The most likely path on series-parallel networks

The Most Likely Path on Series-Parallel Networks

Daniel ReichSchool of Business, Universidad Adolfo Ibáñez, Santiago, Chile

Program in Applied Mathematics, University of Arizona, Tucson, Arizona 85721

Leo LopesSchool of Mathematical Sciences, Monash University, Melbourne, Victoria 3800, Australia

Department of Systems and Industrial Engineering, University of Arizona, Tucson, Arizona 85721

In this article, we present a stochastic shortest pathproblem that we refer to as the Most Likely Path Prob-lem (MLPP). We demonstrate that optimal solutions tothe MLPP are not composed of optimal subpaths, whichlimits the computational tractability of exact solutionmethods. On series-parallel networks, we produce ana-lytical bounds for the MLPP’s optimality indices, theprobabilities of given paths in the network being short-est, and compute these bounds efficiently via numericalintegration. These bounds can also be used indepen-dently of the MLPP to gain further understanding forpaths of interest that are identified by other stochas-tic shortest path frameworks, e.g., robust shortest pathsor expected shortest paths. Additionally, we present aheuristic method that uses dynamic programming andordinal optimization to identify an MLP on series-parallelnetworks. Our computational study shows our boundsto be tight in a majority of test networks and shows ourheuristic to be both efficient and highly accurate for iden-tifying an MLP in all test networks. © 2010 Wiley Periodicals,Inc. NETWORKS, Vol. 58(1), 68–80 2011

Keywords: optimization under uncertainty; shortest path; series-parallel; probability bounds

1. INTRODUCTION

Shortest path problems are of great interest in networktheory, in part because they have numerous practical applica-tions. In many circumstances, it does not suffice to considera network where the edge costs are deterministic. Depending

Received September 2008; accepted June 2010Correspondence to: D. Reich; e-mail: [email protected] grant sponsor: The National Science Foundation; Contract grantnumber: DMS-0602173DOI 10.1002/net.20416Published online 22 December 2010 in Wiley Online Library(wileyonlinelibrary.com).© 2010 Wiley Periodicals, Inc.

on the situation of interest, there are many possible ques-tions that can be asked: What is the path with the shortestexpected cost? Is there a path costing less than L with prob-ability greater than p? Is there a path whose worst case costis less than L? In this research we address the following tworelated questions: (1) What is the probability that a given pathis shortest? This probability is known as an optimality index.(2) Which path is the most likely to be shortest? We call thispath a Most Likely Path (MLP) and the associated problemthe Most Likely Path Problem (MLPP).

In Project Evaluation and Review Technique (PERT)applications [15, 25], one is interested in critical (longest)paths rather than shortest ones. On directed acyclic networks,these problems can be interchanged by negating all edgecosts. As a consequence, in stochastic PERT networks, iden-tifying an MLP, its optimality index and optimality indices forother paths is of great importance. This information enablesmanagers to improve the overall project completion timeby crashing (improving the completion time of) tasks alongpaths that are likely to require the most time.

Aside from PERT, an MLP and optimality indices maybe applicable in other circumstances that require one-timedecisions, as opposed to recurring ones. For example, when aperson is late for an appointment that person prefers a drivingroute that is most likely to be quickest. If several courses fortreating a terminal disease are available, a patient may wishto consider how likely each treatment is to maximize qualityof life.

There are many useful frameworks for posing differentshortest path (or longest path) type questions on networkswith stochastic edge costs. The situations just described areones when the law of large numbers would not be applicableand, as a consequence, where expected values would not bean ideal measure. Our goal in this research is to expand themethodological arsenal available to the operations researchcommunity for solving stochastic shortest path problems.

NETWORKS—2011—DOI 10.1002/net

Recently, robust optimization frameworks have emergedfor modeling shortest paths involving stochasticity [5, 19,21, 28, 29]. In these frameworks, the optimal path is char-acterized by the superiority of its worst-case performanceto that of the other paths in the network. The two perfor-mance measures most commonly considered in the literatureare absolute robustness and robust deviation. Kouvelis and Yu[19] (see also [28]) proved that for both absolute robustnessand robust deviation, the robust shortest path problem witha discrete scenario set is NP-complete, even if there are onlytwo scenarios per edge and the topology is restricted to lay-ered networks of width 2. Yu and Yang [28] further showedthese problems to be strongly NP-hard if the number of sce-narios is unbounded. Bertsimas and Sim [5] showed that witha continuous scenario set and a predefined limit on the num-ber of edges that may experience uncertainty under any givenscenario, the absolute robustness shortest path problem ispolynomially solvable. Zielinski [29] showed that the robustdeviation shortest path problem with a continuous scenarioset is NP-hard and Kasperski and Zielinski [17] proved thatthis problem remains NP-hard on the class of series-parallelnetworks.

Perhaps the most used technique for finding optimal pathsthrough stochastic networks is to maximize the expectedvalue of a utility function. Loui [20] uses this technique toextend the von Neumann-Morgenstern (see [12]) formulationfor identifying ideal methods of evaluation in the presenceof uncertainty. Bard and Miller [4] solved constrained prob-abilistic shortest path problems by using utility functionsto weight dynamic programming solutions to determinis-tic networks. Utility functions that vary inversely with pathcost have also been used to provide dynamic programmingapproaches to stochastic problems by Bard and Bennett [3]and Eiger et al. [11].

Aside from expected value problems, the other mainbranch of research in nondeterministic shortest path prob-lems focuses on the use of a probability measure, that is,distribution functions and joint probability functions. Fora given network, many edges may be present in multiplepaths, resulting in dependent random variables and there-fore complicated probability measures. Frank [14] identifiedan exact method for obtaining the distribution function ofthe minimum cost path through a network with random edgecosts. His solution requires the use of multivariate integra-tion, which in general can be computationally burdensome,if not intractable. Consequently, he provides a sampling basedmethod for statistically analyzing the distribution functions.Burt and Garman [7] develop an alternate sampling methodthat uses conditional Monte Carlo simulation for estimat-ing distribution functions along paths in stochastic networks.Adlakha [1] combines ideas from the cutset approach of Sigalet al. [23] with aspects of the simulation method of Burt andGarman [7] to compute distribution functions of the minimumcost paths in stochastic networks.

Sigal et al. [24] aim to compute optimality indices forpaths in a given network. To reduce the path dimensionsof joint probability functions, which are nontrivial due to

dependence of random path cost variables, Sigal et al. [24]introduce a cutset approach for dividing a network intoindependent and dependent path sets. However, evaluatingthe joint probability function of the remaining dependentpath sets remains a non-trivial and burdensome computation.Their method of solution is theoretically precise, and theirarticle is frequently cited, but rarely extended, due to com-putational difficulties involving multivariate integration andcombinatorial numbers of paths.

Alexopoulos [2] builds upon the work of Sigal et al. [24]in the case where edge costs are discrete independent randomvariables. To reduce the computational burden, Alexopoulos[2] proposes a method for iteratively partitioning the statespace. At each iteration, bounds for the optimality indicesare provided, which eventually converge to the index value.

