Applying Fuzzy Set Approach into Achieving Quality Improvement for Qualitative Quality Response

Transcription

1 Proceedings of the 007 WSES International Conference on Compter Engineering and pplications, Gold Coast, stralia, Janary 17-19, pplying Fzzy Set pproach into chieving Qality Improvement for Qalitative Qality Response Kn-Lin Hsieh Department of Information Management National Taitng University 684, Sec.1, Chng-Ha Rd., Taitng Taiwan 950, R.O.C. bstract: - Improving qality is essential for manfactring organizations competing in the global marketplace. Generally, two forms of qality response are available: a qantitative response and a qalitative response. Most stdies primarily focs on qantitative qality improvement. Qantitative qality improvement has rarely been reported. he qalitative response is generally represented in the percentage form, or it is classified into several categories. Employing the ordered categorical descriptions (or sbjective estimations) to formlate the performance of the qalitative characteristic is also a meaningfl approach. Sbjective estimation may provide more information for analyzing the problem. However, sbjective estimation cannot be directly defined sing the conventional binary set for the ncertainties involved. Experimental design techniqes and the Tagchi method are two primary approaches sed to improve qality. However, these two approaches are inappropriate when the qality response mst be sbjectively estimated. Hence, a novel approach based on a fzzy set is proposed in this stdy to deal with the qality improvement problem of qalitative qality response. he fzzy set is a well-known approach for dealing with the ncertainties of ordered categorical response. n illstrative example, based on the niformity of an ion implantation process in Taiwan s semicondctor indstry, demonstrates the effectiveness of the proposed approach. Key-Words: - Qalitative qality characteristic, lingistic description, fzzy set, membership fnction, parameter optimization 1 Introdction Fierce market competitiveness has driven manfactrers to enhance their prodct s qality. Off-line qality control is a cost-effective means of optimizing the prodct and process design in spport of on-line qality control. Under the off-line qality control approach, design parameters and noise parameters heavily inflence the responses of a prodct or manfactring process. Designers can control design parameters. However, designers cannot always control noise parameters robst design is desired to obtain the optimm design parameter settings for a prodct or an manfactring process, in sch a manner that the prodct attains its desired target with minimm variation. In most prodcts, the qalitative qality response is freqently considered owing to the inherent natre of the qality response. Conventional experimental design techniqes [4] can be employed to investigate the relationship between the qantitative qality response and the design dimensions (or noise parameters). dditionally, Tagchi s method[5], [7] combines experimental design techniqes with qality loss considerations, making it an efficient approach for off-line qality control. In some cases, the qality response of interest may be a qalitative (or categorical) qality response. However, optimization of a qalitative qality characteristic has seldom been reported[1], [5], [7]. To optimize the qalitative qality characteristic problem, the qalitative response is generally represented sing the percentages, or it is classified into several categories. Discriminant analysis[] can be performed to recognize the relevant factors when analyzing a qalitative response problem. The accmlation analysis ()[7] developed by Tagchi can also be performed to optimize the ordered categorical qality response. In a related work, Nair [6] proposed two scoring schemes (SS) to obtain the optimm factor/level combination by compromising location and dispersion effects of each control factor; Jean and Go [] also proposed a weighted probability scoring schemes (WPSS) to obtain an optimal factor/level combination with respect to the minimm mean sqared deviation (MSD), where MSD is the combination of the location and dispersion effects. The qalitative form can also be described lingistically. Using a lingistic description allows

