An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts

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.      

Branch-and-cut algorithms for the split delivery vehicle routing problem

In this paper we present two exact branch-and-cut algorithms for the Split Delivery Vehicle Routing Problem (SDVRP) based on two relaxed formulations that provide lower bounds to the optimum. Procedures to obtain feasible solutions to the SDVRP from a feasible solution to the relaxed formulations are presented. Computational results are presented for 4 classes of benchmark instances. The new approach is able to prove the optimality of 17 new instances. In particular, the branch-and-cut algorithm based on the first relaxed formulation is able to solve most of the instances with up to 50 customers and two instances with 75 and 100 customers.     

A branch-and-cut-and-price algorithm for the cumulative capacitated vehicle routing problem

In this paper we consider the Cumulative Capacitated Vehicle Routing Problem (CCVRP), which is a variation of the well-known Capacitated Vehicle Routing Problem (CVRP). In this problem, the traditional objective of minimizing total distance or time traveled by the vehicles is replaced by minimizing the sum of arrival times at the customers. We propose a branch-and-cut-and-price algorithm for obtaining optimal solutions to the problem. To the best of our knowledge, this is the first published exact algorithm for the CCVRP. We present computational results based on a set of standard CVRP benchmarks and investigate the effect of modifying the number of vehicles available.     

Heuristic Procedures for the Capacitated Vehicle Routing 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 instances. In both heuristics, the initial solution is improved using tabu search techniques. Computational results over a set of known instances, most of them with a proved optimal solution, are given.     

Robust Branch-and-Cut-and-Price for the Capacitated Vehicle Routing Problem

The best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) have been based on either branch-and-cut or Lagrangean relaxation/column generation. This paper presents an algorithm that combines both approaches: it works over the intersection of two polytopes, one associated with a traditional Lagrangean relaxation over q-routes, the other defined by bound, degree and capacity constraints. This is equivalent to a linear program with exponentially many variables and constraints that can lead to lower bounds that are superior to those given by previous methods. The resulting branch-and-cut-and-price algorithm can solve to optimality all instances from the literature with up to 135 vertices. This more than doubles the size of the instances that can be consistently solved.     

Improved lower bounds for the Split Delivery Vehicle Routing Problem

This paper presents an algorithm to obtain lower bounds for the Split Delivery Vehicle Routing Problem. An extended formulation over a large set of variables is provided and valid inequalities are identified. The algorithm combined column and cut generation and improved the best known lower bounds for all instances from the literature. Some reasonably sized instances are solved to optimality for the first time.     

An Ant Colony Algorithm for the Capacitated Vehicle Routing

The Vehicle Routing Problem (VRP) requires the determination of an optimal set of routes for a set of vehicles to serve a set of customers. We deal here with the Capacitated Vehicle Routing Problem (CVRP) where there is a maximum weight or volume that each vehicle can load. We developed an Ant Colony algorithm (ACO) for the CVRP based on the metaheuristic technique introduced by Colorni, Dorigo and Maniezzo. We present preliminary results that show that ant algorithms are competitive with other metaheuristics for solving CVRP.     

A hybrid algorithm for the Heterogeneous Fleet Vehicle Routing Problem

This paper deals with the Heterogeneous Fleet Vehicle Routing Problem (HFVRP). The HFVRP generalizes the classical Capacitated Vehicle Routing Problem by considering the existence of different vehicle types, with distinct capacities and costs. The objective is to determine the best fleet composition as well as the set of routes that minimize the total costs. The proposed hybrid algorithm is composed by an Iterated Local Search (ILS) based heuristic and a Set Partitioning (SP) formulation. The SP model is solved by means of a Mixed Integer Programming solver that interactively calls the ILS heuristic during its execution. The developed algorithm was tested in benchmark instances with up to 360 customers. The results obtained are quite competitive with those found in the literature and new improved solutions are reported.     

BoneRoute: An Adaptive Memory-Based Method for Effective Fleet Management

This paper presents an adaptive memory-based method for solving the Capacitated Vehicle Routing Problem (CVRP), called BoneRoute. The CVRP deals with the problem of finding the optimal sequence of deliveries conducted by a fleet of homogeneous vehicles, based at one depot, to serve a set of customers. The computational performance of the BoneRoute was found to be very efficient, producing high quality solutions over two sets of well known case studies examined.     

