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Decision Making with Second Order Information Granules

  • Chapter
Granular Computing and Intelligent Systems

Part of the book series: Intelligent Systems Reference Library ((ISRL,volume 13))

Abstract

Decision-making under uncertainty has evolved into a mature field. However, in most parts of the existing decision theory, one assumes decision makers have complete decision-relevant information. The standard framework is not capable to deal with partial or fuzzy information, whereas, in reality, decision-relevant information about outcomes, probabilities, preferences etc is inherently imprecise and as such described in natural language (NL). Nowadays, there is no decision theory with second-order uncertainty in existence albeit real-world uncertainties fall into this category. This applies, in particular, to imprecise probabilities expressed by terms such as likely, unlikely, probable, usually etc. We call such imprecise evaluations second-order information granules.

In this study, we develop a decision theory with second-order information granules. The first direction we consider is decision making with fuzzy probabilities. The proposed theory differs from the existing ones as one that accumulates non-expected utility paradigm with NL-described decision-relevant information. Linguistic preference relations and fuzzy utility functions are used instead of their classical counterparts as forming a more adequate description of human preferences expressed under fuzzy probabilities. Fuzzy probability distribution is incorporated into the suggested fuzzy utility model by means of a fuzzy number-valued fuzzy measure instead of a real-valued non-additive probability. We provide representation theorems for a fuzzy utility function described by a fuzzy number-valued Choquet integral with a fuzzy number-valued integrand and a fuzzy number-valued fuzzy measure. The proposed theory is intended to solve decision problems when the environment of fuzzy states and fuzzy outcomes are characterized by fuzzy probabilities. As the second direction in this realm we consider hierarchical imprecise probability models. Such models allow us to take into account imprecision and imperfection of our knowledge, expressed by interval values of probabilities of states of nature and degrees of confidence associated with such values. A decision-making process analysis and a choice of the most preferable alternative subject to variation of intervals at the lower and upper levels of models and the types of distribution on the sets of random values of probabilities of states of nature is also of significant interest.

We apply the developed theories and methodologies to solving real-world economic decision-making problems. The obtained results show validity of the proposed approaches.

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References

  • Ahn, B.S.: Multiattribute Decision aid with Extended ISMAUT. IEEE Trans-actions on Systems, Man and Cybernetics, Part A: Systems and Humans 36(3), 507–520 (2006)

    Article  Google Scholar 

  • Akerlof, G.A., Shiller, R.J.: Animal Spirits, How Human Psychology Drives the Economy, and Why it Matters for Global Capitalism. Princeton University Press, Princeton (2009)

    Google Scholar 

  • Aliev, R.A.: Modeling and stability analysis in fuzzy economics. Applied and Computational Mathematics 7(1), 31–53 (2008)

    MathSciNet  MATH  Google Scholar 

  • Aliev, R.A.: Decision And Stability Analysis in Fuzzy Economics. In: North American Fuzzy Information Processing Society Conference (NAFIPS 2009), Cincinnati, Ohio, USA, pp. 1–2 (2009)

    Google Scholar 

  • Aliev, R.A., Aliev, R.R.: Soft Computing and its applications. World Scien-tific, Singapore (2001)

    Google Scholar 

  • Aliev, R.A., Bonfig, K.W., Aliew, F.T.: Soft Computing. Technik Verlag, Ber-lin (2004)

    MATH  Google Scholar 

  • Aliev, R.A., Fazlollahi, B.: Decision making and stability analysis in fuzzy economics. In: Aliev, R.A., Bonfig, K.W., Jamshidi, M., Pedrycz, W., Turksen, I.B. (eds.) Proceeding of the Seventh International Conference on Applications of Fuzzy Systems and Soft Computing (ICAFS 2008), Helsinki, pp. 9–44 (2008)

    Google Scholar 

  • Aliev, R.A., Fazlollahi, B., Aliev, R.R.: Soft Computing and its Applications in Business and Economics. Springer, Heidelberg (2004)

    MATH  Google Scholar 

  • Aliev, R.A., Jafarov, S.M., Gardashova, L.A., Zeinalova, L.M.: Principles of decision making and control under uncertainty. Nargiz, Baku (1999)

