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 new VRPPD model and a hybrid heuristic solution approach for e-tailing

We analyze a business model for e-supermarkets to enable multi-product sourcing capacity through co-opetition (collaborative competition). The logistics aspect of our approach is to design and execute a network system where “premium” goods are acquired from vendors at multiple locations in the supply network and delivered to customers. Our specific goals are to: (i) investigate the role of premium product offerings in creating critical mass and profit; (ii) develop a model for the multiple-pickup single-delivery vehicle routing problem in the presence of multiple vendors; and (iii) propose a hybrid solution approach. To solve the problem introduced in this paper, we develop a hybrid metaheuristic approach that uses a Genetic Algorithm for vendor selection and allocation, and a modified savings algorithm for the capacitated VRP with multiple pickup, single delivery and time windows (CVRPMPDTW). The proposed Genetic Algorithm guides the search for optimal vendor pickup location decisions, and for each generated solution in the genetic population, a corresponding CVRPMPDTW is solved using the savings algorithm. We validate our solution approach against published VRPTW solutions and also test our algorithm with Solomon instances modified for CVRPMPDTW.     

A library of local search heuristics for the vehicle routing problem

The vehicle routing problem (VRP) is a difficult and well-studied combinatorial optimization problem. Real-world instances of the VRP can contain hundreds and even thousands of customer locations and can involve many complicating constraints, necessitating the use of heuristic methods. We present a software library of local search heuristics that allows one to quickly generate solutions to VRP instances. The code has a logical, object-oriented design and uses efficient data structures to store and modify solutions. The core of the library is the implementation of seven local search operators that share a similar interface and are designed to be extended to handle additional options with minimal code change. The code is well-documented, straightforward to compile, and is freely available online. The code contains several applications that can be used to generate solutions to the capacitated VRP. Computational results indicate that these applications are able to generate solutions that are within about one percent of the best-known solution on benchmark problems.     

Creating very large scale neighborhoods out of smaller ones by compounding moves

This paper discusses neighborhood search algorithms where the size of the neighborhood is “very large” with respect to the size of the input data. We concentrate on such a very large scale neighborhood (VLSN) search technique based on compounding independent moves (CIM) such as 2-opts, swaps, and insertions. We present a systematic way of creating and searching CIM neighborhoods for routing problems with side constraints. For such problems, the exact search of the CIM neighborhood becomes NP-hard. We introduce a multi-label shortest path algorithm for searching these neighborhoods heuristically. Results of a computational study on the vehicle routing problem with capacity and distance restrictions shows that CIM algorithms are very competitive approaches for solving vehicle routing problems. Overall, the solutions generated by the CIM algorithm have the best performance among the current solution methodologies in terms of percentage deviation from the best-known solutions for large-scale capacitated VRP instances.     

Symmetry helps: Bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints

When vehicle routing problems with additional constraints, such as capacity or time windows, are solved via column generation and branch-and-price, it is common that the pricing subproblem requires the computation of a minimum cost constrained path on a graph with costs on the arcs and prizes on the vertices. A common solution technique for this problem is dynamic programming. In this paper we illustrate how the basic dynamic programming algorithm can be improved by bounded bi-directional search and we experimentally evaluate the effectiveness of the enhancement proposed. We consider as benchmark problems the elementary shortest path problems arising as pricing subproblems in branch-and-price algorithms for the capacitated vehicle routing problem, the vehicle routing problem with distribution and collection and the capacitated vehicle routing problem with time windows.     

Saving-based algorithms for vehicle routing problem with simultaneous pickup and delivery

The vehicle routing problem (VRP) with simultaneous pickup and delivery (VRPSPD) is an extension of the classical capacitated VRP (CVRP). In this paper, we present the saving heuristic and the parallel saving heuristic for VRPSPD. Checking the feasibility of a route in VRPSPD is difficult because of the fluctuating load on the route. In the saving heuristic, a new route is created by merging the two existing routes. We use a cumulative net-pickup approach for checking the feasibility when two existing routes are merged. The numerical results show that the performance of the proposed heuristics is qualitatively better than the existing insertion-based heuristics.     

