the use of cutsets in monte carlo analysis of stochastic networks

Mathematics and Computers in Simulation XXI (1979) 376-384

North-Holland Publishing Company

THE USE OF CUTSETS IN MONTE CARLO ANALYSIS OF STOCHASTIC NETWORKS

C.E. S[C;AL*, A.A.B. PRITSKER and J.J. SOLBERG

School of Industrial En,~$mwin~, Purdue University. West I.afa,vrttr. l,Y 4790 7, U.S.A

c’lonte Carlo methods utilizing a new network concept, Uniformly Direcied 0itsets (UL'Cs), are presented for analyzing directed, acyclic networks with probabilistic arc durations. DE procedures involve sampling arc values for arcs not on a UDC and utilizing knowi? probnbilil~~ in"formation for arcs on a UK. This approach results in less sampling effort and Zess associated variance than a straightforward simulation approach. i! proo_f o>f this variance reduction is offered. The i:rocedures provide estimates for project completion time disLribu- tions, criticality indices, minimum time distributions and path optimalitjy iiziices. All of these network per,formance mcaswes are usefu2 to decision makers in project planning. appli- cation areas include .~~?-t~~pe network planning, equipment replacement analysis, reliability modeling, stochastic dynamic progranrming problems and maximal flow problems.

1. INTRODUCTION

Consider a directed, acyclic network with a single source and sink. Let each arc have an associated arc duration, Xi. All Xi's are assumed independent, each with a known cumulative distribution function (cdf), FX.(xi). Let each network path m, m=l,...,M, have'an associated WV-I duration, Y,,,, which is the sum of the arc durations of its constituent arcs.

When such networks are used to model project planning strategies, it is of interest to compute the distribution of the project completion Line. The project completion distribution aids the decision maker in planning to meet project due dates and in identifying potential project delays. Also of interest is the path criticality index, that is, the probability a given path duration is greater than or ecual to all other path durations. Criticality indices allow a project planner to identify project activities that need to be monitored carefully in order to avoid delayins the entire project. In addition, these indices indicate which activities should be expedited if the entire project is to be expedited.

In shortest route analysis, each network path represents an alternative approach to a project. The objective is to minimize ihe time to complete the project. If all approaches are to be pur- sueti, it is of interest to compute the distribution of minimum path durations. If only one alternative is to be selected, however, the path cptimality index is of interest. A path optimality index is the probability a given path has the shortest path duration [ll,lZ].

Analvtic methods for determinina a network's project completicn time distribition often involve evaluation of multivariate integrals [2,7,8,10]. Fionte Carlo methods can be used

to evaluate these integrals. Alternatively, a simulation of the network could be performed. However, Konte Carlo methods result in less variance associated with the estimate than the simulation approach. In addition, the sampling effort is reduced. Burt and Garman's conditional Monte Carlo procedure [l] illustrates these points in the evaluation of the multivariate integrals derived by Martin [B]. The technique requires the identification of arcs belonging to exactly one network patli (called path unique arcs). In Burt and Garman's procedure, only non- unique arcs are sampled.

This paper extends the concent of conditional Monte Carlo by utilizing cutsct arcs in rlact of unique arcs. In particular, the special orooerties of arcs in Uniformlv Directed Lutsets are'exploited to derive multivariate integrals of lower dimensionality. Hence, in the Monie Carlo analysis of these integrals fewer arc values need to be sampled and more known probability iriforna- tion is utilized. This results in a rei!,ction in both the sampling effort and the variance of ihe estimates.

Previous work with conditicnal tlonte Carlo dealt exclusively with project completion time distributions. This paper presents new lmetl-lads for estimating not only completion time distributions but also criticality indices, minimum time distributions and path optimality indices. The first two performance measures are useful in project planning or PERT-type network analysis. The latter two network performance Il;easut-es arc use;ul in stochastic shortest route analysis [ll ,121.

2. PAST RESEARCH

Elmaqhraby [z] provides an excellent review of analytic and llonte Carlo approaches to determining project completion time distributions and

* Currently at the Pritzker School of Medicine, University of Chicago.

