Simultaneous Monitoring of Multivariate-Attribute Process Mean and Variability Using Artificial Neural Networks

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1 Journal of Qualty Engneerng and Producton Optmzaton Vol., No., PP , 05 Smultaneous Montorng of Multvarate-Attrbute Process Mean and Varablty Usng Artfcal Neural Networks Mohammad Reza Malek and Amrhossen Amr Industral Engneerng Department, Shahed Unversty, Tehran, Iran Correspondng Author: Amrhossen Amr (E-mal: Abstract- In some statstcal process control applcatons, the qualty of the product s characterzed by the combnaton of both correlated varable and attrbutes qualty characterstcs. In ths paper, we propose a novel control scheme based on the combnaton of two mult-layer perceptron neural networks for smultaneous montorng of mean vector as well as the covarance matrx n multvarate-attrbute processes whose qualty characterstcs are correlated. The proposed neural network-based methodology not only detects separate mean and varance shfts, but also can effcently detect smultaneous changes n mean vector and covarance matrx of multvarate-attrbute processes. The performance of the proposed neural network-based methodology n detectng separate as well as smultaneous changes n the process s evaluated thorough a numercal example based on smulaton n terms of average run length crteron and the results are compared wth a statstcal method based on the combnaton of two control charts that are developed for montorng the mean vector and covarance matrx of multvarate-attrbute processes, respectvely. The results of model mplementaton on numercal example show the superor detecton performance of the proposed NN-based methodology rather than the developed combned statstcal control charts. Keywords: Average run length, Covarance matrx, Mean vector, Mult-layer perceptron neural network, Multvarate-attrbute process. I. INTRODUCTION In many statstcal control applcatons, the qualty of the product or the process s characterzed n terms of several correlated qualty characterstcs. In order to montor such processes, dfferent multvarate or mult-attrbute control charts are developed separately by qualty engneerng researchers. Some of the newest multvarate and mult-attrbute control schemes are lsted as follows. Yeh et al. (0) proposed a control chart based on the penalzed lkelhood estmaton of the precson matrx n order to montor multvarte process varablty when the ndvdual observatons are avalable. They compared the proposed control chart wth the competng MaxMEWMV, MEWMS and MEWMC control charts n terms of average run length crteron. Shang et al. (03) proposed a new approach n order to model multstage processes wth bnomal data and developed correspondng montorng and dagnoss schemes by utlzng a herarchcal lkelhood approach and drectonal nformaton based on the Bnary State Space Model (BSSM). Apars et al. (04) proposed some new control schemes for smultaneous montrng of several Posson varables. Ther proposed method can use a multple scheme,.e. one control chart for montorng each qualty characterstc, or can use a multvarate scheme, based on montorng all the attrbutes wth a sngle control chart. L et al. (04) nvestgated the use of log-lnear models for characterzng the relatonshp among categorcal factors n multvarate bnomal and multvarate multnomal processes. Bersms et al. (007) have revewed dfferent procedures for constructon of the multvarate control charts. Topaldou and Psaraks (009) have also revewed the multnomal and mult-attrbute control charts. Nowadays, artfcal neural networks are consdered as the effectve alternatves of control charts n montorng dfferent processes. Recently, many researches have been devoted to the applcaton of artfcal neural networks n Manuscrpt Receved: June 04 and revsed: 0 Aug 04 ISSN: Accepted: 8 Dec 04

2 44 M. R. Malek and A. Amr. Smultaneous Montorng of Multvarate-Attrbute Process montorng multvarate and mult-attrbute processes due to the superor performance of artfcal neural networks n comparson wth control charts. Nak and Abbas (005) proposed an artfcal neural network based model for dagnosng faults n out-of-control states as well as dentfyng aberrant varables when multvarate Hotellng's T control chart s used. Apars et al. (006) ntroduced a neural network based procedure n order to dentfy whch varable have been shfted n stuatons where the T control chart shows an out-of-control state. Hwarng (008) presented a neural network based dentfer to detect mean shfts n the multvarate processes as well as ndcate the varable(s) that cause out-of-control sgnals. Nak and Abbas (008) desgned a neural network for detectng out-of-control states n mult-attrbute processes