Resource allocation decisions under various demands and cost requirements in an unreliable flow network

Computers & Operations Research 32 (2005) 2771 – 2784

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Resource allocation decisions under various demands and cost requirements in an unreliable %ow network Chung-Chi Hsieh∗ , Yi-Ting Chen Department of Industrial and Information Management, National Cheng Kung University, 1, University Road, Tainan 70101, Taiwan

Abstract This paper considers resource allocation decisions in an unreliable multi-source multi-sink %ow network, which applies to many real-world systems such as electric and power systems, telecommunications, and transportation systems. Due to uncertainties of components in such an unreliable %ow network, transmitting resources successfully and economically through the unreliable %ow network is of concern to resource allocation decisions at resource-supplying (source) nodes. We study the resource allocation decisions in an unreliable %ow network for a range of demand con1gurations constrained by demand-dependent and demand-independent cost considerations under the reliability optimization objective. Solutions to these problems can be obtained by computing the resource allocation for each demand con1guration independently. In contrast, we pursue an updating scheme that eludes time-consuming enumeration of %ow patterns, which is necessary in independent computation of resource allocations for di6erent demand con1gurations. We show that updating is attainable under both demand-independent and demand-dependent cost constraints when demand incurs an incremental change, and demonstrate the proposed updating scheme with numerical examples. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Minimal path vector; Network reliability; Resource allocation; Unreliable %ow network

1. Introduction Many optimization problems in 1elds such as computer networking, telecommunications, transportation, and electric and power systems are best examined through graphical or network representation. Various network %ow models have been proposed for these problems, and approaches developed ∗

Corresponding author. Fax: +886-6-236-2162. E-mail address: [email protected] (C.-C. Hsieh).

0305-0548/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2004.04.003

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for solving them [1]. This paper considers a class of networks, unreliable %ow networks, in which components (either nodes or arcs) are unreliable, due to uncertainties in operating conditions. When resources are transmitted from resource-supplying (source) node(s) to resource-demanding (sink) node(s) through such unreliable %ow networks, the network reliability issues arise. The literature studying network reliability in an unreliable single-source single-sink %ow network with unreliable arcs is extensive [2–7]. In contrast with the traditional binary-state assumption, the operational state of each unreliable component in the literature is assumed to be a random variable taking on one of the discrete states with some predetermined probability [8,9]. Network reliability, thus, concerns the likelihood that a speci1ed resource demand can be transmitted successfully through the multi-state components in the unreliable %ow network. Computing network reliability generally involves two stages: obtain minimal path vectors or minimal cut vectors to a given level 1 for the %ow network [4,6,10], and utilize them to calculate the network reliability [2,3]. The above-mentioned studies mainly consider unreliable arcs. When both unreliable nodes and unreliable nodes are present in an unreliable %ow network, the evaluation methods of network reliability via minimal path vectors and minimal cut vectors are developed in [11] and [12], respectively; the former is further enhanced in [13]. In an unreliable multi-source multi-sink %ow network, on the other hand, not only network reliability but also resource allocation decisions at source nodes are essential in resource transmission. A resource allocation problem is formulated in [14] which maximizes the network reliability, i.e. the likelihood that resources can be transmitted from source nodes to sink nodes successfully, for a given demand con1guration; and a reliability-maximizing resource allocation is found by integrating the existing network reliability evaluation methods. Yet, how to eIciently obtain the resource allocation decisions for a range of demand con1gurations subject to some cost constraints remains unanswered. This study, therefore, explores an eIcient means of computing resource allocation decisions in an unreliable %ow network for a range of demand con1gurations constrained by demand-dependent and demand-independent cost considerations under the reliability optimization objective. In the aforementioned studies, computation of network reliability necessitates enumeration of %ow patterns, which characterize how resources %ow through a %ow network. Due to the time-consuming nature of enumerating %ow patterns, it is desirable to have some machinery to reduce its practice when seeking for resource allocation decisions for a range of demand con1gurations. This paper strives to unfold the connection of %ow patterns between two similar demand con1gurations under the constraints of demand-independent (constant) and demand-dependent cost functions, and to derive an updating scheme that requires no enumeration of %ow patterns when making successive resource allocation decisions from the current one. Here, two demand con1gurations are said to be similar if one demand con1guration results from a unitary change of the resource quantity at some sink node in the other. We distinguish demand-independent cost constraints from demand-dependent cost constraints in order to simulate two basic decision-making settings: A decision-maker may specify a transmission cost constraint, regardless of the quantity of resources being transmitted, or he or she may have different economic preferences for di6erent resource quantities (intuitively, high cost limits are imposed for larger resource quantities as transmitting more resources costs more). Our formulation provides 1

Minimal path vectors and minimal cut vectors are, respectively, termed as lower boundary points and upper boundary points in [8], and d -MPs and d-MCs in [4]. The de1nition of a minimal path vector will be introduced later in Section 2.

