problem in the T.E. network. 

2. Repeating this flow for successive time periods. 

Ford and Fulkerson showed that this maximal temporally repeated flow is the optimal dynamic 
flow. 

3.3.2. Multicommodity 

Ford and Fulkerson's result cannot be extended to more than one commodity, that is, to 
the case where the upperbounds c y refer not to each commodity separately but rater to the sum 
of the u jjk over all commodities k. Bellmore and Vemuganti [3] (B.V.) have used the fact that, 
while the temporally repeated multicommodity flow need not be optimal, it is feasible. This 
furnishes a lower bound on the maximal dynamic flow. 

B.V. showed that, as the number of time periods T becomes large, an upper bound is fur- 
nished by 

[Multi. Comm. Max. Dyn. Flow for T periods] 
— [Max. temporally repeated flow for T periods] ^ a. 



72 J.H. BOOKBINDER AND S.P. SETHI 

The number a is independent of T, so that for large T, the percentage difference between the 
two terms on the left tends to zero. B.V. show how a can be obtained by solving the mul- 
ticommodity transshipment problem on the static network. 

The bounds a are refined by B.V. through the use of duality. As they note, their method 
and results generalize to the case of the weighted, multicommodity maximal dynamic flow prob- 
lem. Nevertheless, as far as known to the present authors, no exact algorithm has been 
presented for the problem of maximal dynamic flow in the multicommodity case. 

4. A STANDARD MULTIPERIOD, MULTIMODE MODEL 

Our next illustrations for the case of discrete time incorporate uncertainty in the demand 
requirements B r For the single stage problem, treatments by Dantzig [10], Williams [52] and 
Szwarc [43] have employed stochastic linear programming. 

In the multistage case, the probabilistic version of the transportation problem can be 
thought of as involving demand governed by a stochastic process. The paper by Midler [34] 
considers such a multiperiod problem with random demands for the multicommodity case. A 
dynamic programming algorithm is used to decide which mode of transport to employ, and to 
assign commodity classes to various modes and supply points to destinations.* 

The problem is formulated as one of discrete time stochastic optimal control, in which the 
objective is minimization of total expected cost over the planning horizon. The costs con- 
sidered are those of shipping, by commodity and route; costs of rerouting a carrier from a desti- 
nation back to each origin; costs of stockouts and of carrying inventories at destinations; and 
costs involving deviations from port capacities. Midler assumed that the shipping costs are 
linear, with all other costs quadratic. 

The dynamic equations were taken to be linear in the state and control variables, and the 
model solved in closed form. Midler found that the optimal values of the control variables 
could be expressed linearly in terms of the current values of the state variables, with the 
coefficients in this relationship depending upon the number of periods remaining in the hor- 
izon. Midler then furnished an extensive discussion of the solution to the transportation prob- 
lem, including the optimal dependence upon time and upon location of the assignment of each 
commodity class to a mode, and the interpretation of the time-varying stochastic shadow prices. 

5. SOME APPLICATIONS 

We now discuss two problems which have been formulated as applications of one form or 
other of the dynamic transportation problem with discrete time. 

5.1 A Multiperiod Truck Delivery Problem 

The first application concerns the scheduling of truck deliveries. This problem, studied by 
Hausman and Gilmour [25], arose in the analysis of delivery of home heating oil. Deliveries 
are made from a single source to n customers, each of which has a minimum required fre- 
quency of service f,. This frequency f h which may be exceeded to take advantage of truck 
economies of scale, furnishes the time-dependence of the problem at hand. 



*For the case of a dynamic communicalion network with stochastic inputs, Segall and Moss [37] have furnished an 
analysis (in the continuous-lime framework) which uses Hamillon-Jacobi theory (see e.g., Fleming and Risliel (131). 



DYNAMIC TRANSPORTATION PROBLEM 73 

As an operating policy, the customers are classified into k groups, where k is treated as a 
decision variable. When any customer in a group needs service, the whole group is served. 
The objective function, then, is the assignment of each customer to a group, so that annual 
delivery costs are minimized. 

The truck costs involve both a fixed cost per delivery, as well as a variable cost per mile 
travelled. The presence of this variable cost means that evaluation of the objective function 
requires knowledge of the distance travelled in the optimal tour of each group. This, of course, 
amounts to solving the travelling salesman problem, in general a difficult task indeed. 

Hausman and Gilmour took the point of view that the customers in a group were likely to 
be reasonably close geographically. The truck driver would then intuitively choose a sequence 
of deliveries which involved a distance D fairly close to that in the optimal tour, at least if there 
were not an "excessive" number of customers per group. (The latter is guaranteed by the lim- 
ited capacity of the truck.) Accordingly, these authors estimated Dj for each group j by multi- 
ple regression. This heuristic device employed as independent variables three simple statistics 
for the group under consideration. These variables involved the standard deviations of the lati- 
tudes and longitudes, and the number of customers in the group. Parameters were determined 
by comparison with some randomly generated travelling salesman problems and their optimal 
solutions. 

Hausman and Gilmour's algorithm began with this approximation £>, for each group, fol- 
lowed by a search for a re-assignment of customer groupings in an attempt to lower the total 
costs. While the problem at hand is quite difficult, their algorithm does not appear to be very 
efficient, because customers are one-by-one temporarily re-assigned to each of the other groups. 

Nevertheless, there are two interesting aspects to this work. The regression approach (to 
estimate Dj) could, as Hausman and Gilmour point out, be useful in estimating bounds on the 
optimal solution to other problems, or in deciding whether to continue with further searches on 
a given problem. 

Moreover, in the course of their work, Hausman and Gilmour found that they could still 
obtain a significant cost reduction (and, of course, considerable saving in computer time) if the 
area under consideration were partitioned, and these sub-problems treated independently. It 
would be worthwhile to obtain conditions on the validity of this "geographical decomposition." 

5.2. Optimal Rescheduling of Airplanes 

We next turn attention to the possibility of adjusting a previously-scheduled flow, when 
stochastic disturbances warrant such adjustment. Ellis and Rishel [11] have formulated a model 
for the one-way flow of air traffic between two airports, subject to random constraints on the 
takeoff and landing capacities. Their presentation was based upon the state-space approach of 
(stochastic) optimal control theory, but the specific example which they solved employed a 
dynamic programming algorithm. 

Ellis and Rishel considered (n + 1) discrete time intervals, by the end of which all 
scheduled takeoffs from airport 1 will have actually taken off and landed at airport 2. Once 
takeoff occurs, (a deterministic) j intervals of time are required to fly from 1 to 2. During any 
time interval, the number of aircraft movements which a controller can permit is limited by 
runway or airside capacity. Each capacity was treated as a random variable to model delays due 
to weather or mechanical failure. 



74 J.H. BOOKBINDER AND S.P. SETHI 

The objective function is the minimization of total waiting time, with time in the air 
valued higher by a factor a > 1, compared to waiting time on the ground. The control vari- 
ables are the numbers of landings and takeoffs permitted in each time interval. The state vari- 
ables are the numbers of planes currently in the air; aircraft awaiting takeoff; those awaiting 
landing; and the current capacities of the respective airports. 

To ensure with probability one that by time n, the schedule of movements begun at time 
will have been completed, Ellis and Rishel assume that there exist positive upper and lower 
bounds on the capacities at each airport. More importantly, the completion constraints at time n 
were shown to imply constraints on the values of variables at all intermediate times. These 
implied constraints define bounded convex sets giving the feasible values of the state variables. 
Ellis and Rishel use this convexity to prove a theorem which shows the existence of an optimal 
control, and which recursively exhibits an optimal solution to the dynamic programming prob- 
lem. They show the optimal controls for a 5-period numerical example which, although a fairly 
simple problem, involved considerable computation. 

The remainder of the present paper deals with the continuous-time dynamic transportation 
problem with delays. In particular, the following section deals with the minimum-time prob- 
lems, as well as minimum cost problems with delays. Section 7 concludes with a discussion of 
some important extensions of the dynamic transportation problem with delays. 

6. DYNAMIC TRANSPORTATION PROBLEM WITH TIME DELAYS 

An important class of extensions of the classical transportation problem recognizes the 
fact that it takes time for a shipment to reach the sink after it has left the source. An appropri- 
ate framework to deal with such problems is that of optimal control theory. Frank [17] was first 
to recognize this as he attempted to examine a communication network problem with finite 
delays in the discrete-time state-space framework.* 

6.1. Frank 117] and Frank and El-Bardai (18] 

To describe Frank's formulation, we let G denote a directed graph with each arc or branch 
bj of G having an integer (or a rational) branch delay 8,. We can assume 8, = 1, for all /, 
without any loss of generality. t 

Let z,(t) represent the flow in b, at time t and let z(t) = (z\(t), z 2 (t) z m (t)), where 

m denotes the number of branches, be called the flow or state vector. Further, let Uj(t) be the 
external input and yj(t) be the output at node j. With u(t) = {u\{t), u 2 (t), . . ., u n (t)) and 
y(t) = (y\U), }>2U), ■ ■ ., y n U)), where n is the number of nodes, we can write the state equa- 
tions as: 

(1) z(t + \) = f[z(t), u(t)], 

(Equation (1) continues) 



*Note that the first application of optimal control theory to a static transportation problem is due to Fan and Wang 
[12]. They applied a discrete maximum principle to solve a nonlinear transportation problem (of Bellman and Dreyfus 
[1]) with two sources with unlimited supplies available at increasing (concave) cost, and multiple sinks with specified 
demands. Charnes and Kortanek [6] pointed out an inconsistency in the Fan-Wang procedure. The reader is referred 
to Halkin [22] for a precise statement of the discrete maximum principle. 

tlf not, we can construct a new graph G" by replacing the / th branch by 8,/S series branches with common time delay 
8, where 8 is the greatest common divisor of all 8,. Note that 8, in turn, can be considered a unit of time. 



DYNAMIC TRANSPORTATION PROBLEM 



75 



and 
(1) 



y(t + \) = fh(t), «(/)]. 



With this model, Frank relates the idea of satisfying demands at the nodes of the network 
to state reachability. He restricts the system strategy if, f) to be conservative linear. Note that 
if, f) is conservative if at each node j, total inflows equal total outflows, and if, f) is linear if 
f=Az(t) + Bit it) and /= Cz{t) + Du{t). 

Frank shows that a given final state z(T) is reachable from the zero state in time T if and 
only if there exists a state cr ^ z(T) in the subspace spanned by the columns of the controlla- 
bility matrix Q given by 



(2) 



Q = [B,AB,A 2 B, ...,A T ~ X B] 



Furthermore, z(T) is reachable from the zero state (in a finite time) if and only if it is reach- 
able from the zero state in time r, where r is the degree of the minimal polynomial of A. Frank 
goes on to examine the necessary and sufficient conditions for the existence of a linear strategy 
under which a given set of terminal states are reachable (from the zero state) . 

Finally, Frank oulines a procedure to find the set of reachable states for a linear system 
with upperbounds on input u(t), i.e., u(t) < U, Vr. Note that it is no longer possible to set a 
uniform upper limit r on the number of time periods necessary to reach a given state. 

Frank and El-Bardai [18] also impose upperbounds on the flows in branches of the net- 
work. For this, let c = (c,, c 2 , ... ,c,„) denote the branch-capacity vector. We can now express 
the linear strategy with capacity constraint as: 

z(t + 1) = minUz(r) + Bu(t),c] 

(3) and 

v(/ + 1) = Czit) + Du(t). 

Frank and El-Bardai show that with the input constraint u (t) ^ U, the set of reachable states 
(in time T) in system (3) is given by 

d\d ^ £ d(t) 



where 



and 



d{\) = mm[BU,c] 



d(t + 1) = min 



Ad(t),c 



I d(r) 

T=l 



t= 1,2, ...,(T- 1). 



'Frank and El-Bardai also derive a similar result for systems with memory (but without input constraints). That means, 
real number Wj ^ can be associated with node j, representing the maximum amount of flow that can be stored at 
node j. Note that any excess flow to be stored at node j will be lost. Obviously, if a state is reachable in a memoryless 
system (i.e., one in which no flow can be stored), it is reachable in a system with memory. The converse is not neces- 
sarily true. 



76 J.H. BOOKBINDER AND S.P. SETHI 

So far, we have dealt with the question of state reachability which is the same as the ques- 
tion of feasibility in operations research. The answer involved applying a large enough input in 
every period to saturate all branches. It may be that a given demand vector can be satisfied in 
time T without saturating all branches. This raises the question of optimality. The idea is to 
find the 'smallest' inputs with which a demand vector may be satisfied. 

The problem of finding such an input sequence can be easily formulated as a linear pro- 
gramming problem. That is, given z(0) and parameters A, B, z f , C and U, find u(t) and z(r) 
that minimizes 

T-\ n 
/=0 j=\ 

subject to 

z(t + 1) ^ Azit) + BuU) t = 0,1,2 (T - 1) 

z(t) > z f 

^ z(t) < c, t = 1,2 T 

< Uj(t) < U, j = 1,2 n, t =0,1,2 (T - 1) . 

The solution of this linear programming problem will yield a sequence of inputs that takes the 
state z(0) of G to a state z(T) ^ zy in a way that is optimal in r/?e m/wr smse. An important 
property of this sequence is that it is also optimal in the loss sense, i.e., it minimizes loss in the 
network. 

It is the idea of optimality with which we are really concerned in this survey. Tapiero and 
Soliman [51] address themselves to this important problem of optimality in dynamic transporta- 
tion networks with transport delays. They use the continuous-time state-space framework and 
the maximum principle for their analysis.* 

6.2. Tapiero and Soliman (51] and Tapiero [49] 

Tapiero and Soliman formulate a dynamic multi-commodity transportation problem with 
time delays as an optimal control problem. For simplicity in exposition, we will only develop 
the single-commodity case formulated in Tapiero [49] since its extension to the multi- 
commodities case is rather straightforward; all our discussion addressed to the single- 
commodity case is applicable to the multi-commodity version. 

6.2.1. The Model 

To develop the model, let there be m sources and n destinations. Let /' denote source / 
and j denote sink j. Let t denote the time. We can now define the following variables and 
parameters: 

Ujj(t) = rate of flow from source i enroute to j at time r, 

x,(r) = inventory at source i at time f, 



* While we survey only his latter papers, Tapiero's 1971 paper is the first paper applying the maximum principle to the 
dynamic transportation problem. The paper considers the problem without delays. We also note that more recently 
Segall and Moss [37] have dealt with a similar problem for a communication network. 



DYNAMIC TRANSPORTATION PROBLEM 77 

yj(t) = inventory at sink j at time t, 

Ty = transit time from i to j, 

Cy = maximum flow capacity of route (ij), 

K = total flow capacity of the transport system, 

T = planning horizon, 

Aj = initial supply at source i ; i.e., x,(0) = A h 

Bj = demand at sink j at time T; i.e., yj(T) — Bj, 

L{xi(t), yj(t), u,j{t), t) = cost function expressing inventory and 

transportation costs. 

Note that it is necessary to have the total available supplies exceed or equal the total available 
demand. We may assume (without any loss of generality) 

(4) 

E 4 - I ^ 

/=l 7=1 



We can now state the optimal control problem as follows: 
(5) 



min 



J = f L[x,{t), yjit), u u U), t } dt 



subject to 

(6) 

*/(') = £ UuU), x,(0) = A lt Xi(T) = 0,* 
/-i 

(7) 

yjU) = £ UuU - t,j), yj(s) = 0, V5 < 0; yj(T) = B jt 

and the capacity constraints 

0<«yO)<Cy 

and 

(9) 

< £ Z UuU) < K. 



Note that the terminal conditions of x, can be rewritten as x t (T— nun t (> ) = 0. We note that Tapiero's formulation has 

j 
misprints in the specification of terminal conditions of x, and yj. In passing, we may also remark that there are other 
errors in Tapiero [491 and Tapiero and Soliman [51]. 



78 J-H. BOOKBINDER AND S.P. SETHI 

Note that x, and yj are the state variables and u,j are the control variables in the terminology of 
optimal control theory. We refer to (9) as a linking constraint since it introduces interdepen- 
dencies among flows on various routes. 

It is convenient to transform this formulation into an equivalent formulation by defining a 
new variable r)y(/) denoting the cumulative flow (up to time t) which has left source / for sink 
j. The equivalent problem is: 



7* 

J - f A foy(f), riijit-Tij), u u (t), t)dt 



(10) 

min 

.subject to 

(11) Tjy(f) - u u U), -Oij(s) = 0, Vs € [-1-0,0], 

with constraints on terminal conditions 

(12) 

7-1 7=1 

(13) m m 

Em,(r>-E iK/Cr-T^-jy, 

r-i /=i 

and the control constraints (8) and (9). 

6.2.2. Application of the Maximum Principle 

We are now ready to apply the maximum principle with delays in the state variables (see 
Kharatishvili [30]; see also Ray and Soliman [36]). The Hamiltonian is: 

(14) m n 

H = -K{r )ii U),r\ ij U-T iJ ), u u U),t) + £ £ \ tJ (t) u u (t) 

(-1 7=1 

where the adjoint variables Xy(f) satisfy 

with the transversality conditions (see Tapiero and Soliman [51] and Tapiero [49]) 

(16) \ijiT-Tjj) +k mn (T-T mn ) = \ mj (T-T mj ) + \ in (T-T in ), i = \, 2, .... (m-l), 

j= 1, 2, ..., («- 1). 

It is noted that the adjoint variables in optimal control play a role similar to the Lagrange multi- 
pliers in nonlinear programming. For explanation of the transversality conditions, see Fleming 
and Rishel [13], Tapiero [50], and Sethi and Thompson [39]. 



'Note that x,(t) = A, ■ - £ t),,(') and _y y (/) = ^T Tjy(/ — Ty) and, therefore, the new loss function (denoted in ab- 

7=1 /=1 

n m 

7=1 '=1 



breviated form by A(r)) is obtained as \{t } = L 



DYNAMIC TRANSPORTATION PROBLEM 79 

A necessary condition for optimal transportation is that the Hamiltonian be a maximum 
(subject to control constraints (8) and (9)) along the optimal path. This is, in general, a 
difficult problem to solve. However, it may be possible to resolve the problem in some special 
cases. Tapiero takes up the minimum time problem as the special case for consideration. 

6.2.3. Minimum-Time Problem 

The minimum-time transportation problem is concerned with determining a transport 
schedule which will transport supplies at m sources to meet the demands at n sinks in minimum 
time. This problem, especially its multi-commodity version, is extremely important in army 
logistics, in supplying help to disaster areas and, possibly, in transporting perishable goods. For 
the case of discrete time, we have discussed this problem in Section 3. 

For minimum-time problems the objective function (10) is the total time T, thus 

(,7 > i-A-l. 

That means \ i} ■ = in (15), implying that the A.,, are mn (unknown) constants satisfying 
(m — 1) (n — 1) linear equations (16). With this information, we can rewrite the Hamiltonian 
as: where 

// = -l+ I I Xtftyto. 

Since it is easier to deal with the minimum-time problem without the transport system's 
capacity constraint (9), we will take this up next. 



6.2.4. Minimum-time Problem Without (9) 

Since the Hamiltonian (18) is linear in «,-,- and since the only constraints we need to worry 
about are the route capacity constraints (8), which are not of the linking-type, it is easy to see 
that 



(19) X„ < - u„(t) = , \ft, 

UijU) = C/j, t < t u 



X„ > - 1 



, otherwise, 



where 

(20) t tj A min 



A,/ £ c„, Bj/ £ c kj 

/|X„>0) / {k\k kj >0) 



This means that along any arc (/j), either there is no shipment or there is a shipment at 
full capacity from time zero to some time t u ^ T. This observation allows us to set up a linear 
program which will solve the minimum-time problem without (9): 



80 JH. BOOKBINDER AND S.P. SETHI 

Min T 
subject to 

n 

Z Cij t u - ^,, V/, 

(21) m 

/■=1 

r tf ^ 0, v/j. 

This L.P. problem will give the optimal ty and 7 1 provided there is a feasible solution to 

the given problem. We remark that in the optimal solution, T = min (f y + t, 7 ). 

U 

Note that it may not be easy or even possible to modify this program to take into account 
(9) and still retain an L.P. problem. The existence of other ways to find an optimal solution 
with (9) is our next concern. 

6.2.5. Minimum-Time Problem with (9) 

If the linking constraint (9) is binding, there will be some kind of priority assignment in 
the set of routes (/J). Obviously, the Hamiltonian-maximizing condition will imply that this 
priority assignment be reflected in the values of Ay. 

Suppose that we have a given set of values for \y which satisfy (16). There are many 
possible sets that may do this. Note that Ay = 1, V/J always satisifies (16). Given the set of 
values for Ay, the Hamiltonian maximization is a knapsack problem, in fact a parametric knap- 
sack problem. A tentative outline of this procedure is suggested below; see Sethi and Book- 
binder [38] for more details. 

We begin by listing positive \ fJ in decreasing order. Starting with the route associated 
with the largest \y, we go down the list by shipping the maximum flow c u in the associated 
route (ij) until all routes with positive Ay are full or the sum of the flows equals or just 
exceeds the total system capacity K. If it exceeds K, we cut back the flow in the last route just 
enough so that the sum of the flows equals K. 

The flows defined above commence at time / = and continue until the first t^ defined in 
(20) is encountered. Let this be t t j . This t t j can be of one of two types depending upon 
whether the first or the second argument in (20) is smaller. 

a) If t t j is of the first type, set all u t j = 0, Vr > ?,• j . 

b) If t t j is of the second type, set all Uy =0, Vt > t, j . 

We then go back to assigning the flows according to the knapsack procedure by making a 
new list. 

These iterations continue until no further flow can be assigned. At this point, there are 
two possible situations. Either the terminal conditions (12) and (13) (equivalent to terminal 



DYNAMIC TRANSPORTATION PROBLEM 81 

conditions in (6) and (7)) are met in which case we term the set of \y a feasible set, or they 
are not met, in which case the set of X,y is clearly infeasible. 

If infeasible, we have to get another set of X. y satisfying (16) and start over again. If 
feasible, we have obtained a local minimum. It may not be possible to obtain the global 
minimum since this may require obtaining all the local minima, of which there could be a large 
number. 

To conclude this section, we note that it would be more realistic to replace the total sys- 
tem constraint (9) by loading and unloading constraints, 



(22) 

and 
(23) 



0< £ u,j(t) < b„ Vr, 



<K £ Hy(r) < cj, Vr. 

;=1 



The algorithm suggested in this section can be easily modified to deal with (22) and (23). 

6.2.6. The Multicommodity Case 

Tapiero and Soliman [51] have treated the multicommodity version of the minimum-time 
problem. Their paper does not contain a proof of their algorithm. It should, however, be 
noted that in the multicommodity case, an essential feature of the problem is the linking con- 
straint. To state the linking constraint (the analogue of (9)), let there be r commodities and let 
subscript k denote the k lh commodity. The linking constraint can be stated as 

o< £ oc k u ijk (t) ^ c, r \ft, 

where a k is the capacity required per unit of commodity k. These constraints are analogous to 
(9) in so far as their effect in developing an algorithm is concerned. 

We offer the following remarks concerning (24). The constraint states that the total ship- 
ment leaving source / at time t enroute to sink j should not exceed Cy. However, the loads-in- 
transit at any point along arc (ij) need not be less than C y . An example will further clarify 
this issue. Suppose there is a bridge along arc (ij) which is a bottleneck and whose capacity is 
Qj. Imposition of constraint (24) does not guarantee that at some time later than t, there will 
not be a load on the bridge in excess of Cy. Different commodities k may require different 
times T IJk to travel from source / to sink j, and it is possible for them to leave the source /' satis- 
fying (24) and yet exceed Q, on the bridge. Tapiero and Soliman have not recognized this in 
stating the arc-capacity constraints (24). Once again, constraints (24) are more like loading 
constraints associated with arc (ij) rather than the arc-capacity constraints. Note also that the 
arc-capacity constraints, such as in the case of the bridge situation above, will be extremely 
difficult to handle. 



82 



J.H. BOOKBINDER AND S.P. SETHI 



6.2.7. Linear Inventory and Transportation Costs 

Tapiero and Soliman [51] have also considered the problem wherein the inventory and 
transportation costs are assumed linear. In this case, the loss function 

(25) m n n m 

L = £ a,x, + £ djy, + £ £ fly Uy, 

/=1 7=1 7=1 <=1 

where a, and a} are the costs per unit time of holding a unit of inventory at source / and at sink 
j, respectively, and fly is the cost per unit time of transporting the commodity at a unit rate. In 
terms of 7} y , we can write the loss function 



'=1 7=1 7=1 1=1 



(26) 



+ Z I fly "„(')■ 

7=1 '=1 



The Hamiltonian can be written as 



(27) 



( = 1 /-I 7=1 i=\ 

n m 

+ Z £[-% + x, 7 (0]« y (r), 

7=1 /=1 



\, y (r) = \y(r - Ty) + ( fl| . - dj)(T - Ty ~ t) , 



where the adjoint variables A, y (/) can be expressed as 

(28) 

with \jj(T — Ty) satisfying (16). Furthermore, it is obvious that 

(29) a, > dj — <■ XijU) decreases with r, 

a, < dj — \jj(t) increases with t. 

But the Hamiltonian (27) is linear in Uy, implying that the optimal control is bang-bang. 
From (29), we can conclude that for the problem without (9), the form of the optimal policy 
for each are (ij) can be characterized by one number ty if a, ^ a", and ty if a, < a", : 



a, > dj — Ujjit) = 



Cy, ^ t ^ ty 

0, otherwise, 



(30) 



a, < dj — Ujjit) = 



0, < t < ty 

Cy, 7y<t< T-Ty. 



This policy is consistent with our intuition. It states that if the inventory cost at source / 
is higher than that at sink j, then shipment along arc (/J), if any, must commence as early as 
possible. On the other hand, if a, < a} then delay the shipment along arc (i,j), if any, as long 
as possible. 



DYNAMIC TRANSPORTATION PROBLEM 83 

Having characterized the form of the optimal policy, it is possible to formulate a quadratic 
program to solve the linear-costs transportation problem without (9). This is done in a manner 
similar to that in Section 6.2.4. We note that Tapiero and Soliman [51] had earlier made this 
observation. For the actual formulation of the quadratic program, see Sethi and Bookbinder 
[38]. Finally, Sethi and Bookbinder also attempt to obtain an algorithm for the problem with 
(9). This is similar but far more difficult than the procedure in Section 6.2.5. 

6.2.8. Linear Inventory and Quadratic Transportation Costs 

Linear-quadratic control problems have a special place in the optimal control theory. Usu- 
ally these problems yield closed-form solutions. Midler [34] treated a transportation problem 
with linear transportation costs and quadratic inventory costs in the stochastic dynamic pro- 
gramming framework. Tapiero and Soliman [51] treat a similar problem with delays in the 
deterministic framework, but with linear inventory costs and quadratic transportation costs. To 
state the latter problem, we only need to replace the <7 y u ti term in (25) by (7yG/y — w y ) 2 , where 
w y is the most desirable shipment rate from the point-of-view of transportation. Furthermore, 
constraints (8) and (9) are assumed to be no longer acting for this problem. 

The Hamiltonian of this problem can be written as 

m n n m 

H - - £ a\A { - £ ,,„(*)] - £ £ d m {t - r y ) 

HI 7=1 7=1 HI 

/ii\ n in n m 

uu - I E «</ty,(f) - fy) 2 + I EMi/W. 

7 = 1 ( = 1 7 = 1 i-l 

where the adjoint variable A,y(f) satisfies (28) and (16). Furthermore, the Hamiltonian maxim- 
izing condition yields 

(32) 

WjjU)= u v + \ u (t)Hq u , 

where u*j(t) is the optimal control if u*j(t) ^ 0, \/t. We will assume this to be the case; oth- 
erwise the problem becomes more difficult. 

We can now use (32) in (11) to obtain iq^iT - t u ) which must satisfy (12) and (13). 
These conditions can be simplified as: 



(33) 



Z 9 T ° kij(T- t u ) + (a, - dj)(T- t,j)/2\ = A„ / = 1, 2 m, 



7=1 *"*« 

(34) m T—t 

I -r-^ [\,j{T-r ij ) + {a i -d j ){T-r lj )l2] = Bj, j= 1,2,..., (n - 1) . 

HI Za U 

We have not written (34) for j = n since it is redundant on account of condition (4), which 
states that total demand equals total supply. 

The system of Equations (33) and (34) contains m + n - 1 equations. Along with (16) 
which contains (w - 1) (n - 1) equations, we have a total of mn equations which must be 
satisfied by mn variables \ (T - t, 7 ). The solution of this linear system of equations should 
provide the optimal control when substituted in (32). 



84 J H. BOOKBINDER AND S.P. SETHI 

7. SUGGESTED IMPORTANT EXTENSIONS 

A natural extension of the dynamic transportation problem with delays involves time- 
dependent demands at various sinks. If we let ijjU) represent the demand rate at sink j, then 
(7) becomes 

(35) " 

with an additional constraint that 

(36) yjU) > 0, Vr. I 

For this extension, it may be necessary to either assume v 7 (0) > or £,-(/) = for 
< / < min T,, to have feasibility. Another way to handle this problem is to allow shortages 

and incorporate shortage costs in the objective function. See also Midler [34] in this connec- 
tion. 

A simple and interesting extension is the case of perishable goods. If we assume a con- 
stant spoilage rate y in transit, it amounts to replacing (7) by 

(37) m- 1 Uijit-T,)^* . 