Efficient elementary and restricted non-elementary route pricing

Column generation is involved in the current most efficient approaches to routing problems. Set partitioning formulations model routing problems by considering all possible routes and selecting a subset that visits all customers. These formulations often produce tight lower bounds and require column generation for their pricing step. The bounds in the resulting branch-and-price are tighter when elementary routes are considered, but this approach leads to a more difficult pricing problem. Balancing the pricing with route relaxations has become crucial for the efficiency of the branch-and-price for routing problems. Recently, the ng-routes relaxation was proposed as a compromise between elementary and non-elementary routes. The ng-routes are non-elementary routes with the restriction that when following a customer, the route is not allowed to visit another customer that was visited before if they belong to a dynamically computed set. The larger the size of these sets, the closer the ng-route is to an elementary route. This work presents an efficient pricing algorithm for ng-routes and extends this algorithm for elementary routes. Therefore, we address the Shortest Path Problem with Resource Constraint (SPPRC) and the Elementary Shortest Path Problem with Resource Constraint (ESPPRC). The proposed algorithm combines the Decremental State-Space Relaxation technique (DSSR) with completion bounds. We apply this algorithm for the Generalized Vehicle Routing Problem (GVRP) and for the Capacitated Vehicle Routing Problem (CVRP), demonstrating that it is able to price elementary routes for instances up to 200 customers, a result that doubles the size of the ESPPRC instances solved to date.     

A hybrid Granular Tabu Search algorithm for the Multi-Depot Vehicle Routing Problem

In this paper, we propose a hybrid Granular Tabu Search algorithm to solve the Multi-Depot Vehicle Routing Problem (MDVRP). We are given on input a set of identical vehicles (each having a capacity and a maximum duration), a set of depots, and a set of customers with deterministic demands and service times. The problem consists of determining the routes to be performed to fulfill the demand of the customers by satisfying, for each route, the associated capacity and maximum duration constraints. The objective is to minimize the sum of the traveling costs related to the performed routes. The proposed algorithm is based on a heuristic framework previously introduced by the authors for the solution of the Capacitated Location Routing Problem (CLRP). The algorithm applies a hybrid Granular Tabu Search procedure, which considers different neighborhoods and diversification strategies, to improve the initial solution obtained by a hybrid procedure. Computational experiments on benchmark instances from the literature show that the proposed algorithm is able to produce, within short computing time, several best solutions obtained by the previously published methods and new best solutions.     

The Arc-Item-Load and Related Formulations for the Cumulative Vehicle Routing Problem

The Capacitated Vehicle Routing Problem (CVRP) consists of finding the cheapest way to serve a set of customers with a fleet of vehicles of a given capacity. While serving a particular customer, each vehicle picks up its demand and carries its weight throughout the rest of its route. While costs in the classical CVRP are measured in terms of a given arc distance, the Cumulative Vehicle Routing Problem (CmVRP) is a variant of the problem that aims to minimize total energy consumption. Each arc’s energy consumption is defined as the product of the arc distance by the weight accumulated since the beginning of the route.The purpose of this work is to propose several different formulations for the CmVRP and to study their Linear Programming (LP) relaxations. In particular, the goal is to study formulations based on combining an arc-item concept (that keeps track of whether a given customer has already been visited when traversing a specific arc) with another formulation from the recent literature, the Arc-Load formulation (that determines how much load goes through an arc).Both formulations have been studied independently before – the Arc-Item is very similar to a multi-commodity-flow formulation in Letchford and Salazar-González (2015) and the Arc-Load formulation has been studied in Fukasawa et al. (2016) – and their LP relaxations are incomparable. Nonetheless, we show that a formulation combining the two (called Arc-Item-Load) may lead to a significantly stronger LP relaxation, thereby indicating that the two formulations capture complementary aspects of the problem. In addition, we study how set partitioning based formulations can be combined with these formulations. We present computational experiments on several well-known benchmark instances that highlight the advantages and drawbacks of the LP relaxation of each formulation and point to potential avenues of future research.     

On the use of Monte Carlo simulation,cache and splitting techniques to improve the Clarke and Wright savings heuristics

This paper presents the SR-GCWS-CS probabilistic algorithm that combines Monte Carlo simulation with splitting techniques and the Clarke and Wright savings heuristic to find competitive quasi-optimal solutions to the Capacitated Vehicle Routing Problem (CVRP) in reasonable response times. The algorithm, which does not require complex fine-tuning processes, can be used as an alternative to other metaheuristics—such as Simulated Annealing, Tabu Search, Genetic Algorithms, Ant Colony Optimization or GRASP, which might be more difficult to implement and which might require non-trivial fine-tuning processes—when solving CVRP instances. As discussed in the paper, the probabilistic approach presented here aims to provide a relatively simple and yet flexible algorithm which benefits from: (a) the use of the geometric distribution to guide the random search process, and (b) efficient cache and splitting techniques that contribute to significantly reduce computational times. The algorithm is validated through a set of CVRP standard benchmarks and competitive results are obtained in all tested cases. Future work regarding the use of parallel programming to efficiently solve large-scale CVRP instances is discussed. Finally, it is important to notice that some of the principles of the approach presented here might serve as a base to develop similar algorithms for other routing and scheduling combinatorial problems.     

