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We focus on optimization models involving individual chance constraints, in which only the right-hand side vector is random with a finite distribution. A recently introduced class of such models 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. Leveraging recent methodological advances for modeling and solving chance-constrained linear programs with fixed reliability levels, we develop strong mixed-integer programming formulations for this new variant with variable reliability levels. 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 last mile relief network design problem and provide numerical results for a case study based on the real-world data from the 2011 Van earthquake in Turkey.

We apply logic-based Benders decomposition (LBBD) to two-stage stochastic planning and scheduling problems in which the second-stage is a scheduling task. We solve the master problem with mixed integer/linear programming and the subproblem with constraint programming. As Benders cuts, we use simple nogood cuts as well as analytical logic-based cuts we develop for this application. We find that LBBD is computationally superior to the integer L-shaped method, with a branch-and-check variant of LBBD faster by several orders of magnitude, allowing significantly larger instances to be solved. This is due primarily to computational overhead incurred by the integer L-shaped method while generating classical Benders cuts from a continuous relaxation of an integer programming subproblem. To our knowledge, this is the first application of LBBD to two-stage stochastic optimization with a scheduling second-stage problem, and the first comparison of LBBD with the integer L-shaped method. The results suggest that LBBD could be a promising approach to other stochastic and robust optimization problems with integer or combinatorial recourse.

We apply logic-based Benders decomposition (LBBD) to two-stage stochastic planning and scheduling problems in which the second stage is a scheduling task. We solve the master problem with mixed integer/linear programming and the subproblem with constraint programming. As Benders cuts, we use simple no-good cuts as well as analytic logic-based cuts we develop for this application. We find that LBBD is computationally superior to the integer L-shaped method. In particular, a branch-and-check variant of LBBD can be faster by several orders of magnitude, allowing significantly larger instances to be solved. This is due primarily to computational overhead incurred by the integer L-shaped method while generating classic Benders cuts from a continuous relaxation of an integer programming subproblem. To our knowledge, this is the first application of LBBD to two-stage stochastic optimization with a scheduling second-stage problem and the first comparison of LBBD with the integer L-shaped method. The results suggest that LBBD could be a promising approach to other stochastic and robust optimization problems with integer or combinatorial recourse. Summary of Contribution: We study an important class of optimization problems, namely, two-stage stochastic programs with integer recourse, which are known to be extremely difficult to solve in general. We focus on an application in which the second-stage problem is a scheduling problem, a first in the literature to the best of our knowledge. Our study exemplifies how one can exploit the combinatorial structure of the scheduling problem to derive novel analytic Benders cuts and use them within a branch-and-check algorithm. The proposed algorithm solves instances that are intractable for commercial solvers and state-of-the-art decomposition-based methods, such as the integer L-shaped method. We believe that our study will inspire further research in the use of hybrid logic-based optimization methods for solving stochastic combinatorial optimization problems.