state space partitioning methods for stochastic shortest path problems

State Space Partitioning Methods for StochasticShortest Path Problems

Christos Alexopoulos

School of Industrial and Systems Engineering, Georgia Institute of Technology,Atlanta, Georgia 30332-0205

Received 14 June 1993; accepted 16 January 1997

Abstract: This paper describes methods for computing measures related to shortest paths in networkswith discrete random arc lengths. These measures include the probability that there exists a path withlength not exceeding a specified value and the probability that a given path is shortest. The proposedmethods are based on an iterative partition of the network state space and provide bounds that improveafter each iteration and eventually become equal to the respective measure. These bounds can also beused for constructing simple variance reducing Monte Carlo sampling plans, making the proposed algo-rithms useful for large problems where exact evaluations are virtually impossible. The algorithms can beeasily modified to compute performance characteristics in stochastic activity networks. Computationalexperience has been encouraging as we have been able to solve networks that have presented difficultiesto existing methods. q 1997 John Wiley & Sons, Inc. Networks 30: 9–21, 1997

Keywords: shortest path; reliability; stochastic networks; Monte Carlo methods

1. INTRODUCTION link failures and variable travel times due to congestion)in a variety of distribution systems makes a stochasticnetwork model more realistic. These models have applica-This paper describes exact and approximation methodstions in the study of communication systems, emergencyfor computing measures related to shortest paths in proba-service delivery systems, or spread of fire in a buildingbilistic networks. Let G Å (N , A , s , t) be a directed(see [25]) . See Ball et al. [4] for a comprehensive reviewnetwork with node set N Å {1, . . . , t}, arc set A Å {1,of network performability models and related approaches.. . . , a}, source s Å 1, and terminal node t . If each arcSpecifically, we assume that each arc i √ A has a discretehas a deterministic length (or time required for its tra-positive random length Xi taking values xi (1) õ xi (2)versal) , then a basic problem in network optimization isõ rrr õ xi (ni ) with respective probabilities pi (1) ,the determination of an (s , t) path with minimum length.pi (2) , . . . , pi (ni ) . The state space V of the randomAnother problem is the identification of the arcs that be-vector X Å (X1 , . . . , Xa) consists of the a-tuples x(v)long to shortest paths. These problems can be easilyÅ (x1(v1) , . . . , xa(va)) , where the index vi √ {1, . . . ,solved by a variety of algorithms (see [17]) .ni } for i √ A . To simplify the notation, we use the indexUnfortunately, the presence of uncertainties (such ask to designate the value xi (k) for each arc i and level k .Therefore, the state x(v) will be denoted by v Å (v1 ,. . . , va) . We consider subsets of V that are (discrete)Contract grant sponsor: AFOSR

Contract grant number: F49620-93-1-0043 multidimensional intervals in the sense that each such set

q 1997 John Wiley & Sons, Inc. CCC 0028-3045/97/010009-13

9

8U11 764/ 8U11$$0764 06-05-97 14:33:33 netwa W: Networks

10 ALEXOPOULOS

R has lower and upper limiting states a Å a[R] and b t) path in the original network has length L ¢ r . Thus,the ability to evaluate c(e) in the augmented networkÅ b[R] and all states v √ R satisfy ai ° vi ° bi for

all i √ A . We denote R by (a; b) . G * Å (N , A < {e}) implies the ability to evaluate P(L¢ r) in the original network G . Since the evaluation ofLet LP(v) be the length of an (s , t) path P . Also, let

T(v) denote an s-rooted shortest path tree for the state P(L ¢ r) is a #P-hard problem, the evaluation of c(e)is also a #P-hard problem. Finally, the evaluation of c(P0)v. The predecessor of node j on the tree is denoted by

,( j ; v) and the shortest (s , j) path length is denoted by is a #P-hard problem because for the path P0 Å {e} wehave c(P0) Å c(e) . jd( j ; v) . Then, the (s , t) path P(v) with arcs in T(v)

has length d( t ; v) and the shortest path length is theA related problem of considerable interest is the evalu-random variable

ation of the distribution of the length of a longest path in(acyclic) PERT networks. For our setting, Hagstrom [18,L Å L(X ) Å d( t ; X ) . (1)19] showed that this problem is #P-complete and pro-posed a method for solving it.Let f (r) and F(r) denote the probability and cumulative

One possible way for computing the distribution of Ldistribution functions of L . Hereafter, the term path willis by formulating the problem as a stochastic linear pro-denote an (s , t) path unless stated otherwise.gram with random objective coefficients. Bereanu [6, 7]For convenience, we assume that the arc lengths areand Eubank et al. [12] proposed methods for computingindependent random variables. This assumption does notthe distribution of the optimal objective value when theease the computation of the distribution of L Å minP√P

coefficients are continuous random variables. These meth-LP(X ) by path enumeration because the cardinality of theods require the evaluation of the probability that a givenset of paths P can be as large as (n 0 2)!e for abasis is optimal, a task that requires a complicated parti-complete graph on n nodes and the path lengths LP(X )tion of the state space of the objective function. Frankare not generally independent due to shared arcs. Note[15] and Sigal et al. [31] presented exact methods, boththat the probability of an interval R Å (a; b) can beof which rely on the evaluation of multiple integrals.written asBecause of the great complications that arise in thoseevaluations, they suggested Monte Carlo sampling. Kulk-

P(R) å P(X √ R) Å ∏a

iÅ1

∑bi

kÅai

pi (k) . (2) arni [23] considered the case of independent and expo-nentially distributed arc lengths and proposed a methodfor computing the distribution of L that is based on a

We focus on the computation of the following: Markov process with an absorbing state.Several studies have addressed the case of independent

F(r) Å probability that there exists an discrete arc lengths. The approach in Mirchandani [26]starts with sorting all (s , t) paths and creates a disjoint(s , t) path of length ° r for fixed r .

