Exact and heuristic algorithms for the vehicle routing problem with multiple interdependent time windows

In this paper a model and several solution procedures for a novel type of vehicle routing problems where time windows for the pickup of perishable goods depend on the dispatching policy used in the solution process are presented. This problem is referred to as Vehicle Routing Problem with multiple interdependent time windows (VRPmiTW) and is motivated by a project carried out with the Austrian Red Cross blood program to assist their logistics department. Several variants of a heuristic constructive procedure as well as a branch-and-bound based algorithm for this problem were developed and implemented. Besides finding the expected reduction in costs when compared with the current procedures of the Austrian Red Cross, the results show that the heuristic algorithms find solutions reasonably close to the optimum in fractions of a second. Another important finding is that increasing the number of pickups at selected customers beyond the theoretical minimum number of pickups yields significantly greater potential for cost reductions.     

An adaptive guidance approach for the heuristic solution of a minimum multiple trip vehicle routing problem

One of the most important problems in combinatorial optimization is the well-known vehicle routing problem (VRP), which calls for the determination of the optimal routes to be performed by a fleet of vehicles to serve a given set of customers. Recently, there has been an increasing interest towards extensions of VRP arising from real-world applications. In this paper we consider a variant in which time windows for service at the customers are given, and vehicles may perform more than one route within a working shift. We call the resulting problem the minimum multiple trip VRP (MMTVRP), where a “multiple trip” is a sequence of routes corresponding to a working shift for a vehicle. The problem objective is to minimize the overall number of the multiple trips (hence the size of the required fleet), breaking ties in favor of the minimum routing cost.     

A hybrid variable neighborhood tabu search heuristic for the vehicle routing problem with multiple time windows

This paper presents a new hybrid variable neighborhood-tabu search heuristic for the Vehicle Routing Problem with Multiple Time windows. It also proposes a minimum backward time slack algorithm applicable to a multiple time windows environment. This algorithm records the minimum waiting time and the minimum delay during route generation and adjusts the arrival and departure times backward. The implementation of the proposed heuristic is compared to an ant colony heuristic on benchmark instances involving multiple time windows. Computational results on newly generated instances are provided.     

Exact algorithms for the double vehicle routing problem with multiple stacks

The Double Traveling Salesman Problem with Multiple Stacks (DTSPMS) is a one-to-one pickup-and-delivery single-vehicle routing problem with backhaul deliveries. The vehicle carries a container divided into stacks of fixed height, each following a Last-In-First-Out policy, and the aim is to perform pickups and deliveries by minimizing the total routing cost and ensuring a feasible loading/unloading of the vehicle.A realistic generalization of the DTSPMS arises when a single vehicle is not enough to collect all products, and therefore multiple, and possibly heterogeneous vehicles are needed to perform the transportation operations. This paper introduces and formulates this generalization, that we refer as the Double Vehicle Routing Problem with Multiple Stacks. It proposes three models, the first one based on a three-index formulation and solved by a branch-and-cut algorithm, and the other two based on two set partitioning formulations using different families of columns and solved by a branch-and-price and a branch-and-price-and-cut algorithm, respectively.The performance of these algorithms has been studied on a wide family of benchmark test instances, observing that, although the branch-and-cut algorithm shows a better performance on instances with a small number of vehicles, the performance of the branch-and-price and the branch-and-price-and-cut algorithms improves as the number of vehicles grows. Additionally, the first set partitioning formulation yields tighter lower bounds, but the second formulation, because of its simplicity, provides better convergence properties, solving instances with up to fifty vertices to proven optimality.     

A branch-price-and-cut algorithm for the commodity constrained split delivery vehicle routing problem

We consider the Commodity constrained Split Delivery Vehicle Routing Problem (C-SDVRP), a routing problem where customers may request multiple commodities. The vehicles can deliver any set of commodities and multiple visits to a customer are allowed only if the customer requests multiple commodities. If the customer is visited more than once, the different vehicles will deliver different sets of commodities. Allowing the splitting of the demand of a customer only for different commodities may be more costly than allowing also the splitting of each individual commodity, but at the same time it is easier to organize and more acceptable to customers. We model the C-SDVRP by means of a set partitioning formulation and present a branch-price-and-cut algorithm. In the pricing phase, the ng-path relaxation of a constrained elementary shortest path problem is solved with a label setting dynamic programming algorithm. Capacity cuts are added in order to strengthen the lower bound. We solve to optimality within 2 h instances with up to 40 customers and 3 commodities per customer.     