We expand on the work of Sigal et al. [24] and Alex-opoulos [2] by producing analytical lower and upper bounds,which can be computed efficiently via numerical integration,for the optimality index of any given path in a series-parallel network. For the MLPP, we show that optimalityof subpaths does not hold, which limits the computationaltractability of identifying an MLP through exact solutionmethods. We present an efficient heuristic method, which uti-lizes dynamic programming, ordinal optimization and MonteCarlo sampling, for identifying an MLP on the class of series-parallel networks. This heuristic approach avoids the highcomputational cost associated with multivariate integration.

Series-parallel networks have been studied extensively inelectrical engineering and have been applied to network flowproblems [6, 9, 18, 22, 27]. Valdes et al. [26] present a linear-time algorithm for recognizing series-parallel networks.

The remainder of this article is organized as follows. InSection 2, we provide background graph-theoretic conceptsand formulate the MLPP. Section 3 provides examples thatdistinguish the MLPP from other stochastic shortest pathframeworks and show that subpath optimality does not holdfor the MLPP, even on restricted network topologies. Section4 provides a formal definition of series-parallel networks andintroduces a new subclass of essential series-parallel net-works. Section 5 presents analytical lower and upper boundsfor optimality indices on series-parallel networks. Section6 introduces a heuristic method for identifying an MLP onseries-parallel networks. Section 7 summarizes our compu-tational results. Finally, Section 8 presents consequences ofour work and future research directions.

2. FORMULATION OF THE MOST LIKELY PATHPROBLEM

Given a network topology and its corresponding edge cost(or edge length) vector, the deterministic shortest path prob-lem can be solved easily as either a dynamic programmingproblem or as a linear programming problem. However inmany situations, portions of the network are unobservable andthe cost vector is therefore unknown. The MLPP is intendedto analyze cases where at least part of a given cost vector iscomposed of independent random variables with probability

NETWORKS—2011—DOI 10.1002/net 69

density functions. For realizations of these random variables,solutions to the deterministic shortest path problem can beobtained. We wish to find a solution that would be obtainedmost frequently under repeated realizations, that is, a pathwith greatest probability of being shortest.

In this article, we use the notation in Cook et al. [10]whenever possible, augmenting it when necessary for deal-ing with series-parallel networks. Let Gst = (Vst , Est) be adirected network with source s and sink t, where Vst andEst are its vertex and edge sets, respectively. For any nodesu, v ∈ Vst , let Guv = (Vuv, Euv) be the subnetwork of Gst

with Euv = {e ∈ Est : ∃ a u − v path containing e} andVuv = {i ∈ Vst : ∃ a u − v path passing through node i}.Let Kuv be the set of all u − v paths through Guv. For the sakeof brevity, k ∈ Kuv is used to represent not only a u − v pathbut also the edge and vertex sets composed of the edges in kand vertices in k, respectively.

Definition 1 (Most Likely Path Problem). Consider a net-work Gst where at least part of the edge cost vector C iscomposed of continuous independent random variables. Wemodel all deterministic costs as degenerate random vari-ables with Dirac-Delta probability density functions, whichare centered at their respective deterministic costs. (Note:Dirac-Delta functions are not Lebesgue integrable and aretherefore not probability density functions in the strict senseof the terminology. However, throughout the remainder ofthis article, we treat them as probability density functions toavoid burdensome notation, as they satisfy all required prop-erties for any proofs that follow.) Let fC denote the vector ofprobability density functions corresponding to C. Let the costof each path i ∈ Kst be defined as �i = ∑

e∈i Ce.The Most Likely Path Problem is the problem of finding an

s − t path (not necessarily unique) with highest probabilityof being the shortest path, i.e., a path r∗

st satisfying

r∗st ∈ arg max

i∈Kst

P(�i = min

k∈Kst

�k), (1)

where

P(�i = min

k∈Kst

�k) = P

⋂

k∈Kst\{i}{�i ≤ �k}

. (2)

We refer to r∗st as a Most Likely Path.

In the MLPP, deterministic costs represent certainty on thepart of the modeler about parts of a given network, whereasstochastic costs represent uncertainty. When multiple equiv-alent deterministic cost paths remain in any given network, ifany of these equivalent deterministic cost paths has non-zeroprobability of being an MLP, then

∑i∈Kst

P(�i = min

k∈Kst

�k)

> 1 (3)

because the deterministic edges introduce non-disjoint eventswith positive measure. Otherwise, the interpretation of thisprobability is intuitive, i.e.,

FIG. 1. In the network above, the MLP is the path with cost X and is notan expected shortest path. The costs on all edges are independent continuousrandom variables uniformly distributed on the given intervals.

∑i∈Kst

P(�i = min

k∈Kst

�k) = 1,

since: (i) continuous random variables (random costs) are notequal to one another (almost surely); (ii) continuous randomvariables (random costs) are not equal to deterministic costs(almost surely); and (iii) fixed path costs are either not equalto one another or are never optimal.

The probability in (2) is commonly referred to as an opti-mality index for path i [2, 24]. Although the random edgecosts in C are independent of one another, the random pathcosts are not, since edges may appear in multiple paths.Consequently, computing optimality indices is difficult ongeneral networks; known solution methods require evalu-ating the joint probability function in (2), which in turnrequires multivariate integration. An alternative is to usea Monte Carlo method, but even this becomes intractablefor large networks. The bounds we introduce for optimalityindices on series-parallel networks, although these bounds arenot guaranteed to be tight, are easily computed via numer-ical integration. Combining these analytical bounds with aheuristic method for identifying an MLP on series-parallelnetworks, we introduce a computationally efficient methodfor obtaining solutions to the MLPP.

3. IMPORTANT EXAMPLES FORUNDERSTANDING THE MOST LIKELY PATHPROBLEM

This section highlights properties of the MLPP that distin-guish it from both deterministic and other stochastic shortestpath problems. The examples we introduce are intended toprovide insight and intuition into the MLPP while demon-strating in a straightforward manner several key difficultiesin solving this problem.

3.1. MLP versus Expected Shortest Path

Figure 1 provides an example where the MLP has thehighest expected cost of any path in the network. To seethis, let X ∼ Uniform(0, 3) be the cost of edge (s, t),

70 NETWORKS—2011—DOI 10.1002/net

Y ∼ Uniform(0, 1) be the cost of edge (s, u) and Zk ∼Uniform(0, 1), k ∈ 1, . . . , 6, be the cost of the parallel edges(u, t). Every s − t path k passing through subnetwork Gut hasan expected cost of E[Y +Zk] = E[Y ]+E[Zk] = 1, whereasthe s − t path consisting solely of edge (s, t) has an expectedcost of E[X] = 1.5, which is the highest. Consequently, inan expected shortest path problem an optimal solution wouldbe any path passing through subnetwork Gut .

The following calculation shows that the solution to theMLPP on this network is in fact the path transversing edge(s, t). Edge (s, t) will have a lower cost than any competingpath when X ≤ Y . It follows that

P( ∩6

i=1 {X ≤ Y + Zi})

> P(X ≤ Y)

= P(X ≤ Y |X ≤ 1)P(X ≤ 1)

=(

1

2

) (1

3

)= 1

6.

But since there are six equivalent competing s − t paths kcontaining a u − t subpath it follows that

P(∩k �=i{Y + Zk ≤ Y + Zi} ∩ {Y + Zk ≤ X})< P(∩k �=i{Y + Zk ≤ Y + Zi})

= P(∩k �=i{Zk ≤ Zi}) = 1

6.