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a good starting point for the more complex problem of coping with multiple preferences (e.g. for costs, %ow patterns, and durations). The essential part of updating is the attainment of minimal path vectors with respect to some demand, which results from an incremental change in current demand and with which we can, in turn, evaluate how reliable resource allocation decisions are. We show that updating is achievable under a demand-independent cost constraint. We then analyze the applicability of updating under a demand-dependent cost constraint, and design a mechanism so that we can perform updating for a range of demand con1gurations under the demand-dependent cost constraint. The outline of this paper is as follows. Section 2 introduces the model of an unreliable %ow network, and presents the process of computing the optimal resource allocation in the unreliable %ow network for given demand under a prespeci1ed cost constraint. Section 3 proposes an updating scheme when demand incurs an incremental change, and uses the proposed updating scheme iteratively to compute the optimal resource allocation decisions for all possible demand con1gurations under a demand-independent cost constraint. It then analyzes the updating mechanism under a demand-dependent cost constraint. Finally, Section 4 concludes with a brief summary.

2. Unreliable ow networks We let G = (N; A) represent an unreliable multi-source multi-sink %ow network, which consists of a set N of nodes and a set A of n directed arcs, and transmits integral-valued resources of single type. We characterize the nodes as source nodes, at which resources are supplied to G, sink nodes, at which resource demands take place, and intermediate nodes, which are neither source nor sink nodes. We let Ns = {s1 ; : : : ; s } ⊂ N denote the set of source nodes, and Nt = {t1 ; : : : ; t } ⊂ N the set of sink nodes. Each source node si ∈ Ns has an upper bound i and a lower bound i on the resource quantity it supplies. We assume that every node in N is perfect and incurs no transmission cost. We also assume that no %ow will vanish or be created in arcs and at intermediate nodes during %ow transmission. Each arc al ∈ A, joining a pair of nodes, is associated with a cost cl , denoting the cost per unit %ow on al , and a maximum capacity ul , signifying the maximum amount that can %ow through al . Due to uncertainties on the operational states of arcs, arc capacities are assumed to be statistically independent discrete random variables; the capacity of each arc al has a probability distribution over the integers 0; 1; : : : ; ul . We note that if arc capacities are distributed over a set of non-sequential integers (e.g. {0; 2; 3; 5}), some de1nitions in the following sections shall be modi1ed accordingly. An example of an unreliable %ow network G with three source nodes and two sink nodes is illustrated in Fig. 1, with the possible values, which arc capacities can take on, and the corresponding probabilities summarized in Table 1 [15]. The decision variables in the resource allocation problem are the resources allotted at source nodes. We let r = (r1 ; : : : ; r ) be the vector of resources allocated at source nodes, where ri is the resource quantity allocated at source node si ∈ Ns . Similarly, we let d = (d1 ; : : : ; d ) denote given resource demand at sink nodes, where dj is the demanded resource quantity at sink node tj ∈ Nt . In this paper, comparison of vectors is frequently considered, and such comparison is always presumed to be made for vectors of the same size. Given two vectors w and z, we de1ne w = z if wl = zl for all l, w 6 z if wl 6 zl for all l, and w ¡ z if wl 6 zl for all l with wk ¡ zk for some k.

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a1

a7

a2

a8

t1

a9 s2

a3 a4 t2

a5 s3

a10

a6

Fig. 1. A three-source two-sink %ow network. Table 1 Probability mass functions of arc capacities in the %ow network in Fig. 1 Arc

Capacity

pmf

Arc

a1

3 2 1 0 4 3 2 1 0 3 2 1 0 2 1 0 3 2 1 0

0.80 0.14 0.04 0.02 0.65 0.15 0.10 0.06 0.04 0.90 0.05 0.03 0.02 0.87 0.08 0.05 0.85 0.10 0.04 0.01

a6

a2

a3

a4 a5

a7 a8

a9

a10

Capacity

pmf

3 2 1 0 2 1 0 3 2 1 0 3 2 1 0 4 3 2 1 0

0.78 0.15 0.05 0.02 0.85 0.10 0.05 0.75 0.15 0.06 0.04 0.80 0.12 0.05 0.03 0.75 0.15 0.05 0.04 0.01

2.1. State vectors and 5ow patterns When transmitting resources through an unreliable %ow network, we shall be aware of the characteristics of arc capacities, how resources %ow through the unreliable %ow network, and the relationship between arc capacities and %ow patterns. We let x = (x1 ; x2 ; : : : ; x n ) be a state vector of arc capacities, where xl is the capacity of arc al ∈ A. Then, x is a path vector if under x the %ow network can transmit one unit of resource %ow from some source node(s) to some sink node(s); x

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Table 2 An example of a %ow pattern f = (0; 0; 0; 0; 1; 2; 1) in the %ow network in Fig. 1 fijk

pijk

fijk

pijk

0 0 0 0

p1; 1; 1 = {a1 ; a7 } p1; 1; 2 = {a2 ; a9 } p1; 2; 1 = {a1 ; a8 } p2; 1; 1 = {a3 ; a9 }

