Roberto D. Galvão

A tabu search algorithm for the vehicle routing problem with simultaneous pick-up and delivery service

The vehicle routing problem with simultaneous pick-up and delivery (VRP_SPD) is a variant of the classical vehicle routing problem (VRP) where clients require simultaneous pick-up and delivery service. Deliveries are supplied from a single depot at the beginning of the vehicles service, while pick-up loads are taken to the same depot at the conclusion of the service. One important characteristic of this problem is that a vehicles load in any given route is a mix of pick-up and delivery loads. In this paper we develop a tabu search algorithm to solve VRP_SPD. This algorithm uses three types of movements to obtain inter-route adjacent solutions: the relocation, interchange and crossover movements. A 2-opt procedure is used to obtain alternative intra-route solutions. Four types of neighbourhoods were implemented, three of them defined by the use of each of the single inter-route movements and the fourth by using a combination of these movements. Two different search strategies were implemented for selecting the next movement: first admissible movement and best admissible movement. Intensification and diversification of the search were achieved through frequency penalization. Computational results are reported for a set of 87 test problems with between 50 and 400 clients.

A Lagrangean heuristic for the maximal covering location problem

Abstract We develop a Lagrangean heuristic for the maximal covering location problem. Upper bounds are given by a vertex addition and substitution heuristic and lower bounds are produced through a subgradient optimization algorithm. The procedure was tested in networks of up to 150 vertices. A duality gap was generally present at the end of the heuristic for the larger problems. The test problems were run in an IBM 3090-600J ‘super-computer’; the maximum computing time was kept below three minutes of CPU.

A method for solving to optimality uncapacitated location problems

In this paper we develop a method for solving to optimality a general 0–1 formulation for uncapacitated location problems. This is a 3-stage method that solves large problems in reasonable computing times.The 3-stage method is composed of a primal-dual algorithm, a subgradient optimization to solve a Lagrangean dual and a branch-and-bound algorithm. It has a hierarchical structure, with a given stage being activated only if the optimal solution could not be identified in the preceding stage.The proposed method was used in the solution of three well-known uncapacitated location problems: the simple plant location problem, thep-median problem and the fixed-chargep-median problem. Computational results are given for problems of up to the size 200 customers ×200 potential facility sites.

A comparison of Lagrangean and surrogate relaxations for the maximal covering location problem

Abstract We compare heuristics based on Lagrangean and surrogate relaxations of the Maximal Covering Location Problem (MCLP). The Lagrangean relaxation of MCLP used in this paper has the integrality property and the surrogate relaxed problem we solve is the LP relaxation of the original 0−1 knapsack problem. The heuristics were compared using 331 test problems available in the literature, corresponding to networks ranging from 55 to 900 vertices. The gaps obtained with both heuristics were very low and did not differ substantially among themselves for the several problem sets used, in accordance with theoretical results reviewed in the paper. When the initial set of multipliers was determined using a valid bound for MCLP the computing times did not differ significantly between the Lagrangean and surrogate heuristics.

Towards unified formulations and extensions of two classical probabilistic location models

We give a unified view of Daskins Maximum Expected Covering Location Problem (MEXCLP) and ReVelle and Hogans Maximum Availability Location Problem (MALP), identifying similarities and dissimilarities between these models and showing how they relate to each other. These models arise in the location of servers in congested emergency systems. An existing extension of MEXCLP is reviewed; we then develop an extension of MALP and give the corresponding mathematical formulation. These two extensions are obtained when the simplifying assumptions of the original models are dropped and Larsons hypercube model is embedded into local search methods. In this paper these methods are further enhanced by the use of simulated annealing. Computational results are given for problems available in the literature.

A review of congestion models in the location of facilities with immobile servers

Capacity effects are investigated, particularly as regards congestion of facilities with immobile (or fixed) servers. A review is given of research to date in this area, together with some related work. First, general implications of facility capacities being capacitated are discussed with regard to assignment of users to facilities, time (or distance) transformations, integrality, feasibility, equity and load balancing. Then models using both system choice and user choice assignment are presented, and hierarchical problems covered briefly. Finally, there is a summary and a discussion together with some possible future research directions.

A hierarchical model for the location of perinatal facilities in the municipality of Rio de Janeiro

Abstract We present a 3-level hierarchical model for the location of maternal and perinatal health care facilities in Rio de Janeiro. Relaxations and heuristics are developed for this model and computational results are given for problems available in the literature, for networks ranging from 10 to 400 vertices. The quality of the solutions produced by the procedures we developed do not differ significantly among themselves. The model is also applied to a case study corresponding to real 1995 data of the municipality of Rio de Janeiro.

A statistical analysis of simulated annealing applied to the p-median problem

We present a statistical analysis of simulated annealing applied to the p-median problem. The algorithm we use combines elements of the vertex substitution method of Teitz and Bart with the general methodology of simulated annealing. The cooling schedule adopted incorporates the notion of temperature adjustments rather than just temperature reductions. Computational results are given for test problems ranging from 100 to 900 vertices, retrieved from Beasleys OR-Library for combinatorial problems. Each problem was run for a maximum of 100 different streams of random numbers. Optimal solutions were found for 26 of the 40 problems tested, although high optimum hitting rates were obtained for only 20 of them. The worst gap in relation to the optimal solution was 1.62%, after all runs for each of the test problems were computed.

Vehicle Routing Problems with Simultaneous Pick-up and Delivery Service

Variations of the classical Vehicle Routing Problem (VRP) consider clients that require pick-up and delivery service, simultaneously or not. Each variation corresponds to a specific routing policy, applicable to a given context. Although several routing problems of this nature are defined in the literature, there are relationships among them that define one problem as a particular case of another, which allows transformation among problems. We classify these problems on the basis of the characteristics of the service required by the clients.We study in particular the Vehicle Routing Problem with Simultaneous Pickup and Delivery Service. This problem may be solved for example through a tour partitioning heuristic developed for the traveling salesman problem (TSP). The routing in each segment of the partitioned tour is achieved by solving capacitated TSP subproblems with simultaneous pick-up and delivery service. Computational results are given for a set of problems available in the literature.

Load balancing and capacity constraints in a hierarchical location model

Recent work on the location of perinatal facilities in the municipality of Rio de Janeiro resulted in the development of an uncapacitated, three-level hierarchical model. An important issue that arose during the course of the research was the need to include some form of capacity constraints into the model, especially in the higher, resource intensive level of the hierarchy. This need was felt from technical discussions related to the research work and from contacts with municipality health officials. A capacitated model is presented and solved using a Lagrangean heuristic. For smaller problems optimal solutions are found using CPLEX; it is found that the heuristic usually gives high quality solutions. Next a bi-criterion model is formulated in which the additional objective measure is the imbalance in facility loading. This is solved exactly for a small problem and approximately for the 152-vertex Rio de Janeiro data. � 2004 Elsevier B.V. All rights reserved.