### Article

Mathematical Model and Stochastic Multi-Criteria Acceptability Analysis for Facility Location Problem
Selahattin Karabay, Erkan Köse, Mehmet Kabak, Eren Ozceylan
Keywords:case study, facility location problem, mixed integer linear programming, Stochastic Multi-criteria Acceptability Analysis, public sector,

Abstract

This paper studies a real-life public sector facility location problem. The problem fundamentally originated from the idea of downsizing the number of service centres. However, opening of new facilities is also considered in case the current facilities fail to fulfil general management demands. Two operation research methodologies are used to solve the problem and the obtained results are compared. First, a mathematical programming model is introduced to determine where the new facilities will be located, and which districts get service from which facilities, as if there were currently no existing facilities. Second, the Stochastic Multi-criteria Acceptability Analysis-TRI (SMAA-TRI) method is used to select the best suitable places for service centres among the existing facilities. It is noted that the application of mathematical programming model and SMAA-TRI integration approach on facility location problem is the first study in literature. Compression of outcomes shows that mixed integer linear programming (MILP) model tries to open facilities in districts which are favoured by SMAA-TRI solution.

References

Weber A. About the location of industries [in German]. Erster Teil: Reine Theorie des Standortes; 1909.

ReVelle CS, Eiselt HA. Location analysis: A synthesis and survey. Eur J Oper Res. 2005 Aug;165(1):1-19.

Hakimi SL. Optimum locations of switching centres and the absolute centres and medians of a graph. Oper Res. 1964 May;12(3):450-459.

Kariv O, Hakimi SL. An algorithmic approach to network location problems. Part II: The p-median. SIAM

J Applied Math. 1979 Dec;37(3):539-560.

Reese J. Solution methods for the p-median problem: an annotated bibliography. Networks. 2006 Aug;48(3):125-142.

Lim GJ, Reese J, Holder A. Fast and robust techniques for the euclidean p-median problem with uniform weights. Comp & Ind Eng. 2009 Oct;57(3):896-905.

Teitz MB, Bart P. Heuristic methods for estimating the generalized vertex median of a weighted graph. Oper Res. 1968 Oct;16(5):955-961.

Aguado JS, Trandafir PC. Some heuristic methods for solving p-median problems with a coverage constraint. Euro J Oper Res. 2012 July;220(2):320-327.

Hakimi SL. Optimum distribution of switching centres in a communication network and some related graph theoretic problems. Oper Res. 1965 June;13(3):462-475.

Toregas C, Swain R, ReVelle C, Bergman L. The location of emergency services facilities. Oper Res. 1971 Oct;19(6):1363-1373.

Church R, ReVelle C. The maximal covering location problem. Papers Reg Sci Assoc. 1974 Dec;32(1):101-118.

Daskin MS. Network and discrete location: Models, algorithms, and applications. New York, NY: Wiley; 1995.

Alumur S, Kara BY. Network hub location problems: The state of the art. Euro J Oper Res. 2008 Oct;190(1):1-21.

Klose A, Drexl A. Facility location models for distribution system design. Euro J Oper Res. 2004 Apr;162(1):4-29.

Balinski ML. Integer programming: Methods, uses, computation. Manag Sci. 1965 Nov;12(3):253-313.

Andreas K, Görtz S. A branch-and-price algorithm for the capacitated facility location problem. Euro J Oper Res. 2007 June;179(3):1109-1125.

Wu L, Zhang X, Zhang J. Capacitated facility location problem with general setup cost. Comp & Oper Res. 2006 May;33(5):1226-1241.

Melkote S, Daskin MS. Capacitated facility location-network design problems. Euro J Oper Res. 2001 Mar;129(3):481-495.

Kuehn AA, Hamburger MJ. A heuristic program for locating warehouses. Manag Sci. 1963 July;9(4):643-666.

Davis PS, Ray TL. A branch-bound algorithm for capacitated facilities location problem. Naval Res Log Quar. 1969 Oct;16(3):331-344.

Akinc U, Khumawala BM. An efficient branch and bound algorithm for the capacitated warehouse location problem. Manag Sci. 1977 Feb; 23(6):585-594.

Jacobsen SK. Heuristics for the capacitated plant location model. Euro J Oper Res. 1983 Mar;12(3):253-261.

Van Roy TJ. A cross decomposition algorithm for capacitated facility location. Oper Res. 1986 Feb;34(1):145-163.

Beasley JE. An algorithm for solving large capacitated warehouse location problems. Euro J Oper Res. 1988 Feb;33(3):314-325.

Magnanti TL. Wong RT. Decomposition methods for facility location problems [Internet]. MIT: Operations Research Centre; 1986 [cited 2015 Oct 26]. Available from: http://dspace.mit.edu/bitstream/handle/1721.1/5128/OR-153-86-24513129.pdf?

sequence=1.

Delmaire H, Diaz JA, Fernandez E, Ortega M. Reactive GRASP and Tabu search based heuristics for the single source capacitated plant location problem. Infor. 1999 Aug;37(3):194-225.

Agar MC, Salhi S. Lagrangian heuristics applied to a variety of large capacitated plant location problems. J Oper Res Soc. 1998 Oct;49(10):1072-1084.

Hindi KS, Pienkosz K. Efficient solution of large scale, single-source, capacitated plant location problems. J Oper Res Soc. 1999 Mar;50(3):268-274.

Hinojosa Y, Puerto J, Fernandez F. A multi-period two-echelon multi-commodity capacitated plant location problem. Euro J Oper Res. 2000 June;123(2):271-291.

Alfieri A, Brandimarte P, D’Orazio S. LP-based heuristics for the capacitated lot-sizing problem: The interaction of model formulation and solution algorithm. Inter J Prod Res. 2002 Jan;40(2):441-458.

