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MARCH 1980
VOL. 27, NO. 1
OFFICE OF NAVAL RESEARCH
NAVSO P1278
<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, CarnegieMellon University
Jack Borsting, Naval Postgraduate School Charles Kriebel, CarnegieMellon 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 overall 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
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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
P35 (Revised 174).
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 BoxJenkins 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(4924)021 1 Nuclear Regulatory Commission and Program in Logistics Contract
N0001475C0729 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
(23) A> = ^vov,*,.,^*,^....
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 BoxJenkins
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
(HoldenDay, 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, 7079 (1975).
[3] Jenkins, G.M. and D.G. Watts, Spectral Analysis and its Applications (HoldenDay, 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. 107112 (1978).
[6] Singpurwalla, N.D. "Estimating Reliability Growth Using Time Series Analysis," The Naval
Research Logistics Quarterly, Vol. 25, No. 1, pp. 114 (1978).
[7] Willie, R.R. "Everyman's guide to TIMS," ORC 772, Operations Research Center, Univer
sity of California at Berkeley (1977).
OPTIMAL MAINTENANCEREPAIR 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 singlerepairman 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 firstcomefirstserved 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 wellknown 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 /7period 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
wellknown, 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 oneperiod 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 wperiod 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,) + Njigik) + 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 nonnegative. 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 BirthDeath Model of Advertising and Pricing,"
Advances in Applied Probability, 77, 134152 (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. 2663, Control Analysis Corp, Palo Alto, Ca. (1972).
Crabill, T., "Optimal Control of a Maintenance System with Variable Service Rates,"
Operations Research, 22, 736745 (1975).
Denardo, E., "Contraction Mappings in the Theory Underlying Dynamic Programming,"
SIAM Review, 9, 165177 (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, 328339 (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, 484492
(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, 595613 (1973).
Iglehart, D.L., and A. Lemoine, "Approximations for the Repairman Problem with Two
Repair Facilities, II: Spares," Advances in Applied Probability, 6, 147158 (1974).
Kakumanu, P., "Relation Between Continuous and Discrete Time Markovian Decision
Problems," Naval Research Logistics Quarterly, 24, 431439 (1977).
Lippman, S.A., "Applying a New Device in the Optimization of Exponential Queuing Sys
tems," Operations Research, 23, 687710 (1975).
Lippman, S.A., "CountableState, ContinuousTime Dynamic Programming with Struc
ture," Operations Research, 24, 477490 (1976).
Ross, S.M., Applied Probability Models with Optimization Applications (HoldenDay, 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, 259268 (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 01 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
01 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 # ENG7707468
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
/^
fcl m
2.3) Z Z yifki  (kl) x kl = for k = 2 m, /  1 /w
/=i /i
/*/
2.4) Z x '/ = ! for / = 1 m
2.5) Z^/ = 1 for y1
/??
/ = !
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 zeroone 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
:eroone 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 zeroone 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 cutconstraint. 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 yvariables 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
l»£ 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
iI 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 regenerated 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
fci /=i
Note that g u is obtained by solving a linear assignment problem in m1 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 1015% 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
onetoone correspondence with the points in X A . In other words, all integral solutions in A" are
zeroone 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 dualinteger cuts were unable to find an integer primal feasible solution.
Further, in this process, the updated coefficients in the simplex tableau tended to blowup in
magnitude.
We thus resorted to Glover's [12] pseudoprimaldual procedure which iterates between a
lexicographic dual feasible stage related to Gomory's dual allinteger algorithm [13] and a dual
infeasible stage related to Young's primal allinteger 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 revisedsimplex 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 7428/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 7428/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 KuhnTucker 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
(51) 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
REFERENCES
Benders, J.F., "Partitioning Procedures for Solving Mixed Variables Programming Prob
lems," Numerische Mathematik, Vol. 4, pp. 238252 (1962).
Burkard, R.E. and K.H. Stratmann, "Numerical Investigations on Quadratic Assignment
Problem," Naval Research Logistics Quarterly, Vol. 25, No. 1, pp. 129148 (1978).
Cabot, A.V. and R.L. Francis, "Solving Certain NonConvex Quadratic Minimization Prob
lems by Ranking the Extreme Points," Operations Research, Vol. 18, pp. 8286 (1970).
Elshafei, A.N., "Hospital Layout as a Quadratic Assignment Problem," Operational
Research Quarterly, Vol. 28, No. 1, pp. 167179 (1977).
