Evolutionary Algorithms for the Vehicle Routing Problem with Time Windows

This paper surveys the research on evolutionary algorithms for the Vehicle Routing Problem with Time Windows (VRPTW). The VRPTW can be described as the problem of designing least cost routes from a single depot to a set of geographically scattered points. The routes must be designed in such a way that each point is visited only once by exactly one vehicle within a given time interval. All routes start and end at the depot, and the total demands of all points on one particular route must not exceed the capacity of the vehicle. The main types of evolutionary algorithms for the VRPTW are genetic algorithms and evolution strategies. In addition to describing the basic features of each method, experimental results for the benchmark test problems of Solomon (1987) and Gehring and Homberger (1999) are presented and analyzed.     

A multi-shift vehicle routing problem with windows and cycle times

In many applications of the vehicle routing problem with time windows (VRPTW), goods must be picked up within desired time frames. In addition, they have some limitations on their arrival time to the central depot. In this paper, we present a new version of VRPTW that minimizes the total cycle time of the goods. In order to meet the time windows and also minimize the cycle time, the courier’s schedule is allowed to vary. An algorithm, named VeRSA, is proposed to solve this problem. VeRSA employs concepts of scheduling theorems and algorithms to determine feasible routes and schedules of the available couriers. We prove a theoretical lower bound that provides a useful bound on the optimality gap. We also conduct a set of numerical experiments. VeRSA obtains a feasible solution faster than solving the MIP. The optimality gap using our proposed lower bound is smaller than the gap found with the standard LP relaxation.     

Heuristic algorithms for a vehicle routing problem with simultaneous delivery and pickup and time windows in home health care

This paper addresses a vehicle scheduling problem encountered in home health care logistics. It concerns the delivery of drugs and medical devices from the home care company’s pharmacy to patients’ homes, delivery of special drugs from a hospital to patients, pickup of bio samples and unused drugs and medical devices from patients. The problem can be considered as a special vehicle routing problem with simultaneous delivery and pickup and time windows, with four types of demands: delivery from depot to patient, delivery from a hospital to patient, pickup from a patient to depot and pickup from a patient to a medical lab. Each patient is visited by one vehicle and each vehicle visits each node at most once. Patients are associated with time windows and vehicles with capacity. Two mixed-integer programming models are proposed. We then propose a Genetic Algorithm (GA) and a Tabu Search (TS) method. The GA is based on a permutation chromosome, a split procedure and local search. The TS is based on route assignment attributes of patients, an augmented cost function, route re-optimization, and attribute-based aspiration levels. These approaches are tested on test instances derived from existing VRPTW benchmarks.     

An integrated approach for collection network design,capacity planning and vehicle routing in reverse logistics

This study considers network design, capacity planning and vehicle routing for collection systems in reverse logistics. The network design and capacity planning problems are to determine the static locations and capacities of collection points as well as the dynamic allocations of demand points to the opened collection points over a planning horizon, and the vehicle routing problem is to determine the number and routes of vehicles in such a way that each collection point must be visited exactly once by one vehicle starting and terminating at the depot while satisfying the return demands at collection points and the vehicle capacity. The objective is to minimize the sum of fixed costs to open collection points and to acquire vehicles as well as variable costs to transport returns at demand points to the opened collection points and travel the opened collection points by vehicles. Unlike the location-routing problems, the integrated problem considered in this study has several features: multi-period dynamic model, capacity planning for collection points, maximum allowable collection distances, etc. To solve the integrated problem, two types of tabu search algorithms, hierarchical and integrated ones, are suggested, and their test results are reported. In particular, the efficiency of the integrated approach is shown by comparing the two algorithm types.     

A Hybrid Multiobjective Evolutionary Algorithm for Solving Vehicle Routing Problem with Time Windows

Vehicle routing problem with time windows (VRPTW) involves the routing of a set of vehicles with limited capacity from a central depot to a set of geographically dispersed customers with known demands and predefined time windows. The problem is solved by optimizing routes for the vehicles so as to meet all given constraints as well as to minimize the objectives of traveling distance and number of vehicles. This paper proposes a hybrid multiobjective evolutionary algorithm (HMOEA) that incorporates various heuristics for local exploitation in the evolutionary search and the concept of Pareto's optimality for solving multiobjective optimization in VRPTW. The proposed HMOEA is featured with specialized genetic operators and variable-length chromosome representation to accommodate the sequence-oriented optimization in VRPTW. Unlike existing VRPTW approaches that often aggregate multiple criteria and constraints into a compromise function, the proposed HMOEA optimizes all routing constraints and objectives simultaneously, which improves the routing solutions in many aspects, such as lower routing cost, wider scattering area and better convergence trace. The HMOEA is applied to solve the benchmark Solomon's 56 VRPTW 100-customer instances, which yields 20 routing solutions better than or competitive as compared to the best solutions published in literature.     