/-I 

In this case we cannot require condition (4), since that would imply infeasibility. Conse- 
quently, feasibility also becomes an issue in this case. Of course, the cost of spoilage must be 
incorporated in the objective function by modifying unit transportation costs on the arcs of the 
transportation network. 

Another extension is the case in which there are unloading delays at various sinks. Typi- 
cally, the unloading delay at a sink will depend on the inventory at the sink and (possibly) the 
time of unloading. The latter dependence may derive from an expansion program which is in 
progress at the sink under consideration. For the single commodity case, this amounts to 
replacing (7) by 

(38) m 

y^t) = ^u u U-Tjiy j {t),t)) 

where the delay tj may be assumed to satisfy Brj/Byj ^ and Brj/Bt < 0. Ray and Soliman 
[36] have a weak maximum principle for dealing with systems subject to equations of the form 
(38). 

Perhaps the most important extension is the case in which the time required for a ship- 
ment to reach from source i to sink j is a function of the amount shipped. That is 

T,y= Tjjild), dTjj/dUjj ^ 0. 

In this case the transit time increases with the amount shipped. For the single commodity case, 
this amounts to replacing (7) by 

(40) 

1-1 (010+7,^.(0)]=/) 



DYNAMIC TRANSPORTATION PROBLEM 85 

We note that this equation is an essential feature of any dynamic transportation problem in 
which the transit time is a function of the load-in-transit. 

Equation (40) is a state equation in which not only the delays depend on controls at previ- 
ous times but also these delays are defined recursively.* We do not know of any maximum 
principle allowing even for delays depending upon controls at an earlier time, let alone where 
that time is defined recursively. Ray and Soliman's [36] paper contains only a weak maximum 
principle for systems with delays depending on state and control at time t and time t. 

Treatment of state equations of the type (40) will obviously require some theoretical 
developments, including a suitable maximum principle. For the time being, we are working 
with a discrete-time version of (40), employing dynamic programming and/or some theory for 
optimization systems developed by Clarke [8]. 

Finally, the problem could be extended to take into account the stochastic nature of tran- 
sit delays. This would require replacing (7) by a stochastic differential equation. One could 
also consider demands at sinks to be stochastic (see Midler [34] and Segall and Moss [37]). 
Either case results in difficult stochastic optimal control problems.t 

8. ACKNOWLEDGMENT 

This paper is dedicated to the memory of Ray Fulkerson. This research is supported by 
Grant 3-214-385-80 from the University of Toronto/ York University Joint Program in Tran- 
sportation. The authors are grateful to the referees for their instructive comments concerning 
an earlier draft of this paper. 

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86 JH. BOOKBINDER AND S.P. SETHI 

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[24] Hammer, P.L., "Communication on 'The Bottleneck Transportation Problem' and 'Some 
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487-490 (1971). 

[25] Hausman, W.H. and P. Gilmour, "A Multiperiod Truck Delivery Problem," Transportation 
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[26] Hitchcock, F.L., "The Distribution of a Product from Several Sources to Numerous Locali- 
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[27] Holladay, J., "Some Transportation Problems and Techniques for Solving Them," Naval 
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[28] Kantorovich, L.V., "Mathematical Methods in the Organization and Planning of Produc- 
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[29] Kennington, J.L., "A Survey of Linear Cost Multicommodity Network Flows," Operations 
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[32] Kreibel, C.H., "Warehousing and Transshipment under Seasonal Demand," Journal of Re 
gional Science 3, 57-71 (1961). 



DYNAMIC TRANSPORTATION PROBLEM 87 

33] Manitius, A., "On the Optimal Control of Systems with a Delay Depending on State, Con- 
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(1963). 



THE USE OF DYNAMIC PROGRAMMING METHODOLOGY 

FOR THE SOLUTION OF A CLASS OF NONLINEAR 

PROGRAMMING PROBLEMS 

Mary W. Cooper 

Department of Operations Research 

and 

Engineering Management 

Southern Methodist University 

Dallas, Texas 

ABSTRACT 

This paper presents an application of a method for finding the global solu- 
tion to a problem in integers with a separable objective function of a very gen- 
eral form. This report shows that there is a relationship between an integer 
problem with a separable nonlinear objective function and many constraints and 
a series of nonlinear problems with only a single constraint, each of which can 
be solved sequentially using dynamic programming. The first solution to any of 
the individual smaller problems that satisfies the original constraints in addi- 
tion, will be the optimal solution to the multiply-constrained problem. 



INTRODUCTION 

Let us define a nonlinear integer programming problem using the following notation: 

(1) Maxz = £/,(*,) 

7=1 

(2) such that /;, (x) < / = 1,2 m 

(3) Xj € I p y =1,2 n 

where f J: I p — R p j = \,2, ... , n, 

and /?,: I p — ► R p i—1,2, . . . , m and x = (x h x 2i . . . , x n ). 

Additional assumptions are as follows: 

1. The functions /} (•) satisfy a necessary conditon for dynamic programming. 

2. The region defined by (2) and (3) is non-empty, with at least one lattice point in the 
region and the region is bounded. 

In [1] and [2] algorithms are developed to find candidate lattice points on a series of 
hypersurfaces. A dynamic programming technique is used to solve a sequence of problems 

89 



90 M.W. COOPER 

with only a single constraint. Each of these problems corresponds to a value of the objective 
function, and this value is reduced from an upper bound until a feasible lattice point is found. 
In [1] the value of the objective function is reduced by one unit for each successive single con- 
straint problem. In [2] an argument is made for allowing reduction of the trial value by more 
than a unit amount using numbers corresponding to hyperplanes with integer solutions in the 
dynamic programming return function tables. Therefore, only hypersurfaces which contain lat- 
tice points are investigated and the speed of the algorithm is greatly improved. The effort con- 
tained in this paper is an explication of the algorithm developed in [2] and its application to 
problems with a different type of objective function: namely, those for which the separable 
functions are such that 

fj{-):I p ^R p 7-1,2. ...'it 

(It was previously assumed that /}(•): (I p — > I p , j = 1, 2, . . . , n). Such problems would typi- 
cally have an objective function like z = x\ 12 + 3x 2 1/4 + 4x 3 2 . It will be shown that the algo- 
rithm of [2] can be used on problems where the objective function is not restricted to integers. 
The algorithm will find a global optimum, even for these less restricted problems. Therefore, 
this method can be used for nonlinear problems such as the concave transportation problem or 
for discrete problems in which the objective function is neither concave nor convex. 

Summary of the Algorithm 

Let us formulate the following approach for solving the problem described in (l)-(3). In 
this development, we will consider a sequence of hypersurfaces of the following form: 

(4) f,fj(Xj)-Zk. k = 0.1, ... 
7-1 

If we can find an upper bound z on the optimal solution, then we search the hypersurfaces (4) 
corresponding to a sequence of values of z k for lattice points. We choose z so that z 
corresponds to an upper bound on the value of the objective function, and every successive 
value of z k is such that z k < z , and the hyperplane (4) corresponding to z k contains an integer 
point. We will show how this is done in the following sections. The search for lattice points on 
these hyperplanes is performed by a dynamic programming algorithm. Any lattice points so 
found are only candidates for the optimal solution to (l)-(3). They must be tested finally for 
feasibility with the constraints given in (2) -(3). Using this method of examining all lattice 
points which correspond to some hyperplane (4), then testing for feasibility with the original 
constraints (2) -(3), and terminating as soon as a feasible lattice point is found is the sequence 
of steps of the algorithm given in [1] and improved in [2]. The method in both references 
looks at all the lattice points corresponding to one hypersurface at a time and checks any points 
found for feasibility. The improvement given in [2] is that the sequence of values z k can be 
calculated so that only those hypersurfaces are examined that are known to contain lattice 
points. Since the sequence of values z k , k=0, 1, ... , is strictly decreasing, and if we have 
examined all possible surfaces that contain lattice points and check these points for feasibility 
with the original constraints (2) -(3), then we can guarantee that the first feasible lattice point 
will be optimal. In the special case in which some of the constraints are separable it is possible 
to discard or fathom partial solutions thereby improving the speed of the algorithm. This is 
done by calculating 

(5) ^ 

*/ " b, - £ Ay Xj 

J = n-I 

for each partial solution x„, x„_ 1( . . . , x t . If s, < for any i, i = 1, . . . , m, the partial solu- 
tion will only lead to infeasible solutions and can be dropped. 






DYNAMIC PROGRAMMING FOR CLASS OF PROBLEMS 91 

This procedure is able to use several advantageous properties of dynamic programming. 
r irst we may find global optima for nonlinear functions and identify integer points efficiently, 
iecond, it is possible to extend the algorithm to handle even nonseparable constraints (as 
hown in the numerical example (10)). This is done by dropping the feasibility test for con- 
tracts that are nonseparable. So the calculations of s, is made only for constraints that are 
eparable. If all constraints are nonseparable, then feasibility is checked after integer points are 
dentified and there is no possibility of fathoming at the partial solution stage. So it is possible 
o handle highly nonlinear nonseparable constraints. Hence the emphasis in the development 
tf this algorithm has been to utilize the advantages of a dynamic programming approach while 
voiding, as much as possible, the "curse of dimensionality" which has doomed such efforts in 
he past. 

We now present a stepwise description of the algorithm. 

MODIFIED HYPERSURFACE SEARCH ALGORITHM 

Determine upper bounds Uj for each variable. We then have: 
^ xj < Uj for 7 = 1, 2 , . . . , n 

Xj € I p 
Compute z if one is not known a priori by 

n 

Z = Z fj (ty) = Z k 

7=1 

Find all combinations of x ]t j = 1, 2 n which satisfy: 



7=1 


■ z k 




k = 


0,1, 


0< Xj 


< Uj 




j 


= 1,2, 


Xj € /„ 


J = 


1 


2, .. 


. , n. 



Test any integer point found on the hypersurface of step 3 at the partial solution stage 
(for separable constraints) for feasibility with the original constraints (2) -(3). If a feasible 
point is found, then the value of z k is tested against the greatest feasible lower bound on 
z. If it is less than this bound, then the point corresponding to that bound is optimal. 
Otherwise go to step 5. 

Calculate a new z k+x according to logic and notation developed in succeeding sections. 
Return to step 3. 

The process of finding lattice points in step 3 is achieved by using a dynamic programming 
cumulation of the following problem: Find all x jt j=\, ... , n such that: 

n 
7=1 

6) 0<x y < «, j= 1, .... n 

xj € I p j = 1, ... , n 

i [1] and [2] an equivalent formulation of (5) is given: 

n 

Max z = £ fj(xj) 

/-I 



92 M.W. COOPER 

such that 

n 
L fj (Xj) < Z k 
/-l 

(7) < xj < Uj j - 1 /i 

xj e i p j = \ n. 

For every value of z* this gives us the values Xj, j = 1, . . . , n of all lattice points x on the 
objective function hypersurface. Therefore, this problem can be used to identify integer points 
x at any value z k of the objective function. The optimal return functions are given by the fol- 
lowing: 



(8) s,(A) = Max /,(*,) = 

.-,-8, 



X( X=./, (8,), 8, =0,1 
-°°, otherwise 



(9) 



5 = 2,3 n 

g s (K) = Max [/ v (x s ) + g s _, (X - / s (x v )] x _ n i A 

0<a s =8 s A — U, 1, . . . , l\ s 

where 8^ = min (u s , [t; J) 

7=1 

and the notation [b] indicates the largest integer ^ (b). 

Equations (7) and (8) give the dynamic programming recursion formulas for calculating integer 
points in step 3. 

In step 5 a new objective function value z k is calculated. It is not necessary to consider 
every integer value between an upper bound on the problem z and the optimal objective func- 
tion value. In the case in which the separable functions are required to map onto the range I p , 
this would have been a feasible strategy. However, for functions fy. I p — ► R p this method 
would skip any noninteger z. Let us consider a different method of calculating the sequence z , 
Z] . . . , z k in step 5. We know that only values corresponding to integer points will be finite 
entries in the tabulation of the return function g s (\). (Values for noninteger x s (\) are associ 
ated with g s (\) — — <».) Therefore, let 



z k = \ (n n + f„(x„) x„ = u n , u„-\, ... , 0, 

where \ ( " _1) are values of X. entered in the g n -\(-) function tabulation. This formula for calcu- 
lating z k will give all hypersurface level sets which contain lattice points. The interesting fact 
for the current application is that this calculation will give all z k corresponding to hypersurfaces 
containing lattice points even for noninteger values of A.'" - " and /„ (x„). 

A NUMERICAL EXAMPLE 

Let us consider the solution of the following example: 

Maximize 

z = 6x! 2 + 3x 2 1/3 + 2xj /2 



j 



DYNAMIC PROGRAMMING FOR CLASS OF PROBLEMS 



93 



such that 




(10) 3x, + 4x 2 + 3x 3 < 10 


2x, + 3x 2 + 3x 3 < 10 


x x x 2 2 , < 7 


x\, x 2 , x 3 ^ 0, integer 


then 


A Cxi) = 6x, 2 , 


/ 2 (x 2 ) = 3xp 


f 3 (x 3 ) - 2x 3 1/2 . 


from the constraints of (10) we see that: 


^ xj < 3 = U\ 


< x 2 < 2 = u 2 


< x 3 < 3 = u 3 . 


Therefore, / 


*i<X)-' 


— (X 



X = 6 Si 2 , 8j = 0,1,2,3 
otherwise. 
Let us tabulate the return function for the first variable 

TABLE 1 . Optimal Return and Policy 
X g x (X) xf (X) 












6 


6 


1 


24 


24 


2 


54 


54 


3 



For the second stage the dynamic programming recursion is 

g 2 (\) = max [3x 2 1/3 + gl (X - 3x 2 1/3 )] 

0<x 2 <8 2 

for 8 2 = min (2, [f 2 ]) and X = 3£ 1/3 . 
We can thus compile the following table: 

TABLE 2. Optimal Return and Policy 



g 2 (\) x 2 *(X) 



g 2 (X) x 2 *(X) 












24 


24 





3 


3 


1 


27 


27 


1 


3.78 


3.78 


2 


27.78 


27.78 


2 


6 


6 





54 


54 





9 


9 


1 


57 


57 


1 


9.78 


9.78 


2 


57.78 


57.78 


2 



Now we may choose the sequence of values z , z u z 2 , ... by using the following rule: 



z k = k<»- l) +Mx 3 ), 



94 M.W. COOPER 

where \" _1 takes on all values entered for the state variable in Table 2, and x 3 = w 3 , w 3 

— 1 0. It is shown in reference [2], if it is not immediately apparent to the reader, that 

lattice points may lie only on hypersurfaces corresponding to 

t/ / U / )=z, = \ ( "-"+/ / , (x„); 
/=i 

therefore, 

z = \ (2) + / 3 (w 3 ) = 57.78 + 2 • 3 ,/2 = 61.24. 

The corresponding integer solution is x* = 3, x* = 2, x* = 3, which is not feasible for the 
constraints given in (9). Now we choose 

z, = \ (2) + h ("3 - = 57.78 + 2 • 2 1/2 = 60.61 

corresponding to x* = 3, x* = 2, x* = 2. Again the constraints given in (9) are violated. We 
continue the steps of the algorithm until we generate a feasible lattice point: 

z 2 = 57.78 + 2 = 59.78; 
z 3 = 57.78 + = 57.78; 
z 4 = 57 +'3.465 = 60.46; 
z 5 = 57 + 2.828 = 59.828; 
z 6 = 57 + 2 = 59; 
z 7 = 57 + = 57; 
z 8 = 54 + 3.464 = 57.464; 
z 9 = 54 + 2.828 = 56.828; 
z 10 = 54 + = 56 
z,, = 54 + = 54; 

Now we have a bound which is feasible for the integer problem. However, we must continue 
the calculations until no hypersurface with a greater z is unexamined. If we check z 12 , we find 
that 

z 12 = 27.78 + 3.465 < 54 

and all subsequent calculations also have z values less than this bound. Hence 54 is the optimal 
value of the objective function and this value corresponds to a global maximum at x, = 3, 
x 2 = 0, x 3 = 0. 

COMPUTATIONAL RESULTS 

Results from 51 randomly generated problems are reported. The problems were of the 
following form: 

n 
Max z = 52 fA x j) 
/-i 

(11) 2>:/*j < b > ' = 1.2 m 

where 

(12) fj (xj) = «, Xj + /3, xf + y, xf 

and (Xj, j8 y , y , were non-negative integers. Results are given in Table 3. The number of terms 
in the objective function for n = 20 could be as large as 3« = 60. In addition, the problems 
were generated so that the upper bounds Uj for each x, determined from the constraints (15) 
varied among the values 0, 1, 2, 3 so that a maximum of 4 integer values need be considered 



xf-3, 


x 2 *=2, 


x 3 *=l 


(infeasible) 


xf=3, 


x 2 *=2, 


x 3 *=0 


(infeasible) 


xf=3, 


x 2 *=l, 


x 3 *=3 


(infeasible) 


xf=3, 


x 2 *=l, 


x 3 *=2 


(infeasible) 


xf=3, 


x 2 *=l, 


x 3 *=l 


(infeasible) 


xf=3, 


x 2 *=l, 


x 3 *=0 


(infeasible) 


xf=3, 


x 2 *=0, 


x 3 *=3 


(infeasible) 


xf=3, 


x 2 *=0, 


x 3 *=2 


(infeasible) 


xf=3, 


x 2 *=0 ( 


x 3 *=l 


(infeasible) 


xf=3, 


x 2 *=0, 


x 3 *=0 


(feasible). 









DYNAMIC PROGRAMMING FOR CLASS OF PROBLEMS 



95 



TABLE 3 



7 — 7* 


m x n 


m x n 


m x n 


m x n 


m x n 


Z 2 


4 x 10 


4 x 15 


4 x 20 


4 x 25 


4 x 30 


79-38 


.92 


1.35 


32.38 


16.26 


69.86 


101-71 


1.16 


14.83 


45.66 


68.61 


52.28 


129-37 


2.38 


13.19 


16.45 


26.18 


63.77 


102-35 


2.45 


4.84 


25.45 


94.65 


229.74 


75-34 


5.28 


5.39 


47.01 


57.32 




35-16 


1.30 


9.74 


21.06 


27.35 




51-30 


.49 


26.53 


77.69 


35.65 




95-84 


.58 


3.47 


18.69 






62-23 


1.09 


10.20 


22.88 






53-28 


1.99 


3.09 


48.19 






58-29 


.38 


11.14 


68.62 






85-57 


.53 


6.82 








59-36 


.49 


9.27 








169-94 


11.71 


4.51 








61-28 


.76 


7.92 








80-28 


1.00 


18.12 








125-34 


4.52 


7.64 








53-25 


.32 


12.55 








77-23 


1.16 


17.07 








64-30 


.84 


23.81 








Total (sec) 


39.35 


211.48 


424.08 


326.02 


415.65 


Avg. (sec) 


1.97 


10.57 


38.55 


37.26 


103.91 



or each variable. All computations were carried out on a CDC Cyber 70, Model 72, a medium 
peed computer. An important feature of the program when the constraints are separable is 
hat partial solutions may be discarded when they become infeasible by calculating the slack in 
ach constraint as soon as Xj is determined. This procedure is not carried out in the detailed 
xample of the previous section because of the presence of a single nonseparable constraint, 
^ven in this case, when some constraints are separable, partial solutions may still be discarded 
ising only the separable constraints. 

INCLUSION 



This paper presents an exact solution method for an extended class of problems with 

n 

lighly nonlinear objective functions, ]£ fj (xj), that have the property that ff. I P — ► R P . 



i ~j 



lence problems with an extended class of objective functions— namely those that have rational 
alues— may be solved exactly for the global optimum. The method is one example of an 
ffort to utilize the favorable features of dynamic programming while avoiding the storage and 
omputational difficulties associated with multiply-constrained problems. 



1] 



2] 



REFERENCES 

Cooper, L., and M.W. Cooper, "Non-Linear Integer Programming," Computers and 
Mathematics With Applications, /, 215-222 (1975). 

Cooper, M.W., "An Improved Algorithm for Non-Linear Integer Programming," Report 
IEOR 77005, Southern Methodist University (February 1977). 



THE U.S. VERSUS THE SOVIET INCENTIVE MODELS 



Gregory G. Hildebrandt 

Department of Economics, Geography 

and Management 

United States Air Force Academy 

Colorado Springs, Colorado 

ABSTRACT 

This paper is concerned with models of the use of performance incentives 
in the Soviet Union and United States. The principal analytical result is an ex- 
tension of an analysis of the methods whereby Soviet planners make the deci- 
sion about production targets a variable under control of the producer, who is 
the only one possessing a knowledge of the uncertain condition of production. 
It is shown that this device can be viewed as a classical inventory problem. 
There is also an examination of the "U.S. incentive program" referring to 
multi-incentive contracts in which the profits received by the private producers 
are related to performance, outcome and cost. The analysis describes how this 
device can be extended to solve the target output selection problem of the So- 
viet planning system. 



INTRODUCTION 

The central planning organizations of both the Soviet Union and the United States have 
been concerned with influencing the behavior of enterprises in order to achieve improved allo- 
cation of resources. Although one is more likely to associate the central planning task with an 
economy such as the Soviet Union, the provision of many goods is centrally planned in the 
United States. For example, the Planning-Programming-Budgeting System of the United States 
government can be viewed as part of a central planning process. In the analysis I compare the 
incentive systems of the Soviet Union and the United States, thereby clarifying the similarities 
that exist between the two economic systems. A suggestion for improvement to the U.S. 
incentive system is also made. 

The key similarity between planning in the Soviet Union and the United States is that the 
government does not know as much about individual values and technological opportunities as 
do the relevant producers or consumers. For example, in the United States, the Planning- 
Programming-Budgeting System frequently culminates in government acquisition of goods such 
as military hardware and space systems from private enterprises. The production technolgies 
associated with many of these goods are not only highly uncertain, but are also more accurately 
known by the producers than by the government. Such goods as intercontinental ballistic mis- 
siles and manned space vehicles have embodied in them advanced technology and associated 
uncertainty about the conditions of production. This uncertainty is probably most pervasive 
during the engineering development phase of the "production" process when the performance 
characteristics of these goods are determined. The producers of these goods, however, will 

97 



98 G.G. HILDEBRANDT 

typically have a greater knowledge of this advanced technology and its impact on the conditions 
of production than does the government. Thus, the conditions of production are more uncer- 
tain for the government than for the producer at that time. The uneven impact of uncertainty 
implies that it is not possible for the government to specify the best output level. 

A similar situation exists in the Soviet Union. The state enterprise may have better 
knowledge of its production technology than the planners, and the Soviet planners, therefore, 
may be unable to specify the optimal output level. Yet both the U.S. and Soviet decision mak- 
ers attempt to provide appropriate incentives to motivate producers to select the "right" output 
level. 

In the United States this system employs the so-called "contractual incentive function" 
which specifies a mutually acceptable rule connecting the monetary rewards of one decision 
maker to the subsequent performance of another. Numerous enterprises have devised profit- 
sharing formulae to motivate supervisory and managerial personnel, and the Department of 
Defense and NASA have relied on the use of performance incentives to monitor the work of 
major contractors. For example, performance incentives were included in contracts with a total 
value of several billion dollars during the U.S. moon program.* Recent innovations in the use 
of performance incentives have appeared in the new Amtrak contract which provides payments 
to the railroads according to the quality of services they provide [2]. A similar contractual 
arrangement guarantees a one percent increase in the salaries of the policemen of Orange, Cali- 
fornia for every three percent decline in rape, robbery, burglary and auto theft [15]. 

Although the existing literature on economic planning does not specifically mention the 
use of contractual incentive functions, a related concept has arisen in discussions of "success 
indicators" in Soviet planning. The Soviet planning system often rewards enterprise agents 
according to the degree to which producers reach certain planned targets. f Thus, Soviet 
planners have implicitly defined a performance incentive system. In contrast to similar systems 
employed in the West, the Soviet system has not been "contractual" in the sense that it has 
been agreed upon by the planners and the enterprise managers. Instead, the state has unila- 
terally chosen the targets and rewards, and the enterprise managers have been expected to com- 
ply in order to attain their own maximum reward within the confines of the rules laid down by 
the planners. This "non-contractual" incentive system has clearly been an example of the use of 
performance incentives in the implementation of economic planning. 

Recently, the Soviets have experimented with an extended incentive system which pro- 
vides motivation for the state enterprises to select the optimal target output level before the 
determination of the actual output. The importance of this additional incentive stems from the 
fact that if the central planners have a good estimate of the amount of the good which will be 



*Using r. v, and r to represent measures of relative profit, cost, and performance, the functional form which applied to 
several of the large dollar value incentive contracts used during the U.S. moon program is 

: = fix) + ,c(.v) + af(x)f({v) + 13 

where 

and a\, aj, 03, h \ , bj, 63, <*, and p are constants. In addition, between 1967 and 1970, there were approximately $27 
billion of multiple incentive contracts evaluated by a Department of Defense analysis group. 

fit is widely accepted that this is the Soviets' most famous planning problem. In addition to monetary incentives, the 
Soviets have also tried to solve this problem using informational exchange during bargaining with the enterprise. 



U.S. AND SOVIET INCENTIVE MODELS 99 

oduced before it is actually produced, then a better coordinated plan can be achieved. There 
also a need for planners to coordinate outputs that are jointly used. For example, some inter- 
ediate goods are used jointly in the production of final goods, and final goods may be jointly 
nsumed. The reason why an incentive is required to motivate the managers of the state 
iterprises to reveal the optimal target is that the enterprise may also receive a reward based on 
e actual output achieved in relation to the target output level. The existence of this reward 
ay motivate the managers to understate the target output level if they are simply asked its 
ilue. 

Martin Weitzman [16] has analyzed this new incentive system using a model whose pro- 
xies have been investigated by Fan [6] and Bonin [3]. In this model, the planners fix the 
sources or inputs available to the enterprise, but there is uncertainty associated with the out- 
it that can be produced with these fixed inputs. The uncertainty rests with the planners, thus 
stifying the selection of the target output level by the enterprise. Although the output actu- 
ly achieved is not selected by the enterprise, Weitzman shows how the enterprise can use its 
lowledge of the uncertain conditions of production in conjunction with a specified perfor- 
ance incentive to select the best target output. 

This report will first review the Weitzman analysis and then show that the new Soviet 
centive program can be viewed as a classical inventory problem, which is a problem of deter- 
ining how much of product to keep in storage. This interpretation of the incentive program is 
lportant because inventory theory is a well-developed analytical framework and general associ- 
:ons between inventory theory and planning may prove fruitful. 

Recently, the state enterprises in the Soviet Union have been given greater flexibility in 
eir use of inputs. In view of this change, I next show how the new Soviet incentive program 
n be extended to deal with a situation in which the inputs used by the enterprise are choice 
iriables with associated cost. This extension has some similarities to the analyses of Bonin 
id Marcus [4], Snowberger [14], and Miller and Thornton [11] who have investigated the 
iplications of a relationship between output and variations in the producer's effort. 

An analysis of the U.S. incentive program will follow the discussion of the Soviet incen- 
/e system. To ease comparison between the two systems, a cost-effectiveness model will be 
;ed. Thus, I assume that the objective of the goverment is the achievement of some specified 
vel of performance at minimum cost. The large degree of uncertainty that exists during 
lgineering development prevents the government from knowing in advance what performance 
vel will be achieved for any level of expenditure. Furthermore, both the estimate of the 
tual performance level, the target, and the performance level actually achieved have associ- 
ed costs that must be borne by the government rather than the producer. Therefore, the per- 
rmance incentive can be viewed as a method of motivating the producer to take appropriate 
count of these costs during engineering development. This internalization of social costs by 
e producer suggests a relationship not only to the control of environmental externalities, but 
so to the design of a transfer price by one division of an enterprise that is providing some 
termediate good to a second division. The transfer pricing problem has been recently 
talyzed by Ronen [13], and Groves and Loeb [7]. 

The existing incentive system motivates the producer to economize on the costs associ- 
ed with the output level actually achieved. This system can be expanded to solve the target 
tput selection problem. This expansion would lead to a greater compatibility of the various 
terrelated output decisions, thus making the target output itself a product worth paying for in 
e U.S. 



100 G.G. HILDEBRANDT 

I. THE NEW SOVIET INCENTIVE MODEL 

In analyzing the new Soviet incentive system, Weitzman uses a model in which the fac- 
tors of production used by the enterprise are set by the planners, an assumption which is realis- 
tic in the U.S.S.R. where inputs have typically been rationed by the state. A tentative target J 
and a tentative bonus fund B are assigned to the enterprise during the first or preliminary phase. 
The tentative target is the planner's best estimate of the target output level at that time. Dur- 
ing the second, or planning phase, the enterprise has the option of revising the tentative target 
to y which has associated with it a revised bonus fund B computed in accordance with the for- 
mula,* 

5 = B+B(y-y), 

where the constant /3 is proportional to the "real social value of having an extra unit which has 
been pre-planned" [16]. 

In the third or implementation phase, when the enterprise ends up producing amount y, ii 
actually receives the bonus fund 



B = 



B + aiy — y) : y ^ y (overfulfilment) 
£j — y(y — y) : y < y (underfulfillment) 



where a is proportional to the "real social value of having an extra unit unexpectedly 
delivered," and y is proportional to the "real social cost of being unexpectedly caught short bj 
one unit" [16]. Subsequent analysis reveals that correct decision making by the enterprise 
requires that the constant 0, a, and y be in the same proportion to their respective value 
coefficients. Under the "old" Soviet incentive system, B and y were fixed by the planners 
Under the new system, they are set by the enterprise. 