Finkelstein, J.J., "Estimation of the Number of Iterations for Gomory's AllInteger Algo
rithm," Doklady Akademii Nauk. SSSR, Vol. 193, pp. 988992 (1970).
Francis, R.L. and J. A. White, "Facility Layout and Location: An Analytical Approach,"
(PrenticeHall, 1974).
Garfinkel, R.S. and G.L. Nemhauser, "Integer Programming" (John Wiley and Sons, 1972).
Gaschiitz, G.K. and J.J. Ahrens, "Suboptimal Algorithm for the Quadratic Assignment
Problem," Naval Research Logistics Quarterly, Vol. 15, pp. 4962 (1968).
Gavett, J.W. and N.V. Plyter, "The Optimal Assignment of Facilities to Locations by
Branch and Bound," Operations Research, Vol. 14, pp. 210232 (1966).
Geoffrion, A.M. and G.W. Graves, "Multicommodity Distribution System Design by
Benders' Decomposition," Management Science, Vol. 20, No. 5, pp. 822844 (1974).
Gilmore, P.C., "Optimal and Suboptimal Algorithms for the Quadratic Assignment Prob
lem," SIAM Journal on Applied Mathematics, Vol. 10, pp. 305313 (1962).
Glover, F., "A Pseudo Primal Dual Integer Programming Algorithm," Journal of Research
of the National Bureau of Standards, 71B, pp. 187195 (1967).
Gomory, R.E., "AllInteger Programming Algorithm," Industrial Scheduling, J.F. Muth and
G.L. Thompson, eds. (Prentice Hall, 1963).
Graves, G.W. and A.B. Whinston, "An Algorithm for the Quadratic Assignment Problem,"
Management Science, Vol. 17, pp. 453471 (1970).
Hanan, M. and J. Kurtzberg, "A Review of the Placement and Quadratic Assignment Prob
lems," SIAM Review, Vol. 14, pp. 324342 (1972).
Heider, C.H., "An nStep, 2Variable Search Algorithm for the Component Placement
Problem," Naval Research Logistics Quarterly, Vol. 20, pp. 669724 (1973).
Hillier, F.S. and M.M. Connors, "Quadratic Assignment Problem Algorithms and the Loca
tion of Indivisible Facilities," Management Science, Vol. 13, pp. 4257 (1966).
Kaufman, L. and F. Broeckx, "An Algorithm for the Quadratic Assignment Problem Using
Benders' Decomposition," European Journal of Operational Research, Vol. 2, pp. 204
211 (1978).
Koopmans, T.C. and M. Beckmann, "Assignment Problems and the Location of Economic
Activities," Econometrica, Vol. 25, pp. 5376 (1957).
Land, A.M., "A Problem of Assignment with Interrelated Costs," Operational Research
Quarterly, Vol. 14, pp. 185198 (1963).
Lasdon, L.S., Optimization Theory for Large Systems, (The MacMillan Company, New York,
1970).
BENDERS' PARTITIONING FOR QUADRATIC ASSIGNMENT 41
[22] Lawler, E.L., "The Quadratic Assignment Problem," Management Science, Vol. 19, pp.
586590 (1963).
[23] Nugent, C.E., T.E. Vollmann and J. Ruml, "An Experimental Comparison of Techniques
for the Assignment of Facilities to Locations," Operations Research, Vol. 16, pp. 150
173 (1968).
[24] Obata, T. and P. Mirchandani, "Algorithms for a Class of Quadratic Assignment Prob
lems," presented at the Joint National TIMS/ORSA Meeting, New Orleans (May 1979).
[25] Pierce, J.F. and W.B. Crowston, "Tree S;arch Algorithms for Quadratic Assignment Prob
lems," Naval Research Logistics Quarterly, Vol. 18, pp. 136 (1971).
[26] Rardin, R. and V.E. Unger, "Surrogate Constraints and the Strength of Bounds Derived
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[27] Sherali, H.D., "The Quadratic Assignment Problem: Exact and Heuristic Methods,"
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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 longrun 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 N000146703560030 and N0001475
C0797.
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 followup testing, and even the risk of physical harm to
the testee; e.g., from the cumulative effect of xray 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
followup 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 agespecific 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, agespecific 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^jS 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 setup 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 "timeuntildetectability" 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 rcJcy versus test No. 2 applied with frequency r
(actions reflected, respectively, in the left and righthand 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 visavis test No. 2, were the two tests scheduled at their maximal (subject to budget)
frequencies rcJ 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. 107895 (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. 4159 (1961).
