Document Type : Review Paper


1 Department of Mathematics, Ganesh Dutt College, Begusarai, Bihar, India.

2 Department of Mathematics, Lalit Narayan Mithila University, Darbhanga, India


Multiple Criteria Decision Analysis (MCDA) has been widely investigated and successfully applied to many fields,owing to its great capability of modeling the process of actual decision-making problems and establishing proper evaluation and assessment mechanisms. With the development of management and economics, real-world decision-making problem are becoming diversified and complicated to an increasing extent, especially within a changeable and unpredictable enviroment. Multi-criteria is a decision-making technique that explicitly evaluates numerous contradictory criteria. TOPSIS is a well-known multi-criteria decision-making process. The goal of this research is to use TOPSIS to solve MCDM problems in a Pythagorean fuzzy environment. The distance between two Pythagorean fuzzy numbers is utilized to create the model using the spherical distance measure. To construct a ranking order of alternatives and determine the best one,the revised index approach is utilized. Finally, we look at a set of MCDM problems to show how the proposed method and approach work in practice. In addition, it shows comparative data from the relative closeness and updated index methods.


Main Subjects

  • Zadeh, L. A. (1965). Fuzzy sets. Information and control8(3), 338-353.
  • Bhowmik, M., Adak, A. K., & Pal, M. (2011). Application of generalized intuitionistic fuzzy matrices in multi-criteria decision making problem.  math. comput. sci.1(1), 19-31.
  • Adak, A. K., Manna, D., Bhowmik, M., & Pal, M. (2016). TOPSIS in generalized intuitionistic fuzzy environment. In Handbook of research on modern optimization algorithms and applications in engineering and economics(pp. 630-642). IGI Global.
  • Atanassov, K. T. (1993). A second type of intuitionistic fuzzy sets. BUSEFAL56, 66-70.
  • Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy sets and systems, 20(1), 87-96.
  • Ebrahimnejad, A., Adak, A. K., & Jamkhaneh, E. B. (2019). Eigenvalue of intuitionistic fuzzy matrices over distributive lattice. International journal of fuzzy system applications (IJFSA)8(1), 1-18.
  • Manna, D., & Adak, A. K. (2016). Interval-valued intuitionistic fuzzy R-subgroup of near-rings. Journal of fuzzy mathematics24(4), 985-994.
  • Nasseri, S. H., & Mizuno, S. (2010). A new method for ordering triangular fuzzy numbers. Iranian journal of optimization6(1), 720-729.
  • Nasseri, S. H., & Sohrabi, M. (2010). Hadi’s method and its advantage in ranking fuzzy numbers. Australian journal of basic applied sciences4(10), 4630-4637.
  • Nasseri, S. H. (2015). Ranking trapezoidal fuzzy numbers by using Hadi method. Australian journal of basic and applied sciences4(8), 3519-3525.
  • Yager, R. R., & Abbasov, A. M. (2013). Pythagorean membership grades, complex numbers, and decision making. International journal of intelligent systems28(5), 436-452.
  • Yager, R. R. (2013). Pythagorean fuzzy subsets. 2013 joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS)(pp. 57-61). IEEE.
  • Yager, R. R. (2016). Properties and applications of Pythagorean fuzzy sets. In imprecision and uncertainty in information representation and processing: new tools based on intuitionistic fuzzy sets and generalized nets, (pp. 119-136). Cham, Springer
  • Zhang, X., & Xu, Z. (2014). Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. International journal of intelligent systems29(12), 1061-1078.
  • Adak, A. K., & Darvishi Salokolaei, D. (2019). Some properties of Pythagorean fuzzy ideal of near-rings. International journal of applied operational research-an open access journal9(3), 1-9.
  • Biswas, A., & Sarkar, B. (2019). Pythagorean fuzzy TOPSIS for multicriteria group decision‐making with unknown weight information through entropy measure.International journal of intelligent systems34(6), 1108-1128.
  • Gou, X., Xu, Z., & Ren, P. (2016). The properties of continuous Pythagorean fuzzy information. International journal of intelligent systems31(5), 401-424.
  • Li, D., & Zeng, W. (2018). Distance measure of Pythagorean fuzzy sets. International journal of intelligent systems33(2), 348-361.
  • Zeng, W., Li, D., & Yin, Q. (2018). Distance and similarity measures of Pythagorean fuzzy sets and their applications to multiple criteria group decision making. International journal of intelligent systems33(11), 2236-2254.
  • Wang, H., He, S., & Pan, X. (2018). A new bi-directional projection model based on Pythagorean uncertain linguistic variable. Information9(5), 104.
  • Yu, L., Zeng, S., Merigó, J. M., & Zhang, C. (2019). A new distance measure based on the weighted induced method and its application to Pythagorean fuzzy multiple attribute group decision making. International journal of intelligent systems34(7), 1440-1454.
  • Peng, X., & Li, W. (2019). Algorithms for interval-valued Pythagorean fuzzy sets in emergency decision making based on multiparametric similarity measures and WDBA. IEEE access7, 7419-7441.
  • Yoon, K. P., & Hwang, C. L. (1995). Multiple attribute decision making: an introduction. Sage publications.