as well as dagnosng attrbute(s) that cause the sgnals. They presented three numercal examples and compared the performance of the proposed methodology wth mult-attrbute control charts. They found that ther neural network based methodology outperforms the mult-attrbute control charts n detectng dfferent step shfts n the mult-attrbute process mean. Yu and X (009) ntroduced a learnng-based model for montorng and dagnosng out-of-control sgnals n a bvarate process. They proved that the proposed model outperforms the conventonal multvarate control scheme n terms of average run length (ARL) crteron. Yu et al. (009) proposed a method based on the jont use of several selected neural networks to classfy source(s) of out-of-control sgnals n multvarate processes. Hwarng and Wang (00) proposed a neural-network-based dentfer (NNI) for montorng multvarate auto-correlated processes as well as to dentfy the source of the shfts n such processes. Cheng and Cheng (0) presented a neural network-based approach for detectng varance shfts n multvarate processes. They also nvestgated some mportant mplementaton ssues of neural networks such as wndow sze, number of tranng examples, sample sze and tranng algorthm. Ahmadzadeh (0) suggested two approaches ncludng maxmum lkelhood estmator as well as the artfcal neural network n order to estmate the tme of change n the mean parameters of multvarate processes. Saleh et al. (0) proposed a model wth two modules for on-lne analyss of out of control sgnals n multvarate processes. In the frst module, they used a support vector machne classfer to recognze mean and varance shfts. Then n the second module, they appled two neural networks n order to dentfy magntude of mean and varance shfts. Sometmes n real manufacturng systems, the combnaton of both varable and attrbute qualty characterstcs that are correlated expresses the qualty of the product or the process. For example, n the producton process of LED lamps, the number of nonconformtes on a product and ts weght are dscrete and contnuous qualty characterstcs, respectvely that are correlated wth each other. Despte of varous statstcal control schemes as well as artfcal neural networks that are proposed separately for montorng multvarate as well as mult-attrbute processes, only few methods are avalable n the lterature about montorng multvarate-attrbute processes. These few researches are lsted as follows: Kang and Brenneman (0) provded a bootstrap methodology to construct a confdence bound for the overall defect rate of a product whose qualty assessment nvolves multple pass/fal bnary data and multple contnuous data. They supposed that the qualty characterstcs are ndependent. However n most real systems, the ndependence assumpton of multvarate-attrbute qualty characterstcs s volated. Doroudyan and Amr (0) developed a multvarate T control chart based on the root transformaton method for montorng the mean vector of multvarateattrbute qualty characterstcs. Doroudyan and Amr (03) nvestgated the use of four transformaton methods n order to montor the multvarate-attrbute processes. In the frst approach, the dstrbuton of multvarate-attrbute qualty characterstcs s transformed to approxmate multvarate Normal dstrbuton and then the transformed data are montored by multvarate control charts. In the second approach, multvarate-attrbute qualty characterstcs are transformed such a way that the correlaton between the qualty characterstcs becomes roughly equal to zero. Then the unvarate control charts are used n order to montor the transformed qualty characterstcs. In the thrd and fourth approaches, they used a method based on the combnaton of two transformaton technques n order to make the qualty characterstcs ndependent and transform them to Normal dstrbuton. They mentoned that the dfference between the thrd and fourth methods s the order of usng the transformaton technques. Malek et al. (0) desgned an artfcal neural network for detectng mean shfts as well as dagnosng the source of out-of-control sgnals n multvarate-attrbute processes. Malek et al. (03) developed two exponentally weghted movng average (EWMA)-based control charts ncludng