[4] Keller, J., "Optimum Checking Schedules for Systems Subject to Random Failure,"
Management Science, Vol. 21, pp. 256fcO (1974).
[5] Kirch, R. and M. Klein, "Surveillance Schedules for Medical Examinations," Management
Science, Vol. 20, pp. 14039 (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. 30722 (1963).
[9] Roeloffs, R., "Minimax Surveillance Schedules for Replacement Units," Naval Research
Logistics Quarterly, Vol. 14, pp. 46171 (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 £>(•) nonnegative increasing
implies Vj > for J  0, 1, 2, . . . . Then W n = £ V, and
yo
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(700)
m = l
ByEq. (2.1),
A(r,T)N\r J,] D(u)b lT ^ (u) ]J
, •'(« — l)/r *■*■
= 18,
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 / tax 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 righthand inequality follows from /(■) increasing and IT V ax ^ x. To
prove the theorem, it only remains to estabilsih the lefthand 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 Ti
ax + 28
y (/(F8) + /(T + ax +8))
^T I f(s)ds\
+ 28 j ts J J
ax + 16 "z8
which is nonnegative for 8 6 [0,71 by Lemma 1. Therefore, (A. 3) is nonnegative 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 LowLippman 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 Gammadistributed, and the arrival rate to the system is subject to
setting an admission fee p. The arrival rate \(p) is nonincreasing in p. We
prove that the optimal admission fee p* is a nondecreasing 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 LowLippman model to
include a statedependent 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 707708 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 LowLippman model [6] to include
statedependent 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 welldefined (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 inprocess inventories or higher machine (or service facility) maintenance cost.
Finally, the extension of the LowLippman model to accommodate statedependent 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 membersat 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 yo
; 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) = \
(AX(/,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 nondecreasing 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 realvalued 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)(Ati  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 LOWLIPPMAN MODEL WITH STATEDEPENDENT SERVICE RATES
Once again we consider Lippman's model [6, p. 707], this time incorporating a state
dependent service rate and a statedependent 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 nondecreasing in i.
When fij is taken to be nondecreasing 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 nonincreasing 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 nondecreasing in i for
the finite horizon continuous time problem as follows. We first show that p* is nonincreasing
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 LowLippman Model [6, p. 707] with statedependent 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 statedependent service rate model. A few numerical computations have been
made but no counterexamples 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 birthdeath 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, 165177 (1967).
[4] Kakumanu, P., "Relation Between Continuous and Discrete Time Markovian Decision
Problems," Naval Research Logistics Quarterly, 24, 431441 (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, 687710 (1975).
[7] Lippman, S., "On Dynamic Programming With Unbounded Rewards," Management Sci
ence, Vol. 21, 12251233 (1975).
[8] Lippman, S., "CountableState, ContinuousTimeDynamic Programming With Structure,"
Operations Research, Vol. 24, 477490 (1976).
[9] Lippman, S., and S. Stidham, "Individual Versus Social Optimization in Exponential
Congestion Systems," Operations Research, Vol. 25, 233247 (1977).
[10] Low, D., "Optimal Dynamic Pricing Policies for an M/M/S Queue," Operations Research,
Vol. 22, 545561 (1974).
[11] Ross, S., Applied Probability Models HeldenDay, San Francisco, CA (1970).
[12] Serfozo, R.F., "An Equivalence Between Continuous and Discrete Time Markov Decision
Processes," Operations Research, Vol 27, 616620 (MayJune 1976).
[13] Topkis, D.M., "Minimizing a Submodular Function on a Lattice," Operations Research,
Vol. 28, 305321 (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 uptodate
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 minimumtime transportation problem (Szwarc [44,45,46], Hammer
[23], Tapiero and Soliman [51], Tapiero [49], and Srinivasan and Thompson [41]); the
minimumcost 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 multicommodity 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
timedependent 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
minimumtime, minimumcost, and the maximal dynamic flow problems. We consider in Sec
tion 4 a stochastic multiperiod, multimode model (Midler [34]), a problem of discretetime 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 ElBardai [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 wellknown 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 controltheoretic 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 timedependent:
2.1. A, and/or B, Functions of Time
If the supply capabilities or the demand requirements are timedependent, 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 timedependent, 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 MinimumTime 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 minimumtime 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 SzwarcHammer algorithm is a primal algo
rithm, but Garfinkel and Rao [20] employ a "threshold" algorithm that yields a primalfeasible
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 minimumtime transportation problem) that is virtually identical to the
SzwarcHammer algorithm. Srinivasan and Thompson [41] have also shown that their algo
rithm, and hence the SzwarcHammer algorithm, is computationally more efficient than that of
Garfinkel and Rao [20].
