COMPARING THREE APPROACHES TO ROBUST DESIGN: TAGUCHI VERSUS DUAL RESPONSE VERSUS TOLERANCE ANALYSIS. Presented at 1996 Fall Technical Conference

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1 COMPARING THREE APPROACHES TO ROBUST DESIGN: TAGUCHI VERSUS DUAL RESPONSE VERSUS TOLERANCE ANALSIS Presened a 996 Fall Technical Conference Dr. Wayne A. Taylor, Taylor Enerprises, Inc. ( Robusness is a key sraegy for achieving high qualiy - low cos producs and processes. Three differen approaches o robus design are commonly used: he inner/ouer array approach advocaed by Taguchi, he dual response approach using response surfaces and he olerance analysis approach which also uses response surfaces. Each of hese approaches will be explained. The hree approaches will hen be conrased. The hree approaches differ as o how he sudies are run. Requiremens for using he differen approaches will be compared. In robus design, he objecive is o esimae he effec ha he arges of he inpu variables have on he variaion of he oupu and o selec he se of arges ha imize he variaion while achieving he desired average. The hree approaches differ as o he precision and accuracy of he resuling esimaes. The accuracy and precision of he differen approaches will be compared under a variey of circumsances including sudy condiions ha are no represenaive of manufacuring condiions and when some sources of variaion are no included in he sudy. All hree approaches have srenghs and weaknesses. No approach can be said o be universally superior. However, hese comparisons sugges ha in mos cases, he olerance analysis approach is he bes approach o use. The major weakness of he olerance analysis approach is ha i only esimaes he variaion caused by he inpus included in he sudy. I ignores oher sources of variaion. This weakness can be overcome using a combinaion of he olerance analysis and dual-response approaches. KE WORDS Robus Design, Parameer Design, Dual Response, Inner/Ouer Arrays, Taguchi Mehods, Saisical Tolerance Analysis, Tolerance Design INTRODUCTION For many producs and processes, he variaion of he oupu or response variables is affeced by he arges seleced for he inpu variables. For a hea seal machine, adjusing inpus such as emperaure and dwell ime can affec he amoun ha he oupu seal srengh varies. For a pump, adjusing inpus like sroke lengh and moor speed can likewise affec he amoun ha he oupu flow rae varies. When selecing arges for inpu variables, heir effecs on he oupus' variaion should be considered in addiion o heir effecs on he oupus' average. This resuls in wha is called a robus design. There are hree common approaches o robus design: () he dual response approach using response surface sudies, () he inner/ouer array approach advocaed by Taguchi and (3) he olerance analysis approach also using response surface sudies. Each approach is explained. There are imporan differences beween hese hree approaches in erms of heir requiremens and he resuls obained. The requiremens and he resuls are compared o help idenify when each approach should be used. A SIMPLE EAMPLE The following example will be used o compare he hree approaches. Suppose here is a single inpu variable and a single oupu variable and ha: = () A plo of he effec of on is shown in Figure. 4 5 Figure : Effec of on Furher suppose ha varies around is seleced arge wih a sandard deviaion of σ =.5. We can selec any arge or average for over he range of o. We wan o idenify he arge for ha imizes he amoun ha varies.

