Chance-constrained stochastic programming models for humanitarian relief network design

Many engineering applications concerned with issues such as the probability of meeting demand or the reliability of a system give rise to mathematical programming models that involve chance (or probabilistic) constraints. First, we focus on optimization models involving individual chance constraints, in which only the right-hand side vector is random with a finite distribution. We develop strong mixed-integer programming formulations for a recently introduced class of chance-constrained models which treats the reliability levels/ risk tolerances associated with the chance constraints as decision variables and trades off the actual cost / return against the cost of the selected reliability levels in the objective function. In addition, we introduce an alternate cost function type associated with the risk tolerances which requires capturing the value-at-risk (VaR) associated with a variable reliability level. We accomplish this task via a new integer linear programming representation of VaR. Our computational study illustrates the effectiveness of our mathematical programming formulations. We also apply the proposed modeling approach to a new stochastic post-disaster relief network design problem and provide numerical results for a case study. Second, we consider a stochastic pre-disaster relief network design problem in which there is uncertainty in post-disaster demands and transportation network conditions. We develop a risk-averse two-stage chance-constrained stochastic programming model which features a mean-risk objective, and a joint probabilistic constraints enforced on the feasibility of the second-stage problem. We employ an exact Benders decomposition-based branch-and-cut algorithm and our extensive numerical analysis demonstrates the computational effectiveness of the solution algorithm