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A New Method for Triangular Fuzzy Compare Wise Judgment Matrix Process Based on Consistency Analysis

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Abstract

To cope with the uncertainty in the process of decision making, fuzzy preference relations are proposed and commonly applied in many fields. In practical decision-making problems, the decision maker may use triangular fuzzy preference relations to express his/her uncertainty. Based on the row weighted arithmetic mean method, this paper develops an approach for deriving the fuzzy priority vector from triangular fuzzy compare wise judgment matrices. To do this, this paper first analyzes the upper and lower bounds of the triangular fuzzy priority weight of each alternative, which indicates the decision maker’s optimistic and pessimistic attitudes. Based on (acceptably) consistent multiplicative preference relations, the triangular fuzzy priority vector is obtained. Meanwhile, a consistency concept of triangular fuzzy compare wise judgment matrices is defined, and the consistent relationship between triangular fuzzy and crisp preference relations is studied. Different to the existing methods, the new approach calculates the triangular fuzzy priority weights separately. Furthermore, the fuzzy priority vector from trapezoidal fuzzy reciprocal preference relations is considered. Finally, the application of the new method to new product development (NDP) project screening is tested, and comparative analyses are also offered.

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References

  1. Aguarón, J., Moreno-Jiménez, J.M.: The geometric consistency index: approximated thresholds. Eur. J. Oper. Res. 147(1), 137–145 (2003)

    Article  MATH  Google Scholar 

  2. Arbel, A., Vargas, L.G.: The analytic hierarchy process with interval judgements. In: Proceedings of the IXth International Symposium on Multi-criteria Decision Making Fair fax (1990)

  3. Buckley, J.J.: Fuzzy hierarchical analysis. Fuzzy Sets Syst. 17(3), 233–247 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boender, C.G.E., de Graan, J.G., Lootsma, F.A.: Multi-criteria decision analysis with fuzzy pairwise comparisons. Fuzzy Sets Syst. 29(1), 133–143 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brunelli, M.: A note on the article Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean. Fuzzy Sets Syst. 161(2010), 1604–1613; Fuzzy Sets Syst. 176(1), 76–78 (2011)

  6. Chang, D.Y.: Application of the extent analysis method on fuzzy AHP. Eur. J. Oper. Res. 95(3), 649–655 (1996)

    Article  MATH  Google Scholar 

  7. Crawford, G.B., Williams, C.: A note on the analysis of subjective judgment matrices. J. Math. Psychol. 29(4), 387–405 (1985)

    Article  MATH  Google Scholar 

  8. Calantone, R.J., Di Benedetto, C.A., Schmidt, J.B.: Using the analytic hierarchy process in new product screening. J. Prod. Innov. Manag. 16(1), 65–76 (1999)

    Article  Google Scholar 

  9. Chin, K.S., Xu, D.L., Yang, J.B., Lam, J.P.K.: Group-based ER–AHP system for product project screening. Expert Syst. Appl. 35(4), 1909–1929 (2008)

    Article  Google Scholar 

  10. Dubois, D.: The role of fuzzy sets in decision sciences: old techniques and new directions. Fuzzy Sets Syst. 184(1), 3–28 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dubois, D., Prade, H.: A unified view of ranking techniques for fuzzy numbers. In: Proceedings of the 1999 IEEE International Conference on Fuzzy Systems Seoul, pp. 1328–1333 (1999)

  12. Gen, S., Boran, F.E., Akay, D., Xu, Z.S.: Interval multiplicative transitivity for consistency, missing values and priority weights of interval fuzzy preference relations. Inf. Sci. 180(24), 4877–4891 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Haines, L.M.: A statistical approach to the analytic hierarchy process with interval judgments. (I). Distribution on feasible regions. Eur. J. Oper. Res. 110(1), 112–125 (1998)

    Article  MATH  Google Scholar 

  14. Herrera-Viedma, E., Herrera, F., Chiclana, F., Luque, M.: Some issues on consistency of fuzzy preference relations. Eur. J. Oper. Res. 154(2), 98–109 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Islam, R., Biswal, M.P., Alam, S.S.: Preference programming and inconsistent interval judgments. Eur. J. Oper. Res. 97(1), 53–62 (1997)

    Article  MATH  Google Scholar 

  16. Kacprzyk, J., Fedrizzi, M.: Multiperson Decision Making Models Using Fuzzy Sets and Possibility Theory. Kluwer Academic Publishers, Dordrecht (1990)

    Book  MATH  Google Scholar 

  17. Lee, E.S., Li, R.L.: Comparison of fuzzy numbers based on the probability measure of fuzzy events. Comput. Math Appl. 15(10), 887–896 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  18. Leung, L.C., Cao, D.: On consistency and ranking of alternatives in fuzzy AHP. Eur. J. Oper. Res. 124(1), 102–113 (2000)

