The preference order of fuzzy numbers. (English) Zbl 1104.91300
Summary: Many fuzzy number ranking approaches are developed in the literature for multiattribute decision-making problems. Almost all of the existing approaches focus on quantity measurement of fuzzy numbers for ranking purpose. In this paper, we consider the ranking process to determine a decision-maker’s preference order of fuzzy numbers. A new ranking index is proposed to not only take quantity measurement, but incorporate a quality factor into consideration for the need of general decision-making problems. For measuring quantity, several \(\alpha\)-cuts of fuzzy numbers are used. A signal/noise ratio is defined to evaluate quality of a fuzzy number. This ratio considers the middle-point and spread of each \(\alpha\)-cut of fuzzy numbers as the signal and noise, respectively. A fuzzy number with the stronger signal and the weaker noise is considered better. Moreover, the associated \(\alpha\)-levels are treated as the degree of belief about the \(\alpha\)-cut and used as weights in the index for strengthening the influence of an \(\alpha\)-cut with higher \(\alpha\)-levels.
The membership functions of fuzzy numbers are not necessarily to be known beforehand while applying this index. Only a few left and right boundary values of \(\alpha\)-cuts of fuzzy numbers are required. We prove the feature of the proposed index in a particular case. Several examples are also used to illustrate the feature and applicability in ranking fuzzy numbers.
The membership functions of fuzzy numbers are not necessarily to be known beforehand while applying this index. Only a few left and right boundary values of \(\alpha\)-cuts of fuzzy numbers are required. We prove the feature of the proposed index in a particular case. Several examples are also used to illustrate the feature and applicability in ranking fuzzy numbers.
Keywords:
multi-attribute decision making; fuzzy numbers; ranking methods; signal/noise ratio (S/N ratio)References:
| [1] | Chen, S.-J.; Hwang, C.-L., Fuzzy Multiple Attribute Decision Making (1992), Springer: Springer New York · Zbl 0768.90042 |
| [2] | Adamo, J. M., Fuzzy decision trees, Fuzzy Sets and Systems, 4, 207-219 (1980) · Zbl 0444.90004 |
| [3] | Baldwin, J.; Guild, N., Comparison of fuzzy sets on the same decision space, Fuzzy Sets and Systems, 2, 213-233 (1979) · Zbl 0422.90004 |
| [4] | Bass, S.; Kwakernaak, H., Rating and ranking of multiple-aspect alternatives using fuzzy sets, Automatica, 13, 47-58 (1977) · Zbl 0363.90010 |
| [5] | Buckley, J.; Chanas, S., A fast method of ranking alternatives using fuzzy numbers, Fuzzy Sets and Systems, 30, 337-339 (1989) · Zbl 0673.90004 |
| [6] | Delgado, M.; Verdegay, J. L.; Villa, M. A., A procedure for ranking fuzzy numbers using fuzzy relations, Fuzzy Sets and Systems, 26, 49-62 (1988) · Zbl 0647.94026 |
| [7] | Dubois, D.; Prade, H., Ranking of fuzzy numbers in the setting of possibility theory, Inform. Sci., 30, 183-224 (1983) · Zbl 0569.94031 |
| [8] | Kerre, E., (Gupta, M.; Sanchez, E., The use of fuzzy set theory in electrocardiological diagnostics. The use of fuzzy set theory in electrocardiological diagnostics, Approximate Reasoning in Decision Analysis (1982), North-Holland: North-Holland Amsterdam), 277-282 |
| [9] | Kolodziejczyk, W., Orlovsky’s concept of decision-making with fuzzy preference relation further results, Fuzzy Sets and Systems, 19, 11-20 (1986) · Zbl 0597.90004 |
| [10] | Mabuchi, G., An approach to the comparison of fuzzy subsets with an α-cut dependent index, IEEE Trans. Syst. Man and Cybern., 18, 264-272 (1988) |
| [11] | Nakamura, K., Preference relation on a set of fuzzy utilities as a basis for decision making, Fuzzy Sets and Systems, 20, 147-162 (1986) · Zbl 0618.90001 |
