Improving Column Generation for Vehicle Routing Problems via Random Coloring and Parallelization

We consider a variant of the vehicle routing problem (VRP) where each customer has a unit demand and the goal is to minimize the total cost of routing a fleet of capacitated vehicles from one or multiple depots to visit all customers. We propose two parallel algorithms to efficiently solve the column-generation-based linear-programming relaxation for this VRP. Specifically, we focus on algorithms for the “pricing problem,” which corresponds to the resource-constrained elementary shortest path problem. The first algorithm extends the pulse algorithm for which we derive a new bounding scheme on the maximum load of any route. The second algorithm is based on random coloring from parameterized complexity which can be also combined with other techniques in the literature for improving VRPs, including cutting planes and column enumeration. We conduct numerical studies using VRP benchmarks (with 50–957 nodes) and instances of a medical home care delivery problem using census data in Wayne County, Michigan. Using parallel computing, both pulse and random coloring can significantly improve column generation for solving the linear programming relaxations and we can obtain heuristic integer solutions with small optimality gaps. Combining random coloring with column enumeration, we can obtain improved integer solutions having less than 2% optimality gaps for most VRP benchmark instances and less than 1% optimality gaps for the medical home care delivery instances, both under a 30-minute computational time limit. The use of cutting planes (e.g., robust cuts) can further reduce optimality gaps on some hard instances, without much increase in the run time. Summary of Contribution: The vehicle routing problem (VRP) is a fundamental combinatorial problem, and its variants have been studied extensively in the literature of operations research and computer science. In this paper, we consider general-purpose algorithms for solving VRPs, including the column-generation approach for the linear programming relaxations of the integer programs of VRPs and the column-enumeration approach for seeking improved integer solutions. We revise the pulse algorithm and also propose a random-coloring algorithm that can be used for solving the elementary shortest path problem that formulates the pricing problem in the column-generation approach. We show that the parallel implementation of both algorithms can significantly improve the performance of column generation and the random coloring algorithm can improve the solution time and quality of the VRP integer solutions produced by the column-enumeration approach. We focus on algorithmic design for VRPs and conduct extensive computational tests to demonstrate the performance of various approaches.

A branch-and-price algorithm for two-echelon electric vehicle routing problem

Motivated by express and e-commerce companies’ distribution practices, we study a two-echelon electric vehicle routing problem. In this problem, fuel-powered vehicles are used to transport goods from a depot to intermediate facilities (satellites) in the first echelon, whereas electric vehicles, which have limited driving ranges and need to be recharged at recharging stations, are used to transfer goods from the satellites to customers in the second echelon. We model the problem as an arc flow model and decompose the model into a master problem and pricing subproblem. We propose a branch-and-price algorithm to solve it. We use column generation to solve the restricted master problem to provide lower bounds. By enumerating all the subsets of the satellites, we generate feasible columns by solving the elementary shortest path problem with resource constraints in the first echelon. Then, we design a bidirectional labeling algorithm to generate feasible routes in the second echelon. Comparing the performance of our proposed algorithm with that of CPLEX in solving a set of small-sized instances, we demonstrate the former’s effectiveness. We further assess our algorithm in solving two sets of larger scale instances. We also examine the impacts of some model parameters on the solution.