# An approach of fuzzy and TOPSIS to bi-level multi-objective nonlinear fractional programming problem

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## Abstract

This paper proposes a solution technique to bi-level multi-objective nonlinear fractional programming problem which is based on the concept of TOPSIS (technique for order preference by similarity to ideal solution) and fuzzy goal programming approach. Nonlinear polynomial functions are considered as the numerators as well as denominators of the fractional objectives at each level. The concept used implements simultaneous minimization and maximization of the functions (numerators and denominators of fractional objectives, decision variables controlled by the upper level decision makers) from their respective aspired (ideal) and acceptable (anti-ideal) values. Distance functions and their corresponding fuzzy membership functions are constructed at both levels for the objectives. Aspired and acceptable values of the decision variables of upper level are ascertained using a certain process. The sum of only under deviational variables obtained from the fuzzy membership goals of the distance functions and the decision variables controlled by upper level decision maker is minimized to obtain the best compromise solution of the concerned bi-level problem. Some comparative discussions with an existing approach are incorporated, and two illustrative numerical examples are discussed to demonstrate the effectiveness of the proposed method.

## Keywords

Nonlinear fractional programming Bi-level multi-objective optimization Distance functions TOPSIS Fuzzy goal programming## Notes

### Acknowledgements

The authors would like to thank the editor and anonymous referees for their valuable suggestions and comments to improve the quality of the paper.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