In the model developed by Weitzman, there is uncertainty during the planning phase as t( 
the amount of output that will actually be produced with the fixed inputs. Only the produce: 
knows the probability density function /(y). Thus, we have an example of the informationa 
asymmetry which is so prevalent during the planning process and a justification for the produce 
to select the target output level. This uncertainty might in fact persist during the implementa 
tion phase, but with fixed inputs, actual output y is not a choice variable, and thus, the charac 
ter of the uncertainty that applies then is not relevant to this analysis. 

During the planning phase, when y is selected, the problem faced by the enterprisi 
(assumed risk neutral) is to choose y to maximize 

(*' [B+p(y-y)+y{y-y)]f(y)dy 

%/ — oo 

(1) +C [B + B(y - y) + aiy - j»] f(y)dy. 

By differentiating with respect to y, Weitzman shows that the optimal solution to this maximiza 
tion problem is to select y such that 

(2) P{y> y )= ^ZA. i 

y — a 
where 






*If the selection of the target ^influences the allocation of inputs to the enterprise, then the producer will take this int 
account in the selection of y. This raises a serious incentive compatability problem which has been discussed by Magi 
and Loeb [101. We assume that such a relationship does not exist. 



U.S. AND SOVIET INCENTIVE MODELS 1 1 

P(y > y)=Cf(y)dy. 

J y 

Because it is possible to multiply all of the coefficients by a constant without changing (2), 
only the relative magnitudes of the coefficients matter in determining the optimal y. The 
appropriate relative magnitudes are achieved when these coefficients are in the same proportion 
to their respective value coefficients.* Furthermore, in view of the fact that (2) must be posi- 
tive, this incentive system is meaningful only when the coefficients are set such that 
a < B < y. 

An Inventory Theory Interpretation 

The fact that there are costs borne by the center when the actual outcome is both below 
and above target suggests that an inventory theoretic interpretation can be given to Weitzman's 
analysis. To see the classical inventory structure of this problem, rewrite (1) as 

(3) B + B(y-y) + V y(y - y)f(y)dy + C aiy - y)fiy)dy. 

%/ — oo »/ y 

The difference, B — By, is fixed and therefore not relevant when choosing the target output 
level, but we must concern ourselves with the term fly which can be written as 

By = B f +C °yf(y)dy +p[ y iy- y)f(y)dy - B C iy - y)f{y)dy. 

•/-co »/ — oo %/ y 

The expression, 

J» oo 
. iy-y)fiy)dy, 
y 

is proportional to the benefits foregone, weighted by the probabilities, as a result of the economic 
system not being geared to a higher y when the actual output is larger than the target. 

If the target is not achieved, the actual y is less than y, and 
(5) bV (y-y) f(y)dy 

*/ — oo 

is applicable. This expression can be viewed as (proportional to) the benefits still received 
(weighted by probabilities) from having the system geared to y. Now insert (4) and (5) into 
the last two parts of (3), and obtain as the producer's problem the maximization of 

fc " y ) SL { y ~ y)f<y)dy + (« - /3)/.°° (y - y)f(y)Jy 
which is equivalent to the minimization of 

(7 - 0) /' (y- y)fiy)dy + (B - a) f°° (y - y)f(y)dy. 

«/ — OO %} y 

The coefficient y — B can now be identified as (proportional to) the net social cost per 
unit of output actually achieved below the target and B - a as (proportional to) the net social 
cost per unit of output above the target. 



*Each coefficient of the right hand side of (2) can be multiplied by a constant k yielding 

ky - Ar/3 k(y - /3) y - j3 

ky — ka k (y — a ) y — a 
In that the value coefficients are measured in rubles per extra output, multiplying each coefficient by a constant can be 
viewed as a change in the monetary unity which could never affect the selection of y. Also, note that the units associat- 
ed with each coefficient of the right hand side of (2) cancel. As the left hand side of (2) is a probability (a pure 
number), such a cancellation is required to equate both sides of (2). 



102 G.G. HILDEBRANDT 

Let us use the notational convention 
ci-y-0, 

c 2 = /3 - a. 

The problem facing the enterprise can therefore be written 

(y-^)/(v)afy + c 2 ). iy-y)f(y)dy. 

-oo ./^ 

When the producer's maximization problem (1) is rewritten as the minimization problem 
(6), it is possible to view the selection of y as the selection of the amount of a good (the target) 
to be placed in inventory. The coefficient c x can be viewed as the carrying cost per unit of 
unsold inventory and c 2 the per unit shortage cost. Taking the derivative of (6) with respect to 
y (and equating it to zero) we see that for the optimal solution value y y 

(7) _, . M c 2 p- a 

Piy < y) = — ; ^ . 

c i + c 2 y — a 

Equation (7) is a well-known formula from inventory theory [5]. Therefore, 

c i + c 2 y — a 
which is the solution obtained by Weitzman. 

The inventory formula (7) has a simple economic interpretation. Letting P = Piy < y), 
this formula can be rewritten as 

(8) Pc x = (1 - P) c 2 

and indicates that P should be selected through the selection of y, so that the expected net 
social cost of the output produced less than the target and the output produced at least as great 
as the target are equal. The reason an inventory theoretical interpretation is interesting is that 
inventory theory is a well-developed framework and analogies that can be found with the plan- 
ning process might prove fruitful in the development of a theory of economic planning. 

Production Inputs Variable 

Although Weitzman has chosen to view production inputs as fixed, largely because this 
assumption reflects the Soviet planning environment, it is possible to extend his analysis by 
allowing the production inputs used during the implementation phase to be choice variables of 
the enterprise. This extension may have relevance to the Soviet planning problem now that the 
managers of state enterprises are being given greater flexibility in the use of inputs. In order to 
simplify the analysis, I assume that there is the same degree of uncertainty about the conditions 
of production during the planning phase when the enterprise selects the target output level and 
during the implementation phase when a level of cost expenditure is selected. In the view of 
the enterprise, the conditions of production during both phases can be represented by 

y = h(c, 0), 

where 9 is a random variable with density function fid) applicable for both the planning phase 
and the implementation phase, and c represents production costs. Although identical uncer- 
tainty permits one to view the enterprise as selecting y and c simultaneously, a meaningful 
economic interpretation can still be given for the need to select y beforehand by assuming that 



U.S. AND SOVIET INCENTIVE MODELS 103 

the actual output v is not revealed at the time c is selected but rather at some later time which 
can be called the implementation phase. It will also be assumed in this extension that the share 
of cost expenditure borne by the enterprise is equal to s. 

Assuming that the coefficients yS, y, and a continue to apply, and the profits are deter- 
mined by T(y,c), the producer must solve 

M Max T(y,c)= f 9 ' i{yx) [B + /3(j> - y) + y(h(c,9) - y) - sc]f(O)d0 
y.c J -°° 

+ | . [B+fl(y-y)+a(h(c,9)-y)- sc]f(9)d9 

where the inverse function 0~ l (y,c) determines the value of 9 which achieves y = y when the 
production costs are c. The enterprise must set the derivative of this with respect to y equal to 
zero obtaining 

dT/dy = f e ~' 0x) (B - y)f(9)d9 + f " (j8 - a)f(9)d9 = 0. 
It is easy to verify that this equality implies that 

y — a 
similarly, the derivative of (9) with respect to c set equal to zero yields* 

11= [ e ~' 0c) [ y h c (c,9)-s}f(9)d9 
:il) + f °°, [ah c (c,9) - s)f(9)d9 = 0. 

J 6 l (y,c) 

fo obtain qualitative results we require knowledge of the function h{c,9). Assume that the 
incertainty is additive and that h c depends only on c(h c9 = 0).t Then it can be shown that (11) 
mplies that 

12) j»» > r'Oc)) - yh '~ s 



(y - a)h c ' 

7 or both (10) and (12) to be satisfied simultaneously, it must be true that 

y -/3 _ yh c ~ s 
y — a (y — a)h c 

vhich implies that the producer must set 

13) ph c (c) = s. 

his condition implies that when y is optimal, the selection of the level of cost by the enterprise 
an be determined by evaluating the profit from a small adjustment in y. The effect on the 
•rofit obtained from y captured when the optimal value of y is selected. Thus, the producer 
hould vary c until the extra profit associated with a small increase in target output (fih c ) just 
quals the reduced profits from increasing c by one unit (5). 

One can rewrite (13) as 

c= h-Hs/p), 

— 

Notation such as h c represents the partial derivative of the function h with respect to the variable c. 
Although this assumption is strong, it is frequently interesting to know what assumptions are required to obtain a 
narp characterization of an optimal policy. It is not difficult, however, to imagine an interaction between the level of 
)st expenditure and the random variable 9. For example, high levels of cost expenditure might be associated with 
reater uncertainty. Such interactions have been excluded from the analysis. 



104 G.G. HILDEBRANDT 

thereby permitting (10) to be written as 

(14) P{9 > 9- l (y,h c -Hs/p)) = ^^-. 

y — a 

The enterprise must satisfy this condition during the planning phase when selecting y. This 
condition recognizes that during the implementation phase the producer selects the optimal cost 
expenditure. Comparing (14) with (2) shows that the producer must simply account for the 
impact of the additional choice variable (cost) on the likelihood of being over target. However, 
once this adjustment is made, the economic interpretation described by (8) continues to apply. 

II. THE U.S. INCENTIVE MODEL 

The purpose of the U.S. incentive model is to motivate producers to select a performance 
level that is socially optimal. The DOD and NASA Guide states that 

the concept of multiple incentive contracting must quantitatively relate profit motivation 
directly and in accordance with the Government's objectives. ... it establishes the 
contractor's profit in direct relationship to the value of the combined level of performance 
in all areas [5]. 

Furthermore, 

the process of' including performance in an incentive structure must logically begin with 
the determination of the "value" of the characteristics which will be incentivized. The mul- 
tiple incentive contract should reflect the importance to the government of various cost, 
schedule, and performance outcomes, through the profits assigned to each part of the mul- 
tiple incentive structure [5]. 

Cost-Effectiveness Analysis 

One method of describing the U.S. incentive model is to use a cost-effectiveness analysis 
approach.* This approach applies when the government's objective is the achievement of some 
specified level of system performance at minimum cost and it simplifies comparison of the U.S. 
incentive model with the Soviet model. It is assumed that increasing performance level p of 
some component of the system during the engineering developing phase of procurement leads 
to future, or "downstream" cost savings for the government because of reduced acquisition cost, 
maintenance costs, etc. The basic structure of the U.S. incentive model can be most easily 
illustrated if it is assumed that the producer is given a performance reward based on the level of 
p actually achieved and on development costs. Later, a more complicated model will show how 
the U.S. incentive program can be expanded to incorporate the target selection features of the 
new Soviet incentive program. 

I assume that the cost of development function, C(p), is deterministic during the imple- 
mentation phase when the producer actually selects p. This function may, however, be known 
only to the producer. Indeed, in order to justify using a performance incentive in the first 
place, there must be some uncertainty in the government's mind about the cost of development 
at the time the incentive is specified. Otherwise, the government would simply specify p. The 






*As far as I am aware, the first mathematical treatment of multiple incentive contracting using a cost-effectiveness ap- 
proach similar to the one presented here was by Ackerman and Krutz [1]. 



U.S. AND SOVIET INCENTIVE MODELS 1 05 

downstream cost function, Dip), determines the costs borne by the government through the 
dependence on the output level selected by the producer. This function is assumed to be known 
by the government. Total cost is the sum of the development cost and the downstream cost 
and is designated Tip). At the time of producer decision making, the objective of the govern- 
ment is to solve 

Min Tip)= Cip) + Dip). 

p 

The first order condition for this problem is 

(15) C = -D', 

which simply says that the performance level should be increased until the producer's marginal 
development cost expenditure just equals the government's marginal downstream cost reduc- 
tion. 

The profit, or performance incentive function given to the producer under the U.S. incen- 
tive system is typically of the form 

(16) 7T=Gip)-sC 

where Gip) represents dollars of profit earned as a function of the performance level p, and s 
equals the share of the development cost borne by the producer. The relevant first order, or 
profit maximizing condition for the producer is 

G'ip) = sC'ip). 

In view of the government's optimization condition (15), the optimal incentive structure is 
obtained when —D' is substituted for C, and the government constructs the performance 
incentive function such that 

G'ip) = - sD'ip). 

Therefore, with the inclusion of a constant A, a performance incentive function of the form 

77 = - sDip) - sC + A 

will motivate the producer to satisfy (15), thereby satisfying the objectives of the government.* 

Extending the U.S. Incentive Model 

For .selected U.S. procurements in which the producer is the only supplier of a good 
whose performance is rewarded in relation to some target (thereby creating an incentive for the 
producer to understate the target if simply asked its level), there is value in extending the U.S. 
incentive program to include producer target specification. In addition to depending on the 
actual performance level, downstream costs also depend on the target performance level 
because of the time needed to prepare the operational environment (e.g., train maintenance 
people, etc.) for the actual performance level. 

In extending the U.S. incentive model, I assume, for the purpose of comparison with the 
Soviet incentive model, that during the planning phase an incentive function is specified and 
the producer selects a target performance level p. The actual performance level p is not 
achieved until an implementation phase. 



*N. J. Ireland [9] has shown that the "ideal price" is the benefit function of the central planning organization for certain 
types of planning environments. The function Dip) can be interpreted as the benefit function of the planners for the 
problem at hand. Note, however, that in this analysis, the cost sharing ratio is also applied to the function Dip). 



106 G.G. HILDEBRANDT 

During the planning phase, the downstream cost function will be of the form D(p,p). 
Although all costs are variable at that time, certain downstream costs are fixed at the time the 
actual performance is achieved. During the planning phase the dependence of these costs on 
the target performance level can be represented by F(p). Those costs which remain variable 
when the actual performance level is determined can be represented by Dv (p,p). 

In order to parallel the extension of the new Soviet incentive model to the situation where 
inputs are variable, I now assume that the choice variable of the producer is a level of develop- 
ment cost expenditure c. In the cost-effectiveness analysis section above, the performance level 
p was selected as the producer's choice variable. At both the time the producer selects the tar- 
get performance level and the time that a cost expenditure level is selected, the producer's view 
of the conditions of production is represented by 

P = g(c,9), 

where 9 is a random variable which has the same density function at both of the times of pro- 
ducer decision making. It is assumed that the government does not know g during the planning 
phase, thus justifying the selection of p by the producer. As we shall see, the government's 
information about downstream cost is transmitted to the producer in the incentive function. 
This information combined with the producer's information about the conditions of production 
yields, via profit maximization, the best solution to the target selection problem. 

The government in taking a social view is interested in the minimization of 

E(c + Dv(p,p) + F(p)). 
The first order conditions associated with this minimization are 

(17) 1 + E(dDv/dp) (dp/dc) = 

(18) E(dDv/dp) + dF(p)/dp = 0. 
The profit function given to the producer is of the form 

(19) 77 = G(p,p) - sc, 

where 5 again represents the share of the development cost borne by the producer. This func- 
tion has the same basic form as (16) to retain compatibility with what has typically been used 
for the existing U.S. incentive system. The first order conditions which apply for the producer 
are 

(20) EibG/bp) = 0, 

(21) E(BG/Bp) (Bp/dc) = s. 

Comparing (17) and (18) with (19) and (20) shows that the government can achieve its objec- 
tive if it constructs an incentive function such that 

(22) G-=-sU)v- + F-), 

(23) G p = -s(D p ). 

When condition (22) is satisfied, the incentive profit received by the producer from a change in 
the target performance level is just equated to a proportion of the incremental downstream cost 
savings. A similar interpretation applies to (23).* 






*The expectation operators are not required in (22) or (23) because the terms inside the expectation operators of (20) 
and (21) are substituted for the terms of (17) and (18) inside these operators. 



U.S. AND SOVIET INCENTIVE MODELS 107 

Thus, 

G(p,p) = - sD(p,p) + A, 
where A is a constant. 

By linearizing the function D\ (p,p) about p, and F(p) about p, where p is some specified 
performance level, e.g., a government estimate of the target performance level, one can obtain 
a formal equivalence of the U.S. and the Soviet incentive models. Thus, if one approximates 
D\(p,p) by 

K/s + a/s (p - p) when p > p 

K/s + y/s (p — p) when p < p, 



and F(p) by 

then 

- sD(p.p) = 



M/s + 13/ sip- p), 

- (K + M)+a(p-p)+/3(p-p~) :p>p) 

- (K + M) +y(p- p) +/3(p-p):p< p). 



The parameters a, y, and have the same interpretation as in the new Soviet incentive model. 
For example, in that p is a preplanned performance outcome, /3 is simply proportional to the 
social value of having an extra unit which has been preplanned and can be identified as a pro- 
portion of the cost savings achieved when p is varied during the planning phase. 



If inputs are fixed as assumed by Weitzman, the term sc vanishes from the profit function 
(19), and one obtains an equivalence to the new Soviet incentive model. If the inputs are vari- 
able and g(c,0) applies, then one obtains an equivalence to the extended Soviet incentive 
model developed above. The relevant maximization problem that must be solved by the pro- 
ducer is analogous to (9) . 

Note that the cost share 5 is the factor of proportionality that applies to the parameters a , 
/, and |8. If the cost share changes, then so too will the parameters. Thus, there appears to be 
i degree of freedom in the selection of these parameters. However, this factor of proportional- 
ly has distributional significance and, in fact, is related to the distribution of societal profits 
between the center and the enterprise. It is subject to optimization in an analysis of risk shar- 
ng between the center and the producer, and has been discussed by Hildebrandt and Tyson [8] 
.vho show that under certain types of differentiated information structures, it is still possible to 
ichieve both the distributional and allocational objectives of the planner. 

CONCLUSIONS 

Decision makers in both the United States and the Soviet Union face similar problems of 
:orrectly guiding production at the enterprise level. To achieve certain social objectives, the 
United States government has employed the contractual incentive function whereas the Soviet 
planners have used the non-contractual or unilarteral incentive function. 

The new Soviet incentive system provides an incentive for the enterprise to reveal the 
>ocially optimal target output level. My analysis has shown that this system can be expanded to 
ieal with the situation when the enterprise controls the amount of resources utilized, a situa- 
ion which is becoming increasingly typical in the Soviet Union and which continues to be the 



108 G.G. HILDEBRANDT 

norm in the United States. Although the option of placing an incentive on the target output 
level has not yet been used in the United States, the existing U.S. incentive system can be 
expanded to permit that possibility. 

BIBLIOGRAPHY 



[1 
[2 
[3 
[4 

[5 
[6 

[7 

[8 

[9 

[10 

[11 

[12 
[13 

[14 

[15 
[16 



Ackerman, D. and Krutz, R. "Structuring of Multiple Incentive Contracts," United States 
Air Force Academy Working Draft (May 1966). 

Baumol, W.J. "Payment by Performance in Rail Passenger Transportation: An Innovation 
in Amtrak's Operations," Bell Journal of Economics, 281-99 (Spring 1975). 

Bonin, J. P. "On the Design of Managerial Incentive Structures in a Decentralized Planning 
Environment," American Economic Review, 682-87 (September 1976). 

Bonin, J. P. and A. Marcus, "Information, Motivation and Control in Decentralized Plan- 
ning: The Case of Discretionary Managerial Behavior," Unpublished Manuscript, 1-27 
(June 1977). 

DOD and NASA Guide, "Incentive Contracting Guide" (October 1969). 

Fan, Liang-Shin. "On the Reward System," American Economic Review, 226-29 (March 
1975). 

Groves, T. and M. Loeb, "Reflections in "Social Costs and Benefits and the Transfer Pric- 
ing Problem," Journal of Public Economics, 5: 353-59 (1976). 

Hildebrandt, G. and L. Tyson, "Performance Incentives and Planning Under Uncertainty," 
Econometric Research Program Research Memorandum No. 201, Princeton University 
(July 1976). 

Ireland, N.J. "Ideal Prices vs. Prices vs. Quantities," Review of Economic Studies, 44: 
183-86 (1977). 

Loeb, M. and W. Magat, "Success Indicators in the Soviet Union: The Problem of Incen- 
tives and Efficient Allocations," American Economic Review, 173-81 (March 1978). 

Miller, J. and J. Thornton, "Effort, Uncertainty, and the New Soviet Incentive System," 
Southern Economic Journal, 432-60 (October 1978). 

Naddor, E. Inventory Systems, (John Wiley and Sons, Inc., 1966). 

Ronen, J. "Social Costs and Benefits and the Transfer Pricing Problem," Journal of Public 
Economics, 3: 71-82 (1974). 

Snowberger, V. "The New Soviet Incentive Model: Comment", Bell Journal of Econom- 
ics, 591-600 (Autumn 1977). 

The Trenton Times, p. 16 (Sunday, December 15, 1974). 

Weitzman, M. "The New Soviet Incentive Model," Bell Journal of Economics, 251-57 
(Spring 1976). 



THEORETICAL ANALYSIS OF LANCHESTER-TYPE 
COMBAT BETWEEN TWO HOMOGENEOUS FORCES WITH 

SUPPORTING FIRES* 

James G. Taylor 

Department of Operations Research 

Naval Postgraduate School 

Monterey, California 

ABSTRACT 

This paper studies combat between two homogeneous forces modelled with 
variable-coefficient Lanchester-type equations of modern warfare with support- 
ing fires not subject to attrition. It shows that this linear differential-equation 
model for combat with supporting fires may be transformed into one without 
the supporting fires so that all the previous results for variable-coefficient 
Lanchester-type equations of modern warfare (without supporting fires) may be 
invoked. Consequently, new important results for representing the solution 
(i.e. force levels as functions of time) in terms of canonical Lanchester func- 
tions and also for predicting force annihilation are developed for this model 
with supporting fires. Important insights into the dynamics of combat between 
two homogeneous forces with such supporting fires are discussed. 



1. INTRODUCTION 

Today military operations analysts commonly use deterministic Lanchester-typet models 
for developing insights into the dynamics of combat. Militarily realistic computer-based 
Lanchester-type models of quite complex military systems have been developed for almost the 
entire spectrum of combat operations, from combat between battalion-sized units [3], [7] to 
theater-level operations [5], [6]. Nevertheless, a simple combat model may yield a clearer 
understanding of significant interrelationships that are difficult to perceive in a more complex 
model, and such insights can subsequently provide valuable guidance for more detailed compu- 
terized investigations (see [2], [18]). In this paper we consider such a simplified variable- 
coefficient Lanchester-type model of combat between two homogeneous forces with supporting 
fires not subject to attrition, and develop important results concerning the representation and 
behavior of its solution. These theoretical results are shown to provide important insights into 
the dynamics of this combat situation. 

Thus, the model that we study is important because it yields some important insights into 
the effects of supporting fires on the dynamics of combat (see [11], [15], and [17]). Our work 
here extends and unifies previous results of a number of authors [1], [11], and [13] through 



*This research was partially supported by the Office of Naval Research (both through direct funding and also through 
the Foundation Research Program at the Naval Postgraduate School) and partially by the U.S. Army Research Office, 
Durham, North Carolina, under R&D Project No. 1L161 102H57-05 Math (funded with MIPR No. ARO 7-79). 
t So-called after pioneering work by F. W. Lanchester [10]. 

109 



110 J.G.TAYLOR 

[17]. Taylor and Parry [17] have considered the same model and developed more-restrictive 
victory-prediction conditions for fixed-force-ratio-breakpoint battles by considering the force- 
ratio equation (see also [11]). Taylor and Brown [14] have developed a mathematical theory 
for solving variable-coefficient Lanchester-type equations of modern warfare (without support- 
ing fires) and introduced canonical hyperbolic-like Lanchester functions for constructing their 
solution. Taylor and Comstock [16] extended this work by developing theoretical results for 
predicting force annihilation from initial conditions without having to spend the time and effort 
to explicitly compute force-level trajectories for the model without supporting fires. 

In the paper at hand we show that the variable-coefficient model with supporting fires may 
be transformed into the one without them so that all the known results about the latter may be 
applied: representation of solution in terms of canonical Lanchester functions, nonoscillation of 
the solution, force-annihilation-prediction conditions, explicit calculation of annihilation time, 
etc. We consequently can translate all these results to the case of combat with supporting fires. 
As a result of our work here, the theory of the model with supporting fires may be considered 
to be as complete as that of the model without them. Consequently, one can now study this 
variable-coefficient model almost as easily and thoroughly as Lanchester's original simple 
constant-coefficient model without supporting fires. 

2. NOTATION 

The symbols that are used in this paper are defined as follows: 

a, b, a, /3 = constant attrition-rate coefficients, 

a it), bit), _ (time-dependent attrition-rate coefficients, the first two are 
a(t), (lit) {taken to be given in the form ait) = k a g{t) and bit) = k b h(t), 

Ait), Bit) — time-dependent attrition-rate coefficients in the transformed 
model (5.2); given by (5.3), 

Cpit), Spit) = hyperbolic-like general Lanchester functions iGLF) which 
are linearly-independent solutions to the transformed 
P force-level equation (5.5); they are analogous to the hyperbolic 
cosine and hyperbolic sine respectively, and their quotient is denoted as 
Tpit) = Spit)/C P it);C Q it) and S Q it) are similarly defined, 

C x it), S x it) = hyperbolic-like GLF which are linearly-independent solutions 

to the X force-level equation (4.3); they are analogous to the hyperbolic 
cosine and hyperbolic sine respectively, and their quotient is denoted as 
T x it) = S x it)/C x it);C Y it) and S y it) are similarly defined, 

K> k b = positive constants ("scale" factors) used for the 
representation of a it) and bit), 

pit), qit) = transformed "force-level" variables corresponding to xit) and yit), 

respectively and related to them by (5.1); with initial values x and^ , 
p = dp/dt = dp/dt, 

Q* = parity-condition parameter for the model without supporting fires (4.1); 
defined by (4.7), 






LANCHESTER COMBAT WITH SUPPORTING FIRES 1 1 1 

t — battle time, with t = denoting the beginning of battle, 
// = time at which the X force is annihilated, i.e. x Uf) = 0, 

t = largest finite time at which a{t) or b(t) ceases to be defined, 

positive, or continuous; we take t = when no such finite time exists, 

x(t), y(t) = force levels of X and Fat time / for the model with supporting 
fires (3.1); with initial values x and.yo> 

X(t), Y(t) = force levels of A' and Fat time t for the model without 
supporting fires (4.1); with initial values x and v , 

z(t) = x(t)fyU), 

^/ = y/k a k b and is called the combat-intensity parameter, 

^■r = kjk b and is called the relative-fire-effectiveness parameter, 

A * = parity-condition parameter for the model with supporting fires (3.1); defined 
by (6.4), 

9 = -Jab + [(a - /3)/2] 2 . 



3. COMBAT MODELLED BY VARIABLE-COEFFICIENT LANCHESTER-TYPE 
EQUATIONS OF MODERN WARFARE WITH SUPPORTING FIRES 



We consider combat modelled by the following Lanchester-type equations 
(3.1) 



six 

~ = -a (t)y - /3 (t)x with x (0) = x Q , 
at 



^- = -b (t)x - a (t)y with y (0) = v , 
at 

where t = denotes the time at which the battle begins, x(t) and y(t) the numbers of X and Y 
at time r, and a(t), b(t), a(t), and (3(t) denote time-dependent Lanchester attrition-rate 
coefficients, which represent the effectiveness of each side's fire. In any analysis of combat, we 
should use the above equations only for x and v > and, for example, set dx/dt = when 
x = 0, since negative force levels have no physical meaning. However, for studying the 
mathematical properties of the functions defined by these differential equations, we will find it 
more convenient to ignore this restriction and assume that (3.1) holds for all values of x and v. 

Two situations that have been hypothesized to yield the above equations are: 

(51) "aimed-fire" combat between two homogeneous forces with "operational" losses [1], [9], 

(52) "aimed-fire" combat between two homogeneous (primary) forces with superimposed 
effects of supporting fires not subject to attrition [17] (see Figure 1). 

The practical use of such equations in analysis depends on one's ability to obtain realistic values 
for the coefficients: the prediction of the attrition-rate coefficients from weapon-system- 
performance data has been discussed by Bonder and Farrell [2], (see also [14], and [17]). 



112 









J.G. TAYLOR 








a(t) 














1 X 

I INFANTRY 


b(t) ^~~^ 


Y ] 
INFANTRY / 




XFORCE 
ARTILLERY 


Y FORCE 
ARTILLERY 






x(t) "^. 


*. a(t) 


"^ v(t) 


















-—____ fi(t) ____^-^ 





Figure 1. Combat between two homogeneous forces (infantry) with 
supporting weapons (artillery) not subject to attrition. 

In our study here of the behavior of solutions to (3.1), we make the following assump- 
tions about the attrition-rate coefficients: 

(Al) ait) and bit) are defined, positive, and continuous for / < t < + ©o with t < 0, 

(A2)a(r) and/3(r) ^ for t ^ t < + «> , 

(A3) a(t), bit), ait), and /3(f) <E L(t ,T) for any finite T. 

Here we use the notation ait) € Lit Q , T) to mean that y ait) dt exists (and is given by a 
finite quantity). It follows that (and we will assume so below) ait)% Li0,+°°) means that 



lim ait)dt = + oo. 



We further take ait) and bit) to be given in the form ait) = k a git) and bit) = k b hit), 
where k a and k b are positive constants ("scale" factors) chosen so that a it) /bit) = kjk b when 
git) = hit) for all t. We introduce the primary weapon systems' combat-intensity parameter X, 
and the relative-fire-effectiveness parameter X R (for the primary weapon systems) defined by 

(3.2) X/ = yfkjT b , andX/j = kjk b . 



