| Title:
|
Measuring consistency and inconsistency of pair comparison systems (English) |
| Author:
|
Ramík, Jaroslav |
| Author:
|
Vlach, Milan |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
49 |
| Issue:
|
3 |
| Year:
|
2013 |
| Pages:
|
465-486 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we deal with mathematical modeling of real processes that are based on preference relations in the sense that, for every pair of distinct alternatives, the processes are linked to a value of preference degree of one alternative over the other one. The use of preference relations is usual in decision making, psychology, economics, knowledge acquisition techniques for knowledge-based systems, social choice and many other social sciences. For designing useful mathematical models of such processes, it is very important to adequately represent properties of preference relations. We are mainly interested in the properties of such representations which are usually called reciprocity, consistency and transitivity. In decision making processes, the lack of reciprocity, consistency or transitivity may result in wrong conclusions. That is why it is so important to study the conditions under which these properties are satisfied. However, the perfect consistency or transitivity is difficult to obtain in practice, particularly when evaluating preferences on a set with a large number of alternatives. Under different preference representation structures, the multiplicative and additive preference representations are incorporated in the decision problem by means of a transformation function between multiplicative and additive representations. Some theoretical results on relationships between multiplicative and additive representations of preferences on finite sets are presented and some possibilities of measuring their consistency or transitivity are proposed and discussed. Illustrative numerical examples are provided. (English) |
| Keyword:
|
multi-criteria optimization |
| Keyword:
|
pair-wise comparison matrix |
| Keyword:
|
AHP |
| MSC:
|
90B50 |
| MSC:
|
90C29 |
| MSC:
|
91B08 |
| . |
| Date available:
|
2013-07-18T15:40:50Z |
| Last updated:
|
2013-07-31 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143359 |
| . |
| Reference:
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[1] Aguarón, J., Moreno-Jimenéz, J. M.: The geometric consistency index: Approximated thresholds..European J. Oper. Res. 147 (2003), 137-145. Zbl 1060.90657, 10.1016/S0377-2217(02)00255-2 |
| Reference:
|
[2] Boyd, J. P.: Numerical methods for Bayesian ratings from paired comparisons..J. Quantitative Anthropology 3 (1991), 117-133. |
| Reference:
|
[3] Bozóki, S., Rapcsák, T.: On Saaty's and Koczkodaj's inconsistencies of pairwise comparison matrices..WP 2007-1, June 2007, Computer and Automation Research Institute, Hungarian Academy of Sciences, http://www.oplab.sztaki.hu/WP_2007_1_Bozoki_Rapcsak.pdf Zbl 1177.90205 |
| Reference:
|
[4] Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine..Comput. Networks and ISDN Systems 30 (1998), 107-117. 10.1016/S0169-7552(98)00110-X |
| Reference:
|
[5] Crawford, G., Williams, C.: A note on the analysis of subjective judgment matrices..J. Math. Psychol. 29 (1985), 387-405. 10.1016/0022-2496(85)90002-1 |
| Reference:
|
[6] Chiclana, F., Herrera, F., Herrera-Viedma, E.: Integrating three representation models in fuzzy multipur-pose decision making nased on fuzzy preference relations..Fuzzy Sets and Systems 97 (1998), 33-48. MR 1618276 |
| Reference:
|
[7] Chiclana, F., Herrera, F., Herrera-Viedma, E.: Integrating multiplicative preference relations in a multipur-pose decision making model based on fuzzy preference relations..Fuzzy Sets and Systems 112 (2001), 277-291. MR 1854819 |
| Reference:
|
[8] Chiclana, F., Herrera, F., Herrera-Viedma, E., Alonso, S.: Some induced ordered weighted averaging opera-tors and their use for solving group decision-making problems based on fuzzy preference relations..European J. Oper. Res. 182 (2007), 383-399. 10.1016/j.ejor.2006.08.032 |
| Reference:
|