3 JQEPO Vol., No., PP , MEWMS AS as well as MEWMS AT for montorng the covarance matrx of multvarate-attrbute qualty characterstcs usng Normal to anythng (NORTA) nverse technque. Based on the comparson study, they ponted out that the developed control charts outperforms a tradtonal control chart n detectng varance shfts n the process. Amr et al. (04) proposed a neural network-based approach for montorng the varablty of multvarate-attrbute processes as well as dagnosng the qualty characterstc(s) responsble for the out-of-control states. We can conclude from the lterature that there s no method about smultaneous montorng of mean vector as well as covarance matrx of multvarate-attrbute qualty characterstcs. As the man contrbuton, n ths paper frst we desgn two mult-layer perceptron neural networks for montorng mean and varance shfts n multvarate-attrbute processes where the qualty characterstcs are correlated. Then, we use them smultaneously n order to provde a control scheme for smultaneous montorng of the mean vector as well as the covarance matrx of multvarate-attrbute processes. We also extend a combned control chart and use t for smultaneous montorng of the multvarte-attrbute process mean and varablty that s the second contrbuton of our work. The neural network-based method as well as the extended control chart are both proposed under the assumpton that the process s montored n Phase II. Consequently, the dstrbuton parameters of multvarate-attrbute process data ncludng the mean vector, covarance matrx as well as correlaton coeffcent between qualty characterstcs are known based on the Phase I analyss. The rest of ths paper s organzed as follows: In secton, the problem and assumptons of the multvarate-attrbute model are brefly defned. In secton 3, the extended multvarate-attrbute control chart for smultaneous montorng of the process mean and varablty s dscussed. In secton 4, two mult-layer perceptron neural networks are desgned for detectng mean and varance shfts n multvarate-attrbute processes. Secton 5 presents the proposed neural networkbased methodology for smultaneous montorng of mean vector as well as covarance matrx of multvarate-attrbute qualty characterstcs. In secton 6, through a smulated example, the performance of the proposed smultaneous neural networks-based procedure s evaluated and compared wth a statstcal method based on the extenson of two control charts. Fnally, the conclusons and the recommendatons for future study are gven n secton 7. II. PROBLEM STATEMENT AND ASSUMPTIONS Consder a multvarate-attrbute process wth p varable and q attrbute qualty characterstcs, where all qualty characterstcs are correlated and characterzed by column vector of X = ( x, x,..., x, x,..., x ). In the vector X, the p p+ p+ q frst p elements are varable and the last q elements are attrbute qualty characterstcs. We assume that the correlaton between multvarate-attrbute qualty characterstcs s stable durng the process. Both proposed NN-based as well as the extended multvarate-attrbute control chart are nvolved n Phase II. Hence, the mean vector (µ 0 ) and the covarance matrx (Σ 0 ) of qualty characterstcs are known based on the Phase I analyss. The mean vector of the process at n-control state s µ = ( µ,..., µ, µ,..., µ ), where µ ; =,..., p + q are the mean value of p+q qualty 0 p p+ p+ q characterstcs. If a shft n the mean value of at least one qualty characterstc occurs, the multvarate-attrbute process mean consdered to be out-of-control. The covarance matrx of the qualty characterstcs at n-control s determned as follows: σ σ L σ ( p+ σ σ L σ ( p+ Σ 0 =, () M M O M σ σ σ L ( p+ ( p+ p+ q where σ ; =,..., p + q s the varance of th qualty charactrstc and σ ; j s the covarance between th and jth j qualty characterstcs. If a varance shft n at least one qualty characterstc occurs, the covarance matrx goes to an out-of-control state.

4 46 M. R. Malek and A. Amr. Smultaneous Montorng of Multvarate-Attrbute Process III. THE EXTENDED T -MEWMS AS CONTROL CHART In ths secton the extended T -MEWMS AS for smultaneous montorng of mean vector and covarance matrx of correlated multvarate-attrbute data s brefly explaned. The frst multvarate control chart s T control chart that s proposed by Hotellng (947) for montorng the mean vector of the processes wth multvarate Normal data. The statstc of ths control chart n whch the number of samples n each subgroup s equal to n s calculated as follows: T = n( x x) S ( x x). On the other hand, MEWMS AS control chart s proposed by Memar and Nak (0) based on the squared devaton of observatons from target for montorng the covarance matrx of multvarate processes wth Normal data. In order to apply both T and MEWMS AS control charts for montorng the mean vector as well as the covarance matrx of multvarate-attrbute processes, frst usng NORTA Inverse method, we transform the dstrbuton of orgnal data to a multvarate Normal dstrbuton. After usng NORTA Inverse transformaton, the jont dstrbuton of the transformed data follows a standardzed multvarate Normal dstrbuton n whch the qualty characterstcs are correlated. In MEWMS AS control chart, after usng the NORTA Inverse transformaton, the correlated data should be ndependent. For ths