The main points of the SzwarcHammer 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 4step 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 nonbasic 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 nonbasic 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 tradeoff 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
Paretooptimal 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 + yjyj +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 singleperiod 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 recalculate 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 timevarying 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 timedependence 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 continuouslime framework) which uses HamillonJacobi 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 reassignment 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 onebyone temporarily reassigned 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 subproblems 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 previouslyscheduled flow, when
stochastic disturbances warrant such adjustment. Ellis and Rishel [11] have formulated a model
for the oneway flow of air traffic between two airports, subject to random constraints on the
takeoff and landing capacities. Their presentation was based upon the statespace 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 5period numerical example which, although a fairly
simple problem, involved considerable computation.
The remainder of the present paper deals with the continuoustime dynamic transportation
problem with delays. In particular, the following section deals with the minimumtime 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 discretetime statespace framework.*
6.1. Frank 117] and Frank and ElBardai (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 FanWang 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 ElBardai [18] also impose upperbounds on the flows in branches of the net
work. For this, let c = (c,, c 2 , ... ,c,„) denote the branchcapacity 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 ElBardai 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 ElBardai 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 continuoustime statespace framework and
the maximum principle for their analysis.*
6.2. Tapiero and Soliman (51] and Tapiero [49]
Tapiero and Soliman formulate a dynamic multicommodity transportation problem with
time delays as an optimal control problem. For simplicity in exposition, we will only develop
the singlecommodity 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 multicommodity 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 JH. 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), riijitTij), u u (t), t)dt
(10)
min
.subject to
(11) Tjy(f)  u u U), Oij(s) = 0, Vs € [10,0],
with constraints on terminal conditions
(12)
71 7=1
(13) m m
Em,(r>E iK/CrT^jy,
ri /=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 UT 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) \ijiTTjj) +k mn (TT mn ) = \ mj (TT mj ) + \ in (TT in ), i = \, 2, .... (ml),
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. MinimumTime Problem
The minimumtime 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 multicommodity 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 minimumtime problems the objective function (10) is the total time T, thus
(,7 > iAl.
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 minimumtime problem without the transport system's
capacity constraint (9), we will take this up next.
6.2.4. Minimumtime 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 linkingtype, 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 minimumtime 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. MinimumTime 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 Hamiltonianmaximizing 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 minimumtime
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 loadsin
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 arccapacity constraints (24). Once again, constraints (24) are more like loading
constraints associated with arc (ij) rather than the arccapacity constraints. Note also that the
arccapacity 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 bangbang.
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< TTy.
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 linearcosts 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
Linearquadratic control problems have a special place in the optimal control theory. Usu
ally these problems yield closedform 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 pointofview 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 il
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{Tr ij ) + {a i d j ){Tr 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 UijitT,)^* .
/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 UTjiy 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)
11 (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 loadintransit.
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 discretetime 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 321438580 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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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 multiplyconstrained 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 nonempty, 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 71,2. ...'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, ...
71
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 = nI
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 nonnegative 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
xf3,
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
7938
.92
1.35
32.38
16.26
69.86
10171
1.16
14.83
45.66
68.61
52.28
12937
2.38
13.19
16.45
26.18
63.77
10235
2.45
4.84
25.45
94.65
229.74
7534
5.28
5.39
47.01
57.32
3516
1.30
9.74
21.06
27.35
5130
.49
26.53
77.69
35.65
9584
.58
3.47
18.69
6223
1.09
10.20
22.88
5328
1.99
3.09
48.19
5829
.38
11.14
68.62
8557
.53
6.82
5936
.49
9.27
16994
11.71
4.51
6128
.76
7.92
8028
1.00
18.12
12534
4.52
7.64
5325
.32
12.55
7723
1.16
17.07
6430
.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 multiplyconstrained problems.
1]
2]
REFERENCES
Cooper, L., and M.W. Cooper, "NonLinear Integer Programming," Computers and
Mathematics With Applications, /, 215222 (1975).
Cooper, M.W., "An Improved Algorithm for NonLinear 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
multiincentive 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 PlanningProgrammingBudgeting 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
ProgrammingBudgeting 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 socalled "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 "noncontractual" 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 welldeveloped 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 costeffectiveness 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(yy),
where the constant /3 is proportional to the "real social value of having an extra unit which has
been preplanned" [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(yy)+y{yy)]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(yy) + 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
. iyy)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 (yy) 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
ciy0,
c 2 = /3  a.
