### A Novel Technique for Solving Multiobjective Fuzzy Linear Programming Problems

#### Abstract

This study considers multiobjective fuzzy linear programming (MFLP) problems in which the coefficients in the objective functions are triangular fuzzy numbers. The study proposing a new technique to transform MFLP problems into the equivalent single fuzzy linear programming problem and then solving it via linear ranking function using the simplex method, supported by numerical example.

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Allahviranloo, T., Hosseinzadeh, L.F., Kiasary, M.K., Kiani, N.A. and

Alizadeh, L., 2008. Solving full fuzzy linear programming problem by the

ranking function. Applied Mathematical Sciences, 2(1), pp.19-32.

Amid, A., Ghodsypour, S.H. and Obrien, C., 2006. Fuzzy multiobjective

linear model for supplier selection in a supply chain. International journal of Production Economics, 104, pp.394-407.

Amid, A., Ghodsypour, S.H. and Obrien, C., 2011. A weighted maxmin model for fuzzy multi-o´ bjective supplier selection in a supply chain. International Journal of Production Economics, 131, pp.139-145.

Baky, I A., 2009. Fuzzy goal programming algorithm for solving decentralized bi-level multi-objective programming problems. Fuzzy Sets and Systems, 160, pp.2701-2713.

Baky, I.A., 2010. Solving multi-level multi-objective linear programming

problems through fuzzy goal programming approach. Applied Mathematical Modelling, 34, pp.2377-2387.

Bellman, R.F. and Zadeh, L.A., 1970. Decision making in fuzzy environment. Management Science, 17(4), pp.141-146.

Buckley, J.J. and Feuring, T., 2000. Evolutionary algorithm solution to fuzzy problems. Fuzzy Sets and Systems, 109, pp.35-53.

Cadenas, J.M. and Verdegay, J.L., 2000. Using ranking functions in multiobjective fuzzy linear programming. FuzzySets and Systems, 111, pp.47-53.

Chen, L.H. and Ko, W.C., 2009. Fuzzy linear programming models for new product design using QFD with FMEA. Applied Mathematical Modelling, 33(2), pp.633-647.

Chiang, J., 2005. The OS of the transportation problem with fuzzy demand and fuzzy product. Journal of Information Science and Engineering, 21, pp.439-451.

Dantzig, G.B., 1963. Linear programming and extensions. University Press, Princeton.

Dehghan, M., Hashemi, B. and Ghatee, M., 2006. Computational methods for solving fully fuzzy linear system. Applied Mathematics and Computation, 179, pp.328-343.

Dubois, D. and Prade, H., 1978. Operations on fuzzy numbers. International Journal of Systems Science, 9(6), pp.613-626.

Ebrahimnejad, A., 2011. Sensitivity analysis in fuzzy number linear programming problems. Mathematical and Computer Modelling, 53(9-10), pp.1878-1888.

Ebrahimnejad, A. and Tavana, M., 2014. A novel method for solving linear programming problems with symmetrictrapezoidal fuzzy numbers. Applied Mathematical Modelling. Availale from: http://www.dx.doi.org/10.1016/j.apm.2014.02.024.13.

Fang, S.C. and Hu, C.F., 1996. Linear programming with fuzzy coefficients in constraints. Computers and Mathematics with Applications, 37(10), pp.63-76.

Fortemps, P. and Roubens, M., 1996. Ranking and defuzzification methods based on area compensation. Fuzzy Sets Systems, 82(3), pp.319-330.

Ganesan, K. and Veeramani, P., 2006. Fuzzy linear programming with trapezoidal fuzzy numbers. Annals of Operations Research, 143, pp.305-315.

Garcia-Aguado, C. and Verdegay, J.L., 1993. On the sensitivity of membership functions for fuzzy linear programming problems. Fuzzy Sets Systems, 56(1), pp.47-49.

Gupta, A. and Kumar, A., 2012. A new method for solving linear multi-objective transportation problems with fuzzy parameters. Applied Mathematical Modelling, 36, pp.1421-1430.

Hamadameen, A.O. and Zainuddin, Z.M., 2013. Multiobjective fuzzy stochastic linear programming problems in the 21st century. Life Science Journal, 10(4), pp.616-647.

Hashemi, S., Nasrabadi, M.M.E. and Nasrabadi, M., 2006. Fully fuzzified linear programming, solution and duality. Journal of Intelligent and Fuzzy Systems, 17(3), pp.253-261.

Hassanzadeh, A.S., Razmi, J. and Zhang, G., 2011. Supplier selection and order allocation based on fuzzy SWOT analysis and fuzzy linear programming. Expert Systems with Applications, 38(1), pp.334-342.

Hosseinzadeh Lotfi, F., Allahviranloo, T., Alimardani, J.M. and Alizadeh, L., 2009. Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution. Applied Mathematical Modelling, 33(7), pp.3151-3156.

Inuiguchi, M., Ichihashi, H. and Tanaka, H., 1990. Fuzzy programming: A survey of recent developments. In: Slowinski, R. and Teghem, J., editors. Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming Underuncertainty. Kluwer Academic Publishers, Dordrecht.

Iskander, M.G., 2002. Comparison of fuzzy numbers using possibility

programming: Comments and new concepts. Computers and Mathematics with Applications, 43, pp.833-840.