An exact algorithm for solving a capacitated location-routing problem

In location-routing problems, the objective is to locate one or many depots within a set of sites (representing customer locations or cities) and to construct delivery routes from the selected depot or depots to the remaining sites at least system cost. The objective function is the sum of depot operating costs, vehicle acquisition costs and routing costs. This paper considers one such problem in which a weight is assigned to each site and where sites are to be visited by vehicles having a given capacity. The solution must be such that the sum of the weights of sites visited on any given route does not exceed the capacity of the visiting vehicle. The formulation of an integer linear program for this problem involves degree constraints, generalized subtour elimination constraints, and chain barring constraints. An exact algorithm, using initial relaxation of most of the problem constraints, is presented which is capable of solving problems with up to twenty sites within a reasonable number of iterations.     

A synthesis of assignment and heuristic solutions for vehicle routing with time windows

This paper proposes a three-stage method for the vehicle-routing problem with time window constraints (VRPTW). Using the Hungarian method the optimal customer matching for an assignment approximation of the VRPTW, which is a travel time-based relaxation that partially respects the time windows, is obtained. The assignment matching is transformed into feasible routes of the VRPTW via a simple decoupling heuristic. The best of these routes, in terms of travelling and vehicle waiting times, form part of the final solution, which is completed by the routes provided by heuristic methods applied to the remainder of the customers. The proposed approach is tested on a set of standard literature problems, and improves the results of the heuristic methods with respect to total travel time. Furthermore, it provides useful insights into the effect of employing optimal travel time solutions resulting from the assignment relaxation to derive partial route sets of the VRPTW.     

Approximative solutions to the bicriterion Vehicle Routing Problem with Time Windows

The Vehicle Routing Problem with Time Windows (VRPTW) is a combinatorial optimization problem. It deals with route planning and the distribution of goods from a depot to geographically dispersed customers by a fleet of vehicles with constrained capacities. The customers’ demands are known and each customer has a time window in which it has to be supplied. The time windows are assumed to be soft, that means, violations of the time windows are allowed, but associated with penalties. The problem is to organize the vehicle routes optimally, i.e. to minimize the total costs, consisting of the number of used vehicles and the total distance, and the penalties simultaneously. Thus, the problem is formulated as a bicriterion minimization problem and heuristic methods are used to calculate approximations of the Pareto optimal solutions. Experimental results show that in certain cases the allowance of penalties leads to significant savings of the total costs.     

A Branch and Bound Algorithm for a Class of Asymmetrical Vehicle Routeing Problems

This paper describes a branch and bound algorithm for a general class of asymmetrical vehicle routeing problems. Vehicle routes start and end at a central depot. Visits are made to nodes grouped into clusters: every cluster must receive a minimum number of visits. But not all nodes must be visited: there are specified nodes and non-specified nodes. Vehicle routes are also constrained by capacity and distance restrictions. The problem is formulated as an integer linear program. It is then solved by a branch and bound algorithm which exploits the unimodular structure of the subproblems. Computational results are reported.     

A variable neighbourhood search algorithm for the open vehicle routing problem

In the open vehicle routing problem (OVRP), the objective is to minimise the number of vehicles and then minimise the total distance (or time) travelled. Each route starts at the depot and ends at a customer, visiting a number of customers, each once, en route, without returning to the depot. The demand of each customer must be completely fulfilled by a single vehicle. The total demand serviced by each vehicle must not exceed vehicle capacity. Additionally, in one variant of the problem, the travel time of each vehicle should not exceed an upper limit.     

Designing vehicle routes for a mix of different request types,under time windows and loading constraints

This article introduces and solves a new rich routing problem integrated with practical operational constraints. The problem examined calls for the determination of the optimal routes for a vehicle fleet to satisfy a mix of two different request types. Firstly, vehicles must transport three-dimensional, rectangular and stackable boxes from a depot to a set of predetermined customers. In addition, vehicles must also transfer products between pairs of pick-up and delivery locations. Service of both request types is subject to hard time window constraints. In addition, feasible palletization patterns must be identified for the transported products. A practical application of the problem arises in the transportation systems of chain stores, where vehicles replenish the retail points by delivering products stored at a central depot, while they are also responsible for transferring stock between pairs of the retailer network. To solve this very complex combinatorial optimization problem, our major objective was to develop an efficient methodology whose required computational effort is kept within reasonable limits. To this end, we propose a local search-based framework for optimizing vehicle routes, in which feasible loading arrangements are identified via a simple-structured packing heuristic. The algorithmic framework is enhanced with various memory components which store and retrieve useful information gathered through the search process, in order to avoid any duplicate unnecessary calculations. The proposed solution approach is assessed on newly introduced benchmark instances.     

Polyhedral study of the capacitated vehicle routing problem

The capacitated vehicle routing problem (CVRP) considered in this paper occurs when goods must be delivered from a central depot to clients with known demands, usingk vehicles of fixed capacity. Each client must be assigned to exactly one of the vehicles. The set of clients assigned to each vehicle must satisfy the capacity constraint. The goal is to minimize the total distance traveled. When the capacity of the vehicles is large enough, this problem reduces to the famous traveling salesman problem (TSP). A variant of the problem in which each client is visited by at least one vehicle, called the graphical vehicle routing problem (GVRP), is also considered in this paper and used as a relaxation of CVRP. Our approach for CVRP and GVRP is to extend the polyhedral results known for TSP. For example, the subtour elimination constraints can be generalized to facets of both CVRP and GVRP. Interesting classes of facets arise as a generalization of the comb inequalities, depending on whether the depot is in a handle, a tooth, both or neither. We report on the optimal solution of two problem instances by a cutting plane algorithm that only uses inequalities from the above classes.This work was supported in part by NSF grant DDM-8901495.     

