Document Type : Research Paper

Author

Department of Computer Sciences, Javakhishvili Tbilisi State University, Tbilisi.

10.22105/jfea.2021.275084.1080

Abstract

The Ordered Weighted Averaging (OWA) operator was introduced by Yager [57] to provide a method for aggregating inputs that lie between the max and min operators. In this article two variants of probabilistic extensions the OWA operator-POWA and FPOWA (introduced by Merigo [26] and [27]) are considered as a basis of our generalizations in the environment of fuzzy uncertainty (parts II and III of this work), where different monotone measures (fuzzy measure) are used as uncertainty measures instead of the probability measure. For the identification of “classic” OWA and new operators (presented in parts II and III) of aggregations, the Information Structure is introduced where the incomplete available information in the general decision-making system is presented as a condensation of uncertainty measure, imprecision variable and objective function of weights.

Keywords

Main Subjects

[1]       Beliakov, G. (2005). Learning weights in the generalized OWA operators. Fuzzy optimization and decision making4(2), 119-130. https://doi.org/10.1007/s10700-004-5868-3
[2]       Beliakov, G., Pradera, A., & Calvo, T. (2007). Aggregation functions: A guide for practitioners (Vol. 221). Heidelberg: Springer.
[3]            Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment, management science, 17(4), B-141-B-273. https://doi.org/10.1287/mnsc.17.4.B141
[4]            Calvo, T., & Beliakov, G. (2008). Identification of weights in aggregation operators. In Fuzzy sets and their extensions: representation, aggregation and models (pp. 145-162). Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-540-73723-0_8
[5]            De Campos Ibañez, L. M., & Carmona, M. J. B. (1989). Representation of fuzzy measures through probabilities. Fuzzy sets and systems31(1), 23-36. https://doi.org/10.1016/0165-0114(89)90064-X
[6]            Carlsson, C., & Fullér, R. (2012). Fuzzy reasoning in decision making and optimization.  Physica-Verlag Heidelberg-New York.
[7]            Choquet, G. (1954). Theory of capacities. Annals of the Fourier Institute, 5, 131-295. DOI: https://doi.org/10.5802/aif.53
[8]            Denneberg, D. (2013). Non-additive measure and integral (Vol. 27). Springer Science & Business Media.
[9]            Dong, Y., Xu, Y., Li, H., & Feng, B. (2010). The OWA-based consensus operator under linguistic representation models using position indexes. European journal of operational research203(2), 455-463. https://doi.org/10.1016/j.ejor.2009.08.013
[10]         Dubois, D., Marichal, J. L., Prade, H., Roubens, M., & Sabbadin, R. (2001). The use of the discrete Sugeno integral in decision-making: A survery. International journal of uncertainty, fuzziness and knowledge-based systems9(05), 539-561. https://doi.org/10.1142/S0218488501001058
[11]         Dubois, D., & Prade, H. (2007). Possibility theory. Scholarpedia, 2(10), 2074.
[12]         Mesiar, R., Calvo, T., & Mayor, G. (2002). Aggregation operators: new trends and applications. Physica-Verlag.
[13]         Gil-lafuente, A. M., & Merigo-lindahl, J. M. (Eds.). (2010). Computational Intelligence in Business and Economics. Proceedings of The Ms' 10 International Conference (Vol. 3). World Scientific.
[14]         Grabisch, M., Sugeno, M., & Murofushi, T. (2010). Fuzzy measures and integrals: theory and applications. Heidelberg: Physica. http://hdl.handle.net/10637/3294
[15]         Greco, S., Pereira, R. A. M., Squillante, M., & Yager, R. R. (Eds.). (2010). Preferences and Decisions: Models and Applications (Vol. 257). Springer.
[16]         Kacprzyk, J., & Zadrożny, S. (2009). Towards a general and unified characterization of individual and collective choice functions under fuzzy and nonfuzzy preferences and majority via the ordered weighted average operators. International journal of intelligent systems24(1), 4-26. https://doi.org/10.1002/int.20325
[17]         Kandel, A. (1980). On the control and evaluation of uncertain processes. IEEE transactions on automatic control25(6), 1182-1187.DOI: 10.1109/TAC.1980.1102544