2 Proceedings of the 007 WSES International Conference on Compter Engineering and pplications, Gold Coast, stralia, Janary 17-19, s to obtain more information when analyzing the problem. However, the conventional binary set cannot directly deal with the sbjective evalation for the ncertainties involved. Hence, the lingistic description limits the application of the qalitative form [8, 9]. Fzzy set is a well-known approach sed to manage the ncertainties of the qalitative type or lingistic description of response [9, 10]. This stdy proposes a novel viewpoint of applying fzzy sets to optimize qalitative qality response. This stdy focses mainly on providing an approach for applying fzzy sets to improve the qality of the ordered categorical response. The proposed fzzy set approach consider the differences among the nearby lingistic descriptions and analyze the qalitative qality response more flexibly. In addition, the fzzy-qality-loss-fnction (FQLF) proposed in this stdy may be viewed as a criterion for determining the optimm factor/level settings. Qality improvement techniqes for qalitative qality response Tagchi[7] developed accmlation analysis () to effectively resolve the qalitative (categorical) response problems. Tagchi s consists primarily of for steps: (1) define the corresponding cmlative categories, () decide the effects of the factor s levels, (3) plot the cmlative probabilities, and (4) predict the accmlated probabilities of each class nder optimm conditions. Tagchi also recommended sing the Omega ( Ω ) transformation to transfer the accmlated probability of the factor level to a corresponding Ω vale, thereby yielding the predicted accmlated probability of the qalitative response. The optimm factor/level combination can be determined by screening the factor effect diagram. However, Tagchi s might lead to an erroneos reslt nder a sbjective assessment while attempting to determine the optimm level combination from the factor effect diagram. Nair[5] presented two scoring schemes (SS) to recognize the dispersion and location effects. His investigation recommended sing the mean sqare to recognize a prominent effect. The optimal condition of dispersion and location effects can be obtained according to the contribtion of both effects of each control factor. The final optimal control factor/level combination is obtained by adjsting between the dispersion effect and location effect. Jean and Go[1] proposed a weighted probability scoring scheme (WPSS) to conter the drawbacks of Nair s SS. Their approach is simpler and more straightforward than Nair s in that they incorporate the dispersion and location effects into a single mean sqared deviation (MSD). In addition, the expected mean sqare deviation for each class can be obtained according to the definition of the categories. The optimal control factor/level combination is obtained by selecting the minimm mean sqared deviation. 3 Fzzy Set Theory Uncertainties freqently occr in sbjectively estimating the qalitative formof a response. The ncertainties can be well described lingistically. Representing ncertainties in binary form, however, may lead to misnderstandings. Zadeh [8, 9] proposed a method based on fzzy sets, to formalize lingistic evalations. ccordingly, employing the fzzy set concept may prevent these misnderstandings. The lingistic evalation of characteristic can be qantified by sing a membership fnction (MF) in the fzzy set. That is, the lingistic description can be defzzied. The MF can transform the lingistic evalation into a vale in interval [0, 1]. lso, the magnitde of the membership vale is the membership degree of a fzzy term with respect to the set elements in a fzzy set. The major differences between a fzzy set and a traditional set are: the traditional set can only take a characteristic fnction with vales of 0 or 1 to describe a set, however, the fzzy set takes an MF with the interval [0, 1] to describe a set. Therefore, the fzzy set can be regarded as an expansion of the traditional set. niversal set U consists of the possible lingistic set element. Initially, the niversal set U μ ~, ~ mst be defined. The MFs of μ B, lying in the interval [0, 1] are then determined for varios fzzy terms: ~ ~, B, with respect to the possible set elements in the niversal set U. To make the fzzy set more sefl, several operators are developed [8, 9]. The relationship between the lingistic terms and the mathematical form can be represented as follows: [ ] ~ ~ μ ~( ) very = =, U (1) [ ] ~ ~ ( 1 μ ~ ( )) not = \ =, U () ~ ~ ~ ~ max[ ~ ( ), ~ ( )] [ or ] B B B = = μ μ (3) ~ ~ ~ ~ min[ ~ ( ), ~ ( )] [ and ] B B B = = μ μ (4)

