Process Optimization by Soft Computing and Its Application to a Wire Bonding Problem

Transcription

1 Internatonal Journal of Appled Scence and Engneerng , : 59-7 Process Optmzaton by Soft Computng and Its Applcaton to a Wre Bondng Problem Ch-Bn Cheng Department of Industral Engneerng and Management, Chaoyang Unversty of Technology, Wufeng, Tachung country 43, Tawan, ROC Abstract: Modelng and optmzaton of a process wth multple outputs s dscussed n ths paper A neuro-fuzzy system named MANFIS, whch comprses a fuzzy nference structure and neural network learnng ablty, s used to model a multple output process Optmzaton of such a process s formulated as a multple objectve decson makng problem, and a genetc algorthm and a numercal method are ntroduced, respectvely, to solve ths problem based on the MAN- FIS model We have used these two algorthms, respectvely, to solve a chemcal process optmzaton problem, and compared ther performances A combnaton of these two algorthms s also suggested to mprove performances of both algorthms The proposed approach s also appled to a wre-bondng problem n semconductor manufacturng Keywords: process optmzaton; soft computng; neuro-fuzzy system; genetc algorthm, multple objectve decson makng; wre bondng Introducton Process optmzaton s to fnd a settng of the controllable varables, or called nput varables, so as to obtan the best outcomes of a process The outcomes of a process are often referred to as the outputs or responses of a system In ths study, optmzaton of a process wth multple outputs s consdered For example, n a tool lfe problem, we attempt to determne the cuttng speed and depth of cut so as to obtan a maxmal lfe of the tool (a prmary response) and retan a satsfed rate of metal removed (a secondary response) at the same tme Response optmzaton methods are popular tools for process optmzaton Usually, these methods nclude two stages In the frst stage, we use regresson analyss to model a system's responses; that s, we dentfy the relatonshp between responses and nput varables through regresson functons In the second stage, we use optmzaton technques to obtan a settng of system parameters that gve system the most desrable responses Tradtonally, n the frst stage, lnear regresson wth the regressors n a frst-order or second-order polynomal form s used to approxmate the response surfaces However, frequently encountered n practce, systems are complcated and hghly nonlnear, and thus, lnear regresson s not sutable In recent years, nonparametrc regresson approaches, such as neural networks and fuzzy nference systems, are wdely adopted for modelng nonlnear systems These nonparametrc approaches learn the relatons between nput varables and responses drectly from Correspondng e-mal: cbcheng@malcyutedutw Accepted for Publcaton: Dec 23, Chaoyang Unversty of Technology, ISSN Int J Appl Sc Eng, , 59

2 Ch-Bn Cheng the observatons wthout assumng any pre-specfed functonal form The combnaton of fuzzy nference systems and neural networks, together wth genetc algorthms, create a new research area called soft computng Soft computng emerges as a computng approach that tres to mmc human's ablty of reasonng and learnng n an uncertan envronment One representatve technque of soft computng s neuro-fuzzy systems A neuro-fuzzy system s a fuzzy nference system presented n a network structure, and equpped wth neural network learnng abltes In ths study, we use a neuro-fuzzy system, named multple adaptve neuro-fuzzy nference system (MANFIS) [7], to model a system that has multple responses; and furthermore, we also use a genetc algorthm (GA) [5] to optmze ths system s responses based on the model of MANFIS A multple response system has m response y, y 2,, y m, whch are affected by a set of nput varables x = (x, x 2,, x p ) T Tradtonally, the relatons between responses and nput varables are defned through functons: y = f ( x ) + ε,, 2,, m, () = where f s the functonal relaton between x and the -th response y, and ε are d random errors wth zero means and constant varances σ 2, The objectve of the multple response optmzaton s to fnd a soluton x such that each response wll attan a compromsed optmum Many approaches have been proposed to solve the multple response optmzaton problem Derrnger and Such [4] transform each response functon nto a desrablty functon, and then maxmze the geometrc mean of the ndvdual desrablty functons to obtan a compromsed soluton Khur and Conlon [8] presented a procedure based on a dstance functon that calculates the overall closeness where the response functons acheve ther respectve optmum at the same set of condtons; a compromsed soluton s then found by mnmzng ths dstance functon over the expermental regon Pgnatello [3], Ames et al [] and Vnng [6] all propose to mnmze a measure based on a multvarate loss functon, whch evaluates the loss when responses devate from ther targets For the specal case of two responses, Myers and Carter [] ntroduced a dual response approach, whch optmzes the prmary response subject to an approprate constrant on the secondary response The dsadvantage of ther approach s that such an optmzaton scheme can be msleadng due to the unrealstc restrcton of forcng the constraned response to a specfc value [0] To remedy the dsadvantage of the approach of Myers and Carter [], Km and Ln [9] formulate the dual response problem as a multple objectve decson makng (MODM) programmng and ntroduce a fuzzy optmzaton methodology, whch s based on Zmmermann's maxmn approach [7] Ther approach optmzes the prmary response and the secondary response, smultaneously, by maxmzng a compromsed satsfacton degree of both responses The degrees of satsfacton of both the mean response and devaton are defned by membershp functons orgnated n fuzzy set theory Though the prevous approaches vared n ther soluton procedures, they commonly assumed lnear response surfaces In ths study, to deal wth nonlnear responses, the system s modeled by MANFIS, and the multple response optmzaton problem s formulated as an MODM However, snce we use the nonparametrc regresson tool MANFIS to model responses, exact functonal forms of responses are not known and hence dervatve-based optmzaton methods cannot be drectly appled to obtan the optmal soluton Therefore, we wll use a genetc algorthm as well as a numercal method to search optmal solutons on the response surfaces Performances of these two algorthms wll be compared The remander of ths paper s organzed as follows In the next secton, the archtecture 60 Int J Appl Sc Eng, ,

