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In this note, we show by counterexamples that Theorems 9 and 10, and Propositions 7, 18, 19 and 20 in a previous paper by Gera and Dombi (2008) 1 contain some flaws and then we provide the correct versions.

This paper describes a systematic approach to the modeling of engineering systems using a fuzzy formulation that is independent of human knowledge. The computer algorithm described here operates on a set of experimental observations of the system and constructs an optimum fuzzy model for these observations. The program automatically selects membership functions, deduces inference rules, constructs logical relations, and determines the formulae for conducting union and intersection operations. Membership functions, rules, and logical operations are defined parametrically. Model parameters are optimized so that the model can, at least, re-produce with minimum error the data that were used in obtaining the membership functions and rules. Therefore, model parameters are optimized to minimize error or entropy of the back-inferences of the observations from which the model was constructed. To reach the global minimum and avoid entrapment in a local minimum, a random search is carried out, then followed by a systematic Hooke–Jeeves search optimization algorithm. It has been found that this technique is more successful, compared with other statistical techniques.

Axiomatic design theory, in its decoupled form, provides guidance for the design process to result in functionally reliable engineering systems that satisfy the requirements of some safety specification. The decoupled form can conveniently be obtained whenever the design problem is susceptible to dimensional analysis. However, both the ensuing dimensionless groups and the environment in which the system operates are intrinsically fuzzy. A tool for performing arithmetic and logical operations in the fuzzy environment is the t-norm. Experience indicates that only one t-norm, the min-operator originally proposed by L.A. Zadeh and known as Zadeh's Extension Principle, ZEP, is congruent with results of engineering experiments. This extension principle is reviewed in the paper, and its applicability demonstrated on a real-world example.