The use of cutsets in Monte Carlo analysis of stochastic networks

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<ul><li><p>Mathematics and Computers in Simulation XXI (1979) 376-384 </p><p>North-Holland Publishing Company </p><p>THE USE OF CUTSETS IN MONTE CARLO ANALYSIS OF STOCHASTIC NETWORKS </p><p>C.E. S[C;AL*, A.A.B. PRITSKER and J.J. SOLBERG </p><p>School of Industrial En,~$mwin~, Purdue University. West I.afa,vrttr. l,Y 4790 7, U.S.A </p><p>clonte 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&gt;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. </p><p>1. INTRODUCTION </p><p>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. </p><p>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. </p><p>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 distribu- tion of minimum path durations. If only one alternative is to be selected, however, the path cptimality index is of interest. A path optimal- ity index is the probability a given path has the shortest path duration [ll,lZ]. </p><p>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 </p><p>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. </p><p>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. </p><p>Previous work with conditicnal tlonte Carlo dealt exclusively with project completion time distri- butions. 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. </p><p>2. PAST RESEARCH </p><p>Elmaqhraby [z] provides an excellent review of analytic and llonte Carlo approaches to deter- mining project completion time distributions and </p><p>* Currently at the Pritzker School of Medicine, University of Chicago. </p><p>376 </p></li><li><p>criticality indices. Van Slyke [13] was the first to develop a straightforward simulation :pproach to analyzing networks in terms of com- pletion 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 straight- forward simulation. Garman [4] offered an alternative approach to Burt and Garman's pro- cedure but did not address the subject of criticality indices. </p><p>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]. </p><p>3. CONDITIONAL MONTE CARLO AND THE PROJECT COMPLETION TIME DISTRIBUTION </p><p>The distribution of the project completion time, </p><p>YC, is given by </p><p>Fr(Yc </p></li><li><p>378 C.6: S&amp;l et al. / Monte Carlo arzal.vsis ofstochastic rretworks </p><p>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 refer- ence, 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). </p><p>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. </p><p>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. </p><p>Figure 2. Network for illustrating the Cutset Approach </p><p>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. </p><p>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 + </p><p>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. </p><p>Fyc(t) = jj\j Pr(Yc </p></li><li><p>where hi=min(t- x </p><p>k FPlu xi,...,t- z xi, </p><p>kEP k#uQU </p><p>path containing UDC </p><p>STEP 5. If i=N, go to Step 6; otherwise, set i=i+l and go to Step 3. </p><p>STEP 6. Commute the estimate of Fyc(t) as N </p><p>i, (t) = 1 z fi . C N i=l </p><p>STEP 7. If more base points are Step 2. </p><p>5. CRITICALITY INDICES </p><p>desired, go to </p><p>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, </p><p>C, = Pr(Y,s Yl,...,Y,z Y,,...,Y,z YM (8) </p><p>m=l ,...,M; mfa. . In this section we develop a Monte Carlo proce- dure 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 comple- tion time distribution. Additional mathematical details relating to criticality indices have not been included due to space limitations. </p><p>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 criti- cality index for path e, KQ. by treating the UDC arc j and the arcs in y* as constants. </p><p>K, = Pr(Y, &gt;= Yl,. . . ,yQ 5 Y m"" YYl&gt;= YmIY*it'Xj) (9) m=l,.. .,M; mfa. . </p><p>By this conditioning, the path duration y "( </p><p>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. </p><p>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 </p><p>K,= n WYm 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 particu- lar sample values. </p><p>We can now generate Eq. 12 as follows </p><p>K,=b[ II Fx (a,)1 UEX" u </p><p>u $'Path a. </p><p>(14) </p><p>where a is defined as in Eq. 13 and </p><p>0 if the UDC arc which is on path e, say arc u, is on more than one </p><p>b= path a&amp;y, is less than all other (15) durations of paths containing arc u for the given fixed values </p><p>1 otherwise . </p><p>The Monte Carlo algorithm for estimating criti- cality indices is given in Table 2. </p><p>Table 2. Algorithm for Estimating Criticality Indices </p><p>INITIALIZATION. Specify a. Determine N. Set i=l. </p></li><li><p>380 </p><p>STEP 1. </p><p>STEP 2. </p><p>STEP 3. </p><p>STEP 4. </p><p>STEP 5. </p><p>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 </p><p>fi= n </p><p>ucx* FX (a') </p><p>U </p><p>LI f Path n. </p><p>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 </p><p>6. THE STOCHASTIC SHORTEST ROUTE PROBLEM </p><p>In Stochastic Shortest Route Analysis, one is concerned with the distribution of the minimum time through the network and also the proba- bility a given path is shorter than all other paths, that is, a path optimality index [11,12]. </p><p>Let 2 denote the minimum time through the net- work, then </p><p>Z = min(Y1,...,YM) . (15) </p><p>Since Z is a minimum of a set of random variables, </p><p>Pr(Z t) . (18) </p><p>As before, we determine the UDC, x*, and con- dition upon non-UDC arcs, y*. We then rewrite Eq. 18 as </p><p>Pr(Zlt)= 1 -~...~Pr(Y,&gt;t,...,YM&gt;tl~)dFrr* . (19) </p><p>Letting GY(.) and Gxu(.) denote path and arc complementary cdfs respectively, we have </p><p>M </p><p>Pr(Zlt)=l -!.*.I i Gy (tlz*) dFrr* m=l m </p><p>= 1 _I...] n GX (v) dFy* ucrr* u </p><p>where </p><p>v = max(t - z </p><p>k EPlu </p><p>xk,...,t - X </p><p>k EPQu </p><p>xk) </p><p>k#u kfu </p><p>",a: Pqu ;s the q th path to which arc u belongs, ,...I . </p><p>The Monte Carlo Method given for the project </p><p>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 </p><p>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, </p><p>Rp = Pr(Y,~Y,,...,Y~~Y,,...,Y~~YM) (21) </p><p>m=l ,...,M;m#a </p><p>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, </p><p>Re=_/...j e II Gx (d) dFx,(xj)dFT (22) UEX* u J </p><p>u qPath L </p><p>where </p><p>d = max(yi - z </p><p>k ?Plu </p><p>xk,...,y, - z xk) kEP </p><p>Qu kfu k#u </p><p>and </p><p>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 &lt; </p><p>e = of paths containing arc u for the given fixed values </p><p>1 otherwise . The Monte Carlo Method for estimating optimality indices involves changing Step 3 of the algorithm given in Table 2 by defining </p><p>fi </p><p>where e' and </p><p>= e II u Ex* </p><p>GX (di) U </p><p>u +'Path P. </p><p>di are defined in Eq. 22. </p><p>7. VARIANCE REDUCTION OVER STRAIGHTFORWARD SIMULATION </p><p>The procedures presented will have less asso- ciated 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 simula- tion 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 bi- nomial process. The variance, o'$, associated </p></li><li><p>v&amp;it;r;;timating C, is C,(l-C,)/N=(C,-C$_)/N. In 3 whenever one can formulate a problem as </p><p>a multiple integral and solve it with the Monte Carlo Method, the associated variance will be less than that of a straightforward simulation of the problem involving sampling from a binomial distribution [6]. </p><p>The variance, oh, associated with estimating an integral of the form CL =i f dA where A is the </p><p>region of integration is [5] c$,=f [afzdA- </p><p>(if dA)2] where N is the number of sample evalua-...</p></li></ul>

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