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slide 1: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-2 Issue-10 October- 2016 Page | 62 Reliability Evaluation of Multicommodity Limited-Flow Networks with Budget Constraints Jsen-Shung Lin Department of Information Management Central Police University Taiwan E-mail: jslin168mail.cpu.edu.tw Abstract —Many real-world systems such as manufacturing systems transportation systems and logistics/distribution systems that play important roles in our modern society can be regarded as multicommodity flow networks whose arcs have independent finite and multi-valued random capacities. Such a flow network is a multistate system with multistate components and its reliability for level dc i.e. the probability that k different types of commodity can be transmitted from the source node to the sink node such that the demand level ... 2 1 k d d d  d is satisfied and the total transmission cost is less than or equal to c can be evaluated in terms of minimal path vectors to level dc named dc- MPs here. The main objective of this paper is to present an intuitive algorithm to generate all dc-MPs of such a flow network for each level dc in terms of minimal pathsets. Two examples are given to illustrate how all dc-MPs are generated by our algorithm and then the reliability of one example is computed. Keywords — Reliability limited-flow network multicommodity multistate system dc-MP. I. INTRODUCTION Reliability is an important performance indicator in the planning designing and operation of a real-world system. Traditionally it is assumed that the system under study is represented by a probabilistic graph in a binary-state model and the system operates successfully if there exists one or more paths from the source node s to the sink node t. In such a case reliability is considered as a matter of connectivity only and so it does not seem to be reasonable as a model for some real- world systems. Many physical systems such as manufacturing systems transportation systems and logistics/distribution systems can be regarded as flow networks in which arcs have independent finite and integer-valued random capacities. To evaluate the system reliability of such a flow network different approaches have been presented 7 9 14-23 26-28.. However these models have assumed that the flow along any arc consisted of a single commodity only. For such a flow network with multicommodity it is very practical and desirable to compute its reliability for level dc i.e. the probability that k different types of commodity can be transmitted from the source node to the sink node in the way that the demand level ... 2 1 k d d d  d is satisfied and the total transmission cost is less than or equal to c. In general reliability evaluation can be carried out in terms of minimal pathsets MPs in the binary state model case and dc-MPs i.e. minimal path vectors to level dc 3 lower boundary points of level dc 12 or upper critical connection vector to level dc 7 for each level dc in the multistate model case. The multicommodity limited-flow network with budget constraints here can be treated as a multistate system of multistate components and so the need of an efficient algorithm to search for all of its dc-MPs arises. The main purpose of this article is to present a simple algorithm to generate all dc-MPs of such a network in terms of minimal pathsets. Two examples are given to illustrate how all dc-MPs are generated and the reliability of one example is calculated by further applying the state-space decomposition method 4. II. BASIC ASSUMPTIONS Let U A N G  be a directed limited-flow network with the unique source s and the unique sink t where N is the set of nodes 1 | n i a A i    is the set of arcs and ... 2 1 n u u u U  where i u denotes the maximum capacity of each arc i a for . ... 