A New Class of Compact Formulations for Vehicle Routing Problems

TL;DR

A novel compact mixed integer linear programming (MILP) formulation and a discretization discovery-based solution approach for the Vehicle Routing Problem with Time Windows that holds promise for addressing a wide range of routing problems within and beyond the VRPTW domain.

Abstract

This paper introduces a novel compact mixed integer linear programming (MILP) formulation and a discretization discovery-based solution approach for the Vehicle Routing Problem with Time Windows (VRPTW). We aim to solve the optimization problem efficiently by constraining the linear programming (LP) solutions to use only flows corresponding to time and capacity-feasible routes that are locally elementary (prohibiting cycles of customers localized in space). We employ a discretization discovery algorithm to refine the LP relaxation iteratively. This iterative process alternates between two steps: (1) increasing time/capacity/elementarity enforcement to increase the LP objective, albeit at the expense of increased complexity (more variables and constraints), and (2) decreasing enforcement without decreasing the LP objective to reduce complexity. This iterative approach ensures we produce an LP relaxation that closely approximates the optimal MILP objective with minimal complexity, facilitating an efficient solution via an off-the-shelf MILP solver. The effectiveness of our method is demonstrated through empirical evaluations on classical VRPTW instances. We showcase the efficiency of solving the final MILP and multiple iterations of LP relaxations, highlighting the decreased integrality gap of the final LP relaxation. We believe that our approach holds promise for addressing a wide range of routing problems within and beyond the VRPTW domain.

Figures (3)

• Figure 1: We display a route as a sequence of LA-arcs. Each LA-arc describes a sequence of customers near the first customer in the arc, followed by a distant customer. The example uses American universities and LA-neighborhoods defined by university location. LA-arcs are color-coded as follows. Red indicates the first customer (or depot) in an LA-arc, Black indicates intermediate customers in an LA-arc, and Blue indicates the final customer (or depot) in an LA-arc. The first customer of an LA-arc is shared with the last customer of the previous LA-arc. Observe that the class of cycles permitted in a sequence of LA-arcs is highly limited because individual LA-arcs that make up the route must be elementary.
• Figure 2: Combined Figure Set One
• Figure 3: Combined Figure Set Two