×

A two-stage logarithmic goal programming method for generating weights from interval comparison matrices. (English) Zbl 1114.90493

Summary: A two-stage logarithmic goal programming (TLGP) method is proposed to generate weights from interval comparison matrices, which can be either consistent or inconsistent. The first stage is devised to minimize the inconsistency of interval comparison matrices and the second stage is developed to generate priorities under the condition of minimal inconsistency. The weights are assumed to be multiplicative rather than additive. In the case of hierarchical structures, a nonlinear programming method is used to aggregate local interval weights into global interval weights. A simple yet practical preference ranking method is investigated to compare the interval weights of criteria or rank alternatives in a multiplicative aggregation process. The proposed TLGP is also applicable to fuzzy comparison matrices when they are transformed into interval comparison matrices using \(\alpha\)-level sets and the extension principle. Six numerical examples including a group decision analysis problem with a group of comparison matrices, a hierarchical decision problem and a fuzzy decision problem using fuzzy comparison matrix are examined to show the applications of the proposed methods. Comparisons with other existing procedures are made whenever possible.

MSC:

90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
91B06 Decision theory
91B10 Group preferences
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arbel, A., Approximate articulation of preference and priority derivation, Europ. J. Oper. Res., 43, 317-326 (1989) · Zbl 0697.90003
[2] A. Arbel, A linear programming approach for processing approximate articulation of preference, in: P. Korhonen, A. Lewandowski, J. Wallenius, (Eds.), Multiple Criteria Decision Support, Lecture Notes in Economics and Mathematical Systems, vol. 356, Springer, Berlin, 1991, pp. 79-86.; A. Arbel, A linear programming approach for processing approximate articulation of preference, in: P. Korhonen, A. Lewandowski, J. Wallenius, (Eds.), Multiple Criteria Decision Support, Lecture Notes in Economics and Mathematical Systems, vol. 356, Springer, Berlin, 1991, pp. 79-86. · Zbl 0825.90032
[3] A. Arbel, L.G. Vargas, The analytic hierarchy process with interval judgments, in: A. Goicoechea, L. Duckstein, S. Zoints, (Eds.), Multiple criteria decision making, Proc. 9th Internat. Conf. held in Fairfax, Virginia, 1990; Springer, New York, 1992, pp. 61-70.; A. Arbel, L.G. Vargas, The analytic hierarchy process with interval judgments, in: A. Goicoechea, L. Duckstein, S. Zoints, (Eds.), Multiple criteria decision making, Proc. 9th Internat. Conf. held in Fairfax, Virginia, 1990; Springer, New York, 1992, pp. 61-70. · Zbl 0825.90014
[4] Arbel, A.; Vargas, L. G., Preference simulation and preference programmingrobustness issues in priority deviation, Europ. J. Oper. Res., 69, 200-209 (1993) · Zbl 0783.90002
[5] Barzilai, J., Deriving weights from pairwise comparison matrices, J. Oper. Res. Soc., 48, 1226-1232 (1997) · Zbl 0895.90004
[6] Bonder, C. G.E.; de Graan, J. G.; Lootsma, F. A., Multicretia decision analysis with fuzzy pairwise comparisons, Fuzzy Sets and Systems, 29, 133-143 (1989) · Zbl 0663.62017
[7] Bortolan, G.; Degani, R., A review of some methods for ranking fuzzy subsets, Fuzzy Sets and Systems, 15, 1-19 (1985) · Zbl 0567.90056
[8] Bryson, N., A goal programming method for generating priority vectors, J. Oper. Res. Soc., 46, 641-648 (1995) · Zbl 0830.90001
[9] Bryson, N.; Mobolurin, A., An action learning evaluation procedure for multicriteria decision making problems, Europ. J. Oper. Res., 96, 379-386 (1996) · Zbl 0917.90005
[10] Buckley, J. J., Fuzzy hierarchical analysis, Fuzzy Sets and Systems, 17, 233-247 (1985) · Zbl 0602.90002
[11] Buckley, J. J.; Feuring, T.; Hayashi, Y., Fuzzy hierarchical analysis revisited, Europ. J. Oper. Res., 129, 48-64 (2001) · Zbl 1002.93022
[12] Chanas, S., On the interval approximation of a fuzzy number, Fuzzy Sets and Systems, 122, 353-356 (2001) · Zbl 1010.03523
[13] Csutora, R.; Buckley, J. J., Fuzzy hierarchical analysisthe Lamda-Max method, Fuzzy Sets and Systems, 120, 181-195 (2001) · Zbl 0994.90078
[14] Dubois, D.; Prade, H., Additions of interactive fuzzy numbers, IEEE Trans. Automat. Control, 26, 926-936 (1981) · Zbl 1457.68262
[15] Dubois, D.; Prade, H., A unified view of ranking techniques for fuzzy numbers, (IEEE Internat. Fuzzy Systems Conf. Proc.. IEEE Internat. Fuzzy Systems Conf. Proc., Seoul (1991)), 1328-1333
[16] Fortemps, P.; Roubens, M., Ranking and defuzzification methods based on area compensation, Fuzzy Sets and Systems, 82, 319-330 (1996) · Zbl 0886.94025
