Implementing the Method of Imprecision: An Engineering ... - CiteSeerX

Implementing the Method of Imprecision: An Engineering Design Example  William S. Law Erik K. Antonsson

Engineering Design Research Laboratory Division of Engineering and Applied Science California Institute of Technology Pasadena, CA 91125

Abstract

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.

Introduction

Preliminary design is characterized by imprecision : the designer's uncertainty in choosing among alternatives. The Imprecise Design Tool (IDT) introduced in this paper is a program that implements the method of imprecision [1, 2] for the Engine Development Cost Estimator (EDCE) provided by General Electric Aircraft Engines, Cincinnati, Ohio. The method of imprecision, a formal theory that includes imprecision in design calculations, is reviewed brie y, the calculation procedures used by IDT are described, and an example design problem is presented to illustrate how IDT allows the designer to explore interactions in an imprecisely described engineering design.

De nitions and Notation

Otto [2] de nes the design parameter space or DPS as \the set of considered possible alternative con gurations, described using design parameters, which the designer has a direct choice over." Design parameters are denoted di, where i ranges from 1 to n. The whole set of design parameters is an n vector, d~, and the valid design parameter values within the DPS form a subset X . The set of valid values for di is denoted Xi. The performance parameter space or PPS is \the dependent set of evaluated performances determined at each point in the DPS, described using performance parameters" [2]. For each performance parameter pj , where j ranges from 1 to q, there must be a mapping fj such that pj = fj (d~). The set of performance parameters is a q vector, p~ = f~(d~). The subset of valid performance parameter values Y is mapped from X and the set of valid values for pj is denoted Yj . The subjective satisfaction that a designer has for values of di , the ith design parameter, is represented by a membership function on X termed the designer preference : d (di) : Xi ! [0; 1]  R where the Xi are assumed to be compact. The designer's satisfaction with values of pj , the j th performance parameter, is represented by a membership function on Y termed the functional requirement : p (pj ) : Yj ! [0; 1]  R: These preference functions d and p are assumed to be monotonically increasing on their support to a range of values (possibly a single value) with peak preference equal to one, and to be monotonically decreasing after the peak. i

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 Published in Proceedings of the Third IEEE International Conference on Fuzzy Systems (FUZZ-IEEE '94), volume 1, pages 358-363.

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The designer's overall satisfaction with a particular design d~ is represented by an overall preference (d~), which is a function of the designer preferences d (di), and the functional requirements p (pj ):

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(1) (d~) = P d1 (d1); : : :; dn (dn ); p1 (f1 (d~)); : : :; pq (fq (d~)) : The design problem is to maximize  and so we seek design con gurations d~ such that (d~ ) = supf(d~) j d~ 2 Xg. The combination function P must satisfy continuity and annihilation i.e.(P [1 ; : : :; 0; : : :; n+q ] = 0) [2]. Additional restrictions that P should satisfy for engineering design have been proposed in [3]. P re ects the design strategy [3, 4]. Suppose that the designer wishes to maximize his satisfaction with the least satisfactory aspect of the design. This is a conservative or non-compensating design strategy. The design is judged by its lowest preference (d or p ) and P is min:   (d~) = min d1 ; : : :; d ; p1 ; : : :; p : Alternatively, the designer may trade-o dierent aspects of the design, allowing a more satisfactory aspect to partially compensate for a less satisfactory aspect. This is an aggressive or compensating design strategy and P is a normalized product: 1 +1 0 j

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These combination functions satisfy continuity and annihilation as well as the restrictions proposed in [3]. Importance weightings may be speci ed for d and p [4], but they are not relevant to this paper. j

