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Optimization for multi-objective sum of linear and linear fractional programming problem: fuzzy nonlinear programming approach

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Abstract

Multi-objective linear plus linear fractional programming problem is an emerging tool for solving problems in different environments such as production planning, financial and corporate planning and healthcare and hospital planning which has attracted many researchers in recent years. This paper presents a method to find a Pareto optimal solution for the multi-objective linear plus linear fractional programming problem through nonlinear membership function. The proposed approach defines a fuzzy goal for each objective through a nonlinear membership function. By means of nonlinear membership function, the multi-objective linear plus linear fractional programming problem transformed into a multi-objective nonlinear programming problem. Applying the linear approximation method, the nonlinear objectives are converted into linear. In order to solve the multi-objective linear programming problem, the fuzzy goal programming model is formulated by minimizing the negative deviational variables. The proposed procedure is illustrated through numerical examples and a real-life application problem. Further, it is compared with the existing methods. Finally, the Euclidean distance function has been used to prove the efficiency of the proposed method.

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Abbreviations

LPP:

Linear programming problem

LPLFPP:

Linear plus linear fractional programming problem

MOLPLFPP:

Multi-objective linear plus linear fractional programming problem

FMONLFPP:

Fuzzy multi-objective nonlinear fractional programming problem

FMOLPP:

Fuzzy multi-objective linear programming problem

FMOLFPP:

Fuzzy multi-objective linear fractional programming problem

MOLPP:

Multi-objective linear programming problem

FFLFPP:

Fully fuzzy linear fractional programming problem

MOFPP:

Multi-objective fractional programming problem

MOMIFCSTP:

Multi-objective multi-item fixed-charge solid transportation problem

References

  1. Ahmed, Z.H.: An experimental study of a hybrid genetic algorithm for the maximum traveling salesman problem. Math. Sci. 7(1), 10 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Arora, J.: Solving sum of fuzzy linear plus linear fractional programming problem with bounded decision variables. Int. J. Math. Arch. 8(7), 241–245 (2017)

    Google Scholar 

  3. Charnes, A., Cooper, W.W.: Programming with linear fractional functionals. Nav. Res. Logist. 9(34), 181–186 (1962)

    MathSciNet  MATH  Google Scholar 

  4. Chang, C.T.: A goal programming approach for fuzzy multiobjective fractional programming problems. Int. J. Syst. Sci. 40(8), 867–874 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Chen, L.H., Ko, W.C., Yeh, F.T.: Approach based on fuzzy goal programing and quality function deployment for new product planning. Eur. J. Oper. Res. 259(2), 654–663 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Das, S.K., Mandal, T.: A new model for solving fuzzy linear fractional programming problem with ranking function. J. Appl. Res. Ind. Eng. 4(2), 89–96 (2017)

    MathSciNet  Google Scholar 

  7. De, P.K., Deb, M.: Using goal programming approach to solve fuzzy multi-objective linear fractional programming problems. In: Computational Intelligence and Computing Research, pp. 1–5 (2016)

  8. Doke, D.M., Jadhav, V.A.: Approach to solve multi objective linear fractional programming problem. Int. J. Math. 4(5), 1379–1386 (2016)

    Google Scholar 

  9. Ebrahimnejad, A., Ghomi, S.J., Mirhosseini-Alizamini, S.M.: A revisit of numerical approach for solving linear fractional programming problem in a fuzzy environment. Appl. Math. Model. 57, 459–473 (2018)

    MathSciNet  MATH  Google Scholar 

  10. Ebrahimnejad, A., Ghomi, S.J., Mirhosseini-Alizamini, S.M.: A new approach for solving fully fuzzy linear fractional programming problems. In: Knowledge-Based Engineering and Innovation, pp. 0862–0865 (2017)

  11. Ebrahimnejad, A., Nasseri, S.H., Lotfi, F.H., Soltanifar, M.: A primal-dual method for linear programming problems with fuzzy variables. Eur. J. Ind. Eng. 4(2), 189–209 (2010)