(3)

Boolean expression by comparing neighboring paths. Thisc(e) Å probability that arc e is in a shortest path. (4) expression is then used for computing the probability

F(r) or the mean of the shortest path length. Hagstromc(P0) Å probability that the path P0 is shortest. (5)and Kumar [20] considered the model where each archas two modes, an operating mode with known finiteIn addition, we study the evaluation of the distributionlength and a failed mode, and proposed an algorithmfunction F(r) and, therefore, the moments of L .for computing the probability F(r) . At each step, theThe probabilities c(e) and c(P0) are often called thealgorithm conditions (pivots) on the mode of a carefullycriticality indices of the arc i and the path P0 . Note thatchosen arc to partition the current problem into two sim-c(P0) ° c( i) for all i √ P0 . The evaluations of (3) –pler subproblems. In the case of multistate arc lengths,(5) are #P-hard problems. Indeed, the directed (s , t)they recommended the reduction in Mirchandani [26]reliability problem (see [4]) is the evaluation of F(0)which replaces every arc i with ni ú 2 by ni arcs i1 , . . . ,when the arcs assume lengths 0 or 1 with respective prob-ini

, where arc ij is operative with length xi ( j) and probabil-abilities pi and 1 0 pi . The following proposition estab-ity pi ( j) / [1 0 ( j01

kÅ1 pi (k)] or failed. Unfortunately, thislishes the complexity of the last two evaluations:popular reduction increases the numbers of arcs and statesin the network.Proposition 1. The evaluations of c(e) and c(P0) for

Corea and Kulkarni [8, 9] proposed a methodologyfixed arc e and path P0 are #P-hard problems.for computing the distribution of L and criticality indicesof paths. They assumed that the arc lengths are integer-Proof. Assume that the arc lengths are integer-valued.

Add an arc e Å (s , t) with fixed length r . Obviously, e valued, replaced each arc with largest possible length mby a subnetwork with 2m arcs, and constructed Markovbelongs to a shortest path if and only if the shortest (s ,


STATE SPACE PARTITIONING FOR STOCHASTIC SP PROBLEMS 11

chains with absorbing states and binary transition costs. √ S . These sets are called undetermined because theirstates cannot be classified without computing a new short-The above measures are computed by evaluating the dis-

tribution of the total cost incurred until absorption. Unfor- est path tree. Since the sets B *i overlap, we use the well-known approach to partition R 0 D into the intervals Bitunately, their construction limits the applicability of the

methods to problems of small size. Å B *i 0 <kõi(B *i > Bk) orHayhurst and Shier [21] proposed a method for com-

puting the distribution F(r) that is based on structural Bi Å {v √ R : ak ° vk ° gk for k õ i , k √ Sfactoring. At each stage, the method removes a node and

ak ° vk ° bk for k õ i , k √/ Sreplaces its incident arcs by new arcs. This replacementrequires conditioning on the lengths of several incident gi / 1 ° vi ° biarcs, identified as factoring arcs. The approach has been

ak ° vk ° bk for k ú i}, i √ Ssuccessful with problems previously analyzed only byapproximation techniques and will serve for comparisons with gi õ bi .

(7)

with our methods.The intractability of the problems under consideration These intervals are partitioned in subsequent iterations

has motivated the development of approximation meth- and the procedure terminates when no undetermined setsods. Several methods, particularly for the case of acyclic remain. The undetermined intervals are maintained in anetworks, rely on bounds as Dodin [10], Fulkerson [16], list, say L, whose records consist of the boundary statesKleindorfer [22], and Shogan [29]. Other methods as and, possibly, some additional information. Note that theSigal et al. [31] and Fishman [13] rely on Monte Carlo partition of an interval can be thought of as factoringsampling. conditional on the states of the arcs in the corresponding

The proposed methods perform an iterative partition set S .of the network state space V. State space partitioningmethods were originally proposed by Doulliez and Ja-

Remark 1. The set R 0 D was partitioned into the inter-moulle [11] and were designed for the evaluation of per-vals Bi by considering the overlapping sets B*i in as-formance measures for flow networks with random capac-cending order of their indices in S . An alternative partitionities (see also Rueger [28] and Shogan [30]) . Such ais obtained by considering B *i in descending order of imeasure is the probability that a fixed amount of demand√ S . Since the determination of an ordering that resultsat node t can be satisfied. The Doulliez–Jamoulle algo-in improved long-term performance seems to be a hardrithm for the latter measure (with the corrections in Alex-problem, we adopt the ordering in (7) .opoulos [2]) was used by Alexopoulos and Fishman [3]

In view of (7) , the number of resulting undeterminedto design variance reducing Monte Carlo experiments forintervals (during an iteration) is minimized when thethe simultaneous estimation of network performancenumber of arcs with gi Å bi is maximized. Hence, wecharacteristics at several capacity distributions.choose the state g to be an optimal solution to the knap-Let E be an event under consideration. For the shortestsack problempath model, E is one of the events {L(X ) ° r} with

probability F(r) , {LP0(X ) Å L(X )} with probability

maximize É{ i √ S : vi Å bi }É (8)c(P0) , and <P :e√P {LP(X ) Å L(X )} with probabilityc(e) . In short, the probability of E is computed as follows: subject to ∑

i√S

xi (vi ) ° b (9)We start with the interval V. At each iteration, an intervalR Å (a; b) is partitioned by finding an element g of V

vi √ {ai , ai / 1, . . . , bi }, i √ S . (10)and a subset S of A such that the interval

The available space in the knapsack is b . Each arc i √ SD Å {v √ R : ai ° vi ° gi for i √ S} (6) takes up space equal to xi (bi ) 0 xi (ai ) (putting i into

the knapsack means setting gi Å bi ) and all arcs haveequal value, say 1. We solve this problem by renumberingis either a subset of E or does not intersect E . In the

former case, D is called feasible and P(D) is part of the arcs in S such that S Å {1 *, . . . , q *} with xj =(bj =)0 xj =(aj =) ° xk =(bk =) 0 xk =(ak =) for each 1 ° j õ k ° qP(E) , while in the latter case, E is called infeasible. The

states of D satisfy a constraint (i√S xi (vi ) ° b , where S (we break ties by giving preference to larger bj = 0 aj =) .Then, as long as constraint (7) remains valid, for j Å 1,is a subset of a shortest path and b is a positive bound.

