Measurement-theoretic frameworks for fuzzy set theory with applications to preference modelling

摘要

There are three objectives in this thesis: (i) to provide a suitable axiomatization of the fuzzy set theory, (ii) to investigate the appropriateness of those axiomatizations to modelling the vagueness in preferences of decision makers and finally (iii) to apply these findings to real world situations. It is proposed that one should use frameworks provided by measurement theory in order to understand what fuzzy set theory can meaningfully represent. Two different but complementary measurement problems are formulated and critically analyzed. This distinction in formulating the measurement problems highlights some discussions in the foundations of the fuzzy set theory. In order to obtain a complete axiomatization of fuzzy set theory, these two problems must be combined. Three alternative ways of combining the two problems are proposed. Each model results in a possible axiomatization of fuzzy set theory. Recent research interest on representing vague preferences in fuzzy set theory is critically reviewed in the light of an impossibility result. The vagueness of preferences is defined concisely and a resolution to this problem is proposed using the concept of interval-valued fuzzy sets. Relationships of various transitivity conditions are demonstrated and a new transitivity condition with the interval-valued preference structures is proposed. The concept of rationality and scalability of observed choice behaviour is also addressed. It is shown that vague preferences can be viewed as being probabilistically generated from imperfect rankings of human decision makers. The techniques developed in the earlier chapters are applied to the problem of model-based localization of a mobile robot equipped with sonar sensors. Here, the vagueness arises from noisy sonar sensors. A search procedure is suggested in which the preference structure is used as a conflict resolver. In summary, the axiomatizations suggest that fuzzy set theory mostly captures the ordinal aspects of vagueness and this purely ordinal view is adopted in the rest of the thesis. The interval-valued preference structures are shown to be consistent in a well defined sense. It is claimed that the noise associated with sonar sensors defies a probabilistic analysis and using fuzzy set theory based approaches is justified.