The problem facing the enterprise can therefore be written
(y^)/(v)afy + c 2 ). iyy)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 wellknown 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 welldeveloped 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(yy)+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= hHs/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].
CostEffectiveness Analysis
One method of describing the U.S. incentive model is to use a costeffectiveness 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 costeffectiveness 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 costeffectiveness 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(pp)+/3(pp~) :p>p)
 (K + M) +y(p p) +/3(pp):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 noncontractual 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, 28199 (Spring 1975).
Bonin, J. P. "On the Design of Managerial Incentive Structures in a Decentralized Planning
Environment," American Economic Review, 68287 (September 1976).
Bonin, J. P. and A. Marcus, "Information, Motivation and Control in Decentralized Plan
ning: The Case of Discretionary Managerial Behavior," Unpublished Manuscript, 127
(June 1977).
DOD and NASA Guide, "Incentive Contracting Guide" (October 1969).
Fan, LiangShin. "On the Reward System," American Economic Review, 22629 (March
1975).
Groves, T. and M. Loeb, "Reflections in "Social Costs and Benefits and the Transfer Pric
ing Problem," Journal of Public Economics, 5: 35359 (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:
18386 (1977).
Loeb, M. and W. Magat, "Success Indicators in the Soviet Union: The Problem of Incen
tives and Efficient Allocations," American Economic Review, 17381 (March 1978).
Miller, J. and J. Thornton, "Effort, Uncertainty, and the New Soviet Incentive System,"
Southern Economic Journal, 43260 (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: 7182 (1974).
Snowberger, V. "The New Soviet Incentive Model: Comment", Bell Journal of Econom
ics, 591600 (Autumn 1977).
The Trenton Times, p. 16 (Sunday, December 15, 1974).
Weitzman, M. "The New Soviet Incentive Model," Bell Journal of Economics, 25157
(Spring 1976).
THEORETICAL ANALYSIS OF LANCHESTERTYPE
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
variablecoefficient Lanchestertype equations of modern warfare with support
ing fires not subject to attrition. It shows that this linear differentialequation
model for combat with supporting fires may be transformed into one without
the supporting fires so that all the previous results for variablecoefficient
Lanchestertype 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 Lanchestertypet models
for developing insights into the dynamics of combat. Militarily realistic computerbased
Lanchestertype models of quite complex military systems have been developed for almost the
entire spectrum of combat operations, from combat between battalionsized units [3], [7] to
theaterlevel 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 Lanchestertype 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 102H5705 Math (funded with MIPR No. ARO 779).
t Socalled 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 morerestrictive
victoryprediction conditions for fixedforceratiobreakpoint battles by considering the force
ratio equation (see also [11]). Taylor and Brown [14] have developed a mathematical theory
for solving variablecoefficient Lanchestertype equations of modern warfare (without support
ing fires) and introduced canonical hyperboliclike 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 forcelevel trajectories for the model without supporting fires.
In the paper at hand we show that the variablecoefficient 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, forceannihilationprediction 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
variablecoefficient model almost as easily and thoroughly as Lanchester's original simple
constantcoefficient model without supporting fires.
2. NOTATION
The symbols that are used in this paper are defined as follows:
a, b, a, /3 = constant attritionrate coefficients,
a it), bit), _ (timedependent attritionrate 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) — timedependent attritionrate coefficients in the transformed
model (5.2); given by (5.3),
Cpit), Spit) = hyperboliclike general Lanchester functions iGLF) which
are linearlyindependent solutions to the transformed
P forcelevel 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) = hyperboliclike GLF which are linearlyindependent solutions
to the X forcelevel 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 "forcelevel" 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* = paritycondition 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 combatintensity parameter,
^■r = kjk b and is called the relativefireeffectiveness parameter,
A * = paritycondition parameter for the model with supporting fires (3.1); defined
by (6.4),
9 = Jab + [(a  /3)/2] 2 .
3. COMBAT MODELLED BY VARIABLECOEFFICIENT LANCHESTERTYPE
EQUATIONS OF MODERN WARFARE WITH SUPPORTING FIRES
We consider combat modelled by the following Lanchestertype 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 timedependent Lanchester attritionrate
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) "aimedfire" combat between two homogeneous forces with "operational" losses [1], [9],
(52) "aimedfire" 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 attritionrate coefficients from weaponsystem
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 attritionrate 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' combatintensity parameter X,
and the relativefireeffectiveness 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 forcelevel 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 variablecoefficient model (3.1) almost as easily and thoroughly
as Lanchester's simple constantcoefficient one.
