Recent advances in vehicle routing exact algorithms

The capacitated vehicle routing problem (CVRP) is the problem in which a set of identical vehicles located at a central depot is to be optimally routed to supply customers with known demands subject to vehicle capacity constraints. This paper provides a review of the most recent developments that had a major impact in the current state-of-the-art of exact algorithms for the CVRP. The most important mathematical formulations for the problem together with various CVRP relaxations are reviewed. The paper also describes the recent exact methods for the CVRP and reports a comparison of their computational performances.      

Polyhedral study of the capacitated vehicle routing problem

The capacitated vehicle routing problem (CVRP) considered in this paper occurs when goods must be delivered from a central depot to clients with known demands, usingk vehicles of fixed capacity. Each client must be assigned to exactly one of the vehicles. The set of clients assigned to each vehicle must satisfy the capacity constraint. The goal is to minimize the total distance traveled. When the capacity of the vehicles is large enough, this problem reduces to the famous traveling salesman problem (TSP). A variant of the problem in which each client is visited by at least one vehicle, called the graphical vehicle routing problem (GVRP), is also considered in this paper and used as a relaxation of CVRP. Our approach for CVRP and GVRP is to extend the polyhedral results known for TSP. For example, the subtour elimination constraints can be generalized to facets of both CVRP and GVRP. Interesting classes of facets arise as a generalization of the comb inequalities, depending on whether the depot is in a handle, a tooth, both or neither. We report on the optimal solution of two problem instances by a cutting plane algorithm that only uses inequalities from the above classes.This work was supported in part by NSF grant DDM-8901495.     

Solving the capacitated vehicle routing problem using the ALGELECT electrostatic algorithm

The capacitated vehicle routing problem (CVRP) with a single depot is a classic routing problem with numerous real-world applications. This paper describes the design, modelling and computational aspects of ALGELECT (electrostatic algorithm), a new algorithm for the CVRP. After some general remarks about the origin of the algorithm and its parameters, a parameter tuning process is carried out in order to improve its efficiency. The algorithm is then explained in detail and its main characteristics are presented. Thus, ALGELECT develops good-quality solutions to the CVRP, in terms of the number of scheduled routes and the load ratio of the delivery vehicles. Finally, ALGELECT is used to find solutions for some Solomon's and Augerat's instances, which are then compared to solutions generated by other well-known methods.     

物流配送装载率分析与四阶段算法研究

在城市物流配送中,租用车型的选择与车辆平均装载率具有密切的关系。然而,在带能力约束的车辆路径问题(Capacitated Vehicle Routing Problem, CVRP)中, 假设配送车辆装载量为事先已知。在实际物流配送中, 很多配送车辆为租用, 因此需要确定租用的车型大小。本文基于CVRP问题,假设配送车辆载量Q为变量,以车辆平均装载率为优化目标构建了数学模型. 通过数学推导证明了,派送车辆的平均装载率ρ的理论区间为(50%, 100%]。分析得出结论:当顾客需求数据中需求数据大于且接近0.5倍载量Q的越多,车辆平均装载率越低。为了验证分析结论的正确性, 分别设计一个求解CVRP问题的多阶段算法和具有大需求量的CVRP问题算例. 通过求解算例表明:本文理论分析的正确性, 其中四阶段算法的求解结果与当前已知最优解平均偏差仅为0.92%,达到优秀算法水平。     

Internal Twists of <Emphasis Type="Italic">L</Emphasis>-Functions. II

The Capacitated Vehicle Routing Problem (CVRP) is a classic combinatorial optimization problem with a wide range of applications in operations research. Since the CVRP is NP-hard even in a finite-dimensional Euclidean space, special attention is traditionally paid to the issues of its approximability. A major part of the known results concerning approximation algorithms and polynomial-time approximation schemes (PTAS) for this problem are obtained for its particular statement in the Euclidean plane. In this paper, we show that the approach to the development of a PTAS for the planar problem with a single depot proposed by Haimovich and Rinnooy Kan in 1985 can be successfully extended to the more general case, for instance, in spaces of arbitrary fixed dimension and for an arbitrary number of depots.     

A branch-and-cut-and-price algorithm for the cumulative capacitated vehicle routing problem

In this paper we consider the Cumulative Capacitated Vehicle Routing Problem (CCVRP), which is a variation of the well-known Capacitated Vehicle Routing Problem (CVRP). In this problem, the traditional objective of minimizing total distance or time traveled by the vehicles is replaced by minimizing the sum of arrival times at the customers. We propose a branch-and-cut-and-price algorithm for obtaining optimal solutions to the problem. To the best of our knowledge, this is the first published exact algorithm for the CCVRP. We present computational results based on a set of standard CVRP benchmarks and investigate the effect of modifying the number of vehicles available.     

BoneRoute: An Adaptive Memory-Based Method for Effective Fleet Management

This paper presents an adaptive memory-based method for solving the Capacitated Vehicle Routing Problem (CVRP), called BoneRoute. The CVRP deals with the problem of finding the optimal sequence of deliveries conducted by a fleet of homogeneous vehicles, based at one depot, to serve a set of customers. The computational performance of the BoneRoute was found to be very efficient, producing high quality solutions over two sets of well known case studies examined.     

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.     

