Foundations Of Statistics

The main advances regarding the use of the Choquet and Sugeno integrals in multi-criteria decision aid over the last decade are reviewed. They concern mainly a bipolar extension of both the Choquet integral and the Sugeno integral, interesting particular submodels, new learning techniques, a better interpretation of the models and a better use of the Choquet integral in multi-criteria decision aid. Parallel to these theoretical works, the Choquet integral has been applied to many new fields, and several softwares and libraries dedicated to this model have been developed.

A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid

Statistical models in engineering , Statistical models in engineering , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی

Statistical Models in Engineering

The application of multi-attribute utility theory whose aggregation process is based on the Choquet integral requires the prior identification of a capacity. The main approaches to capacity identification proposed in the literature are reviewed and their advantages and inconveniences are discussed. All the reviewed methods have been implemented within the Kappalab R package. Their application is illustrated on a detailed example.

A review of methods for capacity identification in Choquet integral based multi-attribute utility theory: Applications of the Kappalab R package

The arithmetic of discrete Z-numbers

The arithmetic of continuous Z-numbers

Rational decisions are based on information usually uncertain, imprecise and incomplete. The existing decision theories deal with three levels of generalization of decision making relevant information: numerical valuation, interval valuation and fuzzy number valuation. The classical decision theories, such as expected utility theory proposed by von Neumann and Morgenstern, and subjective expected utility theory proposed by Savage use the first level of generalization, i.e. numerical one. These approaches require that the objective probabilities or subjective probabilities and utility values be precisely known. But in real world in many cases it becomes impossible to determine the precise values of needed information. Interval analysis and classical fuzzy set theories have been applied in making decisions and many fruitful results have been achieved. But a problem is that in the mentioned above decision theories the reliability of the decision relevant information is not well taken into consideration. Prof. L. Zadeh introduced the concept of Z-numbers to describe the uncertain information which is more generalized notion closely related with confidence (reliability). Use of Z-information is more adequate and intuitively meaningful for formalizing information structure of a decision making problem. In this chapter we consider two approaches to decision making with Z-information. The first approach is based on reducing of Z-numbers to classical fuzzy numbers, and generalization of expected utility approach and use of Choquet integral with an integrant represented by Z-numbers. A fuzzy measure is calculated on a base of a given Z-information. The second approach is based on direct computation with Z-numbers. To illustrate a validity of suggested approaches to decision making with Z-information the numerical examples are used.

Decision making under Z-information

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.

Decision Making with Second Order Information Granules

Decision making theories are based on decision relevant information much of which is uncertain, imprecise or incomplete. An important qualitative attribute of information on which decisions are based is its reliability. The concept of Z-number relates to the issue of reliability of information, especially in the realms of economics and decision analysis. In this study we suggest Z-information based decision-making method which is more realistic in comparison with the existing methods. An example of investment problem is used to illustrate the proposed approach.

Expected Utility Based Decision Making Under Z-Information

Regression analysis is widely used for modeling of real-world processes in various fields. It should be noted that information relevant to real-world processes is characterized by imprecision and partial reliability. This involves combination of fuzzy and probabilistic uncertainties. Prof.. L. Zadeh introduced the concept of a Z-number as a formal construct for dealing with such information. The present state-of-the-art of regression analysis under Z-number valued information is very scarce. In this paper we consider a Z-number valued multiple regression analysis and its application to a real-world decision-making problem. The obtained results show applicability of the proposed approach.

Lala M. Zeinalova

Papers

The arithmetic of continuous Z-numbers

Decision making under Z-information

Decision Making with Second Order Information Granules

Expected Utility Based Decision Making Under Z-Information

A Z-Number Valued Regression Model and its Application