Performance Measures in Dynamic Parameter Design

Transcription

1 Performance easures n Dynamc Parameter Desgn. Introducton V. Roshan Joseph and C. F. Jeff Wu Department of Statstcs Unversty of chgan Ann Arbor, I , USA Journal of Japanese Qualty Engneerng Socety, pp. 8-86, Dynamc parameter desgn ntroduced by Taguch (987 s one of the most mportant tool n qualty engneerng. It s also known as parameter desgn n sgnal-response systems (ller and Wu, 996; Wu and Hamada,, chapter. The name suggests that the nterest les n a sgnal-response relatonshp rather than a sngle value of the response. Taguch s approach to dynamc parameter desgn can be descrbed as follows. Let Y be the response and the sgnal factor. There exsts an deal relatonshp between the sgnal and the response gven by Y = β I. In realty due to the presence of nose factors, devatons occur from ths deal functon. Therefore a more realstc statstcal model s Y = β + ε, ( where E ( ε =, Var( ε = σ, and ε s a random error caused by the nose factors. The β and σ are functons of the control factors X. The obectve of dynamc parameter desgn s to choose an X to make ( as close to the deal relatonshp as possble. Taguch ntroduced the followng sgnal-to-nose (S rato for evaluatng the performance of the system, β S =, ( σ whch wll be referred to as Taguch s S rato. In ( β and σ are estmated for each control run as J K Y k J K = k = ˆβ = and ˆ J = ( Y ˆ k β JK = k = K = σ, (3 where Y k s the response value at control run, sgnal level, and nose level k. The S rato for the th control run can be estmated as Sˆ ˆ = β ˆ / σ. Then an X s sought that maxmzes the S rato. ext an adustment parameter s used to adust β to ts deal value. The ustfcaton for the S rato was that when an adustment parameter s used to adust the value of β to β I, σ wll change to σ ( β I / β and therefore we should mnmze σ / β rather than σ.

2 In statstcal lterature there was some skeptcsm of usng S rato ndscrmnately to all problems. ller and Wu (996 classfed the dynamc parameter desgn nto measurement systems and multple target systems. A thrd class of dynamc parameter desgn s on the optmzaton wth functonal response. ller and Wu proved that S rato s a meanngful performance measure n measurement systems wth lnear calbraton equaton, but crtczed ts use n multple target systems. Joseph and Wu ( gave a general formulaton for multple target systems and showed that σ s the rght performance measure under model (. They derved a modfed verson of the S rato based on a dfferent statstcal model. Interestngly ths S rato can be ustfed even wthout usng the noton of an deal functon. On the other hand, no reasonable ustfcaton s avalable for the use of S rato n the functonal response problem (ar, Taam, and Ye,. Performance measures and S ratos for parameter desgn wth control systems are developed n Joseph (. In ths artcle we wll descrbe the applcaton of S rato n the analyss of the multple target systems. The exposton s mostly based on the work of Joseph and Wu (. We wll derve an S rato as a performance measure ndependent of adustment under some modelng assumptons. We wll also present two modelng approaches known as response modelng and performance measure modelng for the analyss of dynamc parameter desgn experments. Ths wll be llustrated wth a real experment.. Sgnal-to-ose Rato Let X be the set of control factors and Z the set of nose factors. Assume Y and to be nonnegatve varables takng values n [,. Let Y = f ( X, Z, be the sgnal-response relatonshp. If ths relatonshp passes through the orgn, then as shown n Joseph and Wu (, we can approxmate t as Y = β ( X, Z. Let denote the observable set of nose factors and U the unobservable set of nose factors. So Z = {, U}. The observable nose factors are systematcally vared n the experment. Let E U [ β ( X,, U ] = β ( X, and Var U [ β ( X,, U ] = σ ( X,. Then we can use the followng model Y = β ( X, + ε, (4 where E( ε = and Var( ε = σ ( X,. Consder the qualty loss functon L = c( Y T, where T s the customer ntent. Then E( L = c[ ( β ( X, T + σ ( X, ]. β ( X = E β ( X, and σ ( X Var [ β ( X, ] + E [ σ ( X, ] EL = E E( L = c[ ( β ( X T + σ ( X ]. Let [ ] =. Then For a gven T, we can set the sgnal factor to mnmze the loss. Solvng for from EL = c[ ( β ( X T β ( X + σ ( X ] =,

3 we get Tβ ( X =. (5 β ( X + σ ( X Thus the expected loss at ths optmal sgnal settng s = [ + T σ ( X EL c ( β ( X T σ ( X ]=. β ( X + σ ( X Denote ( β ( X E [ β ( X, ] S X = =, (6 σ ( X Var [ ( X, ] E [ β + σ ( X, ] whch can be called a sgnal-to-nose rato. Then EL = T /( + S( X. Suppose W(T s the dstrbuton of the customer ntent T. Then we want to fnd an X D to mnmze EL averaged over W(T, where D s the feasble regon of X. Thus mn x D EL dw ( T = + T dw ( T. S( X Ths s equvalent of maxmzng S (X. Therefore S (X can be consdered as a performance measure ndependent of adustment (PerIA for sgnal-response systems. Ths s an extenson of PerIA ntroduced by Leon, Shoemaker, and Kacker (987 for statc parameter desgn. See Leon and Wu (99 for a theory on PerIA. Thus the optmzaton procedure for multple target systems can be stated as. Fnd X D to maxmze the sgnal-to-nose rato β ( X S( X = (7 σ ( X. Adust, dependng on T, as Tβ ( X = (8 β ( X + σ ( X The optmzaton n step s sometmes carred out wth some bounds on β (X (see Joseph and Wu ( for detals. The adustment n (8 can be nterpreted as a shrnkage procedure as t results n a lower mean value than the target. The adustment step could be modfed as = T / β ( X, so that the mean wll be on target. It s easy to show that the step remans the same even under ths unbased adustment strategy. The above modelng approach s known as response modelng. Another approach to parameter desgn s the performance measure modelng (see Wu and Hamada,, chapters and. Ths s commonly used n Taguch s approach. It can be formulated as follows. Absorbng the observable nose nto the error term, (4 becomes Y = β ( X + ε, (9 where E ( ε = and Var( ε = σ ( X. ow for a gven control run X, we can estmate β ( X and σ ( X usng weghted least squares as J K Y J K k Y ( ˆ β ˆβ = and ˆ σ =. ( JK JK = k = = k =

