Scatter search for the fleet size and mix vehicle routing problem with time windows

This work proposes a scatter search (SS) approach to solve the fleet size and mix vehicle routing problem with time windows (FSMVRPTW). In the FSMVRPTW the customers need to be serviced in their time windows at minimal costs by a heterogeneous fleet. Computational results on 168 benchmark problems are reported. Computational testing revealed that our algorithm presented better results compared to other methods published in the literature.     

A deterministic tabu search algorithm for the fleet size and mix vehicle routing problem

The fleet size and mix vehicle routing problem consists of defining the type, the number of vehicles of each type, as well as the order in which to serve the customers with each vehicle when a company has to distribute goods to a set of customers geographically spread, with the objective of minimizing the total costs. In this paper, a heuristic algorithm based on tabu search is proposed and tested on several benchmark instances. The computational results show that the proposed algorithm produces high quality results within a reasonable computing time. Some new best solutions are reported for a set of test problems used in the literature.     

The open vehicle routing problem with time windows

In this paper, we consider the open vehicle routing problem with time windows (OVRPTW). The OVRPTW seeks to find a set of non-depot returning vehicle routes, for a fleet of capacitated vehicles, to satisfy customers’ requirements, within fixed time intervals that represent the earliest and latest times during the day that customers’ service can take place. We formulate a comprehensive mathematical model to capture all aspects of the problem, and incorporate in the model all critical practical concerns. The model is solved using a greedy look-ahead route construction heuristic algorithm, which utilizes time windows related information via composite customer selection and route-insertion criteria. These criteria exploit the interrelationships between customers, introduced by time windows, that dictate the sequence in which vehicles must visit customers. Computational results on a set of benchmark problems from the literature provide very good results and indicate the applicability of the methodology in real-life routing applications.     

The vehicle routing problem with coupled time windows

In this article we introduce the vehicle routing problem with coupled time windows (VRPCTW), which is an extension of the vehicle routing problem with time windows (VRPTW), where additional coupling constraints on the time windows are imposed. VRPCTW is applied to model a real-world planning problem concerning the integrated optimization of school starting times and public bus services. A mixed-integer programming formulation for the VRPCTW within this context is given. It is solved using a new meta-heuristic that combines classical construction aspects with mixed-integer preprocessing techniques, and improving hit-and-run, a randomized search strategy from global optimization. Solutions for several randomly generated and real-world instances are presented.     

A cluster-based optimization approach for the multi-depot heterogeneous fleet vehicle routing problem with time windows

This paper presents a novel three-phase heuristic/algorithmic approach for the multi-depot routing problem with time windows and heterogeneous vehicles. It has been derived from embedding a heuristic-based clustering algorithm within a VRPTW optimization framework. To this purpose, a rigorous MILP mathematical model for the VRPTW problem is first introduced. Likewise other optimization approaches, the new formulation can efficiently solve case studies involving at most 25 nodes to optimality. To overcome this limitation, a preprocessing stage clustering nodes together is initially performed to yield a more compact cluster-based MILP problem formulation. In this way, a hierarchical hybrid procedure involving one heuristic and two algorithmic phases was developed. Phase I aims to identifying a set of cost-effective feasible clusters while Phase II assigns clusters to vehicles and sequences them on each tour by using the cluster-based MILP formulation. Ordering nodes within clusters and scheduling vehicle arrival times at customer locations for each tour through solving a small MILP model is finally performed at Phase III. Numerous benchmark problems featuring different sizes, clustered/random customer locations and time window distributions have been solved at acceptable CPU times.     

Scatter search for a real-life heterogeneous fleet vehicle routing problem with time windows and split deliveries in Brazil

In this paper, we consider a real-life heterogeneous fleet vehicle routing problem with time windows and split deliveries that occurs in a major Brazilian retail group. A single depot attends 519 stores of the group distributed in 11 Brazilian states. To find good solutions to this problem, we propose heuristics as initial solutions and a scatter search (SS) approach. Next, the produced solutions are compared with the routes actually covered by the company. Our results show that the total distribution cost can be reduced significantly when such methods are used. Experimental testing with benchmark instances is used to assess the merit of our proposed procedure.     

Tabu search variants for the mix fleet vehicle routing problem

The Mix Fleet Vehicle Routing Problem (MFVRP) involves the design of a set of minimum cost routes, originating and terminating at a central depot, for a fleet of heterogeneous vehicles with various capacities, fixed costs and variable costs to service a set of customers with known demands. This paper develops new variants of a tabu search meta-heuristic for the MFVRP. These variants use a mix of different components, including reactive tabu search concepts; variable neighbourhoods, special data memory structures and hashing functions. The reactive concept is used in a new way to trigger the switch between simple moves for intensification and more complex ones for diversification of the search strategies. The special data structures are newly introduced to efficiently search the various neighbourhood spaces. The combination of data structures and strategic balance between intensification and diversification generates an efficient and robust implementation, which is very competitive with other algorithms in the literature on a set of benchmark instances for which some new best-known solutions are provided.     