Therefore the path consisting of edge (s, t) is the MLP, since

P( ∩6

i=1 {X ≤ Y + Zi})

>1

6

> P(∩k �=i{Y + Zk ≤ Y + Zi} ∩ {Y + Zk} ≤ X)

for every competing s − t path k containing a u − t subpath.A second and related observation from the example in

Figure 1 is that the MLP cannot be found by pairwise com-parisons of paths. When compared with all six competingpaths k containing a u − t subpath, the MLP is less likely tobe shorter, i.e.,

P(X ≤ Y + Zk) = P(X ≤ Y + Zk|X ≤ 2)P(X ≤ 2)

=(

1

2

) (2

3

)= 1

3.

3.2. Lack of Optimality of Subpaths

In deterministic shortest path problems, subpath opti-mality allows for dynamic methods of solution. Unfortu-nately this property does not extend to the MLPP, even onseries-parallel networks.

Consider the network in Figure 2, where the threeindependent (since W = 0) path costs are given byX ∼ Uniform(0, 3 + ε), Y ∼ Uniform(1, 2), and Z ∼Uniform(0, 10). For every ε ≥ 0, the optimality indices forthe three s − t paths are

P(X ≤ Y , X ≤ Z}) = 83

180 + 60ε, (4)

FIG. 2. In the network above, the MLP is the path from s to u to t withcost X . However, on subnetwork Gut , the MLP is the path with cost Y , whichshows that subpath optimality does not hold for the MLPP. The costs X, Y ,and Z are independent continuous random variables uniformly distributedon the given intervals and W = 0 is deterministic.

P(Y ≤ X, Y ≤ Z}) = 77 + 51ε

180 + 60ε(5)

and

P(Z ≤ Y , Z ≤ X}) = 20 + 9ε

180 + 60ε. (6)

So the path with cost X is the unique MLP on Gst for ε ∈[0, 6/51).

However, the optimality indices for the two u− t subpathson Gut are given by

P(X ≤ Y) = 3

6 + 2ε(7)

and

P(Y ≤ X) = 3 + 2ε

6 + 2ε. (8)

So the u − t subpath with cost Y is the unique MLP on sub-network Gut for every ε > 0. This example demonstratesa paradoxical quality of the MLPP and in doing so showsthat subpath optimality does not hold even on series-parallelnetworks with independent uniformly distributed path costs.

The calculations for computing the probabilities in (4),(5), (6), (7) and (8) are given in detail in Appendix 1.

4. SERIES-PARALLEL AND ESSENTIALSERIES-PARALLEL NETWORKS

In this section, we introduce the class of series-parallelnetworks and then define the new specialized class of essen-tial series-parallel networks. We provide an algorithm fortransforming any series-parallel network, with respect to anyof its paths, into an essential series-parallel network. We thenuse the essential networks to provide analytical lower andupper bounds for the optimality indices of paths in the originalnetwork.

4.1. Series-Parallel Networks

Definition 2 (Series-Parallel Networks [6]). A network Gst

is called series-parallel (also known as two-terminal series-parallel, edge series-parallel, strongly series-parallel) if it


FIG. 3. The network on the left is series-parallel. The one on the right is not, i.e., notice edge (1,3).

can be constructed by combinations of the following threerules:

1. Base Network: The network Gst(Vst , Est) with Vst ={s, t}, Est = {(s, t)} is series-parallel.

2. Parallel Composition: Let Gs1t1(Vs1t1 , Es1t1) andGs2t2 (Vs2t2 , Es2t2 ) be two series-parallel networks withsources s1 and s2 and sinks t1 and t2, respectively. Weform the series-parallel network Gsptp(Vsptp , Esptp) by set-ting sp = s1 = s2 and tp = t1 = t2, where Esptp = Es1t1 ∪Es2t2 and Vsptp = {Vs1t1 ∪ Vs2t2 ∪ sp ∪ tp}\{s1, t1, s2, t2}.

3. Series Composition: We form the series-parallel net-work Gs1t2 (Vs1t2 , Es1t2 ) by setting t1 = s2, where Es1t2 =Es1t1 ∪ Es2t2 and Vs1t2 = {Vs1t1 ∪ Vs2t2 }\s2. Alterna-tively by setting t2 = s1, we form series-parallel networkGs2t1(Vs2t1 , Es2t1) in the same way.

Figure 3 provides an example of a series-parallel networkalong with a slightly perturbed network that is not series-parallel.

4.2. Essential Series-Parallel Networks

We obtain bounds for the optimality indices of any path ina series-parallel network via a simplified network topology,which we define as follows.

Definition 3 (Essential Series-Parallel Networks). A net-work Gst(Vst , Est) is essential series-parallel if:

1. There exists an ordering function f : Vst → N, suchthat f (s) = 1, f (t) = |Vst |, f (s) < f (v) < f (t) for allv ∈ Vst\{s, t}, and f (vi) �= f (vj) for all vi �= vj ∈ Vst .

2. If f (vi) − f (vj) = 1, then there exists either one or twoedges from vi to vj for all vi, vj ∈ Vst .

3. If f (vi) − f (vj) �= 1, then there exists at most one edgefrom vi to vj for all vi, vj ∈ Vst .

4. Gst(Vst , Est) is series-parallel.

Figure 4 provides an example of an essential series-parallel network.

4.3. From Series-Parallel to Essential Series-Parallel

We now describe a method for transforming a series-parallel network to an essential series-parallel network basedon any given s − t path in Gst .

4.3.1. Topological Transformation:

Theorem 1. Any series-parallel network Gst can be reducedto an essential series-parallel network Gst via Algorithm 1.

Proof of Theorem 1. First, let us consider the two opera-tions present in Algorithm 1: parallel and series contractions.Parallel contractions occur whenever there are two or moreedges between any two nodes in the network and both thoseedges are not in path k. Series contractions occur wheneverthere is a node in the network not in path k and that nodehas exactly one edge entering it and one edge leaving it.Parallel and series contractions are reversals of parallel andseries compositions (Definition 2), respectively, which areperformed on two base networks so that only one of those basenetworks remain. Accordingly, by Definition 2 neither par-allel nor series contractions compromise the series-paralleltopology.

Observe that the stopping condition for Algorithm 1 guar-antees that upon termination there do not exist any parallel

FIG. 4. The network on the left is series-parallel. The one on the right is essential series-parallel andis formed from the one on the left. [Color figure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]


Algorithm 1 Reduce any series-parallel network to anessential series-parallel network.

Choose any s − t path k ∈ Kst .Vst = Vst .Est = Est .Base Graph Contractions Remain = TRUE.while Base Graph Contractions Remain == TRUE do

Base Graph Contractions Remain = FALSE.for all (u, v) ∈ Est\k do

if there exists more than one edge (u, v) ∈ Est\kthen

Parallel Contraction: Remove from Est all butone edge (u, v) ∈ Est\k.Base Graph Contractions Remain = TRUE.

end ifend forfor all (u, v) ∈ Est\k do

if ∃q ∈ Vst such that (q, u) ∈ Est\k thenif �q′ ∈ Vst such that (q′, u) ∈ Est\(q, u) and�v′ ∈ Vst such that (u, v′) ∈ Est\(u, v) then

Series Contraction: Est ={Est\{(q, u), (u, v)}} ∪ {(q, v)}. Vst = Vst\{u}.Base Graph Contractions Remain = TRUE.

end ifend if

end forend whilereturn Gst

edges between two nodes such that both edges are not in pathk and there does not exist any node outside of path k withexactly one edge entering and one edge leaving.