4 ote that the estmaton procedure n ( s markedly dfferent from that of (3. The sgnal-to-nose rato can be estmated as Sˆ ˆ = β ˆ / σ. Ths can be modeled (after a log transform n terms of X and optmzed. The above sgnal-to-nose rato s dfferent from ( because the underlyng models are dfferent. The two models ( and (9 are pctorally shown n Fgure. In most cases as reduces to, the varance also reduces to and therefore model (9 s more reasonable than model (. Y Y a. model ( b. model (9 Fgure. Sgnal-Response Systems 3. An Example We wll llustrate the approach usng a push-pull cable actuator experment reported n Byrne and Qunlan (993. Ths experment was also analyzed n Tsu (999. There were control factors and one nose factor n the experment. The nput dsplacement s the sgnal factor and the output dsplacement s the response. The data from an OA(, experment s gven n Table. Assume a normal error n model (4. Then usng maxmum lkelhood estmaton we get, log ˆ σ ( X, = x x6, ( ˆ β ( X, = x +.88x +.78x x8. 67x, ( where the two levels of the varables are coded as and. The proposed S rato n (6 becomes ˆ ( x +.88x +.78x x8.67x S( X =. exp( x x6 axmzng ths wth the varables restrcted n the regon D = { X x, =,...,}, we get x =, x =, x6 =, x7 =, x8 =, x =. If we use performance measure modelng, then we wll get the model ˆ ˆ β S = log = x +.596x7, ˆ σ

5 whch on maxmzaton gves x, = x7 =. The results of Taguch s sgnal-to-nose rato analyss from ( s ˆ ˆ β S = log = x x7, ˆ σ whch gves the same conclusons as n the performance measure modelng. The response modelng approach s more nformatve and statstcally vald. We see that the nose factor does not appear n ( and (. Therefore ths was not a useful nose factor to experment wth. The varables x, x 6 affect the varance and are therefore useful to make the cable actuator robust aganst unobserved nose varaton. The varables x, x7, x8, x affect the senstvty. Ther levels are chosen to ncrease the senstvty, thereby ncreasng the S rato. If a hgh senstvty s undesrable, then some of these varables can be manpulated to acheve a desred senstvty. Table. Data of the push-pull cable actuator experment run x x x3x4 x5x6 x7 x8x9x x =8 =6 = Conclusons In ths artcle we have shown that the dynamc sgnal-to-nose rato n (6 can be ustfed as a PerIA under some modelng assumptons. The sgnal-to-nose rato derved s dfferent from Taguch s proposal. We have also descrbed two modelng approaches to parameter desgn. The response modelng s statstcally more effcent than the performance measure modelng, whch has a theoretcal ustfcaton n Berube and ar (998. Joseph and Wu ( descrbed strateges for analyzng nonlnear sgnal-response systems. Taguch treats non-lnearty as an error and tres to mnmze t n hs sgnal-tonose rato optmzaton. However, some of the sgnal-response systems are nherently

6 nonlnear and can work well wth a non-lnear sgnal-response relatonshp. Therefore forcng such systems to behave lke a lnear system wll lead to sub-optmal solutons. Interestngly the model n (4 can stll be used to analyze a nonlnear sgnal-response relatonshp by separatng the lack-of-ft term. References Berube, J., and ar, V.. (998, ``Explotng the Inherent Structure n Robust Parameter Desgn Experments,'' Statstca Snca 8, Byrne, D. and Qunlan, J. (993, Robust Functon for Attanng Hgh Relablty at Low Cost, IEEE Proceedngs of the Annual Relablty and antanablty Symposum, Joseph, V. R. and Wu, C. F. J. (, Robust Parameter Desgn of ultple Target Systems, Technometrcs, to appear. Joseph, V. R. (, Robust Parameter Desgn of Control Systems, Techncal Report 38, Department of Statstcs, Unversty of chgan, Ann Arbor, USA. Leon, R.V., Shoemaker, A.C., and Kacker, R.. (987, ``Performance easures Independent of Adustment: An Explanaton and Extenson of Taguch's Sgnal-to-ose Ratos,'' Technometrcs 9, Leon, R.V. and Wu, C.F.J. (99. ``A Theory of Performance easures n Parameter Desgn''. Statstca Snca, ller, A., and Wu, C.F.J. (996, ``Parameter Desgn for Sgnal-Response Systems: A Dfferent Look at Taguch's Dynamc Parameter Desgn,'' Statstcal Scence, -36. ar, V.., Taam, W., and Ye, K. Q. (, ``Analyss of Functonal Responses from Robust Desgn Studes wth Locaton and Dsperson Effects,'' Journal of Qualty Technology, to appear. Taguch, G. (987, System of Expermental Desgn, Whte Plans, Y: Unpub/Kraus Internatonal. Tsu, K.L. (999, ``odelng and Analyss of Dynamc Robust Desgn Experments,'' IIE Transactons 3, 3-. Wu, C.F.J., and Hamada,. (, Experments: Plannng, Analyss, and Parameter Desgn Optmzaton, ew York: John Wley & Sons.