Taylor and Parry t [17] noted that the force ratio, z = x/y, satisfies the Riccati equation 



(3.3) 



dz 



Xq 



2j ,= bit)z 2 + {ait) - (3it))z - ait) with z(0) = z = — 



dt 



yo 



and used this fact to develop much useful information about the behavior and implications of 
the model (3.1). For the model (3.1) with ait) = /3(/) = for all t ^ [i.e. the model 
without supporting fires (4.1) below], Taylor and Comstock [16] have developed theoretical 
conditions that predict force annihilation without having to spend the time and effort of com- 
puting the entire force-level trajectories. They also briefly considered (3.1) and (3.3) and 
observed that for identically equal fire effectivenesses of the supporting weapons [i.e. 
ait) = pit) for all t ^ 0] the same Riccati equation is satisfied by the force ratio for both (3.1) 
and (4.1). Consequently, in terms of the force ratio, a battle's evolution is the same for the 
two models (3.1) and (4.1), although the force levels initially decay more quickly for (3.1). 
Thus, we are led to conjecture that there must be some kind of intimate relationship between 
the two models with and without supporting fires. 



fit was the author's good fortune to be awarded (jointly with S. Parry) the 1975 MAS Prize by the Military Applica- 
tions Section of the Operations Research Society of America for the three papers Taylor and Parry [171 and Taylor 
[111, 1121. 



LANCHESTER COMBAT WITH SUPPORTING FIRES 1 13 

In this paper, we will show that the model with supporting fires (3.1) may be transformed 
into the one without supporting fires [i.e. (4.1)] and use this fact to develop a fairly thorough 
characterization of the mathematical nature of solutions to (3.1). Our results allow one, in 
theory, to study this particular variable-coefficient model (3.1) almost as easily and thoroughly 
as Lanchester's simple constant-coefficient one. 

4. SUMMARY OF RESULTS FOR VARIABLE-COEFFICIENT LANCHESTER-TYPE 
EQUATIONS OF MODERN WARFARE WITHOUT SUPPORTING FIRES 

In this section we summarize results about the model without supporting fires that we will 
use in our study of the model with supporting fires. These results have appeared in a scattered 
fashion in the literature (see [13], [14], and [16]), and we summarize them here in unified 
form for the reader's convenience. Accordingly, we consider the following variable-coefficient 
Lanchester-type equations of modern warfare without supporting fires 

-^ = - a{t)Y with XiO) = x , 
dt 

(4.1) dy 

2f bit) X with Y(0) = y , 

at 

Here we assume that the attrition-rate coefficients a(t) and bit) satisfy assumptions (Al) and 
(A3) of Section 3. 

Taylor and Brown [14] have shown that the X (without supporting fires) force level as a 
function of time may be written as 

(4.2) Xit) =x {C Y iO)C x it) - S Y iO)S x it)} - y J)^ [C x iO)S x it) - S x iO)C x it)}, 
where the hyperbolic-like general Lanchester functions (GLF) C x it) and S x (t) satisfy 



(4.3) ^f 

dt 2 

with initial conditions 



1 da 



ait) dt 



— - ait)bix)X = 0, 
dt 



(4.4) c x it ) = \, S x it )=0, 



1 dC x i dS x 

it ) = 0, — — — — Oo) 



a(t ) dt ait ) dt 

In other words, these functions are linearly independent solutions to the X force-level equation 
(4.3) and are analogous to the ordinary hyperbolic functions (see [13] and [14] for further 
details). We will refer to any such basic pair of linearly-independent solutions (normalized in 
an appropriate manner) to a force-level equation like (4.3) as general Lanchester functions 
(GLF). Thus, for such linear Lanchester-type differential combat equations, the hyperbolic-like 
GLF play in variable-coefficient combat a role analogous to that which the ordinary hyperbolic 
functions play in constant-coefficient combat. The GLF C Y it) and S Y it) are similarly defined. 
The reader should recall (see Section 3 above) that t denotes the largest finite time at which 
ait) and bit) ceases to be defined, positive, or continuous (see [14] and [15] for further dis- 
cussion). Also, we will set r = if no such finite time exists. 



114 J.G.TAYLOR 

Thus, the hyperbolic-like GLF C x , S x , C Y , and S Y are basic "building blocks" for con- 
structing the solution to (4.1). They depend on only the attrition-rate coefficients ait) and 
bit) and do not depend on the initial force levels x andj'o at all. Taylor and Brown [14] have 
given examples of various such GLF for different attrition-rate coefficients of tactical interest. 
Various numerical examples are also given in their work (see [14] and [15]). We observe that 
(4.2) simplifies to 

(4.5) X(t) = x C x (t) - yoJJ^, S x it), 

when t = 0. However, Taylor and Brown [14] have shown that (4.2) simplifies to (4.5) for 
t Q ^ if and only if a(t)/b(t) = constant = kj k b for all t (constant ratio of attrition-rate 
coefficients). In other words, the hyperbolic-like GLF only possess so-called algebraic addition 
theorems like the hyperbolic functions do only for a constant ratio of attrition-rate coefficients. 

The following results have been developed elsewhere in the literature. They form the 
basis for our subsequent theoretical analysis of the model (3.1). 

RESULT 4.1: At most one of the two force levels X{t) and Y(t) can ever vanish infinite 
time 

RESULTS 4.2: If either a(t) $ LiO, + oo) or bit) <? L (0, + <~), then the X force 
(without supporting fires) will be annihilated infinite time if and only if 

(46) T <ylkR Wcy®)-sm'' 

where 

(47) lim {S x (t)/C x (t)} = 1/0* 

RESULT 4.3: The time of annihilation of the X force (without supporting fires), t*, is given 
by 

(4.8) t a x = T x H{x C Y iO) + y ^S x i0)}/{x S Y i0) + WV^Q(O)}), 

where T x (t) denotes a hyperbolic-like GLF that is analogous to the hyperbolic tangent and is defined 
by T x it) = S x (t)/C x (t). 

Results 4.2 (see [13], however, for the proof of a more general result) and 4.3 were 
developed by Taylor and Comstock [16], although Result 4.2 was given in a slightly more res- 
trictive form (namely, both ait) and bit) GLiO, + «>)). Let us now sketch the proof of 
Result 4.1 (see also [13] for a less detailed proof), which says that all solutions to (4.3) are 
nonoscillatory (see, for example, Ince [8, p. 224]). Multiplying the first equation of (4.1) by K, 
the second by X, adding, and integrating the result between and r, we obtain 

(4.9) Xit)Yit) = xoy - f \ais)Y 2 is) + bis)X 2 is)}ds. 

Considering the standard uniqueness theorem for linear differential equations (see, for exam- 
ple, Coddington and Levinson [4, p. 67]), we see that it is impossible for both Xit) and Yit) 
to be equal to zero at any finite time, since they then would have to be equal to zero for all 
time. Thus, since xoy > and the integral in (4.9) is strictly increasing in t and positive for 
/ > 0, it follows that Xit) Yit) has at most one finite zero for t ^ 0, and Result 4.1 has been 
proven. 



LANCHESTER COMBAT WITH SUPPORTING FIRES 



115 



Result 4.1 is very useful in the theoretical study of the behavior of solutions to (4.1). It 
shows that if there exists a finite f* such that Xit*) = 0, then Yit) > for all t ^ 0. Thus, if 
we can find a zero for X(t), we know that Yit) is guaranteed to be positive for all t. This 
result is useful in proving Result 4.2. Furthermore, when continuous withdrawals for both 
sides (denoted as rit) and s{t)) are added to the model (4.1) (e.g. 
dX/dt = -ait) Y — rit) where rit) > 0), then Result 4.1 no longer holds (at least when 
a it), b it), r it), and sit) are constants). 



5. TRANSFORMATION OF MODEL WITH SUPPORTING 
FIRES TO ONE WITHOUT THEM 

In this section we show that the model with supporting fires (3.1) may be transformed 
into one without them (4.1). This transformation is the key result upon which this paper is 
essentially based. Thus, the substitution 



(5.1) 



pit) = xit) exp 



qit) = yit) exp 



f^is)ds 
f a is)ds 



transforms (3.1) into 



(5.2) 
where 

(5.3) 



-^ Ait)q with/>(0) = x , 

at 



dq = 
I dt 



Bit)p with ? (0) = x , 



and 



Ait) = ait) exp 



Bit) = bit) exp 



f'\pis)-ais) 



ds 



- J" o '[j8(s)-a(s)]<fc 



The above assumptions (Al) through (A3) of Section 3 about ait), bit), ait), and pit) 
imply that A it) and Bit) have the same properties as ait) and bit), i.e. they €L(/ , T) for 
any finite T and are defined, positive, and continuous for t < t < +°°. Thus, we can invoke 
all the results for the model without supporting fires (4.1) and obtain results for the model with 
supporting fires (3.1) by inversion of the transformation (5.1). Before we do this, however, let 
us consider the representation of the solution to (5.2). 



The transformed "force-level" variable pit) satisfies 



(5.4) 



d 2 p 
dt 2 



1 dA 



Ait) dt 



dp 
dt 



-Ait)Bit)p = 0, 



which may be written in the equivalent form 



116 J.G.TAYLOR 



" w -" w + TO-f 



^--aO)bU)p-0, 
dt 



(5.5) A 
rfr 

with initial conditions 

(5.6) /><0)-xo. and — L- i| (0) = - y , 

a (.0) or 

Hence, by the results of Taylor and Brown [14] reported in the previous section (i.e. (4.2)), the 
solution to (5.4) that satisfies the initial conditions (5.6) may be written as 

(5.7) pit) = x {C Q (0)C P (t) - S Q (0)S P (t)} - yJ\^{C P iO)S P it) - S P (0)C P (t)}, 

where the hyperbolic-like GLF C P {t) and S P it) are linearly independent solutions to the P 
force-level equation (5.5) that satisfy the initial conditions 

(5.8) C P it )=\, S p (t ) = 0, 

1 dC P i dS P l 

('o) = 0, T77T-3r('o)- 



a(t ) dt a(t ) dt 

In other words, C P {t) and S P it) play exactly the same role for the equation (5.2) that 
C x it) and S x it) do for (4.1). We have, however, expressed the P force-level equation in 
terms of the attrition-rate coefficients of the model with supporting fires (3.1). We note again 
that t denotes the largest finite time at which a(t) or bit) ceases to be defined, positive, or 
continuous. Also, we will set t = if no such finite time exists. The GLF CqU) and SqU) 
are similarly defined, with the following initial condition worthy of note: 

1 dS Q 

bTo)^r y 



(5.9) * !? Uo) = Jk 



6. BEHAVIOR OF THE MODEL WITH SUPPORTING FIRES 

It follows from (5.1), (5.2), and the results of Section 4 that when the supporting fires are 
present (i.e. for our model (3.1) under study), the X force level as a function of time is given 
by 

(6.1) xit) = [x [C Q iO)C P it) - S Q iO)S P it)} - yoyfc[C P (.0)S P (t) 



-S P iO)C P it)}] exp 



-J> 5) 4 



From (5.1) it follows that for any finite t 

(6.2) pit) =0 if and only if xit) = 0, and qit) = if and only if yit) = 0, 

whence Result 4.1 yields Result 6.1. 

The behavior of the model with supporting fires is then largely described by the following 
results. 

RESULT 6.1: At most one of the two force levels xit) and yit) can even vanish in finite 
time. 

RESULT 6.2: If either A(t) $Li0, + «>) or B(t)4L(0, + °°), then the X force (with X 
force (with supporting fires) will be annihilated in finite time if and only if 



(6.3) 



v 



LANCHESTER COMBAT WITH SUPPORTING FIRES 
Cp(Q)-A*S P (0) 



117 



A*C Q (0) - S Q (0) 



where 
(6.4) 



lim [Sp(t)/C P U)}= 1/A 



t— + oo 



RESULT 6.3: The time of annihilation of the X force ( with supporting fires) , tjj , is given by 
(6.5) tf- T?H[x C Q (0) + yos /)^Sp(0)}/{x Q S Q (0) + W^C^O)}), 



where T P (t) denotes a hyperbolic-like GLF that is analogous to the hyperbolic tangent and is defined 
by Tp(t) = Sp(t)/Cp(t). 

RESULT 6.4: Ifbotha(t) and b(t)0L(O, + oo), then either A (t) <?L(0, + oo) orB(t)4L 
(0, +oo). 

RESULT 6.5: Ifa(t) =*B(t) for all f€ [0, + oo), then 
(6.6) x(t) = [x {C Y (0)C x (t) - S Y (0)S x (t)} - yJT R {C x {Qi)S x (t) 



- S x (0)C x (t)}] exp 



-j'ti(s)ds 



and when either a(t)$L(0, + oo) or b(t)4L(0, + oo) ( then the X force (with supporting fires) will 
be annihilated in finite time if and only if (4.6) holds. 

RESULT 6.6: If a(s) ^ 0(s) for all s € [0,/] with inequality holding for a subinterval of 
positive length, then 



(6.7) 



x(t) . X(t) . n 

v(0 Y(t) 



Results 6.1 through 6.5 follow from the material in Sections 4 and 5. Let us now sketch 
the proof of Result 6.6. Equations (4.3) and (5.4) may be combined to yield 



pX_ pX 
A a 



(6.8) 



- fjB(s) - b(s)}p 2 ds + f^l/Ais) - \/a{s))p 2 ds 

■ds, 



+ 



r< l (pX-pX) 2 

*^o a(s) X 2 



which is known as the Picone formula (e.g. see Ince [8, p. 226]). Next, we observe that the 
initial conditions to (4.3) and (5.4) are given by 

(6 . 9) ,<0)-X<0)-*> and -1^,0) =-^f «,) = -,„. 

Substituting (6.9) into the left-hand side of (6.8) and observing that the right-hand side of 
(6.8) is positive for t > because a (s) ^ /3(s) for all 5 € [0,/] with strict inequality holding 
on a subinterval of positive length implies that < A (/) < a(t) and B(t) > b(t) for t > 0, 
we find that 



X(t) p(t) x(t) 
Y(t) * q(t) " y(t) 6XP1 



f'\B(s)-a(s)]ds 



< 



x(t) 
y(t)' 



and Result 6.6 has been proved. 



118 



J.G. TAYLOR 



7. ILLUSTRATION OF THEORY WITH RESULTS FOR 
CONSTANT-COEFFICIENT MODEL 

In this section we illustrate the above general mathematical theory of the Lanchester-type 
equations (3.1) for the special case in which all the attrition-rate coefficients are constants. 
Hence, we consider 

(7.1) 



dx „ , dy , 

— — = —ay — Bx, and -f- — —bx — ay, 
dt dt 



where a, b, a, and B denote constants, and in this case t — 0. 

Applying the key transformation (5.1) of Section 5, we find that the transformed force 
levels pit) and qit) satisfy (5.2) with 

(7.2) Ait) = ae^- a), t and Bit) = be (a -* )r , 

and consequently (5.4) reads 

4-<*-.>*--,-* 



We then find that 

(7.3) C P U)~ e ,il3 - a)/2 {cosh 9t + 

(7.4) S P it) 



a- B 



29 



= ^L „/</B-«)/2„: 



9 



sin/7 9t), 
sin/? 9t, 



where 9 = -Jab + [(a - B)/2] 2 . It follows that 

1 _ 9-[ja- B)/2] 

A* Vab 



Hence, (6.3) yields the known constant-coefficient result (see Bach, Dolansky, and Stubbs [1] 
or Taylor and Parry [17]) that the X force will be annihilated in finite time if and only if 

(7.5) 



^1 < 1 



9-i^~) 



and also (6.1) reads (since / = 0) 



(7.6) 



xit) = 



xoCpti)- yff yf^S P (ty\e-^ 



which may be written in a more recognizable form as 



(7.7) 



xit) = 



Xq COS/7 9t 



ay Q + 



B - a 



xq 



sin/7 9t 



-I (a +/})/2 



8. DISCUSSION 

The above theoretical results provide many important insights into the dynamics of com- 
bat between two homogeneous forces with supporting fires not subject to attrition, and also 
they are quite useful for facilitating parametric analyses (see [14] and [15]). For example, we 
observe that when each side's supporting fires are always equally effective (i.e. ait) = Bit); 



LANCHESTER COMBAT WITH SUPPORTING FIRES 1 1 9 

see Result 6.5), they cancel out and the battle's outcome in a fight-to-the-finish is the same 
(although the victor suffers greater losses) as when they are not present. Strangely enough, 
although both force levels decay more rapidly in the early stages of battle (see equation (6.6)), 
both the length of battle and the time history of the force ratio are the same both with and 
without the supporting fires. 

Moreover, when the effectiveness of one side's supporting fires always dominates that of 
the other side (e.g. a(t) > fi(t) always; see Result 6.6), then the force ratio is always more 
favorable in this model to the side with superiority in fire support than it is in the correspond- 
ing model without the supporting fires. In other words, gaining superiority in fire support is in 
some sense equivalent to an improvement in the force ratio of primary fighting systems (e.g. 
infantries). Result 6.2 shows, however, that supporting fires alone cannot win a battle of attri- 
tion but that the force ratio of the primary fighting systems must be above a threshold value 
(which, however, does depend on the net effectiveness of the supporting systems; see equa- 
tions (5.5) and (6.4)) in order for a side to win. The main point is that it is the (cumulative) 
net effectiveness of the supporting weapon systems that modifies the relative effectiveness of the 
primary systems in order to determine the outcome of battle (see (5.2), (5.3), (6.3), and (6.4)). 

In this paper we have generalized the results of Taylor and Brown [14] on representing 
the solution in terms of GLF and those of Taylor and Comstock [16] on predicting force 
annihilation for the model (3.1) by means of the key transformation (5.1). Such a close rela- 
tionship between the two models (3.1) and (4.1) was suggested by Taylor and Comstock's 
observation that the Riccati equation (e.g. (3.3)) satisfied by the force ratio is the same for both 
models when each side's supporting fires are always equally effective, i.e. a(t) = /3(f). More- 
over, Result 6.5 means that all the series solutions and GLF developed [14], [15] and [16] for 
the model (4.1) without supporting fires may be used for the model (3.1) in this case. Further- 
more (as observed by Taylor and Comstock's [16]), the force-annihilation-prediction conditions 
are the same for these two models in this special case. However, we have given more general 
force-annihilation-prediction conditions (expressed in terms of special transcendental functions, 
i.e. the GLF), which apply when ait) ^ /3(/) and are complementary to Taylor and Parry's 
[17] conditions (expressed in terms of elementary functions under more restrictive conditions). 

In summary, we have presented new important results about the variable-coefficient 
model (3.1) for Lanchester-type combat between two homogeneous forces with supporting fires 
not subject to attrition (see Figure 1). These results make the mathematical theory about 
representing and analyzing the model's behavior just as complete as that for variable-coefficient 
Lanchester-type equations of modern warfare without supporting fires. In many cases of tacti- 
:al interest (see the time-dependent attrition-rate coefficients considered in [13 through 16]), 
we can now study this variable-coefficient model (even with supporting fires) almost as easily 
and thoroughly as Lanchester's classic constant-coefficient one. Also, our results may be 
viewed as results in the theory of differential equations, and then, of course, they apply to any 
Dther system that can be modelled with (3.1) (e.g. reaction kinetics for chemical or biological 
processes) . 

). CONCLUSIONS 

(I) The net effectiveness of supporting weapons is the parameter that modifies the 
relative effectiveness of primary systems and determines (along with the relative 
effectiveness and the initial force ratio of the primary systems) the outcome of 
combat between two homogeneous forces with supporting fires not subject to 
attrition. 



120 J.G.TAYLOR 

(II) Such combat with supporting fires is equivalent to combat without the supporting 
fires, only the relative effectiveness of the primary systems is modified by the 
cumulative net effectiveness of the supporting systems. 



(Ill) Supporting weapons augment, but do not replace, primary weapon systems 
REFERENCES 
[1 
[2 



[3 
[4 
[5 

[6 

[7 

[8 

[9 

[10 

[11 

[12 
[13 
[14 

[15 

[16 



Bach, R.E., L. Dolansky, and H.L. Stubbs, "Some Recent Contributions to the Lanchester 
Theory of Combat," Operations Research 10, 314-326 (1962). 

Bonder, S. and R.L. Farrell (Editors), "Development of Models for Defense Systems Plan- 
ning," Report No. SRL 2147 TR 70-2 (U), Systems Research Laboratory, The Univer- 
sity of Michigan, Ann Arbor, Michigan (September 1970). 

Bonder, S. and J. Honig, "An Analytic Model of Ground Combat: Design and Applica- 
tion," Proceedings of U. S. Army Operations Research Symposium 10, 319-394 (1971). 

Coddington, E.A. and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, 
New York, 1955). 

Cordesman, A. (Editor), "Developments in Theater Level War Games," Unpublished 
materials for C-5 Working Group of 35th Military Operations Research Symposium 
(1975). 

Farrell, R.L. "VECTOR 1 and BATTLE: Two Versions of a High-Resolution Ground and 
Air Theater Campaign Model," pp. 233-241 in Military Strategy and Tactics, R. Huber, 
L. Jones, and E. Reine (Eds.), (Plenum Press, New York, 1975). 

Hawkins, J. "The AMSAA War Game (AMSWAG) Computer Combat Simulation," 
AMSAA Tech. Report No. 169, U. S. Army Material Systems Analysis Activity, Aber- 
deen Proving Ground, Maryland (July 1976). 

Ince, E.L. Ordinary Differential Equations, Longmans, Green and Co., London, 1927 
(reprinted by Dover Publications, Inc., New York, 1956). 

Isbell, J.R. and W.H. Marlow, "Attrition Games," Naval Research Logistics Quarterly 3 
71-94 (1956). 

Lanchester, F.W. "Aircraft in Warfare: The Dawn of the Fourth Air— No. V., The Princi- 
ple of Concentration," Engineering 98, 422-423 (1914) reprinted on pp. 2138-2148 of 
The World of Mathematics, Vol. IV, J. Newman (Ed.), (Simon and Schuster, New York, 
1956). 

Taylor, J.G., "On the Relationship Between the Force Ratio and the Instantaneous 
Casualty— Exchange Ratio for Some Lanchester-Type Models of Warfare," Naval 
Research Logistics Quarterly 23, 345-352 (1976). 

Taylor, J.G., "Optimal Commitment of Forces in Some Lanchester-Type Combat Models," 
Operations Research 27, 96-114 (1979). 

Taylor, J.G., "Prediction of Zero Points of Solutions to Lanchester-Type Differential Com- 
bat Equations for Modern Warfare," SI AM J. Applied Mathematics 36, 438-456 (1979). 

Taylor, J.G. and G.G. Brown, "Canonical Methods in the Solution of Variable-Coefficient 
Lanchester-Type Equations of Modern Warfare," Operations Research 24, 44-69 
(1976). 

Taylor, J.G. and G.G. Brown, "Numerical Determination of the Parity-Condition Parame- 
ter for Lanchester-Type Equations of Modern Warfare," Computers and Operations 
Research 5, 227-242 (1978). 

Taylor, J.G. and C. Comstock, "Force- Annihilation Conditions for Variable-Coefficient 
Lanchester-Type Equations of Modern Warfare," Naval Research Logistics Quarterly 
24, 349-371 (1977). 



L ANCHESTER COMBAT WITH SUPPORTING FIRES 1 2 1 

[17] Taylor, J.G. and S.H. Parry, "Force-Ratio Considerations for Some Lanchester-Type 

Models of Warfare," Operations Research 23, 522-533 (1975). 
[18] Weiss, H.K. "Some Differential Games of Tactical Interest and the Value of a Supporting 

Weapon System," Operations Research 7, 180-196 (1959). 



ON SINGLE-MACHINE SEQUENCING WITH ORDER CONSTRAINTS 

K. D. Glazebrook 
University of Newcastle upon Tyne, England 

ABSTRACT 

A collection of jobs is lo be processed by a single machine. Each job has a 
cost function associated with it which may be either linear or exponential, costs 
accruing when a job is completed. The machine may be allocated to the jobs 
according to a precedence relation. The problem is to find a strategy for allo- 
cating the machine which minimizes the total cost and which is consistent with 
the precedence relation. The paper extends and simplifies some previous work 
done by Sidney. 

1. INTRODUCTION 

A job-shop consists of one machine and a set J — [\, . . . , n} of jobs to be processed on 
it. Associated with job i is a deterministic processing time X t and a cost Junction C, (.) such that 
if job / is completed at time F, {flow time) its cost is Cj{F,). We usually either assume that 

(1) C,(t) = -C,a', 1 < / < if, 

for positive constants C, and discount rate a,0 < a < 1, in which case the problem is said to 
have discounted costs or we assume that 

(2) C,(t) = C,t, 1 < / < if, 
for positive constants C, in which case the costs are said to be linear. 

A precedence relation R exists on J such that if (/ — » j) € R then job i must precede job j 
in any feasible permutation of J. The objective is to find those permutations a of J which are 

n 

consistent with R and which minimize the total cost TC (a) = £C, [F,(a)}, F,(a) being the 

/-I 
completion time of job i under a. 

Sidney [5] considered the above problem with linear costs. He proposed an algorithm that 
yields all the optimal permutations. This paper modifies and extends Sidney's in the following 
ways: 

(i) Some new results for stochastic versions of this problem are reported in Section 2. 

(ii) Sections 3 to 6 contain a discussion of the difficult deterministic problem with 
discounted costs as well as the rather easier problem with linear costs. An algorithm is 
presented in Section 5 which yields optimal permutations for all of these problems. The proofs 
used to establish the validity of this algorithm are, in the author's opinion, simpler than those 
used by Sidney. 

123 



124 K.D. GLAZEBROOK 

2. STOCHASTIC SCHEDULING 

As Banerjee [1] points out, 'in real life the time taken to complete a job on a machine is 
invariably random'. In the light of this assertion, consider the following stochastic scheduling 
problem: Let job set 7, precedence relation R and cost functions C,, 1 < / < n, be as in Sec- 
tion 1. The processing times [X\, ... , X n \ are independent integer- valued random variables 
with honest distributions. During each time interval [/, / + 1), t a nonnegative integer, just 
one of the unfinished jobs is processed on the machine. 

A feasible strategy tt is any rule for allocating the machine to the jobs which is consistent 
with R. Under strategy tt, job i is completed at random time F^tt). The objective is to find 



those feasible strategies tt that minimize the total expected cost TC (tt) = E 



j:c,[f,(tt)) 



To date, progress has only been made in solving such stochastic problems where the con- 
straint set R satisfies fairly stringent requirements; see, for example, Glazebrook [2] and Meilij- 
son and Weiss [4]. The problem with general R seems very complex. One possible way for- 
ward is to find a large class of problems for which the optimal strategy can be shown to be 
nonpreemptive (that is, given by a permutation of the jobs). 

The following result holds whether the cost structure is discounted or linear. 

THEOREM 1: If the distributions of the processing times X,, 1 ^ / ^ n, have nonde- 
creasing hazard rate there is a nonpreemptive strategy which is optimal among all the feasible 
strategies. 

This result is a consequence of Theorems 3 and 4 in Glazebrook and Gittins [3]. Note 
that the class of probability distributions covered by Theorem 1, namely those with non- 
decreasing hazard rates, contains a large number of examples of interest including the 
geometric distribution. Note also that a similar result to Theorem 1 can be obtained for 
continuous-time problems by considering appropriately chosen sequences of discrete-time prob- 
lems. 

It follows from Theorem 1 that many stochastic scheduling problems are formally 
equivalent to deterministic ones in that they have nonpreemptive optimal strategies. Hence the 
algorithm in Section 5, although given in the context of deterministic problems, also yields 
optimal strategies for a large class of stochastic problems. 

For the rest of the paper we concentrate exclusively on the deterministic case. 

3. ORDERING JOB MODULES: DISCOUNTED COSTS 

Throughout this section we are concerned with the deterministic problem with costs given 
by (1). However, before we can state and prove our main result we require some terminology 
and notation. 

A set of jobs M c 7, to be denoted by {(i,j), 1 < / < m, 1 < j < n,}, forms a job 
module if and only if 

(0 ['(/, j) - (/, /)] <E R, 1 < / < m, 1 < j < I < «,, 
(ii) [(/, J) — (*, /)] <t R, 1 < / * fc < m, 1 < j < n„ 1 < / < n k . 



SINGLE MACHINE SEQUENCING 1 25 

(iii) If a € 7-Mthen precisely one of (a), (b) or (c) holds: 

(a) [a - (/', j)] 6 R, 1 < / < m, 1 < j < nr, 

(b) [(/, j) — a] € R, 1 < / < m, 1 < J < -,; 

(c) {[a - (/, ;)], [(/, y) - a]} n /? = 0, l < / < m, l <.y < »,. 

From (i) and (ii), the jobs in M form disjoint chains {(/', j), 1 < j < «,}, 1 < / < m. 
Condition (iii) indicates that a job in J-M has the same precedence relation to every job in M. 
Hence a job in J-M will either precede every job in M (iii(a)} or be preceded by every job in M 
(iii(b)} or have no relation to any job in M {iii(c)}. 

x 
To simplify the algebraic expressions in what follows we introduce the notation M, = a ' 

and 

-l 



(3) CM\a x , <x 2 a p ] = Y.C., [1 M ai 1 - ftM 

r=\ [/=1 J) I r=\ 

where [a\, a 2 , .... a p } is an ordered subset of J consistent with R. The function CM will play 
an important part in the analysis. 

The following two Lemmas which are both easy to establish are used in the proof of 
Theorem 2. 

