Exact Two-Step Benders Decomposition for Two-Stage Stochastic Mixed-Integer Programs

Many real-life optimization problems belong to the class of two-stage stochastic mixed-integer programming problems with continuous recourse. This paper introduces Two-Step Benders Decomposition with Scenario Clustering (TBDS) as a general exact solution methodology for solving such stochastic programs to optimality. The method combines and generalizes Benders dual decomposition, partial Benders decomposition, and Scenario Clustering techniques and does so within a novel two-step decomposition along the binary and continuous first-stage decisions. We use TBDS to provide the first exact solutions for the so-called Time Window Assignment Traveling Salesperson problem. This is a canonical optimization problem for service-oriented vehicle routing; it considers jointly assigning time windows to customers and routing a vehicle among them while travel times are stochastic. Extensive experiments show that TBDS is superior to state-of-the-art approaches in the literature. It solves instances with up to 25 customers to optimality. It provides better lower and upper bounds that lead to faster convergence than related methods. For example, Benders dual decomposition cannot solve instances of 10 customers to optimality. We use TBDS to analyze the structure of the optimal solutions. By increasing routing costs only slightly, customer service can be improved tremendously, driven by smartly alternating between high- and low-variance travel arcs to reduce the impact of delay propagation throughout the executed vehicle route.