Proceedng of the World Congre on Engneerng 00 Vol II WCE 00, July -, 00, London, U.K. Alpha Rk of Taguch Method wth L Array for NTB Type QCH by Smulaton A. Al-Refae and M.H. L Abtract Taguch method a wdely ued approach for parameter degn to acheve qualty and yeld mprovement for many bune applcaton. Neverthele, there ha been much dcuon n lterature about the nvaldty of the tattcal technque adopted n th method. Th reearch propoe an extenon to ongong reearch by nvetgatng the alpha rk of Taguch method wth L ( 3 7 ) array for the nomnal-the-bet (NTB) type qualty charactertc (QCH) ung mulaton. It aumed that all QCH value are normally dtrbuted wth the ame mean and tandard devaton. Then the null hypothe, that all factor hould be dentfed a ngnfcant, true. Smulaton reult however, howed that the alpha rk very hgh and hence Taguch method may provde a mleadng parameter degn. Th reearch, therefore, recommend relyng on more effcent alternatve. Index Term Alpha rk, Nomnal-the-bet, Smulaton, Taguch method. I. INTRODUCTION Taguch [] conder three tage n product or proce development: ytem degn, parameter degn, and tolerance degn. In ytem degn, the engneer ue centfc and engneerng prncple to determne the bac confguraton. In the parameter degn tage, the pecfc value for the ytem parameter are determned. Fnally, tolerance degn ued to determne the bet tolerance for parameter. In mot lterature revew, the parameter degn, or o-called Taguch method [], receved the mot attenton. Parameter degn an off-lne producton technque for reducng varaton and mprovng qualty by ung the product array. In parameter degn, Taguch focue on determnng the effect of the control factor on the robutne of the product functon. Intead of aumng that the varance of the repone reman contant, t captalze on the change n varance and look for opportunte to reduce the varance by changng the level of the control factor. In Taguch method, orthogonal array (OA) are employed to optmze the amount of nformaton obtaned from a lmted number of experment. The gnal-to-noe (S/N) rato then ued a a qualty meaure to decde optmal factor level. Analy of varance (ANOVA) for S/N rato Manucrpt receved March 6, 00. Th work wa upported by the Department of Indutral Engneerng and Sytem Management n Feng Cha Unverty. A. Al-Refae currently purung the Ph.D. degree n the Dept. of Indutral Engneerng and Sytem Management n Fen Cha Unverty, Tawan, R.O.C. (Correpondng author e-mal: eng_jo_000@ yahoo.com). M.H. L a profeor n the Dept. of Indutral Engneerng and Sytem Management n Fen Cha Unverty, Tawan, R.O.C. follow to determne gnfcant factor effect. In ANOVA, Taguch obtan an approxmate etmate of error varance by poolng-up technque [3]. Then, he adopt F value of four to decde gnfcant factor effect. Accordng to Taguch, the applcaton of the above procedure provde a robut degn. Taguch method ha been adopted for parameter degn n many bune applcaton [-5]. Neverthele, the tattcal technque of Taguch method have been the ubject of debate and much dcuon n dfferent platform. For example, Leon et al. [6] ntroduced the concept of performance meaure ndependent of adjutment a a replacement for S/N rato. Box [7] ued amplng experment wth random number to llutrate the ba produced by poolng. Tu [] mentoned that Taguch analy approach of modellng the S/N rato lead to non-optmal factor ettng due to