LEMMA 1: Let max {CM[a x , a 2 , ... , a,]} = CM[a x , a 2 , ... , a,.] where./* is the 
largest such maximizing index, then 

(4) max {CM[a r , a r+x , ... , a,]} ^ max [CM[a x , a 2 , ... , a 7 ]} ( 1 < r <• j*, 

j — r(\)p j = \(\)p 

and 

(5) max {CM[a r , a r+x , ... , a 7 ]} < max [CM[a x , a 2 , ... , a,]}, j* < r < p. 

j = r(\)p j = \(\)p 

LEMMA 2: If S x = [a x , ... , a p ), S 2 = {(3 X p g ) and S 3 = {y, y r ) are mutu- 
ally disjoint ordered subsets of /consistent with R such that 

(i) [S 2 , S3} = {/3 1( ..., P q , y\, ... , y r ) is also an ordered subset of J consistent with 
R, 



(ii) CM [o, a p ] ^ CA/[/3,, ... , p q ], CM [a, a p ] > CM[y 1( .... y r ] and 

one of these two inequalities is strict, then 

CM[a x , .... a p ] > CA/[j3, /3,, y,, . . . , yj. 

Theorem 2 tells us in which order the jobs in job module M will be processed in an 
optimal permutation for the jobs in J. 

THEOREM 2: There is an optimal permutation for the jobs in J such that the following 
rule is adopted for processing the jobs {(/', ./'), 1 < / < m, 1 <7 < n,) in M: If at some time 
only jobs {(/', y), 1 < / < m, 1 < j ^ 5,} among the jobs in M have been completed, then 
the next job in Mto be processedwill be a job (k, s k + 1) where k is chosen such that s k < n k 
and 

(6) max {CM[(k, s k + \), ..., (k,j)))= max ( max {CM[(i,s, + \) (i, j)]) 

j-(s k +l)(l)n k [f:s f <fl,] ly = (s,+ l)(l)n, 



126 K.D. GLAZEBROOK 

If more than one job satisfies these criteria it is immaterial which of them is chosen next. 

PROOF: Those jobs from {(/, j), 1 < j < «,} still remaining to be processed at any time 
will be referred to as chain i. If at some time chain i consists of the jobs {(/, j), s, + 1 
< j ^ itj] it has associated with it (see (6)) the quantity 

(7) max [CM[(i, s, + \), ... , (/, j)]} 

which will be called the allocation index for chain i. 

The theorem is proved by using an induction on the number of jobs in M. If \M\ = 1 
the result is trivially true. We suppose that the theorem is true for 1 ^ \M\ < n*but not for 
\M\ = «*and show that this leads to a contradiction. Consider a problem in which \M\ = n*. 

We suppose that in an optimal permutation for J the first job to be chosen from M is 
(s, 1) where chain 1, say, has the largest allocation index and chain s does not. Hence, from 

(7) 

max {CMKs, 1) (s, j)]) = CM[{s, 1) (s, j[s})] < CM[(l, 1),..., (\,j{l))) 

i = \(\)n s 

(8) 

= max {CM[(\, 1) (1, j)]} 

y-Ki)/!, 

where j[\} and j[s} are the largest such maximizing indices. From the inductive hypothesis, 
when job (s, 1) has been completed the remaining jobs in M will be processed in the order 
described in the statement of the theorem. It follows from Lemma 1 that the jobs in A/will be 
processed in the order {(5, 1), ..., (s, k{s}), (1, 1), ..., (1, j{\))\ T] for some k[s), 
1 < k[s] < n s , where Tis an ordering of the job set J - {(1, /), 1 < / < j[\); (s, /), 1 < 
/ ^ k{s}}. Consider now the following three cases which are mutually exclusive and exhaus- 
tive. 

The first set of jobs in M to be allocated service before being interrupted by the allocation 
of the machine to a job in J-M: 

CASE 1 contains {(s, 1), .... (s, k[s}), (1, 1), ... , (1, j[l})}\ 

CASE 2 is [(s, 1), .... (5, k(s}), (1, 1), .... (1, p)} for some a 1 ^ p ^ j{\}; 

CASE 3 is {(5, 1), .... (5, q)} for some q, 1 < q ^ k[s}. 

We now consider case 2 in detail. 

CASE 2: Following the processing of jobs {(s, 1), ... , (s, k{s}), (1, 1), ... , (1, p)) 
from M, the first "visit" is made to J-M (that is, the first visit after service is begun on M). 
During this first visit, jobs {(a, v), 1 < v < ^J, .say, are processed in order of increasing v. 
After this visit, jobs {(1, p + 1), .... (1, p{\})} from M are processed where p < p[\} < 
j[\}. During the rth visit to J- M jobs {(a, v), t r _ x < v < t r ) are processed in order of 
increasing v. Assume that in general, after the rth visit to J-M, module jobs 
{(1, p[r - 1} + 1), .... (1, p{r})} are processed and that p{r*} = j[\}. The fact that after 
(1, j{\}) has been processed other jobs from M may also be processed before the (r* + l) sl 
visit to J-M causes no problems. 

Depict the permutation described above as follows: 
({(s,j), l<j^k{s); (1.7). 1 <J<P), {(«. v), Kv^fiMd. j), p + Kj^p{l}}, U) 
where t/ is an appropriate job ordering. Call this permutation 1. 



SINGLE MACHINE SEQUENCING 1 27 

Consider now the following two permutations which, from the definition of a job module, 
are also consistent with R. Permutation 2 is 

({(a, v), 1 < v < t x }, {(s, j), 1 ^ j ^ k{s}; (1, j), 1 ^ j ^ p{\)\, U). 

Permutation 3 is 

(((* j), 1 < j < k{s); (1, j), 1 < j < p{\)}, {(a, v), 1 < v < r,), U). 

It is easy to show that the fact that permutation 2 gives rise to at least as high a cost as 
does permutation 1 implies that 

( 9 ) CMKs, 1), ... , (5, k{s}), (1, 1), ... , (1, p)} > CM[(a, 1), .... (a, /,)]. 

Similarly, the fact that permutation 3 gives rise to at least as high a cost as does permutation 1 
implies that 

(10) CM[(a, I), ... , (a, tj] > CM[{\, p + 1) (1, p{\})]. 

From (9) and (10) it follows that 

(11) CMKs, 1) (5, k[s}), (1, 1) (1, p)) > CM[(\, p + 1) (1, p[l))]. 

Repeating the above argument for all visits to 7-Mup to and including the r*th one, the 
following inequality is obtained: 

(12) CMKs, 1) (5, k{s}), (1, 1) (\,p)]>CM[(\,p(r*-l) + l), ..., (\,j{\})]. 

However, it follows from Lemma 1 that 

CM[(\, p(r*-\) + 1), ... , (1, j{\))} > CMKl, 1), ... , (1, j{l})] 

(13) 

>CM\{\, 1), ••• , (1, p)]. 

It follows from Lemma 1 and the fact that chain 1 has a larger allocation index than chain s that 

CM[{\, p(r* - 1) + 1), .... (1, j[l})] > CMKl, 1), ... , (1, j{\})] 

(14) 

> CMKs, 1), ... , (5, j[s})] > CMKs, 1), .... (5, k[s))}. 

It now follows from (13), (14) and Lemma 2 that 

CMKl, p(r* -I) + 1) (1, 7 {1})] > CMKs, 1) (s, k{s}), (1, 1) (1, p)\, 

vhich contradicts (12). 

We obtain similar contradictions for cases 1 and 3. It may further be shown, using similar 
echniques, that in the event of more than one unfinished chain satisfying (6) it is immaterial 
vhich of the appropriate jobs is chosen next. Hence the induction proceeds and the theorem is 
)roved. 

I. ORDERING JOB MODULES: LINEAR COSTS 

Theorem 3 tells us in which order the jobs in job module M will be processed in an 
optimal permutation for the jobs in J when the costs are linear as in (2). 

THEOREM 3: There is an optimal permutation for the jobs in J such that the following 
ule is adopted for processing the jobs {(/", j), 1 < / < m, 1 < j < n,) in M: If at some time 
mly jobs {(/, j), 1 < / < m, 1 < j < 5,} among the jobs in M have been completed, then the 
lext job in Mto be processed will be a job (k, s k + 1) where k is chosen such that s^ < n k and 



128 



K.D. GLAZEBROOK 



max 

j = (s k + \)(l)n k 



(f J 1 




I 1 


-1 


I c kr 




I x kr 




\r=s k +\ 




r=s k + \ 





[ = max 

[is<n\ 



max 

j = (s l + \)(\)n l 



[ j \ 




j 


-1 




I Q r 




I Xir 






11' 




r=s f + \ 







If more than one job satisfies these criteria it is immaterial which of them is chosen next. Thi; 
result, previously established by Sidney [5], may be deduced from Theorem 2 by considering 
the limit as the discount rate a tends to one. 

5. THE ALGORITHM 

Suppose the jobs {(/', j), 1 < / ^ m, 1 < j < n,} form a job module M within the set o 
jobs J with precedence relation R. A chain transformation C M acts on the pair (J, R 
transforming it to the pair (/, R U S) where 

(i) R D S = 0; 

(ii) with respect to R U S the jobs in Mform a single chain; 

(iii) S contains only elements of the form (a — » /3) a € M, /3 € M. A chain transfor 
mation, then, is a device for changing job modules into chains. (J, R) is chai 
reducible if it can be transformed to a pair (J, R U T) by successive chaii 
transformations, where with respect to precedence relation RUT the jobs in 
form a single chain. 

An optimal permutation for any chain reducible pair (J, R) with discounted or linea 
costs may be obtained by successive applications of whichever is appropriate of Theorem 2 o 
Theorem 3. This is easily seen since those results demonstrate how to process jobs within a joi 
module in an optimal fashion. Such a module, once ordered in this way, effectively becomes 
chain. In this way, using these theorems is seen to be equivalent to applying a chain transfor 
mation. If by successive such applications of the theorems, the pair (J, R) is reduced to a sin 
gle chain of jobs, then this chain must evidently correspond to an optimal permutation. 

However, as can easily be shown, not all pairs (J, R) are chain reducible. The procedur 
for solving general problems that is described in this section utilizes in a helpful way what w 
know about ordering chain reducible pairs. Before presenting this procedure some extra termi 
nology is required. 

Suppose that U is a subset of J then R \ U, the restriction of R to U, is the se 
R Pi {(/ — *j), i 6 U, j € U). If a is a permutation of U consistent with R \ U then a(U) i 
the ordered set obtained by the application of a to U. Further, U is said to be initial if ther 
does not exist / € / — U, j € f/such that (/ — j) € R. An initial permutation a is a permuta 
tion for an initial subset U. 

If a is a permutation for the subset U, then /3 is a sub-ordering of a if and only if /3 is 
permutation for some subset V C U, and there exists a permutation for U — V, y say, sue 
that a(U) = [((S(V), y(U - V)}. When this happens we write a = {p, y). 

The algorithm for solving the problem associated with a general pair (J, R) with eithe 
discounted or linear costs is as follows: 



STEP 1: Compute all the initial permutations a,, 1 < / : < q, say, with the following pre 
perties: 



and where U U, = 7 If a, is an 
1-1 



SINGLE MACHINE SEQUENCING 1 29 

(i) If a, is a permutation for initial subset U t , then (7 - if,, R\J - U,) is chain reduc- 
ible, 1 < / < q\ 

(ii) a, is minimal'm the sense that it has no sub-ordering which satisfies (i), 1 < / < q. 

STEP 2: Compute an optimal permutation fZ t for the pair (7 — £/,7?|7 — U), 1 < / < q. 
Since these pairs are all chain reducible such permutations may be obtained by the successive 
use of whichever is appropriate of Theorem 2 or Theorem 3. 

STEP 3: Select that permutation for /from among the [a,, /3,}, 1 < / < q, which gives 
the smallest total cost. This is an optimal permutation. 

It is clear that this procedure does yield an optimal permutation. Its computational 
efficiency in any problem will be dependent upon the nature of the pair (7, R). In any prob- 
lem with linear deferral costs the option of using Sidney's algorithm is also open to us. 
Sidney's algorithm successively chooses from 7 initial subsets of a particular kind and operates 
as follows: In r steps of the algorithm we choose subsets U,, 1 </<;/', subject to specified 

y-l 7-1 

criteria where Uj is initial in 7 - U U„ R \J — U U, 

i—i i—i 

optimal permutation for Uj, then [a\, . . . , a r ) is optimal for [J, R}. 

Note that it is possible to use Theorem 3 in conjunction with Sidney's algorithm in a 
problem with linear costs. Such an approach is exemplified in the next section. 

6. AN EXAMPLE 

The example discussed here is the one given by Sidney [5]. We seek an optimal permuta- 
tion of the job set 7 = {1, 2, 3, 4, 5, 6, 7}. Each job /has a linear cost function as in (2) with 
C, = 1. The processing times are given by [X x , X 2 , X 3 , X 4 , X 5 , X 6 , Z 7 ) = {5, 8, 3, 5, 3, 7, 
6} and the precedence relation R is { (1 — 3), (1 — 4), (1 — 6), (1 — 7), (2 — 4), (2 — 5), 
(2 - 6), (2 - 7), (3 - 7), (4 - 6), (4 - 7), (5 - 6), (5 - 7), (6 - 7)}. 

METHOD 1 (Glazebrook's algorithm): Adopt the procedure outlined in the previous sec- 
tion. 

STEP 1: Initial permutations a l5 a 2 and a 3 are chosen together with initial subsets 
U h U 2 and £/ 3 , where a,(t/,) = {1}, a 2 (U 2 ) - {2, 1} and a 3 (U 3 ) = (2, 5}. 

STEP 2: Use Theorem 3 to compute /3 b /3 2 and /3 3 . Consider, for example, the computa- 
tion of j8 3 , an optimal permutation for (J - [2, 5}; R \J - {2, 5}) = ({1, 3, 4, 6, 7); {(1 — 3), 
(1 - 4), (1 - 6), (1 - 7), (3 - 7), (4 - 6), (4 - 7), (6 - 7)}). Job set {3, 4, 6} is a job 
module in this new problem since job 1 precedes all of jobs 3, 4 and 6, job 7 is preceded by 
them all and {3} and (4, 6} are disjoint chains. Theorem 3 applied to this new problem indi- 
cates that in an optimal permutation the jobs in the module will be processed in the order 
(3, 4, 6} since 



1 C3 ^ 

- = — > max 



C 4 C4 + C(, 



X 4 X 4 + X 6i 



Hence j8 3 (7 - U 3 ) - {1,3,4,6,7}. Similarly /3,(7 - U x ) = {3,2,5,4,6,7} and 
2 (7 - U 2 ) = {3, 5, 4, 6, 7}. 

STEP 3: Of the permutations {a,, /3,} f [a 2 , /3 2 } and [a 3 , (3 3 ) the first gives rise to the 
smallest total cost. Hence an optimal permutation for (J,R) is given by [a ](£/,), /3j(7 - 
£/,)} = {1, 3, 2, 5, 4, 6, 7}. 



130 K.D. GLAZEBROOK 

METHOD 2 (Sidney's algorithm): According to the criteria which Sidney sets out in [5], 
the subsets U\ = {1, 3}, then U 2 = {2, 4, 5} and then U 3 = {6, 7} are chosen, y x , y 2 and y 3 
are optimal permutations for U x , U 2 , and U 3 respectively. It can be shown that 
yi(^i) = (1. 3}, y 2 (U 2 ) = {2, 5, 4} and y 3 (U 3 ) = {6, 7} so that an optimal permutation for 
(J, R) is given by {1, 3, 2, 5, 4, 6, 7). 

METHOD 3 (Combination of Methods 1 and 2): Use Sidney's algorithm to choose the 
initial subset U x = {1, 3} and note that its optimal permutation y x is such that y x (U x ) = (1, 3). 
(J - (1, 3}, R \J - [I, 3}) or ({2, 4, 5, 6, 7}, {(2 - 4), (2 - 5), (2 - 6), (2 - 7), (4 - 6), 
(4—7), (5 — 6), (5 — 7), (6 — 7)}) is chain reducible with an optimal permutation 8 which 
may be computed by using Theorem 3. It is found that 8(J - [I, 3}) = {2, 5, 4, 6, 7} and so 
an optimal permutation for (J, R) is that implied by the ordered set {1, 3, 2, 5, 4, 6, 7). 



ACKNOWLEDGMENTS 

I should like to thank Drs. J.C. Gittins and P. Nash for many enjoyable and stimulating 
discussions on these and other matters. 

REFERENCES 

[1] Banerjee, B.P., "Single Facility Sequencing with Random Execution Times," Operations 
Research 13, 358-364 (1965). 

[2] Glazebrook, K.D., "Stochastic Scheduling with Order Constraints," International Journal of 
System Science 7, 657-666 (1976). 

[3] Glazebrook, K.D. and J.C. Gittins "On Single-Machine Scheduling with Precedence Rela- 
tions and Linear or Discounted Costs," (submitted) (1979). 

[4] Meilijson, I. and G. Weiss, "Multiple Feedback at a Single Server Station." Stochastic 
Processes and Their Applications 5, 195-205, (1977). 

[5] Sidney, J.B., "Decomposition Algorithms for Single-Machine Sequencing with Precedence 
Relations and Deferral Costs," Operations Research 23, 283-298 (1975). 



STATISTICAL ANALYSIS OF THE OUTPUT DATA 
FROM TERMINATING SIMULATIONS* 

Averill M. Law 

University of Wisconsin 
Madison, Wisconsin 

ABSTRACT 

In this paper we precisely define the two types of simulations (terminating 
and steady-state) with regard to analysis of simulation output and discuss some 
common measures of performance for each type. In addition, we conclude, on 
the basis of discussions with many simulation practitioners, that both types of 
simulations are important in practice. This is contrary to the impression one 
gets from reading the simulation literature, where the steady-state case is al- 
most exclusively considered. 

Although analyses of terminating simulations are considerably easier than 
are those of steady-state simulations, they have not received a careful treat- 
ment in the literature. We discuss and give empirical results for fixed sample 
size, relative width, and absolute width procedures that can be used for con- 
structing confidence intervals for measures of performance in the terminating 
case. 



TYPES OF SIMULATIONS WITH REGARD TO ANALYSIS OF THE OUTPUT 

We begin by giving a precise definition of the two types of simulations with regard to 
nalysis of the output data (cf. Gafarian and Ancker [4] and Kleijnen [6]). A terminating simu- 
ation is one for which the quantities of interest are defined relative to the interval of simulated 
ime [0,7^], where T E , a possibly degenerate random variable (r.v.), is the time that a specified 
vent E occurs. The following are some examples of terminating simulations: 

) Consider a retail establishment (e.g., a bank) which closes each evening (physically ter- 
minating). If the establishment is open from 9 to 5, then the objective of a simulation 
might be to estimate some measure of the quality of customer service over the period 
beginning at 9 and ending when the last customer who entered before the doors closed at 
5 has been served. In this case E ={at least 8 hours of simulated time have elapsed and 
the system is empty}, and reasonable initial conditions for the simulation might be that no 
customers are present at time 0. 

») Consider a telephone exchange which is always open (physically nonterminating). The 
objective of a simulation might be to determine the number of (permanent) telephone 
lines needed to service adequately incoming calls. Since the arrival rate of calls changes 
with the time of day, day of the week, etc., it is unlikely that a steady-state measure of 



This research was supported by the Office of Naval Research under contract N00014-76-C-0403 (NR 047-145) and the 
.rmy Research Office under contract DAAG29-75-C-0024. 

131 



132 A. LAW 

performance (see Section 2), which is defined as a limit as time goes to infinity, will exist. 
A common objective in this case is to study the system during the period of peak loading, 
say, of length t hours, since the number of lines sufficient for this period will also do for 
the rest of the day. In this case, E = {/ hours of simulated time have elapsed}. However, 
care must be taken in choosing the number of waiting calls at time 0, since the actual sys- 
tem will probably be quite congested at the beginning of the period of peak loading. One 
approach would be to collect data from the actual system on the distribution of the 
number of waiting calls at the beginning of the peak period. Then for each simulation run 
of length / hours, a sample is generated from this distribution and used as the number of 
waiting calls at time 0. 

c) Consider a military confrontation between a defensive (fixed position) blue force and an 
offensive (attacking) red force. Relative to some initial force strengths, the objective of a 
simulation might be to estimate some function of the (final) force strengths at the time 
that the red force moves to within a certain specified distance from the blue force. In this 
case, E = (red force has moved to within a certain specified distance from the blue force}. 

A steady-state simulation is one for which the quantities of interest are defined as limits as 
the length of the simulation goes to infinity. Since there is no natural event to terminate the 
simulation, the length of the simulation is made large enough to get "good" estimates of the 
quantities of interest. Alternatively, the length of the simulation could be determined by cost 
considerations; however, this may not produce acceptable results. The following are some 
examples of steady-state simulations: 

a) Consider a computer manufacturer who constructs a simulation model of a proposed com- 
puter system. Rather than use data from the arrival process of an existing computer sys- 
tem as input to the model, he typically assumes that jobs arrive in accordance with a Pois- 
son process with rate equal to the predicted arrival rate of jobs during the period of peak 
loading. He is interested in estimating the response time of a job after the system has 
been running long enough so that initial conditions (e.g., the number of jobs in the sys- 
tem at time 0) no longer have any effect. (Assuming that the arrival rate is constant over 
time allows steady-state measures to exist.) 

b) A chemical manufacturer constructs a simulation model of a proposed chemical process 
operation. The process, when in operation, will be subject to randomly occurring break- 
downs. The input rate of raw materials to the process and the controllable parameters of 
the process are both assumed to be stationary with respect to time. The company would 
like to estimate the production rate after the process has been running long enough so 
that initial conditions no longer have any effect. 

The remainder of this paper is organized as follows. In Section 2 we discuss some com- 
mon measures of performance for both types of simulations and in Section 3 we present our 
findings on the relative occurrence of each type in the real world. A number of procedures 
which can be used to construct confidence intervals for terminating simulations are discussed in 
Section 4 and, finally in Section 5 we summarize our findings. 

2. MEASURES OF SYSTEM PERFORMANCE 

To the best of our knowledge, nowhere in the simulation literature are measures of per- 
formance for terminating simulations explicitly defined. In this section we define and contrast 
several common measures of performance for terminating and steady-state simulations by 



ANALYSIS OF TERMINATING SIMULATIONS 



133 



means of examples. (Because of the diversity of terminating simulations, it is not possible to 
give one definition that fits all cases.) For the simple examples that we consider, it is possible to 
compute analytically measures of performance. This fact will be useful in Section 4 where we study 
stopping rules for terminating simulations. 



A. Averages 

Consider first the stochastic process [D,, i ^ 1} for the M/M/l queue with p < 1, where 
Dj is the delay in queue of the / th customer. The objective of a terminating simulation of the 
M/M/l queue might be to estimate the expected average delay of the first m customers given 
some initial conditions, say, that the number of customers in the system at time 0, MO), is 
zero. The desired quantity, which we denote by d(m |A/(0) = 0), is then given by 



d(m\N(0) = 0) = E 



£ A/m|iV(0)-0 



/=i 



= £ E[D i \N(0) = 0]/m. 

i=\ 

(Alternatively, if one is interested in estimating the expected average delay of all customers 
who arrive and complete their delay in the time interval [0, t], then the desired quantity is 
given by 



d(t\N(0) = 0) = E 



MU) 

£ DfMit)\Ni0)=0 



where M{t) (a r.v.) is the number of customers who arrive and complete their delay in the 
interval [0, t]. Note that in this case the expectation and summation are not interchangeable. 
Thus, the label "expected average delay" is more general than "average expected delay.") Note 
also that the quantity dim |A/(0) = 0), which is often called a transient characteristic of the sto- 
chastic process [D it i ^ 1}, explicitly depends on the state of the system at time 0; i.e., 
d(m\N(0) = /) * d(m\N(0) = j) for i * j. 

The objective of a steady-state simulation of [D,,i ^ 1} for the M/M/l queue would be to 
estimate the steady-state expected average delay d, which is given by 

d = lim d(m\N(0) = /) for any / - 0, 1, .... 

(Under our assumption that p < 1, it can be shown that d exists.) Observe that d is indepen- 
dent of NiO). In Figure 1 we plot dim \ N(0) = 0) as a function of m. (The arrival rate X = 1 
and the service rate p, = 10/9, so p = .9.) The horizontal line that dim\Ni0)=0) asymptoti- 
cally approaches is at height d = 8.1. 

As a second example consider the stochastic process {£,,/ ^ 1} for an (5, S) inventory 
system with zero delivery lag and backlogging, where E, is the expenditure in the i th period. 
This system is described in detail in Law [8]. A possible objective of a terminating simulation 
would be to estimate the expected average cost for the first m periods given that the inventory 
level at the beginning of period 1, I\, is 5 : 



eim\h = S) = E 



£ Ejm\l x = S. 



The objective of a steady-state simulation of [E t ,i ^ 1} would be to estimate the steady-state 
expected average cost: 

e = lim eim\l x = i) for any / = 0, ±1, ±2, .... 



134 



A. LAW 



7 - 



■D 4 _ 



3 - 



2 - 



J I I I I 1_ 



I i i I l_ 



500 1000 1500 2000 

Figure 1. d(m\N(0)=0) as a function of m for the M/M/l queue with p = 0.9 

(It can be shown that e exists.) We plot e(m\l x = S) as a function of m and also e = 112.108 
in Figure 2. 



i on 



50 




Figure 2. e (m \ l x =S) as a function of m for the (s,S) inventory system 

Our third example is quite different from the first two. A reliability model consisting of 
three components will function as long as component 1 works and either component 2 or 3 
works. If T is the time to failure of the whole system and T t the time to failure of component 
/(/ = 1,2,3), then 

r=min[7 , 1 , max(jT 2 , T 3 )]. 

We further assume that the 7)'s are independent r.v.'s and each T, has a Weibull distribution 
with shape parameter a = .5 and scale parameter ft = 1. (A distributional assumption for the 
7^'s is needed in Section 4 where we present simulation results for this model.) The objective 
of a terminating simulation might be to estimate the expected time to failure of the system given 
that all components are new, £"(r|all components are new). If we assume that the system is 
not repaired when it fails, then a steady-state simulation makes no sense for this system. Such 
could be the case if this system were part of a space probe. 



ANALYSIS OF TERMINATING SIMULATIONS . 135 



8. Proportions 



The usual criterion for comparing two or more systems is some sort of average behavior. 
However, different kinds of information may be of more value in some situations. For exam- 
ple, a bank manager may be concerned with estimating the proportion of customers who experi- 
:nce a delay in excess of 5 minutes. Since proportions are really just a special case of averages, 
ve will illustrate them by means of the M/M/l example. 

In a terminating simulation of the M/M/l queue the objective might be to estimate, 
instead of an expected average delay, the expected proportion of the first m customers whose delay 
s less than or equal to x (a specified number) given that N(0) = 0. Denote the desired quan- 
ity by Pirn, x\N(0) = 0) and let 

' 1 if A < x 



Y i (x) = 
hen P(m, x\N(0) = 0) is given by 

P(m, x\N(0) = 0) = E 



if A>x for/=u 



£ ^(x)/m|7V(0) = 

;=1 



■or a steady-state simulation, the objective would be to estimate the steady-state expected propor- 
on of customers whose delay is less than or equal to x : 

P(x) = lim P(m, x\N(0)= i) for any / = 0, 1 



RELATIVE IMPORTANCE OF EACH TYPE OF SIMULATION 

Reading the simulation literature leads one to think that only steady-state simulations are 
nportant; almost every paper written on the analysis of simulation output data deals with the 
eady-state case. This may be a carry-over from mathematical queueing theory where only a 
eady-state analysis is generally possible. However, we believe that terminating simulations are 
so important. We have discovered, by talking to a large number of simulation practitioners, 
lat a significant proportion of simulations in the real world are actually of the terminating type, 
he following are some reasons why a steady-state analysis may not be appropriate: 

The system under consideration is physically terminating. In this case, letting the length 
of a simulation be arbitrarily large makes no sense. 

> The input distributions for the system change over time. In this case, steady-state meas- 
ures of performance will probably not exist. 

One is often interested in studying the transient behavior of a system even if steady-state 
measures of performance exist. 

STOPPING RULES FOR TERMINATING SIMULATIONS 

In the following three subsections we consider procedures that can be used to construct 
■ nfidence intervals (c.i.'s) for measures of performance for terminating simulations. We will 
)t consider the steady-state case since it has been widely discussed in the simulation literature. 
)r surveys of fixed sample size and sequential procedures that can be used to construct c.i.'s 
r steady-state measures of performance, see Law and Kelton [9, 10]. The random numbers 
ed in the remainder of this paper were generated from the generator discussed in [8] . 



136 



A. LAW 



A. Fixed Sample Size Procedures 

Suppose we make n independent replications of a terminating simulation. The indepen- 
dence among replications is accomplished by using different random numbers for each replica- 
tion and by starting each one with the same initial conditions. If X, is the estimator of interest 
from the i th replication (/ = 1, 2, . . . , «), then the X^s are independent identically distributed 

m 

(i.i.d.) r.v's. (For the M/M/l queue, X t might be the average £ Dj/m or the proportion 

m 

£ Yj{x)/m.) If, in addition, the AT/'s are normally distributed, then a 100(l-a)% (0 < a < 1) 

y-i 

c.i. for/Lt = E{X) is given by 

(1) jf(ll) ± ^-l,l-a/ 2 Vs 2 («)//I, 

where X(n) and s 2 (/j) are the usual sample mean and variance, respectively, and t„-\ , i— «/2 > : 
the 1 — a/2 point for a ? distribution with n-\ degrees of freedom. 

