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.     

Metaheuristics for vehicle routing problems with three-dimensional loading constraints

This paper addresses an important combination of three-dimensional loading and vehicle routing, known as the Three-Dimensional Loading Capacitated Vehicle Routing Problem. The problem calls for the combined optimization of the loading of freight into vehicles and the routing of vehicles along a road network, with the aim of serving customers with minimum traveling cost. Despite its clear practical relevance in freight distribution, the literature on this problem is very limited. This is because of its high combinatorial complexity.     

A heuristic algorithm for the symmetric and asymmetric vehicle routing problems with backhauls

We consider an extension of the capacitated Vehicle Routing Problem (VRP), known as the Vehicle Routing Problem with Backhauls (VRPB), in which the set of customers is partitioned into two subsets: Linehaul and Backhaul customers. Each Linehaul customer requires the delivery of a given quantity of product from the depot, whereas a given quantity of product must be picked up from each Backhaul customer and transported to the depot. VRPB is known to be NP-hard in the strong sense, and many heuristic algorithms were proposed for the approximate solution of the problem with symmetric or Euclidean cost matrices. We present a cluster-first-route-second heuristic which uses a new clustering method and may also be used to solve problems with asymmetric cost matrix. The approach exploits the information of the normally infeasible VRPB solutions associated with a lower bound. The bound used is a Lagrangian relaxation previously proposed by the authors. The final set of feasible routes is built through a modified Traveling Salesman Problem (TSP) heuristic, and inter-route and intra-route arc exchanges. Extensive computational tests on symmetric and asymmetric instances from the literature show the effectiveness of the proposed approach.     

An iterated local search algorithm for the vehicle routing problem with backhauls

The Vehicle Routing Problem with Backhauls (VRPB) is an extension of the VRP that deals with two types of customers: the consumers (linehaul) that request goods from the depot and the suppliers (backhaul) that send goods to the depot. In this paper, we propose a simple yet effective iterated local search algorithm for the VRPB. Its main component is an oscillating local search heuristic that has two main features. First, it explores a broad neighborhood structure at each iteration. This is efficiently done using a data structure that stores information about the set of neighboring solutions. Second, the heuristic performs constant transitions between feasible and infeasible portions of the solution space. These transitions are regulated by a dynamic adjustment of the penalty applied to infeasible solutions. An extensive statistical analysis was carried out in order to identify the most important components of the algorithm and to properly tune the values of their parameters. The results of the computational experiments carried out show that this algorithm is very competitive in comparison to the best metaheuristic algorithms for the VRPB. Additionally, new best solutions have been found for two instances in one of the benchmark sets. These results show that the performance of existing metaheuristic algorithms can be considerably improved by carrying out a thorough statistical analysis of their components. In particular, it shows that by expanding the exploration area and improving the efficiency of the local search heuristic, it is possible to develop simpler and faster metaheuristic algorithms without compromising the quality of the solutions obtained.     

A new tabu search algorithm for the vehicle routing problem with backhauls

In the distribution of goods from a central depot to geographically dispersed customers happens quite frequently that some customers, called linehauls, receive goods from that depot while others, named backhauls, send goods to it. This situation is described and studied by the vehicle routing problem with backhauls. In this paper we present a new tabu search algorithm that starting from pseudo-lower bounds was able to match almost all the best published solutions and to find many new best solutions, for a large set of benchmark problems.     

Reactive tabu adaptive memory programming search for the vehicle routing problem with backhauls

A heuristic approach based on a hybrid operation of reactive tabu search (RTS) and adaptive memory programming (AMP) is proposed to solve the vehicle routing problem with backhauls (VRPB). The RTS is used with an escape mechanism which manipulates different neighbourhood schemes in a sophisticated way in order to get a continuously balanced intensification and diversification during the search process. The adaptive memory strategy takes the search back to the unexplored regions of the search space by maintaining a set of elite solutions and using them strategically with the RTS. The AMP feature brings an extra robustness to the search process that resulted in early convergence when tested on most of the VRPB instances. We compare our algorithm against the best methods in the literature and report new best solutions for several benchmark problems.     

Multi-ant colony system (MACS) for a vehicle routing problem with backhauls

The vehicle routing problem with backhaul (VRPB) is an extension of the capacitated vehicle routing problem (CVRP). In VRPB, there are linehaul as well as backhaul customers. The number of vehicles is considered to be fixed and deliveries for linehaul customers must be made before any pickups from backhaul customers. The objective is to design routes for the vehicles so that the total distance traveled is minimized. We use multi-ant colony system (MACS) to solve VRPB which is a combinatorial optimization problem. Ant colony system (ACS) is an algorithmic approach inspired by foraging behavior of real ants. Artificial ants are used to construct a solution by using pheromone information from previously generated solutions. The proposed MACS algorithm uses a new construction rule as well as two multi-route local search schemes. An extensive numerical experiment is performed on benchmark problems available in the literature.     

