×

On the normalization of interval and fuzzy weights. (English) Zbl 1171.68764

Summary: The normalization of interval and fuzzy weights is often necessary in multiple criteria decision analysis (MCDA) under uncertainty, especially in analytic hierarchy process (AHP) with interval or fuzzy judgements. The existing normalization methods based on interval arithmetic and fuzzy arithmetic are found flawed and need to be revised. This paper presents the correct normalization methods for interval and fuzzy weights and offers relevant theorems in support of them. Numerical examples are examined to show the correctness of the proposed normalization methods and their differences from those existing normalization methods.

MSC:

68T37 Reasoning under uncertainty in the context of artificial intelligence
03E72 Theory of fuzzy sets, etc.
91B06 Decision theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Arbel, A.; Vargas, L.G., Preference simulation and preference programming: robustness issues in priority deviation, European J. oper. res., 69, 200-209, (1993) · Zbl 0783.90002
[2] Bonder, C.G.E.; de Graan, J.G.; Lootsma, F.A., Multi-criteria decision analysis with fuzzy pairwise comparisons, Fuzzy sets and systems, 29, 133-143, (1989) · Zbl 0663.62017
[3] Chang, P.T.; Lee, E.S., The estimation of normalized fuzzy weights, Comput. math. appl., 29, 5, 21-42, (1995) · Zbl 0822.90001
[4] Csutora, R.; Buckley, J.J., Fuzzy hierarchical analysis: the lamda – max method, Fuzzy sets and systems, 120, 181-195, (2001) · Zbl 0994.90078
[5] Deng, H., Multicriteria analysis with fuzzy pairwise comparison, Internat. J. approx. reason., 21, 215-231, (1999)
[6] Denoeux, T., Reasoning with imprecise belief structures, Internat. J. approx. reason., 20, 79-111, (1999) · Zbl 0942.68126
[7] Dong, W.M.; Wong, F.S., Fuzzy weighted averages and implementation of the extension principle, Fuzzy sets and systems, 21, 183-199, (1987) · Zbl 0611.65100
[8] Dubois, D.; Prade, H., Additions of interactive fuzzy numbers, IEEE trans. automat. control, 26, 4, 926-936, (1981)
[9] Dubois, D.; Prade, H., The use of fuzzy numbers in decision analysis, (), 309-321
[10] Gogus, O.; Boucher, T.O., A consistency test for rational weights in multi-criterion decision analysis with fuzzy pairwise comparisons, Fuzzy sets and systems, 86, 129-138, (1997) · Zbl 0921.90003
[11] Guh, Y.Y.; Hon, C.C.; Lee, E.S., Fuzzy weighted average: the linear programming approach via charnes and Cooper’s rule, Fuzzy sets and systems, 117, 157-160, (2001) · Zbl 1032.91041
[12] Guh, Y.Y.; Hon, C.C.; Wang, K.M.; Lee, E.S., Fuzzy weighted average: a max – min paired elimination method, Comput. math. appl., 32, 8, 115-123, (1996) · Zbl 0873.90111
[13] Jiménez, A.; Ríos-Insua, S.; Mateos, A., A decision support system for multiattribute utility evaluation based on imprecise assignments, Decision support systems, 36, 65-79, (2003)
[14] Kao, C.; Liu, S.T., Fractional programming approach to fuzzy weighted average, Fuzzy sets and systems, 120, 435-444, (2001) · Zbl 1103.90411
[15] Kwiesielewicz, M., A note on the fuzzy extension of Saaty’s priority theory, Fuzzy sets and systems, 95, 161-172, (1998) · Zbl 0923.90003
[16] Liou, T.S.; Wang, M.J., Fuzzy weighted average: an improved algorithm, Fuzzy sets and systems, 49, 307-315, (1992) · Zbl 0796.90069
[17] Mikhailov, L., Deriving priorities from fuzzy pairwise comparison judgments, Fuzzy sets and systems, 134, 365-385, (2003) · Zbl 1031.90073
[18] Saaty, T.L., Multicriteria decision making: the analytic hierarchy process, (1988), RWS Publications Pittsburgh, PA · Zbl 1176.90315
[19] Sugihara, K., Interval evaluations in the analytic hierarchy process by possibility analysis, Comput. intelligence, 17, 3, 567-579, (2001)
[20] Sugihara, K.; Ishii, H.; Tanaka, H., Interval priorities in AHP by interval regression analysis, European J. oper. res., 158, 745-754, (2004) · Zbl 1056.90093
[21] Tanaka, H.; Sugihara, K.; Maeda, Y., Non-additive measures by interval probability functions, Inform. sci., 164, 209-227, (2004) · Zbl 1056.28010
[22] Triantaphyllou, E.; Lin, C.T., Development and evaluation of five fuzzy multiattribute decision-making methods, Internat. J. approx. reason., 14, 281-310, (1996) · Zbl 0956.68535
[23] Van Laarhoven, P.J.M.; Pedrycz, W., A fuzzy extension of Saaty’s priority theory, Fuzzy sets and systems, 11, 229-241, (1983) · Zbl 0528.90054
[24] Wang, Y.M.; Yang, J.B.; Xu, D.L., Interval weight generation approaches based on consistency test and interval comparison matrices, Appl. math. comput., 167, 252-273, (2005) · Zbl 1082.65525
[25] Y.M. Wang, J.B. Yang, D.L. Xu, K.S. Chin, The evidential reasoning approach for multiple attribute decision making using interval belief degrees, European J. Oper. Res., in press. · Zbl 1137.90568
[26] Xu, R., Fuzzy least-squares priority method in the analytic hierarchy process, Fuzzy sets and systems, 112, 359-404, (2000) · Zbl 0961.90138
[27] Xu, R.; Zhai, X., Fuzzy logarithmic least squares ranking method in analytic hierarchy process, Fuzzy sets and systems, 77, 175-190, (1996) · Zbl 0869.90002
[28] Yager, R.R., Dempster – shafer belief structures with interval valued focal weights, Internat. J. intelligent systems, 16, 497-512, (2001) · Zbl 0990.68087
[29] Zimmermann, H.J., Fuzzy set theory and its applications, (1991), Kluwer-Nijhoff Boston · Zbl 0719.04002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.