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Flow reliability of a probabilistic capacitated-flow network in multiple node pairs case
Computers & Industrial Engineering 45 (2003) 417–428 www.elsevier.com/locate/dsw
Flow reliability of a probabilistic capacitated-flow network in multiple node pairs case Yi-Kuei Lina,*, John Yuanb a
Department of Information Management, Van Nung Institute of Technology, Chung-Li, Tao-Yuan 320, Taiwan, ROC b Department of Industrial Engineering, National Tsing Hua University, Hsin-Chu 300, Taiwan, ROC
Abstract This article mainly generalizes the flow (or transportation) reliability problem for a directed capacitated-flow network in which the capacity of each arc ai has the values 0 , 1 , 2 , · · · , M i from s (source) to t (sink) case to a multiple node pairs case. Given the demands for all specified node pairs simultaneously in the network, a simple algorithm is proposed first to find out the family of all lower boundary points for such demands in terms of minimal paths. The flow reliability, the probability that the system allows the flow satisfying the demands simultaneously, can be calculated in terms of such lower boundary points. The overall-terminal flow reliability, one source to multiple sinks flow reliability and multiple sources to one sink flow reliability can be calculated as special cases. q 2003 Elsevier Ltd. All rights reserved. Keywords: Multiple node pairs; Maximal flow; Capacity; Minimal path; Probability; Reliability
1. Introduction Traditionally, the maximal flow problem was addressed to a directed capacitated-flow (transportation) network (i.e. all arcs are directed and have capacity) in which the capacity of each arc is deterministic. Ford and Fulkerson (1962) developed a labeling procedure to find a flow pattern whose flow Fs;t from s (source) to t (sink) is maximal. Whenever the capacity of each arc is stochastic, the maximal flow is stochastic and so the problem is to study the distribution of maximal flow Fs;t or equivalently to evaluate the probability Rs;t that the maximal flow Fs;t is no less than the given demand ds;t : In order to evaluate such a probability (flow reliability), Xue (1985) and Lin, Jane and Yuan (1995) proposed, respectively, an algorithm to first generate all lower boundary points for ds;t in terms of minimal paths (MPs) under the condition that the capacity of each arc ai is in {0; 1; 2; …; M i }: An MP is a path whose remainder after removing any arc in it is no longer a path (Aggarwal, Chopra, & Bajwa, 1982). Hence, the problem to calculate the flow reliability, i.e. the probability that the flow Fi;j are no less than the given demands di;j * Corresponding author. Tel.: þ886-3-4515811; fax: þ886-3-4621348. E-mail addresses: [email protected] (Y.-K. Lin), [email protected] (J. Yuan). 0360-8352/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0360-8352(03)00070-6
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simultaneously for all node pair ði; jÞ [ K arises. The purpose of this article is to solve this problem by generalizing the studies (Lin et al., 1995; Xue, 1985) also under the condition that the capacity of each arc ai is in {0; 1; 2; …; M i }: We will define the lower boundary points for such demands di;j first and then propose a simple algorithm to generate them also in terms of MPs. Such a probability can then be calculated in terms of them by applying, say the Inclusion– Exclusion rule. One numerical example is presented to illustrate such an algorithm and also how the flow reliability in various cases be computed. The computational time of such an algorithm in the worst case is analyzed. One straight way to solve this problem is to apply the algorithm by either Xue (1985) or Lin et al. (1995) for each node pair ði; jÞ [ K to the demand di;j ) totally in lKl obtain Ri;j (i.e. the probability that the system allows the flow satisfying Q times where lKl is the number Q of node pairs in K; and then let ði;jÞ[K Ri;j to approximate the flow reliability. We will see that ði;jÞ[K Ri;j makes evident error in the numerical example. The overallterminal flow reliability, one source to multiple sinks flow reliability and multiple sources to one sink flow reliability are actually in the special cases that K is the totality of node pairs in the network, K ¼ {ðs; jÞlj [ N1 } and K ¼ {ði; tÞli [ N2 }; respectively, where N1 and N2 are subsets of nodes. 2. Notation and assumptions 2.1. Notation A; n N; lNl M X Y#X Y,X ði; jÞ K Fi;j di;j Eði; jÞ lEði; jÞl m mpk
{ai l1 # i # n} : the set of arcs where ai denotes ith arc in the network; the number of arcs in the network the set of nodes in the network; the number of nodes in the network ðM 1 ; M 2 ; …; M n Þ : the maximal system capacity vector where M i (a positive integer) denotes the maximal capacity of arc ai for each i ¼ 1; 2; …; n ðx1 ; x2 ; …; xn Þ : a (current) system capacity vector where xi [ {0; 1; 2; …; M i } denotes the (current) capacity of ai for each i ¼ 1; 2; …; n yi , xi for each i ¼ 1; 2; …; n Y # X and yi , xi for at least one i (source node, sink node): a node pair a pre-specified set of node pairs the flow from the source node i to the sink node j the required demand for the node pair ði; jÞ [ K the set of all MPs between the node pair ði; jÞ [ K the number of MPs between ði; jÞ [ K P the total number of MPs in all node pairs in K (i.e. ði;jÞ[K lEði; jÞl ¼ m) kth MP for k ¼ 1; 2; …; m
2.2. Assumptions 1. Each node is perfectly reliable. 2. The capacity of each arc ai is an integer-valued random variable which takes integer values from {0; 1; 2; …; M i } according to a given distribution.