(For instance, in Section 2, S is a shortest path, b Å r , . . . , q , we increase the level of arc j * as much as possible.The main advantages of the proposed methods are (1)and the constraint guarantees that for each state in D there

is a path with length at most r .) The remainder R 0 D the use of the simple knapsack problem (8) – (10) tomyopically minimize the number of generated undeter-is the union of the sets B*i Å {v √ R : vi ú gi }, i


12 ALEXOPOULOS

mined intervals ( the Doulliez–Jamoulle methods and gi / 1. The original shortest path tree can then be usedtheir derivatives do not guarantee this minimization); (2) for computing new node labels ,U ( j) and distances dU ( j)their domain of applicability as the methods for the flow- for the state a[Bi ] . Clearly, Bi is infeasible if dU ( t) ú r .related problems use properties of maximum flows; (3) If dU ( t) ° r , the labels ,U ( j) are stored along with thetheir effectiveness for problems that have presented diffi- boundary states of Bi for use during the partition of thisculties to existing methods; (4) their ability to provide set. This approach increases the storage requirements butbounds that improve after each iteration and can be used results in computational savings. It should be mentionedfor designing Monte Carlo sampling plans. These plans here that decomposition methods for flow problems com-are conceptually simple, provide estimates with consider- pute an entirely new flow in each iteration.ably smaller variances, and require less time per replica- Algorithm PARTITION describes the evaluation oftion than do crude Monte Carlo sampling plans. This F(r) . As mentioned previously, each record in the list Lproperty makes the methods beneficial for large-size prob- contains the boundary states of an undetermined set andlems; (5) their flexibility for performing sensitivity analy- the optimal node labels corresponding to the lower bound-sis on the measures of interest with respect to alternative ary point. The maintenance of this list is discussed follow-arc length distributions with common state space; and (6) ing the algorithm. Fl(r) and Fu(r) are lower and uppertheir potential for accommodating statistical dependencies bounds on F(r) and are updated after the partition of anbetween arc lengths (see the method in [24] for flow interval.networks with dependent arc capacities) .

Section 2 describes the computation of F(r) at fixed Algorithm PARTITION(r)r . Section 3 discusses the computation of the distributionfunction F(r) and proposes a separate algorithm for eval- 1. Start with the interval R Å V, empty list L, Fl(r)uating the mean E(L) only. Section 4 describes methods Å 0, and Fu(r) Å 1. Compute an s-rooted shortestfor computing criticality indices and Section 5 applies path tree with arc lengths xi (1) . Let ,( j) be theour ideas to stochastic activity networks. Each of these predecessor of node j on the tree and let d( j) be thesections describes exact algorithm(s) , shows how the al- shortest (s , j) path length. If d( t) ú r , terminategorithms produce bounds, and gives Monte Carlo estima- with F(r) Å 0.tors that are based on a variance reducing sampling 2. Identify a shortest path P .scheme. Section 6 contains results and Section 7 contains

3. Do the following:final remarks and conclusions.

Solve problem (8) – (10) and compute the feasibleinterval D in (6) and the undetermined intervalsBi in (7) .2. COMPUTING THE PROBABILITY F (r )Set Fl(r) Å Fl(r) / P(D) .FOR FIXED rFor i √ P with gi õ bi :

Increase the length of arc i to xi (gi / 1) andThere are numerous ways for decomposing an interval.

compute a shortest path tree with labels ,U ( j)Clearly, there is a trade-off between obtaining a large

and shortest (s , j) path lengths dU ( j) .feasible subset and the required computational effort. WeIf dU ( t) ° r , file the record {a[Bi ] ; b[Bi ] ;propose an approach that is conceptually simple and re-(,U ( j) , j √ N)} in L.quires at most one shortest path evaluation per iteration.If dU ( t) ú r , the set Bi is infeasible; set Fu(r)We then describe the overall algorithm and strategies for Å Fu(r) 0 P(Bi ) .keeping the number of undetermined intervals low and

4. If L is not empty, remove a record {a; b; (,( j) , jproducing tight bounds quickly.√ N)} from it and go to step 2.An undetermined interval R Å (a; b) is partitioned as

5. End with Fl(r) Å Fu(r) Å F(r) .follows: We start with a shortest path tree for the lowerboundary state a with predecessor labels ,( j) . Theselabels are used for computing the shortest path lengths Remark 2. The maintenance of L is an important issue.

If the algorithm is to be carried to completion, then L isd( j) and a shortest path P in O(ÉNÉ) time. If d( t) ú r ,then all states in R are infeasible. If d( t) ° r , then any maintained as a singly linked list where depth-first search

is carried out to keep the number of records stored lowstate v √ R with (i√P xi (vi ) ° r satisfies L(v) ° r .The interval D in (6) obtained by solving problem (8) – at any time. If, on the other hand, the objective is the

computation of bounds, this list is maintained by using a(10) with S Å P and b Å r is therefore feasible whilethe states of the undetermined intervals Bi in (7) cannot heap whose nodes are weighed by the probabilities of the

intervals and the root with the largest weight is removedbe classified without determining a new shortest path.Note that the lower boundary state of Bi is equal to in step 4. We use these strategies for the computation of

the measures in Sections 3–5.that of R except for the i th coordinate of Bi which equals



Remark 3. To minimize the number of undetermined Sj Å ∑m

qÅ1

1(L(X ( j :q ) ) ° r) ,intervals, we attempt to force as many arcs with fixedlength within the present interval in a shortest path aspossible by decreasing their lengths by an e ú 0 chosen is also an unbiased estimator of F(r) with varianceso that the resulting shortest path remains shortest when smaller than that of the crude estimator FV (r) by a factorthese lengths are reset to their original values. of at least

Monte Carlo Sampling S√Fu(r)[1 0 Fl(r)] 0

√Fl(r)[1 0 Fu(r)] D02

.