2 1 n i  Such a flow network is assumed to further satisfy the following assumptions: 1. Each node is perfectly reliable. Otherwise the network will be enlarged by treating each of such nodes as an arc 1. 2. The capacity of each arc i a is an integer-valued random variable that takes integer values from 0 to i u according to a given distribution. slide 2: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-2 Issue-10 October- 2016 Page | 63 3. Every unit flow of commodity  consumes a given amount   of the capacity associated with each arc. 4. The capacities of different arcs are statistically independent. 5. Flow in the network must be integer-valued and satisfy the so-called flow-conservation law 10. This means that no flow will disappear or be created during the transmission. Assumption 4 is made just for convenience. If it fails in practice the proposed algorithm to search for all dc-MPs is still valid except that the reliability computation in terms of such dc-MPs should take the joint probability distributions of all arc capacities into account. Since there are k different types of commodity within the network the system demand level can be represented as a k-tuple vector ... 2 1 k d d d  d where j d is the demand level of commodity j for . ... 2 1 k j  Let ... 2 1 n x x x X  be a system-state vector i.e. the current capacity of each arc i a under X is i x where i x takes integer values i u ... 2 1 0 and ... 2 1 k X V X V X V X V  the system maximal flow vector under X where j X V denotes the maximal flow of commodity j under X. Whenever 2  k there may be more than one maximal flow vector for each X. See the Appendix for more details. Under the system-state vector ... 2 1 n x x x X  the arc set A has the following three important subsets: 0 |    i i X x A a N 0 |    i i X x A a Z and | X V e X V N a S i X i X     where ... 2 1 in i i i e     with 1  ij  if i j  and 0 if i j  . In fact X X X X Z S N S A    \ is a disjoint union of A under X. A system-state vector X is said to be a dc-MP if and only if: 1 its system capacity level is d i.e. VXd 2 each nonzero-capacity arc under X is sensitive i.e. N x S x and 3 the total transmission cost is less than or equal to c. If level dc is given then the probability that k different types of commodity can be transmitted from the source node to the sink node in the way that the demand level ... 2 1 k d d d  d is satisfied and the total transmission cost is less than or equal to c is taken as the system reliability. III. MODEL BUILDING Suppose that m P P P ... 2 1 are the collection of all MPs of the system and let ... ... ... ... 2 1 2 2 2 1 2 1 2 1 1 1 k n n n k k c c c c c c c c c C  denote the transmission cost vector where  i c is the unit transmission cost of commodity  through arc i a . For each j P    i j i i j P a c W |   and | min j i i j P a u L   are taken as the unit transmission cost of commodity  and maximum capacity through it respectively. Under the flow- conservation law any feasible flow pattern from s to t should satisfy that 1 the total flow-in and the total flow-out of each commodity for any given node except for s and t are equal and 2 every unit flow of each commodity from s to t should travel through one of the MPs. Hence under the system -state vector ... 2 1 n x x x X  with ... 2 1 k d d d X V  any feasible flow pattern that the total transmission cost is less than or equal to c can be represented as a flow vector ... ... ... ... 2 1 2 2 2 1 2 1 2 1 1 1 k m m m k k f f f f f f f f f where  j f is the flow of commodity  transmitted through j P such that the following four conditions are satisfied:    m j j d f 1   for each k ... 2 1   1 j k j L f        1 for each m j ... 2 1  2 i j i k m j j u P a f       | 1 1     for each n i ... 2 1  3 c f W j k m j j         1 1 4 slide 3: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-2 Issue-10 October- 2016 Page | 64 Note that | 1 1 j i k m j j P a f          is the least amount of capacity needed for i a under such a flow pattern ... ... ... ... 