Capacitated Vehicle Routing Problem

This paper presents two GRASP metaheuristic algorithms for the vehicle routing problem, considering the capacity and shared demand of the customers. In this paper the solution obtained is compared with a greedy solution and two hybrid solutions (greedy and random). The results obtained show that the GRASP algorithm obtains a better quality solution for this kind of problem.

Rezime: Cilj ovog rada je da na osnovu podataka dobijenih iz JKSP "Komstan" Trstenik izvrši rutiranje vozila u skladu sa postojećim kapacitetima (brojem i nosivošću drumskih vozila -smećara u upotrebi) i na osnovu podataka o količini smeća po ulicama grada koje je potrebno pokupiti. Ovaj problem rutiranja vozila pripada problemima rutiranja vozila sa ograničenim kapacitetom. Klasa problema pripada grupi problema pod nazivom: Kineski poštar. Kako bi se dobila Ojlerova ruta, razvijen je jednostavan ali praktičan pristup za sparivanje čvorova transportne mreže Trstenika. Na kraju, prikazane su rute vozila. Eventualne benefite predloženog modela u odnosu na postojeće stanje nije moguće precizno odrediti zbog načina poslovanja privrednog društva, gde ne postoji ustaljeni plan ruta vozila jer se svakodnevno radi prema stanju na terenu.

This paper presents a methodology based on the Variable Neighbourhood Search metaheuristic, applied to the Capacitated Vehicle Routing Problem. The presented approach uses Constraint Programming and Lagrangean Relaxation methods in order to improve algorithm’s efficiency. The complete problem is decomposed into two separated subproblems, to which the mentioned techniques are applied to obtain a complete solution. With this decomposition, the methodology provides a quick initial feasible solution which is rapidly improved by metaheuristics’ iterative process. Constraint Programming and Lagrangean Relaxation are also embedded within this structure to ensure constraints satisfaction and to reduce the calculation burden. By means of the proposed methodology, promising results have been obtained. Remarkable results presented in this paper include a new best-known solution for a rarely solved 200-customers test instance, as well as a better alternative solution for another benchmark problem.

This paper presents an original hybrid approach to solve the Capacitated Vehicle Routing Problem (CVRP). The approach combines a Probabilistic Algorithm with Constraint Programming (CP) and Lagrangian Relaxation (LR). After introducing the CVRP and reviewing the existing literature on the topic, the paper proposes an approach based on a probabilistic Variable Neighbourhood Search (VNS) algorithm. Given a CVRP instance, this algorithm uses a randomized version of the classical Clarke and Wright Savings constructive heuristic to generate a starting solution. This starting solution is then improved through a local search process which combines: (a) LR to optimise each individual route, and (b) CP to quickly verify the feasibility of new proposed solutions. The efficiency of our approach is analysed after testing some well-known CVRP benchmarks. Benefits of our hybrid approach over already existing approaches are also discussed. In particular, the potential flexibility of our methodology is highlighted.

In this paper, we consider a variant of the transportation problem where any demand may be dropped off elsewhere than at its destination, picked up later by the same or another vehicle, and so on until it has reached its destination. We present two mixed integer linear programming formulations based on a space-time graph. We also develop a branch-and-cut algorithm for the problem and present some computational results.

In this paper we present SR-1, a simulation-based heuristic algorithm for the Capacitated Vehicle Routing Problem (CVRP). Given a CVRP instance, SR-1 uses an initial "good solution", such as the one provided by the classical Clarke and Wright heuristic, in order to obtain observations for the variable "distance between two consecutive nodes in a route". These observations are then fitted by a statistical distribution, which characterizes the inter-node distances in good solutions. Then, the fitted distribution is employed to generate a large number of new random solutions with similar edge-size distribution. Thus, a random but oriented local search of the space of solutions is performed, and a list of "best solutions" is obtained. This list allows considering several properties per solution, not only aprioristic costs, which can be practically used when making multiple-criteria decisions. Several tests have been performed to discuss the effectiveness of this approach.

The aim of this paper is to further study the rich vehicle routing problem (RVRP), which is a well-known combinatorial optimization problem arising in many transportation and logistics settings. This problem is known to be subject to a number of real life constraints, such as the number and capacity limitation of vehicles, time constraints including ready and due dates for each customer, heterogeneous vehicle fleets and different warehouses for vehicles. A Genetic Algorithm (GA)-based approach is proposed to tackle this highly constrained problem. The proposed approach efficiently resolves the problem despite its high complexity. To the best of our knowledge, no GA have been used for solving multi-depot heterogeneous limited fleet VRP with time windows so far. The new algorithm has been tested on benchmark and real-world instances. In fact, promising computational results have shown its good cost-effectiveness.

