@article{giachetti_young_roggatz_eversheim_perrone_1997, title={A methodology for the reduction of imprecision in the engineering process}, volume={100}, ISSN={["0377-2217"]}, DOI={10.1016/S0377-2217(96)00290-1}, abstractNote={Engineering design is characterized by a high level of imprecision, vague parameters, and ill-defined relationships. In design, imprecision reduction must occur to arrive at a final product specification. Few design systems exist for adequately representing design imprecision, and formally reducing it to precise values. Fuzzy set theory has considerable potential for addressing the imprecision in design. However, it lacks a formal methodology for system development and operation. One repercussion of this is that imprecision reduction is, at present, implemented in a relatively ad-hoc manner. The main contribution of this paper is to introduce a methodology called precision convergence for making the transition from imprecise goals and requirements to the precise specifications needed to manufacture the product. A hierarchy of fuzzy constraint networks is presented along with a methodology for creating transitional links between different levels of the hierarchy. The solution methodology is illustrated with an example within which an imprecision reduction of 98% is achieved in only three stages of the design process. The imprecision reduction is measured using the coefficient of imprecision, a new measure introduced to quantify imprecision.}, number={2}, journal={EUROPEAN JOURNAL OF OPERATIONAL RESEARCH}, author={Giachetti, RE and Young, RE and Roggatz, A and Eversheim, W and Perrone, G}, year={1997}, month={Jul}, pages={277–292} }
@article{giachetti_young_1997, title={A parametric representation of fuzzy numbers and their arithmetic operators}, volume={91}, ISSN={["0165-0114"]}, DOI={10.1016/S0165-0114(97)00140-1}, abstractNote={Direct implementation of extended arithmetic operators on fuzzy numbers is computationally complex. Implementation of the extension principle is equivalent to solving a nonlinear programming problem. To overcome this difficulty many applications limit the membership functions to certain shapes, usually either triangular fuzzy numbers (TFN) or trapezoidal fuzzy numbers (TrFN). Then calculation of the extended operators can be performed on the parameters defining the fuzzy numbers, thus making the calculations trivial. Unfortunately the TFN shape is not closed under multiplication and division. The result of these operators is a polynomial membership function and the triangular shape only approximates the actual result. The linear approximation can be quite poor and may lead to incorrect results when used in engineering applications. We analyze this problem and propose six parameters which define parameterized fuzzy numbers (PFN), of which TFNs are a special case. We provide the methods for performing fuzzy arithmetic and show that the PFN representation is closed under the arithmetic operations. The new representation in conjunction with the arithmetic operators obeys many of the same arithmetic properties as TFNs. The new method has better accuracy and similar computational speed to using TFNs and appears to have benefits when used in engineering applications.}, number={2}, journal={FUZZY SETS AND SYSTEMS}, author={Giachetti, RE and Young, RE}, year={1997}, month={Oct}, pages={185–202} }
@article{giachetti_young_1997, title={Analysis of the error in the standard approximation used for multiplication of triangular and trapezoidal fuzzy numbers and the development of a new approximation}, volume={91}, ISSN={["0165-0114"]}, DOI={10.1016/S0165-0114(96)00118-2}, abstractNote={Triangular and trapezoidal fuzzy numbers are commonly used in many applications. It is well known that the operators used for the non-linear operations such as multiplication, division, and inverse are approximations to the actual operators. It is also commonly assumed that the error introduced by the approximations is small and acceptable. This paper examines the error of approximation for repeated use of the multiplication operand and shows it can be sufficiently large in simple circumstances to produce erroneous results. The computational complexity of the multiplication operation is analyzed and shown to be sufficiently complex that a computationally simpler approximation is needed. As a consequence, the error produced by the approximation for the multiplication operation is analyzed and a new approximation developed that is accurate for a large range of problems. An error expression is developed for the new approximation that can be used to determine when it is producing unacceptable results.}, number={1}, journal={FUZZY SETS AND SYSTEMS}, author={Giachetti, RE and Young, RE}, year={1997}, month={Oct}, pages={1–13} }