    Google Scholar 

  12. Garg, H., Rani, M., Sharma, S.P., Vishwakarma, Y.: Intuitionistic fuzzy optimization technique for solving multi-objective reliability optimization problems in interval environment. Expert Syst. Appl. 41(7), 3157–3167 (2014)

    Google Scholar 

  13. Garai, T., Garg, H.: Multi-objective linear fractional inventory model with possibility and necessity constraints under generalised intuitionistic fuzzy set environment. CAAI Trans. Intell. Technol. 4(3), 175–181 (2019)

    Google Scholar 

  14. Jain, S., Lachhwani, K.: Sum of linear and fractional multi objective programming problem under fuzzy rules constraints. Aust. J. Basic Appl. Sci. 2(4), 1204–1208 (2008)

    Google Scholar 

  15. Jain, S., Lachhwani, K.: Linear plus fractional multiobjective programming problem with homogeneous constraints using fuzzy approach. Oper. Res. 2(1), 41–49 (2010)

    MATH  Google Scholar 

  16. Jain, S., Mangal, A., Parihar, P.R.: Solution of a multi objective linear plus fractional programming problem containing non-differentiable term. Int. J. Math. Sci. Eng. Appl. 2(2), 221–229 (2008)

    MATH  Google Scholar 

  17. Kheirfam, B.: Sensitivity analysis in linear-plus-linear fractional programming problems. Iran. J. Optim. 1(1), 1–12 (2009)

    Google Scholar 

  18. Karimi, M., Karimi, B.: Linear and conic scalarizations for obtaining properly efficient solutions in multiobjective optimization. Math. Sci. 11(4), 319–325 (2017)

    MathSciNet  MATH  Google Scholar 

  19. Lachhwani, K., Dwivedi, A.: Bi-level and multi-Level programming problems: taxonomy of literature review and research issues. Arch. Comput. Methods Eng. 25, 1–31 (2017)

    MathSciNet  Google Scholar 

  20. Mangal, ASangeeta: Alternative approach to solve linear plus linear fractional programming problems. Int. J. Math. Sci. Eng. Appl. 2(4), 11–17 (2008)

    MATH  Google Scholar 

  21. Midya, S., Roy, S.K.: Analysis of interval programming in different environments and its application to fixed-charge transportation problem. Discrete Math. Algorithms Appl. 9(03), 1750040 (2017)

    MathSciNet  MATH  Google Scholar 

  22. Najafi, H.S., Edalatpanah, S.A., Dutta, H.: A nonlinear model for fully fuzzy linear programming with fully unrestricted variables and parameters. Alex. Eng. J. 55(3), 2589–2595 (2016)

    Google Scholar 

  23. Nasseri, S.H., Ebrahimnejad, A.: A fuzzy primal simplex algorithm and its application for solving flexible linear programming problems. Eur. J. Ind. Eng. 4(3), 372–389 (2010)

    Google Scholar 

  24. Nishad, A.K., Singh, S.R.: Goal programming for solving fractional programming problem in fuzzy environment. Appl. Math. 6(14), 2360–2374 (2015)

    Google Scholar 

  25. Pal, B.B., Moitra, B.N., Maulik, U.: A goal programming procedure for fuzzy multiobjective linear fractional programming problem. Fuzzy Sets Syst. 139(2), 395–405 (2003)

    MathSciNet  MATH  Google Scholar 

  26. Peri, T., Babi, Z., Rei, S.: A goal programming procedure for solving fuzzy multiobjective fractional linear programming problems. Croat. Oper. Res. Rev. 5(2), 401–414 (2014)

    MathSciNet  Google Scholar 

  27. Pramanik, S., Dey, P.P., Giri, B.C.: Multi-objective linear plus linear fractional programming problem based on Taylor series approximation. Int. J. Comput. Appl. 32(8), 61–68 (2011)