unneceary baed effect etmate. Ben-Gal [9] uggeted the ue of data compreon meaure combned wth S/N rato to ae noe factor effect. Falure to elect the bet condton for proce or product parameter a cotly mtake n today hghly compettve market. L and Al-Refae [0] nvetgated the alpha rk of Taguch method, or the probablty of dentfyng ngnfcant factor a gnfcant, wth L 6 array for the larger-the-better type qualty charactertc (QCH) ung mulaton. The L 6 array contan 5 two-level factor. Occaonally, there nteret n ung an OA that ha ome factor at two level and ome factor at three level. The mot-wdely ued mxed-level OA the L ( 3 7 ) array []. To extend the above reearch for another QCH type, the reearch nvetgate the alpha rk of Taguch method wth L ( 3 7 ) array for the nomnal-the-bet (NTB) type QCH ung mulaton. Further, Dabade et al. [] employed Taguch method ung QCH value ntead of S/N rato. Furthermore, Sun et al. [3] teted factor gnfcance at 5 % gnfcance level ntead of four. In thee regard, the alpha rk of Taguch method wll be alo nvetgated at 5 % gnfcance level and for QCH. The remander of th paper organzed a follow. Secton II outlne reearch methodology. Secton III provde analy and dcuon of alpha rk. Secton IV ummarze concluon. II. METHODOLOGY It aumed that QCH, x, normally dtrbuted wth mean and tandard devaton of μ and σ, repectvely. Let y be a tandardzed random varable of x calculated a (x-μ)/σ; or y ~ NID(0, ). The L ( 3 7 ) array hown n Table. Th array ha row (experment) and nne column, ncludng a hdden column I whch contan A B nteracton. Column A ha two level, wherea column B to I are agned each at three level. All y value wll be generated from NID(0, ). Conequently, the null hypothe, H o, that all the nne factor are ngnfcant, true. The alternatve ISBN:97-9-70-3-7 WCE 00
Proceedng of the World Congre on Engneerng 00 Vol II WCE 00, July -, 00, London, U.K. hypothe, H, wll be that at leat one factor dentfed a gnfcant. Typcally, the alpha rk calculated a the probablty of rejectng H o gven that H o true. Let k repreent the number of pooled-up column nto error term and α k denote alpha rk. The L ( 3 7 ) ha 7 total degree of freedom; one degree of freedom for column A, wherea two degree of freedom aocated wth each of the eght three-level column. If each column agned to a factor, no degree of freedom wll be left for error term. In order to tet factor gnfcance, the um of quare for the bottom fve column; or about half the degree of freedom of L ( 3 7 ) array a uggeted by Taguch, can be pooled-up to obtan an approxmate etmate of error term. Conequently, at mot fve column of L ( 3 7 ) array wll be pooled-up a error term a llutrated n Table. For example, when one column pooled-up nto error term;.e., k equal one, the error um of quare (SS E ) obtaned a follow: a. If the mallet um of quare (SS) correpond to column A, the SS E equal to the SS A. Then, one degree of freedom aocated wth error term (df e ). The MS E equal to SS E. The mean quare (MS) contrbuted by each of three-level factor obtaned from SS dvded by two. b. If the mallet SS correpond to a three-level factor, the SS E equal to the mallet SS, wherea df e equal to two. The SS E obtaned when two to fve column are pooled-up n a mlar manner. The methodology adopted to etmate the alpha rk of Taguch method outlned n the followng tep: Step : Start the frt