The choice of an aggregation function is a common problem in Multi Attribute Decision Making (MADM) systems. The Method of Imprecision (MoI) is a formal theory for the manipulation of preliminary design information that represents preferences among design alternatives with the mathematics of fuzzy sets. The MoI formulates the preliminary design problem as a MADM problem. To date, two aggregation functions have been developed for the MoI, one representing a compensating strategy and one a noncompensating strategy. Much of the prior fuzzy sets research on aggregation functions has been inappropriate for application to engineering design. In this paper, the selection of an aggregation function for MADM schemes is discussed within the context of the MoI. The general restrictions on designappropriate aggregation functions are outlined, and a family of functions, modeling a range of trade-off strategies, is presented. The results are illustrated with an example. Keywords: Fuzzy design; Engineering; Aggregation functions;

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

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...

The Imprecise Design Tool (IDT) presented in this paper implements the method of imprecision, which incorporates the designer's uncertainty in choice into design calculations, using a mathematics derived from fuzzy sets. IDT is intended to be a computational tool for preliminary engineering design.

The Imprecise Design Tool (IDT) presented in this paper is a working computer implementation of the method of imprecision, a formal theory that represents preferences among design alternatives. An aircraft engine design example indicates how the IDT may be applied to support engineering design decisions, using the Engine Development Cost Estimator provided by General Electric Aircraft Engines, Cincinnati, Ohio.

The Method of Imprecision, or M o I, 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: Industrial Applications of DTM; Vehicle Structure Design; Design Methods and Models; Design Representations; Computational Methods of Design; Fuzzy Sets INTRODUCTION Preliminary design is inherently imprecise (Becker, 1973; Blockley, 1980; Gavin, 1994; Yao and Furuta, 1986), and many preliminary design decisions are made informally. Preliminary design has enormous economic importance, as much of the cost of a design is determined by these (often informal) preliminary decisions (Whitney, 1988). A further complication is the difficulty of communicating imprecise information between different...

An extension to the method of imprecision that allows the decomposition or aggregation of imprecise design problems with weighted attributes is presented. The method of imprecision uses the preferences of the designer and customer to de ne fuzzy sets that quantify the imprecision associated with a design attribute. These fuzzy sets, weighted by relative importance, are combined to produce an overall design measure. This paper introduces aggregation operators that can be hierarchically applied to combine weighted design attributes so that the structure of the design problem is appropriately modeled.

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...

iii Frederick Douglass once said “If there is no struggle, there is no progress. ” I think this statement describes my PhD experience here at Caltech well. In ways the past four and half years have been more challenging than I expected yet also more rewarding than I anticipated. This journey would not have been possible without the financial and emotional support of my family and the guidance of my advisors present and past: Professors Fred Culick (Caltech), Victoria Coverstone (University of Illinois at Urbana–Champaign), and Alec Gallimore (University of Michigan). Their support of my education transformed a possibility of a PhD from a dream, to a hope, and now to a reality. I want to thank my Ph. D. advising committee: Professors Erik Antonsson, Jim Beck, and John Ledyard. All three provided invaluable insight and knowledge, especially Professor Beck who assisted with the subset simulation work. Along with Professor Culick, my advising committee gave me the liberty to pursue research ideas and shielded me from many bureaucratic and financial issues a PhD requires. I also thank Dr. Joel Sercel who provided the initial impetus and research ideas that this thesis became and Melinda Kirk for administrative support. This thesis builds upon work by a variety of researchers including Professor Ivan Au

Engineering projects often undergo several design iterations before being completed. Information received from other groups working on a project (analysis, manufacturing, marketing, sales) will often necessitate changes in a design. The interaction between different groups associated with a design project often takes the form of informal “negotiation. ” This form of interaction commonly arises when engineering information is imprecise. The Method of Imprecision (MoI) is a formal method for the representation and manipulation of preliminary and imprecise design information. It provides a mechanism for the formalization of these informal negotiations. The nature and scope of informal negotiation in engineering is explored and discussed, and application of the MoI is illustrated with an example.