3 Process Optmzaton by Soft Computng and Its Applcaton to a Wre Bondng Problem of MANFIS and ts learnng process are summarzed Secton 3 formulates the multple response optmzaton problem as an MODM Two optmzaton algorthms, a genetc algorthm and a numercal method, are presented n Secton 4 to solve the MODM For llustraton, Secton 5 uses the two algorthms, respectvely, to solve a chemcal process optmzaton problem To mprove the performances of both algorthms, we also suggest a combnaton of these two algorthms Computatonal results show ths combned algorthm s promsng Concludng remarks are gven n the last secton 2 Multple adaptve neuro-fuzzy nference system MANFIS s an extenson of the sn gle-output neuro-fuzzy system ANFIS [6], for producng multple outputs A neuro-fuzzy system can serve as a nonparametrc regresson tool, whch model the regresson relatonshp non-parametrcally wthout reference to any pre-specfed functonal form MAN- FIS can be vewed as an aggregaton of many ndependent ANFIS The archtecture of MANFIS s depcted n Fgure x x 2 x p ANFIS ANFIS 2 Fgure Archtecture of MANFI S Every sngle ANFIS n an MANFIS smulates the functonal relatons f, =,, m, n Equaton () ANFIS can be consdered as a network presentaton of a TSK fuzzy nference system [5], and the f-then rules n TSK are comprsed n the network structure To llustrate the archtecture of ANFIS, an example wth a two-dmensonal nput s vsualzed n Fgure 2 ANFIS m y y 2 y m Layer Layer 2 Layer 3 Layer 4 Layer 5 A w N z w x B w 2 N w 2 z 2 (to every node n Layer 4) ŷ x 2 A 2 w 3 N w 3 z 3 B 2 w4 N w 4 z 4 Fgure 2 Archtecture of ANFIS Int J Appl Sc Eng, , 6

