Document Type : Research Paper

**Authors**

Department of Mathematical Sciences, Faculty of Science & Technology, UKM Bangi, Selangor 43600, Malaysia

**Abstract**

This paper presents an efficient and straightforward method with less computational complexities to address the linear fractional programming with fuzzy coefficients (FLFPP). To construct the approach, the concept of $\alpha$-cut is used to tackle the fuzzy numbers in addition to rank them. Accordingly, the fuzzy problem is changed into a bi-objective linear fractional programming problem (BOLFPP) by the use of interval arithmetic. Afterwards, an equivalent BOLFPP is defined in terms of the membership functions of the objectives, which is transformed into a bi-objective linear programming problem (BOLPP) applying suitable non-linear variable transformations. Max-min theory is utilized to alter the BOLPP into a linear programming problem (LPP). It is proven that the optimal solution of the LPP is an $\epsilon$-optimal solution for the fuzzy problem. Four numerical examples are given to illustrate the method and comparisons are made to show the efficiency.

**Keywords**

[1] S. Ahmad, A. Ullah, A. Akgül, D. Baleanu, Analysis of the fractional tumour-immune-vitamins model with MittagLeffler kernel, Results in Physics, 19 (2020), Doi: 10.1016/j.rinp.2020.103559.

[2] T. Allahviranloo, R. Saneifard, Defuzzification method for ranking fuzzy numbers based on center of gravity, Iranian Journal of Fuzzy Systems, 9(6) (2012), 57-67.

[3] C. H. Antunes, M. J. Alves, J. Clímaco, Multiobjective linear and integer programming, Springer, Cham, (2016), Doi: 10.1007/978-3-319-28746-1.

[4] M. Borza, A. S. Rambely, A linearization to the sum of linear ratios programming problem, Mathematics, 9(9) (2021), Doi: 10.3390/math9091004.

[5] M. Borza, A. S. Rambely, A new method to solve multi-objective linear fractional problems, Fuzzy Information and Engineering, (2021), 1-12.

[6] M. Borza, A. S. Rambely, M. Saraj, Solving linear fractional programming problems with interval coefficients in the objective function. A new approach, Applied Mathematical Sciences, 6(69) (2012), 3443-3452.

[7] M. Borza, A. S. Rambely, M. Saraj, Parametric approach for an absolute value linear fractional programming with interval coefficients in the objective function, In AIP Conference Proceedings, 1602(1) (2014), 415-421.

[8] M. Borza, A. S. Rambely, M. Saraj, Fuzzy approaches to the multi objectives linear fractional programming problems with interval coefficients, Asian Journal of Mathematics and Computers Research, 4 (2015), 83-94.

[9] M. Chakraborty, S. Gupta, Fuzzy mathematical programming for multi objective linear fractional programming problem, Fuzzy Sets and Systems, 125(3) (2002), 335-342.

[10] A. Charnes, W. W. Cooper, Programming with linear fractional functionals, Naval Research Logistics Quarterly, 9(34) (1962), 181-186.

[11] S. Chanas, D. Kuchta, Linear programming problem with fuzzy coefficients in the objective function, Fuzzy Optimization, Physica-Verlag, Heidelberg, (1994), 148-157.

[12] V. Chinnadurai, S. Muthukumar, Solving the linear fractional programming problem in a fuzzy environment: Numerical approach, Applied Mathematical Modelling, 40(11) (2016), 6148-6164.

[13] C. Cruz, R. C. Silva, J. L. Verdegay, Extending and relating different approaches for solving fuzzy quadratic problems, Fuzzy Optimization and Decision Making, 10(3) (2011), 193-210.

[14] C. Cruz, R. C. Silva, J. L. Verdegay, A. Yamakami, A survey of fuzzy quadratic programming, Recent Patents on Computer Science, 1(3) (2008), 182-193.

[15] M. Darehmiraki, A novel parametric ranking method for intuitionistic fuzzy numbers, Iranian Journal of Fuzzy Systems, 16(1) (2019), 129-143.

[16] S. K. Das, S. A. Edalatpanah, T. Mandal, Application of linear fractional programming problem with fuzzy nature in industry sector, Filomat, 34(15) (2020), 5073-5084.

[17] S. K. Das, T. Mandal, S. A. Edalatpanah, A new approach for solving fully fuzzy linear fractional programming problems using the multi-objective linear programming, RAIRO-Operations Research, 51(1) (2017), 285-297.

[18] P. K. De, M. Deb, Solution of multi objective linear fractional programming problem by Taylor series approach, In 2015 International Conference on Man and Machine Interfacing (MAMI), 2015, IEEE, 1-5 (2015), Doi: 10.1109/MAMI.2015.7456582.

