Models, relaxations and exact approaches for the capacitated vehicle routing problem (original) (raw)
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Lower Bounds and an Exact Method for the Capacitated Vehicle Routing Problem
2006 International Conference on Service Systems and Service Management, 2006
In this paper we consider the problem in which a fleet of M vehicles stationed at a central depot is to be optimally routed to supply customers with known demands subject to vehicle capacity constraints. This problem is referred as the Capacitated Vehicle Routing Problem (CVRP). We present an exact algorithm for solving the CVRP based on a Set Partitioning formulation of the problem. We describe a procedure for computing a valid lower bound to the cost of the optimal CVRP solution that finds a feasible solution of the dual of the LP-relaxation of the set partitioning formulation without generating the entire set partitioning matrix. The dual solution obtained is then used to limit the set of the feasible routes containing the optimal CVRP solutions. The resulting Set Partitioning problem is solved by using a branch and bound algorithm. Computational results are presented for a number of problems derived from the literature. The results show the effectiveness of the proposed method in solving problems up to about 100 customers.
Operations Research, 2004
The capacitated vehicle routing problem (CVRP) is the problem in which a set of identical vehicles located at a central depot is to be optimally routed to supply customers with known demands subject to vehicle capacity constraints. In this paper, we describe a new integer programming formulation for the CVRP based on a two-commodity network flow approach. We present a lower bound derived from the linear programming (LP) relaxation of the new formulation which is improved by adding valid inequalities in a cutting-plane fashion. Moreover, we present a comparison between the new lower bound and lower bounds derived from the LP relaxations of different CVRP formulations proposed in the literature. A new branch-and-cut algorithm for the optimal solution of the CVRP is described. Computational results are reported for a set of test problems derived from the literature and for new randomly generated problems.
Proceedings of the Institute for System Programming of the RAS, 2018
Vehicle Routing Problem (VRP) is one of the most widely known questions in a class of combinatorial optimization problems. It is concerned with the optimal design of routes to be used by a fleet of vehicles to serve a set of customers. In this study we analyze Capacitated Vehicle Routing Problem (CVRP)-a subcase of VRP, where the vehicles have a limited capacity. CVRP is mostly aimed at savings in the global transportation costs. The problem is NP-hard, therefore heuristic algorithms which provide near-optimal polynomial-time solutions will be considered instead of the exact ones. The aim of this article is to make a survey on mathematical formulations of CVRP and on methods for solving each type of this problem. The first part presents a general information about the problem and restrictions of this work. In the second part, the classical mathematical formulations of CVRP are described. In the third part, a classification of most popular subcases of CVRP is given, including description of additional constraints with their math formulations. This section also includes most perspective methods that can be applied for solving special types of CVRP. The forth part contains an important note about the most powerful algorithm LKH-3. Finally, the fourth part consists of table with solving techniques for each subproblem and of scheme with basic problems of the CVRP class and their interconnections.
A unified exact method for solving different classes of vehicle routing problems
Mathematical Programming, 2009
This paper presents a unified exact method for solving an extended model of the well-known Capacitated Vehicle Routing Problem (CVRP), called the Heterogenous Vehicle Routing Problem (HVRP), where a mixed fleet of vehicles having different capacities, routing and fixed costs is used to supply a set of customers. The HVRP model considered in this paper contains as special cases: the Single Depot CVRP, all variants of the HVRP presented in the literature, the Site-Dependent Vehicle Routing Problem (SDVRP) and the Multi-Depot Vehicle Routing Problem (MDVRP). This paper presents an exact algorithm for the HVRP based on the set partitioning formulation. The exact algorithm uses three types of bounding procedures based on the LP-relaxation and on the Lagrangean relaxation of the mathematical formulation. The bounding procedures allow to reduce the number of variables of the formulation so that the resulting problem can be solved by an integer linear programming solver. Extensive computational results over the main instances from the literature of the different variants of HVRPs, SDVRP and MDVRP show that the proposed lower bound is superior to the ones presented in the literature and that the exact algorithm can solve, for the first time ever, several test instances of all problem types considered.
On the capacitated vehicle routing problem
Mathematical …, 2003
We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Traveling Salesman Problem (TSP) as special cases and conceptually lies at the intersection of these two well-studied problems. The capacity constraints of the integer programming formulation of this routing model provide the link between the underlying routing and packing structures. We describe a decomposition-based separation methodology for the capacity constraints that takes advantage of our ability to solve small instances of the TSP efficiently. Specifically, when standard procedures fail to separate a candidate point, we attempt to decompose it into a convex combination of TSP tours; if successful, the tours present in this decomposition are examined for violated capacity constraints; if not, the Farkas Theorem provides a hyperplane separating the point from the TSP polytope. We present some extensions of this basic concept and a general framework within which it can be applied to other combinatorial models. Computational results are given for an implementation within the parallel branch, cut, and price framework SYMPHONY. *
Heuristic Procedures for the Capacitated Vehicle Routing Problem
Computational Optimization and Applications, 2000
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
An exact solution framework for a broad class of vehicle routing problems
Computational Management Science, 2010
This paper presents an exact solution framework for solving some variants of the vehicle routing problem (VRP) that can be modeled as set partitioning (SP) problems with additional constraints. The method consists in combining different dual ascent procedures to find a near optimal dual solution of the SP model. Then, a column-and-cut generation algorithm attempts to close the integrality gap left by the dual ascent procedures by adding valid inequalities to the SP formulation. The final dual solution is used to generate a reduced problem containing all optimal integer solutions that is solved by an integer programming solver. In this paper, we describe how this solution framework can be extended to solve different variants of the VRP by tailoring the different bounding procedures to deal with the constraints of the specific variant. We describe how this solution framework has been recently used to derive exact algorithms for a broad class of VRPs such as the capacitated VRP, the VRP with time windows, the pickup and delivery problem with time windows, 123 230 R. Baldacci et al. all types of heterogeneous VRP including the multi depot VRP, and the period VRP. The computational results show that the exact algorithm derived for each of these VRP variants outperforms all other exact methods published so far and can solve several test instances that were previously unsolved.
SR2: A Hybrid Algorithm for the Capacitated Vehicle Routing Problem
2008
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.
Mathematical Programming, 2008
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.
Recent advances in vehicle routing exact algorithms
4OR: A Quarterly Journal of Operations …, 2007
5 Comparison of various VRP relaxations 6 Branch-and-cut methods Separation algorithms Branching strategies 7 Branch-and-cut-and-price method R. Baldacci (DEIS) Exact Algorithms for the VRP May 12, 2008 2 / 66 Outline (2) Pricing and cut generation 8 Set partitioning with additional cuts Finding an optimal VRP solution Bounding procedure Route generation algorithm 9 Summary of the computational experiments 10 Appendix 11 References R. Baldacci (DEIS) Exact Algorithms for the VRP May 12, 2008 3 / 66 Problem description