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3. The capacities of different arcs are statistically independent. 4. The flow in the network must satisfy the so-called flow-conservation law (Ford & Fulkerson, 1962). This means that each unit of flow is transmitted through one and only one MP and no flow will disappear or be created during transmission via such a path. Assumption 3 is necessary for evaluating the flow reliability, and assumption 4 implies that the flows in different arcs are statistically dependent. 3. Model to define and generate all lower boundary points for ðdi;j Þði;jÞ[K Throughout the remainder, a flow assignment or flow pattern is denoted by F ¼ ðf1 ; f2 ; …; fm Þ; where fk is presumably to denote the (integer) flow on mpk ; such that fk # min{M i lai [ mpk } for each k ¼ 1; 2; …; m; m X
{fk lai [ mpk } # M i for each i ¼ 1; 2; …; n;
ð1Þ ð2Þ
k¼1
P where m k¼1 {fk lai [ mpk } is the total flow on ai under F: In fact, each F satisfying constraint (2) also satisfies constraint (1) (see the Proof of Lemma 1 in Appendix A). Such two constraints (or constraint (2) alone) are required to mean that each flow pattern cannot violate the maximal capacity of each arc. Similarly, given a system capacity vector X ¼ ðx1 ; x2 ; …; xn Þ; the flow pattern F is called feasible under X (i.e. F cannot violate the capacity requirement of X) if and only if F satisfies m X
{fk lai [ mpk } # xi for each i ¼ 1; 2; …; n
ð3Þ
k¼1
For convenience, we let UX ¼{FlF satisfies constraint (3)}. Whenever the system demand ðdi;j Þði;jÞ[K is given, the flow pattern F satisfies the (resp. exact) system demand ðdi;j Þði;jÞ[K if and only if X X fk $ di;j ðresp: fk ¼ di;j Þ for each node pair ði; jÞ [ K ð4Þ mpk [Eði;jÞ
mpk [Eði;jÞ
That is, Fi;j $ di;j ðresp: Fi;j ¼ di;j Þ for each node pair ði; jÞ [ K: ð5Þ P Note that Fi;j ¼ mpk [Eði;jÞ fk : The flow reliability RK ; the probability that the system allows the flow satisfying the system demand ðdi;j Þði;jÞ[K ; is thus RK ¼ Pr{Xl there exists an F [ UX such that ðFi;j Þði;jÞ[K $ ðdi;j Þði;jÞ[K }. Lemma 2 in Appendix A actually implies that there exist an F [ UX such that ðFi;j Þði;jÞ[K $ ðdi;j Þði;jÞ[K if and only if there exist an F [ UX such that ðFi;j Þði;jÞ[K ¼ ðdi;j Þði;jÞ[K : Hence, if we let V ¼{Xl there exists an F [ UX such that ðFi;j Þði;jÞ[K ¼ ðdi;j Þði;jÞ[K }, then RK ¼ Pr{V}: Let Vmin ¼{XlX [ V and X is minimal in V} ¼ {XlX [ V and Y V for any system capacity vector Y with Y , X}. Such an X is
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called a lower boundary point for ðdi;j Þði;jÞ[K : In particular, for each X [ V; there exists an X p [ Vmin such that X p # X: Hence, RK ¼ Pr{XlX $ X p for a lower boundary point X p for ðdi;j Þði;jÞ[K } RK ¼ Pr{
xi ¼
m X
{fk lai [ mpk } for each i ¼ 1; 2; …; n for each such an F:
ð6Þ
k¼1
GivenPeach flow pattern F such that ðFi;j Þði;jÞ[K ¼ ðdi;j Þði;jÞ[K ; the vector XF ¼ ðx1 ; x2 ; …; xn Þ such that xi ¼ m k¼1 {fk lai [ mpk } for each i ¼ 1; 2; …; n is obviously a system capacity vector as xi [ {0; 1; 2; …; M i } for each i: Let r be the totality of such XF s. By Theorem 1, r contains all lower boundary points for ðdi;j Þði;jÞ[K (i.e. Vmin # r). Refer to Theorem 2 in Appendix A, we will further see that rmin ¼{XlX [ r and X is minimal in r} is the family of all lower boundary points for ðdi;j Þði;jÞ[K : The proposed procedure to search for all lower boundary points for ðdi;j Þði;jÞ[K is shown in Fig. 1. 4. Algorithm to generate all lower boundary points for ðdi;j Þði;jÞ[K The family of all lower boundary points for ðdi;j Þði;jÞ[K can be generated by the following steps. Step 0. List and arrange all MPs between all node pair ði; jÞ [ K orderly. (All MPs between ði; jÞ [ K are required to be known in advance.) Step 1. Generate all flow patterns F ¼ ðf1 ; f2 ; …; fm Þ which satisfies the exact system demand ðdi;j Þði;jÞ[K by solving m X
{fk lai [ mpk } # M i for each i ¼ 1; 2; …; n;
ð7Þ
k¼1
X mpk [Eði;jÞ
fk ¼ di;j for each node pair ði; jÞ [ K:
ð8Þ
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Fig. 1. The procedure to search for all lower boundary points for ðdi;j Þði;jÞ[K :
Step 2. Transform each F ¼ ðf1 ; f2 ; …; fm Þ into XF ¼ ðx1 ; x2 ; …; xn Þ via xi ¼
m X
{fk lai [ mpk } for each i ¼ 1; 2; …; n:
ð9Þ
k¼1
Step 3. Let r ¼ {X 1 ; X 2 ; …; X u } be the family of such generated XF s. Check each X i whether it is a lower boundary point for ðdi;j Þði;jÞ[K in the following (i.e. use the comparison method to delete those non-minimal ones): (3.1) I ¼ f (I is the stack which stores the index of non-lower boundary points for ðdi;j Þði;jÞ[K after checking. Initially, I ¼ f:) (3.2) (Check each X i whether it is a lower boundary points for ðdi;j Þði;jÞ[K :) For i ¼ 1 to u (3.3) For j ¼ i þ 1 to u with j – i and j I (3.4) If X i . X j ; then X i is not a lower boundary point for ðdi;j Þði;jÞ[K : I ¼ I < {i} and go to step (3.7). (3.5) Next j (3.6) X i is a lower boundary point for ðdi;j Þði;jÞ[K : (3.7) Next i (3.8) END. 5. Numerical example The following numerical example is used to illustrate the proposed algorithm. The network configuration and the arc data are shown in Fig. 2 and Table 1, respectively. Three cases shown in Table 2 for system demand and K are discussed. All lower boundary points for ðdi;j Þði;jÞ[K and RK in three cases as follows will be generated and computed, respectively. It is known that lNl ¼ 4; n ¼ 6; and ðM 1 ; M 2 ; M 3 ; M 4 ; M 5 ; M 6 Þ ¼ ð3; 3; 2; 3; 2; 2Þ: Case 1. The approach to search for all lower boundary points for ðdi;j Þði;jÞ[K can be described as follows:
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Fig. 2. The network configuration of numerical example.
Step 0. List all MPs between all node pairs ði; jÞ [ K in the following order: Eð1; 2Þ ¼ {mp1 ; mp2 } where mp1 ¼ {a1 }; mp2 ¼ {a5 ; a4 }: Eð1; 3Þ ¼ {mp3 ; mp4 ; mp5 ; mp6 } where mp3 ¼ {a1 ; a2 }; mp4 ¼ {a1 ; a3 ; a6 }; mp5 ¼ {a5 ; a6 }; mp6 ¼ {a5 ; a4 ; a2 }: Eð4; 3Þ ¼ {mp7 ; mp8 } where mp7 ¼ {a6 }; mp8 ¼ {a4 ; a2 }: Hence, m ¼ 8: Step 1. Generate all flow patterns F ¼ ðf1 ; f2 ; …; f8 Þ which satisfies the exact system demand ðdi;j Þði;jÞ[K by solving 8 f1 þ f3 þ f4 > > > > > f3 þ f6 þ f8 > > > > > < f4 # 2 > > f2 þ f6 þ f8 > > > > > f2 þ f5 þ f6 > > > : f4 þ f5 þ f7
#3 #3 ð10Þ #3 #2 #2
Table 1 The arc data of the example Arc
Capacity (units)
Probability
Arc
Capacity (units)
Probability
a1
0 1 2 3 0 1 2 3 0 1 2
0.05 0.05 0.10 0.80 0.05 0.10 0.10 0.75 0.05 0.05 0.90
a4
0 1 2 3 0 1 2 0 1 2
0.05 0.05 0.05 0.85 0.05 0.05 0.90 0.10 0.05 0.85
a2
a3
a5
a6
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Table 2 Three different cases for the system demand and K d1;2
d1;3
d4;3
K
Case 1 1 3 2 {(1,2), (1,3), (4,3)} Case 2 1 4 2 {(1,2), (1,3), (4,3)} Case 3 (overall-terminal), K {(1,2), (1,3), (1,4), (2,3), (2,4), (3,4), (2,1), (3,1), (4,1), (3,2), (4,2), (4,3)} d1;2 1
d1;3 1
d1;4 1
d2;3 2
d2;4 1
d3;4 0
d2;1 0
d3;1 0
d4;1 0
8 f1 þ f2 ¼ 1 > > < f3 þ f4 þ f5 þ f6 ¼ 3 : > > : f7 þ f8 ¼ 2
d3;2 0
d4;2 1
d4;3 2
ð11Þ
Totally, 17 flow patterns F ¼ ðf1 ; f2 ; …; f8 Þ as follows are generated: (0,1,3,0,0,0,2,0), (1,0,2,0,0,1,2,0), (1,0,2,0,1,0,1,1), (0,1,2,0,1,0,1,1), (1,0,0,1,0,2,1,1), (0,1,1,2,0,0,0,2), (1,0,0,2,0,1,0,2), (1,0,0,1,1,1,0,2).