1 2 1

p2; 2; 1 = {a4 ; a10 } p3; 1; 1 = {a5 ; a9 } p3; 2; 1 = {a6 ; a10 }

is a minimal path vector if x is a path vector and there exists no y ¡ x such that under y the %ow network is able to transmit one unit of resource %ow through. If x is a minimal path vector, the set {al |xl ¿ 0; l = 1; : : : ; n} is called a minimum path set (MP). In Fig. 1, for instance, the set {a1 ; a7 } is an MP that links source node s1 and sink node t1 . We assume in this study that the MPs for an unreliable %ow network are known. (When the MPs are unavailable, we can acquire them, for example, by deploying Xue’s method [7].) We let mij be the number of MPs from source node si to sink node tj , and pijk ; 1 6 k 6 mij , be the kth MP from source node si to sink node tj . We further de1ne a state vector x to be a path vector to demand d if the resource demand d can be transmitted through the %ow network under x; a minimal path vector to demand d (d -MP) if x is a path vector to demand d and there exists no y ¡ x such that under y the %ow network is able to transmit the resource demand d through. Moreover, x is a (d ; c)-MP if x is a d-MP and the transmission cost under x, TC(x), is less than or equal to a predetermined value c. With ordered MPs, we can represent a %ow pattern through any %ow network, including an unreliable %ow network, as f = (f1; 1; 1 ; : : : ; fijk ; : : : ; f m ), where fijk is the non-negative integral-valued %ow transmitted through pijk . Taking the %ow network in Fig. 1, for example, the %ow pattern f = (0; 0; 0; 0; 1; 2; 1) indicates that one unit of resource %ow is transmitted from source node s2 to sink node t2 , two units of resource %ow are transmitted from s3 to t1 , and one unit of resource %ow is transmitted from s3 to t2 , as depicted in Table 2. We note that there may be more than one %ow pattern for a given demand con1guration d subject to a predetermined cost limit c. In order to obtain these %ow patterns, we enumerate all %ow patterns that satisfy the following constraints: mij 



fijk = dj ;

j = 1; : : : ; ;

(1)

i=1 k=1

i 6

mij



fijk 6 i ;

i = 1; : : : ; ;

(2)

j=1 k=1

mij 




{fijk |al ∈ pijk } 6 ul ;

l = 1; : : : ; n;

(3)

i=1 j=1 k=1

mij 


i=1 j=1 k=1

wijk fijk 6 c;

(4)

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 where wijk = l {cl |al ∈ pijk } is the cost of transmitting one unit of fijk through pijk . The constraints in (2) impose the bounds on the resource quantity supplied at source node si , because  mij ri = j=1 k=1 fijk . The constraints in (3) set the maximum amount that can %ow through each arc. Constraint (4) concerns that those %ow patterns which incur transmission costs higher than the prespeci1ed cost value will not be considered. The value of c in (4) can be either demand-independent, i.e. a constant independent of the current demand con1guration, or demand-dependent. For the latter, c will presume a non-decreasing function in total demand quantity. The following proposition, generated from [16], establishes the relationship between (d ; c)-MPs and %ow patterns in acyclic %ow networks. 2 Proposition 1. In acyclic 5ow networks, the path vector to demand d , x, transformed from every feasible 5ow pattern (f1; 1; 1 ; : : :, fijk ,: : :, f m ) via xl =

mij 




{fijk |al ∈ pijk };

l = 1; : : : ; n

(5)

i=1 j=1 k=1

is a (d; c)-MP. Proposition 1 enables us to compute the likelihood that resources can be transmitted successfully through the %ow network with regard to a certain resource allocation; this as to be examined next. 2.2. Computing network reliability The network reliability of an unreliable %ow network for a resource allocation is the probability that resources can be transmitted successfully from source nodes to sink nodes subject to a predetermined cost limit. From Proposition 1, we are certain that feasible %ow patterns can be transformed to (d; c)-MPs. We then suppose that the (d; c)-MPs corresponding to a resource allocation r have been established and are denoted by x1 ; : : : ; x . Letting Bi = {b|xi 6 b 6 u}, where u = (u1 ; : : : ; un ), be the set of the combinations of arc capacities with respect to xi , the network reliability R(r) is the probability that the union of the Bi ’s takes place: R(r) = Pr{B1 ∪ · · · ∪ B }:

(6)

In this study, we adopt the state-space decomposition method [2,3] to evaluate R(r). 2.3. Computing optimal resource allocation for given demand Our resource allocation problem is to 1nd a resource allocation with maximal network reliability, constrained by a predetermined cost limit. The procedure for 1nding the optimal resource allocation decision is outlined below [15]. Procedure 1: Computing the optimal resource allocation in G for demand d under a cost limit c Step 1: Enumerate non-negative integral-valued %ow patterns that satisfy (1)–(4), and transform them into path vectors to demand d under c. 2

In cyclic %ow networks, by contrast, transformed feasible %ow patterns, while containing all (d; c)-MPs, are not necessarily (d ; c)-MPs [16].