Baldacci R, Hadjiconstantinou E, Maniezzo V, Mingozzi A. A new method for solving capacitated location problems based on a set partitioning approach. Comp & Oper Res. 2002 Apr;29(4):365-386.

Diaz JA, Fernandez E. A branch-and-price algorithm for the single source capacitated plant location problem. J Oper Res Soc. 2002 July;53(7):728-740.

Ghiani G, Guerriero F, Musmanno R. The capacitated plant location problem with multiple facilities in the same site. Comp & Oper Res. 2002 Nov;29(13):1903-1912.

Cortinhal MJ, Captivo ME. Upper and lower bounds for the single source capacitated location problem. Euro J Oper Res. 2003 Dec;151(2):333-351.

Ahuja RK, Orlin JB, Pallottino S, Scaparra MP, Scutella MG. A multi-exchange heuristic for the single-source capacitated facility location problem. Manag Sci. 2004 Jun;50(6):749-760.

Contreras, IA, Diaz JA. Scatter search for the single source capacitated facility location problem. Annals Oper Res. 2008 Jan;157(1):73-89.

Chen CH, Ting CJ. Combining Lagrange heuristic and ant colony system to solve the single source capacitated facility location problem. Trans Res Part E. 2008 Nov;44(6);1099-1122.

Lai M, Sohn H, Tseng T, Chiang C. A hybrid algorithm for capacitated plant location problem. Exp Sys with App. 2010 Dec;37(12):8599-8605.

Eiselt HA, Laporte G. Location of a new facility on a linear market in the presence of weights. Asia-Pacific J Oper Res. 1988 Jan;5:160-165.

Guneri AF, Cengiz M, Seker S. A fuzzy ANP approach to shipyard location selection. Exp Sys with App. 2009 May;36(4):7992-7999.

Peer SK. A multi-criteria procedure for the user interface components layout problem, Asia-Pacific J Oper Res. 2009 Apr;26(2):257-284.

Li Y, Liu X, Chen Y. Selection of logistics centre location using axiomatic fuzzy set and TOPSIS methodology in logistics management. Exp Sys with App. 2011 Jun;38(6):7901-7908.

Ozcan T, Celebi N, Esnaf S. Comparative analysis of multi-criteria decision making methodologies and implementation of a warehouse location selection problem. Exp Sys with App. 2011 Aug;38(8):9773-9779.

ReVelle CS, Swain R. Central facilities location. Geog Analy. 1970 Jan;2(1):30-42.

Erlenkotter D. A comparative study of approaches to dynamic location problems. Euro J Oper Res. 1981 Feb;6(2):133-143.

Tervonen T, Figueira J. A survey on stochastic multi-criteria acceptability analysis methods. J Multi-Crit Dec Analy. 2008 Nov;15(1-2):1-14.

ahdelma R, Hokkanen J, Salminen P. SMAA-stochastic multi-objective acceptability analysis. Euro J Oper Res. 1998 Apr;106(1):137-143.

Hokkanen J, Lahdelma R, Miettinen K, Salminen P. Determining the implementation order of a general plan by using a multi-criteria method. J Multi-Crit Dec Analy. 1998 Sep;7(5):273-284.

Lahdelma R, Salminen P. SMAA-2: stochastic multi-criteria acceptability analysis for group decision making. Oper Res. 2001 Jun;49(3):444-454.

Lahdelma R, Miettinen K, Salminen P. Ordinal criteria in stochastic multi-criteria acceptability analysis (SMAA). Euro J Oper Res. 2003 May;147(1):117-127.

Lahdelma R, Miettinen K, Salminen P. Reference point approach for multiple decision makers, Euro J Oper Res. 2005 Aug;164(3):785-791.

Tervonen T, Lahdelma R, Dias JA, Figueira J, Salminen P. SMAA-TRI: A parameter stability analysis method for ELECTRE-TRI, In: Kiker GA, Linkov I, editors. Environmental Security in Harbors and Coastal Areas. Berlin: Springer, 2007; p. 217-231.

Lahdelma R, Salminen P. Stochastic multi-criteria acceptability analysis using the data envelopment model. Euro J Oper Res. 2006 Apr;170(1):241-252.

Hokkanen J, Lahdelma R, Salminen P. A multiple criteria decision model for analyzing and choosing among different development patterns for the Helsinki cargo harbor. Socio-Eco Plan Sci. 1999 Mar;33(1):1-23.

Lahdelma R, Salminen P, Hokkanen J. Locating

a waste treatment facility by using stochastic multi-criteria acceptability analysis with ordinal criteria. Euro J Oper Res. 2002 Oct;142(2):345-356.

Menou A, Benallou A, Lahdelma R, Salminen P. Decision support for centralizing cargo at a Moroccan airport hub using stochastic multi-criteria acceptability analysis. Euro J Oper Res. 2010 Aug;204(3):624-629.

Figueira J, Greco S, Ehrgott M, editors. Multiple criteria decision analysis: State of the art surveys. Springer: Science & Business Media; 2005.

[Tervonen T, Figueira JR, Lahdelma R, Dias JA, Salminen P. A stochastic method for robustness analysis in sorting problems. Euro J Oper Res. 2009 Jan;192(1):236-242.

Tervonen T. SMAA: open source software for SMAA computations. Inter J Sys Sci. 2014 Jan;45(1):69-81.

Ministry of Development [Internet] [cited 2015 Oct 26] Available from: http://www.kalkinma.gov.tr.

Published
24.06.2016
Copyright (c) 2023 Selahattin Karabay, Erkan Köse, Mehmet Kabak, Eren Ozceylan