A technique to perform design calculations on imprecise representations of parameters has been developed and is presented. The level of imprecision in the description of design elements is typically high in the preliminary phase of engineering design. This imprecision is represented using the fuzzy calculus. Calculations can be performed using this method, to produce (imprecise) performance parameters from imprecise (input) design parameters. The Fuzzy Weighted Average technique is used to perform these calculations. A new metric, called the γ-level measure, is introduced to determine the relative coupling between imprecise inputs and outputs. The background and theory supporting this approach are presented, along with one example. 1.

Uncertainty in engineering analysis usually pertains to stochastic uncertainty, i.e.,variance in product or process parameters characterized by probability (uncertainty in truth). Methods for calculating under stochastic uncertainty are well documented. It has been proposed by the authors that other forms of uncertainty exist in engineering design. Imprecision, or the concept of uncertainty in choice, is one such form. This paper considers real-time techniques for calculating with imprecise parameters. These techniques utilize interval mathematics and the notion of α-cuts from the fuzzy calculus. The extremes or anomalies of the techniques are also investigated, particularly the evaluation of singular or multi-valued functions. It will be shown that realistic engineering functions can be used in imprecision calculations, with reasonable computational performance.

In the design and manufacture of mechanical devices, there are parameters whose values are determined by the manufacturing process in response to errors introduced in the device’s manufacture or operating environment. Such parameters are termed tuning parameters, and are distinct from design parameters which the designer selects values for as a part of the design process. This paper introduces tuning parameters into the design methods of: optimization, Taguchi’s method, and the method of imprecision [10]. The details of the mathematical formulation, along with a design example, are presented and discussed. Including tuning parameters in the design process can result in designs that are more tolerant of variational noise. 1

Two methods have been proposed for manipulating uncertainty reflecting designer choice: utility theory and the method of imprecision. Both methods represent this uncertainty across decision making attributes with zero to one ranks, higher preference modeled with a higher rank. The two methods can differ, however, in the combination metrics used to combine the ranks of the incommensurate design attributes. Utility theory resolves the multi-attributes using various well proven additive metrics. In contrast, the method of imprecision resolves by also considering non-additive metrics, such as ranking by the worst case performance or by multiplicative metrics. The axioms of utility theory are appropriate for designs where it is deemed the attributes can always be traded off, even to the point of achieving zero preference in some attributes. In the case of a design with attributes which cannot have zero preference, such as stress limits or maximum allowed cost, the method of imprecision is more appropriate: it trades off attribute levels without permitting any of them to be traded off to zero performance. 1

A formal method to allow designers to explicitly make trade-off decisions is presented. The methodology can be used when an engineer wishes to rate the design by the weakest aspect, or by cooperatively considering the overall performance, or a combination of these strategies. The design problem is formulated with preference rankings, similar to a utility theory or fuzzy sets approach. This approach separates the design trade-off strategy from the performance expressions. The details of the mathematical formulation are presented and discussed, along with two design examples: one from the preliminary design domain, and one from the parameter design domain. 1

Negotiations are common in engineering design, especially on large projects, and are typically conducted informally. Often, negotiation is used to handle the imprecision or uncertainty that is inherent in the design process. Performance targets, initially specified as hard numerical constraints, are adjusted throughout the design process in negotiations between engineers and managers. Crucial unmeasured or unmeasurable aspects of performances, such as aesthetic concerns, are commonly negotiated. Negotiations settle conflicts between engineering groups over values of shared design variables and distribution of limited design resources. In this thesis, a formal description of negotiation in engineering design is presented. This formal model builds on earlier work at Caltech in the modelling of imprecision in engineering design. Negotiation is modelled mathematically as the aggregation of preferences. A complete characterization of the aggregation problem and of the aggregation operators...

iii This document and the work it represents was impossible without the support of my wife Ginger. Often one needs non-technical advice to make clear what one is contemplating. Also one always needs a financial supporter. My thesis advisor Erik Antonsson helped focus many of my thoughts. In addition to providing me with technical assistance, he as well provided instruction on the process of conducting academic research, the communication of ideas both orally and written, and the approach to a developing field. I also owe much to my colleagues in the Engineering and Applied Science Division at Caltech. Their comments and advice maintained my comprehension and rigor. Andrew Lewis in particular provided me with invaluable support. Many of the technical proofs were impossible without him. This material and the work it represented were made possible, in part, by a fellowship from the AT&T-Bell Laboratories Ph.D. scholar program, sponsored by the AT&T foundation. Also, the National Science Foundation provided funding under a Presidential Young

Imprecision is uncertainty that arises because of vague or incomplete information. Preliminary design information is characteristically imprecise: specifications and requirements are subject to change, and the design description is vague and incomplete. Yet many powerful evaluation tools, including finite element models, expect precisely specified data. Thus it is common for engineers to evaluate promising designs one by one. Alternatively, optimization may be used to search for the single "best" design. These approaches focus on individual, precisely specified points in the design space and provide limited information about the full range of acceptable designs. An alternative approach would be to evaluate sets of designs. The method of imprecision uses the mathematics of fuzzy sets in order to represent imprecision as preferences among designs: . Functional requirements model the customer's direct preference on performance variables based on performance considerations: the quantifi...

The Method of Imprecision, or MoI, is a formal method for incorporating imprecise information into a design process. This methodology has been exercised on a problem in preliminary vehicle structure design in collaboration with VW Wolfsburg. Results show that the method is useful in trading off multiple conflicting attributes, including styling preferences and engineering requirements. Keywords:

In the design and manufacture of mechanical devices, there are parameters whose values are determined by the manufacturing process in response to errors introduced in the device’s manufacture or operating environment. Such parameters are termed tuning parameters, and are distinct from design parameters which the designer selects values for as a part of the design process. This paper introduces tuning parameters into the design methods of: optimization, Taguchi’s method, and the method of imprecision [10]. The details of the mathematical formulation, along with a design example, are presented and discussed. Including tuning parameters in the design process can result in designs that are more tolerant of variational noise. 1.