mulaton cycle by generatng two replcate, y and y, from NID(0, ) for each tandardzed QCH, y, n each row ; =,...,. Step : Let y be the average of y and y value and denote the varance. Calculate the S/N rato, η, ung for a factor greater than four, that factor dentfed a gnfcant. Otherwe, t dentfed a ngnfcant. Perform mulaton for everal cycle each of large enough run to enure that p very mall relatve to α k. Etmate the p (, l) value then calculate α ung Eq. (). () α = p (, l ) l = The probablty of dentfyng correctly all the (9-k) factor a ngnfcant, p (, 0), equal to (-α ). Hence, the p of α equal to the p of p (, 0). Step : By mlar mulaton, repeat tep to 3 to etmate the α k for k equal two to fve. Generally, when k column are pooled-up, the SS E calculated a the um of the k mallet SS, whle df e um of the degree of freedom aocated wth the k pooled-up column. Obtan MS E a SS E by df e. Calculate the F rato aocated wth each of the (9-k) remanng factor a MS dvded by MS E. Then, tet factor gnfcance at four. Etmate the p (k, l) value by mlar mulaton for k equal two to fve. Fnally, calculate α k ung Eq. (5). (9 k ) α = p( k, l) k =,..., 5 (5) k l= Step 5: Repeat tep to by mlar mulaton to etmate the alpha rk at 5 % gnfcance level ntead of four, a llutrated n Table. For example, when one column n pooled-up nto error term, factor gnfcance teted at 5 % gnfcance level a follow: a. If column A pooled-up, the F rato for each of the eght remanng three-level factor compared wth F 0.05,, of 99.50. b. If a three-level column pooled-up, then the F rato for the column A compared wth F 0.05,, of.5, wherea the F rato for each of the eght remanng three-level factor compared wth F 0.05,, of 9.00. η log = 0 y =,..., () Step 6: Repeat the above procedure by mlar mulaton whle y ued ntead of S/N rato n tep. where y calculated a y = (/ ) yr r= and gven by ( - ) = r r= y y =,..., () =,..., (3) Step 3: Let l repreent the number of factor dentfed a gnfcant and p (k, l) denote the probablty of dentfyng l factor a gnfcant when k column are pooled-up. Let p (k, l) repreent the average of p (k, l) value, whle p the tandard devaton for everal mulaton cycle. Conduct ANOVA by calculatng the SS contrbuted by each factor. Then, pool-up one column nto error term a hown n Table. Obtan the F rato for each remanng factor a MS dvded by MS E, and then compare t wth four. If the F rato Table. The orthogonal array L ( 3 7 ). III. ANALYSIS AND DISCUSSION Smulaton conducted for ten cycle each of 0000 run. The alpha rk then etmated for S/N rato and y at both F crtera for all k value. A. The Alpha Rk at Four Ung S/N Rato Th part correpond to tep to. S/N rato ued a a qualty meaure. Then, ANOVA for S/N rato conducted at four. The alpha value at four are etmated for one to fve pooled-up column by mulaton. Table 3 dplay the p (k, l) and α k at four for all k value.. ISBN:97-9-70-3-7 WCE 00