In practice the X,'s will not be normally distributed and the c.i. given by (1) will be onlj 
approximate. To investigate the effect of nonnormality, we simulated the three stochastic 
models of Section 2. For the M/M/l queue with p = .9, the (5,5) inventory system, and the 
reliability model, respectively, the quantities of interest were d(25\N(0) = 0) = 2.12 
6(121/! = 5) = 99.52, and £(F|all components are new) = .778. (See [8] for a discussion o! 
how to compute the first two quantities.) For each model we performed 500 independent simu 
lation experiments, for each experiment we considered n = 5, 10, 20, 40, and for each n wf 
used (1) to construct a 90% c.i. for the desired quantity. In Tables 1, 2, and 3 we give the pro 
portion, p, of the 500 c.i.'s that covered the desired quantity, a 90% c.i. for the true coverage 
and the average value of the c.i. half length divided by the point estimate over the 500 experi 
ments for the three models. The 90% c.i. for the true coverage was computed from 



p ± 1.645 y/p(l-p)/500. 



Observe that for the M/M/l queue and the (s,S) inventory system the coverages are quitt 
close to 90%, but for the reliability model there is a significant degradation in coverage 
apparently caused by a severe departure from normality. To see if this is indeed the case, wi 
generated 1000 Aj's for each stochastic model and estimated the skewness and kurtosis. Thesi 
estimates, which are presented in Table 4, indicate that the A'/'s for the reliability model an 
considerably more nonnormal than are those for the other two models. This conclusion wa 
reinforced by plotting histograms for the three sets of data. 



TABLE 1 . Fixed Sample Size Results 

for d(25)\N(0) = 0) = 2.12, 

M/M/l Queue with p = .9 



c.i. half length 

XM 



coverage 



average of 



5 

10 
20 
40 



.880 ± .024 
.864 ± .025 
.886 ± .023 
.914 ± .021 



.672 
.436 
.301 
.212 



ANALYSIS OF TERMINATING SIMULATIONS 



137 



TABLE 2. Fixed Sample Size Results for 
e(12\I x = S)= 99.52, (s,S) Inventory System 



n 


coverage 


c c.i. half length 

average of , x g — 

X(n) 


5 


.908 ± .021 


.048 


10 


.904 ± .022 


.031 


20 


.880 ± .024 


.021 


40 


.894 ± .023 


.014 



TABLE 3. Fixed Sample Size Results for 

E(T\all components new) = . 778, 

Reliability Model 



c.i. half length 
X(n) 



coverage 



average of 



5 

10 
20 
40 



.708 
.750 
.800 
.840 



.033 
.032 
.029 
.027 



1.163 
.820 
.600 
.444 



TABLE 4. Skewness and Kurtosisfor the Three 
Stochastic Models and the Normal Distribution 



Stochastic Model 
or Distribution 


Skewness 


Kurtosis 


Normal Distribution 
M/M/l Queue 
(s,S) Inventory System 
Reliability Model 


0* 
1.66 

.45 
5.18 


3* 

6.43 

3.76 

54.39 



'Theoretical Values 



B. Relative Width Procedures 

One disadvantage of the fixed sample size approach to constructing a c.i. is that the simu- 
lator has no control over the c.i. half length; for fixed n, the half length will depend on the 
population variance o- 2 = Var(X). In this subsection we consider two sequential procedures 
which allow one to specify the "relative precision" of a c.i. Both assume that X\, X 2i ... is a 
sequence of i.i.d. r.v.'s which need not be normal. 

The first procedure has been suggested for use in several different contexts; see Iglehart 
[5], Lavenberg and Sauer [7], and Thomas [13]. The objective of the procedure is to construct 
a 100(l-a)% c.i. for /n_such that the difference between the point estimator X(n) and /a is no 
more than 100 y% of X(n), that is, 



(2) 



\X(n)-ix\ < y\X(n)\ for < y < 1. 



138 



A. LAW 



Choose an initial sample size n ^ 2, let 



8 r/[ (n,a)= ^-i.i-a/2 y/s 2 (n)/n, 



and let 
(3) 



N r \(y,a) = min 



n: n > n , s z (n) > 0, ' < 

\X(n)\ 



(Note that N r A (y, a), which is the required number of replications, is a r.v.). Then use 

(4) / r , 1 (y,a)= [X(N rA (y,a))-b rA (N rA (y,a),a),X(N rA (y,a))+8 rA (N rA (y r a),a)] 

as an approximate 100(l-a)% c.i. for fx. It easily follows from (3) and (4) that I rA (y,a) 
satisfies the criterion given by (2). Furthermore, using an argument similar to the one 
employed by Lavenberg and Sauer in the context of the regenerative method for steady-state 
simulations, we have been able to prove the following theorem. 

THEOREM 1. H> ^ and < o- 2 < oo, then lim P{fi <E I r] (y,a)} = 1 - a. 

The objective of the second procedure, which is due to Nadas [11], is to construct a c.i. 
such that 



(5) 
Let 



\X(n)-n\ < y \fi\ for 0< y < 1. 



vHn) 



l + £ [X,- X(n)Y 



In = (1/w) + (n- \)s 2 (n)/n, 



and 

Then use 
(6) 



8 r2 (n,a) = /„_, ,_ a/2 yfvHnj/n, 
N r2 (y,a) = min 



8 r2 (n,a) 
n:n ^ riQ, - 



Ir,ih><x) = 



X(n)\ 
X(N r _ 2 (y,a)) X(N r2 (y,a)) 



\ + y 1 — y 

as an approximate 100(l-a)% c.i. for/i. From (6) it is easy to show that I r2 (y,a) satisfies th( 
criterion given by (5). Furthermore, the following theorem was proved by Nadas. 

THEOREM 2. If fi j* and < a 2 < oo, then lim P{u € /, 2 (y,a)} = 1 - a. 

In order to compare the two procedures and to determine the effect of non-infinitesimal ■) 
on coverage, we once again simulated the three stochastic models. For each model we per 
formed 500 independent experiments, for the M/M/l queue and the reliability model we con 
sidered y = .2, .1, .05 for each experiment, and for the inventory system we considered y = 
.2, .1, .05, .025, .0125, .00625 for each experiment. In all cases, « = 5. In Tables 5, 6, and 
we give point estimates and 90% c.i.'s for the true coverages, point estimates and 90% c.i.'s f o i 
E{N ri (y,a)} (/' = 1, 2), and the average c.i. half lengths over the 500 experiments. We con 
sidered more values of y for the inventory system because it appeared from our empiric* 



ANALYSIS OF TERMINATING SIMULATIONS 



139 



TABLE 5. Relative Width Results for d(2 5 \N(0) = 0) = 2.12, 
M/M/l Queue with p = .9 





Procedure 1 


Procedure 2 


y 


E{N rA (y,a)} 


coverage 


average c.i. 
half length 


E[N ri2 (y,a)} 


coverage 


average c.i. 
half length 


.2 

.1 

.05 


42.3 ± 0.9 
175.2 ± 1.7 
704.4 ± 3.5 


.842 ± .027 
.860 ± .026 
.884 ± .024 


.414 
.211 
.106 


41.9 ± 0.8 
174.5 ± 1.7 
703.7 ± 3.5 


.862 ± .025 
.868 ± .025 
.882 ± .024 


.437 
.213 
.106 



TABLE 6. Relative Width Results for e(12\l x = S) - 99.52, 
(s, S) Inventory System 





Procedure 1 


Procedure 2 


y 


E{N rA (y,a)} 


coverage 


average c.i. 
half length 


E{N r _ 2 (y,a)} 


coverage 


average c.i. 
half length 


.2 

.1 

.05 

.025 

.0125 

.00625 


5.0 ± 0.0 

5.0 ± 0.0 

5.9 ± 0.1 

13.3 ± 0.4 

51.0 ± 1.0 

206.3 ± 1.8 


.902 ± .022 
.902 ± .022 
.892 ± .023 
.834 ± .027 
.856 ± .026 
.872 ± .025 


4.89 
4.86 
3.97 
2.35 
1.23 
0.62 


5.0 ± 0.0 

5.0 ± 0.0 

5.7 ± 0.1 

12.3 ± 0.4 

49.8 ± 1.0 

205.4 ± 1.8 


1.0 

1.0 
.962 ± .014 
.858 ± .026 
.862 ± .025 
.876 ± .024 


20.74 
10.06 
4.99 
2.48 
1.24 
0.62 



TABLE 7. Relative Width Results for E(T\all components new) = . 778, 

Reliability Model 





Procedure 1 


Procedure 2 


y 


E{N rA (y,a)} 


coverage 


average c.i. 
half length 


E{N r:1 {y,a)) 


coverage 


average c.i. 
half length 


.2 
.1 
.05 


213.7 ± 4.5 

907.4 ± 11.2 

3720.5 ± 23.7 


.876 ± .024 
.898 ± .022 
.882 ± .024 


.152 
.077 
.039 


214.1 ± 4.5 

908.6 ± 10.8 

3720.0 ± 23.7 


.908 ± .021 
.902 ± .022 
.884 ± .024 


.160 
.078 
.039 



rsults that a smaller y is required for the coverage ultimately to converge to the desired level. 
1 smaller y is required for this model to get a large value of N ri (y,a).) Note also that con- 
vergence of coverage does not appear to be monotone. 

We repeated the above 500 experiments using the same random numbers and n Q = 2. 
l>r procedure 2 the results were identical; however, for procedure 1 there was a significant 
(gradation in coverage due to premature stopping on replications 2, 3, or 4. For example, the 
cverage for the M/M/l queue with y = .2 was .798. 

< Absolute Width Procedures 

In this subsection we present two procedures which allow one to construct a 100(1— a) % 
' . for n such that 



j) \X(n) 

uere c is a specified positive number. 



M 



< c, 



140 



A. LAW 



The first procedure, which is due to Chow and Robbins [1], assumes that X h X 2 , 
sequence of i.i.d. r.v's. Choose « ^ 2. Let v 2 (n) be defined as in Subsection 4.B, let 



is a 



N a \(c, a) = min 



n.n ^ n , v 2 (n) ^ 



2 



Vn-XA-all) 



and then use 



I aA (c, a) = [X(N aA (c, a)) - c, X(N aA (c, a)) + c] 



as an approximate 100(1— a)% c.i. for ft. It is clear that I a \{c,a) satisfies the criterion given by 
(7). The following theorem was proved by Chow and Robbins. 

THEOREM 3. If < a 2 < oo, then lim P{ji € l a x (c,a)} = 1 - a. For an empirical 

c— 0+ 

evaluation of the above procedure under the assumption that the X^s are normal, see Starr 
[12]. 

The second procedure, which is due to Dudewicz [2], assumes that the X^s are i.i.d nor- 
mal r.v.'s. Initially make n (n ^ 2) replications of the simulation and compute X(n ) and 
s 2 (a? ). Let 

N a2 (c,a) - max{/i + 1. [w 2 s 2 (n )]}, 



where w — t„ o _ h ]_ a / 2 /c and fzl is the smallest integer ^ z. Make N a2 {c,a) — n additional 
replications of the simulation, let 

N a2 U.a) 

Y(N a , 2 (c,a) - n ) = £ X,/(N a , 2 (c,a) - n Q ), 

i=n +\ 

and let X(N a 2 (c,a)) = a^C^o) + a 2 Y{N a2 {c,a) - « ), where 



a\ = 



N ai2 (c,a) 



l+i/ 1 



N a2 (c,a) ( N a2 (c,a) - /?o 



"o 



1- 



wV(« ) 



and a 2 = 1 — a \- Then use 

/ a . 2 (c,a) = [X{N a2 {c,a)) - c, X(N a _ 2 (c,a)) + c] 
as an approximate 100(1— a)% c.i. for /a. Dudewicz has proved the following theorem. 

THEOREM 4. P{fi € I a , 2 (c,a)} = 1 - a for all c> 0. 

To compare the sequential procedure of Chow and Robbins and the two-stage procedure 
of Dudewicz, we performed 500 independent experiments for each model. To make the abso- 
lute width results somewhat comparable to the relative width results, we chose the values of < ( 
to correspond to the values of y; that is, for each y we chose c = y/x. For the Chow and Rob 
bins procedure we chose n = 5 and for the Dudewicz procedure we considered n = 15, 30 
and 60. (Dudewicz [3] recommended that n be at least 12.) The results of the simulatior 
experiments for the three models are given in Tables 8, 9, 10. 



ANALYSIS OF TERMINATING SIMULATIONS 



141 



TABLE 8. Absolute Width Results for d(25\N(0) = 0) = 2.12, 
MIMI1 Queue with p = .9 





Chow and Robbins 


Dudewicz 


c 


E{N aA (c,a)} 


coverage 


"0 


E[N ai2 (c,a)} 


coverage 


.425 


38.0 ± 1.2 


.800 ± .029 


15 
30 
60 


49.9 ± 2.1 
48.2 ± 1.3 
62.1 ± 0.4 


.850 ± .026 
.912 ± .020 
.926 ± .019 


.212 


173.5 ± 2.5 


.898 ± .022 


15 
30 
60 


196.9 ± 8.5 
185.7 ± 5.6 
182.9 ± 4.0 


.854 ± .026 
.888 ± .023 
.894 ± .023 


.106 


706.8 ± 4.8 


.906 ± .021 


15 
30 
60 


786.1 ± 34.2 

741.1 ± 22.6 

730.2 ± 15.7 


.868 ± .025 
.878 ± .024 
.898 ± .022 



TABLE 9. Absolute Width Results for e(12\l x = S) = 99.52, 
(s,S) Inventory System 





Chow anc 


Robbins 


Dudewicz 


c 


E{N aA (c,oc)) 


coverage 


»o 


E{N ai2 (c,a)} 


coverage 


19.90 


5.0 ± 0.0 


1.0 


15 
30 
60 


16.0 ± 0.0 
31.0 ± 0.0 
61.0 ± 0.0 


.936 ± .018 
.878 ± .024 
.888 ± .023 


9.95 


5.0 ± 0.0 


1.0 


15 
30 
60 


16.0 ± 0.0 
31.0 ± 0.0 
61.0 ± 0.0 


.936 ± .018 
.880 ± .024 
.890 ± .023 


4.98 


5.7 ± 0.1 


.976 ± .011 


15 
30 
60 


16.0 ± 0.0 
31.0 ± 0.0 
61.0 ± 0.0 


.922 ± .020 
.882 ± .024 
.886 ± .023 


2.49 


12.3 ± 0.4 


.880 ± .024 


15 
30 
60 


18.5 ± 0.3 
31.0 ± 0.0 
61.0 ± 0.0 


.908 ± .021 
.894 ± .023 
.882 ± .024 


1.24 


48.3 ± 1.1 


.872 ± .025 


15 
30 
60 


60.8 ± 2.0 
55.0 ± 1.3 

62.9 ± 0.4 


.904 ± .022 
.912 ± .020 
.902 ± .022 


0.62 


204.4 ± 1.8 


.896 ± .022 


15 
30 
60 


241.7 ± 8.1 
217.9 ± 5.1 
211.5 ± 3.4 


.912 ± .020 
.898 ± .022 
.912 ± .020 



5. SUMMARY AND CONCLUSIONS 



We have defined terminating and steady-state simulations and have discussed some com- 
mon measures of performance for each type. In addition, we have concluded from talking with 
simulation practitioners that a significant proportion of real-world simulations are of the ter- 
minating type. This is fortunate because it means that classical statistical analysis for i.i.d. 
observations (e.g., confidence intervals, hypothesis testing, ranking and selection, etc.) is appli- 
cable to analyzing many simulations. On the other hand, in the steady-state case there is still 
not a totally acceptable procedure even for the relatively simple problem of constructing a c.i. 
for a steady-state expected average. 



142 



A. LAW 



TABLE 10. Absolute Width Results for E(T\all components new) = 

Reliability Model 



778, 





Chow and Robbins 


Dudewicz 


c 


E{N aA (c,a)} 


coverage 


"0 


E{N a , 2 (c,a)} 


coverage 


.156 


179.5 ± 7.0 


.774 ± .031 


15 
30 
60 


246.0 ± 27.2 
220.8 ± 17.0 
231.7 ± 14.9 


.704 ± .034 
.772 ± .031 
.812 ± .029 


.078 


888.0 ± 14.5 


.900 ± .022 


15 
30 
60 


981.8 ± 109.0 
880.6 ± 68.0 
922.2 ± 59.6 


.728 ± .033 
.794 ± .030 
.838 ± .027 


.039 


3672.1 ± 32.9 


.884 ± .024 


15 
30 
60 


3925.6 ± 435.8 
3520.9 ± 272.0 
3687.2 ± 238.5 


.772 ± .031 
.788 ± .030 
.832 ± .028 



We have also considered procedures for constructing c.i.'s for terminating simulations. If 
one is performing an exploratory experiment where precision of the c.i. may not be 
overwhelmingly important, then we recommend using a fixed sample size procedure. However, 
if the J/'s are highly nonnormal and if the number of replications n is too small, then the actual 
coverage of the constructed c.i. may be considerably lower than that desired (see Table 3). 

If one wants a c.i. having half length that is small relative to the point estimate, then a 
relative width procedure may be used. We recommend using Procedure 2 (due to Nadas) with 
n ^ 5. Procedure 2 appears to give slightly better coverage; its criterion (see (5)) is more 
intuitive than the criterion of Procedure 1 (see (2)), and Procedure 2 does not seem subject to 
premature stopping even for n = 2. (On the other hand, Procedure 1 uses a more intuitive 
expression to construct a c.i.) 

If one wants a c.i. for which the half length is a specified number, then an absolute width 
procedure may be used. We recommend using the Chow and Robbins procedure with n ^ 5. 
Their procedure generally requires a smaller average sample size, the variance of the sample 
size is smaller, and its coverage seems to be less affected by departures from normality (see 
Table 10). 

In general, we believe that relative width procedures are more useful than absolute width 
procedures due to the difficulty in specifying the absolute width c for most simulation experi- 
ments. When using either the Nadas procedure or the Chow and Robbins procedure, we 
believe that it is advisable to choose ay ore which will cause the procedure to run until the 
sample is at least of moderate size; perhaps, at least 30. (Since both procedures are based on 
the central limit theorem, it is unreasonable to think that they will work well in general for a 
small sample size; see the results for y = .025 in Table 6.) Note that precise c.i.'s may be 
unaffordable in the real world due to the high cost of making a single replication. 

Our conclusions on the efficacy of the procedures are based on only three models and, 
thus, should be considered tentative. However, since the performance of a procedure depends 
only on the distribution of an A", and not on the complexity of a model, we feel that there is no 
particular reason to suspect that the results for real-world models should differ significantly 
from those for the simple models presented here. 

ACKNOWLEDGMENTS 



The author would sincerely like to thank David Kelton for his programming assistance 
and for his reading of the manuscript. The author is also grateful to Stephen Lavenberg (IBM 



ANALYSIS OF TERMINATING SIMULATIONS 143 

^atson Research Center), Richard Smith (IBM Boulder), Charles White (Dupont), U. A. 
'eber (British Columbia Telephone Company), and Barney Watson (U. S. Army TRASANA) 
ir their discussions of simulation at their organizations. 

REFERENCES 

Chow, Y.S. and H. Robbins, "On the Asymptotic Theory of Fixed-Width Sequential 
Confidence Intervals for the Mean," Annals of Mathematical Statistics 36, 457-462 
(1965). 
Dudewicz, E.J., "Statistical Inference with Unknown and Unequal Variances," Transactions 
of the Annual Quality Control Conference of the Rochester Society for Quality Control 
28, 71-85 (1972). 
3] Dudewicz, E.J., Personal Communication (1977). 

4] Gafarian, A.V. and C.J. Ancker, Jr., "Mean Value Estimation from Digital Computer 
Simulation," Operations Research 14, 25-44 (1966). 
Iglehart, D.L., "The Regenerative Method for Simulation Analysis," Technical Report No. 

86-20, Control Analysis Corporation, Palo Alto, California (1975). 
Kleijnen, J.P.C., "The Statistical Design and Analysis of Digital Simulation: A Survey," 

Management Informatics 1, 57-66 (1972). 
Lavenberg, S.S. and C.H. Sauer, "Sequential Stopping Rules for the Regenerative Method 

of Simulation," IBM Journal of Research and Development 21, 545-558 (1977). 
Law, A.M., "Confidence Intervals in Discrete Event Simulation: A Comparison of Replica- 
tion and Batch Means," Naval Research Logistics Quarterly 24, 667-678 (1977). 
Law, A.M. and W.D. Kelton, "Confidence Intervals for Steady-State Simulations, I: A Sur- 
vey of Fixed Sample Size Procedures," Technical Report 78-5, Dept. of Industrial 
Engineering, University of Wisconsin (1978). 
10] Law, A.M. and W.D. Kelton, "Confidence Intervals for Steady-State Simulations, II: A 
Survey of Sequential Procedures," Technical Report 78-6, Dept. of Industrial Engineer- 
ing, University of Wisconsin (1978). 
11] Nadas, A., "An Extension of a Theorem of Chow and Robbins on Sequential Confidence 

Intervals for the Mean," Annals of Mathematic Statistics 40, 667-671 (1969). 
[I] Starr, N., "The Performance of a Sequential Procedure for the Fixed-Width Interval Esti- 
mation of the Mean," Annals of Mathematical Statistics 37, 36-50 (1966). 
Bl Thomas, M.A., "A Simple Sequential Procedure for Sampling Termination in Simulation 
Investigations," Journal of Statist. Comput. Simul. 3, 161-164 (1974). 



. 



COMPUTATIONAL EXPERIENCE ON AN ALGORITHM 

FOR THE TRANSPORTATION PROBLEM WITH 

NONLINEAR OBJECTIVE FUNCTIONS* 

Ram C. Rao 

Purdue University 
West Lafayette, Indiana 

Timothy L. Shaftel 

University of Arizona 
Tucson, Arizona 

ABSTRACT 

This paper explores computational implications of allowing nonlinear objec- 
tive functions in the transportation problem. Two types of nonlinearities, in- 
cluding polynomials, are studied. The choice of these functions resulted from 
our interest in models of integrated water management. Zangwill's convex 
simplex method and the primal method of transportation problem form the 
basis of our algorithm. Innovative features of our work are compact storage 
and efficient computation procedures. We study the effects on compulation 
time of problem size; the density of nonlinear terms; the size of tolerances for 
stopping rules; and rules for choice of new variables to enter the solution. We 
find that problems up to 95 x 95 in size are capable of reasonably fast solution. 
A particularly surprising finding is that one-dimensional search for improving 
solutions performs adequately, at least for the kinds of problems posed in this 
paper. We are encouraged by our results and believe that models involving 
nonlinear objective functions may be tractable even for relatively large prob- 
lems, thus making possible more accurate descriptions of real situations. 



1. INTRODUCTION 

The recent advent of sophisticated codes [3], [7] for solving the transportation problem 
makes it possible to solve extremely large problems of this type. This makes the transportation 
model a powerful modeling device. In some instances, however, a real situation demands aug- 
mentation of the transportation problem. In particular, such effects as economies or 
diseconomies of scale and other cost interactions among variables cannot be modeled in the 
transportation framework. We feel that the ability to solve large transportation problems that 
incorporate nonlinear cost functions will represent an important addition to the usefulness of 
transportation models in many situations. Our own motivation for solving problems such as 



*We would like to thank Professors V. Srinivasan of Stanford University and G.L Thompson of Carnegie-Mellon 
University for making available their code for the Transportation problem. 



145 



146 



R.C. RAO AND T.L. SHAFTEL 



this stems from work in water recycling models [4]. As will be seen later, we attempt to handle 
two types of nonlinearities in the objective function. One of these types consists of polynomials 
and is thus quite general. 

Our algorithm represents a combination of the primal transportation method [1] and the 
convex simplex method of Zangwill [8]. In developing a computational procedure we use as 
the basic building block the methods of Srinivasan and Thompson [6], [7]. Where possible we 
use their efficient coding procedures for the linear problem. Our innovations include the way 
we store the information required for the nonlinear objective function in an efficient manner, 
and in performing the nonlinear calculations in ways we think will minimize their impact on 
total computation time. In developing these techniques we have drawn from some of the ideas 
in Shaftel and Thompson [5]. We report computation times for problems with up to 95 rows 
and 95 columns. Though computation times are larger than those for similar sized linear prob- 
lems, they compare favorably with what was considered acceptable for the linear transportation 
problem only a few years ago. To our knowledge computational experience on transportation 
problems with general nonlinear objective functions is not heretofore available. In this light 
our study is significant. We believe that it will prove useful to those engaged in developing 
models with a view to computational viability. 

We report on computation times for some of the subroutines used in our code in addition 
to overall solution time so as to isolate areas for further improvements. We will also discuss 
our experience in using different parameters, such as stopping rule tolerance, in the code. We 
believe that many improvements to this code can be made, and that even faster times than 
those reported here are possible. 

2. PROBLEM STATEMENT 

The problem that we are interested in can be stated as follows: find a schedule of ship- 
ments or flows of a homogeneous good from a set of m source nodes (often thought of as 
warehouses) to a set of n sink nodes (often thought of as markets) to minimize the total cost of 
effecting the shipments. We denote x (/ to be the flow from / to j, and a, and b t to be the avail- 
ability and demand at nodes / and j, respectively. The objective function to be minimized con- 
sists of a linear component c, n the cost of unit shipment from / to j, and some other costs that 
are nonlinear in the decision variables x, r We note that the usual transportation problem has 
only linear costs. See Dantzig [2], for example. Mathematically, we can write our problem as: 

Minimize 



(1) 



Z Z C U x u 



Linear Terms 



T m n 



+ zan (*</) 



/ = 1 ( = 1 7 = 1 



Polynomial Terms 
(a total of T terms) 



m 


n 


a 


/=i 


Z "•</' X U 
,/ = ' 




n 


m 


'j 


+ T,Sj 


Z hi Xij 




7 = 1 


/=] 





Demand Interdependences 



Supply Interdependencies. 



TRANSPORTATION WITH NONLINEAR OBJECTIVE FUNCTIONS 147 

Subject to 



2) £*y = fl/ for / = 1, 2 



m 



7=1 



3) L *</-*/ for.Z-1, 2, .... n 

(=i 

4) x,/ > for all / and / 

Ve assume that £ a, = £ b r In addition we require the following: 

/ j 

a, ^ for all / 



Pj 


^ 


for all j 




Ki 


hi 


^ for all 


i and j 


h tu 


> 


for all t, i 


and j. 



dually the nonnegativity of k u and I,, may be relaxed if a, and /3 y are restricted to integers, 
his would be the case if we are dealing with quadratics, for example. For convenience we 
itroduce the following additional notation: 

n 

R, = £ kjj Xjj for / = 1, 2, . . . , m 

7 = 1 
m 

Q = Z 'y x v for./ - 1, 2, ... , n 

i=\ 

Although our purpose in this paper is to examine the computational implications of allow- 
g nonlinear objective functions such as (1) in the framework of a transportation problem, it is 
;eful to motivate the choice of such a function. As noted earlier, we first encountered the 
r,ed for it in modeling integrated water management. The polynomials which we have incor- 
prated reflect the fact that the use of some arcs (shipment routes) affects the cost of use of 
• hers. The transshipment type of model used in water management makes this a naturally 
(countered type of objective function. The supply interdependencies involve functions of C y , 
nich is a linear combination of the shipments into node / As an illustration these can be used 
t model a demand curve at each of the markets. The demand interdependencies involve func- 
tus of /?,, which is a linear combination of the shipments emanating from source i. To illus- 
tite the use of this, suppose sink node n is a "dummy" market. Shipments from node / to j, 
j= 1, 2, .... n — 1, can be thought of as the total production at node /. The demand inter- 
c pendencies term can then be used to model the total cost of production at node /. 

We now proceed to discuss the different steps in the algorithm and some features of the 
cmputer code we have written for the algorithm. 

2 DESCRIPTION OF THE ALGORITHM AND THE COMPUTER CODE 

The algorithm for solving the proposed problem involves a marriage of Zangwill's convex 
s iplex method [8] and the primal transportation method [1]. We specialize this algorithm to 
t .5 transportation problem. Our contribution is in developing procedures that render the algo- 
nm computationally viable both in respect of demands on storage as well as computation 
i le. We will therefore discuss the details of our code at length, but before that we briefly out- 
li3 the various steps of the convex simplex algorithm. 



148 R.C. RAO AND T.L. SHAFTEL 

The convex simplex method starts with an initial basic feasible solution which is obtainec 
as for a linear problem. Let fix) be the objective function to be minimized over x in som< 
convex polyhedral set. A straight forward way to understand the nonlinear approach would bt 
to view the objective function coefficient vector c of the linear problem (clearly V/(x) = c) a; 
being replaced by the gradient of the objective function V/(x) at any iteration of the simple; 
tableau. If this is done then the reduced cost calculation yields the rate of change in the objec 
tive function with respect to x. A variable which has a negative reduced cost may be raisei 
causing the objective function to decrease. 

In a linear problem, the identification of a candidate variable is followed by a pivot to i 
new basis. For a nonlinear objective function, the optimum does not necessarily lie on ai 
extreme point of the feasible set. Thus the candidate variable provides a locally useful directioi 
along which we search for a step length. Having found the extent of a desirable movement, thi 
candidate variable is raised to the appropriate value, and a new solution point has been found 
In the nonlinear problem Vf(x) must be calculated at each new solution point. In addition t< 
increasing (raising) variables with negative reduced costs, nonzero (nonbasic) variables witl 
positive reduced costs may be decreased in value in order to reduce the objective function. A 
any iteration, a new solution point may be the result of a basis change or a modification of thi 
values of the basic variables without a change of basis. The convex simplex method then, car 
be viewed as the following: 

At any iteration, 

(i) Calculate V/(x). Using these, calculate the reduced costs associated with the non 
basic variables. 

