Skip to main content
SpringerLink
Log in

A unified method for Pythagorean fuzzy multicriteria group decision-making using entropy measure, linear programming and extended technique for ordering preference by similarity to ideal solution

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

This paper presents an innovative method for solving Pythagorean fuzzy (PF) multicriteria group decision-making problems with completely unknown weight information about criteria using entropy weight model, linear programming (LP) and modified technique for ordering preference by similarity to ideal solution (TOPSIS). At first, a new distance measure for PF sets is defined considering their degree of hesitancy and based on weighted Hamming distance and Hausdorff metric. To handle the fuzziness in criteria weights, PF entropy weight model is used to find the initial weights of the criteria in PF format. Following the concept of TOPSIS, an LP model is constructed on the basis of the view point that the chosen alternative should have the smallest distance from the positive ideal solution and the largest distance from the negative ideal solution. Then, the LP model is utilized to find optimal weights of the criteria. Using the newly defined distance measure, entropy weight model and LP model, TOPSIS is extended in PF environments. The existing methods are able to find criteria weights in the form of crisp values only, whereas proposed method is able to obtain those weights in PF format. Thus, the proposed method can overcome the drawback in computing criteria weight for multicriteria group decision-making in PF environments and reduce the information loss significantly. Several numerical examples are considered and solved to validate the superiority of the proposed methodology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96

    MATH  Google Scholar 

  • Biswas A, Dewan S (2012) Priority based fuzzy goal programming technique for solving fractional fuzzy goals by using dynamic programming. Fuzzy Inf Eng 4:165–180

    MathSciNet  MATH  Google Scholar 

  • Biswas A, Kumar S (2018) An integrated TOPSIS approach to MADM with interval-valued intuitionistic fuzzy settings. Adv Intell Syst Comput 706:533–543

    Google Scholar 

  • Biswas A, Sarkar B (2018) Pythagorean fuzzy multi-criteria group decision making through similarity measure based on point operators. Int J Intell Syst 33(8):1731–1744

    Google Scholar 

  • Biswas A, Sarkar B (2019a) Interval-valued Pythagorean fuzzy TODIM approach through point operator based similarity measures for multi criteria group decision making. Kybernetes 48(3):496–519

    Google Scholar 

  • Biswas A, Sarkar B (2019b) Pythagorean fuzzy TOPSIS for multicriteria group decision making with unknown weight information through entropy measure. Int J Intell Syst 34(6):1108–1128

    Google Scholar 

  • Biswas A, Majumdar D, Sahu S (2011) Assessing morningness of a group of people by using fuzzy expert system and adaptive neuro fuzzy inference model. Commun Comput Inf Sci 140:47–56

    MATH  Google Scholar 

  • Chen TY (2018) Remoteness index-based Pythagorean fuzzy VIKOR methods with a generalized distance measure for multiple criteria decision analysis. Inf Fusion 41:129–150

    Google Scholar 

  • Çoban V, Onar SÇ (2018) Pythagorean fuzzy engineering economic analysis of solar power plants. Soft Comput 22(15):5007–5020

    Google Scholar 

  • Garg H (2018) Some methods for strategic decision making problems with immediate probabilities in Pythagorean fuzzy environment. Int J Intell Syst 33(4):687–712

    Google Scholar 

  • Guleria A, Bajaj RK (2018) On Pythagorean fuzzy soft matrices, operations and their applications in decision making and medical diagnosis. Soft Comput 23(17):7889–7900

    MATH  Google Scholar 

  • Hwang CL, Yoon K (1981) Multiple attribute decision making methods and applications. Springer, New York

    MATH  Google Scholar 

  • Karasan A, Ilbahar E, Kahraman C (2018) A novel Pythagorean fuzzy AHP and its application to landfill site selection problem. Soft Comput. https://doi.org/10.1007/s00500-018-3649-0

    Article  Google Scholar 

  • Khan MSA, Abdullah S, Ali A, Amin F, Hussain F (2019) Pythagorean hesitant fuzzy Choquet integral aggregation operators and their application to multi-attribute decision-making. Soft Comput 23(1):251–267