## References

- Abo-Sinna MA (2000) Extensions of the TOPSIS for multi-objective dynamic programming problems under fuzziness. Adv Model Anal B 43(3/4):1IV–24IVGoogle Scholar
- Abo-Sinna MA (2001) A bi-level non-linear multi-objective decision making under fuzziness. Opsearch 38(5):484–495MathSciNetzbMATHGoogle Scholar
- Abo-Sinna MA, Amer AH (2005) Extensions of TOPSIS for multi-objective large-scale nonlinear programming problems. Appl Math Comput 162(1):243–256MathSciNetzbMATHGoogle Scholar
- Abo-Sinna MA, Amer AH, Ibrahim AS (2008) Extensions of TOPSIS for large scale multi-objective non-linear programming problems with block angular structure. Appl Math Model 32(3):292–302zbMATHGoogle Scholar
- Ahlatcioglu M, Tiryaki F (2007) Interactive fuzzy programming for decentralized two-level linear fractional programming (DTLLFP) problems. Omega 35(4):432–450Google Scholar
- Baky IA (2009) Fuzzy goal programming algorithm for solving decentralized bi-level multi-objective programming problems. Fuzzy Sets Syst 160(18):2701–2713MathSciNetzbMATHGoogle Scholar
- Baky IA (2010) Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach. Appl Math Model 34(9):2377–2387MathSciNetzbMATHGoogle Scholar
- Baky IA (2014) Interactive TOPSIS algorithms for solving multi-level non-linear multi-objective decision-making problems. Appl Math Model 38(4):1417–1433MathSciNetzbMATHGoogle Scholar
- Baky IA, Abo-Sinna MA (2013) TOPSIS for bi-level MODM problems. Appl Math Model 37(3):1004–1015MathSciNetzbMATHGoogle Scholar
- Bialas WF, Karwan MH (1984) Two-level linear programming. Manag Sci 30(8):1004–1020MathSciNetzbMATHGoogle Scholar
- Chen F, Huang GH, Fan YR, Liao RF (2016) A nonlinear fractional programming approach for environmental-economic power dispatch. Int J Electr Power Energy Syst 78:463–469Google Scholar
- Emam OE (2006) A fuzzy approach for bi-level integer non-linear programming problem. Appl Math Comput 172(1):62–71MathSciNetzbMATHGoogle Scholar
- Emam OE (2013) Interactive approach to bi-level integer multi-objective fractional programming problem. Appl Math Comput 223:17–24MathSciNetzbMATHGoogle Scholar
- Hwang CL, Yoon K (2012) Multiple attribute decision making: methods and applications a state-of-the-art survey, vol 186. Springer, New YorkzbMATHGoogle Scholar
- Lachhwani K (2014) On solving multi-level multi objective linear programming problems through fuzzy goal programming approach. Opsearch 51(4):624–637MathSciNetzbMATHGoogle Scholar
- Lachhwani K (2015) Modified FGP approach for multi-level multi objective linear fractional programming problems. Appl Math Comput 266:1038–1049MathSciNetzbMATHGoogle Scholar
- Lachhwani K, Poonia MP (2012) Mathematical solution of multilevel fractional programming problem with fuzzy goal programming approach. J Ind Eng Int 8(1):1–11Google Scholar
- Lai YJ, Liu TY, Hwang CL (1994) Topsis for MODM. Eur J Oper Res 76(3):486–500zbMATHGoogle Scholar
- Malhotra N, Arora SR (2000) An algorithm to solve linear fractional: bi-level programming problem via goal programming. Opsearch 37(1):1–13MathSciNetzbMATHGoogle Scholar
- Miettinen K (2012) Nonlinear multiobjective optimization, vol 12. Springer, New YorkzbMATHGoogle Scholar
- Mishra S (2007) Weighting method for bi-level linear fractional programming problems. Eur J Oper Res 183(1):296–302zbMATHGoogle Scholar
- Mishra S, Ghosh A (2006) Interactive fuzzy programming approach to bi-level quadratic fractional programming problems. Ann Oper Res 143(1):251–263zbMATHGoogle Scholar
- Mohamed RH (1997) The relationship between goal programming and fuzzy programming. Fuzzy Sets Syst 89(2):215–222MathSciNetGoogle Scholar
- Pal BB, Moitra BN (2003) A fuzzy goal programming procedure for solving quadratic bilevel programming problems. Int J Intell Syst 18(5):529–540zbMATHGoogle Scholar
- Pramanik S, Dey PP (2011) Bi-level linear fractional programming problem based on fuzzy goal programming approach. Int J Comput Appl 25(11):34–40Google Scholar
- Pramanik S, Roy TK (2007) Fuzzy goal programming approach to multilevel programming problems. Eur J Oper Res 176(2):1151–1166zbMATHGoogle Scholar
- Roghanian E, Aryanezhad MB, Sadjadi SJ (2008) Integrating goal programming, Kuhn-Tucker conditions, and penalty function approaches to solve linear bi-level programming problems. Appl Math Comput 195(2):585–590MathSciNetzbMATHGoogle Scholar
- Sakawa M, Nishizaki I (2001) Interactive fuzzy programming for two-level linear fractional programming problems. Fuzzy Sets Syst 119(1):31–40MathSciNetzbMATHGoogle Scholar
- Sakawa M, Nishizaki I, Uemura Y (2000) Interactive fuzzy programming for two-level linear fractional programming problems with fuzzy parameters. Fuzzy Sets Syst 115(1):93–103MathSciNetzbMATHGoogle Scholar
- Shih HS, Lai YJ, Lee ES (1996) Fuzzy approach for multi-level programming problems. Comput Oper Res 23(1):73–91MathSciNetzbMATHGoogle Scholar
- Shi X, Xia H (1997) Interactive bilevel multi-objective decision making. J Oper Res Soc 48(9):943–949zbMATHGoogle Scholar
- Sinha S, Biswal MP (2000) Fuzzy programming approach to bi-level linear programming problems. J Fuzzy Math 8(2):337–348MathSciNetzbMATHGoogle Scholar
- Stancu-Minasian IM (1997) Fractional programming: theory, methods and applications, vol 409. Kluwer Academic Publishers, NetherlandszbMATHGoogle Scholar
- Veeramani C, Sumathi M (2016) A new method for solving fuzzy linear fractional programming problems. J Intell Fuzzy Syst 31(3):1831–1843zbMATHGoogle Scholar
- Zahmatkesh F, Cao BY (2016) On the Fuzzy fractional posynomial geometric programming problems. In: Fuzzy systems operations research and management. Springer, Cham, pp 97–108Google Scholar
- Zheng Y, Liu J, Wan Z (2014) Interactive fuzzy decision making method for solving bilevel programming problem. Appl Math Model 38(13):3136–3141MathSciNetzbMATHGoogle Scholar
- Zimmermann HJ (1978) Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst 1(1):45–55MathSciNetzbMATHGoogle Scholar
- Zimmermann HJ (1987) Fuzzy sets, decision making, and expert systems. Kluwer Academic Publisher, NetherlandsGoogle Scholar