4 Ch-Bn Cheng To reflect dfferent adaptve capabltes, the nodes n ANFIS are represented by crcles or squares, n whch, square nodes represent adaptve nodes and crcle nodes represent fxed nodes Adaptve nodes contan parameters that can be adjusted by learnng, whle the fxed nodes do not contan adjustable parameters In ths study, the adaptve nodes n layer of the ANFIS are parameterzed by Gaussan functons wth ther means and devatons Nodes n layer 2 are fxed nodes labeled Π, whch s a fuzzy conjuncton operator Functons of nodes n ths layer are to synthesze the nformaton from the frst layer The operator Π s defned as a multplcaton of all of ts ncomng sgnals, and output the frng strength w j, j=,,4 Nodes n layer 3 labeled by N smply performs a normalzaton of sgnals from layer 2 and output the normalzed frng strength w = w / w 4, j j r= j=,,4 The adaptve nodes n layer 4 of the ANFIS contan lnear functons of the nput varables wth ther coeffcents as the adjustable parameters; that s, z j = a j x +b j x 2 +c j, j=,,4 The sngle node n layer 5 s a fxed node, whch computes the overall output as the summaton of all ncomng sgnals: y ˆ = 4 w j z j j= Assumng that we have conducted an experment wth n runs on an m-response system, n observatons are collected wth the format of (x k, y k,,y k,,y mk ), k=,,n, where x k s the nput condton at the k-th run and y k s the -th response at the k-th run Wth these observatons, MANFIS can approxmate the multple responses y, =,,m, by mnmzng an error measure E defned as E = n m k = = ( y k k ) 2, (2) where ŷ k s the estmate of the -th response for the k-th run The mnmzaton of E s carred out n an teratve manner, whch r s referred to as a learnng process The learnng process of MANFIS termnates when the error measure E reduces to a satsfactory level Snce E s a summaton of the squared errors from m ndependent ANFIS, the learnng of MANFIS can be treated as the learnng of m ndependent ANFIS Furthermore, snce ANFIS s a mult-layered-feed-forward network, backpropagaton learnng algorthms used n neural networks can be drectly appled to ts learnng The detals of ths learnng process can be found n [6] 3 MODM formulaton of multple response optmzaton By means of the learnng process, MAN- FIS obtans an estmaton of desred outputs wth gven nputs Let ŷ, =,,m, be the -th output of MANFIS, and they are estmates of multple responses y,,y m, respectvely To ndcate these estmates are functons of the nput varables x, they wll be denoted as ŷ (x), =,,m Snce the system under dscusson has multple responses, the optmzaton of the system n fact nvolves the optmzaton of several ndvdual responses at the same tme For all the system responses, they can be dvded nto three sets: ) the-larger-the-better, denoted by L; 2) the-smaller-the-better, denoted by S; and 3) the-nomnal-the-best, denoted by N We have formulated ths optmzaton problem as a multple objectve decson makng problem wth the followng form [2]: max l, l L mn s, s S mn t Tt, t N st x B, (3) where T t s the nomnal target of the t-th response; and B s a feasble regon of x To solve the above multple objectve op- 62 Int J Appl Sc Eng, ,

5 Process Optmzaton by Soft Computng and Its Applcaton to a Wre Bondng Problem tmzaton problem, we follow the dea of Zmmermann's maxmn approach [7] Accordng to the maxmn approach, the soluton of (3) can be obtaned by maxmzng an overall satsfactory degree among all ndvdual objectves n (3) That s, for each objectve, t has ts own satsfactory degree, and the overall satsfacton s an ntersecton of all ndvdual satsfactory degrees, where the ntersecton s defned through a mn operator The satsfactory degree for each objectve s evaluated by an user-defned membershp functon µ ˆ ( ˆ y y ) Let λ be the overall satsfactory degree, and then we can convert the orgnal MODM (3) to: max λ st µ ( ˆ ) λ, =,, m, y ˆ y x B, λ [ 0,] (4) Each response's membershp functon µ ˆ ( y ) should be well chosen so as to reflect ts characterstc For the response belonged to the set of the-larger-the-better, ts degree of satsfacton reaches when t s at max { ˆ = x B y } and then decreases monotoncally to 0 at mn { ˆ = x B y } A typcal membershp functon for y ˆ, L, could be stated as, f > ˆ y, ˆ y µ ( ˆ =, f ˆ ˆ ( ) ˆ y ) y, y x y L ˆ y (5) 0, f ( ) < ˆ x y, The above membershp functon s graphcally shown n Fgure 3 For the response belonged to the set of the-smaller-the-better, we set the satsfactory degree to when a response s at ŷ and then t decreases monotoncally to 0 at above membershp functon s depcted n Fgure 4 µ(ŷ ) Fgure 3 Membershp functon of µ ˆ ( y ) : ''the larger the better'' case y ˆ Such type of membershp functons can be expressed as, ˆ y µ ˆ ( y ) =, ˆ y 0, µ(ŷ ) 0 ŷ ŷ ŷ - f <, f, S, f > 0 ŷ ŷ ŷ - Fgure 4 Membershp functon of µ ˆ ( y ) : ''the smaller the better'' case Smlarly, for the response of the set of the-nomnal-the-best, the degree of satsfacton s maxmzed when t s at ts target T, and decreases as t s away from T Membershp functons of ths type can be defned as (6) Int J Appl Sc Eng, , 63