2 1 2 2 2 1 2 1 2 1 1 1 k m m m k k f f f f f f f f f and so under the system-state vector X | 1 1 j i k m j j P a f          does not exceed the current capacity i x of i a . This fact is given in the following theorem. Theorem 1. Let ... 2 1 n x x x X  be any system-state vector for which  X V d. Then the following is a necessary condition for the flow-conservation law to hold under X: | 1 1 j i k m j j i P a f x           for each n i ... 2 1  5 for any ... ... ... ... 2 1 2 2 2 1 2 1 2 1 1 1 k m m m k k f f f f f f f f f which is a feasible flow pattern of flow d under X. Theorem 2. Let X be a dc-MP. Then the following is a necessary condition for the flow-conservation law to hold under X: | 1 1 j i k m j j i P a f x           for each n i ... 2 1  6 for any ... ... ... ... 2 1 2 2 2 1 2 1 2 1 1 1 k m m m k k f f f f f f f f f which is a feasible flow pattern of flow d under X. Proof . By Theorem 1 | 1 1 j i k m j j i P a f x           for each . ... 2 1 n i  1. For each X i Z a  0  i x and so 6 holds. 2. It remains to show that 6 holds for each X i N a  . Suppose on the contrary that there exists an arc X i N a  such that i j i k m j j x P a f       | 1 1     . Then 1 | 1 1        i j i k m j j x P a f     . In particular X V e X V i    d and so X i S a  which contradicts to the fact that X is a d.c-MP. Hence | 1 1 j i k m j j i P a f x           for each . X i N a  The vector ... 2 1 n x x x X  obtained by first solving ... ... ... ... 2 1 2 2 2 1 2 1 2 1 1 1 k m m m k k f f f f f f f f f F  subject to constraints 1 - 4 and then transforming such ... ... ... ... 2 1 2 2 2 1 2 1 2 1 1 1 k m m m k k f f f f f f f f f F  to ... 2 1 n x x x X  by applying the relationship in 6 will be taken as a dc-MP candidate. To make it clearer that all dc-MPs can be generated by the proposed method the following theorem is necessary. Theorem 3. Every dc-MP is a dc-MP candidate. Proof. Let ... 2 1 n x x x X  be any dc-MP. By definition we know that the maximal flow from s to t under X is d i.e. d  X V and the total transmission cost is less than or equal to c. Hence under the system-state vector X there exists at least one feasible flow pattern ... ... ... ... 2 1 2 2 2 1 2 1 2 1 1 1 k m m m k k f f f f f f f f f F  of flow ... 2 1 k d d d  d such that conditions 1 - 4 are satisfied. As ... 2 1 n x x x X  is a dc-MP we thus conclude by Theorem 2 that | 1 1 j i k m j j i P a f x           for each n i ... 2 1  . This means that X is a dc-MP candidate. Hence every dc-MP is a dc-MP candidate. slide 4: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-2 Issue-10 October- 2016 Page | 65 In this article we first find feasible solutions ... ... ... ... 2 1 2 2 2 1 2 1 2 1 1 1 k m m m k k f f f f f f f f f F  subject to constraints 1 - 4 by applying an implicit enumeration method e.g. backtracking or branch-and-bound 11 and then transform such integer-valued solutions into dc-MP candidates ... 2 1 n x x x via the relationship in 6. Each dc-MP candidate X must be checked whether all nonzero-capacity arcs under X i.e. X N arc  belong to X S . If the answer is “yes ” then X is a dc-MP. Otherwise X is not a dc-MP. The following two theorems play the crucial roles in checking whether a dc-MP candidate is a dc-MP. Theorem 4. For each dc-MP candidate X there exists at least one dc-MP Y such that X Y  . In particular X is not a dc-MP if such a Y satisfies Y X where X Y  if and only if i i x y  for i1 2 ... n and X Y  if and only if X Y  and i i x y  for at least one i. Proof. If X is a dc-MP then Y must be taken as X. Suppose that X is not a dc-MP then there exists a nonzero-capacity arc i a . . X i N a e i  such that d    X V e X V i . Let i e X X   1 . Suppose that 1 X is a dc-MP then Y is taken as 1 X . Otherwise the same procedure may be repeated for 1 X . However this procedure will stop in finite steps i.e. there exists an integer p such that X X X X p p      1 1 ... with d  p X V and p p X X S N  . The proof is thus concluded by letting p X Y  . Theorem 5. If the network is acyclic i.e. contains no directed cycle then each dc-MP candidate is a dc-MP. Proof. Let ... 