Next, we show that as a result of the series contractions,Vst consists only of nodes in Vst that are connected to edgesin k. Assume this was not true. Then there must exist somenode v ∈ Vst not on path k such that there exists at least oneedge (u′, v) ∈ Est\k for u′ ∈ Vst and there exists at least oneedge (v, r′) ∈ Est\k for r′ ∈ Vst .

Let Ein.v be the set of edges {(u′, v) ∈ Est\k : u′ ∈ Vst}.

Since Gst is series-parallel, v must be the sink of some series-parallel network G.v whose edge set contains Ein

.v . Since theedges in k are consecutively connected, the sink of G.v wouldhave to be connected to one of the edges in k if G.v containedany edge in k. Therefore, since v is not on path k, G.v must notcontain any edge in k. It follows from the stopping conditionof Algorithm 1 that G.v must be a base network with exactlyone edge.

Similarly, we can identify and reduce the series-parallelnetwork Gv. whose edge set contains {(v, r′) ∈ Est\k : r′ ∈Vst}. Hence, without loss of generality we can consider thecase when there is exactly one edge (u′, v) ∈ Est\k for someu′ ∈ Vst and one edge (v, r′) ∈ Est\k for some r′ ∈ Vst .However, these two edges are replaced by a single edge duringthe series contraction step and node v is removed, which is acontradiction to our assumption.

We have just shown that Vst consists only of nodes in Vst

that are connected to edges in k. Accordingly, we can orderthese nodes by consecutive integers based on the order of theedges in k: f (s) = 1, f (v) = 2 for (s, v) ∈ k, etc.

Let us now show that the four components of Definition 3(essential series-parallel networks) hold.

1. By construction, the node ordering given above satisfiesf : Vst → N, such that f (s) = 1, f (t) = |Vst |, f (s) <

f (v) < f (t) for all v ∈ Vst\{s, t}, and f (vi) �= f (vj) forall vi �= vj ∈ Vst .

2. By construction, whenever f (vi)− f (vj) = 1 there existsan edge (vi, vj) ∈ k. By the parallel contraction step,there cannot exist more than one other edge from vi tovj .

3. By construction, whenever f (vi)− f (vj) �= 1 there existsno edge (vi, vj) ∈ k. By the parallel contraction step,there cannot exist more than one edge from vi to vj .

4. Neither parallel contractions nor series contractionscompromise the series-parallel topology of a network,as shown above. Therefore the resulting network mustbe series-parallel.

Therefore any series-parallel network Gst can be reducedto an essential series-parallel network Gst via Algorithm 1. ■

We have just described the method for obtaining the essen-tial series-parallel network topology of interest. To calculateoptimality indices, we must also ensure that the probabilitymeasure of interest will remain unchanged on this new topol-ogy. Next, we provide the necessary probability backgroundinformation for updating these random costs.

4.3.2. Updating Edge Costs for Parallel Contractions:Let X and Y be independent random costs, with distributionfunctions FX and FY , respectively. The distribution of therandom variable C := min{X, Y} can be found as follows.Since for any c,

P(C > c) = P(X > c)P(Y > c),

we have that

FC(x) = 1 − ([1 − FX(x)][1 − FY (x)]), (9)

where FC is the cumulative distribution of C. Therefore, ateach parallel contraction the cost distribution of the remainingedge is found by applying the multiplication operator (9).

4.3.3. Updating Edge Costs for Series Contractions: LetX and Y be independent random costs, with probability den-sity functions fX and fY . The distribution of the randomvariable C := X + Y is given by

P(X + Y ≤ c) =∫ ∫

x+y≤cfX(x)fY (y)dxdy

=∫ ∞

−∞FX(c − y)fY (y)dy = FX � fY . (10)


Therefore, at each series contraction, the cost distributionof the new edge is found by applying the convolutionoperator (10).

5. LOWER AND UPPER BOUNDS FOROPTIMALITY INDICES

In this section we provide analytical lower and upperbounds for optimality indices of any given path in a series-parallel network. To do so, we use the essential series-parallelnetwork constructed with respect to the given path. Both thelower and upper bounds we provide are computed by one-dimensional integration and therefore avoid the complexityof the multivariate integral required to compute the exactoptimality indices.

5.1. Lower Bound on Essential Series-Parallel Networks

We say that two events A and B are positively correlated ifP(A ∩ B) ≥ P(A)P(B). Consider an essential series-parallelnetwork Gst that was constructed with respect to path rst . Inthis section, we show that all events corresponding to path rst

in Gst are positively correlated. We then use this correlationto separate our objective (2) on the essential network intomultiple one-dimensional integrals, thereby producing a validlower bound.

In order to show the needed inequality, i.e., positive cor-relation, we measure indicator functions of these events viathe expected value, using the results from the following the-orem. The proof of Theorem 2 is analogous to the proof byChayes et al. [8] of the Harris-FKG Inequality [13], which iscommonly used in percolation theory. For the details of thisproof, please refer to Appendix 2.

Theorem 2. Let θn : Rn → R and φn : Rn → R be non-increasing functions, i.e., for every i, θn(x1, . . . , xi, . . . , xn) ≤θn(x1, . . . , xi, . . . , xn) if xi ≥ xi and φn(x1, . . . , xi, . . . , xn) ≤φn(x1, . . . , xi, . . . , xn) if xi ≥ xi. Let X1, . . . , Xn be indepen-dent real-valued random variables with probability densityfunctions f1, . . . , fn, respectively. Then

E[θn(X1, . . . , Xn)φn(X1, . . . , Xn)]≥ E[θn(X1, . . . , Xn)]E[φn(X1, . . . , Xn)]. ■

In the following corollaries, we use Theorem 2 to developa lower bound for the optimality index of a given path rst .In Corollary 1, we prove the validity of this lower boundon essential networks where all aggregate edge costs havedeterministic costs. In Corollary 2, we generalize the resultof Corollary 1 to essential networks with random aggregateedge costs.

Corollary 1. Let E′ be the set of aggregate edges in Gst , i.e.,Est = {e ∈ Est : e ∈ rst} ∪ E′. Let Xi,i+1, i = 1, . . . , |V |, beindependent real-valued random variables with probabilitydensity function fi,i+1, respectively, representing the edge cost

of rst from node i ∈ Vst to node i + 1 ∈ Vst . Let aij ∈ R,(i, j) ∈ E′, be the fixed edge cost of the aggregate path fromnode i ∈ Vst to node j ∈ Vst . Then the probability that rst isa shortest path is given by

P

⋂

(i,j)∈E′

j−1∑k=i

Xk,k+1 ≤ aij

and its corresponding lower bound by

∏(i,j)∈E′

P

j−1∑

k=i

Xi,i+1 ≤ aij

.

