Exact Two-Step Benders Decomposition for the Time Window Assignment Traveling Salesperson Problem

Next-day delivery logistics services are redefining the industry by increasingly focusing on customer service. Each logistics service provider’s challenge is jointly optimizing time window assignment and vehicle routing for such next-day delivery services. To do so in a cost-efficient and customer-centric fashion, real-life uncertainty, such as stochastic travel times, needs to be incorporated into the optimization process. This paper focuses on the canonical optimization problem within this context: the time window assignment traveling salesperson problem with stochastic travel times (TWATSP-ST). It belongs to two-stage, stochastic, mixed-integer programming problems with continuous recourse. We introduce two-step Benders decomposition with scenario clustering (TBDS) as an exact solution methodology for solving such stochastic programs. The method utilizes a new two-step decomposition along the binary and continuous first stage decisions and introduces a new scenario-retention strategy that combines and generalizes state-of-the-art Benders approaches and scenario-clustering techniques. Extensive experiments show that TBDS is superior to state-of-the-art approaches in the literature. It solves TWATSP-ST instances with up to 25 customers to optimality. It provides better lower and upper bounds that lead to faster convergence than existing state-of-the-art methods. 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.