    Google Scholar 

  • Aliev, R.A., Pedrycz, W.: Fundamentals of a fuzzy-logic-based generalized theory of stability. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 39(4), 971–988 (2009)

    Article  Google Scholar 

  • Allais, M.: Le Comportement de l’Homme Rationnel devant le Risque, Criti-que des Postulats et Axiomes de l’Ecole Americaine. Econometrica 21(4), 503–546 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  • Allais, M., Hagen, O.: Expected Utility Hypotheses and the Allais Paradox. D. Reidel Publishing Co., Dordrecht (1979)

    MATH  Google Scholar 

  • Alo, R., Korvin, A., Modave, F.: Fuzzy functions to select an optimal action in decision theory. In: Proceedings of the North American Fuzzy Information Proc-essing Society (NAFIPS), New Orleans, pp. 348–353 (2002)

    Google Scholar 

  • Anscombe, F.J., Aumann, R.J.: A definition of subjective probability. The An-nals of Mathematical Statistics 34(1), 199–205 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  • Augustin, T.: Expected utility within a generalized concept of probability – a comprehensive framework for decision-making under ambiguity. Statistical Pa-pers 43, 5–22 (2002)

    MathSciNet  MATH  Google Scholar 

  • Augustin, T., Miranda, E., Vejnarova, J.: Imprecise probability models and their applications. International Journal of Approximate Reasoning 50(4), 581–582 (2009)

    Article  MathSciNet  Google Scholar 

  • Aven, T.: Foundations of Risk Analysis: A Knowledge and Decision - Ori-ented Perspective. Wiley, England (2003)

    Google Scholar 

  • Baudrit, C., Dubois, D.: Practical representations of incomplete probabilistic knowledge. Computational Statistics and Data Analysis 51, 86–108 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Bede, B., Gal, S.: Generalizations of the differentiability of fuzzy-number val-ued functions with applications to fuzzy differential equations. Fuzzy Sets and Systems 151(3), 581–599 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Berger, J.O.: Statistical Decision Theory and Bayesian Analysis. Springer, New York (1985)

    MATH  Google Scholar 

  • Bernard, J.-M.: Non-parametric inference about an unknown mean using the imprecise Dirichlet model. In: Proceedings of the 2nd Int. Symposium on Impre-cise Probabilities and Their Applications (ISIPTA 2001), Ithaca, USA, pp. 40–50 (2001)

    Google Scholar 

  • Bernard, J.-M.: Implicative analysis for multivariate binary data using an im-precise Dirichlet model. Journal of Statistical Planning and Inference 105(1), 83–104 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Bernard, J.-M.: Implicative analysis for multivariate binary data using an im-precise Dirichlet model. Journal of Statistical Planning and Inference 105(1), 83–104 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Billot, A.: From fuzzy set theory to non-additive probabilities: how have economists reacted. Fuzzy Sets and Systems 49(1), 75–90 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Billot, A.: An existence theorem for fuzzy utility functions: a new elementary proof. Fuzzy Sets and Systems 74(2), 271–276 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  • Borisov, A.N., Alekseyev, A.V., Merkuryeva, G.V., Slyadz, N.N., Gluschkov, V.I.: Fuzzy information processing in decision making systems. Radio i Svyaz (1989) (in Russian)

    Google Scholar 

  • Buckley, J.J.: Fuzzy Probability and Statistics. Studies in Fuzziness and Soft Computing, vol. 196. Springer, Berlin (2006)

    MATH  Google Scholar 

  • Cantarella, G.E., Fedele, V.: Fuzzy Utility Theory for Analyzing Discrete Choice Behaviour. In: Proceedings of the Fourth International Symposium on Uncertainty Modeling and Analysis (ISUMA 2003), pp. 148–154. College Park, Maryland (2003)

