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A complete ranking of trapezoidal fuzzy numbers and its applications to multi-criteria decision making

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Abstract

The problem (or scenario) involving qualitative or imprecise information is not solvable by classical set theory. To overcome the shortcoming of classical set theory, Zadeh (Inf Control 8(3):338–356, 26) introduced the concept of fuzzy sets that generalizes the concept of classical sets. Fuzzy set theory allows modelling and handling of imprecise information in an effective way. As a special class of fuzzy sets, fuzzy numbers (FN) which are very much important in decision making was introduced by Dubois and Prade (Int J Syst Sci 9:631–626, 12). The available methods for solving multi-criteria decision making problems (MCDM) are problem dependent in nature due to the partial ordering on the class of FN. Total ordering on the class of FN by countable number of real-valued parameters was achieved by Wang and Wang (Fuzzy Sets Syst 243:131–141, 21). A complete ranking on the class of trapezoidal fuzzy numbers (TrFNs) using finite number of score functions is achieved in this paper. In this paper, a new ranking procedure (complete) on the class of TrFNs using the concepts of mid-point, radius, left and right fuzziness of TrFN is proposed and further we introduce a method for solving fuzzy multi-criteria decision making (Fuzzy MCDM) problem. Finally, comparisons of our proposed method with familiar existing methods are listed.

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References

  1. Abbasbandy S, Asady B (2006) Ranking of fuzzy numbers by sign distance. Inf Sci 176(16):2405–2416

    Article  MathSciNet  Google Scholar 

  2. Abbasbandy S, Hajjari T (2009) A new approach for ranking of trapezoidal fuzzy numbers. Comput Math Appl 57:413–419

    Article  MathSciNet  Google Scholar 

  3. Asady B, Zendehnam A (2007) Ranking fuzzy numbers by distance minimizing. Appl Math Model 31:2589–2598

    Article  Google Scholar 

  4. Baas SM, Kwakernaak H (1977) Rating and ranking of multiple aspect alternative using fuzzy sets. Automatica 13:47–58

    Article  MathSciNet  Google Scholar 

  5. Baldwin JF, Guild NC (1978) A model for multi-criterial decision-making using fuzzy logic. In: Proceedings of the workshop of fuzzy reasoning. Queen Mary College, University of London, London

  6. Boender CGE, de Graan JG, Lootsma FA (1989) Multi-criteria decision analysis with fuzzy pairwise comparisons. Fuzzy Sets Syst 29:133–143

    Article  MathSciNet  Google Scholar 

  7. Chen SH (1985) Ranking fuzzy numbers with maximizing set and minimizing set. Fuzzy Sets Syst 17(2):113–129

    Article  MathSciNet  Google Scholar 

  8. Cheng CH (1994) A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets Syst 95(1998):307–317

    MathSciNet  MATH  Google Scholar 

  9. Chen SJ, Hwang CL (1992) Fuzzy multiple attribute decision making methods and applications. Lecture notes in economics and mathematical systems. Springer, New York

    Google Scholar 

  10. Chen SM, Sanguansat K (2011) Analyzing fuzzy risk based on a new fuzzy ranking method between generalized fuzzy numbers. Expert Syst Appl 38(3):2163–2171

    Article  Google Scholar 

  11. Deschrijver G (2007) Arithmatic operators in interval valued fuzzy set theory. Inf Sci 177:2906–2924

    Article  Google Scholar 

  12. Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9:613–626

    Article  MathSciNet  Google Scholar 

  13. Dubois D, Prade H (1983) Ranking fuzzy numbers in the setting of possibility theory. Inf Sci 30:183–224

    Article  MathSciNet  Google Scholar 

  14. Ezzati R, Allahviranloo T, Khezerloo S, Khezerloo M (2012) An approach for ranking of fuzzy numbers. Expert Syst Appl 39:690–695

    Article  Google Scholar 

  15. Jain R (1976) Decision making in the presence of fuzzy variables. In: IEEE Transaction on Systems, Man, and Cybernetics, SMC-6. pp 698–703

  16. Jain R (1977) A procedure for multi-aspect decision making using fuzzy sets. Int J Syst Sci 8:1–7

    Article  Google Scholar 

  17. Kerre EE (1982) The use of fuzzy set theory in electrocardiological diagnostics. In: Gupta MM, Sanchez E (eds) Approximate reasoning in decision analysis. North Holland, pp 277–282

  18. Lee ES, Li RL (1988) Comparison of fuzzy numbers based on the probability measure of fuzzy events. Comput Math Appl 15:887–896

    Article  MathSciNet  Google Scholar 

  19. McCahone C (1987) Fuzzy set theory applied to production and inventory control. Ph.D. Thesis, Department of Industrial Engineering, Kansas State university

  20. Nakamura K (1986) Preference relation on a set of fuzzy utilities as a basis for decision making. Fuzzy Sets Syst 20(2):147–162

    Article  MathSciNet  Google Scholar 

  21. Wang W, Wang Z (2014) Total ordering defined on the set of all fuzzy numbers. Fuzzy Sets Syst 243:131–141

    Article  MathSciNet  Google Scholar 

  22. Xu P, Su X, Wu J, Sun X, Zhang Y, Deng Y (2012) A note on ranking generalized fuzzy numbers. Expert Syst Appl 39:6454–6457

    Article  Google Scholar 

  23. Yager RR (1980) On a general class of fuzzy connectives. Fuzzy Sets Syst 4:235–242

    Article  MathSciNet  Google Scholar 

  24. Yager RR (1980) On choosing between fuzzy subsets. Kybernetes 9:151–154

    Article  Google Scholar 

  25. Yager RR (1981) A procedure for ordering fuzzy subsets of the unit interval. Inf Sci 24:143–161

    Article  MathSciNet  Google Scholar 

  26. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–356

    Article  Google Scholar 

  27. Zimmermann HJ (1987) Fuzzy set, decision making and expert system. Kluwer, Boston

    Book  Google Scholar 

Download references

Acknowledgements

Authors are grateful to the anonymous reviewers, Editor and Associate editor for their valuable comments. The corresponding author thanks the Council of Scientific and Industrial Research (CSIR-HRDG), India, for supporting this research under CSIR SRF.

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Tiruchirappalli, Tiruchirappalli, India

    Dhanasekaran Ponnialagan, Jeevaraj Selvaraj & Lakshmana Gomathi Nayagam Velu

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  1. Dhanasekaran Ponnialagan
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  2. Jeevaraj Selvaraj
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  3. Lakshmana Gomathi Nayagam Velu
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Correspondence to Jeevaraj Selvaraj.

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Ponnialagan, D., Selvaraj, J. & Velu, L.G.N. A complete ranking of trapezoidal fuzzy numbers and its applications to multi-criteria decision making. Neural Comput & Applic 30, 3303–3315 (2018). https://doi.org/10.1007/s00521-017-2898-7

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