Homogeneous Fleet Research Articles

A relax‐and‐price heuristic for the inventory‐location‐routing problem

AbstractThis paper considers the problem of designing a supply chain assuming routing decisions. The objective is to select a subset of depots to open from a set of candidates, the inventory policies for a two‐echelon system, and the set of routes to perform distribution from the upper echelon to the next by a homogeneous fleet of vehicles over a finite planning horizon considering deterministic demand. To solve the problem, a partition is proposed using a Dantzig–Wolfe formulation on the routing variables. A hybridization between column generation, Lagrangian relaxation, and local search is presented within a heuristic procedure. Results demonstrate the capability of the algorithm to compute high quality solutions and empirically estimate the improvement in the cost function of the proposed model at up to 9% compared to the sequential approach. Furthermore, the suggested pricing problem is a new variant of the shortest path problem with applications in urban transportation and telecommunications.

A monolithic approach to vehicle routing and operations scheduling of a cross-dock system with multiple dock doors

Cross-docking is a logistic strategy for moving goods from suppliers to customers via a cross-dock terminal with no permanent storage. The operational planning of a cross-dock facility involves different issues such as vehicle routing, dock door assignment and truck scheduling. The vehicle routing problem seeks the optimal routes for a homogeneous fleet of vehicles that sequentially collects goods at pickup points and delivers them to their destinations. The truck scheduling problem deals with the timing of unloading and reloading operations at the cross-dock. This work introduces a mixed-integer linear programming formulation for the scheduling of single cross-dock systems that, in addition to selecting the pickup/delivery routes, simultaneously decides on the dock door assignment and the truck scheduling at the cross-dock. The proposed monolithic formulation is able to provide near-optimal solutions to medium-size problems involving up to 70 transportation orders, 16 vehicles and 7 strip/stack dock doors at acceptable CPU times.

Solving vehicle routing problems with asymmetric costs and heterogeneous fleets

The vehicle routing problem (VRP) is a flourishing research area with clear applications to real-life distribution companies. However, most VRP-related academic articles assume the existence of a homogeneous fleet of vehicles and/or a symmetric cost matrix. These assumptions are not always reasonable in real-life scenarios. To contribute in closing this gap between theory and practice, we propose a hybrid methodology for solving the asymmetric and heterogeneous vehicle routing problem (AHVRP). In our approach, we consider: 1) different types of vehicle loading capacities (heterogeneous fleets); 2) asymmetric distance-based costs. The proposed approach combines a randomised version of a well-known savings heuristic with several local searches specifically adapted to deal with the asymmetric nature of costs. A computational experiment allows us to discuss the efficiency of our approach and also to analyse how routing costs vary when slight departures from the homogeneous fleet assumption are considered.

An Integrated Model for Production and Distribution Planning of Perishable Products with Inventory and Routing Considerations

In many conventional supply chains, production planning and distribution planning are treated separately. However, it is now demonstrated that they are mutually related problems that must be tackled in an integrated way. Hence, in this paper a new integrated production and distribution planning model for perishable products is formulated. The proposed model considers a supply chain network consisting of a production facility and multiple distribution centers. The facility produces a single perishable product that is storable only for predetermined periods. A homogenous fleet of vehicles is responsible for delivering the product from facility to distribution centers. The decisions to be made are the production quantities, the distribution centers that must be visited, and the quantities to be delivered to them. The objective is to minimize the total cost, where the trip minimization is considered simultaneously. As the proposed formulation is computationally complex, a heuristic method is developed to tackle the problem. In the developed method, the problem is divided into production submodel and distribution submodel. The production submodel is solved using LINGO, and a particle swarm heuristic is developed to tackle distribution submodel. Efficiency of the algorithm is proved through a number of randomly generated test problems.

