Optimization of Multiple Responses in the Taguchi Method Using Desirability Function and Fuzzy Regression

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1 Optimization o Multiple Responses in the Taguhi Method Using Desirability Funtion Fuzzy Regression Abbas Al-Reaie, Ibrahim Rawabdeh, Issam Jalham, Nour Bata, Reema Abu-Alha Abstrat This researh proposes an approah or optimizing uzzy multiple responses using desirability untion uzzy regression. Eah response repetition is transormed into signal to noise ratio then modeled using statistial multiple regression. A trapezoidal uzzy regression model is ormulated or eah response utilizing the statistial regression oeiients. The most desirable response alues the deiation untion are determined or eah response. Finally, our optimization models are ormulated or the trapezoidal membership uzzy numbers to obtain optimal ator leel at eah number. A a ase study is employed or illustration. In onlusion, the proposed approah suessully dealt with inherent ariability uzziness in multiple responses. This shall be aluable to proess produt engineers or optimizing uzzy multiple responses in manuaturing appliations on the Taguhi method. Inde Terms Desirability untion, Fuzzy regression, Optimization, Taguhi method I. INTRODUCTION HE Taguhi [] method utilizes a rational atorial Tdesigns, or the so-alled orthogonal arrays (OAs), to redue the number o eperiments under permissie reliability. This method has been widely aepted or obtaining robust designs in many business appliations. Neertheless, most o the published researhes on the appliations o the Taguhi method hae been onduted to optimize a single uality response o a proess or produt [, ]. Reently, seeral optimization approahes hae been proposed or the optimization o multiple responses [4-8]. Statistial regression has many appliations [9-0], in many manuaturing proesses the behaior o proesses is usually ague the obsered data is irregular, hene the statistial regression models hae an unnaturally wide possibility range. A uzzy regression approah in modeling manuaturing proesses, whih has a high degree o uzziness, possesses the distint adantage o being able to Manusript reeied June, 0; Aepted No., 0. This work was supported by the Uniersity o Jordan, Amman. A. Al-Reaie is with the Department o Industrial Engineering, Uniersity o Jordan, Amman, 94, Jordan ( abbas.alreai@u.edu.o). I. Rawabdeh is with the Department o Industrial Engineering, Uniersity o Jordan, Amman, 94, Jordan. Issam Jalham is with the Department o Industrial Engineering, Uniersity o Jordan, Amman, Jordan. alham@u.edu.o N. Bata has reeied her master degree in Industrial Engineering. R. Abu-Alha is with the Department o Industrial Engineering, Uniersity o Jordan, Amman, 94, Jordan. generate models using only a small number o eperimental data sets. The uzzy regression analysis uses uzzy numbers epressed as interals with membership alues as the regression oeiients. Fuzzy linear regression approahes hae been suessully applied in many business appliations [-]. In at, many manuaturing proesses tend to be ery omple in behaior hae inherent system uzziness; suh as, lutuation o proess pressure temperature due to enironmental eets. This researh, thereore, aims at optimizing multiple responses in the manuaturing appliation on the Taguhi method using uzzy regression analysis. II. THE PROPOSED OPTIMIZATION APPROACH The proposed approah or optimizing multiple responses in the Taguhi method is outlined in the ollowing steps: Step : Typially, the uality response, y, is diided into three main types; inoling the smaller-the-better (STB), the nominal-the-best (NTB), the larger-the-better (LTB) response types. The Taguhi's OA onduts n eperiments to inestigate ators onurrently. Let denotes the number o responses o main onern that are measured in eah eperiment. To inlude all uzziness in the obserations alues or eah response, alulate the S/N ratio, ir, at eperiment i or eah repetition o response, using an appropriate euation rom the ollowing ormulas: ir 0 log ( y 0 ir ) or STB, 0 log ( s / ) or NTB,,, 0 ir y i r ir 0 log (/ y ) or LTB, 0 ir where y i s i are the estimated aerage stard deiation o y ir repliates at the ith eperiment. Determine the optimal ator setting or rth repetition o response using S/N ratio. Let lr denotes the aerage o i alues at leel l o ator or the rth repetition. Calulate the lr alues or all ator leels. Identiy the ombination o optimal ator leels or eah repetition as the leels that maimize the lr or this ator. () IMECS 0