2 Figure shows why he arge of effecs he variaion of. When he arge of is, more variaion is ransmied from o due o seeper slope. This makes more sensiive o he variaion of. When he arge is 5, he slope is less making less sensiive or more robus o he variaion of Figure : Achieving Robusness EQUATION KNOWN If he equaion relaing and is known, an exac soluion o he problem can be obained. This is accomplished by deriving an equaion for he sandard deviaion of and hen finding he arge of ha imizes his equaion. The equaion for he sandard deviaion can be obained using olerance analysis. Deails can be found in Taylor (99), Cox (986), Evans (975) and many oher places. These mehods are also referred o as propagaion of error and variaion ransmission analysis. Le represen he seleced arge or average of. Using olerance analysis, he following expression for he sandard deviaion of can be obained: (..8 ) σ σ ( ) = () 4 +.3σ Figure 3 shows a plo of he sandard deviaion of. I is imized when = 5. The imum sandard deviaion is σ ( 5. ) =. 4. σ ( ).4. 5 Figure 3: Plo of σ ( ) Exac soluions or close approximaions can be obained if he equaion relaing and is known. However, in many problems, he equaion relaing he inpus and oupus is no known. I mus be esimaed by collecing and analyzing daa. Assug he equaion is unknown, we will work his problem using each of he hree approaches. Once he hree approaches are undersood, he remainder of he aricle will compare he resuls obained by he hree mehods under a variey of siuaions. While his problem is an over simplificaion of real world problems, he insigh gained will prepare us for dealing wih more complicaed problems. While here are many problems where he equaion is unknown, here are an equal number of problems where he equaion is known. Engineering and science exbooks are full of such equaions. When he equaion is known, none of he 3 approaches exaed in his aricle should be used. Insead a olerance analysis should be performed o obain an exac soluion. All hree approaches involve approximaing he acual equaion wih a low-order polynomial which is an unnecessary sep ha can resul in he loss of informaion. See Lawson and Madrigal (994), Taylor (996) and Bisgaard and Ankenman (996). APPROACH : DUAL RESPONSE APPROACH The dual response approach involves running a response surface sudy where boh he average and sandard deviaion of he oupus are analyzed. The resuling equaions are hen used o imize he variaion while achieving a desirable average. Furher informaion can be found in Vining and Myers (99), and Myers, Khuri and Vining (99). To solve he example problem, hree rials were run a equally spaced arges. Twelve observaions were made per rial. Table conains he daa. This daa was creaed by generaing values for according o he normal disribuion and plugging hese ino Equaion. Table : Daa for Dual Response Approach Targe 5 Values of

3 Once he daa was colleced, he average and sandard deviaion for each arge was calculaed. The resuling values are shown in Table. Table : Esimaes of he Average and Sandard Deviaion of Targe Average Sd. Dev Quadraic polynomials were hen fi o hese values using regression analysis. The coefficiens of an equaion of he form b + b + b are esimaed by: bˆ B ˆ = bˆ = bˆ ( ) (3) is he design marix shown below were =, = 5 and 3 =. = 3 = is he daa marix. For he sandard deviaion, he log of he sandard deviaion is generally fi insead. The resuling equaion can be back-ransformed o obain an equaion for he sandard deviaion. When fiing he log of he sandard deviaion, is: log(.65797) = log(.996) log(.64563) (4) (5) The resuling equaions for he average and sandard deviaion of are: µ ˆ ( ) = (6) ( σ ˆ ( ) = e ) (7) Mos problems involve several inpus. In many cases hose inpus having he larges effec on he variaion are argeed o imize he variaion. The remaining inpus are hen argeed o achieve he desired average. In oher cases, objecive funcions (including signal o noise raios) are opimized o obain he bes combinaion of he average and variaion. To simplify our comparisons, we will ignore he equaion for he average and assume ha he objecive is o simply imize he variaion. This requires finding he value of ha imizes Equaion 7 over he region of sudy. An esimae of he arge imizing he variaion, denoed ˆ can be obained by aking he parial derivaive of Equaion 7 wih respec o, seing i o zero and hen solving for. This resuls in: ˆ bˆ = 5.97 (8) bˆ = The above equaion yields he imum so long as he resul is in he region of sudy and b is posiive, i.e., he poin is a imum and no a maximum. For boh hese excepions, he imum occurs a one of he edges of he sudy region. A he imum arge, he prediced variaion is σˆ (ˆ ) =. 99 APPROACH : TAGUCHI APPROACH The Taguchi approach differs from he dual response approach in he way he variaion is esimaed a he hree seleced arges. The Taguchi approach uses a noise array o simulae he variaion of he inpus. To solve he example problem, 4 samples were colleced a each of he arges 9.5,.,.5, 4.5, 5., 5.5, 9.5,. and.5. This keeps he oal number of samples he same. Table 3 conains he daa. This approach is also referred o as he inner/ouer array approach. These arges represen all possible combinaions of he inner array I and ouer or noise array O shown below. An esimae of σ is required o design he sudy. I = 5 +σ O = + σ Table 3: Daa for Taguchi Approach Targe Values of (9) 3