Proof of Corollary 1. Let Iij(xi,i+1, . . . , xj−1,j) , (i, j) ∈E′ be indicator functions such that

Iij(xi,i+1, . . . , xj−1,j) ={

1 if∑j−1

k=i xk,k+1 ≤ aij

0 otherwise.

Since Iij(xi,i+1, . . . , xj−1,j) are non-increasing functions,we can apply Theorem 2 iteratively to obtain

P

⋂

(i,j)∈E′

j−1∑k=i

Xk,k+1 ≤ aij

= E

∏

(i,j)∈E′Iij(Xi,i+1, . . . , Xj−1,j)

≥∏

(i,j)∈E′E[Iij(Xi,i+1, . . . , Xj−1,j)]

=∏

(i,j)∈E′P

j−1∑

k=i

Xi,i+1 ≤ aij

.

■

In the following corollary, we generalize this probabilitylower bound to essential networks with random aggregateedge costs.

Corollary 2. Let Aij, (i, j) ∈ E′, be independent real-valued random variables with probability density functiongij, respectively, representing the cost of the aggregate edgefrom node i to node j in Gst . Then the probability of rst isgiven by

P

⋂

(i,j)∈E′

j−1∑k=i

Xk,k+1 ≤ Aij

(11)

and its corresponding lower bound by

∏(i,j)∈E′

P

j−1∑

k=i

Xi,i+1 ≤ Aij

.

Proof of Corollary 2. The probability above (11) can bewritten as the following iterated integral:


P

⋂

(i,j)∈E′

j−1∑k=i

Xk,k+1 ≤ Aij

=∫

. . .

∫P

⋂

(i,j)∈E′

j−1∑k=i

Xk,k+1 ≤ aij

∏

(i,j)∈E′gij(aij)daij

≥∫

. . .

∫ ∏(i,j)∈E′

P

j−1∑

k=i

Xi,i+1 ≤ aij

∏

(i,j)∈E′gij(aij)daij

=∫

. . .

∫ ∏(i,j)∈E′

P

j−1∑

k=i

Xi,i+1 ≤ aij

gij(aij)daij

=∏

(i,j)∈E′

∫P

j−1∑

k=i

Xi,i+1 ≤ aij

gij(aij)daij

=∏

(i,j)∈E′P

j−1∑

k=i

Xi,i+1 ≤ Aij

, (12)

where the inequality in (12) is obtained by applying the resultof Corollary 2. ■

It is important to note that the lower bound for theprobability of rst that we have developed in Corollary 2, i.e.,

P(�i = min

k∈Kst

�k) ≥

∏(i,j)∈E′

P

j−1∑

k=i

Xi,i+1 ≤ Aij

,

can be computed efficiently by multiple one-dimensionalintegrals.

5.2. Upper Bound on Essential Series-Parallel Networks

Now that we have obtained a lower bound for rst that canbe computed efficiently, we prove that a corresponding upperbound for (2) can also be computed by solving

mink∈Kst

P(�rst ≤ �k). (13)

The solution to problem (13) is a valid upper bound since forall k ∈ Kst\{rst},⋂

k∈Kst\{rst}{�rst ≤ �k} ⊂ {�rst ≤ �k}.

To solve (13), we replace all aggregate edge costs Aij, (i, j) ∈E′, with

pij = P

j−1∑

k=i

Xk,k+1 ≤ Aij

,

and replace the edge cost of r∗st from node i to node i+1 with

P(Xi,i+1 ≤ Xi,i+1) = 1. Problem (13) can then be restated as

mink∈Kst

∏(i,j)∈k

pij. (14)

Since the logarithm is an increasing function on (0, 1], wecan restate (14) as

exp

min

k∈Kst

log

∏

(i,j)∈k

pij

= exp

min

k∈Kst

∑(i,j)∈k

log(pij)

.

(15)

Problem (15) now has the form of a deterministic shortestpath problem, where the edge costs are

wij = log

P

j−1∑

k=i

Xk,k+1 ≤ Aij

.

Since the probabilities are in (0, 1], all edge costs are non-positive and bounded; and since essential series-parallelnetworks are acyclic, this deterministic shortest path prob-lem is guaranteed to yield a finite solution. Our upper boundcan then be obtained by exponentiating the result.

6. HEURISTIC APPROACH FOR IDENTIFYING ANMLP ON SERIES-PARALLEL NETWORKS

In this section, we introduce our heuristic for identifyingan MLP on the class of series-parallel networks and referto this algorithm as SPMLP. While we have already estab-lished in Section 3 that the MLPP does not have the desirableproperty of subpath optimality, we still expect this prop-erty to often be useful in practice and it serves as the basisfor our heuristic. SPMLP is a dynamic Monte Carlo sam-pling approach, which uses ordinal optimization to increasecomputational efficiency.

Consider a network Gst and two u − v subpaths r1uv and

r2uv on subnetwork Guv. We eliminate either r1

uv or r2uv as a

potential u − v subpath of the s − t solution by measuringwhich has a greater optimality index on subnetwork Guv.

Ho et al. [16] introduce ordinal optimization as a meansof comparing designs rather than estimating accurately theperformance measures of those designs. They argue that ordi-nal optimization provides a basis for efficient preprocessing,which then reduces the computational burden of the solu-tion step. In the context of the MLPP, precise measurementsof optimality indices on subnetworks could be obtained viaextensive Monte Carlo sampling; however, by utilizing ordi-nal optimization, we are able to limit the sampling neededby aiming to identify only which optimality index is greater,rather than measuring both precisely.

For each sample, we run Dijkstra’s algorithm on a realiza-tion of subnetwork Guv and introduce indicator functions to

TABLE 1. The optimality gaps.

Mean Gap Gap Std. Dev. Median Gap Max Gap

0.006 0.032 0 0.230

These gaps are computed as the differences between each probability upperbound computed by SPMLP and its corresponding lower bound.


TABLE 2. The optimality gaps.

Mean Gap Mean Gap Gap Std. Dev. Gap Std. Dev. Median Gap Median Gap Max Gap Max GapMC-LB UB-MC MC-LB UB-MC MC-LB UB-MC MC-LB UB-MC

0.004 0.002 0.020 0.012 0 0 0.150 0.093

These gaps are computed as the differences between each probability bound computed by SPMLP and its corresponding Monte Carlo probability estimate.

keep track of the number of times r1uv or r2

uv are the shortestpaths. Note that these indicator functions are random vari-ables whose expectations are the probabilities that r1

uv andr2

uv are the shortest paths within subnetwork Guv. By onlytracking one of the two indicator functions at each sample,we ensure that the indicator functions are independent fromone another. We use the following statistical analysis to deriveour stopping condition: Our null hypothesis is that the meansof the two indicator functions are equal, but the variancesare both unequal and unknown. After we reach 120 degreesof freedom, we compute a t-statistic at each iteration. Sincethe t distribution is essentially normal at this point, we usethe percentile from the normal distribution (1.960) for thistest. If we are able to establish the desired 95% confidenceinterval, we select the path whose sample mean is greater.If we are unable to establish this confidence interval within104 iterations (2×104 samples), we stop since the differencebetween the two subpaths in probability is insignificant.

By eliminating one of the two subpaths, we in turn elimi-nate from the set of potential solutions all paths containing theeliminated subpath. Therefore when Gst is series-parallel, weidentify a candidate solution to the MLPP in O(|Est |) steps.