2 Step : Let r denotes the S/N ratio or rth repetition o response. Obtain the multiple linear regressions r or all ator leel ombinations using the alues o ir. That is, r 0r r r gr g g <, r,,..., k () where,, g are the independent ator ariables, the oeiients,, g, are risp alues, is rom error obsered in the response alue. Then, determine the best-it models or desribing the untional relationship between the S/N ratio or response proess ators. The uzzy regression epressed as 0 g g g< (), A trapezoidal uzzy number B as shown in Fig. an be deined as (l,b,,u), where l, b,, u are trapezoidal limits. The ( ) B is then deined as: l, l b b l, b ( ) B (4) u, u u 0, l, u proposed data. For instane a large means that the epert has a poor opinion about their repetitieness. Step : Let ( ) be th response alue by substituting the optimal uzzy ator leels o th response. Calulate the ( ) or all alues. Proide the "most desirable" response alues or all response types using 0, min min d ( ), min ma min, ma Let U L ma (6) denote the upper lower limit or the desirability untions, respetiely. Calulate as ollows U U d ( ( X )),,..., (7) L Min d ( ),..., d ( ( X )),,..., (8) Step 4: Let D ( ) denotes the deiation untion to be minimized then alulate the D ( ) using (9). u ( ) ( ) D ( ),,..., ( ) Then, alulate the ollowings L (9) P D (0) ( ) Q Ma D ( ),..., D ( ),,..., () Step : Formulate the inal model as ollows Fig.. Trapezoidal membership untion. Let l b u (,,, ) are trapezoidal uzzy oeiients. l b u Obtain,, as ollows: m Aerage(,..., k) l m s () s u m s b m s m where s is stard deiation o (,..., k ), is a positie onstant 0, hosen by an epert depending on eperiene about the repetitieness o the M a d ( ),..., d ( ),,..., M in D ( ),..., D ( ),,..., [Fator Leels] () Euation () is onerted to single obetie by introduing two untions S ( ) T ( ). Let l b u S ( ) ( S ( ), S ( ), S ( ), S ( )) () l b u T ( ) ( T ( ), T ( ), T ( ), T ( )) (4) IMECS 0

3 where S ( ) T ( ) indiate the degrees o satisation rom desirability robustness, respetiely. Then, estimate S ( ) T ( ) as ollows: 0, d ( ) L d ( ) L S ( ), L d ( ) U U L, d ( ) U, D ( ) P Q D ( ) T ( ), P D ( ) Q Q P 0, D ( ) Q () (6) Conseuently, the obetie is to maimize S ( ) T ( ). That is, M a S ( ),..., M a T ( ),..., (7) [Fator Leels] Zimmerman Ma-Min operator will be applied to onert the two-obetie model to a single obetie, whih maimizes the minimum degree o satisation [4]. Let Min S S (8) Min T T (9) Then, the inal model is ormulated as: M a S M a T Subet to: d L S,then d S( U L ) L, U L,..., Q D T,then D T ( Q P ) Q, Q P,..., [Fator Leels] (0) Finally, let w w indiate the important weights or desirability robustness epressed by user based on ost, uality loss, warranty. The inal model with only one obetie is transormed to Ma w S w T d S ( U L ) L,,..., D ( ) T( Q P ) Q,,..., ww 0S 0T X [Fator Leels] () Sole the models l, b,, u to determine the alues o ator uzzy leels. III. ILLUSTRATIVE CASE STUDY Chen et. al [] inestigated sputtering proess o GZO ilms using the grey-taguhi method. Fie proess ators were studied inluding: R.F. power,, sputtering pressure,, deposition time,, substrate temperature, 4, postannealing temperature,. The deposition rate (DR, y, LTB), eletrial resistiely (ER, y, STB), optial transmittane (OT, y, LTB) were the main responses. Let,, denote the S/N ratio or DR, ER, ir i r i r OT responses at eperiment i ( i =,, 8) with r (r =, ) repetitions, respetiely. The ir, i