376

criticality indices. Van Slyke [13] was the first to develop a straightforward simulation :pproach to analyzing networks in terms of completion time distributions and criticality indices. Martin L8] developed an analytic approach to these two network measures which involved multivariate integrals. Hartley and Wortham [7] and Ringer [lOI employed similar analytic procedures to derive integrals for completion time distributions. The Monte Carlo method [5,6] can be used to solve such integrals. Burt and Garman [1] were the first to apply the Monte Carlo approach to ttre case of project completion time distributions. They called their approach conditional Monte Carlo and reported significant variance reduction over straightforward simulation. Garman [4] offered an alternative approach to Burt and Garman's procedure but did not address the subject of criticality indices.

Little research has been reported on the stochastic shortest route problem. Frank [3] provided an analytic approach and prescribed straightforward simulation for the distribution of the minimum time through a network. The authors have recently considered analyzing stochastic shortest route problems from the standpoint cf optimality indices Lll,lZ]. For further backsround, we refer to Elmaghraby and Sigal [2,11,12].

3. CONDITIONAL MONTE CARLO AND THE PROJECT COMPLETION TIME DISTRIBUTION

The distribution of the project completion time,

YC, is given by

Fr(Yc <= t) = Pr(Yl 2 t, Y2 <- t,...,YM <= t). (1)

The formal evaluation of Eq. 1 is difficult because the random variables Y,,,m=l,...,M are not independent for most networks. This statistical dependence results from the fact that arcs can belcng tc more than one path in the network.

Burt ancl Garman [l] introduced conditional Monte Carlo to estimate the cdf of the completion time of an acyclic network. The procedure requires identifying a set of "unique" arcs that are on exactly one network path. By conditioning on arcs that are not unique, that is by treating them as constants, all path durations c'-; be treated as independent random variables This permits the evaluation of Eq. 1 as a product of cdfs of the unique arcs.

The procedure is Ibest illustrated by example. Consider the network in Figure 1. Let Xi represent the arc ouration for arc ei. Also let Yl = Xl + X4, Y2 = Xl + x3 + x5, Y3 = x2 + x5. The unique arcs are e2, e3 and e4.

From the Theorem of Total Probability [9], we write

Py (t) = Pr(Y, <- t) C

(2)

= I,/ Pr(Yc <= tlxl ,x5)dF~, (xl )dTX5(x5).

Lower case notation is used to denote fixed values for random variables.

el e4

+

e3

e2 5

Figure 1. Network for illustrating conditional Monte Carlo

To illustrate how the Monte Carlo Method (also called nrobabilistic quadrature [5,6]) evaluates this expression, note that for any fixed values of (x1,x5), the three path durations of the network are independent. Hence for any given t

Pr(Yc;tlxl,x5)=Pr(Yllt,Y2<=t,Y3<=tixl,x5

= Pr(X42 t-xl).Pr(Xj$-xl-x5).Pr(X22 t-x5)

= Fx4(t-xl).FX3(t-Xl-X5)+x2(t-x5) (3)

After taking N samples of the pair (X1,X5), an estimate FY,(t) of the project completion distribution function is given by

N FX

i, (t)= Z p4

(t-x+X (t-xi-x+X (t-x;) 3 2

N (4) C i=l

where superscripts are used to denote sample values. In general, one samples from (conditions upon) non-unique arcs and evaluates the path completion time distribution conditioned on these non-unique arcs. This results in a product of cdfs of unique arcs.

It should be noted that the unique arc approach assumes that no two arcs of the network are in series. Two arcs are said to be in "series" if the origin node of one arc is the terminal node of the other and that node has no other inputs or outputs. This requirement is not restricting, Ihowever, in that all arcs in series can be replaced by a single arc with an equivalent arc duration determined by the convolution of the arc times in series [8].

In this paper, we show that the use of network cutset arcs in place of unique arcs decreases the required sampling and increases the number of cdfs utilized in the estimator.