7. COMPUTATIONS

We generated series-parallel networks to test the SPMLPalgorithm. We compared the SPMLP lower and upper boundresults to the approximate probabilities obtained from MonteCarlo sampling, as well as their respective running times. Wetested 10, 50, 100 and 250 edge networks with inputs of 25,50 and 75 percent for (i) the average percentage of edgeswith deterministic costs and (ii) the average percentage ofseries vs. parallel compositions. For each set of parameters,we tested four networks (144 total networks). These com-putations were performed on identical machines with IntelPentium 4 CPUs with Hyper-threading, 3.0 GHz, 4096 MiBRAM.

We designed Perl modules to generate random test net-works. First, we build series-parallel network topologies viarandom combinations of series and parallel compositions.The inputs to the module are the number of edges and theratio of series compositions to parallel ones. A second mod-ule then reads in the topology and randomly assigns eitherfixed or random costs to each edge of the network. The ran-dom costs are uniform random variables with integer lowerbounds uniformly distributed between 0 and 9; and integerupper bounds uniformly distributed between the correspond-ing lower bound and 10. The deterministic costs are integers

uniformly distributed between 2 and 8, so that each deter-ministic cost has an expected value that is both higher thanthe expected lower bound and lower than the expected upperbound of each random cost.

We implemented SPMLP using a combination of Perl andMATLAB. An MLP is identified via the dynamic samplingapproach detailed in Section 6. The network topology is thensimplified with respect to that solution, resulting in the cor-responding essential series-parallel network. The essentialnetwork distributions are represented as discretized approxi-mations to the actual continuous distributions, computed viaFast Fourier Transforms and point-wise products. For the dis-cretized distributions, we use a uniform domain vector withgrid spacing of 10−3, in order to minimize both running timeand computational error introduced by the discretization. Theprobabilities of interest are then computed by Riemann sumapproximations from the discretized distributions. Finally,these probabilities are used to compute our lower and upperbounds.

In order to test the accuracy of the bounds produced,we implemented sequential Monte Carlo sampling with adesired sample standard deviation of 10−3. If the desired sam-ple standard deviation is not obtained within 106 iterations,the sampling is terminated. Out of the 144 networks tested,our probability bounds yielded exact solutions in 134 cases.The ten cases where a gap is present result from conditions

FIG. 5. The plot above shows the running times achieved by both SPMLPand sequential Monte Carlo sampling. [Color figure can be viewed in theonline issue, which is available at wileyonlinelibrary.com.]


FIG. 6. The curve on the left-hand plot shows that the actual running time for the MLPP grows at a rate greaterthan |Est |2, where |Est | is the number of edges in the network. The downward-sloping curve on the right-hand plotshows that the running time for the MLPP grows at a slower rate than |Est |3. [Color figure can be viewed in theonline issue, which is available at wileyonlinelibrary.com.]

where there are nested aggregate paths in the essential net-work whose distributions are highly competitive with that ofthe projected MLP. The optimality gaps between the SPMLPbounds are summarized in Table 1. The gaps between theactual probabilities, computed via sequential Monte Carlo,and the bounds computed by SPMLP, are summarized inTable 2. The latter indicates that the gaps are not gener-ally centered around the actual probability; rather the lowerbounds are on average about twice as far from the actualprobability than the corresponding upper bounds.

Figure 5 compares the running time of SPMLP versus thatof sequential Monte Carlo sampling. The average runningtime of the sequential Monte Carlo sampling is increasing ata greater rate than is the average running time of SPMLP, asis evidenced by the trend curves in Figure 5. Moreover, theaverage running time of Monte Carlo sampling is approxi-mately two orders of magnitude greater than that of SPMLP.Notice that in practical terms, for a 250 node network, this isthe difference between an average of about five minutes andan average of about 10 hours.

The SPMLP has a polynomial running time based onthe running times of Dijkstra’s algorithm, discretized FastFourier Transforms and numerical integration. In order toanalyze the running time that was achieved in our compu-tations, we plot normalized time as a function of networksize. The two plots in Figure 6 show that the practical growthrate is between |Est |2 and |Est |3, where |Est | is the number ofedges in the network.

8. SUMMARY AND CONCLUSIONS

For series-parallel networks having random edge costs,we have established analytical lower and upper bounds foroptimality indices. Through our computational results, wehave verified that these bounds can be computed efficiently.We have demonstrated that in many of the random networks

tested, the lower and upper bounds agree, providing an exactsolution.

Additionally, we have developed a dynamic samplingheuristic for identifying a solution to the MLPP and in doingso, we have utilized ordinal optimization in the context ofstochastic network optimization.

A next step is to approximate more general networkswhich have series-parallel networks as subgraph isomor-phisms. One method for accomplishing this may be tocompute conditional distributions on the small portions ofa network that prevent it from being series-parallel.

In future research, the methodological advances achievedhere can be adapted for problems where one has cost distri-butions on individual locations or events and is interested inanalyzing processes that result from alternative sequences ofthose events.

APPENDIX 1

Detailed Calculation of Probabilities in Subsection 3.2

Let X ∼ Uniform(0, 3 + ε), Y ∼ Uniform(1, 2), Z ∼Uniform(0, 10) and ε ≥ 0.

Equation (4):

P(X ≤ Y , X ≤ Z})= P({X ≤ 1, Z ≥ 1} ∪ {X ≤ Z , X ≤ 1, Z ≤ 1}

∪ {X ≤ Y , X ∈ [1, 2], Z ≥ 2}∪ {X ≤ Y , X ≤ Z , X ∈ [1, 2], Z ∈ [1, 2]})

= P(X ≤ 1, Z ≥ 1) + P(X ≤ Z , X ≤ 1, Z ≤ 1)

+ P(X ≤ Y , X ∈ [1, 2], Z ≥ 2)

+ P(X ≤ Y , X ≤ Z , X ∈ [1, 2], Z ∈ [1, 2])= P(X ≤ 1)P(Z ≥ 1)


+ P(X ≤ Z|X ≤ 1, Z ≤ 1)P(X ≤ 1)P(Z ≤ 1)

+ P(X ≤ Y |X ∈ [1, 2])P(X ∈ [1, 2])P(Z ≥ 2)

+ P(X ≤ Y , X

≤ Z|X ∈ [1, 2], Z ∈ [1, 2])P(X ∈ [1, 2])P(Z ∈ [1, 2])

=(

1

3 + ε

) (9

10

)+

(1

2

) (1

3 + ε

) (1

10

)

+(

1

2

) (1

3 + ε

) (8

10

)+

(1

3

) (1

3 + ε

) (1

10

)

= 83

180 + 60ε.