Variable Neighborhood Search based Set Covering ILP Model for the Vehicle Routing Problem with Time Windows

In this paper we propose a hybrid metaheuristic based on General Variable Neighborhood search and Integer Linear Programming for solving the vehicle routing problem with time windows (VRPTW). The problem consists in determining the minimum cost routes for a homogeneous fleet of vehicles to meet the demand of a set of customers within a specified time windows. The proposed heuristic, called VNS-SCP is considered as a matheuristic where the hybridization of heuristic (VNS) and exact (Set Covering Problem (SCP)) method is used in this approach as an intertwined collaborative cooperation manner. In this approach an initial solution is first created using Solomon route-construction heuristic, the nearest neighbor algorithm. In the second phase the solutions are improved in terms of the total distance traveled using VNS-SCP. The algorithm is tested using Solomon benchmark. Our findings indicate that the proposed procedure outperforms other local searches and metaheuristics.

A successive approximations method for the heterogeneous vehicle routing problem: analysing different fleet configurations

In this paper, we propose a relatively simple-to-implement procedure for solving the heterogeneous-fleet vehicle routing problem (HeVRP), in which different types of vehicle loading capacities are considered. Our approach is based on the so called successive approximations method (SAM), which is a multi-round process. At each round, a new subset of nodes and a new type of vehicle are selected following some specific criteria. Then, assuming an unlimited fleet of vehicles of this type, the associated homogeneous-fleet vehicle routing problem (HoVRP) is solved. After several rounds, a global solution for the HeVRP is obtained by merging routes from different HoVRP solutions. In the first part of the paper, we analyse how distance-based costs vary when slight deviations from the homogeneous fleet assumption are considered. In the second part of the article, the SAM approach is adapted so it can simultaneously deal with both fixed and variable costs in HeVRPs. An experimental comparison is then made with other HeVRP algorithms.

Hybrid heuristic for the inventory location-routing problem with deterministic demand

The Inventory Location-Routing Problem with deterministic demand can be seen as an approach to both optimize a supply chain design and minimize its operational costs. This problem considers that vehicles might deliver products to more than one retailer per route and that inventory management decisions are included for a multi-depot, multi-retailer system with storage capacity over a discrete time planning horizon. The problem is to determine a set of candidate depots to open, the quantities to ship from suppliers to depots and from depots to retailers per period, and the sequence in which retailers are replenished by an homogeneous fleet of vehicles. A mixed-integer linear programming model is proposed to describe the problem and to provide bounds on the solutions. It is strengthened by two sets of valid inequalities with an analysis of their impact. Since the model is not able to solve the targeted instances exactly within a reasonable computation time, a hybrid method, embedding an exact approach within a heuristic scheme, is presented. Its performance is tested over three sets of instances for the inventory location routing, location-routing and inventory-routing problems. Results show important savings achieved when compared to a decomposed approach and the capability of the algorithm to solve the problem.

Resolução de um caso real do problema dial-a-ride multicritério via clustering search

Este artigo apresenta um clustering search (CS) para o problema dial-a-ride (DARP) multicritério presente na cidade de Vitória (ES). Em Vitória, os usuários especificam requisições de transporte entre origens e destinos com janelas de tempo para os horários de embarque e desembarque; o transporte é realizado por uma frota homogênea de veículos localizados inicialmente em uma mesma garagem; busca-se assim definir um conjunto de rotas de atendimento que minimize o custo de transporte (normalmente, tempo ou distância), o número de veículos utilizados e o tempo total de espera dos usuários, respeitando restrições como as de capacidade dos veículos e de precedência (o embarque de um usuário deve preceder o seu destino em uma rota). Dessa forma, o CS aqui proposto foi capaz de tratar as particularidades do caso de Vitória, ES. Bons resultados computacionais são apresentados considerando instâncias reais obtidas na Secretaria de Transportes, Trânsito e Infraestrutura Urbana de Vitória, ES.