A multistart adaptive memory-based tabu search algorithm for the heterogeneous fixed fleet open vehicle routing problem

This paper addressed the heterogeneous fixed fleet open vehicle routing problem (HFFOVRP), in which the demands of customers are fulfilled by a fleet of fixed number of vehicles with various capacities and related costs. Moreover, the vehicles start at the depot and terminate at one of the customers. This problem is an important variant of the classical vehicle routing problem and can cover more practical situations in transportation and logistics. We propose a multistart adaptive memory programming metaheuristic with modified tabu search algorithm to solve this new vehicle routing problem. The algorithmic efficiency and effectiveness are experimentally evaluated on a set of generated instances.     

A variable neighborhood search for the multi-depot vehicle routing problem with loading cost

The purpose of this paper is to propose a variable neighbourhood search (VNS) for solving the multi-depot vehicle routing problem with loading cost (MDVRPLC). The MDVRPLC is the combination of multi-depot vehicle routing problem (MDVRP) and vehicle routing problem with loading cost (VRPLC) which are both variations of the vehicle routing problem (VRP) and occur only rarely in the literature. In fact, an extensive literature search failed to find any literature related specifically to the MDVRPLC. The proposed VNS comprises three phases. First, a stochastic method is used for initial solution generation. Second, four operators are randomly selected to search neighbourhood solutions. Third, a criterion similar to simulated annealing (SA) is used for neighbourhood solution acceptance. The proposed VNS has been test on 23 MDVRP benchmark problems. The experimental results show that the proposed method provides an average 23.77% improvement in total transportation cost over the best known results based on minimizing transportation distance. The results show that the proposed method is efficient and effective in solving problems.     

Adaptive granular local search heuristic for a dynamic vehicle routing problem

The advance of communication and information technologies based on satellite and wireless networks have allowed transportation companies to benefit from real-time information for dynamic vehicle routing with time windows. During daily operations, we consider the case in which customers can place requests such that their demand and location are stochastic variables. The time windows at customer locations can be violated although lateness costs are incurred. The objective is to define a set of vehicle routes which are dynamically updated to accommodate new customers in order to maximize the expected profit. This is the difference between the total revenue and the sum of lateness costs and costs associated with the total distance traveled. The solution approach makes use of a new constructive heuristic that scatters vehicles in the service area and an adaptive granular local search procedure. The strategies of letting a vehicle wait, positioning a vehicle in a region where customers are likely to appear, and diverting a vehicle away from its current destination are integrated within a granular local search heuristic. The performance of the proposed approach is assessed in test problems based on real-life Brazilian transportation companies.     

Two solution representations for solving multi-depot vehicle routing problem with multiple pickup and delivery requests via PSO

Two solution representations for solving the generalized multi-depot vehicle routing problem with multiple pickup and delivery requests (GVRP-MDMPDR) is presented in this paper. The representations are used in conjunction with GLNPSO, a variant of PSO with multiple social learning terms. The computational experiments are carried out using benchmark test instances for pickup and delivery problem with time windows (PDPTW) and the generalized vehicle routing problem for multi-depot with multiple pickup and delivery requests (GVRP-MDMPDR). The preliminary results illustrate that the proposed method is capable of providing good solutions for most of the test problems.     

Packing first,routing second—a heuristic for the vehicle routing and loading problem

The Vehicle Routing and Loading Problem (VRLP) results by combining vehicle routing, possibly with time windows, and three-dimensional loading. Some packing constraints of high practical relevance, among them an unloading sequence constraint and a support constraint, are also part of the VRLP. Different formulations of the VRLP are considered and the issue is discussed under which circumstances routing and packing should be tackled as a combined task. A two-stage heuristic is presented following a “packing first, routing second” approach, i.e. the packing of goods and the routing of vehicles is done in two strictly separated stages. High quality results are achieved in short computation times for the 46 VRLP instances recently introduced by Moura and Oliveira. Moreover 120 new large benchmark instances including up to 1000 customers and 50,000 boxes are introduced and results for these instances are also reported.     