4. UNIFORMLY DIRECTED CUTSETS (UDCs)

As long as the chosen set of arcs are path independent (meaning simply that no two arcs in the set are on the same path), one can resolve the statistical dependence among paths even if one or more arcs in the selected set is on more than one path. Since a network may contain many path independent sets of arcs we specify that we

378 C.6: S&l et al. / Monte Carlo arzal.vsis of’stochastic rretworks

would like a maximum such set. This would mean fewer arcs are held fixed, more probability information is utilized and less sampling is ;:erformed. We let x denote a path independent set of arcs of a network and we let X* denote the set x of the network which has the most elements (arcs). Sigal [ll,lZ] has proved that X* is a cutset of the network. A cutset is used here to mean a set of arcs connecting a set of nodes, W, which contains the network source, with its complement, w in the set of network nodes, which contains the network sink. In the reference, it is further proved that X* is a cutset with the special prcperty that all arcs in X* are directed from W to W. A cutset with this proper- ty is called a Uniformly Directed Cutset (UDC).

Thus if one utilizes the special UDC, x*, in the conditional Monte Carlo formulation in place of the set of unique arcs, one is guaranteed of having the least arcs to condition upon and an improvement over the unique arc approach will result.

4.1 The Use of UDCs in Determininq Preiect Completion Time Distributions The new procedure will be introduced through an example. Consider the network in Figure 2.

Figure 2. Network for illustrating the Cutset Approach

Arcs e3 and e7 are the only unique arcs. In Burt and Garman's procedure, the remaining arcs are conditioned upon by drawing sample values for Xl, X2, X4, X5, X6 and X8. Using the proposed cutset approach, the UDC with the most arcs is X* = {e2,e3,e4,e71. If we conc'ition upon non-UDC arcs, we can treat each network path as an independent random variable and draw samples from only four arcs.

When using the cutset approach, consideration must be given to the case when a UDC arc is on more than one path. We will illustrate how this is handled for the example network. Let Yl = Xl + X7, Y2 = Xl + X4 f X5, Y - Xl + X,1 + X6 +

x8~ Y4 = x2 + x5, Y5 = x2 + xi r X8, Y6 = x3 + X8., By conditioning upon all non-UDC arcs, the proJect completion time is given by.

Fyc(t) = jj\j Pr(Yc <= tlxl.X5’X6’X8) (5) d~Xl(xl)dFX5(x5)d~X6(x6)d~X8[Xgl

The conditional cdf in the integrand can be expressed as

Pr(Yc ~tlxl,x5,x6,x8) = Pr(X7zt-xl,X41t-x1-x5,

x4<, t-xl-x6-x8,X2~t-x5,X2' t-X6-X8,X3$ t-x*)

= Fx (t-xl).FX (h ).Fx (h )*Fx (t-x8) 4l z2 3

(6) 7

where a minimum value is used when evaluating the joint probability associated with the same random variable, that is,

hl = min(t-x1-x5, t-X1-X6-X8)

h2 = min(t-x5, t-x6-x8)

Note that all terms in Eq. 6 can be evaluated for sample valves of non-UDC arcs and $ given value for t. The Monte Carlo estimate, Fy,(t), of the project completion time cdf is given by

,J FX (t-x;).FX (hl).FX (h2)*FX (t-x;)

i, (t)= 1 >___-_4__1____L__ C i=l

(7)

In Eq. 7, the probability information of each arc in the UDC is utilized and sampling is required only for non-UDC arcs. This represents an iln- provement over Burt and Garman's procedure which for this example requires sampling for values of X2 and X4 in addition to Xl, X5, X6, X8. The UDC approach utilizes the information contained in Fx2(t) and Fx4(t). This results in less sampling ana a reduced variance is to be expected.

As mentioned previously, there is often more than one UDC associated with a given network. Select- ing the UDC, x*, with the greatest number of arcs will minimize the number of arcs required for sampling. Sigal [ll] presents four approaches for identifying x*: a mathematical programming approach, a path-arc matrix procedure, a dual network approach, and a node-arc incidence matrix method.

The Monte Carlo method utilizing UDCs for determining the project completion time distribution can now be stated in general as shown in Table 1.

Table 1. Algorithm for Estimating a Pro&t Completion Time Distribution

INITIALIZATION. Determine the sample size N and set i=l.

STEP 1.

STEP 2.

STEP 3. STEP 4.

Identify x*, the UDC with the most arcs. Define r* to be the complement of x* in the set of network arcs. Select a base point t of the project completion time distribution, Fy,!t). Draw random samples for all arcs in y*. Define f'= n Fx (h')

uEx* u

where hi=min(t- x

k FPlu xi,...,t- z xi,

kEP k#uQU

path containing UDC

STEP 5. If i=N, go to Step 6; otherwise, set i=i+l and go to Step 3.