(ii) Find either a raised (nonzero) nonbasic variable which does not have a zero reducei 
cost or a nonbasic variable at zero with a negative reduced cost. If none stop, th' 
optimum is obtained. 

(iii) Raise or lower the value of the chosen variable in the direction which reduces th. 
objective function. Find out how far this variable should be adjusted using a on< 
dimensional unconstrained search. (Of course, the variable is restricted to be nonne 
gative). If a basic variable goes to zero prior to reaching the unconstrained minimun 
go to (iv). Otherwise go to (v). 

(iv) The* candidate variable enters the basis, and the basic variable driven to zero i 
removed from the basis. Go to (i). 

(v) Modify the values of the basic variables and the nonbasic variable being raised o 
lowered. The basis remains the same. Go to (i). 



The stopping rule in (ii) will lead to a global minimum if the objective function is convex 
Otherwise, a local minimum or a stationary point is arrived at. In effect, a Kuhn-Tucker poin 
is obtained. Although we will refer to the optimum solution in the remainder of the paper, th 
above qualifications are to be borne in mind. 

An important aspect of the work we have accomplished is the modification of the prim; 
transportation solution technique to solve a family of nonlinear problems. We now discuss th 
computational procedures in detail. 



TRANSPORTATION WITH NONLINEAR OBJECTIVE FUNCTIONS 149 

The discussion in this section will make use of Figure 1 to a great extent. This figure 
develops a flow chart which shows the modifications necessary for accommodating nonlinear 
objective functions in the transportation problem. The figure is divided into three parts each 
containing different blocks of the flow chart. The first part represents calculations for the linear 
problem which are not used in the nonlinear case. The second third of the figure represents 
those blocks which would be used by both linear and nonlinear problems. The first two parts 
together represent a flow chart for the linear transportation problem. The last part of the figure 
shows the added computation necessary for nonlinear problems. Combining the last two parts 
of Figure 1 yields the flow chart for solving the nonlinear transportation problem. 

The flow chart in Figure 1 closely follows the FORTRAN computer code which we have 
written. Thus, each section below from 3.1 to 3.9 represents a subroutine in our code. Wher- 
ever appropriate, the block number from Figure 1 will be indicated in the body of the text. 

3.1 Input 

The input for the problem consists of (a) the linear cost coefficients, (b) the rim condi- 
tions, i.e., capacities of sources (plants) and demands at sinks (markets) (block A.l) and (c) 
the nonlinear part of the objective function (block B.l). The linear cost coefficients and rim 
conditions for a problem of size m x n are stored in a (m + 1) x (n + 1) matrix in the usual 
way. The storage of the nonlinear part however needs more elaboration. We will address the 
storage of polynomial terms (type 1) and demand and supply interdependencies (type 2) in the 
I objective function separately. 

3.1.1 Type 1 — Objective Function 

For each term t, in this part of the objective function we must keep the following pieces 
of information: (i) d, the constant and (ii) /;,,, the exponent of each variable x, 7 in term t. If a 
particular variable does not occur in a term, clearly the corresponding exponent would be zero. 
Since in most problems not all variables will occur in each term with a nonzero exponent, we 
can take advantage of this to significantly reduce the storage requirements for this part of the 
objective function. To do this, we assign an index, v, to variable x„, where 
v = n x (/ — 1) + j. Then, we create a list, denoted HL, that contains the term in which a 
variable occurs and its nonzero exponent. If it does not occur in a term, it is not stored. The 
HL list is ordered by the index v. We then use a pointer list, HTL, in order to extract informa- 
tion about each variable as it is needed. In order to accomplish this information extraction, the 
HTL list keeps the beginning and ending positions for information regarding each variable. In 
this fashion we need only store the term numbers in which a variable occurs with a nonzero 
exponent, the value of that exponent, and a pointer list of length m x n. This as opposed to an 
m x n matrix for each term of the polynomial. 

3.1.2 Type 2 — Objective Function 

For each row nonlinear term, / = 1, . . . , w, the coefficients fc y , j = 1, .... n and the 
:oefficients a, are stored. The data for the m rows along with the exponent are stored in a 
n x (n + 1) matrix. The data for the column nonlinear terms /.,, /' = 1, . 



if 



m. 



1 = 1, . . . , n. and/3,, are similarly stored in a (m + 1) x n matrix. 
U INSOL 

This step finds an initial feasible solution (Block A. 2). We do this by using the linear cost 
^efficients only. An initial solution is found using the Modified Row Minimum Rule of 
irinivasan and Thompson [7]. 



R.C. RAO AND T.L. SHAFTEL 



150 



lear Problem 
Only 



Both Linear and 
Nonlinear Problems 



Nonlinear Problem 
Only 



( Stafi ] 



! 


1 Find basi 
using line 


A.2 
feasible 


Glutton 



The figure shows necessary modifications to a linear transporta- 
tion problem in order to accommodate non-linear objective func- 
tions. Dashed lines are unique to nonlinear problem. Crossed 
lines are unique to linear problem. Solid lines are mutual. 
Thick-lined boxes must be modified for different types of non- 
linear objective functions. 



A3 
Create basis list 





1IL 




Calculat 
polynor 
and col 


B 2 
e objective function value 


'I 


B 3 
Calculate linear cost equivalent using 
partial derivatives ol objective function 
for each basic variable 


--^- 


~> 


Calculate 
- partial de 
for each r 
variable 


B 4 
mear cost equivalent using 
ivatives of objective function 
on and raised non-basic 





-^ 


Find ra 
lowered 


B.5 
sed nonbasic variable to be 
or raised 












89 




UrxJat 


e nonbasic 


list ll 


necess 


ary and va 


lue of 


basic 


anables 





I 



Figure 1. Flow chart of computer code 



TRANSPORTATION WITH NONLINEAR OBJECTIVE FUNCTIONS 1 5 1 

3 LABEL 1 

This step creates the basis list in the manner of a "pointed list" (Block A. 3). The basis list 
iows the information on current basic cells, i.e., their row /, column j and current value. See 
inivasan and Thompson [7] for more details. 

4 OBJFUN 2 

This step calculates the value of the nonlinear part of the objective function for each of 
e two nonlinear functions (Block B.2). For the polynomials each term value is calculated 
parately. These values will be needed later in the program for finding the partial derivatives 
the objective function with respect to each of the variables. Actually two vectors store these 
'Irm values for the polynomial objective function. The first vector, Z, stores the term values 
'lich when added will give the current value of the nonlinear objective function. The second 
:ctor, Zl, stores the term values in a form that enables the calculation of partial derivatives, 
lis calculation will be discussed in the next section. The objective function value for row and 
dumn nonlinear terms are calculated in two steps. First the row terms ^k u x n for 



./ 



i= 1 m and the column terms £ /„■ x u for j = 1, ... , n are calculated and stored in 

ictors RSUM and CSUM. Then these sums are raised to the appropriate exponent values and 
pred in RTERM and CTERM. Once again these stored values will be extremely useful for 
ciculation of partial derivatives discussed in the next section. 

-•5 LABEL 2 

This subroutine of the code accomplishes three things. First, it calculates the partial 
crivatives of the objective function with respect to each of the basic variables (Block B.3). 
Scond, it labels the basis tree (Block A. 4) using the predecessor successor index method of 
inivasan and Thompson [7]. This tree is used to determine the cycle created by the addition 
ca new cell to the solution, i.e., a new variable entering the basis. Finally, it calculates the 
crrent value of the dual variables (Block A. 4). In the linear primal transportation code only a 
prt of the tree basis must be relabled and only a subset of the dual variables needs to be 
r/ised. For the nonlinear problem, however, we must use completely revised partial deriva- 
t es at each iteration in order to calculate the duals— in doing so we also relabel the entire tree. 

We now turn to the technique we use for calculating the partial derivatives. We attempt 
t' determine the partial derivatives in an efficient way. For each variable, we calculate the 
divative with respect to the linear, polynomial and row and column sum terms separately and 
tl:n add them together. The derivatives associated with the linear costs are of course merely 
tl: initial coefficients. The derivatives for the two types of nonlinear functions are then calcu- 
l»d as follows: 

3.5.1 Derivatives for Polynomial Terms 

We would like to avoid recalculating each polynomial term for each variable whose partial 
divative is calculated. To do this we take advantage of the fact that at each iteration the value 
o each polynomial term is known to be some value Zl,, and that taking the derivative with 

h„. ■ Zl, 
npect to a variable x r . will yield the result, — . This provides us with a very fast tech- 

n ue for finding the derivative of each term since we perform just one multiplication and divi- 
m. 



1 52 R.C. RAO AND T.L. SHAFTEL 

By using the HL and HTL vectors we can quickly find only those polynomial terms whicl 
contain x, r In the use of this technique, problems arise whenever the value of x u is zero anc 
the exponent h, u is less than or equal to one. When the exponent is greater than one or wher 
more than one zero valued variable occurs in a term the partials are zero and offer no computa 
tional difficulty. In the case where a single x rj with an exponent less than or equal to one is a 
the value zero we act as though this variable is bounded to a prespecified small value (EPSL 
until the partials are found. It is then returned to its original value. (The storage of the extr; 
vector Z allows us to perform this modification without losing the true value of the polynomia 
terms.) 

3.5.2 Derivatives for Row and Column Nonlinear Terms 

The derivatives for these terms are found in a similar fashion to the polynomial terms 
Since the row and column sums are known, the derivative with respect to any variable x,-, wil 
k u ■a i ■ RTERM 

be — - — zrzrz: . , and similarly for the column terms. (RTERM and RSUM were calcu 

RSUM 

lated and stored earlier in OBJFUN 2.) In this case, if RSUM is zero the derivative is als< 
zero. 

3.6 MAIN 

This subroutine of the code first calculates partial derivatives with respect to the nonbasi. 
variables (Block B.4) in the same fashion as described in the last section for basic variables. Ii 
fact there will be two types of nonbasic variables, (i) Raised nonbasic variables whose value 
are not zero that will be stored in an array similar to that for basic variables, and (ii) nonbasi< 
variables that are at the value zero (zero nonbasic variables). Once the partial derivative of. 
nonbasic variable is known, its reduced cost may be found using the standard linear transporta 
tion dual equations and the dual variables calculated in LABEL 2 above (Block A. 5). We nov 
attempt to find a raised nonbasic variable to be raised or lowered (Block B.5) or a zero nonbasi 
variable to be raised (Block A. 6). Any nonbasic variable with a negative reduced cost is a can 
didate for being raised above its current value. A raised, nonbasic variable with a positivi 
reduced cost is also a candidate for being modified. In this case the variable must be reducei 
below its current value in order to improve the objective function. Of these candidates w. 
choose a cell according to one of the following rules: 

(i) Matrix-minimum: here the cell that has the smallest reduced cost among all cells i 
chosen, 

(ii) Row-minimum: here cells are examined row by row. As soon as a candidate i 
encountered, the cell with the lowest reduced cost in that row is chosen, 

(iii) Lot-minimum: here cells are examined in specified lots of NP variables. As soon a 
a lot contains a candidate cell, the cell with the lowest reduced cost in that lot i 
chosen. Of course the row-minimum rule is equivalent to the lot minimum rule wit 
NP = n where n is the number of columns. The lowest reduced cost should b 
interpreted as largest in absolute value. 



More will be said later about the effects of the choice of rule for picking candidate cells. 

3.7 CYCLE 

In this step we find the cycle that the candidate cell forms with the current basis usin 
Srinivasan's and Thompson's procedure (Block A. 7). We isolate the cells whose shipments wi 



TRANSPORTATION WITH NONLINEAR OBJECTIVE FUNCTIONS 1 53 

be altered by modifying the candidate cell. We also find the maximum extent that the candi- 
date cell can be modified. This provides an upper bound for a cell if it is to be raised, or a 
lower bound if it is to be lowered (Block B.6). Of course in the linear transportation problem 
there is always a pivot to the new basis which permits one to alter shipments at this stage. In 
the nonlinear case however, before we can alter shipments we must determine the extent to 
which we wish to modify the candidate cell. We do this in the next step. 

3.8 SEARCH 

In this step we search for a value of the candidate variable that lies between its current 
value and the bound obtained from the previous step. We do this with a one-dimensional gol- 
den section search (Block B.7). Although many possible one-dimensional searches could have 
been chosen, we preferred the robust characteristics of golden sections. For this search we 
have to calculate the objective function for each trial value of the candidate variable (Block 
B.8). This is done in two steps. First we calculate the value of the objective function that is 
not affected by changes in the shipments of the cells in the cycle. This fixed part is calculated 
for the linear and the row and column nonlinear portions by setting all variables in the cycle to 
be zero and then calculating the objective function. For the polynomial portion the same pro- 
cedure is used after setting all variables in the cycle to be one. Then, for each trial value we 
determine the shipments of the cells in the cycle and augment the above fixed part of the 
objective function. This process greatly reduces the number of computations needed for each 
trial value. The exit possibilities from SEARCH are: 

(i) we pivot to a new extreme point, 

(ii) we move to an interior point, 

(iii) we are not able to move more than a pre-specified small value (EPS2), i.e., the can- 
didate cell does not lead to a better solution. 

Actually, before we use the method of golden sections we check to see if a pivot will 
result in a lower objective function. If such is the case, we pivot rather than look for an inte- 
rior point with a lower objective. This takes advantage of the speed of the transportation algo- 
rithm, ignoring the difficult nonlinear search as often as possible. 

3.9 UPDATE 

This step updates the solution. For all exits for SEARCH we update the nonbasic list as 
necessary (Block B.9). If we pivot to a new basis then we also update the basis list (Block A. 8). 
We then return to LABEL 2 and begin a new iteration. 

3.10 Stopping Rules 

There are two possible criteria that will terminate the algorithm, (i) If no candidate vari- 
able with a reduced cost above a certain prespecified amount exists, the program stops (Block 
5.1). This stopping rule is the normal rule for linear programs and is invoked most frequently 
n linear or nonlinear problems, (ii) When performing the nonlinear search (Block B.7), it is 
jossible that the candidate variable that is not at its lower bound cannot be moved a greater 
mount than a prespecified tolerance with a resulting improved objective value. When this 
occurs the program searches for another candidate cell among another subset of NP variables. 
The program stops whenever no candidate cell from these subsets leads to an improved objec- 
ive function value (Block S.2). 



1 54 R.C. RAO AND T.L. SHAFTEL 

4. COMPUTATIONAL EXPERIENCE 

Computational experience is based on random problems. Input for each problem 
includes: 

m, the number or rows 

«, the number of columns 

T, the desired number of terms in the polynomial 

AXT, the average number of variables per term 

ADEN, the average number of nonzero row and column coefficients, k rj and /,, 

EPS2, the tolerance for stopping rule (ii) 

EPS3, the tolerance for stopping rule (i) (set at .001 throughout these tests) 

NP, the number of variables whose reduced cost is calculated during any pass of MAIN 
With these inputs, test problems are generated with the following specifications: 

a,,b l : random integer numbers between and 100 

C/j : random numbers between and 100 plus the row number of that variable 

</, : random numbers between and 30 

h U j : random numbers between and 3 for variables chosen at random to be in each f 

Sj,Dj,kij,ljj : random numbers between and 10 

a,,/^ : random numbers between and 3 

Figures 2 through 7 summarize the results of the computation runs. All numbers except 
those with * are based on 3 runs; * indicate a single run only. For each problem type with 
multiple runs, the computation times are the average of total execution times including time for 
generating input and are in seconds. The code itself is written in FORTRAN. All computation 
was performed on CYBER 70 at the University of Arizona computation center. Inspection of 
Figure 7 indicates that CPU time is improved if only a subset of reduced costs are calculated. 
Generally 10 to 20% of the variables seem about optimal in this case. A number in this range 
was chosen for NP in all the computational results presented. As expected, times increased 
with the finer tolerance in the search to a point where too fine a search would not yield any 
results whatever— see Figure 3. (These were the only cases of no consequence.) Times 
increase dramatically as the row and column nonlinear terms have more nonzero coefficients 
and thus become more active— see Figure 4. As the number of variables occurring in each 
polynomial term increased, the computation time is not greatly affected. This is seen in Figure 

5. This situation is not unexpected since a single zero variable in a term gives it a zero value 
(and two zero variables make the derivative of that term zero). Times decreased as the 
number of terms in the polynomial rose— see Figure 6. We suspect that the multiple random 
terms in the polynomial somehow balanced each other so that the cumulative effect was to 
offset each other, thus leading to a solution closer to an extreme point. 

Due to the realities of computer budgets, more extensive tests were not possible. The 
basic results, however, can be found in Figure 2. Total time even for the 95 x 95 problem are 
well within acceptable computation times. Near optimal results (10% from optimum) were 
obtained very quickly. Also obvious is that the bulk of the time spent in solving the nonlinear 



TRANSPORTATION WITH NONLINEAR OBJECTIVE FUNCTIONS 



155 



oblem 
Size 



Average Average Average Average Average 

Total Time to Within Time in Time in No. of Basic 

Time 10% of OPT Search Main Variables 



Average 
No. of Raised 
Non-Basic Variables 
(% increase in non- 
zero variables) 



x 10 


1.52 0(1.02) 


.42 


1.16 


.10 


19 


3.00 (15.7) 


x 20 


4.64 0(4.20) 


2.00 


3.31 


.53 


39 


6.33 (16.2) 


x 30 


21.02 (18.81) 


10.85 


15.03 


3.00 


59 


16.67 (28.3) 


x 50 


56.71 (56.91) 


34.55 


26.74 


25.85 


99 


28.67 (29.0) 


x 70* 


198.74 (00.00) 


56.13 


143.24 


41.72 


139 


42.00 (30.2) 


x 95* 


354.38 (00.00) 


227.73 


255,19 


76.02 


189 


63.00 (33.3) 



Figure 2. Results for a Range of Problem Sizes 
EPS2 = .05; T = 20; AXT = 20; ADEN = .9 
NP = 7 + 20 percent of the total number of variables 
°Number in parenthesis are median times 



EPS2 



Time 



Value of OBJ Function 



.10 


17.07 




3.562 x 10 5 


.05 


21.30 




3.533 x 10 5 


.01 


Solution not found 








Figure 3. EPS2 vs. 


Time 






T = 20, AXT = 2 


ADEN 


= .9 




m = n = 30, NP = 


= 150 





ADEN Time 

.2 4.30 

.5 14.68 

.9 21.30 

Figure 4. ADEN vs. Time 

T = 20, EPS2 = .05, AXT = 2, 

m = n = 30, NP = 150 



156 



R.C. RAO AND T.L. SHAFTEL 



AXT Time 

2.5 20.49 

5.0 20.89 

20.0 21.30 



Figure 5. AXT vs. Time 

T = 20, ESP2 = .05, ADEN = .9, 

m = n = 30, NP = 150 



Time 



10 23.51 
20 20.89 
50 17.89 



Figure 6. T vs. Time 

AXT = 5.0, ESP2 = .05, ADEN = .9, 

m = n = 30, NP = 150 



Problem Size 


30 


x 30 


20 


x 20 


10 


x 10 




NP 


Time 


NP 


Time 


NP 


Time 




30 


20.90 


20 


6.93 


10 


1.70 




60 


2.1.02 


40 


4.64 


20 


1.52 




150 


17.07 


100 


4.80 


40 


1.53 


- 


300 
450 


17.23 
31.88 


200 


7.39 


50 


2.42 



Figure 7. NP vs. Time 

T = 20, AXT = 20, ADEN =? .9, 

EPS2 = .05 ( = .10 for 30 x 30 case) 



TRANSPORTATION WITH NONLINEAR OBJECTIVE FUNCTIONS 1 57 

problem is in MAIN where the reduced costs are calculated, and in SEARCH. One obvious 
area of further research will be to replace the relatively slow (but robust) golden sections search 
with a faster type of linear search. The final column in figure 2 is the number of nonbasic vari- 
ables in the optimum solution at a value other than zero. We use this as one measure of the 
nonlinearity of the problems we have chosen to solve. We also indicate the percent increase in 
nonzero variables caused by raised nonbasic variables. 

5. CONCLUSIONS 

In this paper we have shown that transportation problems with a class of nonlinear objec- 
tive functions can be solved efficiently for problems large enough to make them useful. We 
believe in fact that the results are startling. These results are even more impressive given the 
common belief that one variable at a time searches cannot be useful in solving nonlinear prob- 
lems. In the cases we studied, only a small percentage of nonbasic variables became active so 
that our problems were in fact smallish nonlinear problems— nevertheless there is no way of 
telling a priori which variables will be inactive. Also, at the optimum the number of raised 
nonbasic variables is significant in relation to the number of basic variables. 

Obvious areas of further research remain. In particular, replacing the slow golden section 
search with a more efficient routine will lead to some improvement. It is also very simple in 
the present code to modify the calculations to solve nonlinear problems with any differentiable 
objective function. Finally, attempting to modify more than one variable during the nonlinear 
search presents an intriguing option. 

We hope that this paper will stimulate others to explore using, where appropriate, more 
extensive nonlinear models in their own problem solving studies. Nonlinear problems pose 
interesting challenges. Computational viability uiltimately depends on exploiting special prob- 
lem structures as well as experimentation with computational procedures. This has been the 
primary aim of our research. 

REFERENCES 

[1] Charnes, A. and W.W. Cooper, Management Models and Industrial Applications of Linear Pro- 
gramming, Vol. I (Wiley, New York, 1961). 

[2] Dantzig, G.B., Linear Programming and Extensions (Princeton U. Press, Princeton, 1963). 

[3] Glover, F., D. Karney, D. Klingman and A. Napier, "Computational Study on Basis Change 
Criteria, and Solution Algorithms for Transportation Problems," Management Science, pp. 
793-814 (1974). 

[4]. Pingry, D. and T. Shaftel, "Integrated Water Management: Treatment, Reuse and Dispo- 
sal," Water Resources Research, to appear. 

[5] Shaftel, T. and G.L. Thompson, "A Simplex-Like Algorithm for the Continuous Modular 
Design Problem," Operations Research 25, pp. 788-807 (1977). 

[6] Srinivasan, V. and G.L. Thompson, "Accelerated Algorithms for Labelling and Relabelling 
of Trees, with Applications to Distribution Problems," Journal of the Association for Com- 
puting Machines 19, pp. 712-726 (Oct. 1972). 

[7] Srinivasan, V. and G.L. Thompson, "Benefit-Cost Analysis of Coding Techniques for the 
Primal Transportation Algorithm," Journal of the Association for Computing Machines 20, 
pp.194-213, (April 1973). 

[8] Zangwill, W., Nonlinear Programming: A Unified Approach (Prentice-Hall, New Jersey, 
1969). 



A NOTE ON DETERMINING OPERATING STRATEGIES 
FOR PROBABILISTIC VEHICLE ROUTING 

James R. Yee 

Department of Electrical Engineering 
University of Maryland 
College Park, Maryland 

Bruce L. Golden 

College of Business and Management 
University of Maryland 
College Park, Maryland 

ABSTRACT 

The stochastic vehicle routing problem is a problem of current importance 
and research interest. Applications include schoolbus routing,, municipal waste 
collection, subscription bus scheduling, daily delivery of dairy goods, and a host 
of related transportation and distribution activities. In this paper, we assume 
that routes for vehicles have already been generated and we focus on determin- 
ing operating strategies. That is, under what conditions should a driver return 
to the central depot in order to replenish his supply? We present a dynamic 
programming recursion which addresses this question and we show that the op- 
timal policy is of a rather simple form. Finally, an algorithm and example illus- 
trate the policy. 



JTRODUCTION 

The stochastic vehicle routing problem is a problem of current importance and consider- 
le research interest. Applications include schoolbus routing, municipal waste collection, sub- 
ription bus scheduling, daily delivery of dairy goods, and a host of related transportation and 
itribution activities. The problem that has been considered to date is to determine a fixed set 
vehicle routes of minimal expected total distance. We assume that all vehicles leave from 
d eventually return to a central depot, and that vehicle capacity constraints and probabilistic 
stomer demands must be satisfied (for deterministic demands see Golden, Magnanti and 
'juyen [1]). 

In early work on stochastic vehicle routing, Tillman [4] introduced a heuristic approach to 

termine, under some rather strong assumptions, a fixed set of routes of minimal expected 

al cost. The total cost of traveling a route included a cost associated with not hauling enough 

|ods to satisfy customer demands and a cost associated with carrying unnecessary excess goods 

the route. 

159 



160 J.R. YEE AND B.L. GOLDEN 

Golden and Stewart [2] address the problem of determining a fixed set of routes where 
the demand of customer / is modeled by a Poisson distribution with mean A.,. The objective is 
the minimization of the expected total distance traveled. In Golden and Yee [3], this work is 
extended and a general framework for solving the stochastic vehicle routing problem is pro- 
vided. The authors discuss a solution procedure for correlated demands and develop a genera 
solution technique for the situation where demands are Poisson, negative binomial, binomial, 
or gamma distributed. In addition, they derive analytical results which describe the relation 
ships between design parameters and offer a scheme for performing perturbation analysis. 

In previous work, whenever a vehicle does not have enough goods to satisfy a customer's 
demand, it must immediately return to the central depot for reloading before supplying goods 
to the remaining customers on the route. This, however, is not always the most cost-effective 
strategy. It is sometimes judicious not to wait until a vehicle is empty in order to return to the 
central depot for reloading. In this paper, we determine conditions under which a vehicle 
should return to the central depot. These operating strategies lead to the minimization or 
expected distance traveled. 



DISCUSSION 

Suppose that the routes for vehicles have already been determined by a solution pro 
cedure such as the Clarke-Wright algorithm (see [1] for details). Each vehicle leaves the cen 
tral depot and proceeds to deliver goods to a number of demand points. After delivering good; 
to a demand point on a route, the driver is faced with the decision of whether or not to returr 
to the depot in order to replenish his supply. We shall prove that the optimal decision is basec 
upon whether or not the remaining supply of goods in the vehicle is greater or less than some 
critical value y which must take into account 

(i) the probabilistic demands on the remaining portion of the route, and 
(ii) the distances between customers. 

Let the demand of customer /'(/ = 1, 2, . . . , N) be given by the discrete random variable 
Xj and have some arbitrary mass function P(J, = k)\ we remark that continuous probability 
distributions can be handled analogously. Each vehicle has a capacity of c and starts out on < 
route with a supply of c units. Without loss of generality, we number the customers in reverse 
order from which deliveries are made, i.e., deliveries are made to customer N first, custome 
N - 1 second, and so on (customer denotes the central depot) . The matrix of distance; 
D = [dij] between customers /' and j is known in advance and we will assume that it is sym 
metric in order to simplify computations. 

In general, after delivering goods to customer n + 1, let y„ represent the remaining sup 
ply of goods on the vehicle with n customers still to be visited and let V n (y n ) fo 
n = 0, 1, . . . , N be the expected distance to be traveled in supplying the remaining n custo 
mers given that there are y n (0 < y n < c) remaining units of supply in the vehicle and ar 
optimal policy is followed; for n = 0, V n {y n ) is defined to be d l0 . 

We now present a dynamic programming approach" for determining the optimal operatinj 
policy for a vehicle. After delivering to customer n + 1, we have 



(1) K„(v„) = min 



d n+l0 + d 0j! + Y,p{x n = k)v n _,{c- k) + , 

k=0 



NOTE ON PROBABILISTIC VEHICLE ROUTING 1 6 1 

d n+ i,„ + £p(X tt = k)V„- l (y„-k)+ £ P(X n = k)[2d , n + V^ic-k+y^) 

where V(a>) + = K(max (0,o>)). 

The recursive equation in (1) may be used to determine V n (y n ), the minimal expected 
distance for completing the route with y„ remaining units of supply in the vehicle and n 
demand points still to be serviced. The two options available to the driver are to go back to the 
depot and replenish his supply or to proceed to the next customer on the route. The expres- 
sion representing the expected distance of first returning to the depot includes the distance of 
traveling from customer n + 1 to the depot, then from the depot to customer «, and finally the 
i minimal expected distance with n — 1 customers remaining. On the other hand, if the driver 
I advances to the next node directly, the expected distance includes the distance between custo- 
; mers n + 1 and «, the distance from customer n to the central depot which might be incurred 
twice as a penalty, and the minimal expected distance with n — 1 customers remaining. It is 
assumed that the probability of any customer's demand exceeding vehicle capacity is 
infinitesimal. 

In the remainder of this section we study properties of the optimal solution to our prob- 
! lem. First, we state the following lemma and definition. 

LEMMA 1: 2d 0n + V n _ x {a + b) ^ V n _ x {b), where a and b are positive integers and 
a + b < c. 

PROOF: After supplying customer n and with b units of supply remaining in the vehicle, 
the driver may decide to return to the depot, pick up an additional a units of supply and return 
to customer n. Since this is not necessarily optimal, the result follows. 



DEFINITION: y„ = min 



integer v(0 < y < c) \ d n+h0 + d Q _ n + £ P(X„ = k)V„_ x 



k=0 



y oo 

(c- k) + > 4,+u+E W„ = k)V„_ l (y-k)+ £ P{X n = k)[2d Q , n +V n _,{c- k + y) + ] 



k=0 k=y+l 

or c + 1 if no such y exists. 



We can now state and prove the general structure of the optimal policy. 

THEOREM 1: After supplying customer n + 1, if the remaining supply y n > y n , then the 
optimal policy is to proceed to the next demand point. Otherwise return to the depot. 

