Investments in stochastic maximum flow networks

Abstract

Each are (i, j) of the network has capacity ξ _ij where ξ _ij is a non-negative random variable. The capacity of any arc may be reduced increased by an amountu _ij ≥0 at a cost ofc _ij u _ij . The objective is to maximizev−K∑c _ij u _ij wherev is the expected maximum flow. This problem is formulated as a two-stage linear program under uncertainty. Each feasible$\bar u = ||\bar u_{ij} ||$ generates a constraint$- \Sigma \pi _{ij} (\bar u)u_{ij} + \theta \leqslant \rho (\bar u)$ where$\pi _{ij} (\bar u)$ is the probability arc (i, j) is in the minimum cut set and$\rho (\bar u)$ the expected value of the maximum flow under$u = (\bar u)$. The formulation is later generalized to include certain conditions under which the increase in capacity of an arc may be a non-deterministic function of the investmentc _ij u _ij .