A branch-and-cut algorithm for the capacitated open vehicle routing problem

In open vehicle routing problems, the vehicles are not required to return to the depot after completing service. In this paper, we present the first exact optimization algorithm for the open version of the well-known capacitated vehicle routing problem (CVRP). The algorithm is based on branch-and-cut. We show that, even though the open CVRP initially looks like a minor variation of the standard CVRP, the integer programming formulation and cutting planes need to be modified in subtle ways. Computational results are given for several standard test instances, which enables us for the first time to assess the quality of existing heuristic methods, and to compare the relative difficulty of open and closed versions of the same problem.     

Approximating the chance-constrained capacitated vehicle routing problem with robust optimization

4OR - The Capacitated Vehicle Routing Problem (CVRP) is a classical combinatorial optimization problem that aims to serve a set of customers, using a set of identical vehicles, satisfying the...     

An Ant Colony Algorithm for the Capacitated Vehicle Routing

The Vehicle Routing Problem (VRP) requires the determination of an optimal set of routes for a set of vehicles to serve a set of customers. We deal here with the Capacitated Vehicle Routing Problem (CVRP) where there is a maximum weight or volume that each vehicle can load. We developed an Ant Colony algorithm (ACO) for the CVRP based on the metaheuristic technique introduced by Colorni, Dorigo and Maniezzo. We present preliminary results that show that ant algorithms are competitive with other metaheuristics for solving CVRP.     

A new efficient approach for solving the capacitated Vehicle Routing Problem using the Gravitational Emulation Local Search Algorithm

Capacitated Vehicle Routing Problem (CVRP) is one of the most famous specialized forms of the VRP, which has attracted considerable attention from scientists and researchers. Therefore, many accurate, heuristic, and meta-heuristic methods have been introduced to solve this problem in recent decades. In this paper, a new meta-heuristic optimization algorithm is introduced to solve the CVRP, which is based on the law of gravity and group interactions. The proposed algorithm uses two of the four basic parameters of velocity and gravitational force in physics based on the concepts of random search and searching agents, which are a collection of masses that interact with each other based on Newtonian gravity and the laws of motion. The introduced method was quantitatively compared with the State-of-the-Art algorithms in terms of execution time and number of optimal solutions achieved in four well-known benchmark problems. Our experiments illustrated that the proposed method could be a very efficient method in solving CVRP and the results are comparable with the results using state-of-the-art computational methods. Moreover, in some cases our method could produce solutions with less number of required vehicles compared to the Best Known Solution (BKS) in a very efficient manner, which is another advantage of the proposed algorithm.     

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.      

The pyramidal capacitated vehicle routing problem

This paper introduces the pyramidal capacitated vehicle routing problem (PCVRP) as a restricted version of the capacitated vehicle routing problem (CVRP). In the PCVRP each route is required to be pyramidal in a sense generalized from the pyramidal traveling salesman problem (PTSP). A pyramidal route is defined as a route on which the vehicle first visits customers in increasing order of customer index, and on the remaining part of the route visits customers in decreasing order of customer index.     

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.      

Use of the BATA algorithm and MIS to solve the mail carrier problem

This paper presents a management information system (MIS) related to a new meta-heuristic algorithm used for solving the mail carrier problem. Although the mail carrier problem involves both pick-ups and deliveries, the problem is stated as the capacitated vehicle routing problem (CVRP), using zero customer demands, since their quantity (number of letters) is not taken into consideration, due to negligible volume occupying in the boot of motor scooter used for distribution operations. The proposed meta-heuristic algorithm termed as backtracking adaptive threshold accepting (BATA) was tested on some known benchmark problems extracted from the literature and it was proved to be quite efficient.     

On the use of Monte Carlo simulation,cache and splitting techniques to improve the Clarke and Wright savings heuristics

This paper presents the SR-GCWS-CS probabilistic algorithm that combines Monte Carlo simulation with splitting techniques and the Clarke and Wright savings heuristic to find competitive quasi-optimal solutions to the Capacitated Vehicle Routing Problem (CVRP) in reasonable response times. The algorithm, which does not require complex fine-tuning processes, can be used as an alternative to other metaheuristics—such as Simulated Annealing, Tabu Search, Genetic Algorithms, Ant Colony Optimization or GRASP, which might be more difficult to implement and which might require non-trivial fine-tuning processes—when solving CVRP instances. As discussed in the paper, the probabilistic approach presented here aims to provide a relatively simple and yet flexible algorithm which benefits from: (a) the use of the geometric distribution to guide the random search process, and (b) efficient cache and splitting techniques that contribute to significantly reduce computational times. The algorithm is validated through a set of CVRP standard benchmarks and competitive results are obtained in all tested cases. Future work regarding the use of parallel programming to efficiently solve large-scale CVRP instances is discussed. Finally, it is important to notice that some of the principles of the approach presented here might serve as a base to develop similar algorithms for other routing and scheduling combinatorial problems.     

Heuristic Procedures for the Capacitated Vehicle Routing Problem

In this paper we present two new heuristic procedures for the Capacitated Vehicle Routing Problem (CVRP). The first one solves the problem from scratch, while the second one uses the information provided by a strong linear relaxation of the original problem. This second algorithm is designed to be used in a branch and cut approach to solve to optimality CVRP instances. In both heuristics, the initial solution is improved using tabu search techniques. Computational results over a set of known instances, most of them with a proved optimal solution, are given.