The fleet size and mix problem for capacitated arc routing

There have been several attempts to solve the capacitated arc routing problem with m vehicles starting their tours from a central node. The objective has been to minimize the total distance travelled. In the problem treated here we also have the fixed costs of the vehicles included in the objective function. A set of vehicle capacities with their respective costs are used. Thus the objective function becomes a combination of fixed and variable costs. The solution procedure consists of four phases. In the first phase, a Chinese or rural postman problem is solved depending on whether all or some of the arcs in the network demand service with the objective of minimizing the total distance travelled. It results in a tour called the giant tour. In the second phase, the giant tour is partitioned into single vehicle subtours feasible with respect to the constraints. A new network is constructed with the node set corresponding to the arcs of the giant tour and with the arc set consisting of the subtours of the giant tour. The arc costs include both the fixed and variable costs of the subtours. The third phase consists of solving the shortest path problem on this new network to result in the least cost set of subtours represented on the new network. In the last phase a postprocessor is applied to the solution to improve it. The procedure is repeated for different giant tours to improve the final solution. The problem is extended to the case where there can be upper bounds on the number of vehicles with given capacities using a branch and bound method. Extension to directed networks is given. Some computational results are reported.     

A reactive variable neighborhood tabu search for the heterogeneous fleet vehicle routing problem with time windows

This paper presents a solution methodology for the heterogeneous fleet vehicle routing problem with time windows. The objective is to minimize the total distribution costs, or similarly to determine the optimal fleet size and mix that minimizes both the total distance travelled by vehicles and the fixed vehicle costs, such that all problem’s constraints are satisfied. The problem is solved using a two-phase solution framework based upon a hybridized Tabu Search, within a new Reactive Variable Neighborhood Search metaheuristic algorithm. Computational experiments on benchmark data sets yield high quality solutions, illustrating the effectiveness of the approach and its applicability to realistic routing problems. This work is supported by the General Secretariat for Research and Technology of the Hellenic Ministry of Development under contract GSRT NM-67.     

POPMUSIC for a real-world large-scale vehicle routing problem with time windows

This paper presents a heuristic approach based on the POPMUSIC framework for a large-scale Multi Depot Vehicle Routing Problem with Time Windows derived from real-world data. POPMUSIC is a very powerful tool for tackling large problem instances. A Memetic Algorithm is used as an optimizer in the POPMUSIC framework. It is shown that a population-based search combined with decomposition strategies is a very efficient and flexible tool to tackle real-world problems with regards to solution quality as well as runtime.     

Simulated annealing metaheuristics for the vehicle routing problem with time windows

This paper develops simulated annealing metaheuristics for the vehicle routing and scheduling problem with time window constraints. Two different neighborhood structures, the λ-interchange mechanism of Osman and thek-node interchange process of Christofides and Beasley, are implemented. The enhancement of the annealing process with a short-term memory function via a tabu list is examined as a basis for improving the metaheuristic approach. Computational results on test problems from the literature as well as large-scale real-world problem are reported. The metaheuristics achieve solutions that compare favorably with previously reported results.     

A vehicle routing problem with multi-trips and time windows for circular items

This article presents a vehicle routing problem with time windows, multiple trips, a limited number of vehicles and loading constraints for circular objects. This is a real problem experienced by a home delivery service company. A linear model is proposed to handle small problems and a two-step heuristic method to solve real size instances: the first step builds an initial solution through the modification of the Solomon I1 sequential insertion heuristic, and the second step improves the initial solution through the Tabu search algorithm proposed; in both steps, the problems related to circle packing with different sizes and bin packing are solved jointly with the use of heuristics. Finally, the computing results for two different sets of instances are presented.     

A hybrid search method for the vehicle routing problem with time windows

Vehicle Routing Problems have been extensively analyzed to reduce transportation costs. More particularly, the Vehicle Routing Problem with Time Windows (VRPTW) imposes the period of time of customer availability as a constraint, a common characteristic in real world situations. Using minimization of the total distance as the main objective to be fulfilled, this work implements an efficient algorithm which associates non-monotonic Simulated Annealing to Hill-Climbing and Random Restart. The algorithm is compared to the best results published in the literature for the 56 Solomon instances and it is shown how statistical methods can be used to boost the performance of the method.     