Heuristic algorithms for the vehicle routing problem with simultaneous pick-up and delivery

The vehicle routing problem with simultaneous pick-up and delivery is the problem of optimally integrating goods distribution and waste collection, when no precedence constraints are imposed on the order in which the operations must be performed. The purpose of this paper is to present heuristic algorithms to solve this problem approximately in a small amount of computing time. We present and compare constructive algorithms, local search algorithms and tabu search algorithms, reporting on our computational experience with all of them. In particular we address the issue of applying the tabu search paradigm to algorithms based on complex and variable neighborhoods. For this purpose we combine arc-exchange-based and node-exchange-based neighborhoods, employing different and interacting tabu lists. All the algorithms presented in this paper are applicable to problems in which each customer may have either a pick-up demand or a delivery demand as well as to problems in which each customer may require both kinds of operation.     

Formulations and exact algorithms for the vehicle routing problem with time windows

In this paper we review the exact algorithms proposed in the last three decades for the solution of the vehicle routing problem with time windows (VRPTW). The exact algorithms for the VRPTW are in many aspects inherited from work on the traveling salesman problem (TSP). In recognition of this fact this paper is structured relative to four seminal papers concerning the formulation and exact solution of the TSP, i.e. the arc formulation, the arc-node formulation, the spanning tree formulation, and the path formulation. We give a detailed analysis of the formulations of the VRPTW and a review of the literature related to the different formulations. There are two main lines of development in relation to the exact algorithms for the VRPTW. One is concerned with the general decomposition approach and the solution to certain dual problems associated with the VRPTW. Another more recent direction is concerned with the analysis of the polyhedral structure of the VRPTW. We conclude by examining possible future lines of research in the area of the VRPTW.     

A parallel evolutionary algorithm for the vehicle routing problem with heterogeneous fleet

Nowadays genetic algorithms stand as a trend to solve NP-complete and NP-hard problems. In this paper, we present a new hybrid metaheuristic which uses parallel genetic algorithms and scatter search coupled with a decomposition-into-petals procedure for solving a class of vehicle routing and scheduling problems. The parallel genetic algorithm presented is based on the island model and its performance is evaluated for a heterogeneous fleet problem, which is considered a problem much harder to solve than the homogeneous vehicle routing problem.     

A two-phase metaheuristic for the cumulative capacitated vehicle routing problem

The cumulative capacitated vehicle routing problem, which aims to minimize the total arrival time at customers, is a relatively new variant of vehicle routing problem. It can be used to model many real-world applications, e.g., the important application arisen from the humanitarian aid after a natural disaster. In this paper, an approach, called two-phase metaheuristic, is proposed to deal with this problem. This algorithm starts from a solution. At each iteration, two interdependent phases use different perturbation and local search operators for solution improvement. The effectiveness of the proposed algorithm is empirically investigated. The comparison results show that the proposed algorithm is promising. Moreover, for nine benchmark instances, the two-phase metaheuristic can find better solutions than those reported in the previous literature.     

Solving school bus routing using the multiple vehicle traveling purchaser problem: A branch-and-cut approach

School bus routing problems, combining bus stop selection and bus route generation, look simultaneously for a set of bus stops to pick up students from among a group of potential locations, and for bus routes to visit the selected stops and carry the students to their school. These problems, classified as Location-Routing problems, are of interest in densely populated urban areas.This article introduces a generalization of the vehicle routing problem called the multi-vehicle traveling purchaser problem, modeling a family of routing problems combining stop selection and bus route generation. It discusses a Mixed Integer Programming formulation extending previous studies on the classical single vehicle traveling purchaser problem. The proposed model is based on a single commodity flow formulation combining continuous variables with binary variables by means of coupling constraints. Additional valid inequalities are proposed with the purpose of strengthening its Linear Programming relaxation. These valid inequalities are obtained by projecting out the flow variables.We develop a branch-and-cut algorithm that makes use of the proposed model and valid inequalities. This cutting plane algorithm is implemented and tested on a large family of symmetric and asymmetric instances derived from randomly generated problems, showing the usefulness of the proposed valid inequalities.     

Ant colony optimization for the two-dimensional loading vehicle routing problem

In this paper a combination of the two most important problems in distribution logistics is considered, known as the two-dimensional loading vehicle routing problem. This problem combines the loading of the freight into the vehicles, and the successive routing of the vehicles along the road network, with the aim of satisfying the demands of the customers.     