An exact solution framework for a broad class of vehicle routing problems

This paper presents an exact solution framework for solving some variants of the vehicle routing problem (VRP) that can be modeled as set partitioning (SP) problems with additional constraints. The method consists in combining different dual ascent procedures to find a near optimal dual solution of the SP model. Then, a column-and-cut generation algorithm attempts to close the integrality gap left by the dual ascent procedures by adding valid inequalities to the SP formulation. The final dual solution is used to generate a reduced problem containing all optimal integer solutions that is solved by an integer programming solver. In this paper, we describe how this solution framework can be extended to solve different variants of the VRP by tailoring the different bounding procedures to deal with the constraints of the specific variant. We describe how this solution framework has been recently used to derive exact algorithms for a broad class of VRPs such as the capacitated VRP, the VRP with time windows, the pickup and delivery problem with time windows, all types of heterogeneous VRP including the multi depot VRP, and the period VRP. The computational results show that the exact algorithm derived for each of these VRP variants outperforms all other exact methods published so far and can solve several test instances that were previously unsolved.     

Heuristic solution approaches for the cumulative capacitated vehicle routing problem

Cumulative capacitated vehicle routing problem (CCVRP) is an extension of the well-known capacitated vehicle routing problem, where the objective is minimization of sum of the arrival times at nodes instead of minimizing the total tour cost. This type of routing problem arises when a priority is given to customer needs or dispatching vital goods supply after a natural disaster. This paper focuses on comparing the performances of neighbourhood and population-based approaches for the new problem CCVRP. Genetic algorithm (GA), an evolutionary algorithm using particle swarm optimization mechanism with GA operators, and tabu search (TS) are compared in terms of required CPU time and obtained objective values. In addition, a nearest neighbourhood-based initial solution technique is also proposed within the paper. To the best of authors’ knowledge, this paper constitutes a base for comparisons along with GA, and TS for further possible publications on the new problem CCVRP.     

A self-tuning heuristic for a multi-objective vehicle routing problem

In this study, a heuristic free from parameter tuning is introduced to solve the vehicle routing problem (VRP) with two conflicting objectives. The problem which has been presented is the designing of optimal routes: minimizing both the number of vehicles and the maximum route length. This problem, even in the case of its single objective form, is NP-hard. The proposed self-tuning heuristic (STH) is based on local search and has two parameters which are updated dynamically throughout the search process. The most important advantage of the algorithm is the application convenience for the end-users. STH is tested on the instances of a multi-objective problem in school bus routing and classical vehicle routing. Computational experiments, when compared with the prior approaches proposed for the multi-objective routing of school buses problem, confirm the effectiveness of STH. STH also finds high-quality solutions for multi-objective VRPs.     

A simple and efficient tabu search heuristic for solving the open vehicle routing problem

The open vehicle routing problem (VRP) is an immediate variant of the standard vehicle routing problem where the vehicle need not return to the depot after servicing its last customer. In this paper, we present results on an implementation of the attribute-based hill climber heuristic to the open VRP. The attribute-based hill climber heuristic is a parameter-free variant of the tabu search principle and has shown to be highly effective for the standard vehicle routing problem.     

Local search for the undirected capacitated arc routing problem with profits

This paper deals with a recently introduced routing problem variant called the undirected capacitated arc routing problem with profits (UCARPP). The UCARPP model considered in the present study is primarily aimed at generating the route set which maximizes the profit collected from a set of potential customers, represented by edges of the examined transportation network. The secondary objective is to minimize the total route travel time. The generated routes are subject both to capacity and travel time constraints. To tackle the examined problem, we propose a local search metaheuristic development which explores two solution neighborhood structures. The conducted search is effectively diversified by means of the promises concept which is based on the aspiration criteria used in tabu search approaches. The proposed algorithm was tested on UCARPP benchmark instances taken from the literature. It efficiently produced high-quality results, improving several previously best known solutions.     

Using clustering analysis in a capacitated location-routing problem

The location routing problem (LRP) appears as a combination of two difficult problems: the facility location problem (FLP) and the vehicle routing problem (VRP). In this work, we consider a discrete LRP with two levels: a set of potential capacitated distribution centres (DC) and a set of ordered customers. In our problem we intend to determine the set of installed DCs as well as the distribution routes (starting and ending at the DC). The problem is also constrained with capacities on the vehicles. Moreover, there is a homogeneous fleet of vehicles, carrying a single product and each customer is visited just once. As an objective we intend to minimize the routing and location costs.     

Cyclic-order neighborhoods with application to the vehicle routing problem with stochastic demand

We examine neighborhood structures for heuristic search applicable to a general class of vehicle routing problems (VRPs). Our methodology utilizes a cyclic-order solution encoding, which maps a permutation of the customer set to a collection of many possible VRP solutions. We identify the best VRP solution in this collection via a polynomial-time algorithm from the literature. We design neighborhoods to search the space of cyclic orders. Utilizing a simulated annealing framework, we demonstrate the potential of cyclic-order neighborhoods to facilitate the discovery of high quality a priori solutions for the vehicle routing problem with stochastic demand (VRPSD). Without tailoring our solution procedure to this specific routing problem, we are able to match 16 of 19 known optimal VRPSD solutions. We also propose an updating procedure to evaluate the neighbors of a current solution and demonstrate its ability to reduce the computational expense of our approach.     