A GRASP×ELS for the vehicle routing problem with basic three-dimensional loading constraints

This paper addresses an extension of the capacitated vehicle routing problem where the client demand consists of three-dimensional weighted items (3L-CVRP). The objective is to design a set of trips for a homogeneous fleet of vehicles based at a depot node which minimizes the total transportation cost. Items in each vehicle trip must satisfy the three-dimensional orthogonal packing constraints. This problem is strongly connected to real-life transportation systems where the packing of items to be delivered by each vehicle can have a significant impact on the routes. We propose a new way to solve the packing sub-problem. It consists of a two-step procedure in which the z-constraints are first relaxed to get a (x,y) positioning of the items. Then, a compatible z-coordinate is computed to get a packing solution. Items can be rotated but additional constraints such as item fragility, support and LIFO are not considered. This method is included in a GRASP×ELS hybrid algorithm dedicated to the computation of VRP routes. The route optimization alternates between two search spaces: the space of VRP routes and the space of giant trips. The projection from one to the other is done by dedicated procedures (namely the Split and the concatenation algorithms). Moreover, a Local Search is defined on each search space. Furthermore, hash tables are used to store the result of the packing checks and thus save a substantial amount of CPU time. The effectiveness of our approach is illustrated by computational experiments on 3L-CVRP instances from the literature. A new set of realistic instances based on the 96 French districts are also proposed. They range from 19 nodes for the small instances to 255 nodes for the large instances and they can be stated as realistic since they are based on true travel distances in kilometers between French cities. The impact of the hash tables is illustrated as well.

Mixed integer and heuristics model for the inventory routing problem in fuel delivery

This paper presents solution approaches for the multi-product multi-period Inventory Routing Problem (IRP) in fuel delivery. A homogeneous fleet of vehicles with compartments is used for fuel distribution from one depot to a set of petrol stations that have deterministic fuel consumption. The IRP consists of two mutually dependent sub-problems, those of inventory and routing, in a Vendor Management Inventory (VMI) environment in which suppliers determine the quantities and time periods of the deliveries. For solving the IRP, we propose a Mixed Integer Programming (MIP) model and a heuristic approach with and without fleet size costs, to observe the impact of these costs on the solutions that are obtained. The heuristics model is based on constructive heuristics with two Variable Neighborhood Descent (VND) search types: a local intra-period search and a large inter-period neighborhood search. Both of these approaches were tested on numerical examples for which the results, together with the performances of the proposed models, are presented. A combination of good computational time and good quality solutions suggests the use of the proposed heuristics on problems with realistic dimensions where the MIP model cannot find an optimal solution in a reasonable amount of time.

A path relinking algorithm for a multi-depot periodic vehicle routing problem

In this paper, we consider a multi-depot periodic vehicle routing problem which is characterized by the presence of a homogeneous fleet of vehicles, multiple depots, multiple periods, and two types of constraints that are often found in reality, i.e., vehicle capacity and route duration constraints. The objective is to minimize total travel costs. To tackle the problem, we propose an efficient path relinking algorithm whose exploration and exploitation strategies enable the algorithm to address the problem in two different settings: (1) As a stand-alone algorithm, and (2) As a part of a co-operative search algorithm called integrative co-operative search. The performance of the proposed path relinking algorithm is evaluated, in each of the above ways, based on standard benchmark instances. The computational results show that the developed PRA performs well, in both solution quality and computational efficiency.

A hybrid algorithm for a class of vehicle routing problems

In this work we propose a hybrid algorithm for a class of Vehicle Routing Problems with homogeneous fleet. A sequence of Set Partitioning (SP) models, with columns corresponding to routes found by a metaheuristic approach, are solved, not necessarily to optimality, using a Mixed Integer Programming (MIP) solver, that may interact with the metaheuristic during its execution. Moreover, we developed a reactive mechanism that dynamically controls the dimension of the SP models when dealing with large size instances. The algorithm was extensively tested on benchmark instances of the following Vechicle Routing Problem (VRP) variants: (i) Capacitated VRP; (ii) Asymmetric VRP; (iii) Open VRP; (iv) VRP with Simultaneous Pickup and Delivery; (v) VRP with Mixed Pickup and Delivery; (vi) Multi-depot VRP; (vii) Multi-depot VRP with Mixed Pickup and Delivery. The results obtained were quite competitive with those found by heuristics devoted to specific variants. A number of new best solutions were obtained.