4 Once he daa has been colleced, he average a each arge is calculaed. The resuling values are shown in Table 4. Table 4: Cell Averages Inner Array Ouer Array Esimaes of he average and sandard deviaion of a each of he ouer array arges are obained by calculaing he average and sandard deviaion of he cell averages. The resuling esimaes are shown in Table 5. When he ouer array arge is, daa is colleced a 9.5, and.5. The average of hese hree poins is and he sandard deviaion is σ =.5. Thus he inner array resuls in a sysemaic sample ha mimics he variaion of he inpu. Taking he average and sandard deviaion of he cell averages hen esimaes he behavior of he oupu. Table 5: Esimaes of he Average and Sandard Deviaion of Targe Average Sd. Dev Quadraic polynomials are hen fi o hese resuls using he same equaions as before (3-5). Again he log of he sandard deviaion is analyzed. The resuling equaions for he average and sandard deviaion of are: µ ˆ σˆ ( ) = ( ) = e ( ).3858 () () The value of ha imizes Equaion over he region of sudy is: ˆ bˆ = () bˆ = A he imum arge, he prediced variaion is σˆ (ˆ ).38. A good inroducion o he Taguchi = approach is Taguchi (986). APPROACH 3: TOLERANCE ANALSIS APPROACH The olerance analysis approach sars wih a response surface sudy on he oupu's average. A olerance analysis is hen performed using he prediced equaion for he average. Exacly he same daa is required as for he dual response approach. Everyhing proceeds as before up o he poin ha he equaion for he average is obained. The olerance analysis approach only uses he equaion for he average. The equaion for he sandard deviaion will be ignored. Furher deails and examples can be found in Taylor (99). For he example problem, he prediced equaion for he average was given in Equaion 6. A olerance analysis is hen performed on his equaion. This resuls in he following equaion: ( ) σ σ ˆ ( ) = (3) σ An esimae of σ is required o use his equaion o complee he analysis. This equaion serves he same purpose as Equaion 7 for he dual response approach and Equaion for he Taguchi approach. The value of ha imizes Equaion 3 over he sudy region is:.5 ˆ = = 4.98 (4).8666 A he imum arge, he prediced variaion is σˆ (ˆ. Since he same daa is required for ) =.44 boh he dual response approach and he olerance analysis approach, boh analysis could be performed and he resuls compared. This migh provide beer resuls and insigh han eiher mehod on is own. COMPARISON OF RESULTS FOR EAMPLE PROBLEM All hree approaches esimae he effec of he inpu's arge on he oupu's he variaion (Equaions 7, and 3). However, hey use differen sraegies for obaining hese esimaes. The dual response approach direcly observes he variaion. Taguchi's approach simulaes he variaion using a noise array. The olerance analysis approach predics he variaion based on an equaion for he average. There are procedural differences beween he hree approaches. Boh he Taguchi and olerance analysis approaches require an esimae of σ. This esimae is used o design he Taguchi sudy. This limis he analysis o he value used o design he sudy. The olerance analysis approach does no use he esimae of σ unil he analysis phase. This allows alernae values 4

5 Dual Response Approach Taguchi Approach Tolerance Approach σ ( ) Figure 4: Esimaed Sandard Deviaion of For Example Problem 5 o be explored as is he case when one is considering he ighening of cerain olerances. The dual response approach does no require an esimae of σ. Insead, i direcly observes he variaion of. This requires ha he variaion presen during he sudy period be represenaive of full-scale producion. This can be difficul o ensure and requires special safeguards while he daa is being colleced. Anoher procedural difference is ha he Taguchi approach requires a larger number of adjusmens and finer adjusmens which may be difficul o make and which increase he cos of he sudy. Besides such procedural differences, heir are also differences in he accuracy and precision of he resuling esimaes. To compare he resuling esimaes,, ses of daa were generaed and analyzed using each approach. Figure 4 shows he resuls of esimaing σ ˆ (). The cener line represens he average of σ ˆ () and he band represens ± sandard deviaions. The rue value is also shown as a doed line. The olerance analysis approach has he narrowes band around he rue value indicaing ha i provides he greaes accuracy and precision. Why he difference in curve shapes? Equaion, he rue equaion for, is a quadraic polynomial. However, Equaion, he rue equaion for he sandard