Local Area Routes and Valid Inequalities for Efficient Vehicle Routing

TL;DR

This paper introduces Local Area (LA) routes and a novel column generation stabilization scheme to improve the efficiency and tightness of vehicle routing problem solutions, leveraging pre-computed sub-routes and spatially-aware orderings.

Contribution

The paper presents LA-routes and LA-arcs for more efficient VRP solving, along with a new stabilization method that enhances column generation performance.

Findings

LA-routes improve solution efficiency

LA-arcs tighten the set cover VRP formulation

The stabilization scheme accelerates column generation

Abstract

In this research we introduce Local Area (LA) routes for improving the efficiency and tightness of column generation (CG) methods for solving vehicle routing problems (VRP). LA-routes rely on pre-computing the lowest cost elementary sub-route (called an LA-arc) for each tuple consisting of the following: (1) a (first) customer where the LA-arc begins, (2) a distant customer (from the first) where the LA-arc ends, and (3) a set of intermediate customers near the first customer. LA-routes are constructed by concatenating LA-arcs where the final customer in a given LA-arc is the first customer in the subsequent LA-arc. A Decremental State Space Relaxation method is applied over LA-routes to construct the lowest reduced cost elementary route during the pricing step of CG. LA-route based solvers can be used to efficiently tighten the standard set cover VRP using a variant of subset row inequalities, which do not alter the structure of pricing. We incorporate LA-arcs into a novel CG stabilization scheme. Specifically each column generated during pricing is mapped to an ordered list of customers consistent with that column. An LA-arc is consistent with an ordering if the first/last customer in the arc come before/after all other customers in the LA-arc in the associated ordering respectively. Each such ordering is then mapped to a multi-graph where nodes correspond to (customer/demand) and edges correspond to LA-arcs consistent with that ordering. Hence any path from source to sink on the multi-graph is a feasible elementary route. The ordering for a column places customers spatially nearby in nearby positions on the ordering so that routes can be generated so as to permit spatially nearby customers to be visited without traveling far away first. We solve the restricted master problem over these graphs, which has special structure allowing for fast solution.