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Step 2: Determine the (d; c)-MPs from the path vectors to demand d under c. Step 3: Partition the (d; c)-MPs into sets of (d ; c)-MPs corresponding to di6erent resource allocations. Step 4: Compute network reliabilities by using the sets of (d ; c)-MPs obtained in Step 3, and choose the resource allocation whose corresponding set of (d ; c)-MPs gives the maximal network reliability. Note that if G is acyclic, Step 2 of Procedure 1 will be unnecessary; that is, the resulting path vectors to demand d under c in Step 1 are all (d ; c)-MPs. 3. Updating resource allocation with demand-independent and demand-dependent costs When encountering various demand con1gurations, probably along with varying economic considerations, we may deploy Procedure 1 repetitively to obtain the optimal resource allocation for each demand con1guration. Yet, such an approach is less favorable because enumeration of %ow patterns in Step 1 of Procedure 1, reportedly the most time-consuming task of the procedure, must be repeatedly performed. We therefore propose an updating scheme that requires no enumeration of %ow patterns when making successive optimal resource allocation decisions from the current one. The mechanism of updating relies on the closeness of the %ow patterns between two similar demand con1gurations. As such, the updating scheme will be advantageous in coping with a range of demand con1gurations or an entire set of demand con1gurations in the presence of dramatic demand %uctuation. We model two types of cost constraints that can be speci1ed by a decision-maker in the course of updating: a demand-independent cost constraint, which sets a constant cost limit for all demand con1gurations, and a demand-dependent cost constraint, which places a higher cost limit value for greater demand. We will examine them in sequence, focusing on the attainment of the (d ; c)-MPs. 3.1. Updating resource allocations under a demand-independent cost constraint Let X = {x1 ; : : : ; x } be the set of the (d; c)-MPs for current demand d under a predetermined cost limit c. Assume that new demand d + is derived from a unit increase of resources at some sink + ∗ ∗ node tj∗ in d. That is, d + is such that for all j d+ j = dj + 1 if j = j , and dj = dj if j = j . Let Y + denote the set of the (d ; c)-MPs, Lijk = min16l6n {ul |al ∈ pijk } the maximum %ow allowed through an MP pijk , and vijk =(v1 ; v2 ; : : : ; vn ) a binary state vector associated with pijk : vl =1 if al ∈ pijk for all l; 0, otherwise. It immediately follows that fijk 6 Lijk through any MP pijk . In the following lemma we establish the relationship between X and Y , which will facilitate the updating mechanism. Lemma 2. For every (d + ; c)-MP y ∈ Y , there exist a (d ; c)-MP x ∈ X with TC(x) ¡ c and an MP pij∗ k with (Lij∗ k − fij∗ k ) ¿ 0 under x such that y = x + vij∗ k . Proof. Consider any d + -MP y ∈ Y . Since d+ j ∗ ¿ 0, there exists an MP pij ∗ k with 0 ¡ fij ∗ k 6 Lij ∗ k under y such that the state vector x = y − vij∗ k is a path vector to demand d . Because xl = yl − 1 for any al ∈ pij∗ k , and xl = yl for any al ∈ pij∗ k , x is a d -MP. Furthermore, because of TC(y) 6 c, the transmission cost under x, TC(x), is TC(x) = TC(y) − wij∗ k ¡ c , and because of fij∗ k 6 Lij∗ k

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under y, fij∗ k ¡ Lij∗ k under x. We therefore conclude that there exist a d -MP x with TC(x) ¡ c and an MP pij∗ k with (Lij∗ k − fij∗ k ) ¿ 0 under x such that y = x + vij∗ k . Lemma 2 enables us to obtain (d + ; c)-MPs from current (d ; c)-MPs. Given the (d ; c)-MP set X = {x1 ; : : : ; x }, we 1rst identify the set, Pq , of MPs that are eligible for %ow increase for every (d ; c)-MP xq ∈ X . Then, for every MP pij∗ k ∈ Pq we add one unit of resource %ow to xq through it to form a (d + ; c)-MP candidate whose transmission cost is (TC(xq ) + wij∗ k ). If (TC(xq ) + wij∗ k ) ¡ c, the (d + ; c)-MP candidate is a (d + ; c)-MP. Once we have obtained all the (d + ; c)-MPs, we partition them into sets of (d + ; c)-MPs, each corresponding to a resource allocation, and compute the network reliabilities for all resource allocations, in a similar manner to Steps 3 and 4 of Procedure 1. A procedure for updating the optimal resource allocation when resource demand d is changed to d + under a constant cost limit c is outlined below, followed by two numerical examples. Procedure 2: Updating the optimal resource allocation from d to d + under a constant cost limit c. Step 1: For every (d; c)-MP xq ; 1 6 q 6 , determine the MPs through which resource %ow can be increased: Pq = {pij∗ k |(Lij∗ k − fij∗ k ) ¿ 0;

i = 1; : : : ; ; k = 1; : : : ; mij∗ };