    MATH  Google Scholar 

  • Kumar S, Biswas A (2019a) A Unified TOPSIS approach to MADM problems in interval-valued intuitionistic fuzzy environment. Adv Intell Syst Comput 799:435–447

    Google Scholar 

  • Kumar S, Biswas A (2019b) Linear programming-based TOPSIS method for solving MADM problems with three-parameter IVIFNs. Stud Comput Intell 687:1–25

    Google Scholar 

  • Li DF (2005) Multiattribute decision making models and methods using intuitionistic fuzzy sets. J Comput Syst Sci 70:73–85

    MathSciNet  MATH  Google Scholar 

  • Li DF, Chen GH, Huang ZG (2010) Linear programming method for multiattribute group decision making using IF sets. Inf Sci 180:1591–1609

    MathSciNet  MATH  Google Scholar 

  • Liang D, Zhang Y, Xu Z, Darko AP (2018a) Pythagorean fuzzy Bonferroni mean aggregation operator and its accelerative calculating algorithm with the multithreading. Int J Intell Syst 33(3):615–633

    Google Scholar 

  • Liang D, Xu Z, Liu D, Wu Y (2018b) Method for three-way decisions using ideal TOPSIS solutions at Pythagorean fuzzy information. Inf Sci 435:282–295

    MathSciNet  Google Scholar 

  • Luca AD, Termini S (1972) A definition of nonprobabilistic entropy in the setting of fuzzy set theory. Inf Control 20:311–312

    MathSciNet  MATH  Google Scholar 

  • Mohagheghi V, Mousavi SM, Vahdani B (2017) Enhancing decision-making flexibility by introducing a new last aggregation evaluating approach based on multi-criteria group decision making and Pythagorean fuzzy sets. Appl Soft Comput 61:527–535

    Google Scholar 

  • Peng X, Dai J (2017) Approaches to Pythagorean fuzzy stochastic multi-criteria decision making based on prospect theory and regret theory with new distance measure and score function. Int J Intell Syst 32(11):1187–1214

    Google Scholar 

  • Peng X, Yang Y (2015) Some results for Pythagorean fuzzy sets. Int J Intell Syst 30:1133–1160

    Google Scholar 

  • Peng X, Yuan H, Yang Y (2017) Pythagorean fuzzy information measures and their applications. Int J Intell Syst 32(10):991–1029

    Google Scholar 

  • Ren P, Xu Z, Gou X (2016) Pythagorean fuzzy TODIM approach to multi-criteria decision making. Appl Soft Comput 42:246–259

    Google Scholar 

  • Sarkar A, Biswas A (2019) Multicriteria decision making using Archimedean aggregation operators in Pythagorean hesitant fuzzy environment. Int J Intell Syst 34(7):1361–1386

    Google Scholar 

  • Szmidt E, Kacprzyk J (2000) Distances between intuitionistic fuzzy sets. Fuzzy Sets Syst 114:505–518

    MathSciNet  MATH  Google Scholar 

  • Wei GW (2008) Maximizing deviation method for multiple attribute decision making in intuitionistic fuzzy setting. Knowl-Based Syst 21:833–836

    Google Scholar 

  • Wei GW (2010) GRA method for multiple attribute decision making with incomplete weight information in intuitionistic fuzzy setting. Knowl-Based Syst 23:243–247

    Google Scholar 

  • Wei G, Wei Y (2018) Similarity measures of Pythagorean fuzzy sets based on the cosine function and their applications. Int J Intell Syst 33(3):634–652

    Google Scholar 

  • Wu JZ, Zhang Q (2011) Multicriteria decision making method based on intuitionistic fuzzy weighted entropy. Expert Syst Appl 38:916–922

    Google Scholar 

  • Xia MM, Xu ZS (2012) Entropy/cross entropy-based group decision making under intuitionistic fuzzy environment. Inf Fusion 13:31–47

    Google Scholar 

  • Xu ZS (2007a) Models for multiple attribute decision making with intuitionistic fuzzy information. Int J Uncert Fuzz Knowl-Based Syst 15:285–297