An adaptive guidance meta-heuristic for the vehicle routing problem with splits and clustered backhauls

This paper presents the case study of an Italian carrier, Grendi Trasporti Marittimi, which provides freight transportation services by trucks and containers. Its trucks deliver container loads from a port to import customers and collect container loads from export customers to the same port. In this case study, all import customers in a route must be serviced before all export customers, each customer can be visited more than once and containers are never unloaded or reloaded from the truck chassis along any route. We model the problem using an Integer Linear Programming formulation and propose an Adaptive Guidance metaheuristic. Our extensive computational experiments show that the adaptive guidance algorithm is capable of determining good-quality solutions in many instances of practical or potential interest for the carrier within 10?min of computing time, whereas the mathematical formulation often fails to provide the first feasible solution within 3?h of computing time.     

A deterministic iterated local search algorithm for the vehicle routing problem with backhauls

The vehicle routing problem with backhauls is a variant of the classical capacitated vehicle routing problem. The difference is that it contains two distinct sets of customers: those who receive goods from the depot, who are called linehauls, and those who send goods to the depot, who are referred to as backhauls. In this paper, we describe a new deterministic iterated local search algorithm, which is tested using a large number of benchmark problems chosen from the literature. These computational tests have proven that this algorithm competes with the best known algorithms in terms of the quality of the solutions and at the same time, it is simpler and faster.     

Multi-objective vehicle routing problems

Routing problems, such as the traveling salesman problem and the vehicle routing problem, are widely studied both because of their classic academic appeal and their numerous real-life applications. Similarly, the field of multi-objective optimization is attracting more and more attention, notably because it offers new opportunities for defining problems. This article surveys the existing research related to multi-objective optimization in routing problems. It examines routing problems in terms of their definitions, their objectives, and the multi-objective algorithms proposed for solving them.     

An interactive GRAMPS algorithm for the heterogeneous fixed fleet vehicle routing problem with and without backhauls

In this article, a visual interactive approach based on a new greedy randomised adaptive memory programming search (GRAMPS) algorithm is proposed to solve the heterogeneous fixed fleet vehicle routing problem (HFFVRP) and a new extension of the HFFVRP, which is called heterogeneous fixed fleet vehicle routing problem with backhauls (HFFVRPB). This problem involves two different sets of customers. Backhaul customers are pickup points and linehaul customers are delivery points that are to be serviced from a single depot by a heterogeneous fixed fleet of vehicles, each of which is restricted in the capacity it can carry, with different variable travelling costs.     

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.     

Branch-cut-and-price for the vehicle routing problem with simultaneous pickup and delivery

This work proposes a Branch-cut-and-price (BCP) approach for the Vehicle Routing Problem with Simultaneous Pickup and Delivery (VRPSPD). We also deal with a particular case of the VRPSPD, known as the Vehicle Routing Problem with Mixed Pickup and Delivery. The BCP algorithm was tested in well-known benchmark instances involving up to 200 customers. Four instances were solved for the first time and some LBs were improved.     

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.     

Solving symmetric vehicle routing problems asymmetrically

The vehicle routing problem can be regarded as a traveling salesman problem with additional constraints. Algorithms based on assignment relaxations provide better solutions for the symmetric traveling salesman problem if they are used on an asymmetric transformed distance matrix, as shown by Jonker et al., in a paper in Operations Research. The generalization of such a transformation to the distance matrix of symmetric vehicle routing problems is described. The approach is illustrated within a heuristic algorithm, although it can also be the basis for an exact algorithm. For a number of standard problems computational results are given, that are competitive to results of other algorithms.     

A goal programming approach to vehicle routing problems with soft time windows

The classical vehicle routing problem involves designing a set of routes for a fleet of vehicles based at one central depot that is required to serve a number of geographically dispersed customers, while minimizing the total travel distance or the total distribution cost. Each route originates and terminates at the central depot and customers demands are known. In many practical distribution problems, besides a hard time window associated with each customer, defining a time interval in which the customer should be served, managers establish multiple objectives to be considered, like avoiding underutilization of labor and vehicle capacity, while meeting the preferences of customers regarding the time of the day in which they would like to be served (soft time windows). This work investigates the use of goal programming to model these problems. To solve the model, an enumeration-followed-by-optimization approach is proposed which first computes feasible routes and then selects the set of best ones. Computational results show that this approach is adequate for medium-sized delivery problems.     

Integer programming formulations of vehicle routing problems

Various mathematical formulations are available for situations represented by vehicle routing problems. The assignment-based integer programming formulations of these problems are more common and easy to understand. Such formulations are discussed in this paper and a much simpler formulation for the vechicle routing problem is presented for the case, when all the vehicles have the same load capacity and maximum allowable cost per route.     

A branch-and-cut algorithm for vehicle routing problems

We present a branch-and-cut algorithm for the identical customer Vehicle Routing Problem. Transforming the problem into an equivalent Path-Partitioning Problem allows us to exploit its polyhedral structure and to generate strong cuts corresponding to facet-inducing inequalities. By using cuts defined by multistars, partial multistars and generalized subtour elimination constraints, we are able to consistently solve 60-city problems to proven optimality and are currently attempting to solve problems involving a hundred cities. We also present details of the computer implementation and our computational results.