[18]         Kandel, A. (1978). Fuzzy statistics and forecast evaluation. IEEE transactions on systems, man, and cybernetics, 8(5),396-401. http://pascal-rancis.inist.fr/vibad/index.php?action=getRecordDetail&idt=PASCAL7930008529   
[19]         Kaufman, M. M. (1985). Gupta, Introduction to fuzzy arithmetic. Van Nostrand Reinhold Company.
[20]         Klir, G. J. (2013). Architecture of systems problem solving. Springer Science & Business Media.
[21]         Klir, G. J., & Folger, T. A. (1998). Fuzzy sets, uncertainty and information. Prentice Hall, Englewood Cliffs
[22]         Klir, G. J., & Wierman, M. J. (2013). Uncertainty-based information: elements of generalized information theory (Vol. 15). Physica.    
[23]         Marichal, J. L. (2000). An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria. IEEE transactions on fuzzy systems8(6), 800-807.DOI: 10.1109/91.890347
[24]         Marichal, J. L. (2000). On Choquet and Sugeno integrals as aggregation functions. Fuzzy measures and integrals-theory and applications, 247-272.
[25]         Marichal, J. L. (2000). On Sugeno integral as an aggregation function. Fuzzy sets and systems114(3), 347-365. https://doi.org/10.1016/S0165-0114(98)00116-X
[26]         Merigo, J. M. (2011). The uncertain probabilistic weighted average and its application in the theory of expertons. African journal of business management5(15), 6092-6102.
[27]         Merigó, J. M. (2011). Fuzzy multi-person decision making with fuzzy probabilistic aggregation operators. International journal of fuzzy systems13(3), p163-174.
[28]         Merigó, J. M., & Casanovas, M. (2011). The uncertain induced quasi‐arithmetic OWA operator. International journal of intelligent systems, 26(1), 1-24. https://doi.org/10.1002/int.20444
[29]         Merigo, J. M., & Casanovas, M. (2010). Fuzzy generalized hybrid aggregation operators and its application in fuzzy decision making. International journal of fuzzy systems12(1), 15-24.
[30]    Merigo, J. M., & Casanovas, M. (2010). The fuzzy generalized OWA operator and its application in strategic decision making. Cybernetics and systems: an international journal41(5), 359-370. https://doi.org/10.1080/01969722.2010.486223
[31]    Merigó, J. M., & Casanovas, M. (2009). Induced aggregation operators in decision making with the Dempster‐Shafer belief structure. International journal of intelligent systems24(8), 934-954. https://doi.org/10.1002/int.20368
[32]    Merigo, J. M., Casanovas, M., & Martínez, L. (2010). Linguistic aggregation operators for linguistic decision making based on the Dempster-Shafer theory of evidence. International journal of uncertainty, fuzziness and knowledge-based systems18(03), 287-304. https://doi.org/10.1142/S0218488510006544
[33]    Mesiar, R., & Špirková, J. (2006). Weighted means and weighting functions. Kybernetika42(2), 151-160.
[34]    Shafer, G. (1976). A mathematical theory of evidence (Vol. 42). Princeton university press.
[35]    Sikharulidze, A., & Sirbiladze, G. (2008). Average misbilief criterion on the minimal fuzzy covering problem. Proceedings of the 9th WSEAS international confeerence on fuzzy systems (pp. 42-48).
[36]    Sirbiladze, G. (2012). Extremal fuzzy dynamic systems: Theory and applications (Vol. 28). Springer Science & Business Media.
[37]    Sirbiladze, G. (2005). Modeling of extremal fuzzy dynamic systems. Part III. Modeling of extremal and controllable extremal fuzzy processes. International journal of general systems34(2), 169-198. https://doi.org/10.1080/03081070512331325204
[38]    Sirbiladze, G., & Gachechiladze, T. (2005). Restored fuzzy measures in expert decision-making. Information sciences169(1-2), 71-95. https://doi.org/10.1016/j.ins.2004.02.010
[39]    Sirbiladze, G., Ghvaberidze, B., Latsabidze, T., & Matsaberidze, B. (2009). Using a minimal fuzzy covering in decision-making problems. Information sciences179(12), 2022-2027. https://doi.org/10.1016/j.ins.2009.02.004
[40]    Sirbiladze, G., Sikharulidze, A., & Sirbiladze, N. (2010, February). Generalized weighted fuzzy expected values in uncertainty environment. Recent advances in artificial intelligence, knowledge engineering and data bases: proceedings O the 9th WSEAS international conference on artificial intelligence, knowledge engineering and data bases (AIKED'10) (pp. 54-64).