一种带时间窗和容量约束的车辆路线问题及其Tabu Search算法

本文提出一种带时间窗和容量约束的车辆路线问题（CVRPTW），并利用Tabu Search快速启式算法，针对Solomon提出的几个标准问题，快捷地得到了优良的数值结果。     

Using greedy clustering method to solve capacitated location-routing problem with fuzzy demands

In this paper, the capacitated location-routing problem with fuzzy demands (CLRP-FD) is considered. In CLRP-FD, facility location problem (FLP) and vehicle routing problem (VRP) are observed simultaneously. Indeed, the vehicles and the depots have a predefined capacity to serve the customers that have fuzzy demands. To model this problem, a fuzzy chance constrained programming model of that is designed based upon the fuzzy credibility theory. To solve this problem, a greedy clustering method (GCM) including the stochastic simulation is proposed. To obtain the best value of the dispatcher preference index of the model and to analyze its influence on the final solution, numerical experiments are carried out. Finally, to show the performance of the greedy clustering method, associated results are compared with the lower bound of the solutions.     

A Lagrangean relaxation heuristic for vehicle routing

This paper presents a new heuristic algorithm for the vehicle routing problem (VRP) which makes use of Lagrangean relaxation to transform the VRP into a modified m-traveling salesman problem. Application of the proposed algorithm to test problems from the literature has produced new best-known solutions.     

车辆路径问题的混合优化算法

讨论了一类车辆路径调度问题(VRP)及其数学模型，并且分析了以遗传算法求解该类问题时的染色体表示和有关遗传操作，然后结合2-opt局部优化算法提出了GA with2-opt算法来求解VRP问题，试验结果说明了该算法的有效性和可行性。     

Exact algorithms for the chance-constrained vehicle routing problem

We study the chance-constrained vehicle routing problem (CCVRP), a version of the vehicle routing problem (VRP) with stochastic demands, where a limit is imposed on the probability that each vehicle’s capacity is exceeded. A distinguishing feature of our proposed methodologies is that they allow correlation between random demands, whereas nearly all existing exact methods for the VRP with stochastic demands require independent demands. We first study an edge-based formulation for the CCVRP, in particular addressing the challenge of how to determine a lower bound on the number of vehicles required to serve a subset of customers. We then investigate the use of a branch-and-cut-and-price (BCP) algorithm. While BCP algorithms have been considered the state of the art in solving the deterministic VRP, few attempts have been made to extend this framework to the VRP with stochastic demands. In contrast to the deterministic VRP, we find that the pricing problem for the CCVRP problem is strongly \(\mathcal {NP}\)-hard, even when the routes being priced are allowed to have cycles. We therefore propose a further relaxation of the routes that enables pricing via dynamic programming. We also demonstrate how our proposed methodologies can be adapted to solve a distributionally robust CCVRP problem. Numerical results indicate that the proposed methods can solve instances of CCVRP having up to 55 vertices.     

Lower and upper bounds for location-arc routing problems with vehicle capacity constraints

This paper addresses multi-depot location arc routing problems with vehicle capacity constraints. Two mixed integer programming models are presented for single and multi-depot problems. Relaxing these formulations leads to other integer programming models whose solutions provide good lower bounds for the total cost. A powerful insertion heuristic has been developed for solving the underlying capacitated arc routing problem. This heuristic is used together with a novel location–allocation heuristic to solve the problem within a simulated annealing framework. Extensive computational results demonstrate that the proposed algorithm can find high quality solutions. We also show that the potential cost saving resulting from adding location decisions to the capacitated arc routing problem is significant.     

Exact algorithms for the vehicle routing problem,based on spanning tree and shortest path relaxations

We consider the problem of routing vehicles stationed at a central facility (depot) to supply customers with known demands, in such a way as to minimize the total distance travelled. The problem is referred to as the vehicle routing problem (VRP) and is a generalization of the multiple travelling salesman problem that has many practical applications. We present tree search algorithms for the exact solution of the VRP incorporating lower bounds computed from (i) shortest spanningk-degree centre tree (k-DCT), and (ii)q-routes. The final algorithms also include problem reduction and dominance tests. Computational results are presented for a number of problems derived from the literature. The results show that the bounds derived from theq-routes are superior to those fromk-DCT and that VRPs of up to about 25 customers can be solved exactly.