A Multivariate Process Capability Index

A Multivariate Process Capability Index *WANG Shaoxi' 2 JIA Xinzhang' 2 JIAO Huifang'BIAN Yingchao' SONG Ning ZHAO Luyu+ ,WEN Xin+( 1.School of Microelectronics , Xidian University ,Xi' an 710071 , China ,E-mail :shxwang@ mail. xidian. edu. cn 2. Key Lab ofMinistry of Education for Wide Band-Gap Semiconductor Materials and Devices ,Xi' an 710071，China ;3. CEPREI ,Guangzhou510610 , China 4. School of Electronic Engineering ,Xidian University Xi an 710071 , China ;5. College of Lifte Science ShandongNormal University Shandong 250014 China万. Department of Chemistry and Chemical Engineering Hunan University Changsha410082 China)process precision，process accuracy , and process performance. Capability measures for processes with a single characteristic harebeen investigated extensively. However , capability measures for processes with multiple characteristics are com paratirely neglected.In this paper，inspired by the approach and model of process capability index inestigated by K. S. Chenet al .( 2003 )and A.B.Yeh et al.( 1998), a note model of multivariate process capability index based on non conformity is presented. As for this index ,the dlata of each single characteristic don' t require satisfsing normal distribution , of which its computing is simple and particionerswill not fell too theoretical. At last the application analysis is made.Key words: non conformity ; multivariate ; process capability index ; quality characteristics1 IntroductionA process capability index( PCI ) is a numerical summary that compares the behavior of a product or processcharacteristic to engineering specifications. These measures are also often called capability or performance indicesor ratios ; we use capability index as the generic term. A capability index relates the voice of the customer( speci-fication limits ) to the voice of the process. A large value of the index indicates that the current process is capableof producing parts that , in all likelihood , will meet or exceed the customer' s requirements. Over the last twodecades , most of the development in PCI focuses on univariate process. Such as Kane( 1986 ),Chan et al.( 1988 ) Choi & Ower( 1990 ) ,Boyle( 1991 ) ,Singhal( 1991 ) ,Pearn et al.( 1992 ) ,Boyle( 1994 )and so orf12].However ,it is not uncommon in manufacturing that one often encounters processes which involve several corre-lated variables of interest. In such situation , simply calculating the PCI' s of individual variables and combiningthem together will inevitably fail to value the level of processes. Therefore ,it is more desirable to assess the pro-cess capability using multivariate process capability index( MPCI ).In recent years , several attempts to develop MPCI have been made by various researchers such as Chan ,Cheng and Spiring( 1991 ), Pearn , Kotz and Johnson( 1992 ) , Litting , Lam and Pollock( 1992 ) and Kotz andLovelace( 1998 ) and so on. However , most of the existing MPCI require that the process be normally distribut-ed ,and the index is largely dependent on the variance- covariance structure of the underlying distribution. Theothers are uneasy to apply them into practices due to the difficulty of computing.Inspired by the recent works of K. S. Chen et al.( 2003 )and A.B. Yeh et al.( 1998 ), in which A. B. Yeh( 1998 ) proposes a universal PCI which is directly related to the probability of non-conformance of the processand can be applied to any distribution , we incorporate the approaches in Chen( 2003 ) to propose and study aMPCI based on non-conformity. The new MPCI , which we denote by MCf , is not limited to normal distribu-tions ,and its result is easy to get.2 Univariate PCI中国煤化工It is generally agreed that the original motives underlying.MYHC N M H Gvere related to the pro-portion of non- conformity products 121. So when we have a value of PCI=1 ,it is known that under the ideal sit-uation the yield is 99. 73% ; and if the situation is less than ideal , the yield is less than 99. 