    Google Scholar 

  28. Prakash, K.A., Suresh, M., Vengataasalam, S.: A new approach for ranking of intuitionistic fuzzy numbers using a centroid concept. Math. Sci. 10(4), 177–184 (2016)

    MathSciNet  MATH  Google Scholar 

  29. Rani, D., Gulati, T.R., Garg, H.: Multi-objective non-linear programming problem in intuitionistic fuzzy environment: optimistic and pessimistic view point. Expert Syst. Appl. 64, 228–238 (2016)

    Google Scholar 

  30. Roy, S.K., Midya, S.: Multi-objective fixed-charge solid transportation problem with product blending under intuitionistic fuzzy environment. Appl. Intell. 49(10), 3524–3538 (2019)

    Google Scholar 

  31. Roy, S.K., Midya, S., Weber, G.W.: Multi-objective multi-item fixed-charge solid transportation problem under twofold uncertainty. Neural Comput. Appl. 31(12), 8593–8613 (2019)

    Google Scholar 

  32. Salahi, M., Fallahi, S.: Parametric approach for solving quadratic fractional optimization with a linear and a quadratic constraint. Comput. Appl. Math. 35(2), 439–446 (2016)

    MathSciNet  MATH  Google Scholar 

  33. Saborido, R., Ruiz, A.B., Bermdez, J.D., Vercher, E., Luque, M.: Evolutionary multi-objective optimization algorithms for fuzzy portfolio selection. Appl. Soft Comput. 39, 48–63 (2016)

    Google Scholar 

  34. Sadia, S., Gupta, N., Ali, Q.M., Bari, A.: Solving multi-Objective linear plus linear fractional programming problem. J. Stat. Appl. Probab. 4(2), 253–258 (2015)

    Google Scholar 

  35. Singh, P., Kumar, S.D., Singh, R.K.: Fuzzy method for multi-objective linear plus linear fractional programming problem. Int. Math. Forum 5(60), 2971–2983 (2010)

    MathSciNet  MATH  Google Scholar 

  36. Singh, P., Kumar, S.D., Singh, R.: Fuzzy multi-objective linear plus linear fractional programming problem: approximation and goal programming approach. Int. J. Math. Comput. Simul. 5(5), 395–404 (2011)

    Google Scholar 

  37. Tayebikhorami, S., Nikoo, M.R., Sadegh, M.: A fuzzy multi-objective optimization approach for treated wastewater allocation. Environ. Monit. Assess. 191(7), 468 (2019)

    Google Scholar 

  38. Toksar, M.D.: Taylor series approach to fuzzy multiobjective linear fractional programming. Inf. Sci. 178(4), 1189–1204 (2008)

    MathSciNet  Google Scholar 

  39. Veeramani, C., Sumathi, M.: Solving the linear fractional programming problem in a fuzzy environment: numerical approach. Appl. Math. Model. 40(11–12), 6148–6164 (2016)

    MathSciNet  MATH  Google Scholar 

  40. Veeramani, C., Sumathi, M.: Fuzzy mathematical programming approach for solving fuzzy linear fractional programming problem. RAIRO Oper. Res. 48(1), 109–122 (2014)

    MathSciNet  MATH  Google Scholar 

  41. Yu, P.L.: A class of solutions for group decision problems. Manag. Sci. 19(8), 936–946 (1973)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Science, PSG College of Technology, Coimbatore, 641 004, India

    C. Veeramani & S. Sharanya

  2. Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

    Ali Ebrahimnejad

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  1. C. Veeramani
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  2. S. Sharanya
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  3. Ali Ebrahimnejad
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Correspondence to S. Sharanya.

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Veeramani, C., Sharanya, S. & Ebrahimnejad, A. Optimization for multi-objective sum of linear and linear fractional programming problem: fuzzy nonlinear programming approach. Math Sci 14, 219–233 (2020). https://doi.org/10.1007/s40096-020-00333-w

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