4. SUMMARY OF RESULTS FOR VARIABLECOEFFICIENT LANCHESTERTYPE
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 variablecoefficient
Lanchestertype 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 attritionrate 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 hyperboliclike 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 forcelevel 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 linearlyindependent solutions (normalized in
an appropriate manner) to a forcelevel equation like (4.3) as general Lanchester functions
(GLF). Thus, for such linear Lanchestertype differential combat equations, the hyperboliclike
GLF play in variablecoefficient combat a role analogous to that which the ordinary hyperbolic
functions play in constantcoefficient 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 hyperboliclike 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 attritionrate 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 attritionrate 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 attritionrate
coefficients). In other words, the hyperboliclike GLF only possess socalled algebraic addition
theorems like the hyperbolic functions do only for a constant ratio of attritionrate 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 hyperboliclike 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 "forcelevel" 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 + TOf
^aO)bU)p0,
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 hyperboliclike GLF C P {t) and S P it) are linearly independent solutions to the P
forcelevel 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, T77T3r('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 forcelevel equation in
terms of the attritionrate 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 hyperboliclike 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 (pXpX) 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 lefthand side of (6.8) and observing that the righthand 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
CONSTANTCOEFFICIENT MODEL
In this section we illustrate the above general mathematical theory of the Lanchestertype
equations (3.1) for the special case in which all the attritionrate 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 constantcoefficient 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
9i^~)
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 fighttothefinish 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 forceannihilationprediction conditions
are the same for these two models in this special case. However, we have given more general
forceannihilationprediction 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 variablecoefficient
model (3.1) for Lanchestertype 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 variablecoefficient
Lanchestertype equations of modern warfare without supporting fires. In many cases of tacti
:al interest (see the timedependent attritionrate coefficients considered in [13 through 16]),
we can now study this variablecoefficient model (even with supporting fires) almost as easily
and thoroughly as Lanchester's classic constantcoefficient 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, 314326 (1962).
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ning," Report No. SRL 2147 TR 702 (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, 319394 (1971).
Coddington, E.A. and N. Levinson, Theory of Ordinary Differential Equations (McGrawHill,
New York, 1955).
Cordesman, A. (Editor), "Developments in Theater Level War Games," Unpublished
materials for C5 Working Group of 35th Military Operations Research Symposium
(1975).
Farrell, R.L. "VECTOR 1 and BATTLE: Two Versions of a HighResolution Ground and
Air Theater Campaign Model," pp. 233241 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
7194 (1956).
Lanchester, F.W. "Aircraft in Warfare: The Dawn of the Fourth Air— No. V., The Princi
ple of Concentration," Engineering 98, 422423 (1914) reprinted on pp. 21382148 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 LanchesterType Models of Warfare," Naval
Research Logistics Quarterly 23, 345352 (1976).
Taylor, J.G., "Optimal Commitment of Forces in Some LanchesterType Combat Models,"
Operations Research 27, 96114 (1979).
Taylor, J.G., "Prediction of Zero Points of Solutions to LanchesterType Differential Com
bat Equations for Modern Warfare," SI AM J. Applied Mathematics 36, 438456 (1979).
Taylor, J.G. and G.G. Brown, "Canonical Methods in the Solution of VariableCoefficient
LanchesterType Equations of Modern Warfare," Operations Research 24, 4469
(1976).
Taylor, J.G. and G.G. Brown, "Numerical Determination of the ParityCondition Parame
ter for LanchesterType Equations of Modern Warfare," Computers and Operations
Research 5, 227242 (1978).
Taylor, J.G. and C. Comstock, "Force Annihilation Conditions for VariableCoefficient
LanchesterType Equations of Modern Warfare," Naval Research Logistics Quarterly
24, 349371 (1977).
L ANCHESTER COMBAT WITH SUPPORTING FIRES 1 2 1
[17] Taylor, J.G. and S.H. Parry, "ForceRatio Considerations for Some LanchesterType
Models of Warfare," Operations Research 23, 522533 (1975).
[18] Weiss, H.K. "Some Differential Games of Tactical Interest and the Value of a Supporting
Weapon System," Operations Research 7, 180196 (1959).
