Heuristics for the lexicographic max-ordering vehicle routing problem

In this paper, we propose fast heuristics for the vehicle routing problem (VRP) with lexicographic max-order objective. A fixed number of vehicles, which are based at a depot, are to serve customers with known demands. The lexicographic max-order objective is introduced by asking to minimize lexicographically the sorted route lengths. Based on a model for this problem, several approaches are studied and new heuristic solution procedures are discussed resulting in the development of a sequential insertion heuristic and a modified savings algorithm in several variants. Comparisons between the algorithms are performed on instances of the VRP library VRPLIB. Finally, based on the results from the computational experiments, conclusions about the applicability and efficiency of the presented algorithms are drawn.     

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.     

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.     

An improved ant colony optimization for vehicle routing problem

The vehicle routing problem (VRP), a well-known combinatorial optimization problem, holds a central place in logistics management. This paper proposes an improved ant colony optimization (IACO), which possesses a new strategy to update the increased pheromone, called ant-weight strategy, and a mutation operation, to solve VRP. The computational results for fourteen benchmark problems are reported and compared to those of other metaheuristic approaches.     

A new hybrid genetic algorithm for the capacitated vehicle routing problem

Recently proved successful for variants of the vehicle routing problem (VRP) involving time windows, genetic algorithms have not yet shown to compete or challenge current best search techniques in solving the classical capacitated VRP. A new hybrid genetic algorithm to address the capacitated VRP is proposed. The basic scheme consists in concurrently evolving two populations of solutions to minimize total travelled distance using genetic operators combining variations of key concepts inspired from routing techniques and search strategies used for a time variant of the problem to further provide search guidance while balancing intensification and diversification. Results from a computational experiment over common benchmark problems report the proposed approach to be very competitive with the best-known methods.     

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.     

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.     

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.     

A new tabu search heuristic for the open vehicle routing problem

In this paper, another version of the vehicle routing problem (VRP)—the open vehicle routing problem (OVRP) is studied, in which the vehicles are not required to return to the depot, but if they do, it must be by revisiting the customers assigned to them in the reverse order. By exploiting the special structure of this type of problem, we present a new tabu search heuristic for finding the routes that minimize two objectives while satisfying three constraints. The computational results are provided and compared with two other methods in the literature.     

A parallel improved ant colony optimization for multi-depot vehicle routing problem

This paper presents a method for solving multi-depot vehicle routing problem (MDVRP). First, a virtual central depot is added to transfer MDVRP to the multi-depot vehicle routing problem with the virtual central depot (V-MDVRP), which is similar to a vehicle routing problem (VRP) with the virtual central depot as the origin. An improved ant colony optimization with coarse-grain parallel strategy, ant-weight strategy and mutation operation, is presented for the V-MDVRP. The computational results for 23 benchmark problems are reported and compared to those of other ant colony optimizations.     

A heuristic method for the open vehicle routing problem

The open vehicle routing problem (OVRP) differs from the classic vehicle routing problem (VRP) because the vehicles either are not required to return to the depot, or they have to return by revisiting the customers assigned to them in the reverse order. Therefore, the vehicle routes are not closed paths but open ones. A heuristic method for solving this new problem, based on a minimum spanning tree with penalties procedure, is presented. Computational results are provided.     

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.     

New mathematical models of the generalized vehicle routing problem and extensions

The generalized vehicle routing problem (GVRP) is an extension of the vehicle routing problem (VRP) and was introduced by Ghiani and Improta [1]. The GVRP is the problem of designing optimal delivery or collection routes from a given depot to a number of predefined, mutually exclusive and exhaustive node-sets (clusters) which includes exactly one node from each cluster, subject to capacity restrictions. The aim of this paper is to provide two new models of the GVRP based on integer programming. The first model, called the node formulation is similar to the Kara-Bekta? formulation [2], but produces a stronger lower bound. The second one, called the flow formulation, is completely new. We show as well that under specific circumstances the proposed models of the GVRP reduces to the well known routing problems. Finally, the GVRP is extended for the case in which the vertices of any cluster of each tour are contiguous. This case is defined as the clustered generalized vehicle routing problem and both of the proposed formulations of GVRP are adapted to clustered case.     

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.     

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.     

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.     

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.     

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 reactive tabu search for the vehicle routing problem

The classical vehicle routing problem (VRP) involves determining a fleet of homogeneous size vehicles and designing an associated set of routes that minimizes the total cost. Our tabu search (TS) algorithm to solve the VRP is based on reactive tabu search (RTS) with a new escape mechanism, which manipulates different neighbourhood schemes in a very sophisticated way in order to get a balanced intensification and diversification continuously during the search process. We compare our algorithm with the best methods in the literature using different data sets and report results including new best known solutions for several well-known 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.