Abstract
The final values of alternatives usually are fuzzy numbers in most decision making problems under uncertain environment. These fuzzy numbers need to be defuzzified by a ranking method to assist in decision making. Chen proposed ranking fuzzy numbers with maximizing set and minimizing set as an improvement on Jain’s maximizing set method. Liou and Wang indicated that ranking of any two fuzzy numbers by Chen’s method may be altered when xmax, xmin and k are changed. To solve this problem, Liou and Wang proposed ranking fuzzy numbers with integral value. However, the problem of Chen’s method remains. This work proposes relative maximizing and minimizing sets to improve upon Chen’s method. Comparative examples are provided to demonstrate advantages of the proposed ranking method, and an experiment is conducted to show its robustness. The proposed ranking method is applied to a fuzzy multiple criteria decision making (MCDM) model to present its applicability. Ranking procedure can be clearly displayed to enhance execution efficiency of the suggested fuzzy MCDM model. A numerical example is used to display feasibility of the proposed model. Finally, an experiment is investigated to show that the proposed ranking method improves upon Chen’s method in consistently ranking the final fuzzy numbers in the suggested fuzzy MCDM model.
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References
Chen, S.H.: Ranking fuzzy numbers with maximizing and minimizing set. Fuzzy Sets Syst. 17, 113–129 (1985)
Chi, H.T.X., Yu, V.F.: Ranking generalized fuzzy numbers based on centroid and rank Index. Appl. Soft Comput. 68, 283–292 (2018)
Chu, T.C., Charnsethikul, P.: Ordering alternatives under fuzzy multiple criteria decision making via a fuzzy number dominance based ranking approach. Int. J. Fuzzy Syst. 15(3), 263–273 (2013)
Chu, T.C., Lin, Y.: An extension to fuzzy MCDM. Comput. Math Appl. 57(3), 445–454 (2009)
Das, S., Guha, D.: A centroid-based ranking method of trapezoidal intuitionistic fuzzy numbers and its application to MCDM problem. Fuzzy Inf. Eng. 8(1), 41–74 (2016)
De, S.K., Beg, I.: Triangular dense fuzzy sets and new defuzzification methods. J. Intelligent Fuzzy Syst. 31(1), 469–477 (2016)
De, S.K., Beg, I.: Triangular dense fuzzy neutrosophic sets. Neutrosophic Sets Syst. 13, 24–37 (2016)
De, S.K.: Triangular dense fuzzy lock sets. Soft. Comput. 22(21), 7243–7254 (2018)
Destercke, S., Couso, I.: Ranking of fuzzy intervals seen through the imprecise probabilistic lens. Fuzzy Sets Syst. 278, 20–39 (2015)
Dong, Y., Liu, Y., Liang, H., Chiclana, F., Herrera-Viedma, E.: Strategic weight manipulation in multiple attribute decision making. Omega 75(3), 154–164 (2018)
Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9(6), 613–626 (1978)
Duzce, S.A.: A new ranking method for trapezial fuzzy numbers and its application to fuzzy risk analysis. J. Intell. Fuzzy Syst. 28(3), 1411–1419 (2015)
Gu, Q., Xuan, Z.: A new approach for ranking fuzzy numbers based on possibility theory. J. Comput. Appl. Math. 309, 674–682 (2017)
Hodgett, R.E., Siraj, S.: SURE: a method for decision-making under uncertainty. Expert Syst. Appl. 115, 684–694 (2019)
Jain, R.: Decision making in the presence of fuzzy variables. IEEE Trans. Syst. Man Cybern. 6, 698–703 (1976)
Jain, R.: A procedure for multiple-aspect decision making using fuzzy sets. Int. J. Syst. Sci. 8(1), 1–7 (1977)
Jiang, W., Xie, C., Luo, Y., Tang, Y.: Ranking Z-numbers with an improved ranking method for generalized fuzzy numbers. J. Intell. Fuzzy Syst. 32(3), 1931–1943 (2017)
Jiang, W., Luo, Y., Qin, X.Y., Zhan, J.: An improved method to rank generalized fuzzy numbers with different left heights and right heights. J. Intell. Fuzzy Syst. 28, 2343–2355 (2015)
Kaufmann, A., Gupta, M.M.: Introduction to fuzzy arithmetic: theory and application. Van Nostrand Reinhold, New York (1991)
Kubler, S., Derigent, W., Voisin, A., Framling, K. \& Thomas, A. (2013). Methods of aggregation of expert opinions in the framework of intelligent products. In 11th IFAC workshop on intelligent manufacturing systems, IMS’2013, Brazil (pp. 163–168)
