Classical Capacitated Vehicle Routing Problem Research Articles

An exact algorithm for unpaired pickup and delivery vehicle routing problem with multiple commodities and multiple visits

This paper addresses a new multi-commodity unpaired pickup and delivery vehicle routing problem in which each vehicle is allowed to visit each customer more than once for both pickup and delivery of each commodity. It is a complex variant of the classical capacitated vehicle routing problem and comes from resource transfer among different plants in a distributed manufacturing environment. Notably, the pairing decisions of pickup and delivery for each commodity and the routing decisions among customers are interdependent, and they need to be made simultaneously and collaboratively. Firstly, a mixed integer programming model is developed, and then new valid inequalities are derived to improve the linear programming relaxation. Afterwards, a branch-and-cut algorithm is proposed by exploring different branching strategies and tailored separation algorithms. Computational results based on real-case instances from an automobile manufacturing company demonstrate that the costs are considerably reduced by considering the multiple visits compared to the single visit of each customer. In addition, our proposed branch-and-cut algorithm outperforms a manual solution method adopted by the company in practice and two solution methods in the literature. Furthermore, two adaptations of our branch-and-cut algorithm also perform better than two state-of-the-art algorithms in solving two closely-related problems in terms of solution quality and computation time. Sensitivity analyses with respect to vehicle capacity, vehicle duration, and vehicle speed are conducted to derive several valuable managerial insights.

A hybrid iterated local search algorithm for the multi-compartment vehicle routing problem

Multi-compartment vehicle routing problem (MCVRP) is an extension of the classical capacitated vehicle routing problem where products with different characteristics are transported together in one vehicle with multiple compartments. This paper deals with this problem, whose objective is to minimize the total travel distance while satisfying the capacity and maximum route length constraints. We proposed a hybrid iterated local search metaheuristic (HILS) algorithm to solve it. In the framework of iterated local search, the current solution was improved iteratively by five neighborhood operators. For every obtained neighborhood solution after the local search procedure, a large neighborhood search-based perturbation method was executed to explore larger solution space and get a better neighborhood solution to take part in the next iteration. In addition, the worse solutions found by the algorithm were accepted by the nondeterministic simulated annealing-based acceptance rule to keep the diversification of solutions. Computation experiments were conducted on 28 benchmark instances and the experimental results demonstrate that our presented algorithm finds 17 new best solutions, which significantly outperforms the existing state-of-the-art MCVRP methods.

Bilevel Memetic Search Approach to the Soft-Clustered Vehicle Routing Problem

This work addresses a soft-clustered vehicle routing problem that extends the classical capacitated vehicle routing problem with one additional constraint, that is, customers are partitioned into clusters and all customers of the same cluster must be served by the same vehicle. Its potential applications include parcel delivery in courier companies and freight transportation. Due to its NP-hard nature, solving it is computationally challenging. This paper presents an efficient bilevel memetic search method to do so, which explores search space at both cluster and customer levels. It integrates three distinct modules: a group matching-based crossover (to generate promising offspring solutions), a bilevel hybrid neighborhood search (to perform local optimization), and a tabu-driven population reconstruction strategy (to help the search escape from local optima). Extensive experiments on three sets of 390 widely used public benchmark instances are conducted. The results convincingly demonstrate that the proposed method achieves much better overall performance than state-of-the-art algorithms in terms of both solution quality and computation time. In particular, it is able to find 20 new upper bounds for large-scale instances while matching the best-known upper bounds for all but four of the remaining instances. Ablation studies on three key algorithm modules are also performed to demonstrate the novelty and effectiveness of the proposed ideas and strategies. Funding: This work was supported by the Macau Young Scholars Program [Grant AM2020011], Fundo para o Desenvolvimento das Cienciase da Tecnologia (FDCT) [Grant 0047/2021/A1], the National Natural Science Foundation of China [Grants 61903144, 71871142, and 71931007], and the Open Project of the Shenzhen Institute of Artificial Intelligence and Robotics for Society [Grant AC01202005002]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.1186 .

An adaptive iterated local search heuristic for the Heterogeneous Fleet Vehicle Routing Problem

The Heterogeneous Fleet Vehicle Routing Problem (HFVRP) is an important variant of the classical Capacitated Vehicle Routing Problem (CVRP) that aims to find routes that minimize the total traveling cost of a heterogeneous fleet of vehicles. This problem is of great interest given its importance in many industrial and commercial applications. In this paper, we present an Adaptive Iterated Local Search (AILS) heuristic for the HFVRP. AILS is a local search-based meta-heuristic that achieved good results for the CVRP. The main characteristic of AILS is its adaptive behavior that allows the adjustment of the diversity control of the solutions explored during the search process. The proposed AILS for the HFVRP was tested on benchmark instances containing up to 360 customers. The results of computational experiments indicate that AILS outperformed state-of-the-art metaheuristics on 87% of the instances.