where Pq is the set of the MPs with respect to xq . Step 2: Generate path vectors to demand d + under the constant cost limit c based on each pair of xq and Pq , q = 1; : : : ; . Let ’ be the number of path vectors to demand d + under c. ’ ← 0; for q = 1 to  for every MP pij∗ k ∈ Pq if (TC(xq ) + wij∗ k 6 c) then ’ ← ’ + 1; y’ ← xq + vij∗ k ; Step 3: Partition {y1 ; : : : ; y’ } into sets of (d + ; c)-MPs, each corresponding to a unique resource allocation, and remove the duplicates within them. Step 4: Calculate the network reliabilities for all resource allocations and choose the resource allocation with maximal network reliability. Procedure 2 di6ers from Procedure 1 in that Procedure 2 utilizes (d; c)-MPs to obtain (d + ; c)-MPs, whereas Procedure 1 enumerates the %ow patterns directly to obtain (d + ; c)-MPs. When measured in terms of the number of MPs and the number Step1 of Procedure 1, which is the most of arcs, 'j +d+

j −1 ) time to enumerate all %ow patterns time-consuming step in Procedure 1, takes O j=1 ('j −1    that satisfy (1), and takes O n j 'j time to verify whether each %ow pattern satis1es (2)–(4),      'j +d+

j −1 ( ) n j 'j where 'j = i=1 mij ; j = 1; : : : ; . Thus, Step 1 of Procedure 1 takes O j=1 'j −1 time. In contrast, Step 2 of Procedure 2, which is the most time-consuming step in Procedure 2, takes   'j +d+ −1 ' +d −1 O('j∗ n) time, where  is the number of (d; c)-MPs. Because  6 j=1 ('jj −1j ) ¡ j=1 ('j −1j )  and 'j∗ ¡ j 'j , Procedure 2 will be more eIcient than Procedure 1 when the number of MPs and the number of arcs are suIciently large.

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Table 3 MPs pijk , the costs, wijk , of transmitting one unit of resource %ow through pijk , and the maximum %ows, Lijk , allowed through pijk in the %ow network in Fig. 1 pijk

wijk

Lijk = min16l6n {ul |al ∈ pijk }

p1; 1; 1 = {a1 ; a7 } p1; 1; 2 = {a2 ; a9 } p1; 2; 1 = {a1 ; a8 } p2; 1; 1 = {a3 ; a9 } p2; 2; 1 = {a4 ; a10 } p3; 1; 1 = {a5 ; a9 } p3; 2; 1 = {a6 ; a10 }

w1; 1; 1 = (c1 + c7 ) = 3 w1; 1; 2 = (c2 + c9 ) = 5 w1; 2; 1 = (c1 + c8 ) = 6 w2; 1; 1 = (c3 + c9 ) = 6 w2; 2; 1 = (c4 + c10 ) = 5 w3; 1; 1 = (c5 + c9 ) = 3 w3; 2; 1 = (c6 + c10 ) = 5

L1; 1; 1 = min{u1 ; u7 } = 2 L1; 1; 2 = min{u2 ; u9 } = 3 L1; 2; 1 = min{u1 ; u8 } = 3 L2; 1; 1 = min{u3 ; u9 } = 3 L2; 2; 1 = min{u4 ; u10 } = 2 L3; 1; 1 = min{u5 ; u9 } = 3 L3; 2; 1 = min{u6 ; u10 } = 3

Table 4 (d; c)-MPs xq , corresponding %ow patterns fq , and transmission costs TC(xq ) (Some (d; c)-MPs are omitted for brevity.) q

xq

fq

TC(xq )

1 2 3 4 .. . 40

(0,0,0,1,2,1,0,0,2,2) (0,0,0,2,2,0,0,0,2,2) (0,0,1,0,1,2,0,0,2,2) (0,0,1,1,1,1,0,0,2,2)

(0,0,0,0,1,2,1) (0,0,0,0,2,2,0) (0,0,0,1,0,1,2) (0,0,0,1,1,1,1)

16 16 19 19

(3,0,0,1,0,0,2,1,0,1)

(2,0,1,0,1,0,0)

17

We illustrate Steps 1 and 2 of Procedure 2 by using the %ow network in Fig. 1 with the probability mass functions of arc capacities listed in Table 1, and letting the cost per unit %ow cl on arc al ; l = 1; : : : ; n; be: c1 = 2, c2 = 3, c3 = 4, c4 = 2, c5 = 1, c6 = 2, c7 = 1, c8 = 4, c9 = 2, and c10 = 3. For easier discussion, we summarize MPs pijk , and the costs, wijk , of transmitting one unit of resource %ow and the maximum %ows, Lijk , allowed through these MPs in Table 3. We consider two examples, each with the cost limit c = 20. In the 1rst example, we illustrate Steps 1 and 2 of Procedure 2 by using a (d; c)-MP when current demand d = (2; 2) is changed to d + = (3; 2). In the second example, we compute the resource allocation decisions for the entire set of demand con1gurations. Consider the 1rst example. For current demand d =(2; 2), there are =40(d ; c)-MPs, which can be computed by deploying Procedure 1 [15]. Table 4 lists some of the (d ; c)-MPs. When current demand d is changed to d + = (3; 2), i.e. the demand quantity at sink node t1 is increased by one, we identify the set of MPs that are eligible for %ow increase for every (d ; c)-MP in Step 1 of Procedure 2. Take, for instance, the (d ; c)-MP x1 in Table 4. Four MPs are associated with sink node t1 , and all of them are eligible for %ow increase, as depicted in the 1rst two columns of Table 5. Thus, P1 = {p1; 1; 1 ; p1; 1; 2 ; p2; 1; 1 ; p3; 1; 1 }. In Step 2 of Procedure 2, we verify whether (TC(x1 ) + wi; 1; k 6 20) holds for each of the MPs. As shown in the third column of Table 5, the inequality holds only for p1; 1; 1 and p3; 1; 1 . Thus, we generate path vectors to demand d + under the cost limit c based on x1 and {p1; 1; 1 ; p3; 1; 1 }. The resulting (d + ; c)-MPs are y1 = x1 + v1; 1; 1 = (1; 0; 0; 1; 2; 1; 1; 0; 2; 2) and y2 = x1 + v3; 1; 1 = (0; 0; 0; 1; 3; 1; 0; 0; 3; 2), as depicted in the 1fth column of Table 5. Similarly, we