A Meta-heuristic Algorithm to Solve the Large-Scale School Bus Scheduling Problem

Given school bus trips for each school in a school district, if a school bus can serve multiple trips, the efficiency of school bus service can be improved in terms of the number of buses needed and the total travel cost. The school bus scheduling problem(SBSP), a class of school bus routing problem(SBRP), is concerned with assigning a fleet of buses to serve all the given trips and aims to get optimal bus schedules. Each school has its xed time window within which school bus must arrive at the destination school of the trip. In existing literatures, SBSP is usually formulated as a transportation problem(TP) or an assignment problem(AP). However, many existing algorithms for vehicle routing problem(VRP) have not been fully utilized to solve the problem effectively. This paper proposes a meta-heuristic algorithm for large-scale SBSP. Treating a trip as a virtual stop with time window, the problem can be converted to a vehicle routing problem with time windows(VRPTW). Therefore, the SBSP can be solved in a VRP algorithm framework. After a set of feasible solutions are generated using construction heuristic algorithm, a simulated annealing(SA) algorithm is designed to improve the initial solutions iteratively. Four general operators for VRP, one-point move, two point move, two-opt move and cross-exchange move, are used in the neighborhood search. In addition to the SBSP objectives of minimizing the number of the routes and the total travel distance, the sum of squared number of route stops is added as a new objective. This will guide the neighborhood search toward the situation that deleting some routes more easily. For avoiding local optimum, some worsening neighborhood solutions can be accepted with a certain probability. Computational tests on 15 instances with a homogeneous fleet show the effectiveness of the proposed approaches. Compared with the existing SBSP solutions, the proposed algorithm can solve large-scale SBSP in a reasonable time and find better solutions using fewer buses. In addition, the algorithm can be easily integrated with GIS for solving real world school bus scheduling.

Grouping genetic algorithms for a bi-objective inventory routing problem

Inventory routing problems (IRPs) is one of the versions of vehicle routing problems (VRPs), which retains the attention of the researchers. The main idea is to coordinate the distribution plan with the inventory management in a same model. The problem studied is to determine the multi-tours of a homogeneous fleet of vehicles covering a set of sales-points and minimising the distribution and inventory cost per hour. Of course no stock-outs are acceptable at the sales-points. In this paper, we analyse this problem like a bi-objective inventory routing problem in which the transportation cost and the delivery cost are considered separately. Two approaches are proposed to approximate the Pareto front of this bi-objective problem. Both methods are an adaptation of the hybrid grouping genetic algorithm (HGGA) that we proposed for the single objective problem in which a grouping genetic algorithm is combined with a local search. Computational experiments are reported using eight instances for four groups: (25, 50,100 and 200 sales points).

A Tabu Search Heuristic for the Inland Container Transportation Problem

The Inland Container Transportation Problem describes the movement of full and empty containers among a number of terminals, depots and customers in a hinterland region. A trucking company with a homogeneous fleet of trucks has to serve customers which either receive goods by inbound containers or ship goods by outbound containers. While keeping given hard time constraints the total operating time of all trucks has to be minimized. A comprehensive mathematical formulation which considers vehicle routing and scheduling and empty container repositioning simultaneously is defined. The problem is solved by an efficient Tabu Search Heuristic. Computational experiments carried out on small- and large-sized instances indicate that the proposed algorithm performs well with respect to effectiveness and efficiency.

Reverse multistar inequalities and vehicle routing problems with a lower bound on the number of customers per route