2 1 n x x x X  be any dc-MP candidate. By Theorem 4 we know that there exists a dc-MP ... 2 1 n y y y Y  such that X Y  . Since 0     d d Y X V no flow is transmitted from s to t under ... 2 2 1 1 n n y x y x y x Y X      . Hence in case Y X  0 |    i i y x i I is not empty and so | I i a i  which is a subset of X N must form cycles since the flow conserves at each node except for s and t and there is no other sink except t see Ford and Fulkerson 10 or Ahuja et al. 2 for more details. This means that if the network is acyclic then   I and so Y X  i.e. each dc-MP candidate X is a dc-MP . Suppose that q X X X ... 2 1 are total dc-MP candidates. We can thus conclude by Theorem 4 that j X is a dc-MP if i j X X  for all q j .... 2 1  but i j  . IV. ALGORITHM Suppose that all MPs m P P P ... 2 1 have been stipulated in advance 5-6 24-25 the family of all dc-MPs can then be derived by the following steps: Step 1. For each ... 2 1 m j P j  calculate | min j i i j P a u L   and    i j i i j P a c W |   Step 2. Find all feasible solutions ... ... ... ... 2 1 2 2 2 1 2 1 2 1 1 1 k m m m k k f f f f f f f f f F  subject to the following constraints by applying an implicit enumeration method: 1    m j j d f 1   for each k ... 2 1   2 j k j L f        1 for each m j ... 2 1  3 i j i k m j j u P a f       | 1 1     for each n i ... 2 1  4 c f W j k m j j         1 1 slide 5: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-2 Issue-10 October- 2016 Page | 66 where  j f is a nonnegative integer for m j ... 2 1  and . ... 2 1 k   Step 3. Transform the solutions ... ... ... ... 2 1 2 2 2 1 2 1 2 1 1 1 k m m m k k f f f f f f f f f into dc-MP candidates ... 2 1 n x x x X  via | 1 1 j i k m j j i P a f x           for . ... 2 1 n i  Step 4. Check each candidate X one at a time whether it is a dc-MP: A If the network is acyclic then each candidate is a dc-MP. B If the network is cyclic and suppose ... 2 1 q X X X is the family of all such dc-MP candidates then i X is a dc-MP if i j X X  for all q j ... 2 1  but i j  . V. EXAMPLES The following two examples are used to illustrate the proposed algorithm: Example 1. FIG. 1. A SERIES-PARALLEL NETWORK. Consider the network in Fig. 1. It is known that 1 2 2 2 1 5 4 3 2 1   u u u u u U 3 2 3 2 6 5 3 2 3 2  C 2 1 2 1     ρ and there exists three MPs . 5 4 3 3 2 2 1 1 a a P a P a a P    Given 1 1  d and 12  c the family of dc-MPs is derived as follows: Step 1. 1 2 1 min 1   L 2 2 min 2   L 1 1 2 min 3   L 4 2 2 1 1    W 6 3 3 2 1    W 5 1 2  W 6 2 2  W 4 2 2 1 3    W . 6 3 3 2 3    W Step 2. Find all feasible solutions 2 3 1 3 2 2 1 2 2 1 1 1 f f f f f f subject to the following constraints by applying an implicit enumeration method:          1 1 2 3 2 2 2 1 1 3 1 2 1 1 f f f f f f                  1 2 1 2 2 1 1 2 1 2 3 1 3 2 2 1 2 2 1 1 1 f f f f f f                              1 2 1 2 2 1 2 2 1 2 2 1 1 2 1 2 3 1 3 2 3 1 3 2 2 1 2 2 1 1 1 2 1 1 1 f f f f f f f f f f 12 6 4 6 5 6 4 2 3 1 3 2 2 1 2 2 1 1 1       f f f f f f where  j f is a nonnegative integer for 3 2 1  j and . 2 1   Total feasible solutions are 0 0 1 0 0 1 1  F and . 0 1 1 0 0 0 2  F s slide 6: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-2 Issue-10 October- 2016 Page | 67 Step 3. Transform such feasible solutions into dc-MP candidates 5 4 3 2 1 x x x x x X  via | 3 1 j i j j i P a f x          for . 5 ... 