73%. Let X be the* Contract/ grant sponsor China National Key Laboratory for analog IC( 51439040103DZ0102 )一753一random variable associated with the characteristic of a process under study. Here we assume that X is positiveand has a continuous distribution. Let aσ and aσ be the expected proportions of non- conformity products themanufacturer can tolerate on the lower and upper specification limits respectively. For instance ,a widely usedproportion in industry is 0.135% on each side in ideal situation. These proportions are usually determined basedon the expectation of the manufacturer. Let al =R( X< LSL )( where LSL is lower speification limits of a pro-cess )and au= R( X> USL )( where USL is upper specification limits of a process ) be the actual proportions ofnon-conformance of the process. The question of whether the process is capable can be answered by comparinga你toal and a to au. Thus the index can be defined as 3]aba、Cf = mir(aL 'a∪(1)Based on the definition of the index ,a process is capable only if it produces equal or smaller proportion ofnon- conformity products than expected ,which is Cj≥1. Also , 1/Cj can be interpreted as the increase or reduc-tion in the actual proportion of non-conformity products when compared to the expected proportion. For exam-ple ,1/Cf= 1.50 indicates that the process is producing 50% more non-conformity products than expected ,and1/Cf= 0.7 would indicate that the process is producing 30% less non. conformity products than expected. Notethat since Cf is defined as a ratio of non-conformity probabilities ,it can be used to monitor directly the probabili-ty of non-conformance of a process. On the other hand , unlike other process capability indices , Cf is not limitedto normal distributionS 3].3 MPCK( MCp)Process capability indices with univatiate characteristic have been investigated extensively. However , capa-bility measures for processes with multiple characteristics are comparatively neglected. Similar to the univariatePCI , MPCI should imply the non- conformity of process. Based on the univariate PCI( Cf ) which is presentedby A. B. Yeh and combined with the approach to propose multivariate process capability index which is presentedby Chen , we present a non-conformity multivariate process capability index which don' t require process data tosatisfy normal distribution. The MPCI called MC; is defined as follows :MC;=+q'{[ I[(24(3C;)- 1)+ 1]2}(2)Where Cf; is the univariate process capability index value of ith characteristicfor i=i=1 2.... m ,and nis the number of characteristics. The new index ,MCf , may be viewed as a generalization of the single charac-teristic index Cj. Let MCf= C , hencetψ'{[ I[(2(3Cμ)-1)+1]2}= C→II[ 24(3C;)- 1]= 24(3C)- 1(3)i=1So we can getη= IIn;= II[24(3C;)- 1]= 2Q(3C)- 1(4)Where η is the yield of process. From Eq.( 4 ) we can get the one- to one correspondence relationship be-tween the index MCf and non conformity(1- η). Then the中国煤化工to assessment level ofprocess with non- conformity in multivariate situation.For a process with n characteristics ,if the requirement fMYHC N M !! Gapbility index is MC,≥Co ,a sufficient condition for the requirement to each univriate process capabilty index can be obtained by thefollowing :Let Cmin be the minimum value for each single characteristic , then_752万 方数据.-φ~'{II(2((3C;)- 1)+ 1]2}>→φ'{ II(24(3Cmn)-1)+ 1 ]2}(5)i=li=1If→φl{ I(2(3Cm-1)+ 1]2}> Co(6)Then'min> 1φ(γ20(3C)-1 +1)(7)2So we can get the the requirement to each univriate process capability index.义29(3C)-1+ 1、Cg≥φ'((i=12...m). (8)When Eq.( 8 )is satisfied , then the multivariate process capability index requirement MCj≥Co will be sat-isfied.4 Result and ApplicationFrom Eq.( 2 )and(4 ), we can get the correspondence value of MCf and non-conformity( see Table 1 ).Table 1 Correspondence value of MCg and non-conformityMCJ( % non-conformity )1.01.331.672. 00(0.002 699 76)(0.000066073)(0.000 000 544 )( 0.000 000 002 )1.068 .1.3841.7142. 037(0.001 35081 )(0.000 033 037)( 0.000000 272)(0.000 000001 )1.1071.4141.7392. 