We consider two types of hop-indexed models for the unit-demand asymmetric Capacitated Vehicle Routing Problem (CVRP): (a) capacitated models guaranteeing that the number of commodities (paths) traversing any given arc does not exceed a specified capacity; and (b) hop-constrained models guaranteeing that any route length (number of nodes) does not exceed a given value. The latter might, in turn, be divided into two classes: (b1) those restricting the length of the path from the depot to any node k, and (b2) those restricting the length of the circuit passing through any node k. Our results indicate that formulations based upon circuit lengths (b2) lead to models with a linear programming relaxation that is tighter than the linear programming relaxation of models based upon path lengths (b1), and that combining features from capacitated models with those of circuit lengths can lead to formulations for the CVRP with a tight linear programming bound. Computational results on a small number of problem instances with up to 41 nodes and 440 edges show that the combined model with capacities and circuit lengths produce average gaps of less than one percent. We also briefly examine the asymmetric travelling salesman problem (ATSP), showing the potential use of the ideas developed for the vehicle routing problem to derive models for the ATSP with a linear programming relaxation bound that is tighter than the linear programming relaxation bound of the standard Dantzig, Fulkerson and Johnson [G. Dantzig, D. Fulkerson, D. Johnson, Solution of large-scale travelling salesman problem, Operations Research 2 (1954) 393-410] formulation.

The Traveling Repairman Problem is a customer-oriented routing problem in which a repairman is visiting a set of geographically distributed customers. The objective function is to minimize the total waiting times of all customers. The importance of this problem can be found in its applications in the following areas: blood distributing, manufacturing systems, and transportation and logistics. Apart from its importance, research on this problem is very limited. In this paper a new mixed-integer programming formulation is developed, and several properties of model are studied. Additionally, by developing lower and upper bounds, a branch and bound algorithm is developed to solve the problems with up to 30 nodes. According to the computational experiments, the developed model is very competitive.

Vehicle Routing Problem (VRP) has an important role in logistics distribution from the depot to the customer, to get the minimum cost delivery route. To get optimal results, it is necessary to improve route from the initial solution. Variable Neighbourhood Descent (VND) is one of the metaheuristics that examine of a number of neighbourhood operators to get the optimal route. A VRP route is called optimal if there are no other routes that can be generated from all the neighbourhood operators used in VND. This article describes the application of VND to get the optimal route on CVRP, MDVRP, and VRPTW. The results of the experiment on some test data used indicate that VND can be used to get more optimal length and travel time route.

In this paper we seek to determine optimal routes for a containership fleet performing pickups and deliveries between a hub and several spoke ports. A capacitated vehicle routing problem with pickups , deliveries and time deadlines is formulated and solved using a hybrid genetic algorithm for establishing routes for a dedicated containership fleet. Results on the performance of the algorithm and the feasibility of the approach show that a relatively small fleet of containerships could provide efficient services within deadlines. Moreover, through sensitivity analysis we discuss performance robustness and consistency of the developed algorithm under a variety of problem settings and parameters values.

This paper analyzes the integration of two combinatorial problems that frequently arise in production and distribution systems. One is the Bin Packing Problem (BPP) problem, which involves finding an ordering of some objects of different volumes to be packed into the minimal number of containers of the same or different size. An optimal solution to this NP-Hard problem can be approximated by means of meta-heuristic methods. On the other hand, we consider the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW), which is a variant of the Travelling Salesman Problem (again a NP-Hard problem) with extra constraints. Here we model those two problems in a single framework and use an evolutionary meta-heuristics to solve them jointly. Furthermore, we use data from a real world company as a test-bed for the method introduced here.

The vehicle routing problem consists in determining a group of optimal delivery routes that allow satisfaction of the demand from customers spread along a network, which can be represented as a graph type structure. This paper presents two GRASP metaheuristic algorithms for the above mentioned problem, considering the capacity and shared demand of the customers, solving besides, the 3D bin packing problem within of such vehicles. In this paper the solution obtained is compared with a greedy solution and two hybrid solutions (greedy and random). The results obtained show that the GRASP algorithm obtains a better quality solution for this kind of problem.

This paper presents two GRASP metaheuristic algorithms for the vehicle routing problem, considering the capacity and shared demand of the customers. In this paper the solution obtained is compared with a greedy solution and two hybrid solutions (greedy and random). The results obtained show that the GRASP algorithm obtains a better quality solution for this kind of problem.

