Estructura y Objetivos de los Grupos Organizacionales del Polo PetroSoft

In this chapter, we analyze the Location Routing Problem (LRP). It is an inte-grated problem, combining two difficult ones, the vehicle routing problem and the location problem. The integration of these two important optimization problems to distribution networks can translate into considerable savings. Its standard ver-sion represents an interrelation of a vehicle routing problem, which determines the optimal set of routes to fulfill the demands of a set of customers, and a location problem, as the depots from which the vehicles performing the routes can be as-sociated must be chosen from a set of possible locations. Being a generalization of the VRP it is also a N P -hard problem.

Laporte [130] describes application areas, different formulations and solution methods in a survey of deterministic LRP. Based on the work published up to 1988, the author concludes by proposing promising future research areas for this problem. In [158], the authors propose a taxonomy for the LRP and a classification scheme that categorizes the up to date exact and heuristic approaches in the liter-ature, according to their specific solution method and to eleven other features that characterize the problem’s assumptions, in what the authors call a two-way clas-sification. They also provide an annotated literature review, organized according to the proposed classification. A more recent survey of LRPs is provided in [165].

The authors also propose a classification scheme and a review of deterministic exact and heuristic methods, and also of stochastic and dynamic solution approaches to the problem. Laporte et al. [133] applied an exact algorithm based on an integer programming formulation to the LRP. The procedure consists of a branch-and-cut algorithm where subtour eliminating constraints and chain barring constraints are introduced. It solved instances with up to 20 customers and 8 depots. Although the depots have no capacity, there are lower and upper bounds on the number of depots to consider and on the number of capacitated vehicles to be associated to

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each depot. Laporte et al. [135] solve an asymmetrical LRP with lower and upper bounds on the number of vehicles based at each depot, and a constraint on the total cost of a route. They reformulate the problem as a constrained assignment problem through the use of a graph transformation. They use a variant of the branch-and-bound algorithm proposed in [43] for the traveling salesman problem to solve the integer problem. The authors report on computational results with up to 3 possible depots and 80 customers Berger et al. [32] proposed a branch-and-price algorithm, based on a set-partitioning formulation, with distance constraints and uncapacitated vehicles and depots. They also introduce a set of valid inequal-ities in order to strengthen the proposed formulation. Akca et al. [4] proposed an exact solution method for the LRP with capacities on both vehicles and depots.

It is a branch-and-price algorithm based on a set-partitioning formulation. The proposed model is an adaptation of a previously presented formulation [3] for the location routing and scheduling problem. The authors presented four variants of the algorithm, based on different heuristic approaches to solve the pricing problem of their column generation scheme. Belenger et al.[28] provided a branch-and-cut algorithm for the LRP with capacitated facilities and capacitated vehicles. The integer programming model uses only binary variables and is based on the one proposed in [133], with the additional depots’ capacity constraints. Both heuris-tic and exact separation algorithms are proposed, in order to find violated valid inequalities in the embedded cutting plane scheme.

According to the classification scheme presented in [165], heuristic methods for the LRP can be clustering-based, iterative or hierarchical. Clustering-based heuristics are composed by two phases. In a first phase, the set of customers is grouped in clusters, which are, in the second phase, allocated either to depots or to a vehicle route. In the first case, the algorithm concludes by solving a VRP (or TSP) for each depot and in the second case a TSP is solved for each cluster which is posteriorly associated to a depot. In iterative methods, the subproblems

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of location and routing are solved separately, in an iterative scheme that exchanges information between these two phases, in order to improve the solution. Finally, hierarchical heuristics tackle the problem as a location problem, incorporating the routing problem as a subproblem in some phases of the routine.

Nagy and Salhi [164] propose a hierarchical heuristic method for the LRP, based on a tabu search algorithm. It is a nested method, where the routing problem is considered in the evaluation of possible moves, which can be add moves (opening a closed depot), drop moves (closing an open depot) or shift moves (closing an open depot and opening a closed one). Tuzun et Burke [185] propose a hierar-chical two-phase tabu search for the LRP with capacitated vehicles. These two coordinated phases correspond either to a location or a routing problem. Every time a neighborhood move is performed in the location phase, the routing phase is started. The search in the routing phase is not global. Only the routing sections affected by the previous location move are explored, in order to, according to the authors, explore the solution space efficiently. Wu et al. [202] decompose the LRP into a location-allocation problem and a vehicle routing problem and solve it with an iterative heuristic, on a combined tabu search and simulated annealing frame-work. The authors consider the problem with capacitated heterogeneous vehicles.

Albareda-Sambola et al. [5] approached the LRP with capacitated depots with one single vehicle associated to each depot. The authors solve the LP-relaxation of a compact model and use it as a lower bound and as a base for the first solution in a tabu search framework. The problem is solved with a hierarchical heuristic where the intensification and diversification phases are concerned, respectively, with the location and the routing problems. They allow infeasible solutions in the intensifi-cation phase, which are controlled with a penalty in the objective function. They propose an additional lower bound that presents a better quality than the one pro-vided by the LP-relaxation. Barreto et al. [26] used a clustering-based heuristic for the LRP with capacity constraints both for the vehicles and the depots. Customers

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are grouped in clusters with a capacity limit, by using one of four grouping meth-ods. A TSP is then solved for each of these groups, by using an exact method, if the group has 40 or less elements, or a two stages heuristic procedure, otherwise. After improving the routes with a local search procedure, a location problem is solved in order to assign them to the selected depots. By combining the four clustering methods and six proximity measures, the authors present several versions of this routing-first, location-second heuristic.

We address a variant of this problem, the location routing problem with multiple routes (MLRP), where each vehicle can perform several routes in the same planning period. This variant has been explored exactly in [3], and heuristically in [141], although in the latter the authors also propose a branch-and-bound algorithm for the problem. In [3], the authors propose a three-index commodity flow model and also a branch-and-price algorithm based on a set-partitioning formulation of the problem. The branch-and-price algorithm is based on a column generation framework, where each column represents what the authors define as pairings. The pricing problem is an elementary shortest path problem with resource constraints.

Finally, the authors propose valid inequalities in order to strengthen the model. In [6], the authors define several formulations for the case where only one customer is visited in each route. The problem is addressed as the capacity and distance constrained plant location problem. For one of the formulations, they propose families of valid inequalities in order to strengthen it, and assess its effectiveness when it is directly solved with a commercial solver. As they do not devise any column generation scheme, they propose an extension of the problem, which can have practical applicabilities, as they describe, where there is a constraint on the number of customers that each vehicle can visit during the planning period. This constraint is addressed as the cardinality constraint.

The remainder of this chapter is organized as follows. In section 8.2, we define our problem, along with its notation. Some mathematical formulations from the

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literature are briefly recalled in section 8.3. In section 8.4 we describe our network flow model, as well as some reduction criteria derived for it. Computational ex-periments are reported in section 8.5, while some conclusions are drawn in section 8.6.