The basic Monte Carlo method involves sampling fromthe state space V with probabilities pi (k) for arc i . Sup-

Remark 4. The latter plan requires less mean time perpose that we draw n independent samples X (1) , . . . , X (n ) .replication than does crude Monte Carlo sampling be-Thencause each set Uj has fewer states than does the statespace V. Also, each Uj must be retained only until all mj

FU (r) Å 1n

∑n

jÅ1

1(L(X ( j ) ) ° r) , samples are drawn from it and then can be discarded.Therefore, these sets can be read one-at-a-time from afile.where 1(r) is the indicator function, is an unbiased esti-

mator of F(r) with varianceA confidence interval for F(r) can be computed by

using the central limit theorem or the method in Fish-Var[FU (r)] Å F(r)[1 0 F(r)] /n .man [14].

Algorithm PARTITION provides us with the capabilityof designing a Monte Carlo sampling plan that combinesimportance and stratified sampling. Indeed, suppose that 3. COMPUTING THE DISTRIBUTION OF Lwe decide to exit when the list L contains k sets, say U1 ,. . . , Uk . Let We discuss two approaches for partitioning an interval.

The first applies to the evaluation of the distribution ofL , and therefore its moments, and the second to the evalu-Qij Å ∑

bi[U j ]

lÅai[Uj ]

pi ( l) , i √ Aation of the mean E(L) only.

The first method decomposes an undetermined intervalR Å (a; b) similarly to the method in Section 2 with theand writeexception that the indices gi for arcs in the shortest pathP(a) are set to ai . All states of the intervalpj Å P(Uj) Å ∏

i√A

Qij , j Å 1, . . . , k .

D Å {v √ R : vi Å ai for i √ P(a)} (11)We use the proportional allocation rule to draw mj

Å npj/ (( kjÅ1 pj) , j Å 1, . . . , k , independent samples have P(a) as a shortest path and P(D) is part of the

from each Uj as follows: probability f ( d( t ; a)) Å P[L Å d( t ; a)] . We finish bypartitioning R 0 D into the undetermined intervals Bi

Algorithm SAMPLE (U1 , . . . , Uk) given in (7) for gi Å ai and S Å P(a) . The presentFor j Å 1, . . . , k : shortest path tree T(a) is then used to compute a shortest

For q Å 1, . . . , mj : path tree corresponding to the lower boundary state ofFor i √ A : Sample the index vi with probabilities each Bi and the record {a[Bi ] ; b[Bi ] ; (,U ( j) , j √ N)}{pi ( l) /Qij , ai[Uj] ° l ° bi[Uj]} and assign with the updated labels ,U ( j) is stored for later consider-length X ( j :q )

i Å xi (vi ) to arc i . ation.End After each stage, the decomposition algorithm pro-

duces the lower boundsThen,

fl(r) Å ∑D

P(D) ,FO (r) Å Fl(r) / ∑

k

jÅ1

pj( Sj/mj) ,

where the sum is over all intervals D whose shortest pathlength equals r , and the upper boundswhere


14 ALEXOPOULOS

where yi Å £i / 1, zi Å bi for i √ P with £i õ bi , andfu(r) Å fl(r) / ∑j

P(Uj) ,yi Å ai , zi Å £i 0 1 for i √/ P with ai õ £i . It shouldbe noted here that the sets Ci and R typically have very

where Uj are the remaining undetermined intervals. distinct most probable states. Since the present shortestAlgorithm SAMPLE can be used for estimating f (r) when path tree T(£) contains little information that might be

the partitioning procedure terminates with remaining un- of use in the computation of a shortest path tree for thedetermined intervals U1 , . . . , Uk . The probability f (r) is most probable state of Ci , the records in the list L containestimated by only boundary states and a new shortest path is computed

at the beginning of an interval partition.At each stage, the partitioning algorithm produces thefO (r) Å fl(r) / ∑

k

jÅ1

pj( Sj(r) /mj) ,lower bound

where El(L) Å ∑D

E(LP ; D) ,

Sj(r) Å ∑m

qÅ1

1(L(X ( j :q ) Å r) .where the sum is taken over all sets D that have beenproduced. Further, a Monte Carlo sampling plan based

A special decomposition of an interval can be per- on the remaining undetermined sets U1 , . . . , Uk yieldsformed when we want to compute the mean E(L) only the estimatorand the probabilities pi (k) are not decreasing in k for allarcs i . In this case, we compute a shortest path P at the

EO ( L) Å El(L) / ∑k

jÅ1

pj( Sj/mj) ,most probable state £ in R ( in case of ties, one can choosethe smallest state) and use the fact that each state of theinterval

withD Å {v √ R : ai ° vi ° £i for i √ P ,

£i ° vi ° bi for i √/ P} Sj Å ∑m j

qÅ1

L(X ( j :q ) ) .

has P as a shortest path. Then, the expected length E(LP ;D) of P in the set D is part of E(L) and can be easilycomputed by

4. COMPUTING CRITICALITY INDICES

E(LP ; D) Å ∑v√D

F ∑i√P

xi (vi )GP[X Å x(v)]We first describe a method for computing the probabilityc(P0) that the path P0 is shortest. The evaluation of the