PROOF: We prove the first part of the theorem here; the second part is proved in an 
analogous fashion. Suppose y„ < c. Then, by definition 

<Wo + d 0in + £ P(X n = k) V n _ x {c - k) + > d n+hn + £ P(X„ = k) V n _ x {y n - k) 

k=Q k=0 

(2 ) + £ P(X n = k)[2d 0i „+ V n _ x (c - k + y n ) + ). 

Let y„ = y n + m, where m is a positive integer. For the decision of proceeding to the next cus- 
omer, the expected cost of completing the route is 



162 JR. YEE AND B.L. GOLDEN 

(3) \t m 

d„ +U + £ P(X n = k) V n _ x {y n + m-k) 
fc-0 

k=y n +m+\ 

Subtracting (3) from the right-hand side of (2) results in 
± P(X n = k)[V n _ x (y n -k)- V n _ x (y n + m-k)} 

k=0 

y„+m 
+ £ />(*„- /r) [2 </„.„+ K„_!(c- k+y n ) + - V n _ x {y n + m - k)\ 
k-y n +\ 

oo 

+ Z P(Z„ = ^)[^_,(c-/c+j)„) + -^_ 1 (c-^+j)„ + m) + ] > 0. 

k=y n + m+\ 

The last inequality is obtained from Lemma 1 and the fact that for all n 

V n {a 4- m) < V„(a), where a,w ^ and a + m ^ c. 

If y a = c + l, then the triangle inequality is violated since d n+Xn > d n+]0 + d „ and the 
optimal decision is to return to the central depot. 

From the previous theorem, the following algorithm becomes apparent. 

OPERA TING STRA TEG Y ALGORITHM: 

Step 0. n — 0. V (y) = d l0 (y = 0, 1, . . . , c). 

Step 1. n — n + 1; determine V n (y)(y = 0, 1, ..., c) and y„. 

Step 2. If n < N — 1, go to step 1; otherwise compute V N {c) and stop. 

EXAMPLE: 

In this section, we illustrate the algorithm by presenting data and results for a small sam- 
ple problem in order to gain more insight into the performance of this simple dynamic pro- 
gramming algorithm. We consider a vehicle routing problem with four demand points and the 
distance matrix shown in Figure 1. Suppose that customer demands are Poisson distributed 
with average demands of A.] = 30, X 2 = 10- ^3 = 30, \ 4 = 25 and that vehicle capacity is 100 
units. After some preliminary analysis, the following fixed route emerges: 
— 4— ►3 — 2— ► 1 — ♦ 0. We seek to determine an effective operating strategy. In this case, 
the dynamic programming recursion yields 

^, = 33 

h=o 

j>3=38 

and the expected distance traveled under this optimal policy is 30.59 units. If this operating 
policy were not followed and a vehicle returned to the depot only after it discovered that custo- 
mer demand could not be satisfied, then the expected distance would be 32.04 units. Thus, the 
operating policy results in a 4.53% savings in expected distance traveled. Furthermore, we note 
that if vehicle capacity were infinite, distance traveled would be 29 units. The operating policy 
is straightforward and easy to calculate, intuitively appealing, and very effective in minimizing 
distance covered. 



NOTE ON PROBABILISTIC VEHICLE ROUTING 
12 3 4 



163 




1 
2 
3 

4 





5 


2 


11 


6 


5 




4 


12 


8 


2 


4 




2 


10 


11 


12 


2 




12 


6 


8 


10 


12 


_ 



Figure 1. Distance Matrix 



f 



KNOWLEDGMENT 



A preliminary version of this paper was presented at the 1978 Northeast AIDS Confer- 
ee in Washington, D.C. 

REFERENCES 

II Golden, B., T. Magnanti and H. Nguyen, "Implementing Vehicle Routing Algorithms," Net- 
works, 7(2), 113-148 (1977). 

[| Golden, B. and W. Stewart, "Vehicle Routing with Probabilistic Demands," Computer Sci- 
ence and Statistics: Tenth Annual Symposium on the Interface (D. Hogben, D. Fife, eds.), 
National Bureau of Standards Special Publication 503, Washington, D.C, 252-259 (1978). 

[I Golden B. and J. Yee, "A Framework for Probabilistic Vehicle Routing," American Institute 
of Industrial Engineers Transactions, 77(2), 109-112 (1979). 

['.I Tillman, F., "The Multiple Terminal Delivery Problem with Probabilistic Demands," Tran- 
sportation Science, 3(3), 192-204 (1969). 



A NOTE ON THE "VALUE" OF BOUNDS ON EVPI 
IN STOCHASTIC PROGRAMMING* 

J. G. Morris and H. E. Thompson 

Graduate School of Business 

University of Wisconsin 

Madison, Wisconsin 



ABSTRACT 

The existing literature concentrates on determining sharp upper bounds for 
EVPI in stochastic programming problems. This seems to be a problem 
without an application. Lower bounds, which we view as having an important 
application, are only the incidental subject of study and in the few instances 
that are available are obtained at an extremely high cost. 

In order to suggest a rethinking of the course of this research, we analyze 
the need for bounds on EVPI in the context of its significance in decision prob- 
lems. 

1. INTRODUCTION 

The standard definition of the expected value of perfect information, EVPI, is the max- 
imum amount that the decision maker would pay for perfect prior information as to the realiza- 
tion of the random variable in a decision problem. This definition is contained in much of the 
recent literature that concentrates on determining upper bounds for EVPI in stochastic pro- 
gramming problems. For example, Ziemba and Butterworth [6, pp. 365-66] note that EVPI is 
the upper bound on the "amount that the decision maker would pay a clairvoyant" to provide 
the information but go on to suggest that since the calculations are formidable, "it's of interest 
to find an upper bound on EVPI that requires less computational effort." No attention is paid 
to developing a lower bound other than zero. (See also [1], [2], [4].) That upper bounds on 
EVPI and this standard definition should be juxtaposed is peculiar. There is little managerial 
'value" in providing upper bounds on EVPI within the context of its definition. In contrast, a 
lower bound on EVPI, which has managerial significance, is typically left at zero by authors 
writing on the subject. To illustrate, suppose that we determined an upper bound on EVPI to 
be 55. Armed with this information, how much will we pay a clairvoyant to predict realizations 
of the random variable? Since $5 in an upper bound, it may be unjustifiable to offer $5. To be 
safe, we should pay close attention to the lower bound. And it seems reasonable to offer to pay 
the clairvoyant only up to the lower bound for this information since it is the maximum amount 
that we could pay and still be assured of justifying the expense. 

In the next section, we discuss a stochastic programming problem from an applications 
joint of view. 



Support from the Rennebohm Foundation is acknowledged. 



166 J.G. MORRIS AND HE. THOMPSON 

2. STOCHASTIC PROGRAMMING AND EVPI 

Consider a decision problem given by 

(1) Max {E z u if ix, z))\x € K) 

where the decision, x, must be made prior to the realization of the random variable, z. In (1), 
x and z may be vectors and fix, z) is the accumulated monetary wealth of the decision maker 
subsequent to the random event and u is a utility function; K is the set of feasible decision 
alternatives. 

Problem (1) can be very difficult computationally since it may involve maximizing an, 
expected value of a nonlinear function of both x and the random variable z. One way to ease ' 
this computational burden is to find and solve a computationally easier problem which gives 
approximately the same solution as (1). Another approach would be to expend resources to' 
obtain aforehand knowledge of z, thus at least eliminating the expectation from the problem. 
This latter approach also provides an opportunity to produce a greater expected utility. 

SOLVING AN EASIER PROBLEM 

Suppose we solve an easier problem and obtain the solution x € . Suppose also that the cost 
to solve (1) is C\. Then it would be worthwhile adopting x e , thus avoiding Cj, as long as 

(2) E z uifix e , z)) > Max {E z uifix, z) - C\)\x € K). 

Now if an easily computed upper bound on the RHS of (2), call it U e , can be found, then a 
decision rule is available. It is to adopt x e as the problem solution if E z uifix e , z)) > U e . 

To see how this might work, consider an example where u is linear and / is concave in z. 
Then (2) becomes 

(2a) C, > Max {EJix, z)\x 6 K) - EJix e , z). 

Let the easier problem be 

(3) Max {uifix, fx))\x 6 K) - w(Max {/0c, /u)|x <E K)) 

where /x = Eiz). Avriel and Williams [1] used the following inequalities for determining ar 
upper bound on EVPI: 

(4) EJix, z) ^ Max {EJix, z)\x € K) ^ £. Max{/U z)\x € K] ^ Max [fix, (t)\x € K) 
where x solves (3). For u linear, 

(5) EVPI = E z Max [fix, z)\x e K) - Max {EJix, z)\x € K) 
and 

(6) Max [fix, il)\x € K) - EJix, z) ^ E z Max [fix, z)\x e K) 

- Max {EJix, z)\x e K) = EVPI. 

If we define x e = 3c, then the LHS of (6) is an upper bound on the RHS of (2a) and x e wouli 
be adopted if 

(7) C, > fix e , (jl)- EJix e ,z). 



NOTE ON BOUNDS 167 

; RHS of (7) is relatively easy to compute since it involves solving a deterministic program- 
lg problem and taking an expectation. 

That the RHS of (7) is an upper bound on EVPI is incidental. The essential elements are: 
tling an easier problem to solve and finding an upper bound on (1) which provides the deci- 
n rule (7). Research directed toward finding an upper bound on EVPI such as found in 
pel and Williams [1], Ziemba and Butterworth [6], and Huang, Vertinsky, and Ziemba [4] is 
lrly applicable to the decision problem posed. However, if we were able to discover an 
;-ier problem for which an upper bound on the RHS of (2a) is less than EVPI, we would be 
'leedingly happy. 

QUIRING AFOREHAND PERFECT INFORMATION 

Assume perfect aforehand information can be acquired for a cost of Q < °°. Given this 
ikmation, one can solve the wait-and-see problem 

Max {u (f(x, z))\x € K) 

> that value of z which will be realized. Solving (8) is ordinarily much easier than solving 
j Suppose that the cost to do so is C w < C\. Then it is desirable to acquire the informa- 
I if 

E z Max {u(J(x, z) - C w - C,)\x € K) > Max [E z u(f(x, z) - Cj\x € K). 
\ define an EVPI' implicity in 

E z Max [uif(x, z) - CV- EVPIOIjc € K) = Max [E z u(f(x, z) - C x )\x € K). 

Equation (10) is in contrast with the implicit definition of EVPI usually stated as 

E z Max {u(f(x, z) - EVPDlx € K) = Max {E z u(f(x, z))\x € K). 
is linear, then (10) yields 

EVPI'= E z Max [fix, z)\x € K) - Max {EJ(x, z)\x € K) + C, - C w 
i (11) yields the definition of Equation (5). Therefore, 

EVPI' = EVPI + C, - C w . 

Now if the acquisition cost of the information C, is less than EVPI', then it is worth 
) ining, i.e., if 

C, ^ EVPI + d - C w . 

Hs C, and C w are assumed known, lower bounds on EVPI are significant to the decision 
clem. Upper bounds are not. 

One exception to the concentration on upper bounds appears in Huang, Vertinsky, and 
e'ba [4]. They show that for a linear utility function, "sharp" upper and lower bounds can be 
>ined for both z = E z Max {/(*, z)\x € K) and z n = Max [E z f{x, z)\x € K). Then if 

z p < zj/ and z„ L < z n < z^, we have 

Max [0, z p L - z n u ] < EVPI < z p u - z n L 



168 J.G. MORRIS AND HE. THOMPSON 

since EVPI = z p - z„, and the problem becomes one of finding sharp bounds z p , z p u , zfr, zl 
Considerable computing effort renders these bounds as sharp as desired by the decision make 
Therefore, we could reconstitute the decision inequality of (14) as 

(16) Q < Max [0, z p L - z n u ] + C, - C w . 

When u is strictly increasing and concave, Ziemba and Butterworth [6, p. 375] show that 

(17) < EVPI < fix, n) - u- ] {E z u(f(x, z))}, 

where w _1 is the inverse function of u. The lower bound which is essential for decision makir 
is left at zero. 

Also for the case of a concave utility function, Huang, Vertinsky and Ziemba [4, pp. 13i 
137] show that with additional assumptions about the probability distribution of z, one a 
obtain either an upper or a lower bound for EVPI, but not both. 

Although Huang, Vertinsky, and Ziemba make some inroads into determining lew 
bounds on EVPI, it appears that such a determination is entirely accidental rather than 1 
design. They appear to ignore the decision maker's real problem and concentrate on the upp. 
bound for EVPI. The research direction in this area would benefit from returning to its applic 
tions base initiated by Dantzig [3] and Madansky [5]. 

3. ADDITIONAL COMMENTS ON LOWER BOUNDS 

For the linear utility case, by Equation (5) z p — z„ defines EVPI. In general, z„ is diffici 
to calculate as has been the assumption in motivating discussion of the decision problem, 
the domain of z is R c E s (Euclidean S-space), the evaluation of z p involves maximizing 
function involving multiple integrals. This may be particularly undesirable and resort to nume 
ical methods may be required. Likewise, z p would also involve evaluating a multiple integr 
and again resort to numerical methods may be required. 

It is instructive to return to the paper by Huang, Vertinsky and Ziemba to discuss tl 
nature of the lower bounds on EVPI which they determine. The methods suggested consist i 
approximating the integrals imbedded in z„ and z p with summations by partitioning. For exan 
pie, consider R = [a,b] C E ] . The upper bound on z„ would be found by evaluating 



(18) zy=Max 



2>,/U j8,)|x € K 



where a, is the probability of z belonging to an interval / and /3, is the mean z conditional on 
belonging to interval i [4, pp. 130-131]. As / — <x> finer partitions are made and z„ 6 ' -* i 
However, (18) is merely a discrete version of (1) and solving it approximates the solution 
(1). Thus, we solve a problem almost as computationally undesirable as (1) save for the elirr 
nation of an integral. « 

The lower bounds on z p would be found by evaluating 
(19) *£:- £ 8/ Max [fU, d,)\x € K) 

where 8, is an appropriate probability measure and d, is an appropriate point in interval /' [4, p 
130-131]. Again, as finer partitions are made, z p -+z p . But calculating z p involves solvii 



NOTE ON BOUNDS 169 

/ + 1 programming problems (whose values are then averaged). Equation (19) consists of a 
discrete version of z p and has as its main computational virtue the elimination of an integral. 
Similar results hold for the case of several random variables. 

Since 

(20) Zp~Zn< EVPI = z p - z„, 

we have a lower bound on EVPI. If we now ask the question of how much we would be willing 
to pay a clairvoyant for the benefits of perfect aforehand knowledge of z ignoring the cost of cal- 
culating z^ and zj/ and ignoring elements C\ and C w as factors in the decision, then the lower 
bound from (20) would help to answer the question. 

If, however, the computation costs are not negligible and if the reason for seeking a clair- 
voyant was to ease the computational burden, then we have an interesting situation. It is 
worthwhile employing the clairvoyant if (14) holds. But (14) is predicated on the assumption 
that C\ will be avoided if the clairvoyant is hired. But in finding the lower bound on EVPI, we 
solved a problem which was an approximation to (1) and as that approximation got better, the 
cost of doing it would approach C\. That is, C\ was not entirely avoided. Now if the decision 
rule (14) tells us to avoid solving (1), we have a dilemma since it suggests that we should not 
do what we just (approximately) did! 

REFERENCES 

[1] M. Avriel and A.C. Williams, "The Value of Information and Stochastic Programming," 
Operations Research 18, 947-954 (1970). 

[2] D.P. Baron, "Information in Two-Stage Programming Under Uncertainty," Naval Research 
Logistics Quarterly 18, 169-176 (1971). 

[3] G.B. Dantzig, "Linear Programming Under Uncertainty," Management Science 1, 197-206 
(1955). 

[4] C.C. Huang, I. Vertinsky, and W.T. Ziemba, "Sharp Bounds on the Value of Perfect Infor- 
mation," Operations Research 25, 128-139 (1977). 

[5] A. Madansky, "Inequalities for Stochastic Linear Programming Problems," Management Sci- 
ence 6, 197-204 (1960). 

[6] W.T. Ziemba and J.E. Butterworth, "Bounds on the Value of Information in Uncertain 
Decision Problems," Stochastics 1, 361-378 (1975). 



A NOTE ON INTEGER LINEAR FRACTIONAL PROGRAMMING 

Suresh Chandra and M. Ch.andramoh.an 

Indian Institute of Technology 
Dellii, India 

ABSTRACT 

This note consists of developing a method for enforcing additional 
constraints to linear fractional programs and showing its usefulness in solving 
integer linear fractional programs. 



1. INTRODUCTION 

Fractional cutting plane methods for solving integer linear fractional programs have been 
oroposed, for example, by Swarup [6], Grunspan and Thomas [3] and Granot and Granot [2]. 
In contrast with these, the method proposed in the sequel does not impose any severe restric- 
:ion on the problem as in Swarup [6], does not consist of solving many integer programs as in 
3runspan and Thomas [3] or does not involve complicated computations in deriving the cuts as 
n Granot and Granot [2]. The results to be followed consist of developing a method for 
enforcing additional constraints to a linear fractional program and then using it for solving pure 
ind mixed integer linear fractional programs by cutting plane methods. 

I. METHOD OF ENFORCING ADDITIONAL CONSTRAINTS 

Consider the following linear fractional program: 

n 

Z c j x j + a 
IP): maximize — 

I fa + P 

7 = 1 



subject to £ a,jXj = a l0 , / = 1, 2, . . . , m, 



7 = 1 



Xj > 0, j = 1, 2, .... n, 



vhere it is assumed that £ djXj + fi > for all feasible solutions. 

7 = 1 

Suppose that a basic feasible solution to (P) is known and with reference to this, let the 
constraints be 

171 



172 S. CHANDRA AND M. CHANDRAMOHAN 
(1) x B - b, + £ yu(-Xj), / - 1, 2 m, 

*,- > o, y-i, 2, .... ii, 

where x fi , i = 1, 2, . . . , m are the basic variables, N is the index set of nonbasic variables an 
the basic solution is obtained by putting Xj = for every j£N in (1). Let z 1 , z 2 be respective 

ly £c fl( A + a, Z^^+jS, z/ = £c fl .j;y, z/ = £ </ B/ v, 7 and A, = U 2 (z/ - c,) zKzl 

/=1 ' /-l ' ;-l 7-1 

d)))lz 2 be calculated for all j 6JV . Let the additional constraint to be appended be 



(2) X P/*/ > ?• 

y-i 

Substituting for x g , / = 1, 2, . . . , m from (1) in (2) let (2) take the form 

(3) *„ +1 = b m+x + £ y m +\ (~xj) > 0. 

Let us denote by (Pi) the new problem obtained from (P) by appending (3). Then 



(4) 



*b, "K /= 1, 2 m + 1 

Xj = 0, j £ N 



is a basic solution to (Pi) where we designate n + 1 for P m+] . Let c„ +1 and d n+x be assign 
the values zero. 

The problem of interest here occurs when b m+] < 0. To establish the validity of the su 
cessive steps of the method to be followed in reoptimizing the problem (Pj), we require t 
following lemmas, the proofs of which can be constructed with the help of Hadley [4]. 

LEMMA 1. If b m+x < and if the set of feasible solutions to (P x ) is nonempty, th 
there exists a j 6 yVsuch that y m +\j < 0. 

LEMMA 2. For every j € N, there exists an i € {1, 2, . . . , m) such that y tj > 0. 

LEMMA 3. If b m+] < and if there exists a k € N satisfying y m + l k < and 

b, 



b m +\ b r 

^ — = Mm 



y m +\,k y r k 'e{i72 mi 



* y,k *> o 
y<k 



then by a change of basis, i.e., by replacing x k by x B in the set of basic variables a ba: 
feasible solution to (P x ) is obtained and - 

.V/n+1.* ^T" < Z * ~ ^ < - 

Z 2 

» 

The method of reoptimizing (P,) can now be given as follows: 

ALGORITHM 1. 

STEP 1. Set up a tableau giving x B , b n v y -, z} — c h z 2 — djLj for / = 1, 2, . . . , m and j €J| 
Compute z 1 , z 2 and z. Append the additional constraint in the form (3). 



NOTE ON LINEAR FRACTIONAL PROGRAMMING 



173 



STEP 2. If b m+x < 0, set J = and go to step 3. Otherwise, set J = 1 and go to step 8. 
STEP 3. Set M = {j\j € N, y m +\j < 0). If Mis empty, no feasible solution exists, stop. 



STEP 4. Compute — — 
yr(j)j 



= Min 

/e(l,2 i 



J m+\ 



y m +\j yr(j)j 

go to step 5. 



b, 

—. y,j > o 
y>j 



for every j € N. Set M j = 



y|y e a/, 



If M] is empty go to step 6. Otherwise, set / = 1, r = m + 1 and 



STEP 5. Find k such that 



A* 

r r = Min 

l'm + l>l JS- M \ 



m 2 (zj 2 - </,-) and go to step 7. 



i. I - 'm+lj < ° 

I'm + ljl 



where / m+1J = y m+hj - 



STEP 6. Find k such that ^+1^ = Min y m +\j. If y m +i,k ^ no feasible solution exists; stop. 
Otherwise, find r such that 



b r 
— = Min 

y r k '€{1.2 m 



-, y lk > o 



y,k 



STEP 7. Do a simplex pivoting to obtain a new basic solution by replacing x B by x k in the set 

of basic variables. Modify TV, calculate A 7 , zj — Cj, zj — dj for all j € A^ by modifying 
their definitions to take the summation from 1 to m + 1. Also calculate z 1 , z 2 and z. 
If J = return to step 3. 

STEP 8. If A y ^ for all j, stop; the optimal solution is x fl = 6, for / = 1, 2, . . . , w + 1, and 



b r 
X: = for j € A 7 . Otherwise let At = Min A,, — = Min 

JtN y r k '€(1.2 m+i! 



6/ 

— , v /7t > 

y,k 



and 



return to step 7. 

3. FRACTIONAL CUTTING PLANE METHOD FOR 
INTEGER LINEAR FRACTIONAL PROGRAMS 

As an application of Algorithm 1 we present in this section a cutting plane method for 
solving integer linear fractional programs. The method for (mixed) integer linear fractional 
programs follows closely Gomory's fractional cutting plane method for (mixed) integer linear 
programs and can briefly be described as follows. 

Solve the problem obtained by omitting the integer restrictions. If the solution satisfies 
the integer restrictions, then it is optimal; otherwise, introduce a Gomory's (mixed integer) 
fractional cut and reoptimize by using Algorithm 1 and repeat the process. 



REMARK. Enforcing a single additional constraint to a linear fractional program can be 
done easily by using the dual simplex method after employing the Charnes and Cooper [1] 
transformation. But in the case of integer linear fractional programs, the transformation des- 
troys the integer nature of the variables and Gomory's fractional cuts in their usual forms can- 
not be used. However, cuts can be obtained as given by Granot and Granot [2], but it can be 
observed that such derivations of cuts involve more complicated computations than those in 
our case. 



174 S. CHANDRA AND M. CHANDRAMOHAN 

Since the objective function is not integer constrained, the proof of finiteness of the above 
method cannot possibly be given in a manner similar to that of Gomory's methods. 

ACKNOWLEDGMENT 

The authors wish to thank the referee for his suggestions for improvement of an earlier 
version of this paper. 

REFERENCES 






[1] Charnes, A., and W.W. Cooper, "Programming with Linear Fractional Functionals," Naval 
Research Logistics Quarterly, Vol. 9, pp. 181-186 (1962). 

[2] Granot, D., and F. Granot, "On Integer and Mixed Integer Fractional Programming Prob- 
lems," Annals of Discrete Mathematics 1, Studies in Integer Programming, pp. 221-231, eds. 
P.L. Hammer et al. (North Holland Publishing Company, 1977). 

[3] Granspan, M., and M.E. Thomas, "Hyperbolic Integer Programming," Naval Research 
Logistics Quarterly," Vol. 20 (2), pp. 341-356 (1973). 

[4] Hadley, G., "Linear Programming," (Addison Wesley, Reading, Massachusetts, 1962). 

[5] Swarup, K., "Linear Fractional Functional Programming," Operations Research, 13, pp. 
1029-1036 (1965). 

[6] Swarup, K., "Some Aspects of Linear Fractional Functional Programming," Australian Jour- 
nal of Statistics, Vol. 7, pp. 90-104 (1965). 



NEWS AND MEMORANDA 
THE 1979 LANCHESTER PRIZE 

Call for Nominations 



Each year since 1954 the Council of the Operations Research Society of America has 
offered the Lanchester Prize for the best English-language published contribution in operations 
research. The Prize for 1979 consists of $2,000 and a commemorative medallion. 

The screening of books and papers for the 1979 Prize will be carried out by a committee 
appointed by the Council of the Society. To be eligible for consideration, the book or paper 
must be nominated to the Committee. Nominations may be made by anyone; this notice con- 
stitutes a call for nominations. 

To be eligible for the Lanchester Prize, a book, a paper or a group of books or papers 
must meet the following requirements: 

(1) It must be on an operations research subject, 

(2) It must carry a current award year publication date, or, if a group, or at least one 
member of the group must carry a current award year publication date, 

(3) It must be written in the English language, and 

(4) It must have appeared in the open literature. 

The book(s) or paper(s) may be a case history, a report of research representing new results, or 
primarily expository. 

For any nominated set (e.g., article and/or book) covering more than the most recent 
year, it is expected that each element in the set represents work from one continuous effort, 
such as a multi-year project or a continuously written, multi-volume book. 

Judgments will be made by the Committee using the following criteria: 

(1) The magnitude of the contribution to the advancement of the state of the art of 
operations research, 

(2) The originality of the ideas or methods, 

(3) New vistas of application opened up, 

175 



NEWS AND MEMORANDA 

(4) The degree to which unification or simplification of existing theory or method is 
achieved, and 

(5) Expository clarity and excellence. 

Nominations should be sent to: 

Marshall L. Fisher, Chairman 
1979 Lanchester Prize Committee 
Department of Decision Sciences 
The Wharton School 
University of Pennsylvania 
Philadelphia, PA 19104 

Nominations may be in any form, but must include as a minimum the title (s) of the 
paper(s) or book, author(s), place and date of publication, and six copies of the material. Sup- 
porting statements bearing on the worthwhileness of the publication in terms of the five criteria 
will be helpful; but are not required. Each nomination will be carefully screened by the Com- 
mittee; nominations must be received by May 30, 1980, to allow time for adequate review. 

Announcement of the results of the Committee and ORSA Council action, as well as 
award of any prize(s) approved, will be made at the 58th National Meeting of the Society, 
November 10-12, 1980 in Colorado Springs, Colorado. 



INFORMATION FOR CONTRIBUTORS 

The NAVAL RESEARCH LOGISTICS QUARTERLY is devoted to the dissemination of 
scientific information in logistics and will publish research and expository papers, including those 
in certain areas of mathematics, statistics, and economics, relevant to the over-all effort to improve 
the efficiency and effectiveness of logistics operations. 

Manuscripts and other items for publication should be sent to The Managing Editor, NAVAL 
RESEARCH LOGISTICS QUARTERLY, Office of Naval Research, Arlington, Va. 22217. 
Each manuscript which is considered to be suitable material tor the QUARTERLY is sent to one 
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Manuscripts submitted for publication should be typewritten, double-spaced, and the author 
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A short abstract (not over 400 words) should accompany each manuscript. This will appear 
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There is no authorization for compensation to authors for papers which have been accepted 
for publication. Authors will receive 250 reprints of their published papers. 

Readers are invited to submit to the Managing Editor items of general interest in the held 
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of the QUARTERLY. 



NAVAL RESEARCH 

LOGISTICS 

QUARTERLY 



MARCH 198C 
VOL. 27, NO. 

NAVSO P-1278 



CONTENTS 



ARTICLES 



I 



Analyzing Availability Using Transfer 

Function Models and Cross Spectral Analysis 

Optimal Maintenance-Repair Policies 
for the Machine Repair Problem 

Benders' Partitioning Scheme Applied 
to a New Formulation of the 
Quadratic Assignment Problem 

Test Selection for a Mass Screening Program 

Optimal Admission Pricing Policies 
for M/E k /1 Queues 

The Dynamic Transportation Problem: A Survey 

The Use of Dynamic Programming Methodology 
for the Solution of a Class of Nonlinear 
Programming Problems 

The U.S. Versus the Soviet Incentive Models 

Theoretical Analysis of Lanchester-Type 
Combat Between Two Homogeneous 
Forces With Supporting Fires 

On Single-Machine Sequencing With 
Order Constraints 

Statistical Analysis of the Output Data 
From Terminating Simulations 

Computational Experience on an Algorithm 
for the Transportation Problem 
With Nonlinear Objective Functions 

A Note on Determining Operating Strategies 
for Probabilistic Vehicle Routing 

A Note on the "Value" of Bounds on EVPI 
in Stochastic Programming 

A Note on Integer Linear Fractional Programming 
News and Memoranda 



N. D. SINGPURWALLA 
S. C. ALBRIGHT 

M. S. BAZARAA 
H. D. SHERALI 

J. A. VOELKER 
W. P. PIERSKALLA 

M. Q. ANDERSON 

J. H. BOOKBINDER 
S. P. SETHI 

M. W. COOPER 



G. G. HILDEBRANDT 
J. G. TAYLOR 

K. D. GLAZEBROOK 

A. M. LAW 

R. C. RAO 
T. L. SHAFTEL 

J. R. YEE 
B. L. GOLDEN 

J. G. MORRIS 
H. E. THOMPSON 

S. CHANDRA 
M. CHANDRAMOHAN 



OFFICE OF NAVAL RESEARCH 

Arlington, Va. 22217