Sparse Process Flexibility Structure via Constraint Sampling

Consider the linear optimization problem Z(b) = max{ ∑n i=1 cixi : Ax≤ b; xi ≥ 0, i = 1, . . . , n}, where ci ≥ 0, b ≥ 0, and A is a m × n matrix. If n > m, it is well known that there is an optimal solution x∗ with the property that |{i : xi > 0}| ≤m. In particular, there is no loss in optimality if we discard up to n−m of the variables, retaining only the m variables needed to support the optimal solution x∗. In this paper, we study a generalization of this phenomenon when b is random. We identify a class of condition under which only a sparse support set of variables (much smaller than n) is needed so that the expected value of the optimal solution will be within a factor of (1− ) from Eb(Z(b)). This problem is motivated by the process flexibility problem studied by Jordan and Graves (1995). They showed that by adding a small number of links to the process flexibility structure of a dedicated production system, they can dramatically enhance the ability of the system to match demand with production capacity. This observation has been empirically verified in numerous computational studies, in a variety of production and service systems. Our approach provides a theoretical justification to this phenomenon under appropriate conditions. We also discuss applications of this approach to a class of transshipment problem.