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Forward Calculation

Having speci ed preferences d (di ) on Xi and p (pj ) on Yj , the designer would like to ensure that the design can satisfy the functional requirements by determining the induced values of d on Y . These induced values d (~p) are given by the extension principle [5]: d (~p) = supfd (d~) j f~(d~) = ~pg where sup over the null set is de ned to be zero and d (d~) is the combined designer preference on X . d (d~) arises from splitting the function P into three operations:   (2)  = Pc [d ; p] = Pc Pd (d1 ; : : :; d ); Pp (p1 ; : : :; p ) where Pd combines the designer preferences, Pp combines the functional requirements, and Pc combines these sub-results. For a conservative design strategy, Pd = Pp = Pc = min. Note that Equation (2) applies on both the DPS and the PPS. To calculate d (~p), we use the Level Interval Algorithm, or LIA [1, 4, 6], rst proposed by Dong and Wong [7] as the \Fuzzy Weighted Average" algorithm and also called the \Vertex Method": For n real imprecise design parameters, d1; : : :; dn, consider a performance parameter p = f (d1 ; : : :; dn). The following steps lead to the solution of the induced designer preference d (p): 1. For each di , discretize the preference function d (di) into a number of values, 1; : : :; M ; where M is the number of steps in the discretization. 2. Determine the intervals for each parameter di; i = 1; : : :; n, at each {cut D , k = 1; : : :; M : (3) D = fd~ 2 X j d (d~)  k g = fd~min; d~maxg: j

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3. Using one end point from each of the n intervals for each k , combine the end points into an n-ary array such that 2n distinct permutations exist for the array. 4. For each of the 2n permutations, determine pl = f (d1 ; : : :; dn); l = 1; : : :; 2n. The induced {cut P is then given by: P = fp 2 Y j d (p)  k g = [pmin; pmax] = [min(pl ); max(pl )]: k

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Otto, Lewis, and Antonsson [6] state conditions under which the induced {cut P is a lower or upper bound for the actual {cut for d (p), denoted W . De ne the sets:  = fp 2 Y j p < p g Ymin min   Ymax = fp 2 Y j p > pmaxg: Recall that combination functions must satisfy continuity and annihilation. Let P be a combination function and let f : X ! Y be a function. If f is continuous and the pair (P ; f ) satis es the following two hypotheses:  ) = ; for all    H1: D \ f ?1 (Ymin k and  ) = ; for all    ; H2: D \ f ?1 (Ymax k then P  W and P is an upper bound for W . Let P be a combination function and f : X ! Y be a function. If P is monotonic and idempotent i.e.(P (; : : : ; ) = ) and f is continuous, then P  W and P is a lower bound for W . If both sets of conditions are satis ed: P is monotonic and idempotent, f is continuous, and the pair (P ; f ) satis es H1 and H2, then P = W and the LIA solution is exact. For the particular case P = min, if f is continuous and monotonic the induced {cut P is exact [8]. k

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Backward Path

The peak preference in X is equal to the peak preference in Y [9]:  = supf(d~) j d~ 2 Xg = supf(~p) j ~p 2 Yg: If f~ is continuous on X , and d (d~) is continuous when restricted to f~?1 (~p) for each ~p 2 Y , then the mapped set of peak preference points in the DPS, f~(X  ), is equal to the set of peak preference points in the PPS, Y  [9]. Hence we may search for either: Y  = f~p 2 Y j (~p ) =  g or X  = fd~ 2 X j (d~ ) =  g; knowing that they are related by: f~ (X  ) = Y  : To nd Y  we need to combine d (~p) and p (~p) to obtain (~p). d (~p) was obtained in the forward calculation. To nd X  we need to combine d (d~) and p (d~) to obtain (d~). But we do not know p (d~) and do not necessarily have the inverse mapping f~?1 : Y ! X needed to backward calculate {cuts. Suppose that we have calculated (~p) and found Y  . The range of peak preference performance is given by Y  , but the design problem is to nd the set of peak preference design con gurations X  . Consider a conservative design strategy:  = min(d ; p) and hence d (d~)  (d~). Recall the {cut D from Equation (3). Let  be the largest k such that k   . Then X   D , since for all d~ 2 X , d (d~)  (d~) =    . Thus we need only search for peak preference design con gurations in the n?cube de ned by D . The overall preference (d~) is given by P [d (d~); p (f~(d~))], but we have already calculated p~ = f~(d~) for the M sets of 2n permutations of {cut endpoints that correspond to the corners of the n?cubes de ned by D , k = 1; : : :; M . Let XD be the set of these endpoints, which are also called ridge points . If the forward calculations were saved, we can nd (d~) for any d~ 2 XD without evaluating f~ again. k

