**Solving a Novel Multi-Objective Inventory Location**

Problem by means of a Local Search Algorithm

Problem by means of a Local Search Algorithm

**Carolina LAGOS ^{1}, Jorge VEGA^{2},**

Guillermo GUERRERO^{1}, Jose-Miguel RUBIO^{3 }

^{1 }Pontificia Universidad Católica de Valparaíso, Chile

{carolina.lagos.c; guillermo.guerrero.c}@mail.pucv.cl

^{2 }Universidad de Antofagasta, Chile

jorge.vega@uantofa.cl

^{3 }Universidad de Playa Ancha, Chile

jose.rubio.l@upla.cl

**Abstract: **Problems in operations research are usually modelled as single objective ones even though there exist several goals that should be attained. Multiple reasons of why inherently multi-objective problems are modelled as single objective can be identified: better understanding of the problem features, simplification of the mathematical formulation of the problem, among others. However, summarising (usually conflicting) objectives into one objective function can be a sign of over-simplification in the problem modelling process. In this paper we extend a single objective inventory location problem formulation to a multi-objective one. We consider two conflicting objectives, namely, location cost and inventory cost. As a result, we obtain a complex multi-objective non-linear integer programming problem. Like its single objective formulation, this multi-objective problem cannot be solved by exact methods as the number of decision variables increases. Thus, a simple yet effective multi-objective local search algorithm is implemented in this paper.

**Keywords: **MO Local Search, MO Inventory Location, Combinatorial Optimisation.

**>>Full text****
CITE THIS PAPER AS**: Carolina LAGOS, Jorge VEGA, Guillermo GUERRERO, Jose-Miguel RUBIO,

**Solving a Novel Multi-Objective Inventory Location Problem by means of a Local Search Algorithm**,

*Studies in Informatics and Control*, ISSN 1220-1766, vol. 25(2), pp. 189-194, 2016. https://doi.org/10.24846/v25i2y201606

**Introduction**

Problems in operations research are usually modelled as single objective ones even though there exists, in real world, several goals that should be attained. Multiple reasons of why inherently multi-objective problems are modelled as single objective ones can be outlined: better understanding of the problem features and simplification of the mathematical formulation of the problem, among others. However, summarising (usually conflicting) objectives into one objective function can be a sign of over-simplification in the problem modelling process.

Multi-objective optimisation (MO) aims to optimise two or more conflicting objectives simultaneously. Unlike in single objective optimisation, where solving a problem means to find its optimal solution w.r.t. some objective function, solving a problem in a MO context means to find a set of solutions such that improving an objective without impair at least some other objective is not possible. These solutions are called *efficient solutions*. Mathematically, a general MO problem can be formulated as follows:

In this work a model for a MO inventory location model (ILM) problem is proposed. ILM problems aim to integrate strategic decisions with tactical ones. In particular, the ILM considered in this paper seeks to integrate location decision making (strategic) with inventory policies (tactical). We consider two conflicting objectives: On the one hand, we aim to minimize the location/allocation cost, that is, the cost of installing a warehouse and, on the other hand, we want to minimize the inventory cost, that is, the cost of holding products in our warehouses and the cost of processing an order from our customers.

The problem of locating/allocating customer to distribution centres is one of the most studied problems in logistics. Usually, after decision makers determine the locations to be installed, the inventory policy is defined. However, this sequential approach is sub-optimal in the sense that the inventory policy is restricted by the network design determined in the previous step. Thus, integrating location/allocation decisions and inventory policies lead to solutions that are more efficient as they consider the entire system as a whole. Figure 1 shows a schema of both the sequential and the integrated approaches.

The remaining of this paper is as follows: In Section 2 we present a brief literature review of previously proposed models for the ILM. Although we include some single objective ILMs, the review is mainly focused on the multi-objective models presented so far. In Section 3 we introduce a novel MO-ILM which is an extension of the single objective ILM presented in [18]. In Section 4, the multi-objective local search we consider to solve the MO-ILM is outlined. In Section 5, the computational experiments performed in this paper are described and the obtained results are discussed. Finally, in Section 5, some conclusions and the future work are outlined.

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