Quality Management for an E-Commerce Network Under Budget Constraint Yi-Kuei Lin Department of Information Management Van Nung Institute of Technology Chung-Li, Tao-Yuan, Taiwan yklin@cc.vit.edu.tw Abstract In general, several types of information data are transmitted through an E-Commerce network simultaneously. Each type of information data is set to one type of commodity. Under the budget constraint, this paper studies the probability that a given amount of multicommodity can be transmitted through an E-Commerce network, where each node and each arc has several possible capacities. We may take this probability as a performance index for this network. Based on the properties of minimal paths, a simple algorithm is proposed to generate all lower boundary points for (d 1 ,d 2 ,…,d p ;C) where d i is the demand of commodity i and C is the budget. The probability can then be calculated in terms of such points. 1. Introduction The capacity of each arc (the maximum flow passing the arc per unit time) in a binary-state flow network has two levels, 0 and a positive integer. For perfect nodes case, Aggarwal et al. [1] computed the system reliability, the probability that the maximum flow of the network is not less than the demand, in terms of minimal paths (MPs). A MP is an ordered sequence of arcs from the source s to the sink t that has no cycle. Lee et al. [10] and Rueger [15] extended the system reliability problem to the case that nodes and arcs have a positive-integer capacity and may fail. A stochastic-flow network is a multi-state network in which each arc has several states or capacities. The system reliability is the probability that the maximum flow of single-commodity through the network is not less than the demand d. Without the budget constraint, several authors [11,12,14,19,21] had presented algorithms to generate lower boundary points for d in order to evaluate the system reliability for perfect node case. Lin [13] and Yeh [22] extended the problem to the more general case that nodes have several capacities as arcs do. However, in real world, many stochastic-flow networks allow multicommodity to be transmitted from s to t simultaneously. Assuming the flow network is deterministic (i.e., the capacity of each arc is a constant), many authors [3,4,8,17,18] studied the multicommodity minimum cost flow problem, which is to minimize the total cost of multicommodity. The purpose of this paper is to extend the system reliability problem to a multicommodity case, named multicommodity reliability here, for a stochastic-flow network with node failure under budget constraint. Then a MP is an ordered sequence of arcs and nodes from s to t that has no cycle. The system reliability is the probability that the given demand (d 1 ,d 2 ,…,d p ) can be transmitted through the stochastic-flow network under budget C, where d k , k = 1, 2, …, p, is the required demand of commodity k. A simple algorithm is proposed to generate all lower boundary points for (d 1 ,d 2 ,…,d p ;C), then the multicommodity reliability can be computed in terms of all lower boundary points for (d 1 ,d 2 ,…,d p ;C). 2. Multicommodity Model Under Budget Constraint G = (A, N, M) is a stochastic-flow network with source s and sink t where A = {a i |1 ≤ i ≤ n} the set of arcs, N = {a i |n + 1 ≤ i ≤ n + r} the set of nodes and M = (M 1 , M 2 , …, M n + r ) with M i the maximal capacity of a i . Let x i denote the (current) capacity of a i , and it takes values from {0, 1, 2, …, M i } with a given probability distribution. 2.1 Assumptions and Nomenclature 1. All commodities are transmitted from s to t. 2. The capacities of different arcs are statistically independent. x the smallest integer such that x ≥ x Y ≥ X (y 1 , y 2 , …, y n + r ) ≤ (x 1 , x 2 , …, x n + r ) if and only if y i ≥ x i for i = 1, 2, ..., n + r Y > X (y 1 , y 2 , …, y n + r ) > (x 1 , x 2 , …, x n + r ) if and only if Y ≥ X and y i > x i for at least one i 2.2 Multicommodity Flow Suppose P 1 , P 2 , …, P m are MPs form s to t. The multicommodity flow model for G is described in terms of the capacity vector X = (x 1 , x 2 , …, x n + r ) and the flow assignment (F 1 , F 2 ,…, F p ), where F k = ( k m k k f f f ,..., , 2 1 ) with k j f denoting the flow (integer-value) of commodity k through P j , j = 1, 2, …, m, k = 1, 2, …, p. Such an (F 1 , F 2 ,…, F p ) which is feasible under X satisfies the following condition: The Second International Conference on Electronic Business Taipei, Taiwan, December 10-13, 2002