Two-echelon capacitated vehicle routing problem with grouping constraints and simultaneous pickup and delivery

This paper investigates the two-echelon capacitated vehicle routing problem with grouping constraints and simultaneous pickup and delivery (2E-VRPGS), which is a new variant of the classical two-echelon capacitated vehicle routing problem (2E-VRP). In the 2E-VRPGS, customers from the same administrative region are served by vehicles from the same satellite so as to ensure service consistency, with pickup and delivery being performed simultaneously in the second echelon. To solve this problem to optimality, we formulate it as a path-based model and develop a tailored branch-and-cut-and-price algorithm, which can also exactly solve two closely related variants of 2E-VRPGS in the literature: the 2E-VRP with grouping constraints (2E-VRPG), and the 2E-VRP with simultaneous pickup and delivery (2E-VRPS). In particular, a novel dominance rule in the labeling algorithm, together with several customized valid inequalities, has been put forward to effectively accelerate the solution method by exploiting the problem characteristics. To evaluate the efficacy of the proposed algorithm on the problems 2E-VRPG, 2E-VRPS, and 2E-VRPGS, extensive numerical experiments have been conducted on three types of benchmark instances. Computational results on the 2E-VRPGS show that our dominance rule can significantly reduce the number of generated labels and all families of valid inequalities have a great impact on strengthening the path-based model. The algorithm is found to be highly competitive when compared with the existing exact algorithm for the 2E-VRPG and some new findings and managerial insights are derived from sensitivity analysis.

The Arc-Item-Load and Related Formulations for the Cumulative Vehicle Routing Problem

The Capacitated Vehicle Routing Problem (CVRP) consists of finding the cheapest way to serve a set of customers with a fleet of vehicles of a given capacity. While serving a particular customer, each vehicle picks up its demand and carries its weight throughout the rest of its route. While costs in the classical CVRP are measured in terms of a given arc distance, the Cumulative Vehicle Routing Problem (CmVRP) is a variant of the problem that aims to minimize total energy consumption. Each arc’s energy consumption is defined as the product of the arc distance by the weight accumulated since the beginning of the route.The purpose of this work is to propose several different formulations for the CmVRP and to study their Linear Programming (LP) relaxations. In particular, the goal is to study formulations based on combining an arc-item concept (that keeps track of whether a given customer has already been visited when traversing a specific arc) with another formulation from the recent literature, the Arc-Load formulation (that determines how much load goes through an arc).Both formulations have been studied independently before – the Arc-Item is very similar to a multi-commodity-flow formulation in Letchford and Salazar-González (2015) and the Arc-Load formulation has been studied in Fukasawa et al. (2016) – and their LP relaxations are incomparable. Nonetheless, we show that a formulation combining the two (called Arc-Item-Load) may lead to a significantly stronger LP relaxation, thereby indicating that the two formulations capture complementary aspects of the problem. In addition, we study how set partitioning based formulations can be combined with these formulations. We present computational experiments on several well-known benchmark instances that highlight the advantages and drawbacks of the LP relaxation of each formulation and point to potential avenues of future research.

A POPMUSIC matheuristic for the capacitated vehicle routing problem

This work proposes a partial optimization metaheuristic under special intensification conditions (POPMUSIC) for the classical capacitated vehicle routing problem. The proposed approach uses a branch-cut-and-price algorithm as a powerful heuristic to solve subproblems whose dimensions are typically between 25 and 200 customers. The whole algorithm can be seen as the application of local search over very large neighborhoods, starting from a single initial solution. The main computational experiments were carried out on instances having between 302 and 1000 customers. Using initial solutions generated by some of the best available metaheuristics for the problem, POPMUSIC was able to obtain consistently better solutions for long runs of up to 32 h. Starting from the best known solutions available in CVRP library (CVRPLIB), POPMUSIC was able to find new best solutions for several instances, including some very large ones. In a final experiment, we successfully applied the same POPMUSIC framework to tackle the heterogeneous fleet vehicle routing problem and the vehicle routing problem with backhauls.