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Table 5 Generating (d + ; c)-MPs from the (d; c)-MP x1 MP

Li; 1; k − fi; 1; k

TC(x1 ) + wi; 1; k

vi; 1; k

(d + ; c)-MP y’

p1; 1; 1 p1; 1; 2 p2; 1; 1 p3; 1; 1

2 − 0 = 2¿0 3 − 0 = 3¿0 3 − 0 = 3¿0 3 − 2 = 1¿0

16 + 3 = 19 ¡ 20 16 + 5 = 21 ¿ 20 16 + 6 = 22 ¿ 20 16 + 3 = 19 ¡ 20

(1,0,0,0,0,0,1,0,0,0) — — (0,0,0,0,1,0,0,0,1,0)

(1,0,0,1,2,1,1,0,2,2) — — (0,0,0,1,3,1,0,0,3,2)

can identify the MP sets and generate the corresponding path vectors to demand d + under the cost limit c for the remaining 39 (d ; c)-MPs. Note that when current demand d encounters a unit decrease, resulting in new demand d − , we may not be able to obtain the entire set of (d − ; c)-MPs via updating. The reason is that there may exist a (d − ; c)-MP y with TC(y) + wijk ¿ c for all wijk , and consequently there will not exist any (d; c)-MP x ∈ X and any vijk such that y = x − vijk . Therefore, updating shall proceed in ascending order of demand. Similarly, when computing the resource allocations for a range of demand con1gurations, we shall start with the smallest possible demand and perform updating sequentially. In the second example, we deploy the updating scheme to obtain the resource allocations for the entire set of demand con1gurations in the unreliable %ow network in Fig. 1, starting with the demand d =(0; 0). Clearly, there is only one (d; c)-MP, i.e. (0,0,0,0,0,0,0,0,0,0), the network reliability is 1.0, and no enumeration of %ow patterns is necessary. When demand d = (0; 0) is changed to d + = (1; 0), we can generate  = 4 (d + ; c)-MPs from the (d ; c)-MP through the MPs p1; 1; 1 , p1; 1; 2 , p2; 1; 1 , and p3; 1; 1 by using Steps 1 and 2 of Procedure 2, and obtain the optimal resource allocation (1; 0; 0) with network reliability 0.995253. Similarly, when demand d =(0; 0) is changed to d + =(0; 1), we can generate  = 3 (d + ; 20)-MPs from the (d; c)-MP through the MPs p1; 2; 1 , p2; 2; 1 , and p3; 2; 1 , and obtain the optimal resource allocation (0; 0; 1) with network reliability 0.970200. With sequential updating for incremental changes of demand quantities at sink nodes, we can obtain the optimal resource allocations for all demand con1gurations, as shown in Table 6. In Table 6, each value in the third column indicates the number of (d ; c)-MPs found for the corresponding demand con1guration, and the resource allocations in the fourth column denote the optimal resource allocations, with their network reliabilities listed in the 1fth column. As can be seen in Table 6, the number of demand con1gurations available in the %ow network will eventually decrease, as the total demand quantity increases. It is because either the cost limit or the maximal %ow of the %ow network will be exceeded as the total demand quantity increases. For the same reason, as the total demand quantity increases, there are fewer (d ; c)-MPs and fewer sets of combinations of arc capacities, resulting in resource allocations with lower network reliabilities. Once the resource allocation decisions for all demand con1gurations have been obtained by using the proposed updating method, we can quickly respond to demand changes and make corresponding resource allocation decisions with maximal network reliability. 3.2. Updating resource allocations under a demand-dependent cost constraint We begin by examining the in%uence of demand-dependent cost limits on the updating of resource allocations, and then provide a mechanism for updating (d; c)-MPs. We presume that the cost limit

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Table 6 Illustration of optimal resource allocations r∗ with network reliabilities R(r ∗ ) for all demand con1gurations d under the cost limit c = 20 (: the number of (d; c)-MPs)  j

0 1 2 3

4

5 6

dj

d = (d1 ; d2 )



r∗

R(r ∗ )

(0,0) (1,0) (0,1) (2,0) (1,1) (0,2) (3,0) (2,1) (1,2) (0,3) (4,0) (3,1) (2,2) (1,3) (0,4) (5,0) (4,1) (3,2) (5,1)