    Article  MATH  Google Scholar 

  19. Liu, F.: Acceptable consistency analysis of interval reciprocal comparison matrices. Fuzzy Sets Syst. 160(18), 2686–2700 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, F., Zhang, W.G., Wang, Z.X.: A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making. Eur. J. Oper. Res. 218(3), 747–754 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, F., Zhang, W.G., Zhang, L.H.: A group decision-making model based on a generalized ordered weighted geometric average operator with interval preference matrices. Fuzzy Sets Syst. 246(1), 1–18 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Liu, F., Zhang, W.G., Zhang, L.H.: Consistency analysis of triangular fuzzy reciprocal preference relations. Eur. J. Oper. Res. 235(3), 718–726 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mikhailov, L.: Deriving priorities from fuzzy pairwise comparison judgments. Fuzzy Sets Syst. 134(3), 365–385 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Matarazzo, B., Munda, G.: New approaches for the comparison of L-R fuzzy numbers: a theoretical and operational analysis. Fuzzy Sets Syst. 118(3), 407–418 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Meng, F.Y., Chen, X.H., Zhang, Q., Lin, J.: Two new methods for deriving the priority vector from interval comparison matrices. Inf. Fusion. 26, 122–135 (2015)

    Article  Google Scholar 

  26. Meng, F.Y., Chen, X.H.: A new method for group decision making with incomplete fuzzy preference relations. Knowledge-Based Syst. 73, 111–123 (2015)

    Article  Google Scholar 

  27. Meng, F.Y., Chen, X.H.: An approach to incomplete multiplicative preference relations and its application in group decision making. Inf. Sci. 309, 119–1375 (2015)

    Article  Google Scholar 

  28. Ohnishi, S., Dubois, D., Prade, H., Yamanoi, T.: A fuzzy constraint-based approach to the analytic hierarchy process. In: Bouchon-Meunier, B., Marsala, C., Rifqi, M., Yager, R.R. (eds.) Uncertainty and Intelligent Information Systems, pp. 217–228. World Scientific, Singapore (2008)

    Chapter  Google Scholar 

  29. Orlovsky, S.A.: Decision-making with a fuzzy preference relation. Fuzzy Sets Syst. 1(3), 155–167 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  30. Podinovski, V.V.: Interval articulation of superiority and precise elicitation of priorities. Eur. J. Oper. Res. 180(1), 406–417 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ramik, J., Korviny, P.: Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean. Fuzzy Sets Syst. 161(11), 1604–1613 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rezaei, J., Ortt, R.: Multi-criteria supplier segmentation using a fuzzy preference relation based AHP. Eur. J. Oper. Res. 225(1), 75–84 (2013)

    Article  MATH  Google Scholar 

  33. Saaty, T.L.: The Analytic Hierarchy Process. McGraw-Hill, New York (1980)

    MATH  Google Scholar 

  34. Saaty, T.L., Vargas, L.G.: Uncertainty and rank order in the analytic hierarchy process. Eur. J. Oper. Res. 32(1), 107–117 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  35. Saaty, T.L.: Multicriteria Decision Making: The Analytic Hierarchy Process. RWS Publications, Pittsburgh (1988)

    Google Scholar 

  36. Salo, A.A.: On fuzzy ratio comparisons in hierarchical decision models. Fuzzy Sets Syst. 84(1), 21–32 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. van Laarhoven, P.J.M., Pedrycz, W.: A fuzzy extension of Saaty’s priority theory. Fuzzy Sets Syst. 11(1–3), 229–241 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wagenknecht, M., Hartmann, K.: On fuzzy rank ordering in polyoptimisation. Fuzzy Sets Syst. 11, 253–264 (1983)

    Article  MATH  Google Scholar 

  39. Wang, X.Z., Kerre, E.E.: Reasonable properties for the ordering of fuzzy quantities (I). Fuzzy Sets Syst. 118(3), 375–385 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, X.Z., Kerre, E.E.: Reasonable properties for the ordering of fuzzy quantities (II). Fuzzy Sets Syst. 118(3), 387–405 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang, Y.M., Luo, Y., Hua, Z.: On the extent analysis method for fuzzy AHP and its applications. Eur. J. Oper. Res. 186(2), 735–747 (2008)

    Article  MATH  Google Scholar 

  42. Wang, Y.M., Yang, J.B., Xu, D.L.: A two-stage logarithmic goal programming method for generating weights from interval comparison matrices. Fuzzy Sets Syst. 152(3), 475–498 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wang, Y.M., Elhag, T.M.S.: On the normalization of interval and fuzzy weights. Fuzzy Sets Syst. 157(18), 2456–2471 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wang, Y.M., Elhag, T.M.S., Hua, Z.S.: A modified fuzzy logarithmic least squares method for fuzzy analytic hierarchy process. Fuzzy Sets Syst. 157(18), 3055–3071 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wang, Y.M., Chin, K.S.: A linear goal programming priority method for fuzzy analytic hierarchy process and its applications in new product screening. Int. J. Approx. Reason. 49(2), 451–465 (2008)