(0,1,2,0,0,1,2,0), (1,0,1,1,0,1,1,1), (1,0,1,1,1,0,0,2),
(1,0,1,0,0,2,2,0), (0,1,1,1,0,1,1,1), (0,1,1,1,1,0,0,2),
(0,1,2,1,0,0,1,1), (1,0,1,0,1,1,1,1), (1,0,1,0,2,0,0,2),
Step 2. Transform each F ¼ ðf1 ; f2 ; …; f8 Þ into the corresponding XF ¼ ðx1 ; x2 ; x3 ; x4 ; x5 ; x6 Þ via 8 x1 ¼ f1 þ f3 þ f4 > > > > > x2 ¼ f3 þ f6 þ f8 > > > > > < x3 ¼ f4 ð12Þ > > x ¼ f þ f þ f 4 2 6 8 > > > > > x5 ¼ f2 þ f5 þ f6 > > > : x6 ¼ f4 þ f5 þ f7 Totally, five different XF s are obtained as follows: X 1 ¼ ð3; 3; 0; 1; 1; 2Þ; X 2 ¼ ð2; 3; 0; 2; 2; 2Þ; X 3 ¼ ð3; 3; 1; 2; 1; 2Þ; X 4 ¼ ð2; 3; 1; 3; 2; 2Þ and X 5 ¼ ð3; 3; 2; 3; 1; 2Þ: Step 3. Check each X i whether it is a lower boundary point for ðdi;j Þði;jÞ[K (i.e. to obtain the minimal set in {X 1 ; X 2 ; X 3 ; X 4 ; X 5 }). (3.1) (3.2) (3.3) (3.4) (3.5)
I¼f i ¼ 1 (check X 1 whether it is a lower boundary point for ðdi;j Þði;jÞ[K ) j¼2 X 1 ¼ ð3; 3; 0; 1; 1; 2Þ is not larger than X 2 ¼ ð2; 3; 0; 2; 2; 2Þ: next j
424
(3.3) (3.4) (3.5) (3.3) (3.4) (3.5) (3.3) (3.4) (3.7) (3.2)
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j¼3 X 1 ¼ ð3; 3; 0; 1; 1; 2Þ is not larger than X 3 ¼ ð3; 3; 1; 2; 1; 2Þ: next j j¼4 X 1 ¼ ð3; 3; 0; 1; 1; 2Þ is not larger than X 4 ¼ ð2; 3; 1; 3; 2; 2Þ: next j j¼5 X 1 ¼ ð3; 3; 0; 1; 1; 2Þ is not larger than X 5 ¼ ð3; 3; 2; 3; 1; 2Þ: So X 1 is a lower boundary point for ðdi;j Þði;jÞ[K next i i ¼ 2 (check X 2 whether it is a lower boundary point for ðdi;j Þði;jÞ[K
.. . The final result is listed in Table 3. If Inclusion– Exclusion rule to calculate the flow reliability is applied to case 1, we obtain that RK ¼ Pr{ðX $ X 1 Þ < ðX $ X 2 Þ} ¼ Pr{X $ X 1 } þ Pr{X $ X 2 } 2 Pr{ðX $ X 1 Þ > ðX $ X 2 Þ} ¼ Pr{X $ ð3; 3; 0; 1; 1; 2Þ} þ Pr{X $ ð2; 3; 0; 2; 2; 2Þ} 2 Pr{X $ ð3; 3; 0; 2; 2; 2Þ} ¼ ð0:8 £ 0:75 £ 1 £ 0:95 £ 0:95£ 0:85Þ þ ð0:90 £ 0:75 £ 1 £ 0:9 £ 0:9 £ 0:85Þ 2 ð0:8 £ 0:75 £ 1 £ 0:9 £ 0:9 £ 0:85Þ ¼ 0.5119125. Case 2. No lower boundary point for ðdi;j Þði;jÞ[K can be generated and so RK ¼ 0; i.e. the system cannot allow the flow which satisfies ðd1;2 ; d1;3 ; d4;3 Þ ¼ ð1; 4; 2Þ simultaneously. Case 3. The final result of case 3 by the proposed algorithm is shown in Table 4 and the overallterminal flow reliability RK in this case is 0.538249219 if by applying the Inclusion –Exclusion rule. 6. Computational time complexity of the proposed algorithm in the worst case Same to those by Aven (1985), Xue (1985) and Lin et al. (1995), the proposed algorithm generates all lower boundary points for ðdi;j Þði;jÞ[K in terms of MPs which are set to be known in advance. P For each node pair ði; jÞ [ K; the number of feasible solutions satisfying mpk [Eði;jÞ fk ¼ di;j is at least lEði; jÞl; and is bounded by ! lEði; jÞl þ di;j 2 1 : di;j Table 3 All lower boundary point for ðdi;j Þði;jÞ[K in case 1 of the example Xi X1 X2 X3 X4 X5
Is a lower boundary point for ðdi;j Þði;jÞ[K ? ¼ ð3; 3; 0; 1; 1; 2Þ ¼ ð2; 3; 0; 2; 2; 2Þ ¼ ð3; 3; 1; 2; 1; 2Þ ¼ ð2; 3; 1; 3; 2; 2Þ ¼ ð3; 3; 2; 3; 1; 2Þ
Yes Yes No No No
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Table 4 All lower boundary points for ðdi;j Þði;jÞ[K in case 3 of the example Xi X1 X2 X3 X4 X5
Is a lower boundary point for ðdi;j Þði;jÞ[K ¼ ð2; 3; 1; 1; 1; 2Þ ¼ ð1; 3; 1; 2; 2; 2Þ ¼ ð2; 3; 2; 2; 1; 2Þ ¼ ð1; 3; 2; 3; 2; 2Þ ¼ ð3; 3; 2; 1; 0; 2Þ
Yes Yes No No Yes
Hence, the total number of feasible solutions F ¼ ðf1 ; f2 ; …; fm Þ satisfying Eq. (8) is bounded by ! X lEði; jÞl þ di;j 2 1 : ði;jÞ[K di;j P i Each such a solution needs OðmÞ time to test whether it satisfies m k¼1 {fk lai [ mpk } # M for each arc ai and Oðm·nÞ time for all arcs in the worst case. Hence, it takes 0 !1 X lEði; jÞl þ di;j 2 1 A O@m·n· ði;jÞ[K di;j time to obtain all solutions of step 1 in the worst case. Each solution in step 1 needs Oðm·nÞ time to transform to the system capacity vector in the worst case. Thus, it takes 0 !1 X lEði; jÞl þ di;j 2 1 A O@m·n· d ði;jÞ[K i;j time to obtain all XF of step 2 in the worst case. It further needs 0 !1 X lEði; jÞl þ di;j 2 1 A O@n· d ði;jÞ[K i;j time to test each solution in step 2 whether it is minimal and 0 0 !12 1 X lEði; jÞl þ di;j 2 1 AA O@n·@ d ði;jÞ[K i;j time to test all solutions in step 2 in the worst case. In sum, the proposed algorithm needs 0 0 !1 X lEði; jÞl þ di;j 2 1 X A þ O@m·n· O@m·n· ði;jÞ[K di;j ði;jÞ[K
lEði; jÞl þ di;j 2 1 di;j
!1 A
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0 0 O@n·@
X ði;jÞ[K
lEði; jÞl þ di;j 2 1 di;j
0 0 !12 1 X A A ¼ O@n·@ ði;jÞ[K
lEði; jÞl þ di;j 2 1 di;j
!12 1 AA
time in the worst case, since ! lEði; jÞl þ di;j 2 1 $ lEði; jÞl di;j and so X ði;jÞ[K
lEði; jÞl þ di;j 2 1
!