AbstractThis article analyzes inequalities derived by projecting out the flow variables of a single‐commodity flow model for a vehicle routing problem. These inequalities are called reverse multistar (RMS) inequalities and are related to the MS inequalities analyzed and used in other articles. Although the MS RMS inequalities are irrelevant for some vehicle routing problems, in others they are fundamental. The article presents a vehicle routing problem in which the RMS are of interest. It is called the vehicle routing problem with lower and upper bound capacities (LU‐VRP). It concerns a vehicle routing problem with one depot and a homogeneous fleet of vehicles. All the customers have a unit demand, and there are upper and lower bounds on the demand covered by each vehicle. New families of inequalities are derived by strengthening the RMS inequalities. Computational experiments show that the new inequalities are useful when solving LU‐VRP instances. The experiments are based on variations of symmetric and asymmetric VRP library (VRPLIB) instances with up to 100 customers. The constraint on a minimum number of customers served in each route is implicit in the unit‐demand capacitated vehicle routing problem with a fixed number of vehicles. Therefore, the article also evaluates the impact of using the lower bound inequalities developed in the context of this variant. It is still unknown whether the new inequalities can help solve it or not. Our theoretical analysis suggests that one of the families of developed inequalities is not implied by other standard inequalities. This prompts us to pursue studies in the search for other families of lower bound based inequalities for this variant. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

Ordering Policies for Periodic-Review Inventory Systems with Quantity-Dependent Fixed Costs

We consider a stochastic periodic-review inventory control system in which the fixed cost depends on the order quantity. In particular, we investigate the optimal ordering policies under three fixed cost structures. The first structure is motivated by transportation and production contracts and considers two fixed costs: if the order size is within a specified limit C, then the fixed cost is K1; otherwise, it is K2, where K1 ≼ K2. The second structure contains multiple fixed costs in which the same incremental fixed cost K is incurred for any additional order quantity up to a given identical batch capacity C. In the third structure, in addition to the K incurred as in the previous case, a common fixed cost is charged for any nonzero order size. An example of the former case arises when an order is shipped with a homogeneous fleet of trucks with per-truck fixed costs. A situation in which a fixed administrative cost plus a quantity-dependent trucking cost is incurred for each shipment exemplifies the latter case. For the first cost structure, we separate the analysis according to the conditions (1) K1 ≼ K2 ≼ 2K1 and (2) K1 ≼ K2. Under condition (1), we introduce a new concept called C-(K1, K2)-convexity, which enables us to almost completely characterize the optimal ordering policy. Under the general condition (2), we utilize a modified notion to provide a partial characterization of the optimal policy and propose a heuristic policy that performs well under a wide variety of model parameters. For the second cost structure, we show that it is optimal to order an integer multiple of the batch capacity to raise the inventory level to a specified range or band of length C, and then to order an additional full or partial batch size depending on the cost function, with no ordering required above the band. We also characterize a similar optimal policy for the third cost structure. Using different techniques, our study extends or redevelops several existing results in the literature.

Efficient Heuristics for Large-Scale Vehicle Routing Problems Using Particle Swarm Optimization

This paper addresses the Capacitated Vehicle Routing Problem (CVRP) with a homogenous fleet of vehicles serving a large customer base. The authors propose a multi-phase heuristic that clusters the nodes based on proximity, orients them along a route, and allots vehicles. For the final phase of determining the routes for each vehicle, they have developed a Particle Swarm Optimization (PSO) approach. Benchmark datasets as well as hypothetical datasets have been used for computational trials. The proposed heuristic is found to perform exceedingly well even for large problem instances, both in terms of quality of solutions and in terms of computational effort.

A two-stage tabu search algorithm with enhanced packing heuristics for the 3L-CVRP and M3L-CVRP

The Three-Dimensional Loading Capacitated Vehicle Routing Problem (3L-CVRP) addresses practical constraints frequently encountered in the freight transportation industry. In this problem, the task is to serve all customers using a homogeneous fleet of vehicles at minimum traveling cost. The constraints imposed by the three-dimensional shape of the goods, the unloading order, item fragility, and the stability of the loading plan of each vehicle are explicitly considered. We improved two well-known packing heuristics, namely the Deepest-Bottom-Left-Fill heuristic and the Maximum Touching Area heuristic, for the three-dimensional loading sub-problem and provided efficient implementations. Based on these two new heuristics, an effective tabu search algorithm is given to address the overall problem. Computational experiments on publicly available test instances show our new approach outperforms the current best algorithms for 20 out of 27 instances. Our approach is also superior to the existing algorithm on benchmark data for the closely related problem variant M3L-CVRP (which uses a slightly different unloading order constraint compared to 3L-CVRP).

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.