## Decision making with interval probabilities.(English)Zbl 1177.90215

Summary: Handling uncertainty by interval probabilities is recently receiving considerable attention by researchers. Interval probabilities are used when it is difficult to characterize the uncertainty by point-valued probabilities due to partially known information. Most of researches related to interval probabilities, such as combination, marginalization, condition, Bayesian inferences and decision, assume that interval probabilities are known. How to elicit interval probabilities from subjective judgment is a basic and important problem for the applications of interval probability theory and till now a computational challenge. In this work, the models for estimating and combining interval probabilities are proposed as linear and quadratic programming problems, which can be easily solved. The concepts including interval probabilities, interval entropy, interval expectation, interval variance, interval moment, and the decision criteria with interval probabilities are addressed. A numerical example of newsvendor problem is employed to illustrate our approach. The analysis results show that the proposed methods provide a novel and effective alternative for decision making when point-valued subjective probabilities are inapplicable due to partially known information.

### MSC:

 90B50 Management decision making, including multiple objectives 90C05 Linear programming 90C20 Quadratic programming 91B44 Economics of information

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 [1] Berleant, D.; Cozman, F. G.; Kosheleva, O.; Kreinovich, V., Dealing with imprecise probabilities: interval-related talks at ISIPTA’05, Reliable Computing, 12, 153-165 (2006) [2] Camerer, C.; Weber, M., Recent development in modeling preference: uncertainty and ambiguity, Journal of Risk and Uncertainty, 5, 325-370 (1992) · Zbl 0775.90102 [3] Cano, A.; Moral, S., Using probability trees to compute marginals with imprecise probabilities, International Journal of Approximate Reasoning, 29, 1-46 (2002) · Zbl 1015.68206 [4] Danielson, M.; Ekenberg, L., Computing upper and lower bounds in interval decision trees, European Journal of Operational Research, 181, 808-816 (2007) · Zbl 1131.91016 [6] de Campos, L. M.; Huete, J. F.; Moral, S., Probability intervals: a tool for uncertain reasoning, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2, 167-196 (1994) · Zbl 1232.68153 [7] Dempster, A., Upper and lower probabilities induced by a multivalued mapping, Annals of Mathematical Statistics, 38, 325-339 (1967) · Zbl 0168.17501 [8] Einhorn, H. J.; Hogarth, R. M., Ambiguity and uncertainty in probabilistic inference, Psychological Review, 92, 433-461 (1985) [9] Figueira, J.; Greco, S.; Slowinski, R., Building a set of additive value functions representing a reference preorder and intensities of preference: GRIP method, European Journal of Operational Research, 195, 460-486 (2009) · Zbl 1159.91341 [10] Greco, S.; Mousseau, V.; Slowinski, R., Ordinal regression revisited: Multiple criteria ranking with a set of additive value functions, European Journal of Operational Research, 191, 415-435 (2008) [12] Khouja, M., The single-period (news-vendor) problem: literature review and suggestion for future research, Omega, 27, 537-553 (1999) [13] Klir, G., Uncertainty and Information: Foundations of Generalized Information Theory (2006), Wiley Interscience · Zbl 1280.94004 [15] Tanaka, H.; Lee, H., Interval regression analysis by quadratic programming approach, IEEE Transactions On Fuzzy Systems, 6, 473-481 (1998) [16] Lippman, S. A.; McCardle, K. F., The competitive newsboy, Operations Research, 45, 54-65 (1997) · Zbl 0892.90057 [17] Lodwick, W. A.; Jamison, K. D., Interval-valued probability in the analysis of problems containing a mixture of possibilistic, probabilistic, and interval