di;j
$
X
lEði; jÞl ¼ m:
ði;jÞ[K
7. Conclusions and discussion This article mainly tries to evaluate the flow reliability which satisfies the system demand ðdi;j Þði;jÞ[K simultaneously for multiple node pairs of a directed capacitated-flow network in which the capacity of each arc ai has the value 0 , 1 , 2 , · · · , M i : A simple algorithm is proposed first to generate all lower boundary points for ðdi;j Þði;jÞ[K in terms of MPs. Then the flow reliability can be calculated in terms of such lower boundary points. The proposed algorithm takes 0 0 !12 1 X lEði; jÞl þ di;j 2 1 AA O@n·@ ði;jÞ[K di;j time in the worst case. That is, the computational time in the worst case is bounded by 0 !12 X lEði; jÞl þ di;j 2 1 A k·n·@ ði;jÞ[K di;j for a real number k: One straight method is to apply the Q algorithm by either Xue (1985) or Lin et al. (1995) for each ði; jÞ [ K to obtain Ri;j ; and to let RK ; ði;jÞ[K Ri;j in order to approximate RK where Ri;j ¼ Pr{Xl there exists an F [ UX such thatQFi;j $ di;j }. It is obvious that Pr{Xl there exists an F [ UX such that ðFi;j Þði;jÞ[K $ ðdi;j Þði;jÞ[K } – ði;jÞ[K Pr{Xl there exists an F [ UX such that Fi;j $ di;j }. However, we can see in Table 5 that the values of RK are over-exaggerated in all three cases of our example. Note that in case 1, RK ¼ R1;2 £ R1;3 £ R4;3 ¼ 0:995125 £ 0:85637625 £ 0:971625 ¼ 0:8280202: The overall-terminal flow reliability, one source to multiple sinks flow reliability and multiple sources to one sink flow reliability are actually in the special cases that K are the totality of node pairs in the network, K ¼ {ðs; jÞlj [ N1 } and K ¼ {ði; tÞli [ N2 }; respectively, where N1 and N2 are subsets of nodes.
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Table 5 The values RK vs RK in the example
Case 1 Case 2 Case 3
RK
RK
RK 2 RK (error)
0.8280202 0.6453629 0.8491870
0.5119125 0.0 0.5382492
0.3161077 0.6453629 0.3019378
Appendix A Lemma 1. Any flow pattern F satisfying constraint (2) also satisfies constraint (1). P i i Proof. m k¼1 {fk lai [ mpk } # M For each mpk ; fk # M for each ai [ mpk due to Eq. (2). This implies i that fk # min{M lai [ mpk }: A Lemma 2. Let X be a system capacity vector. Suppose that the flow pattern F [ UX satisfies ðFi;j Þði;jÞ[K $ ðdi;j Þði;jÞ[K ; then there exists a flow pattern H [ UX with H , F such that ðHi;j Þði;jÞ[K ¼ ðdi;j Þði;jÞ[K : Proof. For any ði; jÞ [ K such that Fi;j . di;j ; we may assume Fi;j ¼ di;j þ 1 without loss of generality. Let F ¼ ðf1 ; f2 ; …fm Þ ¼ ðf1 ; f2 ; …; fw21 ; fw 2 1; fwþ1 ; …; fm Þ for an mpw [ Eði; jÞ with fw . 0: Then F , F and Fi;j ¼ Fi;j 2 1 ¼ di;j (note that Fi;j ¼ Fi;j $ di;j for any other ði; jÞ [ K). This implies that we can repeat the above procedure in finite times to deduce F to an H [ UX such that Hi;j ¼ di;j for each ði; jÞ [ K: A Proof of Theorem 1. (1) (2)
It is trivial since X [ V: P Suppose to the contrary that there exists an arc ai s.t. m k¼1 {fk lai [ mpk } # xi 2 1: Hence, F [ UX2ei where X 2 ei ¼ ðx1 ; x2 ; …; xi21 ; xi 2 1; xiþ1 ; …; xn Þ: In particular, X 2 ei , X which contradicts to the requirement that X is a lower boundary point for ðdi;j Þði;jÞ[K :
A Theorem 2. rmin ¼ Vmin Proof. Note that X [ rmin implies that X [ V and that X [ Vmin implies that X [ r (due to Theorem 1). Suppose that X [ rmin but X Vmin ; i.e. there exists a Y [ Vmin such that Y , X: Then Y [ r which contradicts to that X [ rmin : Hence rmin # Vmin : Conversely, suppose that X [ Vmin but X rmin ; i.e. there exist a Y [ rmin such that Y , X: Then Y [ V which contradicts to that X [ Vmin : Hence, Vmin # rmin and so rmin ¼ Vmin : A
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References Abraham, J. A. (1979). An improved algorithm for network reliability. IEEE Transactions on Reliability, 28, 58 – 61. Aggarwal, K. K., Chopra, Y. C., & Bajwa, J. S. (1982). Capacity consideration in reliability analysis of communication systems. IEEE Transactions on Reliability, 31, 177– 180. Aven, T. (1985). Reliability evaluation of multistate systems with multistate components. IEEE Transactions on Reliability, 34, 473– 479. Doulliez, P., & Jamoulle, J. (1972). Transportation networks with random arc capacities. RAIRO, Recherche Operationnelle Operations Research, 3, 45 –60. Evans, J. R. (1976). Maximal flow in probabilistic graphs—the discrete case. Networks, 6, 161– 183. Ford, L. R., & Fulkerson, D. R. (1962). Flows in networks. New Jersey: Princeton University Press. Griffith, W. S. (1980). Multistate reliability models. Journal of Applied Probability, 17, 735– 744. Hudson, J. C., & Kapur, K. C. (1983). Reliability analysis for multistate systems with multistate components. IIE Transactions, 15, 127– 135. Hudson, J. C., & Kapur, K. C. (1985). Reliability bounds for multistate systems with multistate components. Operations Research, 33, 153– 160. Jane, C. C., Lin, J. S., & Yuan, J. (1993). On reliability evaluation of a limited-flow network in terms of minimal cutsets. IEEE Transactions on Reliability, 42(3), 354–361. Lin, J. S., Jane, C. C., & Yuan, J. (1995). On reliability evaluation of a capacitated-flow network in terms of minimal pathsets. Networks, 3, 131– 138. Xue, J. (1985). On multistate system analysis. IEEE Transactions on Reliability, 34, 329– 337.