Proceedng of the World Congre on Engneerng 00 Vol II WCE 00, July -, 00, London, U.K. Exp. Column * Standardzed QCH () A B C D E F G H Replcate y y, y y y, y y 3 3 3 3 3 3 3 y 3, y 3 y 3 3 3 y, y y 5 3 3 y 5, y 5 y 5 6 3 3 y 6, y 6 y 6 7 3 3 3 y 7, y 7 y 7 3 3 3 y, y y 9 3 3 3 y 9, y 9 y 9 0 3 3 y 0,, y 0, y 0 3 3 y,, y, y 3 3 y,, y, y 3 3 3 y 3,, y 3, y 3 3 3 y,, y, y 5 3 3 y 5,, y 5, y 5 6 3 3 3 y 6,, y 6, y 6 7 3 3 3 y 7,, y 7, y 7 S/N rato (η ) η η 3 η 3 η 5 η 5 6 η 6 7 η 7 η 9 η 9 0 η 0 η η 3 η 3 η 5 η 5 6 η 6 7 η 7 3 3 3 y,, y, y η * A B Interacton etmated n a hdden column (I). Table. Illutraton of poolng-up technque and F tet. k value k = k = k = 3 k = k = 5 F value SS E Pooled-up column df e F tet Taguch 5 % gnfcance level Column A Eght remanng 3-level factor F 0.05,, = 99.50 The mallet SS Column A F 0.05,, =.5 Three-level column Seven remanng 3-level factor F 0.05,, = 9.00 3 Seven remanng 3-level factor F one three-level column 0.05,,3 = 9.55 The um of two mallet SS Column A F 0.05,, = 7.7 Two three-level column Sx remanng 3-level factor F 0.05,, = 6.9 The um of three mallet SS The um of four mallet SS The um of fve mallet SS two three-level column Three three-level column 6 three three-level column Four three-level column four three-level column Fve three-level column 0 5 Sx remanng 3-level factor F 0.05,,5 = 5.79 Column A F 0.05,,6 = 5.99 Fve remanng 3-level factor F 0.05,,6 = 5. 7 Fve remanng 3-level factor F 0.05,,7 =.7 Column A F 0.05,, = 5.3 Four remanng 3-level l factor F 0.05,, =.6 9 Four remanng 3-level factor F 0.05,,9 =.6 Column A F 0.05,,0 =.70 Three remanng 3-level l factor F 0.05,,0 =.0 ISBN:97-9-70-3-7 WCE 00
Proceedng of the World Congre on Engneerng 00 Vol II WCE 00, July -, 00, London, U.K. From Table 3, the followng reult are obtaned:. The rato of p relatve to α k very mall and condered neglgble for all k value. Conequently, mulaton for ten cycle each of 0000 run good enough to obtan accurate etmate of the alpha mtake.. The α k very hgh for all k value. Note that the α k lghtly decreae a k value ncreae. Neverthele, the mallet α k (= 0.766), whch correpond to α 5, tll unacceptable. A a reult, Taguch method ung S/N rato at four concluded a rky approach for parameter degn for all k value. 3. Let p (k, l) be the larget p (k, l) for k pooled-up column. In Table 3, the p (k, l) for one pooled-up column correpond to the probablty, p (, ), of dentfyng all the eght remanng factor a gnfcant. Wherea, the p (k, l) for two to fve pooled-up column correpond to dentfyng a gnfcant all the remanng (k, 7-k) factor. Mathematcally, p (k, l) = p (k, 7-k) k =,, 5 (6) In other word, when k column are pooled-up nto error term then factor gnfcance teted at four, Taguch method ung S/N rato tend to mdentfy mot of the remanng factor a gnfcant. Table 3. The p (k, l) and α k at four. Poolng-up k = k = k = 3 k = k = 5 l = 0 0.0579 0.00 0.07 0.06 0.73 l = 0.033 0.06 0.093 0.6359 0.67 l = 0.037 0.0750 0.6 0. 0.7766 l = 3 0.059 0.70 0.967 0.3500 0.95 l = 0.076 0.56 0. 0.969 0.099 l = 5 0.57 0.06 0.979 0.007 l = 6 0.507 0.95 0.059 l = 7 0.07 0.00 l = 0.397 α k 0.9 0.9797 0.9576 0.936 0.766 p 0.00309 0.00395 0.0050 0.007 0.0036 p/α k 00 % 0.3 0.0 0.53 0.9 0. B. The Alpha Rk at 5 % Sgnfcance Level Ung S/N Rato In th part, ANOVA conducted at 5 % gnfcance level ntead of four. In tep 5, the alpha rk etmated by mlar mulaton for all k value. The reult are dplayed n Table, where t noted that:. The α k very hgh for all k value. Note that the mallet α k (= 0.575), whch correpond to α, becaue the F 0.05,,, F 0.05,,, and F 0.05,, value n Table are much larger than four. A a reult, the probablty of dentfyng correctly a ngnfcant ncreae, and hence the α k decreae. Depte that, the α tll unacceptable. A a reult, Taguch method at 5 % gnfcance level tll provde a mleadng