In this paper we present two new heuristic procedures for the Capacitated Vehicle Routing Problem (CVRP). The first one solves the problem from scratch, while the second one uses the information provided by a strong linear relaxation of the original problem. This second algorithm is designed to be used in a branch and cut approach to solve to optimality CVRP

Matrices with distances between pairs of locations are essential for solving vehicle routing problems like the Capacitated Vehicle Routing Problem (CV RP), Traveling Salesman Problem (T SP) and others. This work deals with the complex reality of transportation networks and asymmetry. Through a series of comprehensive and thorough computational and statistical experiments we study the effect that many factors like asymmetry, geographical location of the depot and clients, demand, territory and maximum vehicle capacity have in the solution of CV RP instances. We examine both classical heuristics as well as current state-of-the-art metaheuristics and show that these methods are seriously affected by the studied factors from a solution time and quality of solutions perspective. We systematically compare the solutions obtained in the symmetric scenario with those obtained in the real asymmetric case at a quantitative as well as a qualitative level, with the objective of carefully measuring and understanding the differences between both cases.

Stručni rad Rezime: Cilj ovog rada je da na osnovu podataka dobijenih iz JKSP "Komstan" Trstenik izvrši rutiranje vozila u skladu sa postojećim kapacitetima (brojem i nosivošću drumskih vozila-smećara u upotrebi) i na osnovu podataka o količini smeća po ulicama grada koje je potrebno pokupiti. Ovaj problem rutiranja vozila pripada problemima rutiranja vozila sa ograničenim kapacitetom. Klasa problema pripada grupi problema pod nazivom: Kineski poštar. Kako bi se dobila Ojlerova ruta, razvijen je jednostavan ali praktičan pristup za sparivanje čvorova transportne mreže Trstenika. Na kraju, prikazane su rute vozila. Eventualne benefite predloženog modela u odnosu na postojeće stanje nije moguće precizno odrediti zbog načina poslovanja privrednog društva, gde ne postoji ustaljeni plan ruta vozila jer se svakodnevno radi prema stanju na terenu.

The Capacitated vehicle routing problem (CVRP) is a complex transportation issue that involves designing a set of routes for a group of homogenous vehicles to serve a set of customers at shortest distance travelled. It belongs to the class of NP-hard problems with a high computational complexity, and then it is difficult to solve this problem with traditional optimization methods when the problem size is large. Consequently, in this study, an improved swarm optimization metaheuristic approach has been proposed based on intelligent foraging behavior of honey bees with several specifically designed local search operators and features to solve this problem. The effectiveness of the proposed approach was illustrated on the standard benchmark problems. The results revealed that the proposed approach has fast convergence rate and high computational accuracy. Overall, our experiments show that our approach could be a very efficient approach for solving the CVRP and its experimental results are competitive in terms of the quality of the solutions reported in other well-known methods.

The capacitated vehicle routing problem (CVRP) with a single depot is a classic routing problem with numerous real-world applications. This paper describes the design, modelling and computational aspects of ALGELECT (electrostatic algorithm), a new algorithm for the CVRP. After some general remarks about the origin of the algorithm and its parameters, a parameter tuning process is carried out in order to improve its efficiency. The algorithm is then explained in detail and its main characteristics are presented. Thus, ALGELECT develops good-quality solutions to the CVRP, in terms of the number of scheduled routes and the load ratio of the delivery vehicles. Finally, ALGELECT is used to find solutions for some Solomon's and Augerat's instances, which are then compared to solutions generated by other well-known methods.

Recently proved successful for variants of the vehicle routing problem (VRP) involving time windows, genetic algorithms have not yet shown to compete or challenge current best search techniques in solving the classical capacitated VRP. In this paper, a hybrid genetic algorithm to address the capacitated vehicle routing problem is proposed. The basic scheme consists in concurrently evolving two populations of solutions to minimize total traveled distance using genetic operators combining variations of key concepts inspired from routing techniques and search strategies used for a time-variant of the problem to further provide search guidance while balancing intensification and diversification. Results from a computational experiment over common benchmark problems report the proposed approach to be very competitive with the best-known methods.