Flow reliability of a probabilistic capacitated-flow network in multiple node pairs case Yi-Kuei Lina,*, John Yuanb a
Department of Information Management, Van Nung Institute of Technology, Chung-Li, Tao-Yuan 320, Taiwan, ROC b Department of Industrial Engineering, National Tsing Hua University, Hsin-Chu 300, Taiwan, ROC
Abstract This article mainly generalizes the flow (or transportation) reliability problem for a directed capacitated-flow network in which the capacity of each arc ai has the values 0 , 1 , 2 , · · · , M i from s (source) to t (sink) case to a multiple node pairs case. Given the demands for all specified node pairs simultaneously in the network, a simple algorithm is proposed first to find out the family of all lower boundary points for such demands in terms of minimal paths. The flow reliability, the probability that the system allows the flow satisfying the demands simultaneously, can be calculated in terms of such lower boundary points. The overall-terminal flow reliability, one source to multiple sinks flow reliability and multiple sources to one sink flow reliability can be calculated as special cases. q 2003 Elsevier Ltd. All rights reserved. Keywords: Multiple node pairs; Maximal flow; Capacity; Minimal path; Probability; Reliability
1. Introduction Traditionally, the maximal flow problem was addressed to a directed capacitated-flow (transportation) network (i.e. all arcs are directed and have capacity) in which the capacity of each arc is deterministic. Ford and Fulkerson (1962) developed a labeling procedure to find a flow pattern whose flow Fs;t from s (source) to t (sink) is maximal. Whenever the capacity of each arc is stochastic, the maximal flow is stochastic and so the problem is to study the distribution of maximal flow Fs;t or equivalently to evaluate the probability Rs;t that the maximal flow Fs;t is no less than the given demand ds;t : In order to evaluate such a probability (flow reliability), Xue (1985) and Lin, Jane and Yuan (1995) proposed, respectively, an algorithm to first generate all lower boundary points for ds;t in terms of minimal paths (MPs) under the condition that the capacity of each arc ai is in {0; 1; 2; …; M i }: An MP is a path whose remainder after removing any arc in it is no longer a path (Aggarwal, Chopra, & Bajwa, 1982). Hence, the problem to calculate the flow reliability, i.e. the probability that the flow Fi;j are no less than the given demands di;j * Corresponding author. Tel.: þ886-3-4515811; fax: þ886-3-4621348. E-mail addresses: [email protected] (Y.-K. Lin), [email protected] (J. Yuan). 0360-8352/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0360-8352(03)00070-6
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simultaneously for all node pair ði; jÞ [ K arises. The purpose of this article is to solve this problem by generalizing the studies (Lin et al., 1995; Xue, 1985) also under the condition that the capacity of each arc ai is in {0; 1; 2; …; M i }: We will define the lower boundary points for such demands di;j first and then propose a simple algorithm to generate them also in terms of MPs. Such a probability can then be calculated in terms of them by applying, say the Inclusion– Exclusion rule. One numerical example is presented to illustrate such an algorithm and also how the flow reliability in various cases be computed. The computational time of such an algorithm in the worst case is analyzed. One straight way to solve this problem is to apply the algorithm by either Xue (1985) or Lin et al. (1995) for each node pair ði; jÞ [ K to the demand di;j ) totally in lKl obtain Ri;j (i.e. the probability that the system allows the flow satisfying Q times where lKl is the number Q of node pairs in K; and then let ði;jÞ[K Ri;j to approximate the flow reliability. We will see that ði;jÞ[K Ri;j makes evident error in the numerical example. The overallterminal flow reliability, one source to multiple sinks flow reliability and multiple sources to one sink flow reliability are actually in the special cases that K is the totality of node pairs in the network, K ¼ {ðs; jÞlj [ N1 } and K ¼ {ði; tÞli [ N2 }; respectively, where N1 and N2 are subsets of nodes. 2. Notation and assumptions 2.1. Notation A; n N; lNl M X Y#X Y,X ði; jÞ K Fi;j di;j Eði; jÞ lEði; jÞl m mpk
{ai l1 # i # n} : the set of arcs where ai denotes ith arc in the network; the number of arcs in the network the set of nodes in the network; the number of nodes in the network ðM 1 ; M 2 ; …; M n Þ : the maximal system capacity vector where M i (a positive integer) denotes the maximal capacity of arc ai for each i ¼ 1; 2; …; n ðx1 ; x2 ; …; xn Þ : a (current) system capacity vector where xi [ {0; 1; 2; …; M i } denotes the (current) capacity of ai for each i ¼ 1; 2; …; n yi , xi for each i ¼ 1; 2; …; n Y # X and yi , xi for at least one i (source node, sink node): a node pair a pre-specified set of node pairs the flow from the source node i to the sink node j the required demand for the node pair ði; jÞ [ K the set of all MPs between the node pair ði; jÞ [ K the number of MPs between ði; jÞ [ K P the total number of MPs in all node pairs in K (i.e. ði;jÞ[K lEði; jÞl ¼ m) kth MP for k ¼ 1; 2; …; m
2.2. Assumptions 1. Each node is perfectly reliable. 2. The capacity of each arc ai is an integer-valued random variable which takes integer values from {0; 1; 2; …; M i } according to a given distribution.
