Solving a Novel Multi-Objective Inventory Location
Problem by means of a Local Search Algorithm
Carolina LAGOS1, Jorge VEGA2,
Guillermo GUERRERO1, Jose-Miguel RUBIO3
1 Pontificia Universidad Católica de Valparaíso, Chile
2 Universidad de Antofagasta, Chile
3 Universidad de Playa Ancha, Chile
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.
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.
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 . 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.
- AHMADI, G., S. A. TORABI, R. TAVAKKOLI-MOGHADDAM, A Bi-objective Location-inventory Model with Capacitated Transportation and Lateral Trans-shipments. Intl. J. Prod. Res., 2015, pp. 1-22.
- ANGEL, E., E. BAMPIS, L. GOURVIES, A Dynasearch Neighborhood for the Bicriteria Traveling Salesman Problem. In X. Gandibleux, M. Sevaux, K. Sorensen, V. T’kindt, eds. Metaheuristics for Multiobjective Optimisation, vol. 535 of Lect. Notes in Ec. & Math. Sys., Springer Verlag, 2004, pp. 153-176.
- ARABZAD, S. M., M. GHORBANI, R., TAVAKKOLI-MOGHADDAM, An Evolutionary Algorithm for a New Multi-objective Location-inventory Model in a Distribution Network with Transportation Modes and Third-party Logistics Providers. Intl. J. Prod. Res., vol. 53(4), 2015, pp. 1038-1050.
- ASKIN, R. G., I. BAFFO, M. XIA, Multicommodity Warehouse Location and Distribution Planning with Inventory Consideration. Intl. J. Prod. Res., vol. 52(7), 2014, pp. 1897-1910.
- BHATTACHARYA, R., S. BANDYOPADHYAY, Solving Conflicting Bi-objective Facility Location Problem by NSGA II Evolutionary Algorithm, Intl. J. Adv. Man. Tech., vol. 51(1), 2010, pp. 397-414.
- BRADLEY, J. R., B. C. ARNTZEN, The Simultaneous Planning of Production, Capacity, and Inventory in Seasonal Demand Environments. Op. Res., vol. 47(6), 1999, pp. 795-806.
- CABRERA, G., G., P. A. MIRANDA, E. CABRERA, R. SOTO, L. J. M. RUBIO, B. CRAWFORD, F. PAREDES, Solving a Novel Inventory Location Model with Stochastic Constraints and Inventory Control Policy, Math. Prob. in Eng., vol. 2013, Article ID 670528, 12 p., 2013.
- CABRERA, G., S. NIKLANDER, E. CABRERA, F. JOHNSON, Solving a Distribution Network Design Problem by means of Evolutionary Algorithms, SIC, vol. 25(1), 2016, pp. 21-28.
- COELLO COELLO, C. A., R. BECERRA, Evolutionary Multi-objective Optimization using a Cultural Algorithm. Proc. 2003 IEEE Swarm Intel. Symp., 2003, pp. 6-13.
- DASKIN, M. S., C. R. COULLARD, Z. M. SHEN, An Inventory Location Model: Formulation, Solution Algorithm and Computational Results. Ann. Op. Res., vol. 110, 2002, pp. 83-106.
- FONSECA, C. M., P. J. FLEMING, Genetic Algorithms for multi-objective optimization: Formulation discussion and generalization. Proc. 5th Intl. Conf. on Gen. Alg., Morgan Kaufmann Pub. Inc., 1993, pp. 416-423.
- HAMEDANI, S., M. JABALAMELI, A. BOZORGI-AMIRI, A Location-inventory Model for Distribution Centers in a Three-level Supply chain under Uncertainty. Intl. J. Ind. Eng. Comp., v. 4(1), 2013, pp. 93-110.
- HANSEN, M. P. Tabu Search for Multiobjective Optimization: MOTS. In Proceedings of the Thirteenth International Conference on Multiple Criteria Decision Making, 1997, Springer-Verlag, pp. 6-10.
- KWONLES, J., D. CORNE, The Pareto Archived Evolution Strategy: A New Baseline Algorithm for Pareto Multi-objective Optimisation. Proc. 1999 Cong. Evol. Comp., vol. 1, 1999, pp. 105.
- LAGOS, C., F. PAREDES, S. NIKLANDER, CABRERA, Solving a Distribution Network Design Problem by Combining Ant Colony Systems and Lagrangian Relaxation, SIC, vol. 24 (3), 2015, pp. 251-260.
- LIAO, S., C. HSIEH, Y. LIN, A Multi-objective Evolutionary Optimization Approach for an Integrated Location-inventory Distribution Network Problem under Vendor-managed Inventory Systems. Op. Res., v. 186(1), 2011, pp. 213-229.
- MIRANDA, P. A., R. A. GARRIDO, Incorporating Inventory Control Decisions into a Strategic Distribution Network Design Model with Stochastic Demand. Trans. Res. Part E, pp. Log. & Trans. Rev., vol. 40(3), 2004, pp. 183-207.
- MIRANDA, P. A., R. A. GARRIDO, A Simultaneous Inventory Control and Facility Location Model with Stochastic Capacity Constraints. Networks and Spatial Economics, vol. 6, 2006, pp. 39-53.
- MIRANDA, P. A., R. A. GARRIDO, Valid Inequalities for Lagrangian Relaxation in an Inventory Location Problem with Stochastic Capacity. Trans. Res. Part E: Log. & Trans. Rev., v. 44(1), 2008, pp. 47-65.
- MOURITS, M., J. M. EVERS, Distribution Network Design: An Integrated Planning Support Framework. Intl. J. Phys. Dist. & Log. Manag., vol. 25, 1995, pp. 43-57.
- OZSEN, L., C. R. COULLARD, M. S. DASKIN, Capacitated Warehouse Location Model with Risk Pooling. Nav. Res. Log., vol. 55(4), 2008, pp. 295-312.
- PAQUETE, L., T. SCHIAVINOTTO, T. STUTZLE, On Local Optima in Multi-objective Combinatorial Optimization Problems. Tech. Rep. AIDA-04-11, FG Intellektik, TU Darmstadt, November 2004.
- SCHAFFER, J. D. Multiple Objective Optimization with Vector Evaluated Genetic Algorithms. Proc. 1st Intl. Conf. Gen. Alg., L. Erlbaum Assoc. Inc., 1985, pp. 93-100.
- SERAFINI, P. Simulated Annealing for Multi- Objective Optimization Problems. Tzeng, G. H., Wang, H. F., Wen, U. P., Yu, P. L., eds. Proc. 10th Conf. MCDM: Expand & Enrich Domains of Thinking & Application, 1994, Springer, pp. 283-292.
- SHEN, Z. M., C. R. COULLARD, M. S. DASKIN, A Joint Location Inventory Model. Trans. Sc., v. 37(1), 2003, pp. 40-55.
- SIMCHI-LEVI, D., Y. ZHAO, The Value of Information Sharing in a Two Stage Supply Chain with Production Capacity Constraints. Nav. Res. Log., vol. 50(8), 2003, pp. 888-916.
- SRINIVAS N., K. DEB, Multi-objective Optimization using Non-dominated Sorting in Genetic Algorithms. Ev. Comp. vol. 2(3), 1994, pp. 221-248.
- ZITZLER E., L. THIELE, Multi-objective Optimization Using Evolutionary Algorithms: A Comparative Case Study. Proc. 5th Int.l Conf. Paral. Prob. Solv. from Nature, 1998, pp. 292-304.