2 1  i Then 0 0 2 1 1 1  X and 1 1 2 0 0 2  X are total dc-MP candidates. Step 4. The network is acyclic and 2 1 X X is the family of all dc-MP candidates. Since j i X X  0 0 2 1 1 1  X and 1 1 2 0 0 2  X are total dc-MP Example 2. FIG. 2. A BRIDGE NETWORK. TABLE 1 PROBABILITY DISTRIBUTIONS OF ARC CAPACITIES IN EXAMPLE 2 Arc Capacity Probability Arc Capacity Probability 1 a 3 0.60 4 a 1 0.90 2 0.25 0 0.10 1 0.10 5 a 2 0.80 0 0.05 1 0.15 2 a 2 0.70 0 0.05 1 0.20 6 a 3 0.65 0 0.10 2 0.20 3 a 1 0.90 1 0.10 0 0.10 0 0.05 TABLE 2 UNIT TRANSMISSION COST ON EACH ARC IN EXAMPLE 2 Arc Commodity Cost Arc Commodity Cost 1 a 1 2 4 a 1 1 2 2 2 1 3 3 3 2 2 a 1 2 5 a 1 2 2 3 2 3 3 3 3 3 3 a 1 1 6 a 1 2 2 1 2 2 3 2 3 3 Consider the network in Fig. 2. It is known that 3 2 1 1 2 3 6 5 4 3 2 1   u u u u u u U 1 2 1 3 2 1      ρ and there exists four MPs . 6 5 4 5 4 2 3 6 3 1 2 2 1 1 a a P a a a P a a a P a a P     Given 1 1 1  d and 16  c the family of dc-MPs is derived as follows: s slide 7: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-2 Issue-10 October- 2016 Page | 68 Step 1. 2 2 3 min 1   L 1 3 1 3 min 2   L 1 2 1 2 min 3   L 2 3 2 min 4   L 4 2 2 1 1    W 5 3 2 2 1    W 6 3 3 3 1    W 5 2 1 2 1 2     W 5 2 1 2 2 2     W 8 3 2 3 3 2     W 5 2 1 2 1 3     W 7 3 1 3 2 3     W 8 3 2 3 3 3     W 4 2 2 1 4    W 5 2 3 2 4    W and . 6 3 3 3 4    W Step 2. Find all feasible solutions 3 4 2 4 1 4 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 f f f f f f f f f f f f subject to the following constraints by applying an implicit enumeration method:                  1 1 1 3 4 3 3 3 2 3 1 2 4 2 3 2 2 2 1 1 4 1 3 1 2 1 1 f f f f f f f f f f f f                                2 1 2 1 1 1 2 1 1 1 2 1 2 1 2 1 3 4 2 4 1 4 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 f f f f f f f f f f f f                                                                        3 1 1 2 2 1 1 2 1 1 2 2 1 1 1 1 2 1 1 1 2 1 2 1 1 2 2 1 1 3 1 1 2 2 1 1 3 4 3 2 2 4 2 2 1 4 1 2 3 4 3 3 2 4 2 3 1 4 1 3 3 3 2 3 1 3 3 2 2 2 1 2 3 3 3 1 2 3 2 1 1 3 1 1 3 2 3 1 2 2 2 1 1 2 1 1 f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f 16 6 5 4 8 7 5 8 5 5 6 5 4 3 4 2 4 1 4 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1             f f f f f f f f f f f f where  j f is a nonnegative integer for 4 3 2 1  j and . 3 2 1   Total feasible solutions are 0 1 0 0 0 0 0 0 1 1 0 0 1  F 1 0 1 0 0 0 0 0 0 0 1 0 2  F 1 0 0 0 0 0 0 0 1 0 1 0 3  F and . 0 1 0 0 0 0 0 0 0 1 0 1 4  F Step 3. Transform such feasible solutions into dc-MP candidates 6 5 4 3 2 1 x x x x x x X  via | 3 1 j i j j i P a f x          for . 6 ... 2 1  i Then 3 2 0 1 1 2 1  X 2 2 0 0 2 2 2  X and 2 1 0 1 2 3 3  X are total dc-MP candidates. Step 4. The network is cyclic and 3 2 1 X X X is the family of all dc-MP candidates. Since j i X X  every dc-MP candidate is a dc-MP. The result is listed in Table 3. TABLE 3 LIST OF ALL DC-MPS IN EXAMPLE 2 dc-MP candidate dc-MP 3 2 0 1 1 2 1  X Yes 2 2 0 0 2 2 2  X Yes 2 1 0 1 2 3 3  X Yes slide 8: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-2 Issue-10 October- 2016 Page | 69 VI. RELIABILITY EVALUATION If ... 2 1 c m Y Y Y d are the collection of all dc-MPs then the system reliability for level dc is defined as . | Pr 1 c i m i Y X X R c     d d To compute it several methods such as inclusion-exclusion 8 12 disjoint subset 13 and state-space decomposition 4 are available. Here we apply the state-space decomposition method to Example 2 and obtain that 53235 . 0 | Pr 1 c      i m i Y X X R c d d for demand level 1 1 1  d and 16  c . VII. CONCLUSION Given all MPs that are stipulated in advance the proposed method can generate all dc-MPs of a multicommodity limited-flow network under budget constraints for each level dc. The system reliability i.e. the probability that k different types of commodity can be transmitted from the source node s to the sink node t in the way that the demand level ... 2 1 k d d d  d is satisfied and the total transmission cost is less than or equal to c can then be computed in terms of these dc-MPs. This algorithm can also apply to the limited-flow network with single commodity. Hence earlier algorithm 18 is shown to be a special case of this new one. REFERENCES 1 K. K. Aggarwal J. S. Gupta and K. G. 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