Hybrid particle swarm optimization algorithm for solving the clustered vehicle routing problem

This paper considers a variant of the classical capacitated vehicle routing problem called clustered vehicle routing problem (CluVRP). In CluVRP, customers are grouped into different clusters. A vehicle visiting a cluster cannot leave the cluster until all customers in the same cluster have been served. Each cluster and customer have to be served only once. A new hybrid metaheuristic, combining the particle swarm optimization (PSO) and variable neighborhood search (VNS) for the specific problem, is proposed to solve the CluVRP. In the hybrid PSO, the basic PSO principle ensures the solution diversity and VNS ensures solution intensity to bring the solution to the local optima. Extensive computational experiments have been performed on numerous benchmark instances with various sizes obtained from the CluVRP literature to evaluate the performance of the proposed hybrid PSO. The obtained results of the proposed algorithm are compared with the results found in the literature to validate the effectiveness of the proposed hybrid PSO. The proposed algorithm is proven to be superior to the state-of-the-art algorithms on the CluVRP. The proposed algorithm obtains 138 new best-known solutions among the 293 benchmark instances.

A branch-and-cut algorithm for the soft-clustered vehicle-routing problem

The soft-clustered vehicle-routing problem is a variant of the classical capacitated vehicle-routing problem (CVRP) in which customers are partitioned into clusters and all customers of the same cluster must be served by the same vehicle. We introduce a novel symmetric formulation of the problem in which the clustering part is modeled with an asymmetric sub-model. We solve the new model with a branch-and-cut algorithm exploiting some known valid inequalities for the CVRP that can be adapted. In addition, we derive problem-specific cutting planes and new heuristic and exact separation procedures. For square grid instances in the Euclidean plane, we provide lower-bounding techniques and a reduction scheme that is also applicable to the respective traveling salesman problem. In comprehensive computational test on standard benchmark instances, we compare the different formulations and separation strategies in order to determine a best performing algorithmic setup. The computational results with this branch-and-cut algorithm show that several previously open instances can now be solved to proven optimality.

A Memetic Algorithm for the Cumulative Capacitated Vehicle Routing Problem Including Priority Indexes

This paper studies the Cumulative Capacitated Vehicle Routing Problem, including Priority Indexes, a variant of the classical Capacitated Vehicle Routing Problem, which serves the customers according to a certain level of preference. This problem can be effectively implemented in commercial and public environments where customer service is essential, for instance, in the delivery of humanitarian aid or in waste collection systems. For this problem, we aim to minimize two objectives simultaneously, the total latency and the total tardiness of the system. A Mixed Integer formulation is developed and solved using the AUGMECON2 approach to obtain true efficient Pareto fronts. However, as expected, the use of commercial software was able to solve only small instances, up to 15 customers. Therefore, two versions of a Memetic Algorithm with Random Keys (MA-RK) were developed to solve the problem. The computational results show that both algorithms provided good solutions, although the second version obtained denser and higher quality Pareto fronts. Later, both algorithms were used to solve larger instances (20–100 customers). The results were mixed in terms of quality but, in general, the MA-RK v2 consistently outperforms the first version. The models and algorithms proposed in this research provide useful insights for the decision-making process and can be applied to solve a wide variety of business situations where economic, customer service, environmental, and social concerns are involved.

A Novel Hybrid Metaheuristic for Solving Automobile Part Delivery Logistics With Clustering Customer Distribution

With the continuous maturity of the automobile industry, the cluster network effect has been gradually formed among automobile parts' suppliers. How to carry out reasonable and efficient distribution in the cluster network has become one of the greatest challenges in today's automobile manufacturers. In this paper, we conducted a field survey of a real-world automobile logistics company and collected data. After analyzing the relevant transportation modes, we concluded that the problem could be classified as a clustered vehicle routing problem (CluVRP). A CluVRP considers gathering customers according to geographical proximity on the basis of the classical capacitated vehicle routing problem (CVRP). An accompanying problem is that there is a hard constraint that all customers in a cluster must be served consecutively by the same vehicle. Because the CluVRP is NP-hard, a hybrid metaheuristic solution approach was proposed. First, the shortest distance best fit decreasing (SD-BFD) algorithm was used to construct an initial solution. Second, a clustering feature was considered in order to divide the problem into two levels for optimization: the intra-cluster level and the inter-cluster level. For the intra-cluster level, the variable neighborhood search (VNS) was applied; for the inter-cluster level, the Lin-Kernighan (LK) considering dummy point was employed for optimizing. The computational performances of the hybrid metaheuristic algorithms are tested on three sets of instances and compared with the results from previous papers and the company's manually calculated solutions. The results show that the proposed algorithm is excellent in both the operation time and the final solution value, which is significant to improve distribution efficiency, reduce transportation costs, and improve customer satisfaction.