6 Ch-Bn Cheng T ˆ y, T ˆ y ( x) T ( ( x)) =, T 0, f < T, f T <, N, µ (7) elsewhere Membershp functon of (7) s depcted n Fgure 5 µ(ŷ ) 0 ŷ - T Fgure 5 Membershp functon of µ ˆ ( y ) : ''the nomnal the best'' case The problem formulaton of (4) cannot be drectly solved by the use of dervatve-based methods due to unknown functonal forms of y ˆ Dervatve-free methods such as genetc algorthm and smulated annealng are deally suted for solvng problems where dervatve nformaton s unavalable Alternatvely, we can approxmate the dervatves wth numercal methods In ths study, we wll apply GA and a numercal method, respectvely, to solve (4) 4 Soluton procedures In the prevous secton, we have formulated the multple response optmzaton problem as an MODM In ths secton, we suggest usng two dfferent algorthms, a genetc algorthm and a numercal method, to solve ths MODM ŷ ŷ 4 Genetc algorthm Genetc algorthm frst proposed by Holland [5] s a dervatve-free stochastc optmzaton approach based on the concepts of bologcal evolutonary processes GA encodes each pont n a soluton space nto a bnary bt strng called a chromosome Operatons of chromosomes ncludng selecton, crossover, and mutaton, are used to generate new chromosomes so as to explore the soluton space Each chromosome s evaluated by a ftness functon Such a ftness functon corresponds to the objectve functon of the orgnal problem A great varety of genetc algorthms have been proposed n the lterature In ths study, we wll just use a basc form of GA Nevertheless, t performs well as observed n our computaton results Ths genetc algorthm contans a roulette wheel selecton, a sngle pont crossover, and a random flppng mutaton The ftness of chromosomes s determned va (4) It s not straghtforward to determne the values of λ for a certan soluton by usng (4) Therefore, we rewrte (4) as max λ st λ = mn = x B,, m { µ ( ˆ ) }, y ˆ y (8) By employng the traned MANFIS, the formulaton of (8) s presented n a network form n Fgure 6, and λ can be drectly read from the output end of ths network Genetc algorthm for solvng the multple response optmzaton problem has been formulated n our earler paper [3] In ths paper, we further nvestgate ts performance wth computatonal experments n Secton 5 42 Numercal method A numercal method based on Lagrange relaxaton for solvng the multple response optmzaton problem has been formulated n 64 Int J Appl Sc Eng, ,

7 Process Optmzaton by Soft Computng and Its Applcaton to a Wre Bondng Problem [2] Though ths earler method has the ANFIS µ(ŷ ) x ANFIS 2 µ(ŷ 2 ) x 2 mn λ x p ANFIS m µ(ŷ m ) Fgure 6 Network presentaton of formulaton (8) advantage of provdng an upper bound of the optmal soluton, t s rather complcated In ths paper, a smpler numercal method, whch drectly solves the prmal problem, s formulated as the follows Recallng the formulaton of (8), to ndcate λ beng a functon of x we denote t as λ(x) The gradent of λ(x) s defned as λ( x) x λ = : (9) λ( x) x p By fxng the values of all x k j, and by gvng a small ncrement x j on x j, the partal dervatve can be approxmated through λ(x) x j λ ( x ) λ j + j p λ j p (0) x j ( x,, x x,, x ) x j ( x,, x,, x ) After the gradent s determned, the maxmzaton of λ(x) can be done by an tera- tve manner through the updatng of x, subject to the feasble regon constrant The rule of ths updatng s x new = x old + s λ(x old ), () where s s a step sze The steps of ths numercal method are summarzed below Step 0 Intalzaton: set the teraton counter r = 0, the accuracy requrement τ, and the step sze s; arbtrarly choose ntal value x 0 wthn the feasble regon Step Gradent calculaton: calculate the gradent of λ through Eq (0) wth x r Step 2 Updatng of x: x r+ = x r + s λ(x r ) Step 3 If λ( x r+ ) - λ(x r ) τ, stop; otherwse, go to Step 4 Step 4 Increase the teraton counter r r + Go to Step Int J Appl Sc Eng, , 65