[9] Chiclana, F., Herrera-Viedma, E., Alonso, S.: A note on two methods for estimating missing pairwise preference values..IEEE Trans. Systems, Man and Cybernetics - Part B: Cybernetics 39 (2009), 6, 1628-1633. 10.1109/TSMCB.2009.2023923 |
| Reference:
|
[10] Chiclana, F., Herrera-Viedma, E., Alonso, S., Herrera, F.: Cardinal consistency of reciprocal preference relations: A characterization of multiplicative transitivity..IEEE Trans. Fuzzy Systems 17 (2009), 1, 14-23. 10.1109/TFUZZ.2008.2008028 |
| Reference:
|
[11] Dopazo, E., Gonzales-Pachón, J.: Consistency-driven approximation of a pairwise comparison matrix..Kybernetika 39 (2003), 5, 561-568. MR 2042341 |
| Reference:
|
[12] Fiedler, M., Nedoma, J., Ramík, J., Rohn, J., Zimmermann, K.: Linear Optimization Problems with Inexact Data..Springer, Berlin - Heidelberg - New York - Hong Kong - London - Milan - Tokyo 2006. Zbl 1106.90051, MR 2218777 |
| Reference:
|
[13] Fishburn, P. C.: Utility Theory for Decision Making..Wiley, New York 1970. Zbl 0213.46202, MR 0264810 |
| Reference:
|
[14] Fishburn, P. C.: Binary choice probabilities: On the varieties of stochastic transitivity..J. Math. Psychol. 10 (1973), 329-352. Zbl 0277.92008, MR 0327330, 10.1016/0022-2496(73)90021-7 |
| Reference:
|
[15] Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support..Kluwer, Dordrecht 1994. Zbl 0827.90002 |
| Reference:
|
[16] Gass, S. I., Rapcsák, T.: Singular value decomposition in AHP..European J. Oper. Res. 154 (2004), 573-584. Zbl 1146.90445, MR 2025802, 10.1016/S0377-2217(02)00755-5 |
| Reference:
|
[17] Golany, B.: A multicriteria evaluation methods from obtaining weights from ratio scale matrices..European J. Oper. Res. 69 (1993), 210-220. 10.1016/0377-2217(93)90165-J |
| Reference:
|
[18] Herrera-Viedma, E., Herrera, F., Chiclana, F., Luque, M.: Some issues on consistency of fuzzy preference relations..European J. Oper. Res. 154 (2004), 98-109. Zbl 1099.91508, MR 2025664, 10.1016/S0377-2217(02)00725-7 |
| Reference:
|
[19] Herrera-Viedma, E., Chiclana, F., Herrera, F., Alonso, S.: Group decision-making model with incomplete fuzzy preference relations based on additive consistency..IEEE Trans. Systems, Man and Cybernetics, Part B - Cybernetics 37 (2007), 1, 176-189. 10.1109/TSMCB.2006.875872 |
| Reference:
|
[20] Krantz, D. H., Luce, R. D., Suppes, P., Tversky, A.: Foundations of Measurement. Vol. I..Academic Press, New York 1971. Zbl 1118.91359, MR 0459067 |
| Reference:
|
[21] Mareš, M.: Coalitional fuzzy preferences..Kybernetika 38 (2002), 3, 339-352. Zbl 1265.91012, MR 1944314 |
| Reference:
|
[22] Mareš, M.: Fuzzy coalitional structures..Fuzzy Sets and Systems 114 (2000), 3, 23-33. Zbl 1153.91325 |
| Reference:
|
[23] Ramík, J., Korviny, P.: Inconsistency of pairwise comparison matrix with fuzzy elements based on geo-metric mean..Fuzzy Sets and Systems 161 (2010), 1604-1613. MR 2608264 |
| Reference:
|
[24] Ramík, J., Vlach, M.: Generalized Concavity in Optimization and Decision Making..Kluwer Publ. Comp., Boston - Dordrecht - London, 2001. |
| Reference:
|
[25] Roberts, F. S.: Measurement theory: with application to decisionmaking, utility and the social sciences..In: Encyklopedia of Mathematics and its Applications, Vol. 7, Addison-Wesley, Reading 1979. MR 0551364 |
| Reference:
|
[26] Saaty, T. L.: Fundamentals of Decision Making and Priority Theory with the AHP..RWS Publications, Pittsburgh 1994. |
| Reference:
|
[27] Saaty, T. L.: Multicriteria Decision Making - The Analytical Hierarchy Process. Vol. I..RWS Publications, Pittsburgh 1991. |
| Reference:
|
[28] Stein, W. E., Mizzi, P. J.: The harmonic consistency index for the analytic hierarchy process..European J. Oper. Res. 117 (2007), 488-497. Zbl 1111.90057, 10.1016/j.ejor.2005.10.057 |
| Reference:
|
[29] Tanino, T.: Fuzzy preference orderings in group decision making..Fuzzy Sets and Systems 12 (1984), 117-131. Zbl 0567.90002, MR 0734944, 10.1016/0165-0114(84)90032-0 |
| Reference:
|
[30] Tanino, T.: Fuzzy preference relations in group decision making..In: Non-Conventional Preference Relations in Decision Making (J. Kacprzyk and M. Roubens, eds.), Springer-Verlag, Berlin 1988, pp. 54-71. Zbl 0652.90010, MR 1133648 |
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