Equation (5):

P(Y ≤ X, Y ≤ Z})= P({Z ≥ 2, X ≥ 2} ∪ {Y ≤ Z , Z ∈ [1, 2], X ≥ 2}

∪ {Y ≤ X , X ∈ [1, 2], Z ≥ 2}∪ {Y ≤ X, X ∈ [1, 2], Y ≤ Z , Z ∈ [1, 2]})

= P(Z ≥ 2, X ≥ 2) + P(Y ≤ Z , Z ∈ [1, 2], X ≥ 2)

+ P(Y ≤ X , X ∈ [1, 2], Z ≥ 2)

+ P(Y ≤ X, X ∈ [1, 2], Y ≤ Z , Z ∈ [1, 2])= P(Z ≥ 2)P(X ≥ 2)

+ P(Y ≤ Z|Z ∈ [1, 2])P(Z ∈ [1, 2])P(X ≥ 2)

+ P(Y ≤ X|X ∈ [1, 2])P(X ∈ [1, 2])P(Z ≥ 2)

+ P(Y ≤ X , Y ≤ Z|X ∈ [1, 2],Z ∈ [1, 2])P(X ∈ [1, 2])P(Z ∈ [1, 2])

=(

8

10

) (1 + ε

3 + ε

)+

(1

2

) (1

10

) (1 + ε

3 + ε

)

+(

1

2

) (1

3 + ε

) (8

10

)+

(1

3

) (1

3 + ε

) (1

10

)

= 77 + 51ε

180 + 60ε.

Equation (6):

P(Z ≤ Y , Z ≤ X})= P({Z ≤ 1, X ≥ 1} ∪ {Z ≤ X , Z ≤ 1, X ≤ 1}

∪ {Z ≤ Y , Z ∈ [1, 2], X ≥ 2}∪ {Z ≤ Y , Z ≤ X , Z ∈ [1, 2], X ∈ [1, 2]})

= P(Z ≤ 1, X ≥ 1) + P(Z ≤ X , Z ≤ 1, X ≤ 1)

+ P(Z ≤ Y , Z ∈ [1, 2], X ≥ 2)

+ P(Z ≤ Y , Z ≤ X , Z ∈ [1, 2], X ∈ [1, 2])= P(Z ≤ 1)P(X ≥ 1)

+ P(Z ≤ X|Z ≤ 1, X ≤ 1)P(Z ≤ 1)P(X ≤ 1)

+ P(Z ≤ Y |Z ∈ [1, 2])P(Z ∈ [1, 2])P(X ≥ 2)

+ P(Z ≤ Y , Z ≤ X|Z ∈ [1, 2],X ∈ [1, 2])P(Z ∈ [1, 2])P(X ∈ [1, 2])

=(

1

10

) (2 + ε

3 + ε

)+

(1

2

) (1

10

) (1

3 + ε

)

+(

1

2

) (1

10

) (1 + ε

3 + ε

)+

(1

3

) (1

10

) (1

3 + ε

)

= 20 + 9ε

180 + 60ε.

Equation (7):

P(X ≤ Y) = P({X ≤ 1} ∪ {X ≤ Y , X ∈ [1, 2]})= P(X ≤ 1) + P(X ≤ Y |X ∈ [1, 2])P(X ∈ [1, 2])

= 1

3 + ε+

(1

2

) (1

3 + ε

)= 3

6 + 2ε.

Equation (8):

P(Y ≤ X) = P({X ≥ 2} ∪ {Y ≤ X, X ∈ [1, 2]}= P(X ≥ 2) + P(Y ≤ X|X ∈ [1, 2])P(X ∈ [1, 2])

= 1 + ε

3 + ε+

(1

2

) (1

3 + ε

)= 3 + 2ε

6 + 2ε.

APPENDIX 2

Proof of Theorem 2

Base case n = 1:Since both θ1 and φ1 are non-increasing, for any x1, x1 the

quantities θ1(x1)−θ1(x1) and φ1(x1)−φ1(x1) are either bothnon-negative, or both negative. Thus:

0 ≤∫ ∫

[θ1(x1)−θ1(x1)][φ1(x1)−φ1(x1)]f1(x1)f1(x1)dx1dx1

=∫ ∫

θ1(x1)φ1(x1)f1(x1)f1(x1)dx1dx1

−∫ ∫

θ1(x1)f1(x1)φ1(x1)f1(x1)dx1dx1

−∫ ∫

θ1(x1)f1(x1)φ1(x1)f1(x1)dx1dx1

+∫ ∫

θ1(x1)φ1(x1)f1(x1)f1(x1)dx1dx1

=∫

θ1(x1)φ1(x1)f1(x1)dx1

∫f1(x1)dx1

−∫

θ1(x1)f1(x1)dx1

∫φ1(x1)f1(x1)dx1

−∫

θ1(x1)f1(x1)dx1

∫φ1(x1)f1(x1)dx1

+∫

θ1(x1)φ1(x1)f1(x1)dx1

∫f1(x1)dx1


= E[θ1(X1)φ1(X1)] − E[θ1(X1)]E[φ1(X1)]− E[θ1(X1)]E[φ1(X1)] + E[θ1(X1)φ1(X1)]

= 2E[θ1(X1)φ1(X1)] − 2E[θ1(X1)]E[φ1(X1)].So,

E[θ1(X1)φ1(X1)] ≥ E[θ1(X1)]E[φ1(X1)].Induction hypothesis Assume the result is true for n ≤ k:Induction step n = k + 1:The proof of the induction step is accomplished via the

following four steps:

1. We condition on X1, . . . , Xk to obtain an outer expecta-tion over a random variable of dimension k and an innerexpectation over a random variable of dimension 1.

2. Since the inner expectation is one-dimensional, i.e., overthe k + 1-st dimension, we apply the result of the basecase to obtain a product of one-dimensional expectations.Each of the two resulting one-dimensional expectationsis a random variable of dimension k.

3. We apply the induction hypothesis to the product ofrandom variables of dimension k.

4. We use the definition of conditional expectation to obtainthe desired result.

Step 1. We condition on X1, . . . , Xk to obtain an outer expec-tation over a random variable of dimension k and an innerexpectation over a random variable of dimension 1.

E[θk+1(X1, . . . , Xk+1)φk+1(X1, . . . , Xk+1)] =E[E[θk+1(X1, . . . , Xk+1)φk+1(X1, . . . , Xk+1)|X1, . . . , Xk]],

(16)

where the inner expectation,

E[θk+1(X1, . . . , Xk+1)φk+1(X1, . . . , Xk+1)|X1, . . . , Xk]

=∫

R

θk+1(X1, . . . , Xk , xk+1)

× φk+1(X1, . . . , Xk , xk+1)fk+1(xk+1)dxk+1. (17)

Step 2. Since the inner expectation is one-dimensional, i.e.,over the k + 1-st dimension, we apply the result of the basecase to obtain a product of one-dimensional expectations.Each of the two resulting one-dimensional expectations is arandom variable of dimension k.

Since by assumption θk+1(x1, . . . , xk+1) and φk+1(x1, . . . ,xk+1) are non-increasing functions of xk+1, by the base casen = 1, we have

E[θk+1(X1, . . . , Xk+1)φk+1(X1, . . . , Xk+1)|X1, . . . , Xk]≥ E[θk+1(X1, . . . , Xk+1)|X1, . . . , Xk]

× E[φk+1(X1, . . . , Xk+1)|X1, . . . , Xk]. (18)

Notice that the resulting product of one-dimensionalexpectations (18) is in fact a product of random variablesof dimension k.

Step 3. We apply the induction hypothesis to the product ofrandom variables of dimension k.

