Heuristics for vehicle routing on tree-like networks

This paper presents two new heuristics for the vehicle routing problem on tree-like road networks. These networks occur, for example, in rural road systems where the supply (or delivery) nodes are located on rural roads leading off from a few highways which form the ‘trunks’ of a tree-like network. The heuristics have the conventional objective of minimising the total distance travelled by the vehicles. The development of the heuristics takes advantage of the tree-like structure of the network. These two new heuristics and two other heuristics from the published literature are applied to some test problems and computational results are presented. The computational experience indicates that one of the new heuristics provides superior solutions to the existing heuristics and in reasonable computing time. It therefore appears suitable for large-scale practical routing problems.     

Heuristics and memetic algorithm for the two-dimensional loading capacitated vehicle routing problem with time windows

This work deals with a new combinatorial optimization problem, the two-dimensional loading capacitated vehicle routing problem with time windows which is a realistic extension of the well known vehicle routing problem. The studied problem consists in determining vehicle trips to deliver rectangular objects to a set of customers with known time windows, using a homogeneous fleet of vehicles, while ensuring a feasible loading of each vehicle used. Since it includes NP-hard routing and packing sub-problems, six heuristics are firstly designed to quickly compute good solutions for realistic instances. They are obtained by combining algorithms for the vehicle routing problem with time windows with heuristics for packing rectangles. Then, a Memetic algorithm is developed to improve the heuristic solutions. The quality and the efficiency of the proposed heuristics and metaheuristic are evaluated by adding time windows to a set of 144 instances with 15–255 customers and 15–786 items, designed by Iori et al. (Transport Sci 41:253–264, 2007) for the case without time windows.     

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.     

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.     

The savings algorithm for the vehicle routing problem

Survey is given concerning the savings method for the vehicle routing problem. Results for several methods and data sets are compared. Furthermore, modifications of the savings method are presented which show less CPU time and reduced storage requirements. Therefore, the savings method can be implemented on microcomputers.     

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.     

On the capacitated vehicle routing problem

 We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Traveling Salesman Problem (TSP) as special cases and conceptually lies at the intersection of these two well-studied problems. The capacity constraints of the integer programming formulation of this routing model provide the link between the underlying routing and packing structures. We describe a decomposition-based separation methodology for the capacity constraints that takes advantage of our ability to solve small instances of the TSP efficiently. Specifically, when standard procedures fail to separate a candidate point, we attempt to decompose it into a convex combination of TSP tours; if successful, the tours present in this decomposition are examined for violated capacity constraints; if not, the Farkas Theorem provides a hyperplane separating the point from the TSP polytope. We present some extensions of this basic concept and a general framework within which it can be applied to other combinatorial models. Computational results are given for an implementation within the parallel branch, cut, and price framework SYMPHONY. Received: October 30, 2000 / Accepted: December 19, 2001 Published online: September 5, 2002 Key words. vehicle routing problem – integer programming – decomposition algorithm – separation algorithm – branch and cut Mathematics Subject Classification (2000): 20E28, 20G40, 20C20     

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.     

The multi-trip vehicle routing problem

The basic vehicle routing problem is concerned with the design of a set of routes to serve a given number of customers, minimising the total distance travelled. In that problem, each vehicle is assumed to be used only once during a planning period, which is typically a day, and therefore is unrepresentative of many practical situations, where a vehicle makes several journeys during a day. The present authors have previously published an algorithm which outperformed an experienced load planner working on the complex, real-life problems of Burton's Biscuits, where vehicles make more than one trip each day. This present paper uses a simplified version of that general algorithm, in order to compare it with a recently published heuristic specially designed for the theoretical multi-trip vehicle routing problem.     

Relation between vehicle routing problems: an insertion heuristic for the vehicle routing problem with simultaneous delivery and pick-up applied to the vehicle routing problem with backhauls

Recently, an increasing number of papers on vehicle routing problems with backhauling has been published. Different types of backhauling problems are discussed. Two of them—the vehicle routing problem with backhauls and so-called ‘mixed loads’ (VRPBM) and the vehicle routing problem with simultaneous delivery and pick-up (VRPSDP)—are closely related. In this paper, we discuss that relationship. Our findings are that previously published results for VRPSDP instances obtained by using a heuristic suggested for the VRPBM do not take into account specific properties of the VRPSDP. As a result of the analysis of the relation between both problem types the possibility of solving the VRPBM by applying an insertion heuristic based on the concept of ‘residual capacities’ originally designed for the VRPSDP is investigated. Numerical results indicate that, for certain instances, this approach is more favourable than the application of a heuristic suggested for the VRPBM in the literature.     

Probabilistic time-dependent vehicle routing problem

Central European Journal of Operations Research - The probabilistic time-dependent vehicle routing problem is presented in this paper. It is a novel variant of the vehicle routing problem. The...     

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.     

New assignment algorithms for the multi-depot vehicle routing problem

This paper considers the design and analysis of algorithms for the multi-depot vehicle routing problem with time windows (MDVRPTW). Given the intrinsic difficulty of this problem class, approximation methods of the type ‘cluster first, route second’ (two-step approaches) seem to offer the most promise for practical size problems. After describing six heuristics for the cluster part (assignment of customers to depots) an initial computational study of their performance is conducted. Finding, as expected, that the heuristics with the best results (in terms of the routing results) are those with the largest computational efforts.     

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.     

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.     

Disruption management of the vehicle routing problem with vehicle breakdown

This paper introduces a new class of problem, the disrupted vehicle routing problem (VRP), which deals with the disruptions that occur at the execution stage of a VRP plan. The paper then focuses on one type of such problem, in which a vehicle breaks down during the delivery and a new routing solution needs to be quickly generated to minimise the costs. Two Tabu Search algorithms are developed to solve the problem and are assessed in relation to an exact algorithm. A set of test problems has been generated and computational results from experiments using the heuristic algorithms are presented.     

On a periodic vehicle routing problem

This paper deals with a study on a variant of the Periodic Vehicle Routing Problem (PVRP). As in the traditional Vehicle Routing Problem, customer locations each with a certain daily demand are given, as well as a set of capacitated vehicles. In addition, the PVRP has a horizon, say T days, and there is a frequency for each customer stating how often within this T-day period this customer must be visited. A solution to the PVRP consists of T sets of routes that jointly satisfy the demand constraints and the frequency constraints. The objective is to minimize the sum of the costs of all routes over the planning horizon. We develop different algorithms solving the instances of the case studied. Using these algorithms we are able to realize considerable cost reductions compared to the current situation.     

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.