8 Ch-Bn Cheng 5 Computatonal comparson To llustrate our approach, a chemcal process optmzaton problem taken from Myers and Montgomery [2] s reproduced as the follows A chemcal process has three controllable varables: reacton tme (x ), temperature (x 2 ), and percent catalyst (x 3 ); and ts responses are percent converson (y ), and thermal actvty (y 2 ) For ths process, t s mportant to maxmze y whle y 2 s held between 55 and 60 wth a nomnal target 575 Experments are conducted wth dfferent setups of reacton tme, temperature, and percent catalyst to collect data of ths chemcal process 5 Modelng by MANFIS MANFIS s employed to model the response surfaces of the above chemcal process The MANFIS for ths problem has two output nodes correspondng to the two responses of ths process, and hence ths MANFIS conssts of two ndependent ANFIS The Fuzzy Toolbox n MATLAB software provdes functons of constructng, edtng and tranng of ANFIS We use ths software to construct two ndependent ANFIS networks and tran them by the expermental data The convergence behavor of one of the learnng process s shown n Fgure 7 Fgure 7 Convergence of the learnng of y 2 After completng the tranng of MANFIS, the multple response problem s solved by usng the formulaton of (8) Snce the response y belongs to the set of the-larger-the better, ts membershp functon should take the form of (5); and the response y 2 has a nomnal target, so t wll take the membershp functon (7) In order to determne these membershp functons, the maxmum and thermal actvty to be held between 55 and 60, therefore, t s reasonable to set 55 and 60 as mnmum for ndvdual response must be obtaned Maxmum and mnmum of responses can be obtaned by formulatng sngle objectve programmng problems for ndvdual responses, and solvng the problems wth any dervatve-free algorthm Alternatvely, they can also be subjectvely determned accordng to users' judgment or ther expectaton In our example, t s desred that the response of the mnmum and maxmum of ths response, respectvely Smlarly, the mnmum and ma- 66 Int J Appl Sc Eng, ,

9 Process Optmzaton by Soft Computng and Its Applcaton to a Wre Bondng Problem xmum of the response of percent converson are set as 50 and 00, respectvely The possble ranges for x, x 2 and x 3 are set as [-2, 2] 52 Solvng by GA The genetc algorthm s mplemented on the MATLAB platform and run on an IBM compatble PC wth Pentum III-800 CPU Ten trals are conducted, wth the parameters n GA settng as: populaton sze = 24, cross-over rate = 07, and mutaton rate = 02 The results are lsted n Table, n whch, the second column s the tme (n seconds) consumed by each tral to obtan the best soluton, and the last two columns are the responses yelded by the best solutons From Table we can see that all the ten trals produce hgh qualty solutons, e all trals except the thrd tral obtan optmal solutons Nevertheless, they usually consume a lot of computaton tmes Table Results of chemcal process optmzaton by GA Tral Tme λ Responses y y Solvng by the numercal method The numercal method s also mplemented on the MATLAB platform and run on the same machne Ten trals are also carred out and ther results are shown n Table 2 The startng ponts n these trals are arbtrarly chosen We found that the qualty of solutons obtaned by ths numercal method cannot compete wth those obtaned by GA, and the startng ponts crtcally affect the results In number method should reach the optmal soluton very fase On the other hand, though partcular, four (Tral 3, 6, 8 and 9) out of ten trals are faled because of the startng ponts are fallng n a flat area of the response surface and hence no gradent can be found 54 Combnng GA and numercal method Though the numercal method faled to produce hgh qualty soluton, t can fast solve the problem If we can provde a startng pont n the vcnty of the optmal soluton, the GA usually takes a long tme to fnd a hgh qualty soluton, n the frst few generatons Int J Appl Sc Eng, , 67

10 Ch-Bn Cheng Table 2 Results of chemcal process optmzaton by the numercal method Tral Tme λ Responses y y Table 3 Results of chemcal process optmzaton by the combned algorthm Tral Alg Tme λ Responses y y 2 GA NM GA NM GA NM GA NM GA NM GA NM GA NM GA NM : no mprovement 68 Int J Appl Sc Eng, ,