STEP 6. Commute the estimate of Fyc(t) as N

i, (t) = 1 z fi . C N i=l

STEP 7. If more base points are Step 2.

5. CRITICALITY INDICES

desired, go to

The criticality index, CQ, for path a. is the probability that path duration Ye is greater than or equal to all other path durations, that is,

C, = Pr(Y,s Yl,...,Y,z Y,,...,Y,z YM (8)

m=l ,...,M; mfa. . In this section we develop a Monte Carlo procedure for estimatinq C,. We will explain the procedure without theluse of multivariate integrals although the approach is similar to the one presented for obtaining the project completion time distribution. Additional mathematical details relating to criticality indices have not been included due to space limitations.

In estimating C, for path p. we introduce the UDC, x*, and sample values for arcs in p and also for the UDC arc j on path e. For each set of sample values we compute a conditional criticality index for path e, KQ. by treating the UDC arc j and the arcs in y* as constants.

K, = Pr(Y, >= Yl,. . . ,yQ 5 Y m"" YYl>= YmIY*it'Xj) (9)

m=l,.. .,M; mfa. . By this conditioning, the path duration y

"(

is a constant since there is only one UDC arc arc j) on each path [11,12] and ye is the sum of x' and values of arcs in y*. From the theorem of $ otal probability the criticality index can be computed by summing Kp over all possible fixed values for xj and for arcs in r*. Each term of the sum must be weighted by the probability the corresponding set of fixed values will occur. Hence, to F$timate Ct. we draw N samples of xj and arcs in X , evaluate Kk for each pf the N set of samples, and compute an estimate, C,, for C, as ihe average of the N values of Ke.

Our problem thus reduces to one of eva!uating Eq. 9. This is straightforward when each arc on the UDC, x*, is on exactly one path. In this case, the random variables Y, are independent for fixed values of arcs in i* and we write

K,= n WYm<=yLIr*.xj) . mfa.

(10)

Now, each network path has exactly one arc in any given UDC [11,12]. Therefore, by fixing arcs in r* each path duration Y, can be represented as

ym = x, + z k fPath m

Xk (11)

kfu where arc u belongs to path m and UDC, x*. Thus, we can rewrite Eq.10 as

K,= n

uY P% 9.

Pr(X,: ya - k CPath &taining u 'k)

II uEx*

FX,(Ya. - x k FPath containing u

'k)

ufpath a. (12)

Hence, K, and, therefore, Cl can be computed from a product of cdfs of arcs in x* evaluated for sampled values for Xj and for arcs in y*.

We assumed above that each arc on the UDC, Y*, was on exactly one path. We now address the situation where one or more UDC arcs are on more than one path. The resulting formulation will be a modification of Eq. 12. For a UDC arc u that is on more than one path, we consider two cases: (1) arc u belonqinq to oath p. for which Co is computed; and (2) arc u not belonging to Path e. Taking case 2 first, let Poll denote the qth path containing arc u, q=l,....Q: Then the term containing the cdf of arc u in Eq. 12 must be modified to FXu(a) where

a=min(ye- z xk,...,ye- X (13) k EPlu

xk) .

k#u kE pQu kfu

For case 1, where arc u belongs to Q-l paths (Q> 1) other than path 1, an extra term in the cdf product of Eq. 12 is required to represent the probability that these path durations are less than or equal to the constant path duration ye. In this case, arc u and arc j are the same arc. Hence, all values on these paths are fixed and the path time is a constant. Therefore, the probability of ye being larger than these Q-l paths is either 0 or 1 depending on the particular sample values.

We can now generate Eq. 12 as follows

K,=b[ II Fx (a,)1 UEX" u

u $'Path a.

(14)

where a is defined as in Eq. 13 and

0 if the UDC arc which is on path e, say arc u, is on more than one

b= path a&y, is less than all other (15) durations of paths containing arc u for the given fixed values

1 otherwise .

The Monte Carlo algorithm for estimating criticality indices is given in Table 2.

Table 2. Algorithm for Estimating Criticality Indices

INITIALIZATION. Specify a. Determine N. Set i=l.

380

STEP 1.

STEP 2.

STEP 3.

STEP 4.

STEP 5.