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3. The capacities of different arcs are statistically independent. 4. The flow in the network must satisfy the so-called flow-conservation law (Ford & Fulkerson, 1962). This means that each unit of flow is transmitted through one and only one MP and no flow will disappear or be created during transmission via such a path. Assumption 3 is necessary for evaluating the flow reliability, and assumption 4 implies that the flows in different arcs are statistically dependent. 3. Model to define and generate all lower boundary points for ðdi;j Þði;jÞ[K Throughout the remainder, a flow assignment or flow pattern is denoted by F ¼ ðf1 ; f2 ; …; fm Þ; where fk is presumably to denote the (integer) flow on mpk ; such that fk # min{M i lai [ mpk } for each k ¼ 1; 2; …; m; m X
{fk lai [ mpk } # M i for each i ¼ 1; 2; …; n;
ð1Þ ð2Þ
k¼1
P where m k¼1 {fk lai [ mpk } is the total flow on ai under F: In fact, each F satisfying constraint (2) also satisfies constraint (1) (see the Proof of Lemma 1 in Appendix A). Such two constraints (or constraint (2) alone) are required to mean that each flow pattern cannot violate the maximal capacity of each arc. Similarly, given a system capacity vector X ¼ ðx1 ; x2 ; …; xn Þ; the flow pattern F is called feasible under X (i.e. F cannot violate the capacity requirement of X) if and only if F satisfies m X
{fk lai [ mpk } # xi for each i ¼ 1; 2; …; n
ð3Þ
k¼1
For convenience, we let UX ¼{FlF satisfies constraint (3)}. Whenever the system demand ðdi;j Þði;jÞ[K is given, the flow pattern F satisfies the (resp. exact) system demand ðdi;j Þði;jÞ[K if and only if X X fk $ di;j ðresp: fk ¼ di;j Þ for each node pair ði; jÞ [ K ð4Þ mpk [Eði;jÞ
mpk [Eði;jÞ
That is, Fi;j $ di;j ðresp: Fi;j ¼ di;j Þ for each node pair ði; jÞ [ K: ð5Þ P Note that Fi;j ¼ mpk [Eði;jÞ fk : The flow reliability RK ; the probability that the system allows the flow satisfying the system demand ðdi;j Þði;jÞ[K ; is thus RK ¼ Pr{Xl there exists an F [ UX such that ðFi;j Þði;jÞ[K $ ðdi;j Þði;jÞ[K }. Lemma 2 in Appendix A actually implies that there exist an F [ UX such that ðFi;j Þði;jÞ[K $ ðdi;j Þði;jÞ[K if and only if there exist an F [ UX such that ðFi;j Þði;jÞ[K ¼ ðdi;j Þði;jÞ[K : Hence, if we let V ¼{Xl there exists an F [ UX such that ðFi;j Þði;jÞ[K ¼ ðdi;j Þði;jÞ[K }, then RK ¼ Pr{V}: Let Vmin ¼{XlX [ V and X is minimal in V} ¼ {XlX [ V and Y V for any system capacity vector Y with Y , X}. Such an X is
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called a lower boundary point for ðdi;j Þði;jÞ[K : In particular, for each X [ V; there exists an X p [ Vmin such that X p # X: Hence, RK ¼ Pr{XlX $ X p for a lower boundary point X p for ðdi;j Þði;jÞ[K } RK ¼ Pr{
xi ¼
m X
{fk lai [ mpk } for each i ¼ 1; 2; …; n for each such an F:
ð6Þ
k¼1
GivenPeach flow pattern F such that ðFi;j Þði;jÞ[K ¼ ðdi;j Þði;jÞ[K ; the vector XF ¼ ðx1 ; x2 ; …; xn Þ such that xi ¼ m k¼1 {fk lai [ mpk } for each i ¼ 1; 2; …; n is obviously a system capacity vector as xi [ {0; 1; 2; …; M i } for each i: Let r be the totality of such XF s. By Theorem 1, r contains all lower boundary points for ðdi;j Þði;jÞ[K (i.e. Vmin # r). Refer to Theorem 2 in Appendix A, we will further see that rmin ¼{XlX [ r and X is minimal in r} is the family of all lower boundary points for ðdi;j Þði;jÞ[K : The proposed procedure to search for all lower boundary points for ðdi;j Þði;jÞ[K is shown in Fig. 1. 4. Algorithm to generate all lower boundary points for ðdi;j Þði;jÞ[K The family of all lower boundary points for ðdi;j Þði;jÞ[K can be generated by the following steps. Step 0. List and arrange all MPs between all node pair ði; jÞ [ K orderly. (All MPs between ði; jÞ [ K are required to be known in advance.) Step 1. Generate all flow patterns F ¼ ðf1 ; f2 ; …; fm Þ which satisfies the exact system demand ðdi;j Þði;jÞ[K by solving m X
{fk lai [ mpk } # M i for each i ¼ 1; 2; …; n;
ð7Þ
k¼1
X mpk [Eði;jÞ
fk ¼ di;j for each node pair ði; jÞ [ K:
ð8Þ
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Fig. 1. The procedure to search for all lower boundary points for ðdi;j Þði;jÞ[K :
Step 2. Transform each F ¼ ðf1 ; f2 ; …; fm Þ into XF ¼ ðx1 ; x2 ; …; xn Þ via xi ¼
m X
{fk lai [ mpk } for each i ¼ 1; 2; …; n:
ð9Þ
k¼1
Step 3. Let r ¼ {X 1 ; X 2 ; …; X u } be the family of such generated XF s. Check each X i whether it is a lower boundary point for ðdi;j Þði;jÞ[K in the following (i.e. use the comparison method to delete those non-minimal ones): (3.1) I ¼ f (I is the stack which stores the index of non-lower boundary points for ðdi;j Þði;jÞ[K after checking. Initially, I ¼ f:) (3.2) (Check each X i whether it is a lower boundary points for ðdi;j Þði;jÞ[K :) For i ¼ 1 to u (3.3) For j ¼ i þ 1 to u with j – i and j I (3.4) If X i . X j ; then X i is not a lower boundary point for ðdi;j Þði;jÞ[K : I ¼ I < {i} and go to step (3.7). (3.5) Next j (3.6) X i is a lower boundary point for ðdi;j Þði;jÞ[K : (3.7) Next i (3.8) END. 5. Numerical example The following numerical example is used to illustrate the proposed algorithm. The network configuration and the arc data are shown in Fig. 2 and Table 1, respectively. Three cases shown in Table 2 for system demand and K are discussed. All lower boundary points for ðdi;j Þði;jÞ[K and RK in three cases as follows will be generated and computed, respectively. It is known that lNl ¼ 4; n ¼ 6; and ðM 1 ; M 2 ; M 3 ; M 4 ; M 5 ; M 6 Þ ¼ ð3; 3; 2; 3; 2; 2Þ: Case 1. The approach to search for all lower boundary points for ðdi;j Þði;jÞ[K can be described as follows:
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Fig. 2. The network configuration of numerical example.
