CONCURRENT OPTIMIZATION OF MULTI RESPONCE QUALITY CHARACTERISTICS BASED ON TAGUCHI METHOD. Ümit Terzi*, Kasım Baynal

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1 CONCURRENT OPTIMIZATION OF MUTI RESPONCE QUAITY CHARACTERISTICS BASED ON TAGUCHI METHOD Ümt Terz*, Kasım Baynal *Department of Industral Engneerng, Unversty of Kocael, Vnsan Campus, Kocael, Turkey / 96, umtterz@kou.edu.tr Abstract Taguch method s one of the most frequently used tools for mprovng qualty. However, most publshed applcatons have been concerned wth the optmzaton of a sngle performance characterstc. In ths study, three dfferent methods are used to mprove a producton prosess wth multple performance characterstcs and results are evaluated. Keywords: Taguch Method; Goal Programmng; Fuzzy ogc; Mult-Response Characterstcs. Introducton Qualty s a key factor for sustanable success n global economy. Taguch method (Taguch, 986 and Ross, 988) s one of the most frequently used tools for mprovng qualty. Ths expermental desgn technque acqures data n a plan of experments, usng orthogonal arrays. The orthogonal array produces smaller, less costly experments that have hgh rates of reproducblty (Peace, 99), whch facltates expermental desgn to be used n ndustry. Taguch focused on nformaton of both the mean and the varablty of a qualty characterstc usng the sgnal to nose (SN) rato. Hence, optmal factor/level combnaton obtaned from the Taguch method can be determned to smultaneously reduce the qualty varaton and brng the mean value close to the target value (Tong, Wang, Chen, & Chen, 004) The method has been succesfully used n many areas. However most publshed applcatons have been concerned wth the optmzaton of a sngle performance characterstc. Generally qualty of products or servces are measured wth more than one characterstc. Although mportance and types of characterstcs vary accordng to the problem, all of the performance goals must be acheved as much as possble. There are numerous methods for solvng multple objectve problems. Selecton of the method for handlng mult-response characterstcs s crucal, because of the varance of characterstcs must be mnmzed. Besdes, easness of the method s also mportant that t would desgnate f t wll be used commonly. In ths study three methods; fuzzy logc, addtve methods and goal programmng that we propose are used to solve a mult-response problem and the methods are compared. A cable producton process wth 7 control factors and 3 characterstcs was chosen as the mplementaton area of these methods. Addtve method and fuzzy logc are used to obtan dfferent Multple Indexes (MI) to be able to analyse multple qualty characterstcs together. Goal programmng method whch uses lnear regresson to formulate objectves s presented as an alternatve. Expermental study s used as a tool to present steps of the tecnques and the results are evaluated to compare the effectveness of the approaches.. Expermental Study The alumnum wre producton process steps are as seen n Fg.. Alumnum bllets are heated through four statons (T, T, T 3, T 4 ) and extruded wth a velocty of V. After extruson the wres are exposed to agng at T o C for H hours. There are 3 mportant responses of qualty. These are Strenght, Percentage of Extenson and Conductvty. 35 th Internatonal Conference on Computers and Industral Engneerng 99

2 Bllets T T T 3 T 4 Heated Bllets Agng V Fg.. Wre producton process In ths secton, an expermental study was made to determne the characterstc of the producton process by usng taguch method. Frst of all, approprate orthogonal array must be selected. The degree of freedom for the orthogonal array should be greater than or equal to those for the process parameters. In ths study an 8 orthogonal array was used, snce t has 7 degrees of freedom whch s equal to degree of the process parameters. There are 7 parameters wth levels, as seen n Table. evel shows the ntal condtons. Table. Producton parameters and ther levels Factors T(A) T(B) T3(C) T4(D) Extruson Velocty (E) Unts 0 C 0 C 0 C 0 C m / mn. Agng Temperature (F) Agng Duraton(G) 0 C hours evel evel Interacton between producton parameters s neglected n the present study. Fve replcatons are made for each expermental combnaton n 8 orthogonal array. Sgnal to nose rato s computed accordng to qualty loss. Qualty loss for each response s calculated usng followng three formulas: n n n k n k y k y k Wrappng, for the smaller the better response, (), for larger the better response, () s, for nomnal the best response, (3) y where qualty loss for th response at j th tral y k observed data for th response at j th tral, k th replcatons n replcatons for th response n n y yk, s ( yk y ) n k n k then sgnal to nose s calculated as below: (4),(5) SN 0 log0 In the present study, Percentage of Extenson and Conductvty are larger the better type characterstcs, whle Strenght s a nomnal better type characterstc. Target value for strenght s 33 N/mm. Computed qualty loss and sgnal to nose values for these responses are shown n Table th Internatonal Conference on Computers and Industral Engneerng