1 4 3 10 12 6 19 30 24 9 13 42 40 18 2 4 14 11 1

(0,0,0) (1,0,0) (0,0,1) (2,0,0) (1,0,1) (1,0,1) (2,0,1) (2,0,1) (1,1,1) (1,1,1) (2,1,1) (2,1,1) (2,0,2) (1,1,2) (0,2,2) (3,1,1) (2,1,2) (2,0,3) (2,1,3)

1.000000 0.995253 0.970200 0.979050 0.994270 0.912764 0.931879 0.982441 0.976220 0.832091 0.811530 0.942914 0.971501 0.873490 0.606825 0.595346 0.808507 0.812534 0.510992

is a non-decreasing function of the total demand quantity. Again, let new demand d + be derived from a unit increase of the resource quantity at sink node tj∗ in d , and let c and c+ ¿ c be the cost limits for d and d + , respectively. Note that the set of (d ; c)-MPs is a subset of the set of (d; c+ )-MPs, which is, in turn, a subset of the set of (d ; ∞)-MPs (or, equivalently, d -MPs). We observe that there may exist a (d + ; c+ )-MP y such that for every vijk , x = y − vijk is a d -MP and the transmission cost under x is c ¡ TC(x) ¡ c+ − wijk . That is, those d -MPs that could be used to generate such y all have transmission costs greater than c. We therefore conclude that we may not be able to generate all (d + ; c+ )-MPs directly from the (d ; c)-MPs. In order to overcome this diIculty, we intend to perform the updating of (d + ; c+ )-MPs indirectly by constructing a kernel set, which is a superset of the set of (d; c)-MPs. We choose the kernel set to be the set of (d ; c)-MPs, ˆ cˆ ¿ c+ , because from this kernel set we can obtain (d ; c)-MPs by identifying those (d ; c)-MPs ˆ with transmission costs less than or equal to c, and with Lemma 2 we can generate a new kernel set, i.e. the set of (d + ; c)-MPs, ˆ from the set of (d; c)-MPs ˆ and then induce (d + ; c+ )-MPs from the new kernel set. Compared with the updating scheme under a constant cost constraint, the updating mechanism here utilizes and updates sets of (d; c)-MPs, ˆ rather than sets of (d ; c)-MPs, for di6erent demand con1gurations d and di6erent cost limits c. A proper choice of the value of cˆ is the maximum of the cost limits for the demand con1gurations considered. Yet, prior to resolving the maximal cost limit, we need to determine the maximal amount of demand that can %ow through the %ow network. Such a problem is referred to as a maximum %ow problem, covered by many Operations Research textbooks; the solution to the problem can be obtained eIciently, for instance, by deploying the strongly polynomial dual simplex methods in [17].

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Table 7 Illustration of (d + ; 45)-MPs xq and the corresponding transmission costs TC(xq ), where d + = (1; 0), and (d + ; 5)-MPs are marked “†” q

xq

TC(xq )

1 2 3 4

(1; 0; 0; 0; 0; 0; 1; 0; 0; 0)† (0; 1; 0; 0; 0; 0; 0; 0; 1; 0)† (0; 0; 1; 0; 0; 0; 0; 0; 1; 0) (0; 0; 0; 0; 1; 0; 0; 0; 1; 0)†

3 5 6 3

Table 8 Resource allocations r and the corresponding network reliabilities R(r) under d + = (1; 0) with cost limit c = 5 r

(d + ; c)-MPs

R(r)

(1,0,0) (0,0,1)

x1 x2 x4

0.995253 0.960300

For illustrative  purposes, we continue to use the %ow network in Fig. 1, assuming that the cost limit c = 5 × j dj is a linear function of the total demand quantity. Since the maximum %ow of this %ow network is 9, the value of cˆ is 5 × 9 = 45. For demand d = (0; 0), there is only one (d ; c)-MP, ˆ i.e. (0,0,0,0,0,0,0,0,0,0). When demand d = (0; 0) is changed to d + = (1; 0), we use the (d ; c)-MP ˆ to generate (d + ; c)-MPs. ˆ As shown in Table 7, there are four (d + ; c)-MPs, ˆ and three + of them are (d ; 5)-MPs (marked “†”) because their transmission costs are less than or equal to c = 5 × j d+ j = 5. Based on Table 7, we compute the resource allocations and their corresponding network reliabilities in Table 8, and conclude that r = (1; 0; 0) is the optimal resource allocation for d + = (1; 0). With successively updating (d ; c)-MPs ˆ for incremental changes in demand d , we are able to compute the optimal resource allocations and their corresponding network reliabilities for the entire set of demand con1gurations. The results are summarized in Table 9.

4. Summary Motivated by the need of obtaining optimal resource allocation decisions for a range of demand con1gurations under cost constraints, this paper proposes a simple updating mechanism that avoids enumeration of %ow patterns, which is required in computing the optimal resource allocation for a single demand con1guration. Essential to the updating scheme is the establishment of the relationship of (d; c)-MPs between two similar demand con1gurations under a demand-independent (constant) cost constraint. Such a relationship can be incorporated into the updating of (d ; c)-MPs under a demand-dependent cost constraint. While this paper has focused on the updating of resource allocation decisions under two types of cost constraints, more decision-related factors could be used to model a decision-maker’s preferences towards resource allocation decisions.