deviaion, is no a quadraic polynomial nor can i be closely approximaed by one. The dual response approach and Taguchi approach aemp o fi a quadraic polynomial o he sandard deviaion. As a resul, here is significan lack of fi in beween he hree fied poins. The olerance analysis approach fis a quadraic polynomial o he average which resuls in an accurae fi. This example illusraes ha equaions for sandard deviaions end o be more complex han equaions for he average. In general, he order of he polynomial required o fi he sandard deviaion is double ha for he average. Thus if a nd-order polynomial fis he average well, a 4h-order polynomial may be required for he sandard deviaion. This is a resul of he fac ha he sandard deviaion of depends on he E{ }. All hree mehods, will be effeced by lack of fi due o incorrec models. However, he olerance analysis approach will generally be effeced less since i fis he average insead of he variaion. For each of he hree approaches, Table 6 compares he esimaes of boh and he variaion a. All values are repored o he number of digis esimaed o be accurae. The rue values are shown a he op of he able. All hree approaches provide an accurae (unbiased) esimae of. However, he olerance analysis approach is more precise (smaller sandard deviaion) a esimaing han he oher wo approaches. The dual response approach is second. Boh he dual response approach and Taguchi approach underesimae he variaion a. The olerance analysis approach provides an accurae esimae ha is en imes more precise han he oher wo approaches. Table 6: Esimaes of Minimum Targe and Sandard Deviaion a Minimum Targe For Example Problem Approach = 5 σ ( ) =.4 Average Sd. Dev. Average Sd. Dev. Dual Response Taguchi Tolerance Analysis

6 Dual Response Approach Taguchi Approach Tolerance Approach σ ( ) Figure 5: Esimaed Sandard Deviaion of For Three Approaches When Sudy Condiions Are No Represenaive of Manufacuring STUD CONDITIONS NOT REPRESENTATIVE OF MANUFACTURING In he example problem, i was assumed ha he variaion of during he sudy was represenaive of is variaion during acual manufacuring. This is frequenly no he case. I is no uncommon for he variaion during he sudy o be less han wha will be experienced under more exended periods of producion. Frequenly, limied maerial is available, he process is beer conrolled, and componens of variaion such as roll o roll variaion and ank o ank variaion are missed. To deere wha effec running he sudy under condiions no represenaive of manufacuring has on he resuls,, new daa se ses were generaed and analyzed. A value of σ =.5 was used o generae daa represening sudy condiions. I is sill assumed ha σ =.5 under long erm manufacuring so ha he correc answers are he same as before. Furher, a value of σ =.5 was sill used o design he Taguchi sudy and o perform he olerance analysis. The resuls are shown in Figure 5 and Table 7. No oo surprisingly, he dual response approach ends up badly underesimaing he variaion of across he enire curve. However, he abiliy o esimae does no suffer. In fac i improves slighly. This migh no be he case if oher inpus were also included in he sudy. Suppose a sudy were run where he inpu conribuing he mos variaion only varied over a narrow range while a less imporan inpu varied over is full range. The dual response approach could end up making he process robus o he second inpu a he expense of making i more sensiive o he firs. While he seleced arges migh opimize he process for he sudy condiions, hey migh perform poorly in manufacuring. When using he dual response approach, sudy condiions should be as represenaive of long erm manufacuring as possible. This can significanly add o he cos of he sudy. Boh he Taguchi and olerance analysis approaches benefi from he reduced variaion during he sudy. In boh case he sandard deviaion of he esimae is reduced by a leas 5% compared o Table 6. When using he Taguchi and olerance analysis approaches, variaion during he sudy should be kep o a imum. The olerance analysis approach mainains is advanage over he oher wo approaches wih respec o precision of he esimae. Boh he dual response approach and Taguchi approach underesimae he variaion a. Again he olerance analysis approach provides an unbiased esimae ha is en imes more precise han he oher wo approaches. Table 7: Esimaes of Minimum Targe and Sandard Deviaion a Minimum Targe When Sudy Condiions Are No Represenaive of Manufacuring Approach = 5 σ ( ) =.4 Average Sd. Dev. Average Sd. Dev. Dual Response Taguchi Tolerance Analysis