11 Process Optmzaton by Soft Computng and Its Applcaton to a Wre Bondng Problem The dea s to combne these two algorthms together That s, use GA to fnd a startng pont for the numercal method We have conducted 8 trals to justfy ths dea In each tral, GA s run frst for 20 generatons to obtan a soluton, and ths soluton wll serve as a startng pont for the numercal method Computatonal results are shown n Table 3, n whch, the second column s the algorthm used and NM denotes the numercal method In Tral, 3 and 4, the numercal method provdes sgnfcant mprovement of the startng solutons, and among whch, Tral and 4 produce hgh qualty solutons Though Tral 2, 5, 6, and 7 do not sgnfcantly mprove ther startng solutons, they all produce hgh qualty solutons It s observed that the startng solutons of these trals are already n good shapes; and ths may be why t s dffcult for the NM to mprove much on these startng solutons The fnal tral found no mprovement for ts startng soluton, possbly caused by the startng soluton fallng on a plateau of the response surface Though the results n Table 3 show mperfecton of the combned algorthm, we stll consder the combnaton of GA and numercal method s promsng for two reasons: ) ths combned algorthm consumes much less tme than GA to fnd a satsfactory soluton and, 2) t s possble to fnd a hgh qualty soluton n a moderate number of trals 6 Applcaton to a wre bondng problem Wre bondng s a weldng process, n whch wre and pad surface are brought nto ntmate contact by usng thn wre and a combnaton of heat, pressure and ultrasonc energy Dynamc random access memory (DRAM) chps and most commodty chps n plastc packages are assembled by wre bondng About 2-4 trllon wre nterconnectons are produced annually Wre bondng falures nclude bond off center, bond not stckng on de, wre breakng and so on In a producton envronment, wre pull strength s usually montored to mnmze process drft To acheve a stable performance of the wre bondng process, the operatng varables such as bondng parameters need to be strctly regulated Crtcal bondng parameters nclude bondng force, bondng tme, and ultrasonc power To fnd optmal setups of bondng parameters, tradtonally, a serous of bondng tests s performed by varyng bondng parameters to draw out the optmal bondng condtons Evaluaton of wre pull strength s used to defne the optmalty of bondng parameters In the evaluaton, three sets of curves of wre pull strength versus ultrasonc power, bondng tme, or bondng force can be obtaned by varyng one of these parameters whle holdng the other two constant at ther optmum In such an approach, the search s less effcent and frequently falls n a local optmum especally when the response surface s hghly nonlnear To demonstrate the potental usage of our approach n real-world problems, the proposed approach n Secton 4 s appled to the optmzaton of a wre bondng process n an IC packagng company n Tawan By fxng parameters of cut mode, heat, and loopng of the wre bonder, and varyng the parameters of bondng force, bondng tme, and ultrasonc power, a desgn of experment s conducted to collect data of the process Each of the three varable parameters s set wth three levels, and hence the experment results n a combnaton of 27 trals of bondng tests Each tral contans 00 replcatons, and the two concernng responses are average wre pull and ts devaton, where average wre pull s a the-larger-the-better response, and devaton s a the-smaller-the-better response Frstly, MANFIS s employed to model the responses of ths process To ensure the generalty of the MANFIS model, and to fully utlze the lmted number of expermental data, an extreme cross-valdaton technque called leave one-out cross-valdaton [4] s used The dea of cross-valdaton s to dvde the sample data nto a constructon Int J Appl Sc Eng, , 69

12 Ch-Bn Cheng sub-sample, whch forms the tranng data set, and a valdaton sub-sample, whch forms the test data set The leave one-out cross-valdaton s to dvde the sample sze n nto a tranng data set contanng n- observatons, and leave the rest sngle observaton as the test datum Such a technque consders the dvson of the observatons n all n possble ways The cross- valdaton crteron s defned as n m CV (P) = ( yk k [ O \ ]) 2, (2) nm = k = where P s the set of crtcal factors that affect the accuracy of MANFIS, and k [ O \ ] s an estmate of y k and t s obtaned from an MANFIS that s traned by the sample data excludng the -th datum The set of crtcal factors P contans only one factor, the number of nodes (n layer ) assocated wth an nput varable To fnd the best setup of P, CV(2), CV(3), and CV(4) s compared, and we found that CV(3) s the mnmum Wth the result of cross-valdaton, the MANFIS for modelng the wre bondng process s constructed as: two ndependent ANFIS, and n each ANFIS there are 3 nodes assocated wth each nput and hence resultng n 9 nodes n layer, 27 nodes n layer 2, 27 nodes n layer 3, and 27 nodes n layer 4 The optmzaton of the wre bondng process s modeled by the formulaton of (3), wth a prmary objectve of maxmzng the wre pull and a secondary objectve of mnmzng process varaton To construct membershp functons for these two objectves we need to know ther respect mnmum and maxmum as defned n Secton 3 The mnmum of wre pull s set as ts specfcaton (e ts mnmal requrement), and the maxmum of wre pull s determned accordng to past experence of runnng test on bondng The mnmum and maxmum of the process varaton are determned through a process performance ndex (Ppk) The defnton of Ppk for one-sded specfcaton (lower lmt only) s Ppk = µ p LSL, (3) 3σ p where µ p s the mean of the process, σ p s the devaton of the process, and LSL s the lower lmt of the process Snce the company s pursung sx-sgma process capablty, we use ths performance goal to determne the expected mnmum of the process varaton; that s, we set Ppk = 2 and nduce the mnmum of σ p as (µ p - LSL)/6 Furthermore, a company usually needs to reach a process capacty hgher than four-sgma to satsfy most customers, and hence we can determne the maxmum for δ p as (µ p - LSL)/399 By employng the genetc algorthm to solve the wre bondng optmzaton problem based on the formulaton of (8), we obtan a soluton that s better than the company s current software can fnd 7 Concludng remarks Ths study used a neuro-fuzzy network, MANFIS, to model a multple response system, and optmzes the system by a genetc algorthm and a numercal method respectvely MANFIS provdes the advantage of modelng a nonlnear and complcated system wthout the need of fndng sutable functonal forms for the system, and ts neural network learnng ablty also equps MANFIS wth hgh effcency n system modelng A chemcal process optmzaton problem s used to llustrate our approach From the computatonal results, t s found that GA always fnds process condtons that yeld very satsfed responses, though t consumes much computatonal tme On the other hand, the numercal method s fast but ts soluton qualty cannot compete wth GA's To mprove performances of these two algorthms, we combne GA and the numercal method by runnng GA frst for few generatons to obtan a 70 Int J Appl Sc Eng, ,