Identify x*, the UDC with the most arcs. Let k* represent the non-UDC arcs. Draw random samoles for all arcs in U* and for the UDC'arc j on path L. " Compute bi as defined in Eq. 15. If bi=D, set fi=O and go to Step 4. Other- wise, compute

fi= n

ucx* FX (a')

U

LI f Path n.

where ai is given by Eq. 13. If i=N, go to Step 5. Otherwise set i=i+l and go to Step 2. Compute the estimate of C, as

6. THE STOCHASTIC SHORTEST ROUTE PROBLEM

In Stochastic Shortest Route Analysis, one is concerned with the distribution of the minimum time through the network and also the probability a given path is shorter than all other paths, that is, a path optimality index [11,12].

Let 2 denote the minimum time through the network, then

Z = min(Y1,...,YM) . (1’5)

Since Z is a minimum of a set of random variables,

Pr(Z<t)=Pr(Y,<=t or Y21t or... YM zt) . (17)

This is equivalent to

Pr(Z<=t)= 1 - Pr(Y1> t,Y2> t,...,YM> t) . (18)

As before, we determine the UDC, x*, and condition upon non-UDC arcs, y*. We then rewrite Eq. 18 as

Pr(Zlt)= 1 -~...~Pr(Y,>t,...,YM>tl~)dFrr* . (19)

Letting GY(.) and Gxu(.) denote path and arc complementary cdfs respectively, we have

M

Pr(Zlt)=l -!.*.I i Gy (tlz*) dFrr* m=l m

= 1 _I...] n GX (v) dFy* ucrr* u

where

v = max(t - z

k EPlu

xk,...,t - X

k EPQu

xk)

k#u kfu

",a: Pqu ;s the q th path to which arc u belongs, ,...I .

The Monte Carlo Method given for the project

completion time distribution can be used to determine the minimum time.distribution by. defining f1 in Step 4 as fi = 1 - II *GXu(vl) where vi is defined as in Eq. 20: EX

Similarly, the development of criticality indices can be modified to produce an expression-for RQ, the optimality index for path L. Since RP is the probability path 9. is as short or shorterthan all other paths,

Rp = Pr(Y,~Y,,...,Y~~Y,,...,Y~~YM) (21)

m=l ,...,M;m#a

The development for Re follows the one given for Cl by reversing inequality signs, replacing path and arc cdfs with complementary cdfs, and using maximization operators instead of minimization operators. Therefore,

Re=_/...j e II Gx (d) dFx,(xj)dFT (22) UEX* u J

u qPath L

where

d = max(yi - z

k ?Plu

xk,...,y, - z xk) kEP

Qu kfu k#u

and

0 if the UDC arc which is on path t, say arc u, is on more than one path and yk is areater than all other durations <

e = of paths containing arc u for the given fixed values

1 otherwise . The Monte Carlo Method for estimating optimality indices involves changing Step 3 of the algorithm given in Table 2 by defining

fi

where e' and

= e’ II

u Ex* GX (di)

U u +'Path P.

di are defined in Eq. 22.

7. VARIANCE REDUCTION OVER STRAIGHTFORWARD SIMULATION

The procedures presented will have less associated variance than that of straightforward simulation. The reason for this is best shown by considering the case of criticality indices although the reasoning is the same for all the above procedures. The straightforward simulation approach would be to draw samples of each arc time, compute and record the critical path and repeat the procedure N times. The number of times path a. is the critical path, divided by N, is an estimate of path e's criticality index. In effect, this involves sampling from a binomial process. The variance, o'$, associated

v⁢r;;timating C, is C,(l-C,)/N=(C,-C$_)/N. In 3 whenever one can formulate a problem as

a multiple integral and solve it with the Monte Carlo Method, the associated variance will be less than that of a straightforward simulation of the problem involving sampling from a binomial distribution [6].

The variance, oh, associated with estimating an integral of the form CL =i f dA where A is the

region of integration is [5] c$,=f [afzdA-

(if dA)2] where N is the number of sample evalua- A

tions of the integrand f. To show CI; is always we show that the ditference,

This difference is

O's - 'Ji= [Ce-C;]/N- [/f*dA- (if dA)*]/N . A

(24)

Since C, = if dA, we have

cl’s - $,=& [j,f dA- 'if dA)' -df2dAt(,f dA)z]

=a [JAf dA- if7 dA] = k ;f(l-f) dA (25)

Since f is a product of cdfs, the integrand in the last term of Eq. 25 is positive (except in degenerate ca;es where arc times are constant). Hence, US - oM > 0.

