A branch-and-cut algorithm for vehicle routing problems with three-dimensional loading constraints

Abstract

This paper presents a new branch-and-cut algorithm based on infeasible path elimination for the three-dimensional loading capacitated vehicle routing problem (3L-CVRP) with different loading problem variants. We show that a previously infeasible route can become feasible by adding a new customer if support constraints are enabled in the loading subproblem and call this the incremental feasibility property. Consequently, different infeasible path definitions apply to different 3L-CVRP variants and we introduce several variant-depending lifting steps to strengthen infeasible path inequalities. The loading subproblem is solved exactly using a flexible constraint programming model to determine the feasibility or infeasibility of a route. An extreme point-based packing heuristic is implemented to reduce time-consuming calls to the exact loading algorithm. Furthermore, we integrate a start solution procedure and periodically combine memoized feasible routes in a set-partitioning-based heuristic to generate new upper bounds. A comprehensive computational study, employing well-known benchmark instances, showcases the significant performance improvements achieved through the algorithmic enhancements. Consequently, we not only prove the optimality of many best-known heuristic solutions for the first time but also introduce new optimal and best solutions for a large number of instances.

Figures (13)

• Figure 1: Definition of the coordinate system, the relative directions, and the access direction for unloading.
• Figure 2: Visualization of the incremental feasibility property in the 3L-CVRP.
• Figure 3: Augmentation of placement points by extreme points and normal patterns. Normal patterns (dashed, blue lines and crosses, cf. Beasley1985a) and extreme points (dotted, red lines and circles, cf. Crainic2008). Five useful normal pattern points that are not in the set of extreme points are shown. Two extreme points on the container walls are not in the dynamically generated set of normal pattern points; they have no supporting item directly below, but they can be useful if partial support is sufficient.
• Figure 4: Tournament tail inequalities for infeasible path $P = \left\{ 1,2,3,4,0\right\}$ and depot node 0. Solid lines have a multiplier of 1, while dashed lines have a multiplier of 0.5.
• Figure 5: Undirected infeasible tail path inequalities for infeasible path $P = \left\{ 1,2,3,4,0\right\}$ and depot node 0. Solid lines have a multiplier of 1, while dashed lines have a multiplier of 0.5.

Theorems & Definitions (1)

• Definition 1