Li, C.C., Dong, Y., Herrera, F., Herrera-Viedma, E., Martínez, L.: Personalized individual semantics in computing with words for supporting linguistic group decision making. An application on consensus reaching. Inform. Fusion 33, 29–40 (2017)
Li, C.C., Rodríguez, R.M., Martínez, L., Dong, Y., Herrera, F.: Personalized individual semantics based on consistency in hesitant linguistic group decision making with comparative linguistic expressions. Knowl. Based Syst. 145, 156–165 (2018)
Liou, T.S., Wang, M.J.J.: Ranking fuzzy numbers with integral value. Fuzzy Sets Syst. 50, 247–255 (1992)
Liu, Y., Dong, Y., Liang, H., Chiclana, F., Herrera-Viedma, E.: Multiple attribute strategic weight manipulation with minimum cost in a group decision making context with interval attribute weights information. IEEE Trans. Syst. Man Cybern. Syst. 34, 2247 (2018)
Nayagam, V.L.G., Jeevaraj, S., Dhanasekaran, P.: An improved ranking method for comparing trapezoidal intuitionistic fuzzy numbers and its applications to multicriteria decision making. Neural Comput. Appl. 30(2), 671–682 (2018)
Nie, R.X., Tian, Z.P., Wang, J.Q., Hu, J.H.: Pythagorean fuzzy multiple criteria decision analysis based on Shapley fuzzy measures and partitioned normalized weighted Bonferroni mean operator. Int. Intell. Syst. 34(2), 297–324 (2019)
Ramalho, F.D., Ekel, P.Y., Pedrycz, W., Pereira Júnior, J.G., Luís Soares, G.: Multicriteria decision making under conditions of uncertainty in application to multiobjective allocation of resources. Inform. Fusion 49, 249–261 (2019)
Ramalingam, S.: Fuzzy interval-valued multi criteria based decision making for ranking features in multi-modal 3D face recognition. Fuzzy Sets Syst. 337, 25–51 (2018)
Salehi, K.: A hybrid fuzzy MCDM approach for project selection problem. Decis. Sci. Lett. 4(1), 109–116 (2015)
Shen, K.Y., Hu, S.K., Tzeng, G.H.: Financial modeling and improvement planning for the life insurance industry by a rough knowledge based hybrid MCDM model. Inf. Sci. 375, 296–313 (2017)
Thorani, Y.L.P., Ravi, S.N.: Ranking generalized LR fuzzy numbers using area, mode, spreads and weights. Appl. Math. Sci. 11(39), 1943–1953 (2017)
Tian, Z.P., Wang, J., Zhang, H.Y., Wang, J.Q.: Multi-criteria decision-making based on generalized prioritized aggregation operators under simplified neutrosophic uncertain linguistic environment. Int. J. Mach. Learn. Cybern. 9(3), 523–539 (2018)
Torfi, F., Farahani, R.Z., Mahdavi, I.: Fuzzy MCDM for weight of object’s phrase in location routing problem. Appl. Math. Modell. 40(1), 526–541 (2016)
Van Laarhoven, P.J.M., Pedrycz, W.: A fuzzy extension of Saaty’s priority theory. Fuzzy Sets Syst. 11, 229–241 (1983)
Wang, X., Kerre, E.E.: Reasonable properties for the ordering of fuzzy quantities (I). Fuzzy Sets Syst. 118(3), 375–385 (2001)
Wang, X., Kerre, E.E.: Reasonable properties for the ordering of fuzzy quantities (II). Fuzzy Sets Syst. 118(3), 387–405 (2001)
Wu, D., Liu, X., Xue, F., Zheng, H., Shou, Y., Jiang, W.: Fuzzy risk analysis based on a new method for ranking generalized fuzzy numbers. Iran. J. Fuzzy Syst. 15(3), 117–139 (2018)
Wu, Y., Xu, C., Zhang, T.: Evaluation of renewable power sources using a fuzzy MCDM based on cumulative prospect theory: a case in China. Energy 147, 1227–1239 (2018)
Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)
Zadeh L.A. (1975). The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci. 8, 199–249(I), 301–357(II) (1975)
Zadeh, L.A.: Fuzzy logic = computing with words. IEEE Trans. Fuzzy Syst. 4(2), 103–111 (1996)
Zadeh, L.A.: Fuzzy computing with numbers to computing with words-from manipulation of measurements to manipulation of perceptions. In: Wang, P.P. (ed.) the dynamics of judicial proof. Wiley series on intelligent systems, pp. 35–68. Wiley, London (2002)
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Chu, TC., Nguyen, H.T. Ranking Alternatives with Relative Maximizing and Minimizing Sets in a Fuzzy MCDM Model. Int. J. Fuzzy Syst. 21, 1170–1186 (2019). https://doi.org/10.1007/s40815-019-00637-7
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DOI: https://doi.org/10.1007/s40815-019-00637-7