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Table 9 Illustration of optimal resource allocations  r∗ and corresponding network reliabilities R(r∗ ) with respect to demand con1gurations d under the cost limit c = 5 × j dj (: the number of (d; c)-MPs)  j

dj

c

0 1

0 5

2

10

3

15

4

20

5

25

6

30

7

35

8

40

9

45

d = (d1 ; d2 )



r∗

R(r ∗ )

(0,0) (1,0) (0,1) (2,0) (1,1) (0,2) (3,0) (2,1) (1,2) (0,3) (4,0) (3,1) (2,2) (1,3) (0,4) (5,0) (4,1) (3,2) (2,3) (1,4) (5,1) (4,2) (3,3) (2,4) (1,5) (5,2) (4,3) (3,4) (2,5) (1,6) (5,3) (4,4) (3,5) (2,6) (5,4) (4,5) (3,6)

1 3 2 8 8 3 16 23 15 3 13 42 40 18 2 7 33 67 46 16 16 49 70 37 8 19 44 48 16 3 14 25 17 3 5 5 2

(0,0,0) (1,0,0) (0,0,1) (2,0,0) (1,0,1) (0,1,1) (2,0,1) (2,0,1) (1,0,2) (0,1,2) (2,1,1) (2,1,1) (2,0,2) (1,1,2) (0,2,2) (2,2,1) (2,1,2) (2,1,2) (2,1,2) (2,1,2) (3,1,2) (3,1,2) (3,1,2) (2,1,3) (2,1,3) (3,2,2) (3,2,2) (3,1,3) (3,1,3) (2,2,3) (3,2,3) (3,2,3) (3,2,3) (3,2,3) (3,3,3) (3,3,3) (3,3,3)

1.000000 0.995253 0.970200 0.979050 0.994270 0.884450 0.931879 0.982441 0.959231 0.795150 0.811530 0.942914 0.971501 0.873490 0.606825 0.601168 0.826617 0.935120 0.942468 0.786514 0.623894 0.820739 0.908211 0.858764 0.693500 0.617453 0.792452 0.856327 0.729529 0.492993 0.584303 0.737813 0.731225 0.569356 0.512201 0.574208 0.455773

Acknowledgements The authors would like to thank the referees for their valuable comments and suggestions that signi1cantly enhanced the paper.

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References [1] Ahuja RK, Magnanti TL, Orlin JB. Network %ows—theory, algorithms, and applications. Englewood cli6s, NJ: Prentice-Hall; 1993. [2] Aven T. Reliability evaluation of multistate systems with multistate components. IEEE Transactions on Reliability 1985;34(5):473–9. [3] Doulliez P, Jamoulle J. Transportation networks with random arc capacities. RAIRO, Recherche Operationnelle Operations Research 1972;3:45–60. [4] Jane CC, Lin JS, Yuan J. Reliability evaluation of a limited-%ow network in terms of minimal cutsets. IEEE Transactions on Reliability 1993;42(3):354–61. [5] Lee SH. Reliability evaluation of a %ow network. IEEE Transactions on Reliability 1980;29(1):24–6. [6] Lin JS, Jane CC, Yuan J. On reliability evaluation of a capacitated-%ow network in terms of minimal pathsets. Networks 1995;3:131–8. [7] Xue J. On multistate system analysis. IEEE Transactions on Reliability 1985;34(4):329–37. [8] Hudson JC, Kapur KC. Reliability analysis for multistate systems with multistate components. IIE Transactions 1983;15(2):127–35. [9] Hudson JC, Kapur KC. Reliability bounds for multistate systems with multistate components. Operations Research 1985;33(1):153–60. [10] Yeh WC. A simple approach to search for all d-MCs of a limited-%ow network. Reliability Engineering and System Safety 2001;71(1):15–9. [11] Lin YK. A simple algorithm for reliability evaluation of a stochastic-%ow network with node failure. Computers & Operations Research 2001;28(13):1277–85. [12] Lin YK. Using minimal cuts to evaluate the system reliability of a stochastic-%ow network with failures at nodes and arcs. Reliability Engineering and System Safety 2002;75(1):41–6. [13] Yeh WC. A simple algorithm to search for all d-MPs with unreliable nodes. Reliability Engineering and System Safety 2001;73(1):49–54. [14] Hsieh CC, Lin MH. Reliability-orinted multi-resource allocation in a stochastic-%ow network. Reliability Engineering and System Safety 2003;81(2):155–61. [15] Hsieh CC, Chen YT. Reliable and economic resource allocation in an unreliable %ow network. Computers and Operations Research, in press. doi:10.1016/j.cor.2003.08.008 [16] Lin JS. Reliability evaluation of capacitated-%ow networks with budget constraints. IIE Transactions 1998;30: 1175–80. [17] Armstrong RD, Chen W, Goldfarb D, Jin ZY. Strongly polynomial dual simplex methods for the maximum %ow problem. Mathematical Programming 1998;80(1):17–33.