Note that∫

Rθk+1(x1, . . . , xk+1)fk+1(xk+1)dxk+1 and∫

Rφk+1(x1, . . . , xk+1)fk+1(xk+1)dxk+1 are non-increasing

functions, since θk+1(x1, . . . , xk+1) and φk+1(x1, . . . , xk+1)

are non-increasing functions. Therefore, by the inductionhypothesis,

E[E[θk+1(X1, . . . , Xk+1)|X1, . . . , Xk]E[φk+1(X1, . . . , Xk+1)|X1, . . . , Xk]]

= E

[(∫R

θk+1(X1, . . . , xk+1)fk+1(xk+1)dxk+1

)

×(∫

R

φk+1(X1, . . . , xk+1)fk+1(xk+1)dxk+1

)]

≥ E

[(∫R

θk+1(X1, . . . , xk+1)fk+1(xk+1)dxk+1

)]

× E

[(∫R

φk+1(X1, . . . , xk+1)fk+1(xk+1)dxk+1

)]

= E[E[θk+1(X1, . . . , Xk+1)|X1, . . . , Xk]]× E[E[φk+1(X1, . . . , Xk+1)|X1, . . . , Xk]]. (19)

Step 4. We use the definition of conditional expectation toobtain the desired result.

E[E[θk+1(X1, . . . , Xk+1)|X1, . . . , Xk]]× E[E[φk+1(X1, . . . , Xk+1)|X1, . . . , Xk]]

= E[θk+1(X1, . . . , Xk+1)]E[φk+1(X1, . . . , Xk+1)]. (20)

Combining equations (16), (18), (19) and (20), we obtain

E[θk+1(X1, . . . , Xk+1)φk+1(X1, . . . , Xk+1)]= E[E[θk+1(X1, . . . , Xk+1)φk+1(X1, . . . , Xk+1)|X1, . . . , Xk]]≥ E[E[θk+1(X1, . . . , Xk+1)|X1, . . . , Xk]

× E[φk+1(X1, . . . , Xk+1)|X1, . . . , Xk]]≥ E[E[θk+1(X1, . . . , Xk+1)|X1, . . . , Xk]]

× E[E[φk+1(X1, . . . , Xk+1)|X1, . . . , Xk]]= E[θk+1(X1, . . . , Xk+1)]E[φk+1(X1, . . . , Xk+1)].

Acknowledgments

The authors would like to thank Robert Indik for his assis-tance in developing computational methods for continuousdistributions. The authors would also like to thank the anony-mous reviewers for their detailed feedback, which we arecertain has allowed us to greatly improve this paper. Thismaterial is based upon work supported by the National Sci-ence Foundation under Grant DMS-0602173. This work wasalso partially funded by ANILLO Grant ACT-88 and Basalproject CMM, Universidad de Chile.


REFERENCES

[1] V.G. Adlakha, An improved conditional Monte Carlo tech-nique for the stochastic shortest path problem, Manage Sci32 (1986), 1360–1367.

[2] C. Alexopoulos, State space partitioning methods for stochas-tic shortest path problems, Networks 30 (1997), 9–21.

[3] J.F. Bard and J.E. Bennett, Arc reduction and path prefer-ence in stochastic acyclic networks, Manage Sci 37 (1991),198–215.

[4] J.F. Bard and J.L. Miller, Probabilistic shortest path prob-lems with budgetary constraints, Comput Oper Res 16 (1989),145–159.

[5] D. Bertsimas and M. Sim, Robust discrete optimization andnetwork flows, Math Program 98 (2003), 49–71.

[6] H. Booth and R.E. Tarjan, Finding the minimum-cost max-imum flow in a series-parallel network, J Algorithms 15(1993), 416–446.

[7] J.M. Burt Jr. and M.B. Garman, Conditional Monte Carlo:A simulation technique for stochastic network analysis,Manage Sci 18 (1971), 207–217.

[8] J.T. Chayes, A. Puha, and T. Sweet, Independent and depen-dent percolation, Probability Theory Appl, IAS/Park CityMath Ser 6 (1999), 51–118.

[9] D.W. Coit and A.E. Smith, Reliability optimization of series-parallel systems using a genetic algorithm, IEEE Trans Reliab45 (1996), 254–260.

[10] W.J. Cook, W.H. Cunningham, W.R. Pulleyblank, and A.Schrijver, Combinatorial Optimization, Wiley, New York,1998.

[11] A. Eiger, P.B. Mirchandani, and H. Soroush, Path preferencesand optimal paths in probabilistic networks, TransportationSci 19 (1985), 75–84.

[12] P.C. Fishburn, Utility theory, Manage Sci 14 (1968),335–378.

[13] C.M. Fortuin, P.W. Kasteleyn, and J. Ginibre, Correlationinequalities on some partially ordered sets, Commun in MathPhys 22 (1971), 89–103.

[14] H. Frank, Shortest paths in probabilistic graphs, Oper Res 17(1969), 583–599.

[15] W. Herroelen and R. Leus, Project scheduling under uncer-tainty: Survey and research potentials, Eur J Oper Res 165(2005), 289–306.

[16] Y.C. Ho, R.S. Sreenivas, and P. Vakili, Ordinal optimizationof DEDS, Discr Event Dynamic Syst 2 (1992), 61–88.

[17] A. Kasperski and P. Zielinski, The robust shortest path prob-lem in series–parallel multidigraphs with interval data, OperRes Lett 34 (2006), 69–76.

[18] B. Klinz and G.J. Woeginger, Minimum-cost dynamic flows:The series-parallel case, Networks 43 (2004), 153–162.

[19] P. Kouvelis and G. Yu, Robust discrete optimization and itsapplications, Kluwer Academic Pub, The Netherlands, 1997.

[20] R.P. Loui, Optimal paths in graphs with stochastic or multi-dimensional weights, Commun ACM 26 (1983), 670–676.

[21] R. Montemanni, L.M. Gambardella, and A.V. Donati, Abranch and bound algorithm for the robust shortest path prob-lem with interval data, Oper Res Lett 32 (2004), 225–232.

[22] J.B. Sidney, The two-machine maximum flow time problemwith series parallel precedence relations, Oper Res 27 (1979),782–791.

[23] C.E. Sigal, A.A.B. Pritsker, and J.J. Solberg, The use of cut-sets in Monte Carlo analysis of stochastic networks, MathComput in Simulation 21 (1979), 376–384.

[24] C.E. Sigal, A.A.B. Pritsker, and J.J. Solberg, The stochasticshortest route problem, Oper Res 28 (1980), 1122–1129.

[25] L.V. Tavares, A review of the contribution of operationalresearch to project management, Eur J Oper Res 136 (2002),1–18.

[26] J. Valdes, R.E. Tarjan, and E.L. Lawler, The recognitionof series parallel digraphs, Proceedings of the EleventhAnnual ACM Symposium on Theory of Computing, Atlanta,Georgia, United States (1979), 1–12.

[27] J.A. Ward, Minimum-aggregate-concave-cost multicom-modity flows in strong series-parallel networks, Math OperRes 24 (1999), 106–129.

[28] G. Yu and J. Yang, On the robust shortest path problem,Comput Oper Res 25 (1998), 457–468.

[29] P. Zielinski, The computational complexity of the relativerobust shortest path problem with interval data, Eur J OperRes 158 (2004), 570–576.


The most likely path on series-parallel networks

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