ON SINGLEMACHINE 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 jobshop 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
continuoustime problems by considering appropriately chosen sequences of discretetime 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 € 7Mthen 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 JM has the same precedence relation to every job in M.
Hence a job in JM 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)]}
yKi)/!,
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 JM:
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 JM (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 JM, 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 JM 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 7Mup 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 subordering 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
11
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 subordering 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
yl 71
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, 358364 (1965).
[2] Glazebrook, K.D., "Stochastic Scheduling with Order Constraints," International Journal of
System Science 7, 657666 (1976).
[3] Glazebrook, K.D. and J.C. Gittins "On SingleMachine 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, 195205, (1977).
[5] Sidney, J.B., "Decomposition Algorithms for SingleMachine Sequencing with Precedence
Relations and Deferral Costs," Operations Research 23, 283298 (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 steadystate) 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 steadystate case is al
most exclusively considered.
Although analyses of terminating simulations are considerably easier than
are those of steadystate 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 steadystate measure of
This research was supported by the Office of Naval Research under contract N0001476C0403 (NR 047145) and the
.rmy Research Office under contract DAAG2975C0024.
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 steadystate 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 steadystate 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 steadystate 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 steadystate 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/miV(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 steadystate simulation of [D,,i ^ 1} for the M/M/l queue would be to
estimate the steadystate 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 steadystate simulation of [E t ,i ^ 1} would be to estimate the steadystate
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, £"(rall components are new). If we assume that the system is
not repaired when it fails, then a steadystate 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)/m7V(0) =
;=1
■or a steadystate simulation, the objective would be to estimate the steadystate 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 steadystate simulations are
nportant; almost every paper written on the analysis of simulation output data deals with the
eadystate case. This may be a carryover from mathematical queueing theory where only a
eadystate 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 steadystate 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, steadystate meas
ures of performance will probably not exist.
One is often interested in studying the transient behavior of a system even if steadystate
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 steadystate 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 steadystate 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(la)% (0 < a < 1)
yi
c.i. for/Lt = E{X) is given by
(1) jf(ll) ± ^l,la/ 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 £(Fall 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(lp)/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(la)% 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.ia/2 y/s 2 (n)/n,
and let
(3)
N r \(y,a) = min
n: n > n , s z (n) > 0, ' <
\X(n)\
(Note that N r A (y, a), which is the required number of replications, is a r.v.). Then use
(4) / r , 1 (y,a)= [X(N rA (y,a))b rA (N rA (y,a),a),X(N rA (y,a))+8 rA (N rA (y r a),a)]
as an approximate 100(la)% c.i. for fx. It easily follows from (3) and (4) that I rA (y,a)
satisfies the criterion given by (2). Furthermore, using an argument similar to the one
employed by Lavenberg and Sauer in the context of the regenerative method for steadystate
simulations, we have been able to prove the following theorem.
THEOREM 1. H> ^ and < o 2 < oo, then lim P{fi <E I r] (y,a)} = 1  a.
The objective of the second procedure, which is due to Nadas [11], is to construct a c.i.
such that
(5)
Let
\X(n)n\ < y \fi\ for 0< y < 1.
vHn)
l + £ [X, X(n)Y
In = (1/w) + (n \)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(la)% 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 noninfinitesimal ■)
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
VnXAall)
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 twostage 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 steadystate 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 realworld 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 steadystate case there is still
not a totally acceptable procedure even for the relatively simple problem of constructing a c.i.
for a steadystate 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 realworld 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 FixedWidth Sequential
Confidence Intervals for the Mean," Annals of Mathematical Statistics 36, 457462
(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, 7185 (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, 2544 (1966).
Iglehart, D.L., "The Regenerative Method for Simulation Analysis," Technical Report No.
8620, Control Analysis Corporation, Palo Alto, California (1975).
Kleijnen, J.P.C., "The Statistical Design and Analysis of Digital Simulation: A Survey,"
Management Informatics 1, 5766 (1972).
Lavenberg, S.S. and C.H. Sauer, "Sequential Stopping Rules for the Regenerative Method
of Simulation," IBM Journal of Research and Development 21, 545558 (1977).
Law, A.M., "Confidence Intervals in Discrete Event Simulation: A Comparison of Replica
tion and Batch Means," Naval Research Logistics Quarterly 24, 667678 (1977).
Law, A.M. and W.D. Kelton, "Confidence Intervals for SteadyState Simulations, I: A Sur
vey of Fixed Sample Size Procedures," Technical Report 785, Dept. of Industrial
Engineering, University of Wisconsin (1978).