The periodic vehicle routing problem with driver consistency

The Periodic Vehicle Routing Problem is a generalization of the classical capacitated vehicle routing problem in which routes are determined for a planning horizon of several days. Each customer has an associated set of allowable visit schedules, and the objective of the problem is to design a set of minimum cost routes that give service to all the customers respecting their visit requirements. In this paper we study a new variant of this problem in which we impose that each customer should be served by the same vehicle/driver at all visits. We call this problem the Periodic Vehicle Routing Problem with Driver Consistency. We present an integer linear programming formulation for the problem and derive several families of valid inequalities. We solve it using an exact branch-and-cut algorithm, and show computational results on a wide range of randomly generated instances.

A New TS Algorithm for Solving Low-Carbon Logistics Vehicle Routing Problem with Split Deliveries by Backpack-From a Green Operation Perspective.

In order to promote the development of low-carbon logistics and economize logistics distribution costs, the vehicle routing problem with split deliveries by backpack is studied. With the help of the model of classical capacitated vehicle routing problem, in this study, a form of discrete split deliveries was designed in which the customer demand can be split only by backpack. A double-objective mathematical model and the corresponding adaptive tabu search (TS) algorithm were constructed for solving this problem. By embedding the adaptive penalty mechanism, and adopting the random neighborhood selection strategy and reinitialization principle, the global optimization ability of the new algorithm was enhanced. Comparisons with the results in the literature show the effectiveness of the proposed algorithm. The proposed method can save the costs of low-carbon logistics and reduce carbon emissions, which is conducive to the sustainable development of low-carbon logistics.

Enhanced Branch-Cut-and-Price algorithm for heterogeneous fleet vehicle routing problems

This paper considers a family of Vehicle Routing Problem (VRP) variants that generalize the classical Capacitated VRP by taking into account the possibility that vehicles differ by capacity, costs, depot allocation, or even by the subset of customers that they can visit. This work proposes a Branch-Cut-and-Price algorithm that adapts advanced features found in the best performing exact algorithms for homogeneous fleet VRPs. The original contributions include: (i) the use of Extended Capacity Cuts, defined over a pseudo-polynomially large extended formulation, together with Rank-1 Cuts, defined over the Set Partitioning Formulation; (ii) the concept of vehicle-type dependent memory for Rank-1 Cuts; and (iii) a new family of lifted Extended Capacity Cuts that takes advantage of the vehicle-type dependent route enumeration. The algorithm was extensively tested in instances of the literature and was shown to be significantly better than previous exact algorithms, finding optimal solutions for many instances with up to 200 customers and also for some larger instances. A new set of benchmark instances is also proposed.

Bilayer Local Search Enhanced Particle Swarm Optimization for the Capacitated Vehicle Routing Problem

The classical capacitated vehicle routing problem (CVRP) is a very popular combinatorial optimization problem in the field of logistics and supply chain management. Although CVRP has drawn interests of many researchers, no standard way has been established yet to obtain best known solutions for all the different problem sets. We propose an efficient algorithm Bilayer Local Search-based Particle Swarm Optimization (BLS-PSO) along with a novel decoding method to solve CVRP. Decoding method is important to relate the encoded particle position to a feasible CVRP solution. In bilayer local search, one layer of local search is for the whole population in any iteration whereas another one is applied only on the pool of the best particles generated in different generations. Such searching strategies help the BLS-PSO to perform better than the existing proposals by obtaining best known solutions for most of the existing benchmark problems within very reasonable computational time. Computational results also show that the performance achieved by the proposed algorithm outperforms other PSO-based approaches.

Skewed general variable neighborhood search for the cumulative capacitated vehicle routing problem

AbstractThe cumulative capacitated vehicle routing problem (CCVRP) is a relatively new version of the classical capacitated vehicle routing problem, and it is equivalent to a traveling repairman problem with capacity constraints and a homogeneous vehicle fleet, which aims to minimize the total arrival time at customers. Many real‐world applications can be modeled by this problem, such as the important application resulting from the humanitarian aid following a natural disaster. In this paper, two heuristics are proposed. The first one is a constructive heuristic to generate an initial solution and the second is the skewed variable neighborhood search (SVNS) heuristic. The SVNS algorithm starts with the initial solution. At each iteration, the perturbation phase and the local search phase are used to improve the solution of the CCVRP, and the distance function in acceptance criteria phase is used to improve the exploration of faraway valleys. This algorithm is applied to a set of benchmarks, and the comparison results show that the proposed algorithms provide better solutions than those reported in the previous literature on memetic algorithms and adaptive large neighborhood search heuristics.