3 Proceedings of the 007 WSES International Conference on Compter Engineering and pplications, Gold Coast, stralia, Janary 17-19, Where ~ ~, B μ ~, ~, denote the fzzy terms; μ B, denote the membership fnction for the fzzy terms ~ ~, B, ; \ ~ represents the complement set of set ~ ; ~ ~ B and ~ B ~ stands for the intersection and the nion, respectively, of set ~ and set B ~ ; and max[ ~, B ~ ] and min[ ~, B ~ ] are meant to take the largest and the smallest set vale between set ~ and set B ~. The lingistic qality characteristic can be described qite flexibly by employing the above rles. 4 Propose pproach The classifications or categorical types of the qalitative response characteristic mst be initially recognized before applying the fzzy set to optimize the qalitative qality response, that is, the lingistic descriptions of the qalitative response. The definition of the lingistic description or the categories of the qalitative response can be determined sing engineering knowledge and experience. To perform off-line qality improvement, a sitable evalation criterion, sch as Tagchi s signal-to-noise ratio (SN) [6, 7], mst be constrcted. The qality loss is the most widely sed index for evalating qality performance [3]. The concept of the Tagchi s loss fnction is applied to formlate the qality loss of the lingistic data sing the MF. Leon et al. [3] demonstrated that maximizing SN is the same as minimizing the qality loss. In this stdy, we se Leon s concept to develop a qality loss fnction based on the fzzy set. The detailed procedre of the proposed approach is smmarized as follows: Step 1. Define the niversal set U, the set element, and the target target of the qalitative response. The niversal set U is a set consisting of the elements affecting the performance of the qalitative response, that is, U ={0, 1,, }, where = 0, 1, represents the coded vales of the possible elements affecting the qalitative response s performance. The target vale target of the qalitative response can be determined with respect to the ser s reqirement. Step. Determine the MF of the elementary fzzy terms. The elementary fzzy terms can be defined as the terms constrcting other possible fzzy terms. For example, the niformity of the ion implanting can be described as Very Good, Good, Not good and not bad, Bad, Very Bad. The Good and Bad can be chosen as the elementary fzzy terms in this example. ccording to the set element, the MFs of the elementary fzzy terms can be determined. Step 3. Constrct the MF for evalating the performance of each category or each classification of the qalitative response. Each class or each classification of the qalitative response can be represented by the elementary fzzy terms. Thereby, the MF of each category or each classification will then be determined according to the MF of the elementary fzzy terms. Eqations (1)~(4) can be employed to bild the MF for evalating the performance of each class or each classification of the qalitative response. Step 4. Perform the designed experiments and accmlate the experimental data. Step 5. Compte the fzzy-qality-loss-fnction (FQLF) for each experimental rn. The FQLF vale is calclated based on the qality loss of each of the class or the classifications of the qalitative response. The FQLF is obtained by the following eqation: FQLF = [( T arg et) μ() ] U ~ ~ ~ B ~ B μ( ) = r μ ( ) + r μ ( ) + ni ri = ni i, i fzzy term where ni are the cont owing to the ith fzzy term, r ~, ~, ~ rb r C, are the related freqencies of fzzy ~ ~ ~ terms, B, C, in the experimental rn, and T arg et denotes the target vale or the desired vale of the qalitative response. Step 6. Compte the FQLF vale for each factor/level and decide the optimm factor/level settings. The FQLF vale of each factor s level can be compted by taking the average FQLF vale with respect to the experimental rns involving the corresponding factor level. Then, plotting the FQLF vale of each factor s level on the diagram. The optimm factor/level settings can be determined by selecting the settings with the minimm FQLF vale on the response diagram. The following example illstrates the comptation of FQLF. Sppose a designed experiment is given as follows: If the cont of the experimental rn for level-1 of factor is, the FQLF vale of the level-1 for factor can be compted as 7(=(5+9)/). If the cont of the

4 Proceedings of the 007 WSES International Conference on Compter Engineering and pplications, Gold Coast, stralia, Janary 17-19, experimental rn for level- of factor B is also, the FQLF vale of the level- for factor B can be compted as 6(=(9+3)/). No B FQLF 1 Level-1 Level-1 5 Level-1 Level- 9 3 Level- Level Level- Level- 3 Step 7. Perform the confirmed experiments and compte the improvement contribtion ratio (ICR) of the qality loss for the qality response. The ICR of the qality loss is designed as: LOSS Initial setting LOSS Optimal setting ICR = 100% LOSS Optimal setting The larger the ICR vale, the better is the qality improvement. n ICR vale is pre-determined as the reqirement of the lowest acceptance criteria. If the ICR vale can not satisfy the reqirement, go back to Step to re-define the MF and repeat the procedre ntil the ICR vale achieves the reqired vale. 5 Illstrative Example niformity optimization example