7 Dual Response Approach Taguchi Approach Tolerance Approach Adjused Tol. Anal σ ( ) Figure 7: Esimaed Sandard Deviaion of When Measuremen and Oher Variaion MEASUREMENT ERROR AND OTHER VARIATION In he example problem, all variaion is due o. There are no oher sources of variaion including measuremen. In his secion we will add addiional variaion o our model and repea he comparison. Suppose we add o Equaion a second inpu E represening he oher sources of variaion The new model becomes: = E (5) Assume E has average µ E = and sandard deviaion σ E =.. Then he rue formula for he sandard deviaion changes o: (..8 ) σ σ ( ) = (6) 4 +.3σ + σ E Figure 6 shows a plo of he sandard deviaion of. I is sill imized when = 5. The resuling imum sandard deviaion changes o.5. To compare he resuling esimaes, anoher, ses of daa were generaed and analyzed using each approach. Like in he previous secion, a value of σ =.5 was used o generae daa. The precision and accuracy of he esimaed variaion of is shown in σ ( ).4. 5 Figure 6: Plo of σ ( ) When Oher Variaion Figure 6. The olerance analysis approach has he narrowes band indicaing ha i provides he greaes precision. However his approach badly underesimaes 's variaion. This is because i only esimaes he variaion resuling from. I ignores variaion from oher sources. The Taguchi approach also underesimaes 's variaion for he same reason, alhough no nearly as badly. The dual response approach coninues o underesimae he variaion due o sudy condiions no represenaive of manufacuring. The reason he olerance analysis approach underesimaes 's variaion is ha i ignores he measuremen variaion and he variaion resuling from variables no included in he sudy. This can be fixed by being sure o include all possible sources of variaion in Table 8: Esimaes of Minimum Targe and Sandard Deviaion a Minimum Targe When Measuremen and Oher Variaion Approach = 5 σ ( ) =.5 Average Sd. Dev. Average Sd. Dev. Dual Response Taguchi Tolerance Analysis Adjused Tol. Anal

8 he sudy. I may also be necessary o adjus he esimaed variaion o accoun for he measuremen variaion. Suppose ˆσ is an esimae of he measuremen E variaion. One could hen esimaes 's variaion as follows: σˆ ( ) = σˆ ( ) + σˆ (7) Adjused Original E Assug σˆ E =., he adjused resuls are also shown in Figure 7. This resuls in a highly accurae and precise esimae of 's variaion. A similar adjusmens can be made for he Taguchi approach. Table 8 shows he average and sandard deviaion of he resuling esimaes of. The sandard deviaion of all hree mehods increased from he previous secion (Table 7). The presence of addiional variaion only makes i more difficul o esimae. However, he sandard deviaion of he olerance analysis approach only increases 3% while he oher wo approaches increase o 3 fold. Wih variaion from oher sources presen, he olerance analysis approach has a sandard deviaion /5 ha of he Taguchi approach and /3 ha of he dual response approach. This confirms wha he auhor has experienced in pracice ha he olerance analysis approach is able o repeaedly discover effecs missed by he oher wo approaches. Table 8 also indicaes ha all hree mehods underesimae he imum sandard deviaion. The olerance analysis approach is worse. However, by adjusing for measuremen error and being sure o include all possible sources of variaion, he olerance analysis approach gives amazingly accurae and precise esimaes. SUMMAR All hree approaches o robusness have been highly successful. Numerous case sudies aes o his fac. Any mehod of using designed experimens and addressing robusness is beer han none. A issue is wheher one approach offers significanly greaer benefis han he ohers and can be said o be he bes demonsraed pracice. Robusness applies o a variey of producs and processes including elecronic hardware, plasics manufacuring, chemical processes and meal processing. Such applicaions vary grealy in he cos of collecing daa, ype of objecives, and so on. There may no be a single mehod ha is bes for all hese applicaions. Wha has worked well in one indusry may no be he bes approach in anoher. This aricle aemps o compare he hree approaches under a variey of circumsances o help idenify differences in he performance beween he hree approaches. In addiion, here are procedural differences ha mus be considered. These differences are summarized in Table 9. The olerance analysis approach is frequenly he bes approach. I far ouperforms he oher approaches in erms of accuracy and precision of is esimaes. I is also is he leas expensive approach. Furher, i has he enormous advanage ha he variaion or olerances for he inpu do no have o be specified unil he analysis phase allowing alernae designs or condiions o be analyzed as well. Robusness alone migh no be enough o achieve your objecives. Tighening of some olerances may be necessary. The olerance analysis approach allows alernae olerances o be explored and opimized wihou requiring furher collecion of daa. The single weakness of he olerance analysis approach is ha i requires ha all imporan sources of variaion o be included in he sudy. Special care should be aken o consider all variables ha migh effec he produc or process. This generally requires running screening experimens or fracional facorial designs firs on a large number of inpus and hen augmening i ino a response surface sudy on he key inpus. If one is sudying a small number of