uncertainty, Fuzzy Sets and Systems, 159, 2845-2858 (2008) · Zbl 1171.60303 [18] Moore, R., Methods and Applications of Interval Analysis (1979), SIAM: SIAM Philadelphia [19] Mustajoki, J.; Hamalainen, R. P.; Salo, A., Decision support by interval SMART/SWING-Incorporating imprecision in the SMART and SWING methods, Decision Science, 36, 317-339 (2005) [20] Neumaier, A., Clouds, fuzzy sets, and probability intervals, Reliable Computing, 10, 249-272 (2004) · Zbl 1055.65062 [21] Petruzzi, N. C.; Dada, M., Pricing and the newsvendor problem: a review with extensions, Operations Research, 47, 183-194 (1999) · Zbl 1005.90546 [22] Raz, G.; Porteus, E. L., A fractiles perspective to the joint price/quantity newsvendor model, Management Science, 52, 1764-1777 (2006) · Zbl 1232.90086 [23] Saaty, T. L., The Analytic Hierarchy Process (1980), McGraw-Hill · Zbl 1176.90315 [24] Salo, A. A., Inconsistency analysis by approximately specified priorities, Mathematical and Computer Modelling, 17, 123-133 (1993) · Zbl 0768.90002 [25] Shafer, G., A Mathematical Theory of Evidence (1976), Princeton University Press: Princeton University Press Princeton · Zbl 0359.62002 [26] Silver, E. A.; Pyke, D. F.; Petterson, R., Inventory Management and Production Planning and Scheduling (1998), Wiley: Wiley New York [28] Sugihara, K.; Ishii, H.; Tanaka, H., Interval priorities in AHP by interval regression analysis, European Journal of Operational Research, 158, 745-754 (2004) · Zbl 1056.90093 [29] Tanaka, H.; Guo, P., Possibilistic Data Analysis for Operations Research (1999), Physica-Verlag, Heidelberg: Physica-Verlag, Heidelberg New York · Zbl 0931.91011 [30] Tanaka, H.; Sugihara, K.; Maeda, Y., Non-additive measure by interval probability functions, Information Sciences, 164, 209-227 (2004) · Zbl 1056.28010 [31] Tenekedjiev, K., Fuzzy-rational explanation of the Ellsberg paradox, Notes on Intuitionistic Fuzzy Sets, 12, 39-52 (2006) [32] Tenekedjiev, K.; Nikolova, N. D., Ranking discrete outcome alternatives with partially quantified uncertainty, International Journal of General Systems, 37, 249-274 (2008) · Zbl 1274.62084 [33] Troffaes, M. C.M., Decision making under uncertainty using imprecise probabilities, International Journal of Approximate Reasoning, 45, 7-29 (2007) · Zbl 1119.91028 [34] Walley, P., Statistical Reasoning with Imprecise Probability (1991), Chapman and Hall: Chapman and Hall London · Zbl 0732.62004 [35] Walley, P., Inferences from multinomial data: learning about a bag of marbles, Journal of the Royal Statistical Society B, 58, 3-57 (1996) · Zbl 0834.62004 [36] Wang, Y. M.; Elhag, T. M.S., A goal programming method for obtaining interval weights from an interval comparison matrix, European Journal of Operational Research, 177, 458-471 (2007) · Zbl 1111.90058 [37] Weichselberger, K.; Pohlmann, S., A Methodology for Uncertainty in Knowledge-Based Systems, Lecture Notes in Artificial Intelligence (1990), Springer-Verlag [38] Weichselberger, K., The theory of interval-probability as a unifying concept for uncertainty, International Journal of Approximate Reasoning, 24, 149-170 (2000) · Zbl 0995.68123 [39] Yager, R. R., On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE Transactions on Systems Man and Cybernetics, 18, 183-190 (1988) · Zbl 0637.90057 [40] Yager, R. R.; Kreinovich, V., Decision making under interval probabilities, International Journal of Approximate Reasoning, 22, 195-215 (1999) · Zbl 1041.91500 [41] Zadeh, L. A., The concept of a linguistic variable and its approximate reasoning, Part 1 and 2, Information Sciences, 8, 199-249 (1975) · Zbl 0397.68071 [42] Zaffalon, M.; Wesnes, K.; Petrini, O., Reliable diagnoses of dementia by the naive credal classifier inferred from incomplete cognitive data, Artificial Intelligence in Medicine, 29, 61-79 (2003)
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