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Compositions and fuzzy compositions in decision-making models

In decision-making models, the compositions (Aitchison, 1986) are employed in various forms. They canrepresent the normalized weights of criteria in multiple-criteria decision-making models, or the probabilitiesof states of the world in the models of decision making under risk. The normalized weights express therelative information about the importance of criteria, while the probabilities reflect the expected incidence ofthe states of the world. The compositions are also the evaluations of objects according to their contributionsto a common goal.This paper addresses the situation when the compositions are set expertly. This is the typical way ofobtaining the normalized weights of criteria. Various methods of setting the weights were introduced in orderto properly reflect the preferences of the decision maker. Direct methods where the decision maker sets theweights directly can be distinguished from indirect ones, e.g., the method of pair-wise comparison, orSaaty’s AHP.The weights of criteria represent the parameters to the models of multiple-criteria evaluation or decisionmaking. Statistical methods are applied to these models to aggregate the information about criteriapreferences within a group of decision makers. However, in such a case, the final weights are usuallyobtained by the standard arithmetic mean, i.e., the data obtained from decision makers are not treated ascompositional data. On the other hand, the weights from the Saaty matrix of pair-wise comparison, where thei-th row expresses the importance of particular criteria with respect to the i-th criterion, are calculated as thegeometric mean of rows, which corresponds well to the concept of expected value of compositional data.In principle, there is no difference between compositional data obtained by measuring and compositionaldata that are set expertly (the measuring instrument here is a human brain). But the latter method isburdened with uncertainty. Therefore, the data values should be modeled by the tools of the fuzzy sets theory.The special structure of fuzzy numbers was proposed for modeling fuzzy compositions that represent theweights of criteria – the structure of normalized fuzzy weights (Pavlačka & Talašová, 2006). In the paper, weshow how to compute the fuzzy weighted average of fuzzy numbers with normalized fuzzy weights (the factthat normalized fuzzy weights represent a fuzzy composition must be taken into consideration by thecalculation). We also describe the special ways of setting the normalized fuzzy weights including those basedon the verbal description of preferences among criteria.For modeling fuzzy data defined on the simplex, a more general mathematical object was proposed – afuzzy vector of normalized weights (Pavlačka & Talašová, 2010). It is capable of expressing the uncertainweights even under further interactions among the weights, besides the fact that they are defined on thesimplex.These methods can also be applied for describing subjective probabilities of states of the worldin the models of decision-making under risk

Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada

Other contributions: Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada
Author: Talasova, J.
Pavlacka, O.
Abstract: In decision-making models, the compositions (Aitchison, 1986) are employed in various forms. They canrepresent the normalized weights of criteria in multiple-criteria decision-making models, or the probabilitiesof states of the world in the models of decision making under risk. The normalized weights express therelative information about the importance of criteria, while the probabilities reflect the expected incidence ofthe states of the world. The compositions are also the evaluations of objects according to their contributionsto a common goal.This paper addresses the situation when the compositions are set expertly. This is the typical way ofobtaining the normalized weights of criteria. Various methods of setting the weights were introduced in orderto properly reflect the preferences of the decision maker. Direct methods where the decision maker sets theweights directly can be distinguished from indirect ones, e.g., the method of pair-wise comparison, orSaaty’s AHP.The weights of criteria represent the parameters to the models of multiple-criteria evaluation or decisionmaking. Statistical methods are applied to these models to aggregate the information about criteriapreferences within a group of decision makers. However, in such a case, the final weights are usuallyobtained by the standard arithmetic mean, i.e., the data obtained from decision makers are not treated ascompositional data. On the other hand, the weights from the Saaty matrix of pair-wise comparison, where thei-th row expresses the importance of particular criteria with respect to the i-th criterion, are calculated as thegeometric mean of rows, which corresponds well to the concept of expected value of compositional data.In principle, there is no difference between compositional data obtained by measuring and compositionaldata that are set expertly (the measuring instrument here is a human brain). But the latter method isburdened with uncertainty. Therefore, the data values should be modeled by the tools of the fuzzy sets theory.The special structure of fuzzy numbers was proposed for modeling fuzzy compositions that represent theweights of criteria – the structure of normalized fuzzy weights (Pavlačka & Talašová, 2006). In the paper, weshow how to compute the fuzzy weighted average of fuzzy numbers with normalized fuzzy weights (the factthat normalized fuzzy weights represent a fuzzy composition must be taken into consideration by thecalculation). We also describe the special ways of setting the normalized fuzzy weights including those basedon the verbal description of preferences among criteria.For modeling fuzzy data defined on the simplex, a more general mathematical object was proposed – afuzzy vector of normalized weights (Pavlačka & Talašová, 2010). It is capable of expressing the uncertainweights even under further interactions among the weights, besides the fact that they are defined on thesimplex.These methods can also be applied for describing subjective probabilities of states of the worldin the models of decision-making under risk
Document access: http://hdl.handle.net/2072/273442
Language: eng
Publisher: Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada
Rights: Tots els drets reservats
Subject: Estadística matemàtica -- Congressos
Mathematical statistics -- Congresses
Decisió, Presa de -- Congressos
Decision-making -- Congresses
Sistemes borrosos -- Congressos
Fuzzy systems -- Congresses
Title: Compositions and fuzzy compositions in decision-making models
Type: info:eu-repo/semantics/conferenceObject
Repository: Recercat

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