A heuristic algorithm for the asymmetric capacitated vehicle routing problem

We consider the Asymmetric Capacitated Vehicle Routing Problem (ACVRP[, a particular case of the standard asymmetric Vehicle Routing Problem arising when only the vehicle capacity constraints are imposed. ACVRP is known to be NP-hard and finds practical applications, e.g. in distribution and scheduling. In this paper we describe the extension to ACVRP of the two well-known Clarke-Wright and Fisher-Jaikumar heuristic algorithms. We also propose a new heuristic algorithm for ACVRP that, starting with an initial infeasible solution, determines the final set of vehicle routes through an insertion procedure as well as intea-route and inter-route arc exchanges. The initial infeasible solution is obtained by using the additive bounding procedures for ACVRP described by Fischetti, Toth and Vigo in 1992. Extensive computational results on several classes of randomly generated test problems involving up to 300 customers and on some real instances of distribution problems in urban areas, are presented. The results obtained show that the proposed approach favourably compares with previous algorithms from the literature.     

Lower and upper bounds for the m-peripatetic vehicle routing problem

The m-Peripatetic Vehicle Routing Problem (m-PVRP) consists in finding a set of routes of minimum total cost over m periods so that two customers are never sequenced consecutively during two different periods. It models for example money transports or cash machines supply, and the aim is to minimize the total cost of the routes chosen. The m-PVRP can be considered as a generalization of two well-known NP-hard problems: the Vehicle Routing Problem (VRP or 1-PVRP) and the m-Peripatetic Salesman Problem (m-PSP). In this paper we discuss some complexity results of the problem before presenting upper and lower bounding procedures. Good results are obtained not only on the m-PVRP in general, but also on the VRP and the m-PSP using classical VRP instances and TSPLIB instances.     

Approximating the chance-constrained capacitated vehicle routing problem with robust optimization

4OR - The Capacitated Vehicle Routing Problem (CVRP) is a classical combinatorial optimization problem that aims to serve a set of customers, using a set of identical vehicles, satisfying the...     

A branch-and-price algorithm for the Vehicle Routing Problem with Deliveries,Selective Pickups and Time Windows

In the Vehicle Routing Problem with Deliveries, Selective Pickups and Time Windows, the set of customers is the union of delivery customers and pickup customers. A fleet of identical capacitated vehicles based at the depot must perform all deliveries and profitable pickups while respecting time windows. The objective is to minimize routing costs, minus the revenue associated with the pickups. Five variants of the problem are considered according to the order imposed on deliveries and pickups. An exact branch-and-price algorithm is developed for the problem. Computational results are reported for instances containing up to 100 customers.     

An exact solution framework for a broad class of vehicle routing problems

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, 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.     

Multistars, partial multistars and the capacitated vehicle routing problem

 In an unpublished paper, Araque, Hall and Magnanti considered polyhedra associated with the Capacitated Vehicle Routing Problem (CVRP) in the special case of unit demands. Among the valid and facet-inducing inequalities presented in that paper were the so-called multistar and partial multistar inequalities, each of which came in several versions. Some related inequalities for the case of general demands have appeared subsequently and the result is a rather bewildering array of apparently different classes of inequalities. The main goal of the present paper is to present two relatively simple procedures that can be used to show the validity of all known (and some new) multistar and partial multistar inequalities, in both the unit and general demand cases. The procedures provide a unifying explanation of the inequalities and, perhaps more importantly, ideas that can be exploited in a cutting plane algorithm for the CVRP. Computational results show that the new inequalities can be useful as cutting planes for certain CVRP instances. Received: January 9, 1999 / Accepted: June 17, 2002 Published online: September 27, 2002 Key Words. vehicle routing – valid inequalities – cutting planes     

A note on branch-and-cut-and-price

In this paper, we propose a methodology for branch-and-cut-and-price when cuts and columns are generated simultaneously. The methodology is illustrated with two application cases: the Split Delivery Vehicle Routing Problem (SDVRP) and the Bus Rapid Transit Route Design Problem (BRTRDP).