Å ∏i√/ P

qiF ∑i√P

hi S ∏j√D0{i }

qjDG criticality index of an arc will follow.The undetermined list L contains records of the form

{a; b; (,( j ; a) , j √ N)}. The partition of the intervalÅ P(D) ∑

i√P

hi /qi , R Å (a; b) proceeds as follows: If P0 is a shortest pathfor the state a [equivalently, if d( j) Å d( i) / xk(ak) forall k Å ( i , j) √ P0] , we compute the length of the secondwhereshortest (s , t) path, say d2( t) . Note that P0 is a shortestpath for each state v √ R such that LP0

(v) ° d2( t) . Weqi Å ∑

£i

,Åai

pi (,) and hi Å ∑£i

,Åai

xi (,)pi (,) . obtain a feasible interval D Å {v √ R : ai ° vi ° gi

for i √ P0} by solving problem (8) – (10) with S Å P0

and bÅ d2( t) . The probability of D is added to the currentThe difference R 0 D is partitioned into the following value of c(P0) , the set R0D is decomposed into intervals

undetermined intervals: Bi in (7) , and the records {a[Bi ] ; b[Bi ] ; (,U ( j) , j√ N)} with the updated labels ,U ( j) for the state a[Bi ]

Ci Å {v √ R : ak ° vk ° £k for k õ i , k √ P are filed in L.Now suppose that a path P x P0 is shortest. In this case,

£k ° vk ° bk for k õ i , k √/ Pany state v √ R satisfying (i√P0P0

xi (vi ) õ (i√P00P xi (ai )yi ° vi ° zi causes LP(v)õ LP0

(v) . Therefore, a solution to problem(8) – (10) with S Å P 0 P0 and b Å (i√P00P xi (ai ) 0 eak ° vk ° bk for k ú i},



where the first (second) sum is over all feasible (infeasi-ble) intervals that have been obtained. Also, Monte Carlosampling based on the undetermined sets U1 , . . . , Uk

produces the estimator

cP (P0) Å cl(P0) / ∑k

jÅ1

pj( Sj/mj) ,

where Sj increases by one at a trial from Uj only if P0 isa shortest path.

The evaluation of the criticality index of an arc e pro-ceeds similarly to the evaluation of the criticality index

Fig. 1. Network with 10 nodes and 23 arcs.of a path. The partition of the interval R Å (a; b) startswith the identification of a shortest path P correspondingto the state a. To force the arc e in P , the shortest path

(for appropriate e ú 0) yields an infeasible interval D tree is computed with the length xe(ae) reduced by e. IfÅ {v √ R : ai ° vi ° gi for i √ P 0 P0}. The remainder e √ P , we solve problem (8) – (10) with S Å P and bR 0 D is partitioned and the new records are filed in L. Å d2( t) . Each state in the interval D Å {v √ R : ai

After each iteration, the overall algorithm produces the ° vi ° gi for i √ P} has P as a shortest path and thenbounds P(D) is part of c(e) . If e √/ P , we solve this problem

with b Å d2( t) 0 e. In this case, no state in the resultinginterval D has e in a shortest path, making the intervalcl(P0) Å ∑

F

P(F) and cu(P0) Å 1 0 ∑I

P(I) ,

TABLE I. Input distribution for Example 1; there are 87.07 1 109

states

Arc Lengths Probabilities

(1, 2) 7.0 7.3 9.4 0.2 0.5 0.3

(1, 3) 2.5 3.5 8.2 0.5 0.4 0.1

(1, 4) 4.2 4.8 6.1 0.2 0.3 0.5

(2, 5) 2.6 3.1 5.5 8.8 9.0 0.1 0.2 0.4 0.2 0.1

(2, 6) 5.8 7.0 9.5 0.3 0.3 0.4

(3, 2) 1.5 7.3 0.4 0.6

(3, 7) 6.5 7.4 7.5 0.4 0.5 0.1

(3, 8) 5.9 7.2 9.8 0.6 0.3 0.1

(4, 3) 2.1 3.2 8.5 9.8 0.3 0.2 0.3 0.2

(4, 9) 8.9 9.6 0.7 0.3

(5, 7) 3.2 4.8 6.7 0.2 0.2 0.6

(5, 10) 6.3 9.9 0.5 0.5

(6, 3) 6.6 8.5 9.8 0.8 0.1 0.1

(6, 5) 0.6 1.5 3.9 5.8 0.1 0.4 0.3 0.2

(6, 7) 0.2 4.8 0.4 0.6

(7, 6) 6.1 6.3 8.5 0.2 0.3 0.5

(7, 8) 1.6 1.8 4.0 5.2 0.2 0.3 0.3 0.2

(7, 10) 1.6 3.4 7.1 0.1 0.5 0.4

(8, 4) 9.0 9.6 0.5 0.5

(8, 7) 2.1 4.6 8.5 0.3 0.4 0.3

(8, 9) 1.7 4.9 5.3 6.5 0.1 0.4 0.4 0.1

(7, 9) 0.3 3.0 5.0 0.1 0.4 0.5

(9, 10) 0.6 1.2 5.4 6.6 0.1 0.1 0.3 0.5


16 ALEXOPOULOS

TABLE IV. Computation of arc criticality indices forTABLE II. Computation of P(L ° r ) for Example 1Example 1; in each case, at most nine undetermined