he mos imporan inpus, he dual response approach should be used. I is he only approach ha achieve robusness agains sources of variaion no formally included in he sudy. Because of he poor precision of his approach, i would be bes o ake 5 o samples per rial. I should be rouine pracice o analyze he sandard deviaion anyime he average is analyzed. While he dual response approach may miss some inpus due o is poor precision, inpus ha i finds can lead o significan improvemens wihou he need for addiional daa. Since he dual response approach and olerance analysis approach can use he same daa, he wo approaches can be used ogeher. Such sudies should be run under condiions as represenaive of manufacuring as possible. While his is no he ideal condiions for he olerance analysis approach, he olerance analysis approach sill has good precision. This allows he prediced performance from he olerance analysis approach o be compared o he observed variaion from he dual response approach and serves as a check and balance. Finally, no mehod is foolproof. None of he hree mehods works when he major source of variaion has no ye been idenified as an inpu and hus is no included in he sudy and his inpu does no vary much during he period he sudy is performed. An example migh be a seasonal variable no considered. 8

9 Table 9: Comparison of Three Approaches o Robus Design Dual Response Taguchi Tolerance Analysis Mehod of Esimaing Variaion Direc Observaion Uses noise array o simulae variaion of inpus Predics variaion using equaion for average Requires sudy condiions o be represenaive of manufacuring Works bes when he variaion presen during he sudy is imized Works bes when he variaion presen during he sudy is imized Procedural Requiremens Requires muliple observaions per rial Requires numerous fine adjusmens o be made Requires esimaes of variaion of inpus o design sudy Requires esimaes of variaion of inpus o analyze sudy Can be run wih observaion per rial Can be run wih one observaion per cell Cos of Running Experimen Cos are increased due o exra precauions o insure sudy condiions are represenaive of manufacuring Coss are increased due o need o make numerous fine adjusmens Lowes cos approach Under esimaes variaion if sudy condiions are no represenaive of manufacuring. Under esimaes variaion if imporan sources of variaion are no included in he sudy Under esimaes variaion if imporan sources of variaion are no included in he sudy Esimaed Variaion Leas precise mehod Low precision bu beer han dual response approach Including all sources of variaion and possible adjusing for measuremen error resuls in accurae predicions of he variaion High precision Targe Minimizing Variaion If sudy condiions are no represenaive of manufacuring, will opimize for sudy condiions and may fail in manufacuring Leas precise mehod. However his lack of precision can be parially overcome by aking larger number of samples per rial. Sample sizes of 5- are more appropriae If an imporan source of variaion is no included, will opimize only for sources of variaion included in he sudy and may fail in manufacuring Low precision bu beer han he dual response approach If an imporan source of variaion is no included, will opimize only for sources of variaion included in he sudy and may fail in manufacuring High precision. Sandard deviaion /5 ha of he Taguchi approach and /3 ha of he dual response approach Oher Only mehod ha can achieve robusness agains unidenified source of variaion Only mehod ha can explore alernae olerances for inpus wihou requiring addiional daa 9

10 Mos debaes abou Taguchi Mehods cener around he dual response approach versus Taguchi. While care should be aken in concluding oo much from hese limied examples, he resuls presened indicae ha a hird lesser known approach called he olerance analysis approach is in many if no mos cases he bes approach o use. Even beer resuls can be obained by combining he dual response and olerance analysis approaches. Finally, all hree of hese approaches are only appropriae when he equaion relaing he inpus and he oupu are unknown. If he equaion is known, none of hese approaches should be used. Insead exac soluions can be obained. REFERENCES Bisgaard, Soren and Ankenman, Bruce (996), "Analyic Parameer Design", Qualiy Engineering, Vol. 8, No., pp Cox, Neil D. (986), How o Perform Saisical Tolerance Analysis, ASQC, Milwaukee, WI. Evans, David H. (975), "Saisical Tolerancing: The Sae of he Ar," Journal of Qualiy Technology, Vol. 7, No., pp. -. Lawson, John S. and Madrigal, J. L. (994), "Robus Design Through Opimizaion Techniques", Qualiy Engineering, Vol. 6, No. 4, pp Taguchi, Genichi (986), Inroducion o Qualiy Engineering - Designing Qualiy ino Producs and Processes, Asian Produciviy Organizaion, Tokyo. Taylor, Wayne A. (99), Opimizaion and Variaion Reducion in Qualiy, Taylor Enerprises Inc, Liberyville, IL. Taylor, Wayne A. (996), VarTran, Taylor Enerprises Inc, Liberyville, IL.