The Vehicle Routing Problem (VRP) deals with the assignment of a set of transportation orders to a fleet of vehicles, and the sequencing of stops for each vehicle to minimize transportation costs. In this paper we study the Capacitated VRP (CVRP), which is mainly characterized by using vehicles of the same capacity. Taking a basic GA to solve the CVRP, we propose a new problem dependent recombination operator, called Best Route Better Adjustment recombination (BRBAX). A comparison of its performance is carried out with respect to other two classical recombination operators. Also we conduct a study of different mutations in order to determine the best combination of genetic operators for this problem. The results show that the use of our specialized BRBAX recombination outperforms the others more generic on all problem instances used in this work for all the metrics tested.

The Vehicle Routing Problem (VRP) deals with the assignment of a set of transportation orders to a fleet of vehicles, and the sequencing of stops for each vehicle to minimize transportation costs. In this paper we study the Capacitated VRP (CVRP), which is mainly characterized by using vehicles of the same capacity. Taking a basic GA to solve the CVRP, we propose a new problem dependent recombination operator, called Best Route Better Adjustment recombination (BRBAX). A comparison of its performance is carried out with respect to other two classical recombination operators. Also we conduct a study of different mutations in order to determine the best combination of genetic operators for this problem. The results show that the use of our specialized BRBAX recombination outperforms the others more generic on all problem instances used in this work for all the metrics tested.

The Capacitated Vehicle Routing Problem (CVRP) plays an important role in the logistics transportation sector. Determining the proper route will reduce the company's operational costs. In CVRP, a number of vehicles have a capacity limit that can serve all customers. This research completes a real case study on a bottled drinking water company where the company still uses the subjective method of the driver to determine the transportation route. Based on the conditions in the company, the selection of the best route will consider vehicle capacity and demand to determine the shortest route. The execution of this case study uses the Football Game Algorithm (FGA) which was first initiated by Fadakar & Ebrahimi which proved promising and had the strongest performance in all cases. FGA is expected to be able to determine the shortest distribution route from the existing cases to reduce the distribution costs incurred. This study takes data from 4 days of delivery that served 78 customers. The average daily transportation cost savings result is 42%. This amount indicates that the FGA algorithm is effective for completing a real case study in CVRP.

In this paper we review the exact algorithms based on the branch and bound approach proposed in the last years for the solution of the basic version of the vehicle routing problem (VRP), where only the vehicle capacity constraints are considered. These algorithms have considerably increased the size of VRPs that can be solved with respect to earlier approaches. Moreover, at least for the case in which the cost matrix is asymmetric, branch and bound algorithms still represent the state-of-the-art with respect to the exact solution. Computational results comparing the performance of di erent relaxations and algorithms on a set of benchmark instances are presented. We conclude by examining possible future directions of research in this ÿeld.

The aim of this paper is to further study the rich vehicle routing problem (RVRP), which is a well-known combinatorial optimization problem arising in many transportation and logistics settings. This problem is known to be subject to a number of real life constraints, such as the number and capacity limitation of vehicles, time constraints including ready and due dates for each customer, heterogeneous vehicle fleets and different warehouses for vehicles. A Genetic Algorithm (GA)-based approach is proposed to tackle this highly constrained problem. The proposed approach efficiently resolves the problem despite its high complexity. To the best of our knowledge, no GA have been used for solving multi-depot heterogeneous limited fleet VRP with time windows so far. The new algorithm has been tested on benchmark and real-world instances. In fact, promising computational results have shown its good cost-effectiveness.

The Capacitated General Routing Problem (CGRP) on mixed graphs is one of the more complex combinatorial optimization problems on vehicle routing. It consists basically of finding a set of routes on a mixed graph, beginning and ending at the same vertex (depot), with minimum total cost, satisfying demands located at links and vertices and with a capacity restriction on the demand satisfied by each route. Several particular cases of this problem have deeply been studied in the operational research literature but, in order to solve the general problem, we only have found a heuristic procedure based on route-first-partition-next. We present here a new heuristic that seems to work much better according to our computational results.

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-®rst-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 ®nal set of feasible routes is built through a modi®ed 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 eectiveness of the proposed approach.

The Capacitated General Routing Problem (CGRP) on mixed graphs is one of the more complex combinatorial optimization problems on vehicle routing. It consists basically of finding a set of routes on a mixed graph, beginning and ending at the same vertex (depot), with minimum total cost, satisfying demands located at links and vertices and with a capacity restriction on the demand satisfied by each route. Several particular cases of this problem have deeply been studied in the operational research literature but, in order to solve the general problem, we only have found a heuristic procedure based on route-first-partition-next. We present here a new heuristic that seems to work much better according to our computational results.