Iskander, M.G., 2008. A computational comparison between two evaluation criteria in fuzzy multiobjective linear programs using possibility programming. Computers and Mathematics with Applications, 55, pp.2506-2511.

Kumar, A., Kaur, J. and Singh, P., 2011. A new method for solving fully fuzzy linear programming problems. Applied Mathematical Modelling, 35, pp.817-823.

Lai, Y.J. and Hwaang, C.L., 1992. Fuzzy Mathematical Programming Methods and Applications. Springer, Berlin. Luhandjula, M.K., 1989. Fuzzy optimization: An appraisal. Fuzzy Sets and System, 30(3), pp.257-282.

Luhandjula, M.K. and Rangoaga, M.J., 2014. An approach for solving a fuzzy multiobjective programming problem. European Journal of Operational Research, 232, pp.249-255.

Mahdavi-Amiri, N. and Nasseri, S.H., 2006. Duality in fuzzy number linear programming by use of a certain linear ranking function. Applied Mathematics and Computation, 180(1), pp.206-216.

Mahdavi-Amiri, N. and Nasseri, S.H., 2007. Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables. Fuzzy Sets and Systems, 158(17), pp.1961-1978.

Maleki, H.R., 2003. Ranking functions and their applications to fuzzy linear programming. Far East Journal of Mathematical Sciences, 4(3), pp.283-301.

Maleki, H.R., Tata, M. and Mashinchi, M., 2000. Linear programming with

fuzzy variables. Fuzzy Sets and Systems, 109(1), pp.21-33.

Nasseri, S.H., Ardil, E., Yazdani, A. and Zaefarian, R., 2005. Simplex method for solving linear programming problem with fuzzy number. World Academy of Science, Engineering and Technology, 10, pp.284-288.

Negi, D.S. and Lee, E.S., 1993. Possibility programming by the comparison of fuzzy numbers. Computers and Mathematics with Application, 25, pp.43-50.

Peidro, D., Mula, J., Jimenez, M. and Botella, M., 2010. A fuzzy linear

programming based approach for tactical supply chain planning in an uncertainty environment. European Journal of Operational Research, 205(1), pp.65-80.

Rong, A. and Lahdelma, R., 2008. Fuzzy chance constrained linear programming model for optimizing the scrap charge in steel production. European Journal of Operational Research, 186(3), pp.953-964.

Roubens, M. and Jacques, T.J., 1991. Comparison of methodologies for fuzzy and stochastic multi-objective programming. Fuzzy Sets and Systems, 42(1), pp.119-132.

Sakawa, M., 1993. Fuzzy Sets and Interactive Multiobjective Optimization. Plenum Press, New York.

Sakawa, M., Nishizaki, I. and Uemura, Y., 2000. Interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters. Fuzzy Sets and Systems, 109, pp.3-19.

Sharma, S.D., 2012. Operations Research. Kedar Nath Ram Nath, Meerut, New Delhi, India.

Shoacheng, T., 1994. Interval number and fuzzy number linear programming. Fuzzy Sets and Systems, 66(3), pp.301-306.

Stanciulescu, C., Fortemps, P., Installe, M. and Wertz, V., 2003. Multiobjective fuzzy linear programming problems with fuzzy decision variables. European Journal of Operational Research, 149, pp.654-672.

Tanaka, H., Okuda, T. and Asai, K., 1974a. On fuzzy mathematical programming. Journal of Cybernetics, 3(4),pp.37-46.

Tanaka, H., Okuda, T. and Asai, K., 1974b. On fuzzy mathematical programming. Journal of Cybernetics, 3(4), pp.131-141.

Ullah Khan, I., Ahmad, T. and Maan, N., 2013. A simplified novel Technique for solving fully fuzzy linear programming problems. Journal of Optimization Theory and Applications, 159(2), pp.536-546.

Wang, L.X., 1997. A Course in Fuzzy Systems and Control. Prentice-Hall, Inc., USA.

Wang, X. and Kerre, E., 2001. Reasonable properties for the ordinary of fuzzy quantities (part II). Fuzzy Sets and Systems, 118(3), pp.375-405.

Wu, H., 2008a. Optimality conditions for linear programming problems with fuzzy coefficients. Computers and Mathematics with Applications, 55, pp.2807-2822.

Wu, H.C., 2008b. Using the technique of scalarization to solve the multiobjective programming problems with fuzzy coefficients. Mathematical and Computer Modelling, 48, pp.232-248.

Yager, R.R., 1981. A procedure for ordering fuzzy subsets of the unit interval. Information Sciences, 24(2), pp.143-161.

Yager, R.R. and Filev, D.P., 1994. Essentials of Fuzzy Modeling and Control. John Wiley and Sons, Inc., USA.

Zadeh, L.A., 1965. Fuzzy Sets. Information and Control, 8, pp.338-353.

Zhang, C., Yuan, X.H. and Lee, E.S., 2005. Duality theory in fuzzy mathematical programming problems with fuzzy coefficients. Computers and Mathematics with Applications, 49(11), pp.1709-1730.

Zimmermann, H.J., 1978. Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems, 1, pp.45-55.

Zimmermann, H.J., 1987. Fuzzy Sets, Decision Making and Expert Systems. Kluwer Academic Publishers, Boston.

DOI: http://dx.doi.org/10.14500/aro.10064

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