[41]    Sirbiladze, G., & Sikharulidze, A. (2003). Weighted fuzzy averages in fuzzy environment: Part I. Insufficient expert data and fuzzy averages. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems11(02), 139-157.   
[42]    Sirbiladze, G., Sikharulidze, A., Ghvaberidze, B., & Matsaberidze, B. (2011). Fuzzy-probabilistic aggregations in the discrete covering problem. International journal of general systems40(02), 169-196. https://doi.org/10.1080/03081079.2010.508954
[43]    Sirbiladze, G., & Zaporozhets, N. (2003). About two probability representations of fuzzy measures on a finite set. Journal of fuzzy mathematics11(3), 549-566.
[44]    Sugeno, M. (1974). Theory of fuzzy integrals and its applications (Doctoral Thesis, Tokyo Institute of technology). (In Japenees). https://ci.nii.ac.jp/naid/10017209011/
[45]    Torra, V. (1997). The weighted OWA operator. International journal of intelligent systems12(2), 153-166. https://doi.org/10.1002/(SICI)1098-111X(199702)12:23.0.CO;2-P
[46]    Torra, V., & Narukawa, Y. (2007). Modeling decisions: information fusion and aggregation operators. Springer Science and Business Media.
[47]    Yager, R. R. (2009). Weighted maximum entropy OWA aggregation with applications to decision making under risk. IEEE transactions on systems, man, and cybernetics-part a: systems and humans39(3), 555-564.DOI: 10.1109/TSMCA.2009.2014535
[48]    Yager, R. R. (2009). On the dispersion measure of OWA operators. Information sciences179(22), 3908-3919. https://doi.org/10.1016/j.ins.2009.07.015
[49]    Yager, R. R. (2007). Aggregation of ordinal information. Fuzzy optimization and decision making6(3), 199-219. https://doi.org/10.1007/s10700-007-9008-8
[50]    Yager, R. R. (2004). Generalized OWA aggregation operators. Fuzzy optimization and decision making3(1), 93-107. https://doi.org/10.1023/B:FODM.0000013074.68765.97
[51]    Yager, R. R. (2002). On the evaluation of uncertain courses of action. Fuzzy optimization and decision making1(1), 13-41. https://doi.org/10.1023/A:1013715523644
[52]         Yager, R. R. (2002). Heavy OWA operators. Fuzzy optimization and decision making1(4), 379-397. https://doi.org/10.1023/A:1020959313432
[53]         Yager, R. R. (2002). On the cardinality index and attitudinal character of fuzzy measures. International journal of general systems31(3), 303-329. https://doi.org/10.1080/03081070290018047
[54]         Yager, R. R. (2000). On the entropy of fuzzy measures. IEEE transactions on fuzzy systems8(4), 453-461.doi: 10.1109/91.868951
[55]         Yager, R. R. (1999). A class of fuzzy measures generated from a Dempster–Shafer belief structure. International journal of intelligent systems14(12), 1239-1247. https://doi.org/10.1002/(SICI)1098-111X(199912)14:123.0.CO;2-G
[56]         Yager, R. R. (1993). Families of OWA operators. Fuzzy sets and systems59(2), 125-148. https://doi.org/10.1016/0165-0114(93)90194-M
[57]         Yager, R. R. (1988). On ordered weighted averaging aggregation operators in multicriteria decisionmaking. IEEE transactions on systems, man, and cybernetics18(1), 183-190.doi: 10.1109/21.87068
[58]         Yager, R., Fedrizzi, M., & Kacprzyk, J. (1994). Advances in the Dempster-Shafer theory of evidence. New York: John Wiley & Sons.
[59]         Yager, R. R., & Kacprzyk, J. (Eds.). (2012). The ordered weighted averaging operators: theory and applications. Springer Science & Business Media. Norwell: Kluwer Academic Publishers.
[60]         Yager, R. R., Kacprzyk, J., & Beliakov, G. (Eds.). (2011). Recent developments in the ordered weighted averaging operators: theory and practice (Vol. 265). Springer. DOI: 10.1007/978-3-642-17910-5
[61]         Wang, Z., & Klir, G. J. (2009). Generalized measure theory. IFSR international series of systems science and engineering 25, 1st edition. Springer. DOI: 10.1007/978-0-387-76852-6
Xu, Z., & Da, Q. L. (2003). An overview of operators for aggregating information. International journal of intelligent systems18(9), 953-969. https://doi.org/10.1002/int.10127