is illstrated here; the example is taken from the ion implantation process of a Taiwanese manfactrer of integrated circits (IC). The qality response of interest is the degree of niformity of the implanted ion. Sbjective estimation are employed in light of the difficlties of qantifying the niformity of the ion implantation process. The qality response has a qalitative qality characteristic, e.g. the niformity can be referred as [very][good], [good], [not][good][and][not][bad],[bad], [very][bad]. From engineering knowledge, the niformity is freqently inflenced by the defect grade on the sensitivity area following the ion implanting. To simplify the analysis, the performance of the qalitative response (the niformity of the ion implantation on the sensitivity area) is divided into five classes (represented as I ~ V); these classes are listed in Table 1. Thirty-six sensitivity areas are assigned on each wafer in the ion implantation process. The engineer expects that the conts lying in the class I~V are from the largest to the smallest, that is, the ideal reslt of the conts lying in each category is (36, 0, 0, 0, 0). Six control factors (~F) are considered in this process (they can not be clearly described here for reasons of bsiness secrecy). mong these control factors, only one factor (factor ) has two levels; the other factors have three levels. The initial parameter settings are 1B1C3D3E1F. The engineer expects to achieve an improvement in qality of 50% or more, the larger the better. The example is analyzed sing the proposed approach. In Step1, a niversal set U is initially constrcted from engineering experience: U = {0, 1,, 3, 4, 5}; where represents the different defect grade, and = 0 denotes the worst defect grade and = 5 denotes the best defect grade. The best grade of defect represents the best niformity of ion implanting. Hence, the target vale, or the desired vale, of the lingistic description is eqal to 5 from the definition. In Step, we decide the elementary fzzy terms. There are five categories of the qalitative characteristic. The terms of Good and Bad can be sed to bild the other fzzy terms from the qality characteristic, therefore, they are chosen as the elementary fzzy terms. Throgh a brainstorming discssion with process engineers, the MFs of the elementary fzzy terms are determined. ccordingly, the MFs of both elementary fzzy terms ~ = [ Good] = are defined as ~ B = [ Bad ] = and In Step 3, the MFs of the lingistic description of the five categories are then determined according to the elementary fzzy terms ~ and B ~. Table lists the membership vale of each category corresponding to each element in the niversal set U. The detailed comptation procedres for each category are given as follows. μ good ( ) I = [ very][ good] = = μ good ( ) II = [ Good] = = [1 μ good ( )] [1 μ bad ( )] III= [ not ][ goodand ][ ][ not ][ bad] = min{, } = min{, } = μ bad ( ) IV = [ bad] = = μ bad ( ) V= [ verybad ][ ] = =

5 Proceedings of the 007 WSES International Conference on Compter Engineering and pplications, Gold Coast, stralia, Janary 17-19, To ct the experimental time and cost, an L18 orthogonal array (O) is sed. The FQLF of each rn is then compted. Table 3 lists the reslts of FQLF vales. The FQLF vale of the factor/level effect can be obtained, and the reslts are smmarized in Table 4. The response diagram of factor/level effect is given in Figre 1. the accmlated cont and the accmlated probability (in parenthesis) of each category denoted by <> ( e.g. <III> represents the smmation of the probabilities for I, II and III). The accmlated probabilities of each factor s levels are listed in Table 7. From this table 7, the optimm factor/level combination can be obtained as B1C1D3EF1. Figre 1. The response diagram of the factor effect. Examining the response diagram of factor/level effect, the optimm parameter setting is obtained by choosing the factor level with minimm FQLF vale. The optimm setting is B1CD3E1F1. Confirmed experiments are performed to verify the effectiveness of the fond optimm parameter condition. Reslts of the confirmed experiments for both of the initial settings and the optimm settings are listed in Table 5. The average FQLF vale of the initial settings and the optimm settings are fond to be and 7.5, respectively. The improvement contribtion (ICR vale) is abot 5.3% (=[( )/15.19] 100%). This is qite a large improvement. In addition, the engineering reqirement (the improvement contribtion ratio 50%) is achieved so no other improvement activities are needed. This case can be also viewed as an ordered categorical problem in Tagchi s experiment and the Tagchi s can be employed to perform the analysis. ccording to Tagchi s, the accmlated probability