    Chapter  Google Scholar 

  • Chateauneuf, A., Dana, R.-A., Tallon, J.M.: Optimal risk-sharing rules and equilibria with Choquet-expected-utility. Journal of Mathematical Economics 34(2), 191–214 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Chateauneuf, A., Eichberger, J.: A Simple Axiomatization and Constructive Representation Proof for Choquet Expected Utility. Economic Theory 22(4), 907–915 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, T.-Y., Wang, J.-C., Tzeng, G.-H.: Identification of General Fuzzy Measures By Genetic Algorithms Based on Partial Information. IEEE Transactions on Systems, Man, And Cybernetics – Part B: Cybernetics 30(4), 517–528 (2000)

    Article  Google Scholar 

  • Choquet, G.: Theory of capacities. Annales de l’Institut Fourier 5, 131–295 (1953)

    Article  MathSciNet  Google Scholar 

  • Cooman, G., Walley, P.: A possibilistic hierarchical model for behaviour un-der uncertainty: Theory and Decision, vol. 52(4), pp. 327–374 (2002)

    Google Scholar 

  • De Cooman, G.: Possibilistic previsions. In: Proceedings of IP-MU 1998, pp. 2–9. Editions EDK, Paris (1998)

    Google Scholar 

  • Denneberg, D.: Non-additive Measure and Integral. Kluwer Academic Pub-lisher, Boston (1994)

    MATH  Google Scholar 

  • Diamond, P., Kloeden, P.: Metric spaces of fuzzy sets. Theory and applica-tions. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  • Eichberger, J., Grant, S., Kelsey, D.: Updating Choquet beliefs. Journal of Mathematical Economics 43(7-8), 888–899 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Ekenberg, L., Thorbiornson, J.: Second–order decision analysis. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 9(1), 13–37 (2001)

    MathSciNet  MATH  Google Scholar 

  • Ellsberg, D.: Risk, Ambiquity and the Savage Axioms. Quarterly Journal of Economics 75, 643–669 (1961)

    Article  Google Scholar 

  • Ferson, S., Ginsburg, L., Kreinovich, V., et al.: Uncertainty in risk analysis: Towards a general second-order approach combining interval, probabilistic, and fuzzy techniques. In: Proceedings of FUZZ-IEEE 2002, Hawaii, vol. 2, pp. 1342–1347 (2002)

    Google Scholar 

  • Gajdos, T., Tallon, J.-M., Vergnaud, J.-C.: Decision making with imprecise probabilistic information. Journal of Mathematical Economics 40(6), 647–681 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Gil, M.A., Jain, P.: Comparison of Experiments in Statistical Decision Prob-lems with Fuzzy Utilities. IEEE Transactions On Systems, Man, And Cyber-neticts 22(4), 662–670 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Gilbert, W., Bassett, J., Koenker, R., Kordas, G.: Pessimistic Portfolio Alloca-tion and Choquet Expected Utility. Journal of Financial Econometrics 2(4), 477–492 (2004)

    Article  Google Scholar 

  • Gilboa, I.: Expected utility with purely subjective non additive probabilities. Journal of Mathematical Economics 16(1), 65–88 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  • Gilboa, I.: Theory of Decision under Uncertainty. Econometric Society Monographs, Cambridge (2009)

    MATH  Google Scholar 

  • Giraud, R.: Objective imprecise probabilistic information, second order beliefs and ambiquity aversion: an axiomatization. In: Proceedings of ISIPTA-2005, Pittsburg, USA, pp. 183–192 (2005)

    Google Scholar 

  • Good, I.J.: Subjective probability at the measure of non-measurable set. In: Nagel, E., Suppes, P., Tarski, A. (eds.) Logic, Methodology and Philosophy of Science, pp. 319–329. Stanford University Press, Stanford (1983)

    Google Scholar 

  • Grabisch, M.: The application of Fuzzy integrals in multicriteria decision making. European Journal of Operational Research 89(3), 445–456 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Grabisch, M.: Alternative representations of discrete fuzzy measures for deci-sion making. International Journal of Uncertainty, Fuzziness and Knowledge Based Systems 5(5), 587–607 (1997)

    Article  MathSciNet  Google Scholar 

  • Grabisch, M., Murofushi, T., Sugeno, M.: Fuzzy Measures and Integrals: The-ory and Applications. Physica-Verlag, Berlin (2000)

