International Journal of Fuzzy Systems

, Volume 19, Issue 5, pp 1290–1299 | Cite as

A Mathematical Model for Subjective Evaluation of Alternatives in Fuzzy Multi-Criteria Group Decision Making Using COPRAS Method

  • K. Rathi
  • S. Balamohan


Multi-criteria group decision-making approaches under fuzzy environment find their attention in the recent works in the area of decision science. In a real-life decision-making situation, decision makers express their opinions often linguistically. Such linguistic expressions are then captured by fuzzy set theory and represented in the present study by heptagonal fuzzy numbers for analytic purpose. Firstly, the concept, the arithmetic operations and a centroid-based ranking method of symmetric heptagonal fuzzy numbers (symHFN) are introduced, and then, some operators are proposed for aggregating the linguistic information. Secondly, the complex proportional assessment (COPRAS) method used for group decision making is extended within the context of the proposed symHFNs to develop an algorithm for fuzzy multi-criteria group decision-making (FMCGDM) method. The proposed method incorporates an efficient computational technique which handles the degree of satisfaction of the decision maker in their judgement. SymHFNs are used to handle the linguistic evaluation and assessment of alternatives by the decision makers. As it records the primary and secondary degree of satisfaction of decision makers’ opinion, symHFN is more suitable and realistic in decision-making problems. Finally, by virtue of the proposed FMCGDM method, fire emergency alternative evaluation is carried out to illustrate the practicality and the effectiveness of the proposed method.


Fuzzy decision making Centroid of fuzzy numbers Fire emergency alternative problem COPRAS method Linguistic variable Degree of satisfaction 