No. intervals were stored at any timeCPU Partitioned

r Seconds Sets P(L ° r) No.CPU Partitioned Criticality

10.6 0.01 4 0.02000 Arc Seconds Sets Index11.0 0.01 10 0.02288

(1, 3) 1.55 6366 0.875612.0 0.02 17 0.07505(1, 4) 1.77 9978 0.094613.0 0.03 47 0.20859(2, 5) 1.70 9595 0.096814.0 0.12 281 0.45722(3, 2) 2.05 11,501 0.065515.0 0.24 568 0.61813(3, 7) 1.60 9026 0.773316.0 0.30 619 0.65679(4, 9) 1.67 9458 0.083317.0 0.28 534 0.82858(7, 10) 1.59 8945 0.786918.0 0.14 286 0.93966(9, 10) 1.68 9449 0.131619.0 0.28 573 0.97381

20.0 0.22 306 0.98498

21.0 0.12 187 0.99348clearly more time-consuming than is the evaluation of22.0 0.13 156 0.99922F(r) at a single r . The examples in Section 6 will show22.3 1that the former evaluations have time requirements of the

In each case, at most 11 undetermined intervals were stored at any same order of magnitude as that of the computation oftime. The computation of E(L)Å 14.70832 by the first method in Section the distribution F(r) . Lower bounds for criticality indices3 required the partition of 12,330 intervals and took 1.52 seconds of can also be obtained during the computation of F(r) byCPU time.

TABLE V. Input distribution for Example 2; there areinfeasible. The partition of R ends with the decomposition70.37 1 1012 states

of the remainder R 0 D into the intervals Bi in (7) andthe records {a[Bi ] ; b[Bi ] ; (,U ( j) , j √ N)} with the up- Arc Lengths Probabilitiesdated labels ,U ( j) are filed in L.

(1, 2) 3.0 5.3 7.4 9.4 0.2 0.2 0.3 0.3Finally, the estimation of c(e) by Monte Carlo simula-(1, 3) 3.5 6.2 7.9 8.5 0.3 0.3 0.2 0.2tion proceeds similarly to the estimation of c(P0) .(1, 4) 4.2 6.1 6.9 8.9 0.2 0.3 0.2 0.3

(2, 5) 2.6 4.1 5.5 9.0 0.2 0.2 0.4 0.2Remark 5. If the network is acyclic, we can obtain alarger infeasible subset of R when e is not in a shortest (2, 6) 5.8 7.0 8.5 9.6 0.3 0.3 0.2 0.2path at the cost of an additional shortest path evaluation. (3, 2) 1.5 2.3 3.6 4.5 0.2 0.2 0.3 0.3In this case, we let e Å ( i , j) and compute a shortest ( j , (3, 7) 6.5 7.2 8.3 9.4 0.5 0.2 0.2 0.1t) path with length h( j) . Then, ue Å d( i) / xe(ae) (3, 8) 5.9 7.8 8.6 9.9 0.4 0.3 0.1 0.2/ h( j) is the length of a shortest (s , t) path containing (4, 3) 2.1 3.2 4.5 6.8 0.2 0.2 0.3 0.3e , all states v √ R with LP(v) õ ue are infeasible, and

(4, 9) 1.1 2.2 3.5 4.3 0.2 0.3 0.4 0.1the infeasible set results by solving problem (8) – (10)

(5, 7) 3.2 4.8 6.7 8.2 0.2 0.2 0.3 0.3with S Å P and b Å ue 0 e.(5, 10) 6.3 7.8 8.4 9.1 0.2 0.2 0.4 0.2

(6, 3) 6.8 7.7 8.5 9.6 0.4 0.1 0.1 0.4Remark 6. The evaluations of criticality indices are(6, 5) 0.6 1.5 3.9 5.8 0.2 0.2 0.3 0.3

(6, 7) 2.1 4.8 6.6 7.5 0.2 0.4 0.2 0.2TABLE III. Computation of path criticality indices for (7, 6) 4.1 6.3 8.5 9.7 0.2 0.3 0.4 0.1Example 1; in each case, at most nine undetermined

(7, 8) 1.6 2.8 5.2 6.0 0.2 0.3 0.3 0.2intervals were stored at any time(7, 10) 1.6 3.4 8.2 9.3 0.2 0.3 0.3 0.2

No. (8, 4) 7.0 8.0 8.8 9.4 0.2 0.2 0.4 0.2CPU Partitioned Criticality (8, 7) 2.1 4.6 8.5 9.6 0.4 0.2 0.2 0.2

Path Seconds Sets Index(8, 9) 1.7 4.9 6.5 7.8 0.2 0.2 0.2 0.4

(7, 9) 3.5 4.0 5.0 7.7 0.1 0.2 0.4 0.31 r 3 r 7 r 10 0.24 2323 0.7578

(9, 10) 4.6 6.4 7.6 8.9 0.4 0.1 0.2 0.31 r 4 r 9 r 10 0.51 4749 0.0762



TABLE VI. Computation of P(L ° r ) for Example 2 resent projects with tasks corresponding to arcs. All tasksdirected into a node must be completed before any task

No. directed out of it can be started. The project is completeCPU Partitioned when all tasks directed into the terminal node t are fin-

r Seconds Sets P(L ° r)ished and the duration of the project is the length of thelongest (s , t) path. The nodes of an activity network can9.90 0.02 4 0.01600be labeled so that i õ j for each arc ( i , j) . For fixed11.0 0.02 25 0.04297durations ,1 , . . . , ,a , the project duration d( t) can be12.0 0.08 149 0.11058computed by the following recursion in time O(ÉAÉ) :13.0 0.26 551 0.22208

14.0 0.73 1381 0.34606

15.0 1.45 2930 0.54963 1. Set d(s) Å 0.16.0 3.24 6443 0.67617 2. For j Å 2, . . . , t : Set d( j) Å max(i ,j )√A{d( i) / , (i ,j )} .17.0 3.17 6137 0.82010

18.0 4.08 7108 0.90792We briefly discuss the evaluation of P(L¢ r) for fixed19.0 3.58 5761 0.95920

r , where now L denotes the project duration. To partition20.0 1.36 2240 0.98964an undetermined interval R Å (a; b) , we start with its