Branch and Cut methods have shown to be very successful in the resolution of some hard combinatorial optimization problems. The success has been remarkable for the Symmetric Traveling Salesman Problem (TSP). The crucial part in the method is the cutting plane algorithm: the algorithm that looks for valid inequalities that cut off the current nonfeasible linear program (LP) solution. In turn this part relies on a good knowledge of the corresponding polyhedron and our ability to design algorithms that can identify violated valid inequalities. This paper deals with the separation of the capacity constraints for the Capacitated Vehicle Routing Problem (CVRP). Three algorithms are presented: a constructive algorithm, a randomized greedy algorithm and a very simple tabu search procedure. As far as we know this is the first time a metaheuristic is used in a separation procedure. The aim of this paper is to present this application. No advanced tabu technique is used. We report computational results with these heuristics on difficult instances taken from the literature as well as on some randomly generated instances. These algorithms were used in a Branch and Cut procedure that successfully solved to optimality large CVRP instances.

Collaborative transportation, as an emerging new mode, represents one of the major developing trends of transportation systems. Focusing on the full truckloads multi-depot capacitated vehicle routing problem in carrier collaboration, this paper proposes a mathematical programming model and its corresponding graph theory model, with the objective of minimizing empty vehicle movements. A two-phase greedy algorithm is given to solve practical large-scale problems. In the first phase, a set of directed cycles is created to fulfil the transportation orders. In the second phase, chains that are composed of cycles are generated. Furthermore, a set of local search strategies is put forward to improve the initial results. To evaluate the performance of the proposed algorithms, two lower bounds are developed. Finally, computational experiments on various randomly generated problems are conducted. The results show that the proposed methods are effective and the algorithms can provide reasonable solutions within an acceptable computational time.

In this paper, we study a non-capacitated Vehicle Routing Problem (VRP) where not necessarily all clients need to be visited and the goal is to minimize the length of the longest vehicle route. An Integer Programming Formulation, a Branch-and-cut (BC) method and a Local Branching (LB) framework that uses BC as the inner solver are presented. Sharper upper bounds are obtained by LB, when the same time limit was imposed on the execution times of both approaches. Our results also suggest that the min-max nature of the objective function combined with the fact that not all vertices need to be visited make such problem very difficult to solve.

In this paper we present two new heuristic procedures for the Capacitated Vehicle Routing Problem (CVRP). The first one solves the problem from scratch, while the second one uses the information provided by a strong linear relaxation of the original problem. This second algorithm is designed to be used in a branch and cut approach to solve to optimality CVRP

This paper presents a methodology based on the Variable Neighbourhood Search metaheuristic, applied to the Capacitated Vehicle Routing Problem. The presented approach uses Constraint Programming and Lagrangean Relaxation methods in order to improve algorithm’s efficiency. The complete problem is decomposed into two separated subproblems, to which the mentioned techniques are applied to obtain a complete solution. With this decomposition, the methodology provides a quick initial feasible solution which is rapidly improved by metaheuristics’ iterative process. Constraint Programming and Lagrangean Relaxation are also embedded within this structure to ensure constraints satisfaction and to reduce the calculation burden. By means of the proposed methodology, promising results have been obtained. Remarkable results presented in this paper include a new best-known solution for a rarely solved 200-customers test instance, as well as a better alternative solution for another benchmark problem.

This paper presents a methodology based on the Variable Neighbourhood Search metaheuristic, applied to the Capacitated Vehicle Routing Problem. The presented approach uses Constraint Programming and Lagrangean Relaxation methods in order to improve algorithm’s efficiency. The complete problem is decomposed into two separated subproblems, to which the mentioned techniques are applied to obtain a complete solution. With this decomposition, the methodology provides a quick initial feasible solution which is rapidly improved by metaheuristics’ iterative process. Constraint Programming and Lagrangean Relaxation are also embedded within this structure to ensure constraints satisfaction and to reduce the calculation burden. By means of the proposed methodology, promising results have been obtained. Remarkable results presented in this paper include a new best-known solution for a rarely solved 200-customers test instance, as well as a better alternative solution for another benchmark problem.

This paper investigates the capacitated green vehicle routing problem (GVRP) with time-varying vehicle speed and soft time windows. The GVRP is developed as a multi-objective mixed integer nonlinear programming (MINLP) model that incorporates a fuel consumption calculation algorithm. The proposed model considers the vehicle load and capacity as well as time-varying speed in order to account for traffic congestion. An improved non-dominated sorting genetic algorithm (NSGA-II) with adaptive strategies and greedy strategies is developed to solve the GVRP. The results of numerical experiments show that the consumption of fuel in a supply chain can be decreased sharply without any significant loss in customer satisfaction. The proposed NSGA-II has a better capability and efficiency than the original NSGA-II. Our experiments also indicate that the proposed model outperforms most of the best-known solutions obtained from the traditional modeling approaches.