parameter degn for all k value.. Obervng the p (k, l) value, t noted that when one and two column are pooled-up, the p (k, l) correpond to the probablty, p (k, 0), of dentfyng correctly a ngnfcant all the (9-k) remanng factor. However, when three to fve column are pooled-up, the p (k, l) correpond to dentfyng a gnfcant all the remanng (k, 6-k) factor, or p (k, l) = p (k, 6-k) k =,, 5 (7) Compare the above reult wth alpha rk at four, t noted that Taguch method tend to dentfy a gnfcant le number of factor at 5 % gnfcance level. 3. Comparng the α k at the ame k value, t clear that the α k at 5 % gnfcance level maller than the α k at four for all k value. The reaon that all the value of 5 % gnfcance level n Table are larger than four. Table. The p (k, l) and α k at 5 % gnfcance level. Poolng-up k = k = k = 3 k = k = 5 l = 0 0.56 0.75 0.30 0.59 0. l = 0.36 0.7 0.690 0.07 0.77 l = 0.063 0.3 0.35 0.99 0.576 l = 3 0.073 0.03 0.0 0.07 0.775 l = 0.065 0.3350 0.603 0.9 0.073 l = 5 0.0609 0.506 0.50 0.063 l = 6 0.0577 0.0906 0.055 l = 7 0.05609 0.0570 l = 0.0673 α k 0.575 0.77 0.660 0.706 0.7906 p 0.003 0.005 0.0037 0.006 0.00 p/α k 00 % 0.37 0. 0. 0.3 0.5 C. The Alpha Rk for A tandardzed QCH Step 6 conducted ung a tandardzed QCH ntead of S/N rato n tep. The p (k, l) and α k value are etmated at both F crtera by mlar mulaton and hown n Table 5. It noted that the α k very hgh at both F crtera for all k value. Comparng the p (k, l) and α k value between S/N rato and a tandardzed QCH at the ame F and k value, t obvou that the p (k, l) and α k are almot the ame for both qualty meaure for all k value. The man concluon made that Taguch method ung a tandardzed QCH tll rky for parameter degn at both F crtera for all k value. Accordngly, the ue of S/N rato unneceary complcate the data analy n parameter degn. To verfy the robutne of alpha rk to ncreang the number of replcate for a tandardzed QCH, four replcate are generated from NID(0, ) for each row. S/N rato then calculated ung Eq. (). ANOVA for S/N rato then conducted at both F crtera for all k value. The p (k, l) and α k value are etmated at both F crtera by mlar mulaton for all k value and dplayed n Table 6. Clearly, at the ame F and k value, the α k wth four QCH replcate almot the ame a the α k wth two replcate lted n Table 3 and. Conequently, the alpha rk concluded nentve to ncreang the number of QCH replcate. ISBN:97-9-70-3-7 WCE 00
Proceedng of the World Congre on Engneerng 00 Vol II WCE 00, July -, 00, London, U.K. Table 5. p (k, l) and α k value at both F crtera ung a tandardzed QCH. four 5 % gnfcance level k = k = k = 3 k = k = 5 k = k = k = 3 k = k = 5 l = 0 0.00973 0.09 0.06 0.0595 0.6757 0.39930 0.369 0.0935 0.57 0.5 l = 0.0933 0.035 0.075 0.999 0.60 0.507 0.57 0.559 0.0 0.773 l = 0.035 0.065 0.365 0.75 0.7 0.093 0.355 0.96 0.373 0.5733 l = 3 0.056 0.073 0.6 0.0 0.07 0.07 0.3576 0.9330 0.965 0.73 l = 0.07 0.556 0.333 0. 0.09376 0.0695 0.7 0.730 0.553 0.079 l = 5 0.07 0.77 0.79 0.7 0.0639 0.66 0. 0.0696 l = 6 0.537 0.0 0.36 0.066 0.006 0.069 l = 7 0.53 0.9773 0.0663 0.06 l = 0.3366 0.0653 α k 0.9907 0.9 0.9737 0.905 0.33 0.60070 0.653 0.9065 0.73 0.7 Table 6. The alpha rk at both F crtera ung S/N rato wth four QCH replcate. four 5 % gnfcance level k = k = k = 3 k = k = 5 k = k = k = 3 k = k = 5 l = 0 0.030 0.055 0.0373 0.0779 0.65 0.356 0.6700 0.60 0.537 0.055 l = 0.03 0.09 0.06 0.590 0.66 0.090 0.703 0.6650 0.0979 0.7766 l = 0.0355 0.0733 0.39 0.99 0.7069 0.050 0.3 0.0 0.9 0.573 l = 3 0.0536 0.0 0.5 0.353 0.06 0.0690 0.06 0.30 0.05 0.797 l = 0.076 0.5356 0.6 0.939 0.0936 0. 