  1. 1.
    Hwang, C.L., Yoon, K.: Multiple Attribute Decision Making: Methods and Applications A State-of-the-Art Survey. Springer, Berlin (1981)CrossRefMATHGoogle Scholar
  2. 2.
    Jiang, Y., Xu, Z.S., Yu, X.H.: Compatibility measures and consensus models for group decision making with intuitionistic multiplicative preference relations. Appl. Soft Comput. 13(4), 2075–2086 (2013)CrossRefGoogle Scholar
  3. 3.
    Tian, J., Yu, D., Bing, Y., Shilong, M.: A fuzzy TOPSIS model via Chi square test for information source selection. Knowl.-Based Syst. 37, 515–527 (2013)CrossRefGoogle Scholar
  4. 4.
    Wu, Z., Xu, J.: A concise consensus support model for group decision making with reciprocal preference relations based on deviation measures. Fuzzy Sets Syst. 206, 58–73 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Zhongliang, Y.: Approach to group decision making based on determining the weights of experts by using projection method. Appl. Math. Model. 36(7), 2900–2910 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Zeng, S.Z., Merigo, J.M., Su, W.H.: The uncertain probabilistic OWA distance operator and its application in group decision making. Appl. Math. Model. 37(9), 6266–6275 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Zhou, L., Chen, H., Liu, J.: Generalized logarithmic proportional averaging operators and their applications to group decision making. Knowl.-Based Syst. 36, 268–279 (2012)CrossRefGoogle Scholar
  8. 8.
    Herrera, F., Alonso, S., Chiclana, F., Herrera-Viedma, E.: Computing with words in decision making: foundations, trends and prospects. Fuzzy Optim. Decis. Making 8, 337–364 (2009)CrossRefMATHGoogle Scholar
  9. 9.
    Liu, P., Jin, F.: Methods for aggregating intuitionistic uncertain linguistic variables and their applications to group decision making. Inf. Sci. 205(1), 58–71 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Pang, J.F., Liang, J.Y.: Evaluation of the results of multi-attribute group decision-making with linguistic information. Omega 40(3), 294–301 (2012)CrossRefGoogle Scholar
  11. 11.
    Tan, C.Q., Wu, D.D., Ma, B.J.: Group decision making with linguistic preference relations with application to supplier selection. Expert Syst. Appl. 38(12), 14382–14389 (2011)CrossRefGoogle Scholar
  12. 12.
    Zhou, L.G., Chen, H.Y.: A generalization of the power aggregation operators for linguistic environment and its application in group decision making. Knowl.-Based Syst. 26, 216–224 (2012)CrossRefGoogle Scholar
  13. 13.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)CrossRefMATHGoogle Scholar
  14. 14.
    Cakir, O., Canbolat, M.S.: A web-based decision support system for multi-criteria inventory classification using fuzzy AHP methodology. Expert Syst. Appl. 35, 1367–1378 (2008)CrossRefGoogle Scholar
  15. 15.
    Rathi, K., Balamohan, S.: Representation and ranking of fuzzy numbers with Heptagonal membership function using value and ambiguity index. Appl. Math. Sci. 8(87), 4309–4321 (2014)Google Scholar
  16. 16.
    Li, D.F., Yang, J.B.: Fuzzy linear programming technique for multiattribute group decision making in fuzzy environments. Inf. Sci. 158, 263–275 (2004)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Yeh, C.-H., Chang, Y.-H.: Modeling subjective evaluation for fuzzy group multicriteria decision making. Eur. J. Oper. Res. 194(2), 464–473 (2009)CrossRefMATHGoogle Scholar
  18. 18.
    Ma, J., Lu, J., Zhang, G.: Decider: a fuzzy multi-criteria group decision support system. Knowl.-Based Syst. 23(1), 23–31 (2010)CrossRefGoogle Scholar
  19. 19.
    Kaya, T., Kahraman, C.: Multicriteria decision making in energy planning using a modified fuzzy TOPSIS methodology. Expert Syst. Appl. 38(6), 6577–6585 (2011)CrossRefGoogle Scholar
  20. 20.
    Alipour, M.H., Shamsai, A., Ahmady, N.: A new fuzzy multicriteria decision making method and its application in diversion of water. Expert Syst. Appl. 37(12), 8809–8813 (2010)CrossRefGoogle Scholar
  21. 21.
    Dalman, H., Guzel, N., Sivri, M.: A fuzzy set-based approach to multi-objective multi-item solid transportation problem under uncertainty. Int. J. Fuzzy Syst. 18(4), 716–729 (2016)Google Scholar
  22. 22.
    Dalman, H.: Uncertain programming model for multi-item solid transportation problem. Int. J. Mach. Learn. Cybern. (2016). doi: 10.1007/s13042-016-0538-7
  23. 23.
    Zavadskas, E.K., Kaklauskas, A., Turskis, Z., Tamošaitiene, J.: Selection of the effective dwelling house walls by applying attributes values determined at intervals. J. Civil Eng. Manag. 14(2), 85–93 (2008)CrossRefGoogle Scholar
  24. 24.
    HajiaghaRazavi, S.H., Hashemi, S.S., Zavadskas, E.K.: A complex proportional assessment method for group decision making in an interval-valued intuitionistic fuzzy environment. Technol. Econ. Dev. Econ. 19(1), 22–37 (2013)CrossRefGoogle Scholar
  25. 25.
    Ghorabaee, M.K., Amiri, M., Sadaghiani, J.S., Goodarzi, G.H.: Multiple criteria group decision-making for supplier selection based on COPRAS method with interval type-2 fuzzy sets. Int. J. Adv. Manuf. Technol. 75(5), 1115–1130 (2014). doi: 10.1007/s00170-014-6142-7
  26. 26.
    Cheng, C.H.: A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets Syst. 95, 307–317 (1998)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Chu, T.C., Tsao, C.T.: Ranking fuzzy numbers with an area between the centroid point and original point. Comput. Math Appl. 43, 111–117 (2002)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Abbasbandy, S., Asady, B.: Ranking of fuzzy numbers by sign distance. Inf. Sci. 176, 2405–2416 (2006)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Wang, Y.M., Yang, J.B., Xu, D.L., Chin, K.S.: On the centroids of fuzzy numbers. Fuzzy Sets Syst. 157, 919–926 (2006)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Pan, H., Yeh, C.H.: A metaheuristic approach to fuzzy project scheduling. In: Palade, V., Howlett, R.J., Jain, L.C. (eds.) KES 2003 LNAI 2773, pp. 1081–1087. Springer, Berlin (2003)Google Scholar
  31. 31.
    Bih-Sheue, S.: An approach to centroids of fuzzy numbers. Int. J. Fuzzy Syst. 9(1), 51–54 (2007)MATHGoogle Scholar
  32. 32.
    Liu, X., Ju, Y., Yang, S.: Some generalized interval-valued hesitant uncertain linguistic aggregation operators and their applications to multiple attribute group decision making. Soft. Comput. 20, 495–510 (2016)CrossRefMATHGoogle Scholar

Copyright information

© Taiwan Fuzzy Systems Association and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsVelalar College of Engineering and TechnologyErodeIndia
  2. 2.SSM College of EngineeringKomarapalayam, NamakkalIndia

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