    MathSciNet  MATH  Google Scholar 

  • Xu ZS (2007b) Multi-person multi-attribute decision making models under intuitionistic fuzzy environment. Fuzzy Optim Decis Making 6:221–236

    MathSciNet  MATH  Google Scholar 

  • Xu ZS (2010) A deviation-based approach to intuitionistic fuzzy multiple attribute group decision making. Group Decis Negot 19:57–76

    Google Scholar 

  • Xu ZS (2012) Intuitionistic fuzzy multiattribute decision making: an interactive method. IEEE Trans Fuzzy Syst 20:514–525

    Google Scholar 

  • Xu ZS, Cai XQ (2010) Nonlinear optimization models for multiple attribute group decision making with intuitionistic fuzzy information. Int J Intell Syst 25:489–513

    MATH  Google Scholar 

  • Xu ZS, Cai X (2012) Intuitionistic fuzzy information aggregation: theory and applications. Springer, Berlin, pp 182–183

    MATH  Google Scholar 

  • Xu ZS, Liao HC (2015) A survey of approaches to decision making with intuitionistic fuzzy preference relations. Knowl-Based Syst 80:131–142

    Google Scholar 

  • Xue W, Xu Z, Zhang X, Tian X (2018) Pythagorean fuzzy LINMAP method based on the entropy theory for railway project investment decision making. Int J Intell Syst 33(1):93–125

    Google Scholar 

  • Yager RR (2013) Pythagorean fuzzy subsets. Proceeding of the joint IFSA world congress and NAFIPS annual meeting. Edmonton, Canada, pp 57–61

    Google Scholar 

  • Yager RR (2014) Pythagorean membership grades in multicriteria decision making. IEEE Trans Fuzzy Syst 22:958–965

    Google Scholar 

  • Ye J (2010) Fuzzy decision-making method based on the weighted correlation coefficient under intuitionistic fuzzy environment. Eur J Oper Res 205:202–204

    MATH  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    MATH  Google Scholar 

  • Zeng S, Chen J, Li X (2016) A hybrid method for Pythagorean fuzzy multiple-criteria decision making. Int J Inf Tech Decis Making 14:1–20

    Google Scholar 

  • Zeng S, Mu Z, Baležentis T (2018) A novel aggregation method for Pythagorean fuzzy multiple attribute group decision making. Int J Intell Syst 33(3):573–585

    Google Scholar 

  • Zhang XL (2016a) A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making. Int J Intell Syst 31:593–611

    Google Scholar 

  • Zhang XL (2016b) Multicriteria Pythagorean fuzzy decision analysis: a hierarchical QUALIFLEX approach with the closeness index-based ranking methods. Inf Sci 330:104–124

    Google Scholar 

  • Zhang XL (2018) Pythagorean fuzzy clustering analysis: a hierarchical clustering algorithm with the ratio index-based ranking methods. Int J Intell Syst 33(9):1798–1822

    Google Scholar 

  • Zhang XL, Xu ZS (2014) Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. Int J Intell Syst 29:1061–1078

    MathSciNet  Google Scholar 

Download references

Acknowledgements

Dr. Animesh Biswas is thankful to University of Kalyani for providing partial financial assistance in carrying out the research work through DST-PURSE Programme of the University. The authors also remain grateful to the reviewers for their valuable comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Basirhat College, Basirhat, 743412, India

    Biswajit Sarkar

  2. Department of Mathematics, University of Kalyani, Kalyani, 741235, India

    Animesh Biswas

Authors
  1. Biswajit Sarkar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Animesh Biswas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Animesh Biswas.

Ethics declarations

Conflict of interest

No.

Additional information

Communicated by V. Loia.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sarkar, B., Biswas, A. A unified method for Pythagorean fuzzy multicriteria group decision-making using entropy measure, linear programming and extended technique for ordering preference by similarity to ideal solution. Soft Comput 24, 5333–5344 (2020). https://doi.org/10.1007/s00500-019-04282-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-019-04282-y

Keywords

Search

Navigation