    Google Scholar 

  • Grabisch, M., Roubens, M.: Application of the Choquet Integral in Multicrite-ria Decision Making. In: Grabisch, M., Murofushi, T., Sugeno, M. (eds.) Fuzzy Measures and Integrals: Theory and Applications, pp. 348–374. Physica-Verlag, Berlin (2000)

    Google Scholar 

  • Grabisch, M.: Fuzzy integral in multicriteria decision making. Fuzzy Sets and Systems, Special issue on fuzzy information processing 69(3), 279–298 (1995)

    MathSciNet  MATH  Google Scholar 

  • Grabish, M., Kojadinovic, I., Meyer, P.: A review of methods for capacity identification in Choquet integral based multi-attribute utility theory. European Journal of Operational Research 186(2), 766–785 (2008)

    Article  MathSciNet  Google Scholar 

  • Grabish, M., Labreuche, C.: A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. A Quarterly Journal Of Operations Research 6(1), 1–44 (2008)

    Article  MathSciNet  Google Scholar 

  • Hable, R.: Data-based decisions under imprecise probability and least favor-able models. International Journal of Approximate Reasoning 50(4), 642–654 (2009a)

    Article  MathSciNet  MATH  Google Scholar 

  • Hable, R.: Finite approximations of data-based decision problems under im-precise probabilities. International Journal of Approximate Reasoning 50(7), 1115–1128 (2009b)

    Article  MathSciNet  MATH  Google Scholar 

  • Helzner, J.: On the Application of Multi-attribute Utility Theory to Models of Choice. Theory and Decision 66(4), 301–315 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Hey, J.D., Lotito, G., Maffioletti, A.: Choquet OK? Discussion Papers of De-partment of Economics, University of York, December 7 (2010), http://www.york.ac.uk/depts/econ/documents/dp/0712.pdf (accessed September 22, 2010)

  • Huede, F.L., Grabisch, M., Labreuche, C., Saveant, P.: MCS-A new algorithm for multi-criteria optimisation in constraint programming. Annals of Operations Research 147(1), 143–174 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Jaffray, J.Y.: Rational decision making with imprecise probabilities. In: Cooman, G., Cozman, F.G., Moral, S., Walley, P. (eds.) Proceedings of the First International Symposium on Imprecise Probabilities and their Applications (ISIPTA), Ghent, Belgium, pp. 324–332 (1999)

    Google Scholar 

  • Jamison, K.D., Lodwick, W.A.: The construction of consistent possibility and necessity measures. Fuzzy Sets and Systems 132(1), 1–10 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Jeleva, M.: Background Risk, Demand for Insurance, and Choquet Expected Utility Preferences. The Geneva Papers on Risk and Insurance 25(1), 7–28 (2000)

    Article  Google Scholar 

  • Kacprzyk, J., Zadeh, L.A.: Computing with Words in Information. In: Intelligent Systems Part 1. Foundations, Physica-Verlag, Heidelberg (1999a)

    Google Scholar 

  • Kacprzyk, J., Zadeh, L.A.: Computing with Words in Information. In: Intelligent Systems Part 2, vol. 609. Physica-Verlag, Heidelberg (1999b)

    Google Scholar 

  • Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under un-certainty. Econometrica 47(2), 263–291 (1979)

    Article  MATH  Google Scholar 

  • Kobberling, V., Wakker, P.P.: Preference Foundations for Nonexpected Util-ity: A Generalized and Simplified Technique. Mathematics of Operations Re-seach 28(3), 395–423 (2003)

    Article  MathSciNet  Google Scholar 

  • Lakshmikantham, V., Mohapatra, R.: Theory of fuzzy differential equations and inclusions. Taylor & Francis, London (2003)

    Book  MATH  Google Scholar 

  • Levi, I.: On indeterminate probabilities. Journal of Philosophy 71, 391–418 (1974)

    Article  Google Scholar 

  • Lukacs, G.: Towards an interactive query environment for a multi-attribute decision model with imprecise data and vague query conditions. In: Proceedings of the Eleventh International Workshop on Database and Expert Systems Appli-cations, pp. 708–712. IEEE Computer Society, London (2000)