8. COMPUTATIONAL RESULTS

In this section we present computational results for the project comoletion time distribution, F(t) for the network in Figure 2. For this example, each arc time Xi, i=1,8 is assumed to be exponentially distributed with a

p.d.f. of the form fX,(xi)=xie-xixi. The ex-

pected values (E(Xi) L 1) of each arc time Xi, hl

i=1,8 are 10, 15, 18, 6, 7, 3, 10, and 2 respectively.

Three methods for computing F(t) are compared: a straightforward simulation approach, Burt and Garman's procedure and the UDC method. In the straightforward simulation approach each arc time is sampled, each path time computed and the longest path time is determined and compared with a specified base point t. This process is repeated N times. The fraction of times the longest path is less than t is the estimate of F(t). The variance of this estimator is computed with the knowledge that this is a binomial sampling process. The Burt and Garman method was proqrammed as described in reference [l]. The unique arcs in the network of Figure 2 are arcs 3 and 7. Therefore, the non-unique arcs 1, 2, 4, 5, 6, and 8 were sampled. The UDC procedure detailed in Table 1 was programmed for the UDC defined by arcs 2, 3, 4, and 7. The procedure calls for sampling non-UDC arcs 1, 5, 6, and 8.

Table 3 presents results for each of the three methods for the basepoint t=30. To observe the effect of varying the sample size, N, each procedure was executed for different values of N ranging from 100 to 10000 as shown. For each procedure and for each value of N (column (l)), the followinq quantities were computed: the estimate i(tj and F(t) (column(2)), the variance of the estimate divided by the sample size, N, that is. the variance of the samole mean (column(3)), and the execution time of the procedure (column(5)). Also computed is the standard deviation of the variance of the sample mean. This quantity, recorded in column 4, represents the standard error of the estimate F(t) and as such is useful in computing confidence'intervals for each result. For example, the 95% confidence interval for the straiahtforward simulation estimate of F(30) after 700 network simulations is defined by the endpoints .2944 and .4857. qiven by the expression .39 i 1.96* .0488. Inthis way the standard errors in column 4 provide an indication of the precision of each estimate.

Table 4 gives the variance reduction computations for the three procedures for each sample size. For example, for a sample size of 100, the straightforward method (SS) has an associated variance of 1.87 times larger than that of Burt and Garman's (BG) procedure. This ratio varies little with increasing sample size N.

The UDC method has approximately one-fifth of the variance of straightforward simulation. The UDC procedure resulted in an associated variance of nearly one-third that of the Burt and Garman procedure.

To speak of relative efficiencies we need to consider the differences in computing times associated with each orocedure. To comoare variance reduction techniques, Hammersley and Handscomb [6] and Elmaqhraby r21 use an improvement ratio

"1 Ll IR=rxx

2 2

where Vi/V:, is the ratio of variances of method 1 and 2 and tl/L2 is the ratio of computing times. Table 4 gives the improvement ratios for each of the comparisons described above. The reader will note that these ratios do not differ drastically from the ratio of variances. This follows from the fact that there is little variation in computer time among procedures as revealed by column (5) of Table 3. Althouqb both the UDC method and the Burt and Garman' method involve a greater number of computations to be performed on the sampled arc times, they both require less sampling. For the example network the difference in computation and samplinq balances out so that each procedure requires approximately the same computer time. Of course the degree to which this will be the case in general varies with network configuration.

Table 5 gives results similar to those in Table

382

Table 3. Comparison of Methods for Estimating Project Completion Time Distribution, F(t), where t=30 for Network of Figure 2.

