Loss-constrained minimum cost flow under arc failure uncertainty with applications in risk-aware kidney exchange

In this article, we study a Stochastic Minimum Cost Flow (SMCF) problem under arc failure uncertainty, where an arc flow solution may correspond to multiple path flow representations. We assume that the failure of an arc will cause flow losses on all paths using that arc, and for any path carrying positive flows, the failure of any arc on the path will lose all flows carried by the path. We formulate two SMCF variants to minimize the cost of arc flows, while respectively restricting the Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) of random path flow losses due to uncertain arc failure (reflected as network topological changes). We formulate a linear program to compute possible losses, yielding a mixed-integer programming formulation of SMCF-VaR and a linear programming formulation of SMCF-CVaR. We present a kidney exchange problem under uncertain match failure as an application and use the two SMCF models to maximize the utility/social welfare of pairing kidneys subject to constrained risk of utility losses. Our results show the efficacy of our approaches, the conservatism of using CVaR, and optimal flow patterns given by VaR and CVaR models on diverse instances.