Self-organizing feature maps for the vehicle routing problem with backhauls

In the Vehicle Routing Problem with Backhauls (VRPB), a central depot, a fleet of homogeneous vehicles, and a set of customers are given. The set of customers is divided into two subsets. The first (second) set of linehauls (backhauls) consists of customers with known quantity of goods to be delivered from (collected to) the depot. The VRPB objective is to design a set of minimum cost routes; originating and terminating at the central depot to service the set of customers. In this paper, we develop a self-organizing feature maps algorithm, which uses unsupervised competitive neural network concepts. The definition of the architecture of the neural network and its learning rule are the main contribution. The architecture consists of two types of chains: linehaul and backhaul chains. Linehaul chains interact exclusively with linehaul customers. Similarly, backhaul chains interact exclusively with backhaul customers. Additonal types of interactions are introduced in order to form feasible VRPB solution when the algorithm converges. The generated routes are then improved using the well-known 2-opt procedure. The implemented algorithm is compared with other approaches in the literature. The computational results are reported for standard benchmark test problems. They show that the proposed approach is competitive with the most efficient metaheuristics.     

A hybrid approach for the vehicle routing problem with three-dimensional loading constraints

This paper addresses a recently practical combinatorial problem named Three-Dimensional Loading Capacitated Vehicle Routing Problem, which combines three-dimensional loading problem and vehicle routing problem in distribution logistics. The problem requires a combinatorial optimization of a feasible loading and successive routing of vehicles to satisfy customer demands, where all vehicles must start and finish at a central depot. The goal of this combinatorial problem is to minimize the total transportation cost while serving customers. Despite its clearly practical significance in the real world distribution management, for its high combinatorial complexity, published papers on this problem in literature are very limited.     

Local search based metaheuristics for the robust vehicle routing problem with discrete scenarios

The Capacitated Vehicle Routing Problem (CVRP) is extended here to handle uncertain arc costs without resorting to probability distributions, giving the Robust VRP (RVRP). The unique set of arc costs in the CVRP is replaced by a set of discrete scenarios. A scenario is for instance the travel time observed on each arc at a given traffic hour. The goal is to build a set of routes using the lexicographic min–max criterion: the worst cost over all scenarios is minimized but ties are broken using the other scenarios, from the worst to the best. This version of robust CVRP has never been studied before. A Mixed Integer Linear Program (MILP), two greedy heuristics, a local search and four metaheuristics are proposed: a Greedy Randomized Adaptive Search Procedure, an Iterated Local Search (ILS), a Multi-Start ILS (MS-ILS), and an MS-ILS based on Giant Tours (MS-ILS-GT) converted into feasible routes via a lexicographic splitting procedure. The greedy heuristics provide the other algorithms with good initial solutions. Tests on small instances (10–20 customers, 2–3 vehicles, 10–30 scenarios) show that the four metaheuristics retrieve all optima found by the MILP. On larger cases with 50–100 customers, 5–20 vehicles and 10–20 scenarios, MS-ILS-GT dominates the other approaches. As our algorithms share the same components (initial heuristic, local search), the positive contribution of using the giant tour approach is confirmed on the RVRP.     

Waste collection vehicle routing problem with time windows

In this paper, we address a real life waste collection vehicle routing problem with time windows (VRPTW) with consideration of multiple disposal trips and drivers’ lunch breaks. Solomon's well-known insertion algorithm is extended for the problem. While minimizing the number of vehicles and total traveling time is the major objective of vehicle routing problems in the literature, here we also consider the route compactness and workload balancing of a solution since they are very important aspects in practical applications. In order to improve the route compactness and workload balancing, a capacitated clustering-based waste collection VRPTW algorithm is developed. The proposed algorithms have been successfully implemented and deployed for the real life waste collection problems at Waste Management, Inc. A set of waste collection VRPTW benchmark problems is also presented in this paper.Waste collection problems are frequently considered as arc routing problems without time windows. However, that point of view can be applied only to residential waste collection problems. In the waste collection industry, there are three major areas: commercial waste collection, residential waste collection and roll-on-roll-off. In this paper, we mainly focus on the commercial waste collection problem. The problem can be characterized as a variant of VRPTW since commercial waste collection stops may have time windows. The major variation from a standard VRPTW is due to disposal operations and driver's lunch break. When a vehicle is full, it needs to go to one of the disposal facilities (landfill or transfer station). Each vehicle can, and typically does, make multiple disposal trips per day. The purpose of this paper is to introduce the waste collection VRPTW, benchmark problem sets, and a solution approach for the problem. The proposed algorithms have been successfully implemented and deployed for the real life waste collection problems of Waste Management, the leading provider of comprehensive waste management services in North America with nearly 26,000 collection and transfer vehicles.