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The Imprecise Design Tool

IDT is a C program that implements the method of imprecision with a conservative design strategy (P = min) for a \black box" calculation f : X ! Y . Currently, this black box is the Engine Development Cost Estimator (EDCE) provided by General Electric Aircraft Engines, Cincinnati, Ohio. While EDCE requires crisp inputs and produces a single crisp output, IDT allows the designer to specify imprecise inputs and obtain an imprecise output. Eight of EDCE's inputs which represent the degree of innovation in eight subsystems of the new engine to be developed were chosen to be design parameters d1; : : :; d8. The parameter d8 , for 3

example, corresponds to the fan system. A value of 0% change indicates that the engine to be developed does not possess a fan system and 10% change indicates that only support-engineering will be required. At the other extreme, 200% change indicates a new fan with similar or existing technology, tted to a new engine design. The ten levels of percent change have de nitions which are speci c to each input. Intermediate values are unde ned, and hence the eight inputs are eectively discrete. EDCE produces a single output p: an estimate of the development cost for the new engine. p = f (d~) is continuous and monotonic: an incrementally larger percentage change in a subsystem results in an incrementally larger development cost. Recall that for P = min and f continuous and monotonic, the {cuts given by the LIA are exact. IDT uses the LIA to forward calculate d (p) on Y . The user speci es designer preferences (d ) at each point in Xi, i = 1; : : :; 8, as an array of numbers. Since the LIA discretizes d (d~) into M levels 1 ; : : :; M , only these values should be speci ed. For IDT, M = 10 and k = 1:0; 0:9; : : :; 0:1: it is questionable whether a designer could distinguish more than ten levels of preference. The functional requirement on Y is speci ed as an ordered list of pairs (p; p), which de ne a piecewise linear preference function. The rst and last pairs in the list are extended to in nite values of p. For two {cuts D and D +1 where k > k+1, D  D +1 , since for all d~ 2 D , d (d~)  k > k+1. But if D = D +1 , the forward calculation need only be performed once, to obtain P , since P +1 contains the same points but has lower preference. Hence D +1 may be ignored. IDT uses a lookup table for f (d~) to avoid repeated EDCE evaluations for the current design calculation and subsequent iterations. The EDCE program is called only once during the forward calculation for all the required points. The calculated P , which represent d (p), are combined with p (p) to produce (p), which is saved as an ordered list of pairs (p; ) de ning a piecewise linear preference function on Y . In the same step, the peak preference  and the peak preference set of development costs Y  are also found. At this point, a problem arises: when Y  is mapped back to the DPS, X  may not contain any of the de ned points that are meaningful to the designer. If we restrict the set of valid points in the DPS, X , to these de ned points, then  0   , where  0 is the peak preference in the restriction of X . In practice, the set of de ned points in the DPS is suciently dense that the dierence between  0 and  is not signi cant. After nding  , IDT determines , the largest k   , so that X   D , as discussed in the backward path. (d~) is calculated at every d~ 2 (D \ XD ) and any (d~) =  are identi ed as peak preference design con gurations. Where d~ 2= XD and f (d~) is not immediately available, d (d~) provides an upper bound since, for a conservative strategy,   d . We now know (d~) or an upper bound for (d~) at every d~ 2 D that could potentially be a peak preference design con guration. A designer may also wish to visualize the variation of  on the DPS, so IDT allows the user to specify points about which eight 2D cross{sections of  in one design parameter, or four 3D cross{sections of  in two design parameters, are generated. i