Routing relatively few customers per route

This paper presents a modification of the well-known Solomon (1987) sequential insertion heuristic I1 for the Vehicle Routing Problem with Time Windows (VRPTW). VRPTW involves servicing customers within a pre-specified service time window by homogeneously capacitated vehicles from a single depot. By using two new measures for the additional time needed to insert a customer in the route, significantly better solutions are obtained for relatively short routes compared to the original Solomon (1987) sequential insertion heuristic I1. Because high-quality initial heuristics often allow local searches and metaheuristics to achieve better solutions more quickly, the new approach is likely to help generating better solutions to practical routing problems.     

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.     

Optimizing departure times in vehicle routes

Most solution methods for the vehicle routing problem with time windows (VRPTW) develop routes from the earliest feasible departure time. In practice, however, temporary traffic congestion make such solutions non-optimal with respect to minimizing the total duty time. Furthermore, the VRPTW does not account for driving hours regulations, which restrict the available travel time for truck drivers. To deal with these problems, we consider the vehicle departure time optimization (VDO) problem as a post-processing of a VRPTW. We propose an ILP formulation that minimizes the total duty time. The results of a case study indicate that duty time reductions of 15% can be achieved. Furthermore, computational experiments on VRPTW benchmarks indicate that ignoring traffic congestion or driving hours regulations leads to practically infeasible solutions. Therefore, new vehicle routing methods should be developed that account for these common restrictions. We propose an integrated approach based on classical insertion heuristics.     

A perturbation metaheuristic for the vehicle routing problem with private fleet and common carriers

The purpose of this article is to propose a perturbation metaheuristic for the vehicle routing problem with private fleet and common carrier (VRPPC). This problem consists of serving all customers in such a way that (1) each customer is served exactly once either by a private fleet vehicle or by a common carrier vehicle, (2) all routes associated with the private fleet start and end at the depot, (3) each private fleet vehicle performs only one route, (4) the total demand of any route does not exceed the capacity of the vehicle assigned to it, and (5) the total cost is minimized. This article describes a new metaheuristic for the VRPPC, which uses a perturbation procedure in the construction and improvement phases and also performs exchanges between the sets of customers served by the private fleet and the common carrier. Extensive computational results show the superiority of the proposed metaheuristic over previous methods.     

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.     

Exact algorithms for routing problems under vehicle capacity constraints

The solution of a vehicle routing problem calls for the determination of a set of routes, each performed by a single vehicle which starts and ends at its own depot, such that all the requirements of the customers are fulfilled and the global transportation cost is minimized. The routes have to satisfy several operational constraints which depend on the nature of the transported goods, on the quality of the service level, and on the characteristics of the customers and of the vehicles. One of the most common operational constraint addressed in the scientific literature is that the vehicle fleet is capacitated and the total load transported by a vehicle cannot exceed its capacity.     

Solving a vehicle routing problem by balancing the vehicles time utilization

Various vehicle routing problems (VRP) appear in the literature due to their important applications in the area of transportation and distribution.A VRP is characterized by the constraints that the involved factors must satisfy and by an optimality goal.In this paper, we develop a heuristic algorithm that

• (i)partitions suitably a distribution network into subnetworks. A single depot complies with every subnetwork, where a fleet of identical vehicles will start their itinerary. The nodes of the corresponding subnetwork are demand nodes that require a onetime visit by one only vehicle.
• (ii)Determine the routes of k vehicles, k=2,3,…, for every subnetwork so to minimize the visiting time of the corresponding demand nodes. Consequently the method computes the necessary vehicle number for each subnetwork so as to complete promptly the visiting requirement of all the demand nodes of the whole network. The main strategy of the algorithm for designing the vehicle routes consists of balancing the time utilization of the used vehicles. The paper is integrated by an application of the presented algorithm to the center of the city of Thessaloniki.

     

An approximation algorithm for the pickup and delivery vehicle routing problem on trees

This paper presents an approximation algorithm for a vehicle routing problem on a tree-shaped network with a single depot where there are two types of demands, pickup demand and delivery demand. Customers are located on nodes of the tree, and each customer has a positive demand of pickup and/or delivery.Demands of customers are served by a fleet of identical vehicles with unit capacity. Each vehicle can serve pickup and delivery demands. It is assumed that the demand of a customer is splittable, i.e., it can be served by more than one vehicle. The problem we are concerned with in this paper asks to find a set of tours of the vehicles with minimum total lengths. In each tour, a vehicle begins at the depot with certain amount of goods for delivery, visits a subset of the customers in order to deliver and pick up goods and returns to the depot. At any time during the tour, a vehicle must always satisfy the capacity constraint, i.e., at any time the sum of goods to be delivered and that of goods that have been picked up is not allowed to exceed the vehicle capacity. We propose a 2-approximation algorithm for the problem.