Linear programming problems with some multi-choice fuzzy parameters
DOI:
https://doi.org/10.2298/YJOR160520004PKeywords:
linear programming, triangular fuzzy number, trapezoidal fuzzy number, multi-choice programming, fuzzy programmingAbstract
In this paper, we consider some Multi-choice linear programming (MCLP) problems where the alternative values of the multi-choice parameters are fuzzy numbers. There are some real-life situations where we need to choose a value for a parameter from a set of different choices to optimize our objective, and those values of the parameters can be imprecise or fuzzy. We formulate these situations as a mathematical model by using some fuzzy numbers for the alternatives. A defuzzification method based on incentre point of a triangle has been used to find the defuzzified values of the fuzzy numbers. We determine an equivalent crisp multi-choice linear programming model. To tackle the multi-choice parameters, we use Lagranges interpolating polynomials. Then, we establish a transformed mixed integer nonlinear programming problem. By solving the transformed non-linear programming model, we obtain the optimal solution for the original problem. Finally, two numerical examples are presented to demonstrate the proposed model and methodology.References
Kaufmann, A., and Gupta, M.M., Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold, New York, 1985.
Saneifard, R., and Saneifard, R., ”A modified method for defuzzification by probability density function”, Journal of Applied Sciences Research, 7 (2) (2011) 102-110.
Yoon, K.P., ”A probabilistic approach to rank complex fuzzy numbers”, Fuzzy Sets and Systems, 80 (2) (1996) 167-176.
Atkinson, K.E., An Introduction to Numerical Analysis, John Wiley & Sons, UK, 2008.
Li, Duan, and Xiaoling, Sun, Nonlinear integer programming, Springer Science & Business Media, New York, USA, 84, 2006.
Barik, S.K., and Biswal, M.P., ”Probabilistic Quadratic Programming Problems with Some Fuzzy Parameters”, Advances in Operations Research, Article ID 635282 (2012) 13.
Dubois, D., Prade, H., ”Operations on fuzzy numbers”, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980.
Shang Gao and Zaiyue Zhang, ”Multiplication Operation on Fuzzy Numbers”, Journal of Software, 4 (4) (2009) 331-338.
Taleshian, A. and Rezvani, S., ”Multiplication Operation on Trapezoidal Fuzzy Numbers”, Journal of Physical Sciences, 15 (2011) 17-26.
Biswal, M.P., Acharya, S., ”Transformation of multi-choice linear programming problem”, Applied Mathematics and Computation, 210 (2009) 182-188.
Biswal, M.P., Acharya, S., ”Solving multi-choice linear programming problems by interpolating polynomials”, Mathematical and Computer Modelling 54 (2011) 1405-1412.
Sarker, Ruhul, Amin, and Charles, S. Newton., Optimization modelling: a practical approach, CRC Press, Florida, USA, 2007.
Rommelfanger, H., ”Fuzzy linear programming and its applications”, European Journal of Operational Research, 92 (1996) 512-527.
Buckley, J.J., ”Solving possibilistic linear programming”, Fuzzy Sets and Systems, 31 (1989) 329-341.
Zimmermann, H.J., ”Fuzzy programming and linear programming with several objective functions”, Fuzzy Sets and Systems, 1 (1978) 45-55.
Tong, S., ”Interval number and fuzzy number linear programming”, Fuzzy Sets and Systems, 66 (1994) 301-306.
Zadeh, L.A., ”Fuzzy sets”, Information and Control, 8 (1965) 338-353.
Tanaka, H., Ichihashi, H., Asai, K., ”A formulation of fuzzy linear programming problems based on comparison of fuzzy numbers”, Control and Cybernet, 13 (1984) 185-194.
Delgado, M., Verdegay, J.L., Vila, M.A., ”A general model for fuzzy linear programming”, Fuzzy Sets and Systems, 29 (1989) 21-29.
Nasseri, S.H., and Behmanesh, E., ”Linear Programming with Triangular Fuzzy Numbers A Case Study in a Finance and Credit Institute”, Fuzzy Information and Engineering, 5 (3) (2013) 295-315.
Schrage, L., LINGO release 11.0, LINDO System, Inc, 2008.
Shashi, A., and Uday, S., ”Fully Fuzzy Multi-Choice Multi-Objective Linear Programming Solution via Deviation Degree”, International Journal of Pure and Applied Sciences and Technology, 19 (1) (2013) 49-64.
Dutta, D., and Murthy, A.S., ”Multi-choice goal programming approaches for a fuzzy transportation problem”, IJRRAS, 2 (2) (2010) 132-139.
Rouhparvar, H., and Panahi, A., ”A new definition for defuzzification of generalized fuzzy numbers and its application”, Applied Soft Computing, 30 (2015) 577-584.
Stephen Wolfram, The Mathematica Book, Fifth Edition-Wolfram Media, 2003.
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