3 Table. Computed qualty loss and sgnal to nose values Strenght Conductvty Per. of Ext. Strenght Conductvty Per. of Ext. Exp. No. j j 3j SN j SN j SN 3j 0, , , ,98 34,603 3,773 0, , , ,90 34,540 4, , , , ,964 34,69 3, , , , ,96 34,674 3,53 5 0, , , ,70 34,737 3, , , , ,796 34,60 3,46 7 0, , , ,36 34,6 3,78 8 0,000 0, , ,66 34,536 3, Optmzaton of Characterstcs In ths study three optmzaton methods, whch are used for solvng the mult objectve problem of reachng qualty goals, are dscussed. All of the methods, use the results of the experments that are realzed accordng to an orthogonal array. Addtve method and fuzzy logc method beneft from these data to obtan dfferent Multple Indexes (MI) to be able to analyse multple qualty characterstcs together. For goal programmng method, lnear regresson equatons are acqured from SN ratos and orthogonal arrays. 3.. Addtve Method Addtve method transforms multple responses to sngle response by smply addng normlzed qualty loss values. Addtve method whch was proposed by Tong, Su, Wang (997) and Anthony (00) has the followng phases:. Compute the qualty loss. Determne the mult-reponse sgnal to nose (MRSN) rato: Ths phase has three steps. Step : Normalze the qualty loss of each tral for each response * C, where {,,..., } max * (6) Step : Compute the total normalzed qualty loss (TNQ) of each tral. TNQ j m m w C (7) where w the weght of th normalzed response (,,...,m) In ths case each response has equal mportance and weght. Total normalzed qualty loss s dvded wth m to prevent MRSN values to be negatve. Step 3: Determne the MRSN rato for each tral MRSN j log ( TNQ ) (8) 0 0 j Table 3. MRSN values for the experments Exp. No MRSN j 0,495,046 0,6 0,47 0,975 0,05 0,574,50 35 th Internatonal Conference on Computers and Industral Engneerng 93

4 3. Determne the optmal factor/level combnaton: Regardless of the category of the performance characterstcs, a larger SN corresponds better performance characterstc. The factor effects on MRSN values are plotted and levels wth hghest SN rato for each factor are selected as the optmal levels. Effects of factors on MRSN are shown n fg. for the wre producton process.,00 0,80 0,60 0,40 0,0 0,00 Fg.. Effects of factors on MRSN for the alumnum wre producton. A B C D E F G factor level combnaton s selected by addtve method. 4. Conduct confrmaton experment: Results of the confrmaton experments are n Table Fuzzy ogc Method The theory of fuzzy logcs, ntated by Zadeh (965) has proven to be useful for dealng wth uncertan and vague nformaton. In fact, the defnton of performance characterstcs such as lower-the-better, hgher-thebetter, and nomnal-the-better contans a certan degree of uncertanty and vagueness (n & n, 003). Therefore, fuzzy logc becomes useful for the optmzaton of multple characterstcs. Fuzzy logc method whch was proposed by (n, Wang, Yan, & Tarng, 000) has the followng phases:. Compute SN values A A B B C C D D E E F F G G. Transform SN values to MRPI (Mut-Response Performance Index) usng fuzzy model: A fuzzy model consst of a fuzzfer, a rule base, an nference engne, a defuzzfer, nput and output varables. Rule base defnes the relaton between nput an output wth f then rules. By takng the max mn compostonal operaton (Zmmmerman,985), fuzzy reasonng of these rules yelds a fuzzy output. In our model there are three nput varables: Strenght, Conductvty and Percentage of Extenson. Each nput varable has 3 trangular subsets. Input varables are bounded by mnmum and maxmum values of SN ratos. MRPI s output varable and has 7 trangular subsets that restrcted between 0 and. 7 fuzzy rules are derved based on the fact that; f the SN rato s hgher, than the performance must be better (Table 4). Fuzzy output results are defuzzfed by usng the center of gravty rule. Table 4. Fuzzy rule table MRPI Percentage of Extenson Strenght Conductvty Small Mddle arge Small Mnmum Very small Small Small Mddle Very small Small Mddle arge Small Mddle arge Small Very small Small Mddle Mddle Mddle Small Mddle arge arge Mddle arge Very large Small Small Mddle arge arge Mddle Mddle arge Very large arge arge Very large Maxmum th Internatonal Conference on Computers and Industral Engneerng