During the eigthies and early nineties, the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) utilized lower bounds obtained by Lagrangean relaxation or column generation. Next, the advances in the polyhedral description of the CVRP yielded branch-andcut algorithms giving better results. However, several instances in the range of 50-80 vertices, some proposed more than 30 years ago, can not be solved with current known techniques. This paper presents an algorithm utilizing a lower bound obtained by minimizing over the intersection of the polytopes associated to a traditional Lagrangean relaxation over q-routes and the one defined by bounds, degree and the capacity constraints. This is equivalent to a linear program with an exponential number of both variables and constraints. Computational experiments show the new lower bound to be superior to the previous ones, specially when the number of vehicles is large. The resulting branch-and-cut-and-price could solve to optimality almost all instances from the literature up to 100 vertices, nearly doubling the size of the instances that can be consistently solved. Further progress in this algorithm may be soon obtained by also using other known families of inequalities. A landmark exact algorithm for the CVRP was presented in 1981 by Christofides, Mingozzi and Toth [16] using a Lagrangean bound obtained by solving minimum q-route problems. A q-route starts at the depot and traverses a sequence of clients limiting the demand accumulated to at most C and returns to the depot. It is not necessarily a simple path, for some clients may be visited more than once. Therefore, the set of valid CVRP routes is contained in the set of q-routes. The resulting branch-and-bound could solve instances up to 25 vertices, a quite respectful size at that time. Several other branch-and-bound algorithms using Lagrangean bounds appear in the literature. This same article [16] also describes a lower bound based on k-degree center trees, minimum spanning trees having degree K ≤ k ≤ 2K on the depot, plus 2K − k least cost edges. Other authors propose Lagrangean bounds based on K-trees, which are sets of n + K edges spanning G, like Fisher [21] and Martinhon, Lucena and Maculan [29]. There is also an algorithm based on minimum b-matchings having degree 2K at the depot and 2 on the remaining vertices by Miller [30]. The Lagrangean bounds can be improved by dualizing capacity inequalities [21, 30] and also combs and multistar inequalities [29]. Another kind of exact algorithms have its stem on the formulation of the CVRP as a set partition problem by Balinsky and Quandt [9]. In that formulation, a column covers a set of vertices S with total demand not exceeding C and have the cost of a minimum route over {0} ∪ S. Bramel and Simchi-Levi [13] proved that for certain natural classes of instances, the ratio between the lower bounds given by that formulation and the optimal solution values asymptotically approaches 1 as the number of clients grows. However, that formulation in itself is not practical because pricing over the exponential number of columns require the solution of capacitated prize-collecting TSPs, a problem almost as difficult as the CVRP itself. Agarwal, Marthur and Salkin [3] proposed a column generation algorithm on a modified set partition where column costs are given by a linear function over the vertices yielding a lower bound on the actual route cost. Columns with the modified cost can be priced by solving easy knapsack problems. Hadjconstantinou et al. [23] derive lower bounds from heuristic solutions to the dual of the set partitioning formulation. Those dual solutions are obtained by the so-called additive approach, combining the q-path with the K-shortest path relaxations. For further information and some comparative results on the above mentioned algorithms, we refer the reader to the survey by Toth and Vigo [37]. Starting in the 90's, most of the research effort on the CVRP is now concentrated on the polyhedral description of convex hull of the edge incidence vectors that correspond to K feasible routes and on the development of effective separation algorithms for the families of valid inequalities identified (for instance, [4, 14, 18, 6, 7, 2, 31, 27]). In particular, Araque et al. [5], Augerat et al. [8], Blasum and Hochstattler [12], Ralphs et al. [36], Achutan, Caccetta and Hill [1] and Lysgard, Letchford and Eglese [28] describe complete branch-and-cut algorithms, some including sophisticated inequalities such as framed capacity, strengthened combs, multistar, among others. (The taxonomy and nomenclature of valid inequalities for the CVRP is not uniform in the literature, we are following [28] in this article.) Although those are the best exact algorithms currently available for the CVRP, the lower bounds obtained at the root nodes, even after adding all those cuts, are not very tight for many instances with as little as 40 vertices. The quality of those bounds is specially problematic for larger values of K, say K ≥ 7. Many nodes in a branch-and-cut tree may have to be explored in order to close the resulting duality gaps. Even resorting to massive computational power (up to 80 processors running in parallel in a recent work by Ralphs [36, 35]) several instances with less than 80 vertices, including some proposed more than 30 years ago by Christofides and Eilon [15], can not be solved at all. In fact, it seems that branch-and-cut algorithms for the CVRP are experimenting a "diminishing returns" phenomenon, where substantial theoretical and implementation efforts leads to practical results that are only marginally better than those of previous works. This work presents a new exact algorithm for the CVRP that seems to break through this situation. The main idea is to combine the branch-and-cut approach with the old q-routes approach (which we interpret as a column generation instead of the original Lagrangean relaxation)