0593(0.000 900 743 )( 0.000022 025 )(0.000000 181 )( 0.000000001 )1. 1331.4361.7572. 0744(0.000 675 633)(0.000 016519)(0.000 000 136 ) .(0.000000 000 )1.153I.4521.7702. 0855(0.000540 543)(0.000 013215)( 0.000000109 )(0.000 000 000 )1.1701. 4651.7812. 0956(0.000 450 473 )(0.00011 013)(0.000 000 091 )( 0.000000 000 )1. 1831.4771.7912. 1037(0.000 386 132 )(0.000 009 439 )( 0.000000 078 )(0.000000000)1.1951. 4861.7992. 1108(0.000 337 874)(0.000 008 259 )(0.000 000 068 )1.2051.4951. 8062.1169(0.000 300 338 )(0.000 007 342)(0.000 000 060 )1.2141. 5021.8122. 12110(0.000 270308 )( 0.000006 608)(0.000000054)1.2221.509 .1.8182. 1261(0.000 245 738 )(0.000006 007 )(0.000 000 049 )1.2301.5151. 8232. 130(0.000 225 262 )(0.000 005 506 )(0.000 000 045)1.2361.520 .1. 8282.13513(0.000 207 936)( 0.000005 083)( 0.000000 042 )141.2431.526中国煤化工2. 138.(0.000 193 085 )(0.000 004 720)1.2481.530YHCNMHG2. 1421:(0.000 180 214 )(0.000004405)(0.000000036)Table 1 lists the correspondence value of MCf and non- conformity for the number characteristics from n =1 ton=15. From the Table 1 , the correspondence relationship between MPCI and non- conformity is similar to-755-univriate situation , so we can know application of the MCf model is practicable.In fact ,in order to meet customers' requirement , products need to be classed. When products are qualifiedto the specification ,it is an important item how to precisely monitor variation of process and how to class theproducts. For the reason , we set up process capability zone of products according to specification and then classthem to conformity or unconformity or super- conformity( see Table 2 ).Table 2 Class of products' MPCIClassMPCISuggestionUnconformityMCjz2Taking measurements to save the cost of the processWhen we know the value of specification( z1 and 22 ), the value of univariate PCI for single characteristiccan be obtained by Eq.( 8 ). Now we give an example to show this point. The specification for a product is 1.0≤MC≤1.67 which is quite common in most factories. The process capability zone for single characteristic isfollowed( see Table 3 ).Table3 Limit value of CnNumber ofLimit value of C;characteristicValue of upper limitValue of lower limit1.0001.6701.0681.7141.1071.7391.1331.7571.1531.7701.1701.7811.1831.791 .1.1951.7991.2051. 806101.214.1.812111. 2221.818121.2301. 823131.236 .1. 836141.2431. 840151.2481.8435 ConclusionsProcess capability indices have been widely used in the manufacturing industry , providing numerical mea-sures on process precision , process accuracy , and process performance. Capability measures for processes with asingle characteristic have been investigated extensively. However , capability measures for processes with multi-ple characteristics are comparatively neglected. In this paper we have proposed and studied a new MPCI. Theproposed index ,MCf , can be applied to a variety of specification zones under non-normal distribution. There-fore ,it has greater flexibility than the existing MPCI' s. Furthermore , the propose index is directly related tothe yield of process. So , unlike other MPCI' s , the index MCf can be used to assess to what extent the processis producing non-conformity products.Acknowledgement中国煤化工The authors express their sincere thanks to Prof. Yeh ，1YHC NMH G'nce and suggestions.References[1] Kane V. E. Process Capability Indices. Journal of Quality Technology ,1986 ,18 1 )41-52.[2] Kotz S ,Johson N. Process Capability Indices-A Review. Journal of Quality Technology ,2002 ,34( 1 ):2-19.- 758万左数据[3] Yeh A. B. Bhattacharya S. A Robust process Capability Index. Communications in Statistics -Simulation and Computation ,1998 ,26 :565-589.[4] ChenK. S. Pearn W. L. LinP. C. Capability Measures for Processes with Multiple Characteristics. Quality and ReliabilityEngineering International ,2003 ,19 :101-110.[5] Palmer K. Tsui K. L. A Review and Interpretation of Process Capability Indices. Annals of Operations Research ,1999 ,87 :31-47[6] Bothe ,D. R. 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