    Article  MATH  Google Scholar 

  46. Wang, T.C., Chen, Y.H.: Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP. Inf. Sci. 178(19), 3755–3765 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  47. Wu, J., Chiclana, F.: Visual information feedback mechanism and attitudinal prioritisation method for group decision making with triangular fuzzy complementary preference relations. Inf. Sci. 279, 716–734 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  48. Xu, Z.S.: A practical method for priority of interval number complementary judgment matrix. Oper. Res. Manag. Sci. 10(1), 16–19 (2001)

    Google Scholar 

  49. Xu, Z.S.: Consistency of interval fuzzy preference relations in group decision making. Appl. Soft Comput. 11(5), 3898–3909 (2011)

    Article  Google Scholar 

  50. Xu, Z.S.: A method for priorities of triangular fuzzy number complementary judgement matrices. Fuzzy Syst. Math. 16(1), 47–50 (2002)

    MathSciNet  MATH  Google Scholar 

  51. Xu, Z.S.: A priority method for triangular fuzzy number complementary judgement matrix. Syst. Eng. Theory Pract. 23(10), 86–89 (2003)

    Google Scholar 

  52. Xu, Y.J., Wang, H.M.: Eigenvector method, consistency test and inconsistency repairing for an incomplete fuzzy preference relation. Appl. Math. Model. 37(7), 5171–5183 (2013)

    Article  MathSciNet  Google Scholar 

  53. Xu, Y.J., Patnayakuni, R., Wang, H.M.: The ordinal consistency of a fuzzy preference relation. Inf. Sci. 224(1), 152–164 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  54. Xu, Y.J., Li, K.W., Wang, H.M.: Incomplete interval fuzzy preference relations and their applications. Comput. Ind. Eng. 67(1), 93–103 (2014)

    Article  Google Scholar 

  55. Xu, Y.J., Gupta, J.N.D., Wang, H.M.: The ordinal consistency of an incomplete reciprocal preference relation. Fuzzy Sets Syst. 246(1), 62–77 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  56. Xu, Y.J., Li, K.W., Wang, H.M.: Consistency test and weight generation for additive interval fuzzy preference relations. Soft. Comput. 18(8), 1499–1513 (2014)

    Article  MATH  Google Scholar 

  57. Xia, M.M., Xu, Z.S.: Interval weight generation approaches for reciprocal relations. Appl. Math. Model. 38(3), 828–838 (2014)

    Article  MathSciNet  Google Scholar 

  58. Xu, R.: Fuzzy least-squares priority method in the analytic hierarchy process. Fuzzy Sets Syst. 112(3), 359–404 (2000)

    Article  MathSciNet  Google Scholar 

  59. Yao, J.S., Lin, F.T.: Fuzzy critical path method based on signed distance ranking of fuzzy numbers. IEEE Trans. Syst. Man Cybern. 30(1), 76–82 (2000)

    Article  MathSciNet  Google Scholar 

  60. Yager, R.R.: In a general class of fuzzy connectives. Fuzzy Sets Syst. 4(3), 235–242 (1980)

    Article  MATH  Google Scholar 

  61. Yager, R.R., Dimitar, F.: On ranking fuzzy numbers using valuations. Int. J. Intell. Syst. 14(12), 1249–1268 (1999)

    Article  MATH  Google Scholar 

  62. Yager, R.R., Detyniecki, M., Meunier, B.B.: A context-dependent method for ordering fuzzy numbers using probabilities. Inf. Sci. 138(1–4), 237–255 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  63. Yu, D.J., Merigo, J.M., Zhou, L.G.: Interval-valued multiplicative intuitionistic fuzzy preference relations. Int. J. Fuzzy Syst. 15(4), 412–422 (2013)

    MathSciNet  Google Scholar 

  64. Zhang, Z., Guo, C.H.: Consistency-based algorithms to estimate missing elements for uncertain 2-tuple linguistic preference relations. Int. J. Comput. Intell. Syst. 7(5), 924–936 (2014)

    Article  Google Scholar 

  65. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  66. Zeng, X.L., Meng, F.Y.: On ranking methods for fuzzy numbers based on possibility degree. Oper. Res. Manag. Sci. Fuzzy 7, 66–72 (2007)

    Google Scholar 

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Acknowledgments

The authors first gratefully thank the Area Editor and three anonymous referees for their valuable and constructive comments which have much improved the paper. This work was supported by the State Key Program of National Natural Science of China (No. 71431006), the National Natural Science Foundation of China (Nos. 71571192, 71501189), the China Postdoctoral Science special Foundation (2015T80901), the China Postdoctoral Science Foundation (2014M560655), and the Innovation-Driven Planning Foundation of Central South University (2015CX010).

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  1. Business School, Central South University, Changsha, 410083, China

    Fanyong Meng & Xiaohong Chen

  2. School of Accounting, Hunan University of Commerce, Changsha, 410205, China

    Xiaohong Chen

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  1. Fanyong Meng
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  2. Xiaohong Chen
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Correspondence to Fanyong Meng.

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Meng, F., Chen, X. A New Method for Triangular Fuzzy Compare Wise Judgment Matrix Process Based on Consistency Analysis. Int. J. Fuzzy Syst. 19, 27–46 (2017). https://doi.org/10.1007/s40815-016-0150-8

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