    Chapter  Google Scholar 

  • Machina, M.J.: Non-Expected Utility Theory. In: Teugels, J., Sundt, B. (eds.) Encyclopedia of Actuarial Science, vol. 2, pp. 1173–1179. John Wiley and Sons, Chichester (2004)

    Google Scholar 

  • Malak, R.J., Aughenbaugh, J.J., Paredis, C.J.J.: Multi-attribute utility analysis in set-based conceptual design. Computer-Aided Design 41(3), 214–227 (2009)

    Article  Google Scholar 

  • Mangelsdorff, L., Weber, M.: Testing Choquet expected utility. Journal of Economic Behavior and Organization 25, 437–457 (1994)

    Article  Google Scholar 

  • Marichal, J.L.: An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria. IEEE Transactions on Fuzzy Systems 8(6), 800–807 (2000)

    Article  MathSciNet  Google Scholar 

  • Marichal, J.L., Meyer, P., Roubens, M.: Sorting Multiattribute Alternatives: The Tomaso method. Computers and Operations Research 32(4), 861–877 (2005)

    Article  MATH  Google Scholar 

  • Marichal, J.L., Roubens, M.: Dependence between criteria and multiple crite-ria decision aid. In: 2nd Int. Workshop on Preferences and Decisions, Trento, Italy, pp. 69–75 (1998)

    Google Scholar 

  • Markowitz, H.: The Utility of Wealth. Journal of Political Economy 60, 151–158 (1952)

    Article  Google Scholar 

  • Mathieu-Nicot, B.: Fuzzy Expected Utility. Fuzzy Sets and Systems 20(2), 163–173 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  • Mendel, J.M.: Computing with words and its relationships with fuzzistics. In-formation Sciences 179(8), 988–1006 (2007)

    Google Scholar 

  • Modave, F., Grabisch, M.: Preference representation by the Choquet integral: The commensurability hypothesis. In: Proceedings of the 7th international con-ference on information processing and management of uncertainty in knowledge-based systems (IPMU), EDK Editions, Paris, France, pp. 164–171 (1998)

    Google Scholar 

  • Modave, F., Grabisch, M., Dubois, D., Prade, H.: A Choquet Integral Repre-sentation in Multicriteria Decision Making. In: Association for the Advancement of Artificial Intelligence (AAAI) Fall Symposium, Boston (1997)

    Google Scholar 

  • Mordeson, J.N., Nair, P.S.: Fuzzy mathematics: an introduction for engineers and scientists. Springer, Heidelberg (2001)

    MATH  Google Scholar 

  • Murofushi, T.: Semiatoms in Choquet Integral Models of Multiattribute Deci-sion Making. Journal of Advanced Computational Intelligence and Intelligent In-formatics 9(5), 477–483 (2005)

    Google Scholar 

  • Nanda, S.: Fuzzy linear spaces over valued fields. Fuzzy Sets and Systems 42(3), 351–354 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Narukawa, Y., Murofushi, T.: Logic of Programs 1981. LNCS, vol. 131, pp. 183–193 (2004)

    Google Scholar 

  • Nau, R.F.: Indeterminate probabilities on finite sets. The Annals of Statistics 20, 1737–1767 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Nguyen, H.T., Walker, E.A.: A first course in Fuzzy Logic. CRC Press, Boca Raton (2000)

    MATH  Google Scholar 

  • Pedrycz, W.: Fuzzy sets and neurocomputation: knowledge-based networks. In: Bezdek, J. (ed.) Proceedings of SPIE Applications of Fuzzy Logic Technology II, Orlando, Florida, USA, vol. 2493, pp. 2–20 (1995)

    Google Scholar 

  • Pedrycz, W., Gomide, F.: Fuzzy Systems Engineering. In: Toward Human-Centric Computing. John Wiley & Sons, Hoboken (2007)

    Google Scholar 

  • Pedrycz, W., Peters, J.F.: Computational Intelligence in Software Engineering. In: Advances in Fuzzy Systems, Applications and Theory, vol. 16. World Scientific, Singapoure (1998)

    Google Scholar 

  • Quiggin, J.: A theory of anticipated utility. Journal of Economic Behavioral and Organization 3(4), 323–343 (1982)