Multi-objective optimization using genetic algorithm based clustering for multi- depot heterogeneous fleet vehicle routing problem with time windows

Efficient routing and scheduling of vehicles has significant economic implications for both the public and private sectors. For this purpose, we propose in this study a decision support system which aims to optimise the classical capacitated vehicle routing problem by considering the existence of different vehicle types (with distinct capacities and costs) and multiple available depots, that we call the multi-depot heterogeneous vehicle routing problem with time window (MDHVRPTW) by respecting a set of criteria including: schedules requests from clients, the heterogeneous capacity of vehicles...., and we solve this problem by proposing a new scheme based on the application of the bio-inspired genetic algorithm heuristics and by embedding a clustering algorithm within a VRPTW optimisation frame work, that we will specify later. Computational experiments with the benchmark test instances confirm that our approach produces acceptable quality solutions compared with the best previous results in similar problems in terms of generated solutions and processing time. Experimental results prove that our proposed genetic algorithm is effective in solving the MDHVRPTW problem and hence has a great potential.

Efficient Golden-Ball Algorithm Based Clustering to solve the Multi-Depot VRP With Time Windows

In this article, the authors propose a decision support system which aims to optimize the classical Capacitated Vehicle Routing Problem by considering the existence of multiple available depots and a time window which must not be violated, that they call the Multi-Depot Vehicle Routing Problem with Time Window (MDVRPTW), and with respecting a set of criteria including: schedules requests from clients, the capacity of vehicles. The authors solve this problem by proposing a recently published technique based on soccer concepts, called Golden Ball (GB), with different solution representation from the original one, this technique was designed to solve combinatorial optimization problems, and by embedding a clustering algorithm. Computational results have shown that the approach produces acceptable quality solutions compared to the best previous results in similar problem in terms of generated solutions and processing time. Experimental results prove that the proposed Golden Ball algorithm is efficient and effective to solve the MDVRPTW problem.

A novel two-level optimization approach for clustered vehicle routing problem

Abstract In this paper, we are addressing the clustered vehicle routing problem (CluVRP) which is a variant of the classical capacitated vehicle routing problem (CVRP). The following are the main characteristics of this problem: the vertices of the graph are partitioned into a given number of clusters and we are looking for a minimum-cost collection of routes starting and ending at the depot, visiting all the vertices exactly once, except the depot, and with the additional constraint that once a vehicle enters a cluster it visits all the vertices within the cluster before leaving it. We describe a novel two-level optimization approach for CluVRP obtained by decomposing the problem into two logical and natural smaller subproblems: an upper-level (global) subproblem and a lower-level (local) subproblem, and solving them separately. The goal of the first subproblem is to determine the (global) routes visiting the clusters using a genetic algorithm, while the goal of the second subproblem is, to determine for the above mentioned routes, the visiting order within the clusters. The second subproblem is solved by transforming each global route into a traveling salesman problem (TSP) which then is optimally computed using the Concorde TSP solver. Extensive computational results are reported and discussed for an often used set of benchmark instances. The obtained results show an improvement of the quality of the achieved solutions and prove the efficiency of our approach as compared to the existing methods from the literature.

A branch-and-cut algorithm for the two-echelon capacitated vehicle routing problem with grouping constraints

In this paper, we present a branch-and-cut algorithm for the two-echelon capacitated vehicle routing problem with grouping constraints (2E-CVRPGC), which is a new problem deriving from the classical two-echelon capacitated vehicle routing problem (2E-CVRP) by considering grouping constraints in the second echelon. Customers in the 2E-CVRPGC are divided into several disjoint groups, and the grouping constraints ensure that customers from the same group are served by vehicles from the same satellite. We formulate the problem as a mixed 0–1 linear program and propose five families of valid inequalities to strengthen the model. Based on the model and the inequalities, we implement a branch-and-cut algorithm to solve the problem. The proposed branch-and-cut algorithm was evaluated on two classes of randomly generated instances. The computational results show that the five families of valid inequalities can substantially improve the lower bounds yielded by the LP relaxation of the model, and the branch-and-cut algorithm can solve more instances to optimality than CPLEX. We also conduct additional experiments to analyze the impacts of the grouping constraints on the problem, and illustrate the differences between the 2E-CVRPGC and the 2E-CVRP.