21.0 0.54 866 0.99439upper boundary state and compute a longest path P . If

22.0 0.27 407 0.99943 LP(b) õ r , the interval R is infeasible; otherwise, we22.1 1 find an optimal solution {di , i √ P} to the problem

In each case, at most 12 undetermined intervals were stored at anytime. The computation of E(L)Å 14.95739 by the first method in Section

maximize É{ i √ P : di Å ai }É3 required the partition of 66,476 intervals and took 8.31 seconds ofCPU time. subject to ∑

i√P

xi (di ) ¢ r (12)

noting that the probability of the interval D in (11) is di √ {ai , ai / 1, . . . , bi }, i √ P ,part of both c(P(a)) and c( i) for i √ P(a) .

which is similar to problem (8) – (10). The interval

5. APPLICATIONS TO STOCHASTICACTIVITY NETWORKS D Å {v √ R : di ° vi ° bi for i √ P}

Appropriate modifications make the methods in Sections2–4 applicable to problems in activity networks with dis- is obviously feasible and the difference R 0 D is parti-

tioned into the intervalscrete random task durations. These acyclic networks rep-

Fig. 2. Network with 15 nodes and 42 arcs.


18 ALEXOPOULOS

TABLE VII. Input distribution for Example 3; there areHi Å {v √ R : dk ° vk ° bk for k õ i5.07 1 1019 states

ai ° vi ° di 0 1Arc Lengths Probabilities

ak ° vk ° bk for k ú i},(1, 2) 16 25 36 0.6 0.3 0.1i √ P with di ú ai .(1, 3) 21 24 25 39 0.5 0.2 0.2 0.1

(1, 4) 11 13 26 0.4 0.4 0.2Since a longest path can be computed in time linear in (2, 11) 24 28 31 0.5 0.3 0.2the number of arcs, the records in the list L contain only

(2, 5) 11 30 0.7 0.3the boundary states of the respective intervals.

(2, 6) 13 37 39 0.6 0.2 0.2

(3, 2) 11 20 24 0.6 0.3 0.1

(3, 7) 23 30 34 0.4 0.3 0.3

(3, 8) 14 23 34 0.5 0.4 0.16. EXAMPLES(4, 3) 22 30 0.7 0.3

(4, 9) 35 40 0.6 0.4The programs used in the following experiments were

(4, 12) 16 25 37 0.5 0.4 0.1written in FORTRAN 77 and were run on a SUN SPARC-(5, 13) 28 35 37 40 0.4 0.3 0.2 0.1STATION 5/85, and the subroutine L2QUE from Gallo(5, 15) 25 32 0.7 0.3and Pallottino [17] was used for computing shortest paths.(5, 10) 27 33 40 0.4 0.3 0.3

(5, 7) 15 17 19 26 0.3 0.3 0.3 0.1Example 1. Figure 1 shows a network with 10 nodes,

(6, 5) 18 25 29 0.5 0.3 0.223 arcs, sÅ 1, and tÅ 10. Consider the arc length distribu-

(6, 13) 21 23 0.5 0.5tions in Table I. For instance, arc (1, 2) has length 7.0,(6, 7) 11 31 37 0.5 0.4 0.17.3, or 9.4 with probabilities 0.2, 0.5, or 0.3, respectively.(6, 3) 18 24 0.7 0.3Table II lists the results relative to the evaluation of the(7, 10) 19 23 37 0.6 0.2 0.2distribution of L . Note that the evaluation of the P(L(7, 8) 12 15 22 24 0.3 0.3 0.2 0.2° 13) took 0.03 seconds and required 47 set partitions.(7, 6) 12 23 31 0.5 0.3 0.2In general, all the CPU times in column 2 are less than

1 second. Also, the computation of the distribution of L (8, 7) 14 34 39 0.6 0.2 0.2required 12,330 set partitions and took only 1.52 seconds. (8, 14) 14 15 27 32 0.3 0.3 0.2 0.2

The method in Hayhurst and Shier [21] computed this (8, 9) 13 31 32 0.8 0.1 0.1distribution in time that was larger by several orders of (8, 4) 13 23 34 0.4 0.3 0.3magnitude (their algorithm was written in PASCAL and (9, 7) 10 17 20 0.6 0.3 0.1run on a different computer) . This method is based on

(9, 10) 16 18 36 39 0.3 0.3 0.2 0.2factoring and performs convolutions between arc lengths

(9, 15) 12 13 25 32 0.4 0.3 0.2 0.1which are often time-consuming. The method in Corea(9, 14) 19 24 29 0.4 0.3 0.3and Kulkarni [8, 9] is practically nonapplicable to this(10, 13) 14 20 25 32 0.3 0.3 0.2 0.2problem because it requires integer-valued arc lengths(10, 15) 15 19 25 0.4 0.3 0.3and replaces every arc with maximum possible length m(10, 14) 23 34 0.9 0.1by a subnetwork with m / 1 nodes and 2m arcs. For(11, 13) 13 31 35 0.6 0.3 0.1example, arc (1, 2) would be replaced by 94 / 1 Å 95

nodes and 2 1 94 Å 188 arcs. Then, the approach con- (11, 5) 18 19 20 33 0.3 0.3 0.3 0.1structs a Markov chain with absorbing states. These re- (11, 6) 10 19 39 0.5 0.4 0.1placements result in Markov chains with an astronomical (12, 8) 15 36 39 0.5 0.3 0.2number of states. However, it should be noted that the (12, 9) 16 22 0.7 0.3methods of Corea and Kulkarni [8, 9] appear to be as (12, 14) 10 13 18 34 0.3 0.3 0.3 0.1efficient as our method when the arc lengths are i.i.d.