r, ir are alulated or eah repetition using the appropriate ormula in (). The obtained ir, i r, ir are then summarized or both repetitions in Table I. The lr alues in eah response repetition are alulated or all ator leels or the DR, ER, OT responses. Table II displays the lr alues or DR, the ombination o optimal ator leels or the two repetitions are identiied as () () () 4 (). In a () similar manner, the ombination o optimal ator leels or the irst seond repetitions o ER are ound respetiely as () () () 4 () () () () () 4 (). () Finally, the ombination o optimal ator leels is () () () 4 () or both OT repetitions. It is () notied that there is a onlit among the ombinations that optimize the three responses onurrently. Moreoer, there are two distint ombinations o optimal ator leels or ER repetitions. This shows the uzziness eet on ER response. The multiple linear regression euations or are respetiely written as: Whereas, the multiple linear regression euations or are respetiely represented by: IMECS 0

4 ; R-S (ad) = 9.4% ; R-S (ad) = 9.% 4 Finally, the multiple linear regression euations or are respetiely epressed as: ; R-S (ad) = 84.9% ; R-S (ad) = 84% 4 4 Using (), the l b u (,,, ) alues are obtained. The trapezoidal uzzy regression, are then ormulated or the three responses as ollows: (0.8, 0., 0.40, 0.44) + ( , , , 0.080) + (0.709, 0.77, 0.897, 0.96) + (0.0006, , 0.00, 0.00) + (0.0000, , 0.00, 0.00) 4 + (0.0007, , , ) (-8.04, -7.88, -7.7, -7.4) + (0.047, 0.0, 0.06, 0.06) + (-0.06, 0.07, 0.8, 0.478) + (0.004, 0.00, 0.008, 0.009) + (0.04, 0.07, 0.009, 0.07) 4 + (0.0, 0.0, 0.0, 0.0) (9.999, 9.00, 9.09, 9.4) + (-0.009, , -0.00, -0.00) + (-0.060, , ,-0.046) + ( , , , ) + (0.00, 0.000, , ) 4 + (0.000, , , ) From Tables III to V, the alues o the ombinations o optimal uzzy ator leels or DR are gien alulated as = (00, 00, 00, 00), = (0.67, 0.67, 0.67, 0.67), = (60, 60, 60, 60), 4 = (00, 00, 00, 00), = (00, 00, 00, 00). The orresponding alues o are (6.4, 6.706, 7.09, 7.76). Similarly, or ER the alues o the ombinations o optimal uzzy ator leels are = (00, 00, 00, 00), = (0.67, 0.67, 0.67, 0.67), = (.79, 64.9, 8.6, 96.), 4 = (00, 00, 00, 00), = (00, 00, 00, 00). The orresponding alues o are (-.80, -., ,-0.4). Finally, the alues o the ombinations o optimal uzzy ator leels or OT are = (0, 0, 0, 0), = (0., 0., 0., 0.), = (0, 0, 0, 0), 4 = (00, 00, 00, 00), = (00, 00, 00, 00). The orresponding alues o are (9.86, 9.496, 9.679, 9.769). Assuming that the aeptable ranges o min ma or DR alues are eual to (,, ) (0, 0, 0), respetiely. Similarly, the alues o min ma or ER are (0, 0, 0) (-4, -4, -4), respetiely. Finally, the alues o min ma or OT are determined as (8, 8, 8) (40, 40, 40), respetiely. The desirability untion or eah response is alulated using (6) (7) then the ollowing alues are determined: U (0.8080,0.870,0.8,0.8460), L (0.64,0.44,0.600,0.684) U (0.88,0.90,0.9649,0.99), L (0.00,0.60,0.69,0.849) U (0.69,0.748,0.840,0.88), L (0.490,0.60,0.00,0.67) Net, the uzzy deiation untions or DR, ER, OT are epressed respetiely as: D D D Using (0) (), the ound respetiely eual to P P (0.768,0.768,0.768,0.768), Q (0.844,0.886,0.97,0.40) P (.044,.066,.0408,.049), Q (.044,.066,.0408,.049) P (0.08,0.08,0.08,0.08), Q (0.00,0.0,0.0,0.04) Q alues are alulated By applying Zimmerman Ma-Min operator, the inal model is ategorized to our models l, b,, u. Table III displays the obtained results. For illustration, the models b are ormulated as ollows. The alues o DR, ER, OT (db) at the optimal uzzy ators leels gien in Table VI are alulated ound eual to (.84,.6,.7,.94), (.9,.9,.9, 4.), (86.48, 86., 87.67, 87.80), respetiely. It is ound that the trapezoidal membership untion inreases the leibility o the uzzy models by proiding ranges o optimal solution. There are IMECS 0