Step 0. List all MPs between all node pairs ði; jÞ [ K in the following order: Eð1; 2Þ ¼ {mp1 ; mp2 } where mp1 ¼ {a1 }; mp2 ¼ {a5 ; a4 }: Eð1; 3Þ ¼ {mp3 ; mp4 ; mp5 ; mp6 } where mp3 ¼ {a1 ; a2 }; mp4 ¼ {a1 ; a3 ; a6 }; mp5 ¼ {a5 ; a6 }; mp6 ¼ {a5 ; a4 ; a2 }: Eð4; 3Þ ¼ {mp7 ; mp8 } where mp7 ¼ {a6 }; mp8 ¼ {a4 ; a2 }: Hence, m ¼ 8: Step 1. Generate all flow patterns F ¼ ðf1 ; f2 ; …; f8 Þ which satisfies the exact system demand ðdi;j Þði;jÞ[K by solving 8 f1 þ f3 þ f4 > > > > > f3 þ f6 þ f8 > > > > > < f4 # 2 > > f2 þ f6 þ f8 > > > > > f2 þ f5 þ f6 > > > : f4 þ f5 þ f7
#3 #3 ð10Þ #3 #2 #2
Table 1 The arc data of the example Arc
Capacity (units)
Probability
Arc
Capacity (units)
Probability
a1
0 1 2 3 0 1 2 3 0 1 2
0.05 0.05 0.10 0.80 0.05 0.10 0.10 0.75 0.05 0.05 0.90
a4
0 1 2 3 0 1 2 0 1 2
0.05 0.05 0.05 0.85 0.05 0.05 0.90 0.10 0.05 0.85
a2
a3
a5
a6
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Table 2 Three different cases for the system demand and K d1;2
d1;3
d4;3
K
Case 1 1 3 2 {(1,2), (1,3), (4,3)} Case 2 1 4 2 {(1,2), (1,3), (4,3)} Case 3 (overall-terminal), K {(1,2), (1,3), (1,4), (2,3), (2,4), (3,4), (2,1), (3,1), (4,1), (3,2), (4,2), (4,3)} d1;2 1
d1;3 1
d1;4 1
d2;3 2
d2;4 1
d3;4 0
d2;1 0
d3;1 0
d4;1 0
8 f1 þ f2 ¼ 1 > > < f3 þ f4 þ f5 þ f6 ¼ 3 : > > : f7 þ f8 ¼ 2
d3;2 0
d4;2 1
d4;3 2
ð11Þ
Totally, 17 flow patterns F ¼ ðf1 ; f2 ; …; f8 Þ as follows are generated: (0,1,3,0,0,0,2,0), (1,0,2,0,0,1,2,0), (1,0,2,0,1,0,1,1), (0,1,2,0,1,0,1,1), (1,0,0,1,0,2,1,1), (0,1,1,2,0,0,0,2), (1,0,0,2,0,1,0,2), (1,0,0,1,1,1,0,2).
(0,1,2,0,0,1,2,0), (1,0,1,1,0,1,1,1), (1,0,1,1,1,0,0,2),
(1,0,1,0,0,2,2,0), (0,1,1,1,0,1,1,1), (0,1,1,1,1,0,0,2),
(0,1,2,1,0,0,1,1), (1,0,1,0,1,1,1,1), (1,0,1,0,2,0,0,2),
Step 2. Transform each F ¼ ðf1 ; f2 ; …; f8 Þ into the corresponding XF ¼ ðx1 ; x2 ; x3 ; x4 ; x5 ; x6 Þ via 8 x1 ¼ f1 þ f3 þ f4 > > > > > x2 ¼ f3 þ f6 þ f8 > > > > > < x3 ¼ f4 ð12Þ > > x ¼ f þ f þ f 4 2 6 8 > > > > > x5 ¼ f2 þ f5 þ f6 > > > : x6 ¼ f4 þ f5 þ f7 Totally, five different XF s are obtained as follows: X 1 ¼ ð3; 3; 0; 1; 1; 2Þ; X 2 ¼ ð2; 3; 0; 2; 2; 2Þ; X 3 ¼ ð3; 3; 1; 2; 1; 2Þ; X 4 ¼ ð2; 3; 1; 3; 2; 2Þ and X 5 ¼ ð3; 3; 2; 3; 1; 2Þ: Step 3. Check each X i whether it is a lower boundary point for ðdi;j Þði;jÞ[K (i.e. to obtain the minimal set in {X 1 ; X 2 ; X 3 ; X 4 ; X 5 }). (3.1) (3.2) (3.3) (3.4) (3.5)
I¼f i ¼ 1 (check X 1 whether it is a lower boundary point for ðdi;j Þði;jÞ[K ) j¼2 X 1 ¼ ð3; 3; 0; 1; 1; 2Þ is not larger than X 2 ¼ ð2; 3; 0; 2; 2; 2Þ: next j
424
(3.3) (3.4) (3.5) (3.3) (3.4) (3.5) (3.3) (3.4) (3.7) (3.2)
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j¼3 X 1 ¼ ð3; 3; 0; 1; 1; 2Þ is not larger than X 3 ¼ ð3; 3; 1; 2; 1; 2Þ: next j j¼4 X 1 ¼ ð3; 3; 0; 1; 1; 2Þ is not larger than X 4 ¼ ð2; 3; 1; 3; 2; 2Þ: next j j¼5 X 1 ¼ ð3; 3; 0; 1; 1; 2Þ is not larger than X 5 ¼ ð3; 3; 2; 3; 1; 2Þ: So X 1 is a lower boundary point for ðdi;j Þði;jÞ[K next i i ¼ 2 (check X 2 whether it is a lower boundary point for ðdi;j Þði;jÞ[K
.. . The final result is listed in Table 3. If Inclusion– Exclusion rule to calculate the flow reliability is applied to case 1, we obtain that RK ¼ Pr{ðX $ X 1 Þ < ðX $ X 2 Þ} ¼ Pr{X $ X 1 } þ Pr{X $ X 2 } 2 Pr{ðX $ X 1 Þ > ðX $ X 2 Þ} ¼ Pr{X $ ð3; 3; 0; 1; 1; 2Þ} þ Pr{X $ ð2; 3; 0; 2; 2; 2Þ} 2 Pr{X $ ð3; 3; 0; 2; 2; 2Þ} ¼ ð0:8 £ 0:75 £ 1 £ 0:95 £ 0:95£ 0:85Þ þ ð0:90 £ 0:75 £ 1 £ 0:9 £ 0:9 £ 0:85Þ 2 ð0:8 £ 0:75 £ 1 £ 0:9 £ 0:9 £ 0:85Þ ¼ 0.5119125. Case 2. No lower boundary point for ðdi;j Þði;jÞ[K can be generated and so RK ¼ 0; i.e. the system cannot allow the flow which satisfies ðd1;2 ; d1;3 ; d4;3 Þ ¼ ð1; 4; 2Þ simultaneously. Case 3. The final result of case 3 by the proposed algorithm is shown in Table 4 and the overallterminal flow reliability RK in this case is 0.538249219 if by applying the Inclusion –Exclusion rule. 6. Computational time complexity of the proposed algorithm in the worst case Same to those by Aven (1985), Xue (1985) and Lin et al. (1995), the proposed algorithm generates all lower boundary points for ðdi;j Þði;jÞ[K in terms of MPs which are set to be known in advance. P For each node pair ði; jÞ [ K; the number of feasible solutions satisfying mpk [Eði;jÞ fk ¼ di;j is at least lEði; jÞl; and is bounded by ! lEði; jÞl þ di;j 2 1 : di;j Table 3 All lower boundary point for ðdi;j Þði;jÞ[K in case 1 of the example Xi X1 X2 X3 X4 X5
Is a lower boundary point for ðdi;j Þði;jÞ[K ? ¼ ð3; 3; 0; 1; 1; 2Þ ¼ ð2; 3; 0; 2; 2; 2Þ ¼ ð3; 3; 1; 2; 1; 2Þ ¼ ð2; 3; 1; 3; 2; 2Þ ¼ ð3; 3; 2; 3; 1; 2Þ
Yes Yes No No No
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425
Table 4 All lower boundary points for ðdi;j Þði;jÞ[K in case 3 of the example Xi X1 X2 X3 X4 X5
Is a lower boundary point for ðdi;j Þði;jÞ[K ¼ ð2; 3; 1; 1; 1; 2Þ ¼ ð1; 3; 1; 2; 2; 2Þ ¼ ð2; 3; 2; 2; 1; 2Þ ¼ ð1; 3; 2; 3; 2; 2Þ ¼ ð3; 3; 2; 1; 0; 2Þ
Yes Yes No No Yes