    Article  Google Scholar 

  • Savage, L.J.: The Foundations of Statistics. Wiley, New York (1954); 2nd ed. Dover, New York (1972)

    MATH  Google Scholar 

  • Schmeidler, D.: Integral representation without additivity. Proceedings of the American Mathematical Society 97(2), 255–261 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  • Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57(3), 571–587 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Setnes, M.: Compatibility-Based Ranking of Fuzzy numbers. In: Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS 1997), Syracuse, New York, pp. 305–310 (1997)

    Google Scholar 

  • Sims, J.R., Zhenyuan, W.: Fuzzy measures and fuzzy integrals. International Journal of General Systems 17(2-3), 157–189 (1990)

    Article  MATH  Google Scholar 

  • Troffaes, M.C.M.: Decision making under uncertainty using imprecise prob-abilities. International Journal of Approximate Reasoning 45(1), 17–29 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Troffaes, M.C.M., De Cooman, G.: Uncertainty and conflict: A behavioral approach to the aggregation of expert opinions. In: Vejnarova, J. (ed.) Proceedings of the 6th Workshop on Uncertainty VSE, pp. 263–277. Oeconomica Publishers (2003)

    Google Scholar 

  • Tversky, A., Kahneman, D.: Advances in Prospect theory: Cumulative Repre-sentation of Uncertainty. Journal of Risk and Uncertainty 5(4), 297–323 (1992)

    Article  MATH  Google Scholar 

  • Utkin, L.V.: Imprecise second-order hierarchical uncertainty model. Interna-tional Journal of Uncertainty, Fuzziness and Knowledge-Based System 13(2), 177–193 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Utkin, L.V.: Analysis of risk and decision making under incomplete information. Nauka, Saint Petersburg (2007)

    Google Scholar 

  • Utkin, L.V., Augustin, T.: Decision making with imprecise second-order prob-abilities. In: Bernard, J.M., Seidenfeld, T., Zaffalon, M. (eds.) Proceedings of the Third International Symposium on Imprecise Probabilities and their Applications (ISIPTA 2003), Lugano, Switzerland, pp. 545–559. Carleton Scientific, Waterloo (2003)

    Google Scholar 

  • Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behav-iour. Princeton University Press, Princeton (1944)

    Google Scholar 

  • Wakker, P.P., Zank, H.: State Dependent Expected Utility for Savage’s State Space. Mathematics of Operations Reseach 24(1), 8–34 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Walley, P.: Statistical Reasoning with Imprecise Probabilities. In: Monographs on Statistics and Applied Probability. Chapman and Hall, London (1991)

    Google Scholar 

  • Walley, P.: Statistical inferences based on a second-order possibility distribu-tion. International Journal of General Systems 26(4), 337–383 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Walley, P., De Cooman, G.: A behavioral model for linguistic uncertainty. In-formation Sciences 134(1), 1–37 (2001)

    MATH  Google Scholar 

  • Wang, G., Li, X.: On the convergence of the fuzzy valued functional defined by μ-integrable fuzzy valued functions. Fuzzy Sets and Systems 107(2), 219–226 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, Z., Leung, K.-S., Wang, J.: A genetic algorithm for determining nonad-ditive set functions information fusion. Fuzzy Sets and Systems 102(3), 463–469 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Wilde, P.D.: Fuzzy Utility and Equilibria. IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics 34(4), 1774–1785 (2004)

    Article  Google Scholar 

  • Yaari, M.E.: The dual theory of choice under risk. Econometrica 55(1), 95–115 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  • Yager, R.R.: Decision Making with Fuzzy Probability Assessments. IEEE Transactions on Fuzzy Systems 7(4), 462–467 (1999)

    Article  MathSciNet  Google Scholar 

  • Yager, R.R.: A general approach to uncertainty representation using fuzzy measures. In: Proceedings of the Fourteenth International Florida Artificial Intel-ligence Research Society Conference, pp. 619–623. AAAI Press, Menlo Park (2001)