[19] W. Dinkelbach, On nonlinear fractional programming, Management Science, 13(7) (1967), 492-498.

[20] N. Güzel, A proposal to the solution of multi-objective linear fractional programming problem, In Abstract and Applied Analysis, (2013), Doi: 10.1155/2013/435030.

[21] A. Kaufmann, M. M. Gupta, Fuzzy mathematical models in engineering and management science, Elsevier Science Inc. 655 Avenue of the Americas New York, NYUnited States, 1988.

[22] B. Kheirfam, J. L. Verdegay, Strict sensitivity analysis in fuzzy quadratic programming, Fuzzy Sets and Systems, 198 (2012), 99-111.

[23] X. Liu, Y. L. Gao, B. Zhang, F. P. Tian, A new global optimization algorithm for a class of linear fractional programming, Mathematics, 7(9) (2019), 1-21.

[24] A. Mehra, S. Chandra, C. R. Bector, Acceptable optimality in linear fractional programming with fuzzy coefficients, Fuzzy Optimization and Decision Making, 6 (2007), 5-16.

[25] R. E. Moore, R. B. Kearfott, M. J. Cloud, Introduction to interval analysis, Society for Industrial and Applied Mathematics, 2009.

[26] S. Nayak, A. K. Ojha, Solution approach to multi-objective linear fractional programming problem using parametric functions, Opsearch, 56(1) (2019), 174-190.

[27] S. Nayak, A. K. Ojha, Multi-objective linear fractional programming problem with fuzzy parameters, In Soft Computing for Problem Solving, Springer, Singapore, (2019), 79-90.

[28] B. B. Pal, B. N. Moitra, U. Maulik, A goal programming procedure for fuzzy multi objective linear fractional programming problem, Fuzzy Sets and Systems, 139 (2003), 395-405.

[29] F. A. Pramy, M. A. Islam, Determining efficient solutions of multi-objective linear fractional programming problems and application, Open Journal of Optimization, 6 (2017), 164-175.

[30] R. E. Precup, R. C. David, E. M. Petriu, A. I. Szedlak-Stinean, C. A. Bojan-Dragos, Grey wolf optimizer-based approach to the tuning of PI-fuzzy controllers with a reduced process parametric sensitivity, IFAC-Papers OnLine, 49(5) (2016), 55-60.

[31] B. Radhakrishnan, P. Anukokila, Fractional goal programming for fuzzy solid transportation problem with interval cost, Fuzzy Information and Engineering, 6(3) (2014), 359-377.

[32] H. Rashmanlou, R. A. Borzooei, Vague graphs with application, Journal of Intelligent and Fuzzy Systems, 30(6) (2016), 3291-3299.

[33] I. M. Stancu-Minasian, Fractional programming: Theory, methods and applications, Kluwer Academic Publishers, Springer Netherlands, 1997.

[34] B. Stanojevic, M. Stanojevic, Solving method for linear fractional programming problem with fuzzy coefficients in the objective function, International Journal of Computers and Communications Control, 8 (2013), 146-152.

[35] M. D. Toksari, Taylor series approach to fuzzy multi objective linear fractional programming, Information Sciences, 178 (2008), 1189-1204.

[36] C. Veeramani, M. Sumathi, Fuzzy mathematical programming approach for solving fuzzy linear fractional programming problem, RAIRO-Operations research, 48(1) (2014), 109-122.

[37] L. X. Wang, A course in fuzzy systems and control, Prentice-Hall, International, Inc., 1996.

[38] C. Wang, J. Li, Periodic solution for a max-type fuzzy difference equation, Journal of Mathematics, (2020), Doi: 10.1155/2020/3094391.

[39] C. Wang, J. Li, L. Jia, Dynamics of a high-order nonlinear fuzzy difference equation, Journal of Applied Analysis and Computation, 11(1) (2021), 404-421.

[40] Y. Wang, L. Liu, S. Guo, Q. Yue, P. Guo, A bi-level multi-objective linear fractional programming for water consumption structure optimization based on water shortage risk, Journal of Cleaner Production, 237 (2019), Doi: 10.1016/j.jclepro.2019.117829.

[41] H. Zapata, N. Perozo, W. Angulo, J. Contreras, A hybrid swarm algorithm for collective construction of 3D structures, International Journal of Artificial Intelligence, 18(1) (2020), 1-18.

[42] H. J. Zimmermann, Fuzzy set theory and its applications, Springer Science and Business Media, 2011.

January and February 2022

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