| [12] | Tsukamoto, Y.; Nikiforuk, P. N.; Gupta, M. M., (Akashi, H., On the comparison of fuzzy sets using fuzzy chopping. On the comparison of fuzzy sets using fuzzy chopping, Control Science and Technology for Progress of Society (1983)), 46-51, New York |
| [13] | Watson, S. R.; Weiss, J. J.; McDonnell, M. L., Fuzzy decision analysis, IEEE Trans. Syst. Man Cybern., 9, 1-9 (1979) |
| [14] | Yager, R. R., On choosing between fuzzy subsets, Kybernetes, 9, 151-154 (1980) · Zbl 0428.03050 |
| [15] | Lee, E. S.; Li, R.-J., Comparison of fuzzy numbers based on the probability measure of fuzzy events, Computers Math. Applic., 15, 10, 887-896 (1988) · Zbl 0654.60008 |
| [16] | Chen, S.-H., Ranking fuzzy numbers with maximizing set and minimizing set, Fuzzy Sets and Systems, 17, 113-129 (1985) · Zbl 0618.90047 |
| [17] | Choobineh, F.; Li, H., An index for ordering fuzzy numbers, Fuzzy Sets and Systems, 54, 287-294 (1993) |
| [18] | Huang, C. C., A study on the fuzzy ranking and its application on the decision support systems, (Ph.D. Dissertation (1989), Tamkang Univ: Tamkang Univ Taiwan, R.O.C) |
| [19] | Jain, R., Decision making in the presence of fuzzy variables, IEEE Trans. Syst. Man Cybern., 6, 698-703 (1976) · Zbl 0337.90005 |
| [20] | McCahone, C., Fuzzy set theory applied to production and inventory control, (Ph.D. Thesis (1987), Department of Industrial Engineering: Department of Industrial Engineering Kansas State University) |
| [21] | Yager, R. R., On a general class of fuzzy connectives, Fuzzy Sets and Systems, 4, 235-242 (1980) · Zbl 0443.04008 |
| [22] | Yager, R. R., A procedure for ordering fuzzy subsets of the unit interval, Information Sciences, 24, 143-161 (1981) · Zbl 0459.04004 |
| [23] | Efstathiou, J.; Tong, R., Ranking fuzzy sets using linguistic preference relations, (Northwestern University, Evanston, IL. Northwestern University, Evanston, IL, Proc. \(10^{th}\) International Symposium on Multiple-Valued Logic (1980)), 137-142 |
| [24] | Tong, R. M.; Bonissone, P. P., (Zimmermann, H. J., Linguistic solutions to fuzzy decision problems. Linguistic solutions to fuzzy decision problems, TIMS/Studies in the Management Science, Volume 20 (1984)), 323-334 · Zbl 0538.90003 |
| [25] | Bortolan, G.; Degani, R., A review of some methods for ranking fuzzy subsets, Fuzzy Sets and Systems, 15, 1-19 (1985) · Zbl 0567.90056 |
| [26] | Fortemps, P.; Roubens, M., Ranking and defuzzification methods based on area compensation, Fuzzy Sets and Systems, 82, 319-330 (1996) · Zbl 0886.94025 |
| [27] | Kim, K.; Park, K., Ranking fuzzy numbers with index of optimism, Fuzzy Sets and Systems, 35, 143-150 (1990) |
| [28] | Chen, C.-B.; Klein, C. M., A simple approach to ranking a group of aggregated fuzzy utilities, IEEE Trans. Syst. Man. Cybern.—Part B: Cybernetics, 27, 26-35 (1997) |
| [29] | Liou, T.-S.; Wang, M.-J., Ranking fuzzy numbers with integral value, Fuzzy Sets and Systems, 50, 247-255 (1992) · Zbl 1229.03043 |
| [30] | Tseng, T.-Y.; Klein, C. M., New algorithm for the ranking procedure in fuzzy decision making, IEEE Trans. Syst. Man Cybern, 19, 1289-1296 (1989) |
| [31] | Chen, L.-H.; Lu, H.-W., An approximate approach for ranking fuzzy numbers based on left and right dominance, Computers Math. Applic., 41, 12, 1589-1602 (2001) · Zbl 0984.03041 |
| [32] | Taguchi, G.; Clausing, D., Introduction to Quality Engineering (1986), Asian Productivity Organization |
| [33] | Dubois, D.; Prade, H., Operations on fuzzy numbers, Int. J. Systems Sci., 9, 613-626 (1978) · Zbl 0383.94045 |
| [34] | Zimmermann, H. J., Fuzzy Set Theory and Its Applications (1991), Kluwer Academic: Kluwer Academic Boston, MA · Zbl 0719.04002 |
| [35] | Cheng, C.-H., A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems, 95, 307-317 (1998) · Zbl 0929.91009 |
| [36] | Buckley, J., Ranking alternatives using fuzzy numbers, Fuzzy Sets and Systems, 15, 21-31 (1985) · Zbl 0567.90057 |
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