Recent exact algorithms for solving the vehicle routing problem under capacity and time window constraints

This paper provides a review of the recent developments that had a major impact on the current state-of-the-art exact algorithms for the vehicle routing problem (VRP). The paper reviews mathematical formulations, relaxations and recent exact methods for two of the most important variants of the VRP: the capacitated VRP (CVRP) and the VRP with time windows (VRPTW). The paper also reports a comparison of the computational performances of the different exact algorithms for the CVRP and VRPTW.     

Approximation algorithms for a vehicle routing problem

In this paper we investigate a vehicle routing problem motivated by a real-world application in cooperation with the German Automobile Association (ADAC). The general task is to assign service requests to service units and to plan tours for the units such as to minimize the overall cost. The characteristics of this large-scale problem due to the data volume involve strict real-time requirements. We show that the problem of finding a feasible dispatch for service units starting at their current position and serving at most k requests is NP-complete for each fixed k ≥ 2. We also present a polynomial time (2k − 1)-approximation algorithm, where again k denotes the maximal number of requests served by a single service unit. For the boundary case when k equals the total number |E| of requests (and thus there are no limitations on the tour length), we provide a -approximation. Finally, we extend our approximation results to include linear and quadratic lateness costs.     

On the complexity of the k-customer vehicle routing problem

We investigate the complexity of the k-CUSTOMER VEHICLE ROUTING PROBLEM: Given an edge weighted graph, the problem requires to compute a minimum weight set of cyclic routes such that each contains a distinguished depot vertex and at most other k customer vertices, and every customer belongs to exactly one route.     

Gossip algorithms for heterogeneous multi-vehicle routing problems

In this paper we address a class of heterogeneous multi-vehicle task assignment and routing problems. We propose two distributed algorithms based on gossip communication: the first algorithm is based on a local exact optimization and the second is based on a local approximate greedy heuristic. We consider the case where a set of heterogeneous tasks arbitrarily distributed in a plane has to be serviced by a set of mobile robots, each with a given movement speed and task execution speed. Our goal is to minimize the maximum execution time of robots.     

A computational comparison of algorithms for the inventory routing problem

The inventory routing problem is a distribution problem in which each customer maintains a local inventory of a product such as heating oil and consumes a certain amount of that product each day. Each day a fleet of trucks is dispatched over a set of routes to resupply a subset of the customers. In this paper, we describe and compare algorithms for this problem defined over a short planning period, e.g. one week. These algorithms define the set of customers to be serviced each day and produce routes for a fleet of vehicles to service those customers. Two algorithms are compared in detail, one which first allocates deliveries to days and then solves a vehicle routing problem and a second which treats the multi-day problem as a modified vehicle routing problem. The comparison is based on a set of real data obtained from a propane distribution firm in Pennsylvania. The solutions obtained by both procedures compare quite favorably with those in use by the firm.Part of this work was performed while this author was visiting the University of Waterloo.     

New exact method for large asymmetric distance-constrained vehicle routing problem

In this paper we revise and modify an old branch-and-bound method for solving the asymmetric distance–constrained vehicle routing problem suggested by Laporte et al. in 1987. Our modification is based on reformulating distance–constrained vehicle routing problem into a travelling salesman problem, and on using assignment problem as a lower bounding procedure. In addition, our algorithm uses the best-first strategy and new tolerance based branching rules. Since our method is fast but memory consuming, it could stop before optimality is proven. Therefore, we introduce the randomness, in case of ties, in choosing the node of the search tree. If an optimal solution is not found, we restart our procedure. As far as we know, the instances that we have solved exactly (up to 1000 customers) are much larger than the instances considered for other vehicle routing problem models from the recent literature. So, despite of its simplicity, this proposed algorithm is capable of solving the largest instances ever solved in the literature. Moreover, this approach is general and may be used for solving other types of vehicle routing problems.     