[17] Grzegorzewski, P., Nearest interval approximation of a fuzzy number, Fuzzy Sets and Systems, 130, 321-330 (2002) · Zbl 1011.03504
[18] Haines, L. M., A statistical approach to the analytic hierarchy process with interval judgments. (I). Distributions on feasible regions, Europ. J. Oper. Res., 110, 112-125 (1998) · Zbl 0934.91014
[19] Ishibuchi, H.; Tanaka, H., Multiobjective programming in optimization of the interval objective function, Europ. J. Oper. Res., 48, 219-225 (1990) · Zbl 0718.90079
[20] Islam, R.; Biswal, M. P.; Alam, S. S., Preference programming and inconsistent interval judgments, Europ. J. Oper. Res., 97, 53-62 (1997) · Zbl 0923.90008
[21] Jiménez, M., Ranking fuzzy numbers through the comparison of its expected intervals, Internat. J. Uncertainty Fuzziness Knowledge-Based Systems, 4, 379-388 (1996) · Zbl 1232.03040
[22] Kress, M., Approximate articulation of preference and priority derivation—A comment, Europ. J. Oper. Res., 52, 382-383 (1991)
[23] Kundu, S., Preference relation on fuzzy utilities based on fuzzy leftness relation on intervals, Fuzzy Sets and Systems, 97, 183-191 (1998)
[24] Leung, L. C.; Cao, D., On consistency and ranking of alternatives in fuzzy AHP, Europ. J. Oper. Res., 124, 102-113 (2000) · Zbl 0960.90097
[25] McCahon, C. S.; Lee, E. S., Comparing fuzzy numbersthe proportion of the optimum method, Internat. J. Approx. Reason., 4, 159-181 (1990) · Zbl 0709.03534
[26] Mikhailov, L., Fuzzy analytical approach to partnership selection in formation of virtual enterprises, Omega, 30, 393-401 (2002)
[27] Mikhailov, L., Deriving priorities from fuzzy pairwise comparison judgments, Fuzzy Sets and Systems, 134, 365-385 (2003) · Zbl 1031.90073
[28] Mikhailov, L., Group prioritization in the AHP by fuzzy preference programming method, Comput. Oper. Res., 31, 293-301 (2004) · Zbl 1057.90547
[29] Moreno-Jiménez, J. M., A probabilistic study of preference structures in the analytic hierarchy process with interval judgments, Math. Comput. Modelling, 17, 4/5, 73-81 (1993) · Zbl 0780.90002
[30] Moreno-Jiménez, J. M.; Aguarón, J.; Escobar, M. T.; Turón, A., The multicriteria procedural rationality on SISDEMA, Europ. J. Oper. Res., 96, 379-386 (1999)
[31] Saade, J. J.; Schwarzlander, H., Ordering fuzzy sets over the real linean approach based on decision making under uncertainty, Fuzzy Sets and Systems, 50, 237-246 (1992)
[32] Saaty, T. L.; Vargas, L. G., Uncertainty and rank order in the analytic hierarchy process, Europ. J. Oper. Res., 32, 107-117 (1987) · Zbl 0632.90002
[33] A. Salo, R.P. Hämäläinen, Processing interval judgments in the analytic hierarchy process, in: A. Goicoechea, L. Duckstein, S. Zoints, (Eds.), Multiple Criteria Decision Making, Proc. 9th Internat. Conf. held in Fairfax, Virginia, 1990; Springer, New York, 1992, pp. 359-372.; A. Salo, R.P. Hämäläinen, Processing interval judgments in the analytic hierarchy process, in: A. Goicoechea, L. Duckstein, S. Zoints, (Eds.), Multiple Criteria Decision Making, Proc. 9th Internat. Conf. held in Fairfax, Virginia, 1990; Springer, New York, 1992, pp. 359-372.
[34] Salo, A.; Hämäläinen, R. P., Preference programming through approximate ratio comparisons, Europ. J. Oper. Res., 82, 458-475 (1995) · Zbl 0909.90006
[35] Stam, A.; Duarte Silva, A. P., On multiplicative priority rating methods for the AHP, Europ. J. Oper. Res., 145, 92-108 (2003) · Zbl 1026.90508
[36] Tseng, T. Y.; Klein, C. M., New algorithm for the ranking procedure in fuzzy decisionmaking, IEEE Trans. Systems Man Cybernet., 19, 1289-1296 (1989)
[37] 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
[38] Wang, X.; Kerre, E. E., Reasonable properties for the ordering of fuzzy quantities (I), Fuzzy Sets and Systems, 118, 375-385 (2001) · Zbl 0971.03054
[39] Wang, X.; Kerre, E. E., Reasonable properties for the ordering of fuzzy quantities (II), Fuzzy Sets and Systems, 118, 387-405 (2001) · Zbl 0971.03055
[40] Wang, Y. M.; Xu, N. R., Linear programming method for solving group-AHP, J. Southeast Univ., 20, 6, 58-63 (1990) · Zbl 0729.90856
[41] Y.M. Wang, J.B. Yang, D.L. Xu, A preference aggregation method through the estimation of utility intervals, Comput. Oper. Res. 2004, in press.; Y.M. Wang, J.B. Yang, D.L. Xu, A preference aggregation method through the estimation of utility intervals, Comput. Oper. Res. 2004, in press. · Zbl 1068.90075
[42] Xu, R., Fuzzy least-squares priority method in the analytic hierarchy process, Fuzzy Sets and Systems, 112, 359-404 (2000) · Zbl 0961.90138
[43] 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
[44] Yuan, Y., Criteria for evaluating fuzzy ranking methods, Fuzzy Sets and Systems, 44, 139-157 (1991) · Zbl 0747.90003
[45] Zimmermann, H. J., Fuzzy Set Theory and its Applications (1991), Kluwer-Nijhoff: 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.