Soft Computing

, Volume 23, Issue 14, pp 5605–5618 | Cite as

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

  • Suvasis NayakEmail author
  • Akshay Ojha
Methodologies and Application


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.


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



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.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Basic SciencesIndian Institute of Technology BhubaneswarBhubaneswarIndia

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