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Example

As an example design problem, suppose that $145 million has been allocated towards the development of a new aircraft engine, however, it is desired that as much of the budget as possible should be used. Such a functional requirement could be represented by a ramp rising linearly from p = 0 at p = $0 million to p = 1 at p = $145 million, with p = 0 elsewhere (see Figure 1). Given the needs that the new engine is intended to satisfy, the design team has agreed on a set of designer preferences for the degree of innovation in each of its eight subsystems (Figure 2). It is expected that only support-engineering will be required for the bearing and lubrication subsystem, and hence d3 is non-zero only for a change value of 10%. Greater imprecision is associated with the high pressure turbine (d5 ). The design team has also decided to pursue a conservative design strategy. d (p) and (p) obtained from the forward calculation are shown in Figure 1. Note that the development costs shown are representative and were not calculated using actual cost data. The peak preference in the PPS  = 1:000, at a development cost of $145 million. The corresponding point in the DPS is not de ned, and  0 = 0:998, at d~ = (4; 2; 2; 3; 3; 4; 4; 3), the peak preference design. Other designs (d~ 2 D ) with near peak preference are listed in Table 1. Points indicated with asterisks are also on the 2D and 3D cross{ sections in Figures 3 and 5. The ten points in Table 1 and d~ are the highest preference designs and may be considered to be the solution set. All other designs d~ 2= D have (d~)  d (d~)  0:9, since  = 1:0 and the next highest value of d speci ed is 0:9. The design team may now introduce other considerations to reduce i

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Figure 1.

Control and Accessories

Preferences , d , and p on the PPS.

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(d~) d~ (3; 2; 2; 4; 3; 4; 4; 3) 0:985 (3; 2; 2; 4; 4; 4; 3; 3) 0:983 (4; 2; 2; 3; 3; 4; 3; 3) 0:979 (4; 2; 2; 3; 4; 4; 3; 3) 0:996 (4; 2; 2; 4; 3; 4; 3; 3) 0:996 Points in D with  close to  = 0:988.

d~ (3; 2; 2; 3; 3; 4; 3; 3) (3; 2; 2; 3; 3; 4; 4; 3) (3; 2; 2; 3; 4; 4; 3; 3) (3; 2; 2; 3; 4; 4; 4; 3) (3; 2; 2; 4; 3; 4; 3; 3) Table 1.

(d~) 0:949 0:968 0:966 0:985 0:966

the number of design solutions, or adjust the designer preferences and functional requirement preferences and run IDT again, or both.

Conclusion

This paper has summarized the method of imprecision, described the calculation procedures used by IDT, and presented a contemporary commercial engineering design example. IDT applies this formal method for representing and manipulating imprecison to a real engineering design problem, allowing the designer to explore and evaluate a design while it is still imprecisely described.

Acknowledgments

This material is based upon work supported, in part, by: The National Science Foundation under a Presidential Young Investigator Award, Grant No. DMC-8552695, and NSF Grant No. DDM-9201424. Any opinions, ndings, conclusions or recommendations expressed in this publication are those of the authors and do not necessarily re ect the views of the sponsors.

References

[1] K. L. Wood, K. N. Otto, and E. K. Antonsson. Engineering Design Calculations with Fuzzy Parameters. Fuzzy Sets and Systems, 52(1):1{20, November 1992. [2] K. N. Otto. A Formal Representational Theory for Engineering Design. PhD thesis, California Institute of Technology, Pasadena, CA, June 1992. [3] K. N. Otto and E. K. Antonsson. Trade-O Strategies in Engineering Design. Research in Engineering Design, 3(2):87{104, 1991. [4] K. N. Otto and E. K. Antonsson. Trade-O Strategies in the Solution of Imprecise Design Problems. In T. Terano et al., editors, Fuzzy Engineering toward Human Friendly Systems: Proceedings of the International Fuzzy Engineering Symposium '91, Volume 1, pages 422{433, Yokohama Japan, November 1991. LIFE, IFES. [5] L. A. Zadeh. Fuzzy sets. Information and Control, 8:338{353, 1965. [6] K. N. Otto, A. D. Lewis, and E. K. Antonsson. Approximating -cuts with the Vertex Method. Fuzzy Sets and Systems, 55(1):43{50, April 1993. [7] W. M. Dong and F. S. Wong. Fuzzy weighted averages and implementation of the extension principle. Fuzzy Sets and Systems, 21(2):183{199, February 1987. [8] D. Dubois and H. Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980. [9] K. N. Otto, A. D. Lewis, and E. K. Antonsson. Determining optimal points of membership with dependent variables. Fuzzy Sets and Systems, 60(1), November 1993. 6