    Google Scholar 

  • Yang, R., Wang, Z., Heng, P.-A., Leung, K.-S.: Fuzzy numbers and fuzzifica-tion of the Choquet integral. Fuzzy Sets and Systems 153(1), 95–113 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Yoneda, M., Fukami, S., Grabisch, M.: Interactive Determination of a Utility Function Represented as a Fuzzy Integral. Information Sciences 71(1-2), 43–64 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh, L.A.: The concept of a linguistic variable and its application to ap-proximate reasoning. Part I Information Sciences, 8, 199-249; Part II Information and Sciences 8, 301–357; Part III Information and Sciences, 9, 43–80 (1975)

    Google Scholar 

  • Zadeh, L.A.: Fuzzy logic = Computing with Words. IEEE Transactions on Fuzzy Systems 4(2), 103–111 (1996)

    Article  MathSciNet  Google Scholar 

  • Zadeh, L.A.: Toward a theory of fuzzy Information Granulation and its central-ity in human reasoning and fuzzy logic. Fuzzy Sets and Systems 90(2), 111–127 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh, L.A.: From computing with numbers to computing with words – from manipulation of measurements to manipulation with perceptions. IEEE Transac-tions on Circuits and Systems – I: Fundamental Theory and its Applications 45(1), 105–119 (1999)

    Article  MathSciNet  Google Scholar 

  • Zadeh, L.A.: A new direction in AI – toward a computational theory of per-ceptions. AI Magazine 22(1), 73–84 (2001)

    Google Scholar 

  • Zadeh, L.A.: Generalized theory of uncertainty (GTU) – principal concepts and ideas. Computational Statistics and Data Analysis 51, 15–46 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh, L.A.: Computation with imprecise probabilities. In: Proceedings of the Seventh International Conference on Applications of Fuzzy Systems and Soft Computing (ICAFS 2008), Helsinki, pp. 1–3 (2008a)

    Google Scholar 

  • Zadeh, L.A.: Is there a need for fuzzy logic? Information Sciences, vol. 178(13), pp. 2751–2779 (2008b)

    Google Scholar 

  • Zadeh, L.A., Aliev, R.A., Fazlollahi, B., Alizadeh, A.V., Guirimov, B.G., Huseynov, O.H.: Decision Theory with Imprecise Probabilities. Report on the contract “Application of Fuzzy Logic and Soft Computing to Communications, Planning and Management of Uncertainty”. Berkeley, USA, Baku, Azerbaijan (2009), http://www.raliev.com/report.pdf (accessed September 21, 2010)

  • Zhang, G.-Q.: Fuzzy number-valued fuzzy measure and fuzzy number-valued fuzzy integral on the fuzzy set. Fuzzy Sets and Systems 49(3), 357–376 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhong, Q.: On fuzzy measure and fuzzy integral on fuzzy set. Fuzzy Sets and Systems 37(1), 77–92 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Computer-Aided Control Systems, Azerbaijan State Oil Academy, Azerbaijan, Baku, Azadlyg ave. 20, AZ, 1010

    R. A. Aliev, O. H. Huseynov & L. M. Zeinalova

  2. Department of Electrical & Computer Engineering, University of Alberta, Edmonton, AB, T6R 2G7, Canada

    W. Pedrycz

  3. System Research Institute, Polish Academy of Sciences Warsaw, Poland

    W. Pedrycz

Authors
  1. R. A. Aliev
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  2. W. Pedrycz
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  3. O. H. Huseynov
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  4. L. M. Zeinalova
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Editors and Affiliations

  1. Department of Electrical & Computer Engineering, University of Alberta, T6G 2G6, Edmonton, AB, Canada

    Witold Pedrycz

  2. Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology, , 43, Section 4, Keelung Road, 106, Taipei, Taiwan

    Shyi-Ming Chen

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© 2011 Springer-Verlag Berlin Heidelberg

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Aliev, R.A., Pedrycz, W., Huseynov, O.H., Zeinalova, L.M. (2011). Decision Making with Second Order Information Granules. In: Pedrycz, W., Chen, SM. (eds) Granular Computing and Intelligent Systems. Intelligent Systems Reference Library, vol 13. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19820-5_7

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  • DOI: https://doi.org/10.1007/978-3-642-19820-5_7

  • Publisher Name: Springer, Berlin, Heidelberg

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  • Online ISBN: 978-3-642-19820-5

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