0636 0.3 0.637 0.706 0.075 l = 5 0.076 0.6 0.036 0.07 0.060 0.6 0.06 0.0605 l = 6 0.65 0.950 0.303 0.0590 0.09370 0.05 l = 7 0. 0.3 0.05750 0.0599 l = 0.305 0.0607 α k 0.9660 0.95 0.9657 0.9 0.375 0.565 0.3300 0.7396 0.563 0.795 IV. CONCLUSIONS One may ak doe t matter f ome ngnfcant factor effect are pronounced gnfcant ung the Taguch method?. It ometme argued that for dentfyng the combnaton of bet factor level t of no mportance whether or not a factor effect tattcally gnfcant. However, f we are to ue tattc to catalyze the creatvty of engneer and centt they hould know what factor to reaon about. Tryng to argue why ngnfcant factor effect have an effect wll merely confue and lead a proce/product engneer atray. One nteretng apect of the Taguch method that t ha been qute ucceful depte t hortcomng. Apparently any reaonable ytematc expermentaton, however flawed, may convey mportant nformaton on how to degn a new product or proce and on how to mprove extng product and procee. It our belef that the Taguch trategy ound and hould be ncluded n any qualty mprovement attempt. However, the Taguch method neffcent to carry out h trategy nto practce. Th reearch recommend the ue of mpler and more modern data analytc method for parameter degn. REFERENCES [] G. Taguch, Taguch Method. Reearch and Development. Vol.. Dearborn, MI: Amercan Suppler Inttute Pre, 99. [] N. Belavendram. Qualty by Degn-Taguch Technque for Indutral Expermentaton, Prentce Hall Internatonal, 995. [3] P.J. Ro, Taguch Technque for Qualty Engneerng. McGraw Hll, 996. [] M.S. Phadke, Qualty Engneerng Ung Robut Degn. Englewood Clff, NJ: Prentce-Hall, 99. [5] S. Me, J. Yang, J.M.F. Ferrera, and R. Martn, Optmaton of parameter for aqueou tape-catng of corderte-baed gla ceramc by Taguch method, Materal Scence and Engneerng, vol. A33, 00, pp.. [6] R.V. Leon, A.C. Shoemaker and R.N. Kacker, Performance meaure ndependent of adjutment: An explanaton and extenon of Taguch gnal-to-noe rato, Technometrc, vol. 9(3), 97, pp. 53 65. [7] G. Box, Sgnal-to-noe rato, performance crtera, and tranformaton, Technometrc, vol. 30(), 9, pp. -7. [] K.L. Tu, A crtcal look at Taguch modellng approach for robut degn, Journal of Appled Stattc, vol. 3(), 996, pp. 95. [9] Ben-Gal, On the ue of data compreon meaure to analyze robut degn, IEEE Tranacton on Relablty, vol. 5(3), 005, pp. 3 3. [0] L M.H. and A. Al-Refae, The alpha error of Taguch method wth L 6 array for the LTB repone varable ung mulaton Journal of Stattcal Computaton and Smulaton, to be publhed. [] M.H. L and S.M. Hong, Optmal parameter degn for chp-on-board technology ung the Taguch method, Internatonal Journal of Advanced Manufacturng Technology, vol. 5, 005, pp. 5 53. [] U.A. Dabade, S.S. Joh, and N. Ramakrhnan, Analy of urface roughne and chp cro-ectonal area whle machnng wth elf-propelled round nert mllng cutter, Journal of Materal Proceng Technology, vol. 3, 003, pp. 305 3,. [3] H.W. Sun, J.Q. Lu, D. Chen and P. Gu, Optmzaton and expermentaton of nanomprnt lthography baed on FIB fabrcated tamp, Mcroelectronc Engneerng, vol., 005, pp.75 79. ISBN:97-9-70-3-7 WCE 00