Method

(1) (2) (3) (4) (5)

Sample Size,N F(t)

VAR[[(t)]

Straightforward 100 .3900 .0023790 .0488 0.18 Simulation 500 .3840 .0004731 .0218 0.42

(SS) 1000 .3880 .0002375 .0154 0.74 5000 3844

:3868 .0000473 .0069 3.26

10000 .0000237 .0049 6.40

100 .3672 .0012745 .0357 0.17 Burt & Garman 500 .3672 .0002530 .0159 0.38 Procedure [l] 1000 .3698 .0001268 .0113 0.65

(BG) 5000 .3857 .0000249 .0050 2.91 10000 .3873 .0000124 .0035 5.93

100 .3928 .0004584 .0214 0.18 UDC 500 .3859 .0000881 .0094 0.40

Procedure 1000 .3910 .0000436 .0066 0.69 (UDC) 5000 .3857 .0000090 .0030 2.98

10000 .3879 .0000045 .0021 5.68

*Execution time on the DEC 20 system, University of Chicago

Table 4. Variance Reduction (VR)* and Improvement Ratio (IR)** Computations Based on Results from Table 3

Methods Compared Ratios 100

Sample Size, N

500 1000 5000 10000

SS:BG VR 1.87 1.87 1.87 1.90 1.91

IR 1.98 2.07 2.13 2.13 2.06

SS:UDC VR 5.19 5.37 5.45 5.26 5.27

IR 5.19 5.64 5.84 5.75 5.94

BG:UDC VR 2.78 2.87 2.91 2.77 2.76

IR 2.63 2.73 2.74 2.70 2.88

*Ratio of respective entries in column (3), Table 3.

**Variance ratio multiplied by ratio of respective entries in column (5), Table 3.

C.E. SiRal ct 01. / Monte Carlo anal_vsis ofstochastic nctl~~orks 383

Table 5. Comparison of Methods for Estimating F(t) for Network of Figure 2 for Various Base Points

(1) (2) (3) (4) (5) (6)

Method Base Point, t Sample Size, N F(t) Var[i(t)] J

Var[i(t)] CPU Time N N (Sec.)

15 100 5000

30 100 Straightforward 5000

Simulation 40 100 (SS) 5000

50 100 5000

70 100 5000

.0400 .0003840

.0402 .0000077 .0196 0.16 .0028 3.16 .0488 0.18 .0069 3.26 .0499 0.17 .0068 3.23 .0466 0.20 .0056 3.25 .0313 0.17 .0033 3.26

.3900 .0023790 .0000473 .0024910 .0000461 .0021760 .0000317 .0009790 .0000107

.3844

.5300

.6392

.6800

.8028

.8900

.9432

Burt & Garman Procedure [l]

(BG)

15 100 5000

30 100 5000

40 100 5000

50 100 5000

70 100 5000

.0408 .0001282 .0113

.0412 .0000026 .0016 0.15 2.48 0.17 2.91 0.17 2.97 0.20 3.07 0.18 3.12

0.16 2.35 0.18 2.98 0.17 2.99 0.20 3.06 0.18 3.08

.3672

.3857

.6247

.6382

.7718

.8018

.9119

.9438

.0012745

.0000249

.0013464

.0000250

.0010197

.0000170

.0005379

.0000053

.0357

.0050

.0367

.0050

.0319

.0041

.0232

.0023

15 100 5000

.0444

.0410

.3928

.3857

.6179

.6362 .0000087 .0029

.7655 .0005372 .0232

.0000382 .0062

.0000007 .0008

.0004584 .0214

.0000090 .0030

.0005959 .0244

30 100 UDC 5000

'rocedure 40 100 (UDC) 5000

50 100 5000

70 100 5000

.8015

.9286

.9459

.0000053 .0023 .0129 .OOlO

.0001674

.0000011

Table 6. Variance Reduction (VR)* and Improvement Ratio (IR)** Computations Based on Results from Table 5.

Base Point, t 15 30 40 50 70

Sample Size, N 100 5000 100 5000 100 5000 100 5000 100 5000 Average

SS:BG VR 3.00 2.96 1.87 1.90 1.85 1.84 2.13 1.86 1.82 2.02 2.13 IR 3.20 3.77 1.98 2.13 1.85 2.00 2.13 1.97 1.72 2.11 2.26

SS:UDC VR 10.05 11.00 5.19 5.26 4.18 5.30 4.05 5.98 5.85 9.73 6.66 IR 10.05 14.79 5.19 5.75 4.18 5.73 4.05 6.35 5.53 10.30 7.19

BG:UDC VR 3.36 3.71 2.78 2.77 2.26 2.87 1.90 3.21 3.21 4.82 3.09 19 3.15 3.92 2.63 2.70 2.26 2.85 1.90 3.27 3.21 4.88 3.08

*Ratio of respective entries in column (4), Table 5. **Variance ratio multiplied by ratio of respective entries in column (6), Table 5.

