The mathematics of triangular fuzzy numbers to support a model of imprecision in design

Design is characterized by high levels of imprecision, vague parameters, and ill-defined relationships. Few design systems exist for adequately representing design imprecision. Fuzzy set theory has considerable potential for addressing the imprecision in design. Engineering design is usually described in a domain of equations and other mathematical relationships. Problems where fuzzy sets have previously been applied are usually described by rules, not mathematical expressions. While the mathematics of fuzzy sets has been developed theoretically, there are few implementations. One obstacle of implementing fuzzy sets in design is implementing the nonlinear operators of multiplication and division. Direct calculation is computationally complex and the existing approximations result in large errors. This research begins by establishing the requirement for modeling imprecision in design. This research presents a model for using fuzzy mathematics in design. Implementation of the design model requires a review of fuzzy mathematics and the approximations used. Analysis of the mathematical operators and the approximation errors involved reveals that the existing approaches are either computationally complex or may result in significant errors. A new approximation is developed which is computationally feasible for implementation in a design support system. The new approximation can be used in a system of algebraic constraint and maintains some useful arithmetic properties. The modeling techniques are demonstrated with a design example which is solved and analyzed using the new approximation. This approach appears to have utility in solving concurrent engineering problems.