Hence, the total number of feasible solutions F ¼ ðf1 ; f2 ; …; fm Þ satisfying Eq. (8) is bounded by ! X lEði; jÞl þ di;j 2 1 : ði;jÞ[K di;j P i Each such a solution needs OðmÞ time to test whether it satisfies m k¼1 {fk lai [ mpk } # M for each arc ai and Oðm·nÞ time for all arcs in the worst case. Hence, it takes 0 !1 X lEði; jÞl þ di;j 2 1 A O@m·n· ði;jÞ[K di;j time to obtain all solutions of step 1 in the worst case. Each solution in step 1 needs Oðm·nÞ time to transform to the system capacity vector in the worst case. Thus, it takes 0 !1 X lEði; jÞl þ di;j 2 1 A O@m·n· d ði;jÞ[K i;j time to obtain all XF of step 2 in the worst case. It further needs 0 !1 X lEði; jÞl þ di;j 2 1 A O@n· d ði;jÞ[K i;j time to test each solution in step 2 whether it is minimal and 0 0 !12 1 X lEði; jÞl þ di;j 2 1 AA O@n·@ d ði;jÞ[K i;j time to test all solutions in step 2 in the worst case. In sum, the proposed algorithm needs 0 0 !1 X lEði; jÞl þ di;j 2 1 X A þ O@m·n· O@m·n· ði;jÞ[K di;j ði;jÞ[K
lEði; jÞl þ di;j 2 1 di;j
!1 A
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0 0 O@n·@
X ði;jÞ[K
lEði; jÞl þ di;j 2 1 di;j
0 0 !12 1 X A A ¼ O@n·@ ði;jÞ[K
lEði; jÞl þ di;j 2 1 di;j
!12 1 AA
time in the worst case, since ! lEði; jÞl þ di;j 2 1 $ lEði; jÞl di;j and so X ði;jÞ[K
lEði; jÞl þ di;j 2 1
!
di;j
$
X
lEði; jÞl ¼ m:
ði;jÞ[K
7. Conclusions and discussion This article mainly tries to evaluate the flow reliability which satisfies the system demand ðdi;j Þði;jÞ[K simultaneously for multiple node pairs of a directed capacitated-flow network in which the capacity of each arc ai has the value 0 , 1 , 2 , · · · , M i : A simple algorithm is proposed first to generate all lower boundary points for ðdi;j Þði;jÞ[K in terms of MPs. Then the flow reliability can be calculated in terms of such lower boundary points. The proposed algorithm takes 0 0 !12 1 X lEði; jÞl þ di;j 2 1 AA O@n·@ ði;jÞ[K di;j time in the worst case. That is, the computational time in the worst case is bounded by 0 !12 X lEði; jÞl þ di;j 2 1 A k·n·@ ði;jÞ[K di;j for a real number k: One straight method is to apply the Q algorithm by either Xue (1985) or Lin et al. (1995) for each ði; jÞ [ K to obtain Ri;j ; and to let RK ; ði;jÞ[K Ri;j in order to approximate RK where Ri;j ¼ Pr{Xl there exists an F [ UX such thatQFi;j $ di;j }. It is obvious that Pr{Xl there exists an F [ UX such that ðFi;j Þði;jÞ[K $ ðdi;j Þði;jÞ[K } – ði;jÞ[K Pr{Xl there exists an F [ UX such that Fi;j $ di;j }. However, we can see in Table 5 that the values of RK are over-exaggerated in all three cases of our example. Note that in case 1, RK ¼ R1;2 £ R1;3 £ R4;3 ¼ 0:995125 £ 0:85637625 £ 0:971625 ¼ 0:8280202: The overall-terminal flow reliability, one source to multiple sinks flow reliability and multiple sources to one sink flow reliability are actually in the special cases that K are the totality of node pairs in the network, K ¼ {ðs; jÞlj [ N1 } and K ¼ {ði; tÞli [ N2 }; respectively, where N1 and N2 are subsets of nodes.
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Table 5 The values RK vs RK in the example
Case 1 Case 2 Case 3
RK
RK
RK 2 RK (error)
0.8280202 0.6453629 0.8491870
0.5119125 0.0 0.5382492
0.3161077 0.6453629 0.3019378
Appendix A Lemma 1. Any flow pattern F satisfying constraint (2) also satisfies constraint (1). P i i Proof. m k¼1 {fk lai [ mpk } # M For each mpk ; fk # M for each ai [ mpk due to Eq. (2). This implies i that fk # min{M lai [ mpk }: A Lemma 2. Let X be a system capacity vector. Suppose that the flow pattern F [ UX satisfies ðFi;j Þði;jÞ[K $ ðdi;j Þði;jÞ[K ; then there exists a flow pattern H [ UX with H , F such that ðHi;j Þði;jÞ[K ¼ ðdi;j Þði;jÞ[K : Proof. For any ði; jÞ [ K such that Fi;j . di;j ; we may assume Fi;j ¼ di;j þ 1 without loss of generality. Let F ¼ ðf1 ; f2 ; …fm Þ ¼ ðf1 ; f2 ; …; fw21 ; fw 2 1; fwþ1 ; …; fm Þ for an mpw [ Eði; jÞ with fw . 0: Then F , F and Fi;j ¼ Fi;j 2 1 ¼ di;j (note that Fi;j ¼ Fi;j $ di;j for any other ði; jÞ [ K). This implies that we can repeat the above procedure in finite times to deduce F to an H [ UX such that Hi;j ¼ di;j for each ði; jÞ [ K: A Proof of Theorem 1. (1) (2)
It is trivial since X [ V: P Suppose to the contrary that there exists an arc ai s.t. m k¼1 {fk lai [ mpk } # xi 2 1: Hence, F [ UX2ei where X 2 ei ¼ ðx1 ; x2 ; …; xi21 ; xi 2 1; xiþ1 ; …; xn Þ: In particular, X 2 ei , X which contradicts to the requirement that X is a lower boundary point for ðdi;j Þði;jÞ[K :
A Theorem 2. rmin ¼ Vmin Proof. Note that X [ rmin implies that X [ V and that X [ Vmin implies that X [ r (due to Theorem 1). Suppose that X [ rmin but X Vmin ; i.e. there exists a Y [ Vmin such that Y , X: Then Y [ r which contradicts to that X [ rmin : Hence rmin # Vmin : Conversely, suppose that X [ Vmin but X rmin ; i.e. there exist a Y [ rmin such that Y , X: Then Y [ V which contradicts to that X [ Vmin : Hence, Vmin # rmin and so rmin ¼ Vmin : A
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