A well known transformation by Pearn, Assad and Golden reduces a Capacitated Arc Routing Problem (CARP) into an equivalent Capacitated Vehicle Routing Problem (CVRP). However, that transformation is regarded as unpractical, since an original instance with r required edges is turned into a CVRP over a complete graph with 3r + 1 vertices. We propose a similar transformation that reduces this graph to 2r + 1 vertices, with the additional restriction that r edges are already fixed to 1. Using a recent branch-and-cut-and-price algorithm for the CVRP, we observed that it yields an effective way of attacking the CARP, being significantly better than the exact methods created specifically for that problem. Computational experiments obtained improved lower bounds for almost all open instances from the literature. Several such instances could be solved to optimality.

One of the main characteristics of transportation problems is not only the hard design model process; but also, in the characterization of the elements that make up the supply chain. A more detailed analysis will grant undoubtedly a greater knowledge of the system. However, the use of regular networks for the development of models can eliminate key characteristics for correct decision making. Faced with this fact, the characterization of transport models is proposed through the conceptualization of complex networks. This new proposal will allow to apply metrics and perform more detailed analysis to the distribution network; such as vulnerability analysis, fault propagation, element grouping identification, among other metrics. The objective of this research is to show a methodology to build the Vehicle Routing Problem models and analyze the distribution network using the corresponding metrics of complex networks.

In this work, a new variant of the Capacitated Vehicle Routing Problem (CVRP) is presented where the vehicles cannot perform any route leg longer than a given length L (although the routes can be longer). Thus, once a route leg length is close to L, the vehicle must go to a stop node to end the leg or return to the depot. We introduce this condition in a variation of the CVRP, the Split Delivery Vehicle Routing Problem, where multiple visits to a customer by different vehicles are allowed. We present two formulations for this problem which we call Split Delivery Vehicle Routing Problem with Stop Nodes: a vehicle flow formulation and a commodity flow formulation. Because of the complexity of this problem, a heuristic approach is developed. We compare its performance with and without the stop nodes

In this paper we present two new heuristic procedures for the Capacitated Vehicle Routing Problem (CVRP). The first one solves the problem from scratch, while the second one uses the information provided by a strong linear relaxation of the original problem. This second algorithm is designed to be used in a branch and cut approach to solve to optimality CVRP

During the last decades a lot of work has been devoted to develop algorithms that can provide nearoptimal solutions for the Capacitated Vehicle Routing Problem (CVRP). Most of these algorithms are designed to minimize an objective function, subject to a set of constraints, which typically represents aprioristic costs. This approach provides adequate theoretical solutions, but they do not always fit reallife needs since there are some important costs and some routing constraints or desirable properties that cannot be easily modeled. In this paper, we present a new approach which combines the use of Monte Carlo Simulation and Parallel and Grid Computing techniques to provide a set of alternative solutions to the CVRP. This allows the decision-maker to consider multiple solution characteristics other than just aprioristic costs. Therefore, our methodology offers more flexibility during the routing selection process, which may help to improve the quality of service offered to clients.

In this paper we present SR-1, a simulation-based heuristic algorithm for the Capacitated Vehicle Routing Problem (CVRP). Given a CVRP instance, SR-1 uses an initial "good solution", such as the one provided by the classical Clarke and Wright heuristic, in order to obtain observations for the variable "distance between two consecutive nodes in a route". These observations are then fitted by a statistical distribution, which characterizes the inter-node distances in good solutions. Then, the fitted distribution is employed to generate a large number of new random solutions with similar edge-size distribution. Thus, a random but oriented local search of the space of solutions is performed, and a list of "best solutions" is obtained. This list allows considering several properties per solution, not only aprioristic costs, which can be practically used when making multiple-criteria decisions. Several tests have been performed to discuss the effectiveness of this approach.

