6th International Conference of Iranian Operations Research Society May 8-9, 2013 Research Center of Operations Research 1 Multi-commodity Unreliability Evaluation for a Stochastic-Flow Network M. Forghani-elahabad 1 , Ph. D. student of Faculty of Mathematical Sciences, Sharif University of Technology, forghanimajid@mehr.sharif.ir N. Mahdavi-Amiri, Academic member of Faculty of Mathematical Sciences, Sharif University of Technology, nezamm@sharif.ir Abstract: An upper boundary point for a demand vector is defined to be a system state vector for which the system capacity is equal to the demand vector, and an increase in its unsaturated components leads to at least one unit of increase in system capacity. System capacity of a single- commodity stochastic-flow network is the maximum flow from the source node to the sink node of the network. Extending the notion of system capacity to a multi-commodity stochastic-flow network, several algorithms have been proposed to evaluate system reliability or unreliability of the network in terms of minimal cuts. Here, by presenting and using some new useful results, an efficient algorithm is proposed for finding all the upper boundary points of a multi-commodity stochastic-flow network. Two examples are worked out to illustrate comparative efficiency of the proposed algorithm. Keywords: Unreliability, Multi-commodity, Stochastic flow network, System capacity, Minimal cut. 1 Corresponding Author 1. INTRODUCTION System capacity of a single-commodity network flow is defined to be the maximum flow from a source node s to a sink node t. While the system capacity of a deterministic flow network is known to be fixed, but in a stochastic-flow network (SFN) with arcs and nodes having random integer capacities, the system capacity is not fixed. System unreliability, ur, the probability of the system capacity being less than or equal to a given demand level, and reliability, 1-ur, are two effective quantities useful for quality management and decision making. Several algorithms have been proposed to evaluate system unreliability of a single-commodity SFN in terms of Minimal Cuts (MCs) [2], [6]-[10]. An MC is a set of arcs with none of its proper subsets being a cut. Yan and Qian [8] presented an algorithm that obtained all the d-MC candidates in order to find all the d-MCs. They lessened the number of candidates by finding the Lower Capacity Limits (LCLs). Yeh [10] in developing an efficient algorithm provided a novel technique to avoid the production of duplicates and a new technique to obtain LCLs. In [6], Salehi-Fathabadi and Forghani presented a simple algorithm working merely on the definition. By introducing a new data structure to eliminate the duplicates and a simple technique to find the LCLs, Forghani and Mahdavi-Amiri [2] improved the proposed algorithm in [6] and demonstrated the efficiency of their algorithm in comparison with other proposed algorithms in [8] and [10]. Yang and Chen employed the Monte Carlo simulation to approximately evaluate the reliability of an SFN. However, there are a number of real systems with two or more types of transmitting commodities, and thus make the p-commodity SFN attractive (see [3] and [4]). Lin [3] considered a two- commodity SFN and computed the probability of the system capacity of the network being less than or equal to a given demand vector (d 1 ,d 2 ). In [4], the author extended his proposed algorithm to a p-commodity case along with budget constraint. Here, we consider a p-commodity network flow problem with a vector of demand levels d=(d 1 , d 2 , …, d p ). By extending some results on the single-commodity case, an algorithm is proposed and its efficiency is illustrated. 2. STOCHASTIC-FLOW NETWORK A. Preliminaries and results Let G(A, M) be an SFN with the set of arcs A={a 1 , a 2 , …, a m }and M=(M 1 , M 2 , …, M m ) with M i =M(a i ) denoting the max capacity of arc a i , for i=1,2,…, m. Denote the current capacity of a i by x i , and let X= (x 1 , x 2 , …, x m ) be a system-state vector representing the current capacity of all the arcs. Also, let U(X)= {a i  A| x i < M i } be the set of unsaturated arcs under X. We say X= (x 1 ,…,x m ) ≥ Y= (y 1 ,…,y m ), when x i ≥ y i , for i=1,2,…,m, and X > Y, when X ≥ Y and there exists at least one 1 ≤ j ≤ m such that x j >y j . Let α i be the consumed amount of capacity on each arc by one unit of commodity i, for i=1, 2, …, p. Without loss of generality, assume that α 1 ≤ α 2 ≤ … ≤ α p . Corresponding to an MC, C i ={ i n i i i a a a ,..., , 2 1 }, a flow assignment F= ) ,..., ,..., ,..., , ,..., ( 1 1 1 2 2 1 1 p i p i i i i i i n i n i n f f f f f f is