13 Process Optmzaton by Soft Computng and Its Applcaton to a Wre Bondng Problem startng soluton for the numercal method Computatonal results show that ths combned algorthm s promsng The proposed approach of process optmzaton s appled to a wre-bondng problem n IC manufacturng References: [ ] Ames, A E, Mattucc, N S, Mac- Donald, G Szony, and Hawkns, D M 997 Qualty loss functons for optmzaton across multple response surfaces Journal of Qualty Technology, 29: [ 2] Cheng, C B 2000 Mult-response optmzaton based on a neuro-fuzzy system Neural Network World, 0: [ 3] Cheng, C B, Cheng, C J, and Lee, E S 2002 Neuro-fuzzy and genetc algorthm n multple response optmzaton Computers and Mathematcs wth Applcatons, 44: [ 4] Derrnger, G and Such, R 980 Smultaneous optmzaton of several response varables Journal of Qualty Technology, 2: [ 5] Holland, J H 975 Adaptaton n natural and artfcal systems Unversty of Mchgan Press, Mchgan [ 6] Jang, J S R 993 ANFIS: adaptve-network-based fuzzy nference system IEEE Transactons on Systems, Man and Cybernetcs, 23: [ 7] Jang, J S R, Sun, C T, and Mzutan, E 997 Neuro-Fuzzy and Soft Computng: a Computatonal Approach to Learnng and Machne Intellgence Prentce-Hall New Jersey [ 8] Khur, A I and Conlon, M 98 Smultaneous optmzaton of multple responses represented by polynomal regresson functons Technometrcs, 23: [ 9] Km, K J and Ln, D 998 Dual response surface optmzaton: a fuzzy modelng approach Journal of Qualty Technology, 30: -0 [0] Ln, D and Tu, W 995 Dual response surface optmzaton Journal of Qualty Technology, 27: [] Myers, R H and Carter, W H 973 Response surface technques for dual response systems Technometrcs, 5: [2] Myers, R H and Montgomery, D C 995 Response Surface Methodology John Wley and Sons, Inc New York [3] Pgnatello, J J Jr 993 Strateges for robust multresponse qualty engneerng IIE Transactons, 25: 5-5 [4] Stone, M 974 Cross-valdatory choce and assessment of statstcal predctons Journal of the Royal Statstcal Socety Seres B, 36: -47 [5] Takag, T and Sugeno, M 985 Fuzzy dentfcaton of systems and ts applcaton to modelng and control IEEE Transactons on Systems, Man, and Cybernetcs,5: 6-32 [6] Vnng, G G A 998 Compromse approach to multresponse optmzaton Journal of Qualty Technology, 30: [7] Zmmermann, H J 978 Fuzzy programmng and lnear programmng wth several objectve functons Fuzzy Sets and Systems, : Int J Appl Sc Eng, , 7