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

TL;DR

This paper introduces an exact two-step Benders decomposition method with scenario clustering to efficiently solve the stochastic Time Window Assignment Traveling Salesperson Problem, improving solution quality and convergence speed.

Contribution

It presents a novel two-step decomposition approach with a new scenario-retention strategy for solving stochastic TSP problems with time windows and travel time uncertainty.

Findings

TBDS outperforms existing methods in solving TWATSP-ST instances.

The method achieves faster convergence and better bounds.

Optimal solutions reveal that slight cost increases can significantly enhance customer service.

Abstract

Next-day delivery logistics services are redefining the industry by increasingly focusing on customer service. A challenge each logistics service provider faces is to jointly optimize 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 need to be incorporated in 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 the class of 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.