Tabu Search Heuristics for the Arc Routing Problem with Intermediate Facilities under Capacity and Length Restrictions

This paper deals with the Arc Routing Problem with Intermediate Facilities under Capacity and Length Restrictions(CLARPIF), a variant of the classical Capacitated Arc Routing Problem(CARP), in which vehicles may unload or replenish at intermediate facilities and the length of any route may not exceed a specified upper bound. Three heuristics are developed for the CLARPIF: the first is a constructive procedure based on a partitioning approach while the second and the third are tailored Tabu Search procedures. Computational results on a set of benchmark instances with up to 50 vertices and 92 required edges are presented and analyzed.     

A Branch-and-Price Algorithm for the Capacitated Arc Routing Problem with Stochastic Demands

We address the Capacitated Arc Routing Problem with Stochastic Demands (CARPSD), which we formulate as a Set Partitioning Problem. The CARPSD is solved by a Branch-and-Price algorithm, which we apply without graph transformation. The demand’s stochastic nature is incorporated into the pricing problem. Computational results are reported.     

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.     

Using mixed integer programming for solving the capacitated arc routing problem with vehicle/site dependencies with an application to the routing of residential sanitation collection vehicles

We present the Composite Approach for solving the Capacitated Arc Routing Problem with Vehicle/Site Dependencies (CARP-VSD). We also present two mixed integer programs, the Initial Fleet Mix Generator (IFM) and the Mathematical Programming Procedure (MPP), and a multi-criterion function called the Measure of Goodness. The IFM, MPP and Measure of Goodness are critical components of the Composite Approach. A key application area of the CARP-VSD is the routing of residential sanitation vehicles; throughout this paper, we derive parameters specific to this problem. In this paper, we describe the Composite Approach, the IFM, the MPP, the Measure of Goodness and work out several examples in detail.     

A way to optimally solve a time-dependent Vehicle Routing Problem with Time Windows

In this paper we deal with a generalization of the Vehicle Routing Problem with Time Windows that considers time-dependent travel times and costs. Through several steps we transform this extension into an Asymmetric Capacitated Vehicle Routing Problem, so it can be solved both optimally and heuristically with known codes.     

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.     

Competitive Memetic Algorithms for Arc Routing Problems

The Capacitated Arc Routing Problem or CARP arises in applications like waste collection or winter gritting. Metaheuristics are tools of choice for solving large instances of this NP-hard problem. The paper presents basic components that can be combined into powerful memetic algorithms (MAs) for solving an extended version of the CARP (ECARP). The best resulting MA outperforms all known heuristics on three sets of benchmark files containing in total 81 instances with up to 140 nodes and 190 edges. In particular, one open instance is broken by reaching a tight lower bound designed by Belenguer and Benavent, 26 best-known solutions are improved, and all other best-known solutions are retrieved.     

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.     

A generalized variable neighborhood search heuristic for the capacitated vehicle routing problem with stochastic service times

This paper describes a generalized variable neighborhood search heuristic for the Capacitated Vehicle Routing Problem with Stochastic Service Times, in which the service times at vertices are stochastic. The heuristic is tested on randomly generated instances and compared with two other heuristics and with an alternative solution strategy. Computational results show the superiority and effectiveness of the proposed heuristic.     

A unified exact method for solving different classes of vehicle routing problems

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.      

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.     

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.     

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 Guided Tabu Search for the Vehicle Routing Problem with two-dimensional loading constraints

We present a metaheuristic methodology for the Capacitated Vehicle Routing Problem with two-dimensional loading constraints (2L-CVRP). 2L-CVRP is a generalisation of the Capacitated Vehicle Routing Problem, in which customer demand is formed by a set of two-dimensional, rectangular, weighted items. The purpose of this problem is to produce the minimum cost routes, starting and terminating at a central depot, to satisfy the customer demand. Furthermore, the transported items must be feasibly packed into the loading surfaces of the vehicles. We propose a metaheuristic algorithm which incorporates the rationale of Tabu Search and Guided Local Search. The loading aspects of the problem are tackled using a collection of packing heuristics. To accelerate the search process, we reduce the neighbourhoods explored, and employ a memory structure to record the loading feasibility information. Extensive experiments were conducted to calibrate the algorithmic parameters. The effectiveness of the proposed metaheuristic algorithm was tested on benchmark instances and led to several new best solutions.     

A mixed integer formulation and an efficient metaheuristic procedure for the <Emphasis Type="Italic">k</Emphasis>-Travelling Repairmen Problem

In this paper, we study a k-Travelling Repairmen Problem where the objective is to minimize the sum of clients waiting time to receive service. This problem is relevant in applications that involve distribution of humanitarian aid in disaster areas, delivery and collection of perishable products and personnel transportation, where reaching demand points to perform service, fast and fair, is a priority. This paper presents a new mixed integer formulation and a simple and efficient metaheuristic algorithm. The proposed formulation consumes less computational time and allows solving to optimality more than three times larger data instances than the previous formulation published in literature, even outperforming a recently published Branch and Price and Cut algorithm for this problem. The proposed metaheuristic algorithm solved to optimality 386 out of 389 tested instances in a very short computational time. For larger instances, the algorithm was assessed using the best results reported in the literature for the Cumulative Capacitated Vehicle Routing Problem.     

Zigzag inequalities: a new class of facet-inducing inequalities for Arc Routing Problems

In this paper we introduce a new class of facet-inducing inequalities for the Windy Rural Postman Problem and the Windy General Routing Problem. These inequalities are called Zigzag inequalities because they cut off fractional solutions containing a zigzag associated with variables with 0.5 value. Two different types of inequalities, the Odd Zigzag and the Even Zigzag inequalities, are presented. Finally, their application to other known Arc Routing Problems is discussed.     

A Scatter Search for the periodic capacitated arc routing problem

This paper considers the Periodic Capacitated Arc Routing Problem (PCARP), a natural extension of the well-known CARP to a multi-period horizon. Its objective is to assign a set of service days to each edge in a given network and to solve the resulting CARP for each period, in order to minimize the required fleet size and the total cost of the trips on the horizon. This new and very hard problem has various applications in periodic operations on street networks, like waste collection and sweeping. A greedy heuristic and a Scatter Search (SS) are developed and evaluated on two sets of PCARP instances derived from classical CARP benchmarks. The results show that the SS strongly improves its initial solutions and clearly outperforms the greedy heuristic. Preliminary lower bounds are also provided. As they are not sufficiently tight, the SS is also tested in the single-period case (CARP) for which tight bounds are available: in fact, it competes with one state-of-the-art metaheuristic proposed for the CARP.     

Constant-factor approximations for Capacitated Arc Routing without triangle inequality

Given an undirected graph with edge costs and edge demands, the Capacitated Arc Routing problem (CARP) asks for minimum-cost routes for equal-capacity vehicles so as to satisfy all demands. Constant-factor polynomial-time approximation algorithms were proposed for CARP with triangle inequality, while CARP was claimed to be NP-hard to approximate within any constant factor in general. Correcting this claim, we show that any factor  α$\alpha$ approximation for CARP with triangle inequality yields a factor  α$\alpha$ approximation for the general CARP.     

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.