A unified exact method for solving different classes of vehicle routing problems

This paper presents a unified exact method for solving an extended model of the well-known Capacitated Vehicle Routing Problem (CVRP), called the Heterogenous Vehicle Routing Problem (HVRP), where a mixed fleet of vehicles having different capacities, routing and fixed costs is used to supply a set of customers. The HVRP model considered in this paper contains as special cases: the Single Depot CVRP, all variants of the HVRP presented in the literature, the Site-Dependent Vehicle Routing Problem (SDVRP) and the Multi-Depot Vehicle Routing Problem (MDVRP). This paper presents an exact algorithm for the HVRP based on the set partitioning formulation. The exact algorithm uses three types of bounding procedures based on the LP-relaxation and on the Lagrangean relaxation of the mathematical formulation. The bounding procedures allow to reduce the number of variables of the formulation so that the resulting problem can be solved by an integer linear programming solver. Extensive computational results over the main instances from the literature of the different variants of HVRPs, SDVRP and MDVRP show that the proposed lower bound is superior to the ones presented in the literature and that the exact algorithm can solve, for the first time ever, several test instances of all problem types considered.      

The complexity of branch-and-price algorithms for the capacitated vehicle routing problem with stochastic demands

The capacitated vehicle routing problem with stochastic demands (CVRPSD) is a variant of the deterministic capacitated vehicle routing problem where customer demands are random variables. While the most successful formulations for several deterministic vehicle-routing problem variants are based on a set-partitioning formulation, adapting such formulations for the CVRPSD under mild assumptions on the demands remains challenging. In this work we provide an explanation to such challenge, by proving that when demands are given as a finite set of scenarios, solving the LP relaxation of such formulation is strongly NP-Hard. We also prove a hardness result for the case of independent normal demands.     

Solving multiobjective vehicle routing problem with stochastic demand via evolutionary computation

This paper considers the routing of vehicles with limited capacity from a central depot to a set of geographically dispersed customers where actual demand is revealed only when the vehicle arrives at the customer. The solution to this vehicle routing problem with stochastic demand (VRPSD) involves the optimization of complete routing schedules with minimum travel distance, driver remuneration, and number of vehicles, subject to a number of constraints such as time windows and vehicle capacity. To solve such a multiobjective and multi-modal combinatorial optimization problem, this paper presents a multiobjective evolutionary algorithm that incorporates two VRPSD-specific heuristics for local exploitation and a route simulation method to evaluate the fitness of solutions. A new way of assessing the quality of solutions to the VRPSD on top of comparing their expected costs is also proposed. It is shown that the algorithm is capable of finding useful tradeoff solutions for the VRPSD and the solutions are robust to the stochastic nature of the problem. The developed algorithm is further validated on a few VRPSD instances adapted from Solomon’s vehicle routing problem with time windows (VRPTW) benchmark problems.     

On the tour partitioning heuristic for the unit demand capacitated vehicle routing problem

The tour partitioning heuristic for the vehicle routing problem assumes an unlimited supply of vehicles. If the number of vehicles is fixed, this heuristic may produce infeasible solutions. We modify the heuristic to guarantee feasibility in this situation and we analyze the worst-case performance of the modified heuristic.     

Heuristics for multi-attribute vehicle routing problems: A survey and synthesis

The attributes of vehicle routing problems are additional characteristics or constraints that aim to better take into account the specificities of real applications. The variants thus formed are supported by a well-developed literature, including a large variety of heuristics. This article first reviews the main classes of attributes, providing a survey of heuristics and meta-heuristics for Multi-Attribute Vehicle Routing Problems (MAVRP). It then takes a closer look at the concepts of 64 remarkable meta-heuristics, selected objectively for their outstanding performance on 15 classic MAVRP with different attributes. This cross-analysis leads to the identification of “winning strategies” in designing effective heuristics for MAVRP. This is an important step in the development of general and efficient solution methods for dealing with the large range of vehicle routing variants.     