5 Table 5. MRPI values for the experments Exp. No MRPI j 0,333 0,446 0,435 0,344 0,609 0,6 0,34 0, Determne the optmal factor/level combnaton: Factor effects on MRPI values are plotted and levels wth hghest PI value for each factor are selected as the optmal levels. 0,470 0,40 0,370 0,30 0,70 A A B B C C D D E E F F G G Fg. 3. Effects of factors on MRPI for the alumnum wre producton. A B C D E F G factor level combnaton s selected as optmal by fuzzy logc method. 4. Conduct confrmaton experment: Results of the confrmaton experments for 5 replcatons are presented n Table Goal Programmng Method near soluton methods are used n dfferent areas. Schmt and Mele (989) used lnear programmng for mult-response desgn problem. Chen (997) transformed SN rato to degree of satsfacton for a lnear model. Goal programmng method has the followng steps.. Compute multple regresson model that consder qualty response as dependent varable and producton parameters as ndependent varables. SN ratos of each characterstcs are used for dependent values and assgned orthogonal array values are used for ndependent varables. In our problem there are only 8 case datas to obtan regresson models that have 7 ndependent varable. To be able to calculate correlaton matrx, one of the dependent varables whch has a beta coeffcent closer to 0 must be omtted. Models calculated for ths case are below. Y 55, ,03594 A + 0,05036 B - 0,47 C + 0,08558 D - 0,477 F -,66905 G (R 0,99) Y 4, ,03340 A + 0,0980 B - 0,5760 C - 0,0970 D + 0,858 E + 0,060 F (R 0,98) Y 3 0, ,00364 A + 0,0470 B + 0,0673 C - 0,03760 D + 0,054 F + 0,6845 G (R 0,99) Then the goal programmng model s constructed as below: Mn { P d - + P d - + P d 3 - } 0,03594 A + 0,05036 B - 0,47 C + 0,08558 D - 0,477 F -,66905 G + d - - d + M 0,03340 A + 0,0980 B - 0,5760 C - 0,0970 D + 0,858 E + 0,060 F + d - - d + M 0,00364 A + 0,0470 B + 0,0673 C - 0,03760 D + 0,054 F + 0,6845 G + d d 3 + M where M s a very large number and varables has upper and lower bounds as n ther factor levels. A B C D E F G factor level combnaton s selected as optmal by goal programmng method. 35 th Internatonal Conference on Computers and Industral Engneerng 933

6 4. Results Results of the confrmaton experment (Table 6) show that all of the methods mproved nal results. But fuzzy logc and goal programmng methods presents better results than the addtve method. Table 6. Intal condtons and confrmaton experment results Factor/evel Mean Standart Error Combnaton Streng. Cond. Exten. Streng. Cond. Exten. Intal Condtons A B C D E F G 30,96 53,74 4,9 0,78 0,89 0,39 Addtve A B C D E F G 3,0 53,98 5,34 0,6 0,46 0,34 Fuzzy ogc, Goal Prog. A B C D E F G 3,36 54,34 5,50 0,3 0,59 0,9 5. Concluson Optmzaton of mult-response performance characterstcs s stll an mportant research area. In ths paper three dfferent methods are compared on a case study. Although, fuzzy logc method recommends the same beter soluton we propose goal programmng method because of easness. Constructng a fuzzy model would be dffcult, f dfferent weghts of characterstcs are necessary. Importance relatons between qualty responses n a real problem would be consdered more effcently by usng goal programmng method. Determnng the mportance weghts s another sgnfcant ssue. In our opnon The Analtc Herarchy Method, ntated by Saaty (983) s useful for ths matter. In the case study, control factors have two levels and nteracton between factors s neglected due to experment number constrants of mplementaton area. Effectveness of the approach can be tested wth multple factor levels consderng nteracton n future researches. References. Antony, J., (00). Smultaneous optmsaton of multple qualty characterstcs n manufacturng processes usng Taguch s loss functon. Internatonal Journal of Advanced Manufacturng Technology,, Chen.H., (997).Desgnng robust products wth multple qualty characterstcs. Computers Operatons Research, Vol.4, No.0, n, J.., n, C.., (003). The use of grey-fuzzy logc for the optmzaton of the manufacturng process. Journal of Materals Processng Technology, 60, n, J.., Wang, K.S., Yan B.H., Tarng Y.S. (000). Optmzaton of the electrcal dscharge machnng process based on the Taguch method wth fuzzy logcs. Journal of Materals Processng Technology, 0, Peace G.S. (99). Taguch Methods. Addson-Wesley Publshng Company. 6. Ross, P.J. (988). Taguch Technques for Qualty Engneerng. McGraw-Hll. 7. Saaty, T.. (983). The Analytc Herarchy Process. McGraw-Hll. 8. Schmt, M.S., Mele,., (989) Taguch Desgns and near Programmng Speed New Formulaton.The Insttute of Management Scences, 9, Taguch, G., (986). Introducton to Qualty Engneerng: Desgnng Qualty nto Products and Processes. Asan Productvty Organzaton, Tokyo. 0. Tong,.I., Su, C.T., Wang, C.H., (997). The optmzaton of mult-response problems n the Taguch method. Internatonal Journal of Qualty &Relablty Management, Vol.4, No.4, Tong,.I., Wang, C.H., Chen C.C., Chen C.-T. (004). Dynamc multple responses by deal soluton analyss. European Journal of Operatonal Research, 56, Zadeh,., (965). Fuzzy Sets. Informaton and Control, 8, Zmmerman, H.J.,(985). Fuzzy Set Theory and ts Applcatons. Kluwer Academc Publshers th Internatonal Conference on Computers and Industral Engneerng