10] Law, A.M. and W.D. Kelton, "Confidence Intervals for SteadyState Simulations, II: A
Survey of Sequential Procedures," Technical Report 786, 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, 667671 (1969).
[I] Starr, N., "The Performance of a Sequential Procedure for the FixedWidth Interval Esti
mation of the Mean," Annals of Mathematical Statistics 37, 3650 (1966).
Bl Thomas, M.A., "A Simple Sequential Procedure for Sampling Termination in Simulation
Investigations," Journal of Statist. Comput. Simul. 3, 161164 (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 onedimensional 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 CarnegieMellon
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.Z1, 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 KuhnTucker 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 nonlinear objective func
tions. Dashed lines are unique to nonlinear problem. Crossed
lines are unique to linear problem. Solid lines are mutual.
Thicklined 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 nonbasic
^
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) Matrixminimum: here the cell that has the smallest reduced cost among all cells i
chosen,
(ii) Rowminimum: 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) Lotminimum: 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 rowminimum 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 onedimensional gol
den section search (Block B.7). Although many possible onedimensional 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 prespecified 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
NonBasic 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.
793814 (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 SimplexLike Algorithm for the Continuous Modular
Design Problem," Operations Research 25, pp. 788807 (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. 712726 (Oct. 1972).
[7] Srinivasan, V. and G.L. Thompson, "BenefitCost Analysis of Coding Techniques for the
Primal Transportation Algorithm," Journal of the Association for Computing Machines 20,
pp.194213, (April 1973).
[8] Zangwill, W., Nonlinear Programming: A Unified Approach (PrenticeHall, 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 costeffective
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 ClarkeWright 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^ick+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 (yk)+ £ 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 + mk)
fc0
k=y n +m+\
Subtracting (3) from the righthand side of (2) results in
± P(X n = k)[V n _ x (y n k) V n _ x (y n + mk)}
k=0
y„+m
+ £ />(*„ /r) [2 </„.„+ K„_!(c k+y n ) +  V n _ x {y n + m  k)\
ky 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), 113148 (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, 252259 (1978).
[I Golden B. and J. Yee, "A Framework for Probabilistic Vehicle Routing," American Institute
of Industrial Engineers Transactions, 77(2), 109112 (1979).
['.I Tillman, F., "The Multiple Terminal Delivery Problem with Probabilistic Demands," Tran
sportation Science, 3(3), 192204 (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. 36566] 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 waitandsee 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 Sspace), 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. 130131]. 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
130131]. 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, 947954 (1970).
[2] D.P. Baron, "Information in TwoStage Programming Under Uncertainty," Naval Research
Logistics Quarterly 18, 169176 (1971).
[3] G.B. Dantzig, "Linear Programming Under Uncertainty," Management Science 1, 197206
(1955).
[4] C.C. Huang, I. Vertinsky, and W.T. Ziemba, "Sharp Bounds on the Value of Perfect Infor
mation," Operations Research 25, 128139 (1977).
[5] A. Madansky, "Inequalities for Stochastic Linear Programming Problems," Management Sci
ence 6, 197204 (1960).
[6] W.T. Ziemba and J.E. Butterworth, "Bounds on the Value of Information in Uncertain
Decision Problems," Stochastics 1, 361378 (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, yi, 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 71
d)))lz 2 be calculated for all j 6JV . Let the additional constraint to be appended be
(2) X P/*/ > ?•
yi
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 =
yy 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. 181186 (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. 221231, 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. 341356 (1973).
[4] Hadley, G., "Linear Programming," (Addison Wesley, Reading, Massachusetts, 1962).
[5] Swarup, K., "Linear Fractional Functional Programming," Operations Research, 13, pp.
10291036 (1965).
[6] Swarup, K., "Some Aspects of Linear Fractional Functional Programming," Australian Jour
nal of Statistics, Vol. 7, pp. 90104 (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 Englishlanguage 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.
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must meet the following requirements:
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(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
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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 1012, 1980 in Colorado Springs, Colorado.
INFORMATION FOR CONTRIBUTORS
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NAVAL RESEARCH
LOGISTICS
QUARTERLY
MARCH 198C
VOL. 27, NO.
NAVSO P1278
CONTENTS
ARTICLES
I
Analyzing Availability Using Transfer
Function Models and Cross Spectral Analysis
Optimal MaintenanceRepair 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 LanchesterType
Combat Between Two Homogeneous
Forces With Supporting Fires
On SingleMachine 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