of each category mst be compted first. Next, the accmlated probability of varios factor levels will be obtained. Table 6 lists This illstrative example is re-analyzed by employing Jean and Go s WPSS method. ccording to their method, two scores [1] indicating the location and dispersion effects of each experimental rn mst be initially compted. Herein, the dispersion effect is regarded as the discrepancy between the location effect and the target of each category. In addition, the weight vale of each category mst be assigned at the same time, and it is regarded as the location effect in their method. There are five categories in this illstrative example, the location score of each category is given as 5, 4, 3, and 1 for I to V, respectively. In this example, the target of the weight vales can be given as (5, 0, 0, 0, 0) for categories (I, II, III, IV, V), respectively. The formlas for calclating the location score,

6 Proceedings of the 007 WSES International Conference on Compter Engineering and pplications, Gold Coast, stralia, Janary 17-19, dispersion score and the performance measre are given as follows: 5 Wn = wi pi, n = 1,,, 18 The location score: i = 1 5 d n = ( wi pi Targ eti), n = 1,,, 18 The dispersion score: i = 1 The performance measre: E MSD 1 d i ( ) ( + ) 1 3 W i W i The E(MSD) reslts of factor/levels are smmarized in Table 8. From these reslts, the optimm factor/level combination can be determined by selecting the minimm E(MSD) vale. Therefore, the optimm parameter setting is B1CD3EF1. To make the comparisons, we performed the confirmed experiments of the optimm condition fond by Tagchi s and Jean and Go s WPSS. Table 1 lists the reslts of the confirmed experiments employing Tagchi s, Jean and Go s WPSS, and the proposed approach. ccording to Table 1, the optimm settings are different for each of the above three methods, and the ICR vales of Tagchi s, Jean and Go s WPSS, and the proposed approach are approximately 49.5% (=[( )/15.19] 100%), 51.7% (=[( )/15.19] 100%) and 5.3%, respectively. lthogh these improvement ratios obtained sing all three methods are rather close, the proposed approach has the lowest FQLF. In addition, the proposed approach is more flexiable than the other two methods, becase it takes the lingistic description or the sbjective estimation into accont. 6 Conclding Remarks Most stdies of off-line qality control have largely focsed on the optimization of the qantitative qality response. The qalitative characteristic has rarely been reported. The qalitative form can be described by means of lingistic description. Lingistic description can provide more information to analyze the problem. However, the conventional binary set cannot directly deal with sbjective evalation for the ncertainties involved, and therefore, the conventional experimental design techniqes and the Tagchi method cannot be directly applied. Fzzy set is a well-known approach to managing the ncertainties of the qalitative type or lingistic description [8, 9]. This stdy present a novel approach based on fzzy set techniqes, to improve qality when the qality response needs to be sbjectively estimated. From experience demonstrating an illstrative example, the following two conclding remarks can be made: 1.The proposed approach can effectively describe the difference among the nearby lingistic description when we rate the qalitative qality response. ccording to the finding of the illstrative example, the effectiveness of employing fzzy set to analyze the qalitative qality response problem can be verified;.sbjective engineering evalation for evalating the qalitative qality response sing lingistic description can be inclded in the proposed approach. Hence, the decision selecting of the optimm parameter setting will be more precise. References: [1] Jean, Y.-C. and Go, S.-M. Qality improvement for RC60 chip resistor, Qality and Reliability Engineering International, 1, 1996, pp [] Johnson, R.. and Wichern, D. W., 199, pplied Mltivariate Statistical nalysis, Prentice Hall, Inc. [3] Leon, R. V., Shoemaker,. C. and Kacker, R. N., Performance measres independent of adjstment (with discssion), Technometrics, 9, 1987, pp [4] Montgomery, D. C., Design and nalysis of Experiments, John Wiley & Sons, Inc., [5] Nair, V. N. Testing in indstrial experiments with ordered categorical data, Technometrics, 8, 1986, pp [6] Phadke, M. S., Qality Engineering Using Robst Design, Prentice Hall, New Jersey, [7] Tagchi, G., Statistical nalysis (in Japanese), Marzen, Tokyo, [8] Zadeh, L.., Otline of a new approach to the analysis of complex systems and decision processes, IEEE Transaction on System, Man and Cybernetics, SMC-3, 1973, pp [9] Zadeh, L.., Fzzy sets, Information and Control, 8, 1965, pp