3 but for 5 different base points. Since varying the sample size does not provide much added information, results for only two values of N are presented. Table 6 summarizes the variance and improvement ratios for the results of Table 5. The last column gives an average of the results in each row. From the averages of. the improvement ratios one concludes that, for this example, the Burt and Garman procedure is more than twice as efficient as straiahtforward simulation. The UDC method is three times as efficient as Burt and Garman's procedure and seven times as efficient as straightforward simulation.

One final advantage of the UDC approach should be noted. The variance reduction is achieved by sampling fewer arcs. It follows that one would like to sample arcs that have the least variance. Since there are usually many UDCs in a network one can choose which one to use such that one samples from arcs that have the least variabili-

ty. Since there is only one set of unique arcs in a network, this choice is not available within the Burt and Garman procedure.

9. RECOMMENDATIONS FOR FURTHER RESEARCH

The conditional Monte Carlo approach involves the multiplication of cdfs and the utilization of oath relationshios to obtain estimates with a smailer variance based on a reduced sampling effort. The cost of findinq unique arcs and identifying paths in the network has not been addressed, since it varies with each network configuration. The authors of this paper see the need for a more comprehensive comparison study designed to evaluate the total efficiency of straiqhtforward simulation. Burt and Garman's procedure, and the UDC method;. An interesting project would be to develop rules for evaluating when a particular variance reduction procedure is worthwhile and how much of an improvement can be expected for a particular network configuration.

Another recommended study involves the use of antithetic sampling, control variates, and stratified sampling in combination with the UDC aDDrOaCh. These procedures can effect siqnifi- cant variance reductions when Monte Carlo- analysis is used to solve multivariate integrals [5,6] and should augment variance reductions obtained through the use of the UDC approach. The development of network reduction techniques to combine series and oarallel arcs before making estimates is also considered to be a fertile area for future research (see Sigal [ill). The stochastic shortest route problem and the use of path optimality indices are new network concepts. Examples of applications explored to date include equipment replacement analysis, reliability modeling, maximal flow problems and stochastic dynamic programming [ll]. The further exploration of the use of these new network concepts in decision sciences is warranted and should be expected.

10. REFERENCES

i 11 J. M. Burt and M. B. Garman, Conditional Monte Carlo: A simulation technique for stochastic network analysis, Management Science, 18 (November 1971).

[ 21 S. E. Elmaghraby, Activity Networks: Project Planning and Control by Network Models, (John Wiley & Sons, Inc, New York, 1977).

[ 31 H. Frank, Shortest paths in probabilistic graphs, Operations Research, 17 (July-August 1969).

[ 41 M. B. Garman, More on conditioned sampling in the simulation of stochastic networks, Manage- ment Science, 19 (September 1972).

[ 5] S. Haber, Numerical evaluation of multiple integrals, SIAM Review, 12 (October 1970).

[ 61 J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods, (John Wiley & Sons, Inc., New York, 1964).

r 71 H. 0. Hartlev and A. W. Wortham. A statistical theory for PERT critical path analysis, Manaqement Science, 12 (June 1966), B-469 - B-482.

[ 81 J. J. Martin, Distribution of the time through a directed acyclic network, Operations Research, 13 (January-February 1965), 46-66.

[ 91 A. Papoulis, Probability Random Variables and Stochastic Processes, (McGraw-Hill Book Company, New York, 1964).

[lo] L. Ringer, Numerical operators for statistical PERT critical path analysis, Management Science, 16 (October 1969).

[ll] C. E. Sigal, The Stochastic Shortest Route Problem, Ph.D. Dissertation, Purdue University, (December 1977).

[12] C. E. Sigal, A. A. B. Pritsker and J. J. Solberg, The stochastic shortest route problem, Operations Research, in press (1980).

[13] R. M. Van Slyke, Monte Carlo methods and the PERT problem, Operations Research, 11 (1963), 839-860.

the use of cutsets in monte carlo analysis of stochastic networks

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