The Capacitated Vehicle Routing Problem (CVRP) is a well known problem which has long been tackled by researchers for several decades now, not only because of its potential applications but also due to the fact that CVRP can be used to test the efficiency of new algorithms and optimization methods. The objective of our work is to present SR-GCWS, a hybrid algorithm that combines a CVRP classical heuristic with Monte Carlo simulation using state-of-the-art random number generators. The resulting algorithm is tested against some well known benchmarks. In most cases, our approach is able to compete or even outperform much more complex algorithms, which is especially interesting if we consider that our algorithm does not require any previous parameter fine-tuning or setup process. Moreover, our algorithm has been able to produce high-quality solutions almost in real-time for most tested instances. Another important feature of the algorithm worth mentioning is that it uses a randomized constructive heuristic, capable of generating hundreds or even thousands of alternative solutions with different properties. These alternative solutions, in turn, can be really useful for decisionmakers in order to satisfy their utility functions, which are usually unknown by the modeler. The presented methodology may be a fine framework for the development of similar algorithms for other complex combinatorial problems in the routing arena as well as in some other research fields.

The design of course timetables for academic institutions is a very difficult job due to the huge number of possible feasible timetables with respect to the problem size. This process contains lots of constraints that must be taken into account and a large search space to be explored, even if the size of the problem input is not significantly large. Different heuristic approaches have been proposed in the literature in order to solve this kind of problem. One of the efficient solution methods for this problem is tabu search. Different neighborhood structures based on different types of move have been defined in studies using tabu search. In this paper, different neighborhood structures on the operation of tabu search are examined. The performance of different neighborhood structures is tested over eleven benchmark datasets. The obtained results of every neighborhood structures are compared with each other. Results obtained showed the disparity between each neighborhood structures and another in terms of penalty cost.

Problem usmjeravanja vozila je problem kombinatorne optimizacije nastao na temelju potrebe za optimizacijom transporta u suvremenom svijetu. Jedan od načina za rješavanje ovog problema primjena je algoritma optimizacije kolonijom mrava kao popularne metaheuristike. Detaljnije je promatran kapacitativni problem usmjeravanja vozila zajedno s elitističkim mravljim sustavom na kojima se temelji ostvareno programsko rješenje problema. Programsko rješenje kreirano je u programskom jeziku C#. Objašnjen je način rada programskog rješenja i njegova uporaba. Provedena eksperimentalna analiza prikazala je kako različite vrijednosti parametara EAS algoritma utječu na kvalitetu pronađenih rješenja kapacitativnog problema usmjeravanja vozila.The vehicle routing problem is a combinatorial optimization problem based on the need for transport optimization in the modern world. One way of solving this problem is the use of ant colony optimization algorithm as a popular metaheuristic. Capacitated vehic...

Order picking consists in retrieving products from storage locations to satisfy independent orders from multiple customers. It is generally recognized as one of the most significant activities in a warehouse (Koster et al, 2007). In fact, order picking accounts up to 50% (Frazelle, 2001) or even 80% (Van den Berg, 1999) of the total warehouse operating costs. The critical issue in today's business environment is to simultaneously reduce the cost and increase the speed of order picking. In this paper, we address the order picking process in one of the Portuguese largest companies in the grocery business. This problem was proposed at the 92 nd European Study Group with Industry (ESGI92). In this setting, each operator steers a trolley on the shop floor in order to select items for multiple customers. The objective is to improve their grocery e-commerce and bring it up to the level of the best international practices. In particular, the company wants to improve the routing tasks in order to decrease distances. For this purpose, a mathematical model for a faster open shop picking was developed. In this paper, we describe the problem, our proposed solution as well as some preliminary results and conclusions.

This paper presents a new exact algorithm for the Capacitated Vehicle Routing Problem (CVRP) based on the set partitioning formulation with additional cuts that correspond to capacity and clique inequalities. The exact algorithm uses a bounding procedure that finds a near optimal dual solution of the LP-relaxation of the resulting mathematical formulation by combining three dual ascent heuristics. The first dual heuristic is based on the q-route relaxation of the set partitioning formulation of the CVRP. The second one combines Lagrangean relaxation, pricing and cut generation. The third attempts to close the duality gap left by the first two procedures using a classical pricing and cut generation technique. The final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between an upper bound and the lower bound achieved. The resulting problem is solved by an integer programming solver. Computational results over the main instances from the literature show the effectiveness of the proposed algorithm.