A new exact algorithm for the vehicle routing problem based onq-paths andk-shortest paths relaxations

We consider the basic Vehicle Routing Problem (VRP) in which a fleet ofM identical vehicles stationed at a central depot is to be optimally routed to supply customers with known demands subject only to vehicle capacity constraints. In this paper, we present an exact algorithm for solving the VRP that uses lower bounds obtained from a combination of two relaxations of the original problem which are based on the computation ofq-paths andk-shortest paths. A set of reduction tests derived from the computation of these bounds is applied to reduce the size of the problem and to improve the quality of the bounds. The resulting lower bounds are then embedded into a tree-search procedure to solve the problem optimally. Computational results are presented for a number of problems taken from the literature. The results demonstrate the effectiveness of the proposed method in solving problems involving up to about 50 customers and in providing tight lower bounds for problems up to about 150 customers.     

Metastrategy simulated annealing and tabu search algorithms for the vehicle routing problem

The vehicle routing problem (VRP) under capacity and distance restrictions involves the design of a set of minimum cost delivery routes, originating and terminating at a central depot, which services a set of customers. Each customer must be supplied exactly once by one vehicle route. The total demand of any vehicle must not exceed the vehicle capacity. The total length of any route must not exceed a pre-specified bound. Approximate methods based on descent, hybrid simulated annealing/tabu search, and tabu search algorithms are developed and different search strategies are investigated. A special data structure for the tabu search algorithm is implemented which has reduced notably the computational time by more than 50%. An estimate for the tabu list size is statistically derived. Computational results are reported on a sample of seventeen bench-mark test problems from the literature and nine randomly generated problems. The new methods improve significantly both the number of vehicles used and the total distances travelled on all results reported in the literature.     

An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts

This paper presents a new exact algorithm for the Capacitated Vehicle Routing Problem (CVRP) based on the set partitioning formulation with additional cuts that correspond to capacity and clique inequalities. The exact algorithm uses a bounding procedure that finds a near optimal dual solution of the LP-relaxation of the resulting mathematical formulation by combining three dual ascent heuristics. The first dual heuristic is based on the q-route relaxation of the set partitioning formulation of the CVRP. The second one combines Lagrangean relaxation, pricing and cut generation. The third attempts to close the duality gap left by the first two procedures using a classical pricing and cut generation technique. The final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between an upper bound and the lower bound achieved. The resulting problem is solved by an integer programming solver. Computational results over the main instances from the literature show the effectiveness of the proposed algorithm.      

Improved bounds for vehicle routing solutions

We present lower bounds for the vehicle routing problem (VRP) with and without split deliveries, improving the well known bound of Haimovich and Rinnooy Kan. These bounds are then utilized in a design of best-to-date approximation algorithms.     

Path relinking for the vehicle routing problem

This paper describes a tabu search heuristic with path relinking for the vehicle routing problem. Tabu search is a local search method that explores the solution space more thoroughly than other local search based methods by overcoming local optima. Path relinking is a method to integrate intensification and diversification in the search. It explores paths that connect previously found elite solutions. Computational results show that tabu search with path relinking is superior to pure tabu search on the vehicle routing problem.     

Branch-and-cut algorithms for the split delivery vehicle routing problem

In this paper we present two exact branch-and-cut algorithms for the Split Delivery Vehicle Routing Problem (SDVRP) based on two relaxed formulations that provide lower bounds to the optimum. Procedures to obtain feasible solutions to the SDVRP from a feasible solution to the relaxed formulations are presented. Computational results are presented for 4 classes of benchmark instances. The new approach is able to prove the optimality of 17 new instances. In particular, the branch-and-cut algorithm based on the first relaxed formulation is able to solve most of the instances with up to 50 customers and two instances with 75 and 100 customers.     

An approximation algorithm for vehicle routing with compatibility constraints

We study a multiple-vehicle routing problem with a minimum makespan objective and compatibility constraints. We provide an approximation algorithm and a nearly-matching hardness of approximation result. We also provide computational results on benchmark instances with diverse sizes showing that the proposed algorithm (i) has a good empirical approximation factor, (ii) runs in a short amount of time and (iii) produces solutions comparable to the best feasible solutions found by a direct integer program formulation.