A greedy look-ahead heuristic for the vehicle routing problem with time windows

In this paper we consider the problem of physically distributing finished goods from a central facility to geographically dispersed customers, which pose daily demands for items produced in the facility and act as sales points for consumers. The management of the facility is responsible for satisfying all demand, and promises deliveries to the customers within fixed time intervals that represent the earliest and latest times during the day that a delivery can take place. We formulate a comprehensive mathematical model to capture all aspects of the problem, and incorporate in the model all critical practical concerns such as vehicle capacity, delivery time intervals and all relevant costs. The model, which is a case of the vehicle routing problem with time windows, is solved using a new heuristic technique. Our solution method, which is based upon Atkinson's greedy look-ahead heuristic, enhances traditional vehicle routing approaches, and provides surprisingly good performance results with respect to a set of standard test problems from the literature. The approach is used to determine the vehicle fleet size and the daily route of each vehicle in an industrial example from the food industry. This actual problem, with approximately two thousand customers, is presented and solved by our heuristic, using an interface to a Geographical Information System to determine inter-customer and depot–customer distances. The results indicate that the method is well suited for determining the required number of vehicles and the delivery schedules on a daily basis, in real life applications.     

Heuristic solutions for the vehicle routing problem with time windows and synchronized visits

We present a simulated annealing based algorithm for a variant of the vehicle routing problem (VRP), in which a time window is associated with each client service and some services require simultaneous visits from different vehicles to be accomplished. The problem is called the VRP with time windows and synchronized visits. The algorithm features a set of local improvement methods to deal with various objectives of the problem. Experiments conducted on the benchmark instances from the literature clearly show that our method is fast and outperforms the existing approaches. It produces all known optimal solutions of the benchmark in very short computational times, and improves the best results for the rest of the instances.     

A powerful route minimization heuristic for the vehicle routing problem with time windows

We suggest an efficient route minimization heuristic for the vehicle routing problem with time windows. The heuristic is based on the ejection pool, powerful insertion and guided local search strategies. Experimental results on the Gehring and Homberger’s benchmarks demonstrate that our algorithm outperforms previous approaches and found 18 new best-known solutions.     

A column generation algorithm for the vehicle routing problem with soft time windows

The Vehicle Routing Problem with Time Windows consists of computing a minimum cost set of routes for a fleet of vehicles of limited capacity visiting a given set of customers with known demand, with the additional constraint that each customer must be visited in a specified time window. We consider the case in which time window constraints are relaxed into “soft” constraints, that is penalty terms are added to the solution cost whenever a vehicle serves a customer outside of his time window. We present a branch-and-price algorithm which is the first exact optimization algorithm for this problem.     

Integrated machine scheduling and vehicle routing with time windows

This paper integrates production and outbound distribution scheduling in order to minimize total tardiness. The overall problem consists of two subproblems. The first addresses scheduling a set of jobs on parallel machines with machine-dependent ready times. The second focusses on the delivery of completed jobs with a fleet of vehicles which may differ in their loading capacities and ready times. Job-dependent processing times, delivery time windows, service times, and destinations are taken into account. A genetic algorithm approach is introduced to solve the integrated problem as a whole. Two main questions are examined. Are the results of integrating machine scheduling and vehicle routing significantly better than those of classic decomposition approaches which break down the overall problem, solve the two subproblems successively, and merge the subsolutions to form a solution to the overall problem? And if so, is it possible to capitalize on these potentials despite the complexity of the integrated problem? Both questions are tackled by means of a numerical study. The genetic algorithm outperforms the classic decomposition approaches in case of small-size instances and is able to generate relatively good solutions for instances with up to 50 jobs, 5 machines, and 10 vehicles.     

An exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles

The vehicle routing problem with multiple use of vehicles is a variant of the classical vehicle routing problem. It arises when each vehicle performs several routes during the workday due to strict time limits on route duration (e.g., when perishable goods are transported). The routes are defined over customers with a revenue, a demand and a time window. Given a fixed-size fleet of vehicles, it might not be possible to serve all customers. Thus, the customers must be chosen based on their associated revenue minus the traveling cost to reach them. We introduce a branch-and-price approach to address this problem where lower bounds are computed by solving the linear programming relaxation of a set packing formulation, using column generation. The pricing subproblems are elementary shortest path problems with resource constraints. Computational results are reported on euclidean problems derived from well-known benchmark instances for the vehicle routing problem with time windows.     

A branch-price-and-cut algorithm for the vehicle routing problem with time windows and multiple deliverymen

We address a variant of the vehicle routing problem with time windows that includes the decision of how many deliverymen should be assigned to each vehicle. In this variant, the service time at each customer depends on the size of the respective demand and on the number of deliverymen assigned to visit this customer. In addition, the objective function consists of minimizing a weighted sum of the total number of routes, number of deliverymen and traveled distance. These characteristics make this variant very challenging for exact methods. To date, only heuristic approaches have been proposed for this problem, and even the most efficient optimization solvers cannot find optimal solutions in a reasonable amount of time for instances of moderate size when using the available mathematical formulations. We propose a branch-price-and-cut method based on a new set partitioning formulation of the problem. To accelerate the convergence of the method, we rely on an interior-point column and cut generation process, a strong branching strategy and a mixed-integer programming-based primal heuristic. Additionally, a hierarchical branching strategy is used to take into account the different components of the objective function. The computational results indicate the benefits of using the proposed exact solution approach. We closed several instances of the problem and obtained upper bounds that were previously unknown in the literature.