Set Mapping in the Method of Imprecision

Citation

Wang, Xiaoou (2003) Set Mapping in the Method of Imprecision. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/J0JH-M190. https://resolver.caltech.edu/CaltechETD:etd-10032002-214953

Abstract

The Method of Imprecision, or MoI, is a semi-automated set-based approach which uses mathematics of fuzzy sets to aid the designer making decisions with imprecise information in the preliminary design stage.

The Method of Imprecision uses preference to represent the imprecision in engineering design. The preferences are specified both in the design variable space (DVS) and the performance variable space (PVS). To reach the overall preference which is needed to evaluate designs, the mapping between the DVS and the PVS should be explored. Many engineering design tools can only produce precise results with precise specifications, and usually the cost is high. In the preliminary stage, the specifications are imprecise and resources are limited. Hence, it is not cost-effective nor necessary to use these engineering design tools directly to study the mapping between the DVS and the PVS. An interpolation model is introduced to the MoI to construct metamodels for the actual mapping function between the DVS and the PVS. Due to the nature of engineering design, multistage metamodels are needed. Experimental design is used to choose design points for the first metamodel. In order to find an efficient way to choose design points when a priori information is available, many sampling criteria are discussed and tested on two specific examples. The difference between different sampling criteria when the number of added design points is small, while more design points do improve the accuracy of the metamodel substantially.

The metamodels can be used to induce preferences in the DVS or the PVS according to the extension principle. The Level Interval Algorithm (LIA) is a discrete approximate implementation of the extension principle. The resulting preference by the LIA is presented as an alpha-cut, which is the set of designs or performances with a certain level of preference. There are some limitations of the LIA, especially for multidimensional DVS and PVS. A new extension of the LIA is proposed to compute alpha-cuts with more accuracy and less limitations. The designers have more control over the trade-off between the cost and accuracy of the computation with the new extension of the LIA.

The results of the Method of Imprecision should be the set of alternative designs in the DVS at a certain preference level, and the set of achievable performances in the PVS. The information about preferences in the DVS and the PVS is needed to transfer back and forth. Usually the mapping from the PVS to the DVS is unavailable, while it is needed to induce preference in the DVS from the PVS. A new method is constructed to compute the alpha-cuts in both spaces from preferences specified in the DVS and the PVS.

Finally, a new measure is proposed to find the most cost-effective sampling region of new design points for a metamodel. Also, the full implementation of the Method of Imprecision is listed in detail. Then it is applied to an example of the structure design of a passenger vehicle, and comparisons are made between the new results and previous results.

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