5 wide ranges between the lower maimal DR ER alues, whereas a tight range is noted or OT. Eluding the uzziness will result in misleading response alues i soled by the traditional regression tehniue. Thus, by onsidering eah response repetition separately, the inherent ariability in the olleted data is minimized. Model b b b Ma 0.S 0. T b b S b 4 b S S T T T 0.0 ww 0 S, T b X,,,, [Fator Leels] 4 Ma 0.S 0.T Model S S S T T T 0.04 w w 0 S, T 4 4 X,,,, [Fator Leels] TABLE I The S/N ratios or the repetitions o sputtering eperiment DR ER OT Ep. i i i i i i i IMECS 0

6 TABLE II THE OPTIMAL FACTOR LEVELS FOR THE TWO REPETITIONS OF DR Fator Repliate r = Repliate r = ( ) Leel Leel Leel Leel Leel Leel TABLE III THE OPTIMAL FACTOR LEVELS FOR THE SPUTTERING PROCESS Model l Model b Model Model u IV. CONCLUSIONS This researh proposed an optimization approah using desirability untion uzzy regression to deal with uzzy multiple responses. A ase study rom preious literature are employed or illustration. It is ound that the proposed approah () suessully deals with inherent ariability uzziness in multiple responses by onsidering trapezoidal membership or eah response, () proides ranges or optimal solution in ontrast with traditional optimization tehniues, () deals with response repetition rather than aerage alue o response repetitions hene proides reliable results, () onsiders uzzy proess ator leels rather than risp settings, hene allows leibility in hanging ator leels that may be aeted during operation by unontrollable ators. In onlusion, the proposed approah shall proide great assistane to proess engineers in minimizing deiation obtain desirable response alues in manuaturing appliations on the Taguhi method. REFERENCES [] G. Taguhi, Taguhi Methods, Researh Deelopment Dearborn, MI: Amerian Suppliers Institute Press, ol., 99. [] M. H. Li, A. Al-Reaie, C. Y. Yang, DMAIC approah to improe the apability o SMT solder printing proess, IEEE Transations on Eletronis Pakaging Manuaturing, ol., no., pp. 6-, 008. [] A. Al-Reaie, M. H. Li, R. Fouad, Applying DMAIC proedure to improe the perormane o LGP printing proess ompanies, Adanes in Prodution Engineering Management, ol. 6, no., pp. 4-6, 0. [4] A. Al-Reaie, Optimizing SMT perormane using omparisons o eiieny between dierent systems tehniue in DEA, IEEE Transations on Eletronis Pakaging Manuaturing, ol., no. 4, pp. 6-64, 009. [] A. Al-Reaie, T. H. Wu, M. H. Li, DEA approahes or soling the multi-response problem in Taguhi method, Artiiial Intelligent or Engineering Design, Analysis Manuaturing, ol., pp. 9-7, 009. [6] A. Al-Reaie, Super-eiieny DEA approah or optimizing multiple uality harateristis in parameter design, International Journal o Artiiial Lie Researh, ol., no., pp. 8-7, 00. [7] A. Al-Reaie, D. M. Al-Tahat, Soling the multi-response problem in Taguhi method by beneolent ormulation in DEA, Journal o Intelligent Manuaturing, ol., pp. 0-4, 0. [8] A. Al-Reaie, M. H. Li, K. C. Tai, Optimizing SUS 04 wire drawing proess by proess by grey analysis utilizing Taguhi method, Journal o Uniersity o Siene Tehnology Beiing, Mineral, Metallurgy Material, ol., no. 6, pp. 74-7, 009. [9] Y. T. Shu, D. J. Ming, T. W. Jen, S. W. Chun, A robust design in hardaing using a plasma transer ar, International Journal o Adaned Manuaturing Tehnology, ol. 7, pp , 006. [0] B. M. Gopalsamy, B. Mondal, S. Ghosh, Taguhi method ANOVA: An approah or proess parameters optimization o hard mahining while mahining hardened steel, Journal o Sientii Industrial Researh, ol. 68, pp , 009. [] H. Tanaka, S. Ueima, K. Asai, Linear Regression Analysis with Fuzzy Model, IEEE Transations on Systems, Man, Cybernetis SMC, ol., pp , 98. [] C. Wen, C. Lee, Deelopment o a ost untion or wastewater treatment systems with uzzy regression, Fuzzy Set System, ol. 06, pp 4, 999. [] S. B. Oh, W. Kim, J. K. Lee, An approah to ausal modeling in uzzy enironment its appliation, Fuzzy Set System, ol., pp. 4, 990. [4] H. J. Zimmermann, Fuzzy Sets, Deision Making Epert Systems. Kluwer, Boston, 987. [] C. C. Chen, C. C. Tsao, Y. C. Lin, C. Y. Hsu, Optimization o the sputtering proess parameters o GZO ilms using the Grey Taguhi method, Ceramis International, ol. 6, pp , 009. IMECS 0