(13, 15) 12 31 0.9 0.1discrete uniform random variables with range {1, . . . , k}

(14, 15) 14 19 32 0.5 0.3 0.2for k ° 3. A clear advantage of our approaches is theability to produce bounds.

Tables III and IV list the criticality indices of selectedpaths and arcs. A path is denoted by its sequence of nodes. Example 2. Table V displays alternative arc length distri-

butions for the network in Figure 1. Every arc has fourThe paths and arcs that are not listed have small criticalityindices. possible lengths. Note in Table VI that the CPU times



TABLE VIII. Computation of P(L ° r ) for Example 3

No. Partitioned Lower Bound Upper Boundr CPU Seconds Sets P(L ° r) Fl(r) Fu(r)

51 0.03 5 0.03

52 0.04 17 0.31518

54 0.04 26 0.35754

56 0.07 74 0.43363

58 0.09 92 0.52863

60 0.14 187 0.70651

62 0.41 531 0.84400

64 0.59 802 0.89552

66 0.37 598 0.92236

68 0.70 1130 0.94890

70 1.17 1700 0.96466

75 7.91 10,072 0.99469 0.89326 0.99950

80 11.36 11,310 0.99940 0.98803 0.99992

85 9.37 13,834 0.99991 0.99245 0.999997

90 24.49 30,345 0.99999 0.99496 0.99999997

95 18.52 15,703 1.9999998 0.99853 0.99999996

98 1

In each case, at most 12 undetermined intervals were stored at any time. The bounds in columns 5 and 6 were computed after 1000 intervalswere partitioned by using a heap. The computation of E(L) Å 57.81583 by the first method in Section 3 required the partition of 298,918 intervalsand took 68.72 seconds of CPU time.

for computing P(L ° r) for small r remain significantly noted that the latter number is a small fraction (5.911 10015) of the total number of states. For this problem,shorter than the times required for r in the middle of the

range of L . The small size of the lists of undetermined the time requirements for computing F(r) for fixed rincrease significantly for r ¢ 75 but the number of setsrectangles is attributable to the effectiveness of the LIFO

maintenance. remains very small relative to the cardinality of the sam-

Example 3. We now consider the network in Figure 2with 15 nodes and 42 arcs with the arc length distributions TABLE X. Computation of arc criticality indices for

Example 3; in each case, at most 14 undeterminedin Table VII. The probabilities were selected from {0.1,intervals were stored at any time0.2, . . . , 0.9} and were ranked so that the smaller length

has the largest probability. The results relative to the dis-No.tribution of L are listed in Table VIII. For instance, the

CPU Partitioned Criticalitycomputation of the distribution of L took 68.72 seconds Arc Seconds Sets Indexand required the partition of 298,918 sets. It should be

(1, 2) 92.02 297,473 0.5239

(1, 4) 89.72 291,035 0.4406TABLE IX. Computation of path criticality indices for (2, 5) 91.18 295,571 0.4743Example 3; in each case, at most 13 undetermined

(2, 11) 91.85 298,029 0.0100intervals were stored at any time(4, 9) 91.44 296,785 0.1597

No. (4, 12) 88.73 288,358 0.2863CPU Partitioned Criticality (5, 15) 91.08 295,571 0.4743

Path Seconds Sets Index(9, 15) 90.42 293,935 0.3179

(11, 13) 91.86 298,029 0.01001 r 2 r 5 r 15 14.94 89,446 0.4654

(12, 14) 89.37 289,615 0.18221 r 4 r 9 r 15 18.39 103,650 0.1572

(13, 15) 92.41 300,064 0.05461 r 4 r 12 r 14 r 15 14.30 85,533 0.1815

(14, 15) 88.06 286,501 0.19481 r 2 r 11 r 13 r 15 10.58 67,060 0.0092


20 ALEXOPOULOS

TABLE XI. Monte Carlo estimation of the entire distribution of the shortest path length for Example 3

r Fl(r) Fu(r) FO (r) Variance 1 107 Variance Reduction

51 0.03 0.03

52 0.3152 0.3152

54 0.3575 0.3575

56 0.4282 0.4534 0.4334 0.463 321.15

58 0.5105 0.5408 0.5291 0.806 365.52

60 0.6583 0.7991 0.7066 2.981 50.36

62 0.7388 0.9087 0.8446 3.775 29.29

64 0.7596 0.9661 0.8965 3.983 17.45

66 0.7623 0.9749 0.9226 3.280 16.75

68 0.7747 0.9904 0.9491 2.623 14.05

70 0.7793 0.9957 0.9657 2.194 10.41

75 0.7829 0.9997 0.9948 0.496 5.62

80 0.8785 0.99997 0.9994 0.063 5.46

85 0.9782 0.999994 0.9998 0.016 5.16

The bounds in columns 2 and 3 were computed after 10,000 intervals were partitioned by using a heap. The sample size was n Å 20,000 andthe entries in columns 5 and 6 were estimated.

ple space. On the other hand, 1000 partitions with the of the system state space and gain their effectiveness fromtheir ability to generate bounds and information that canundetermined sets maintained with the use of heaps pro-

duced the reasonably tight bounds in columns 5 and 6. be used for constructing simple and efficient Monte Carlosampling plans. These methods can also be applied toTables IX and X contain selected criticality indices.

As with Tables III and IV, the evaluation of a path criti- the computation of other system characteristics as theconditional probability that the shortest path length ex-cality index requires roughly three to four times as few

partitions as does the evaluation of an arc criticality index. ceeds a given value given that a specified arc belongs toa shortest path.The results in Table XI illustrate the method in Section

3 for estimating the entire distribution of L . The boundsin columns 2 and 3 resulted after only 10,000 intervalswere partitioned with the undetermined intervals pro- REFERENCEScessed via a heap with a root corresponding to the mostprobable set. Note that the computation of F(r) for r

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