purpose, we use a transformaton that s proposed by Golnab and Houshmand (999). The ndependent standardzed Normal qualty characterstcs are computed accordng to Equaton (3): / x = Σ ( y µ ), tk 0 tk 0 (3) where µ 0 and Σ 0 are the mean vector and covarance vector of transformed qualty characterstcs, respectvely and obtaned n Phase I analyss,,,, s the tth transformed observaton (based on NORTA Inverse transformaton) n kth subgroup where,, and,,. Note that, s jth element (qualty characterstc) of matrx where.,,, As noted, the MEWMS AS control chart s based on the St statstc that s proposed by Yeh et al. (005). Hence, n ths paper we use the S t statstc based on transformed data wth smoothng parameter of λ accordng to the followng equaton: n n λ S = ( λ) S +, S =. t t 0 n x x k k n x x k k (4) k= k= () The control statstc of MEWMS AS statstc s the sum of the elements of matrx lmts: p + q UCL = χ ( ν ), MEWMSAS αas ν St wth the followng control (5) LCL MEWMSAS = p + q χ ν α AS ( ν ), (6) where p and q are the number of varables and attrbute qualty characterstcs, respectvely and ν = ( n( λ)) λ. The extended T -MEWMS AS control chart alarms an out-of-control sgnal when at least one statstc falls outsde ts correspondng control lmts. Based on smulaton, the control lmts of the extended T and MEWMS AS control charts based on the transformed data wth standardzed multvarate normal dstrbuton are determned such that: () the value of ARL 0 obtaned separately for each control chart be the same, () The desred overall ARL 0 s obtaned by smultaneous applcaton of these control charts. Note that after transformng the orgnal data nto multvarate Normal dstrbuton by NORTA Inverse method, the mean vector and the covarance matrx of qualty characterstcs are ndependent. Consequently, after transformng the data, the control lmts of the extended control charts are determned based on the mutvarte Normal dstrbuton. IV. MONITORING MULTIVARIATE-ATTRIBUTE PROCESSES USING NEURAL NETWORKS In ths secton, the structure and the tranng procedure of two artfcal neural networks proposed for montorng the mean vector and covarance matrx of multvarate-attrbute processes are llustrated.

5 JQEPO Vol., No., PP , A. Detectng mean shfts n multvarate-attrbute processes In order to montor the mean vector of multvarate-attrbute processes, a three-layer feed-forward neural network whch uses back-propagaton tranng algorthm wth followng structure s suggested: The number of the nodes n the nput layer of neural network A that s suggested for detectng mean shfts n multvarate-attrbute processes s consdered equal to total qualty characterstcs. For nstance, n a process whose qualty s represented by the combnaton of p varable and q attrbutes wth the correlaton matrx of ρ = [ ρ ]( p+ q ) ( p+ q, ) p+q nodes wll be consdered n the nput layer of neural network A. The nput vector of the neural network A s the column vector of X = [ x j, x j,..., x p+ q, j ] T, where xj, =,,..., p + q s the mean value of th qualty characterstcs n j jth subgroup. The neural network A has one node n ts output layer, where ts observed value determnes the process mean state. It s ponted out n the lterature that one or two hdden layer n most engneerng applcatons wll be enough (Cheng, 995). It s also mentoned that only one hdden layer can properly approxmate any contnuous mappng from the nput patterns to the output patterns n a back propagaton network. Because of lackng any systematc method, the number of nodes n the hdden layer that s hghly problem-dependent s set based on tral and error experments. The sgmod functon s used as the transfer functon n the desgned neural network A for detectng mean shfts n the process. The sgmod transfer functon that s the mostly used functons put the outputs values of the neural network A n the range of [0,]. Fgure () depcts the proposed neural network for detectng mean shfts: In order to tran the neural network A after desgnng ts structure, the proper tranng data sets whch have a crucal effect on the performance of the neural network should be collected. There are several methods for generatng multvarate random numbers n the lterature. The proposed methods n whch the qualty characterstcs are correlated and follow a known margnal dstrbuton are categorzed nto three man types ncludng analytc, numerc and smulaton based methods. In the analytc and numerc approaches, t s assumed that the jont dstrbuton of qualty characterstcs s known. However, n most stuatons ths assumpton s volated. Moreover, these methods are manly applcable n bvarate stuatons. In the smulaton based methods, the random vectors can be generated only by havng the margnal dstrbutons of the qualty characterstcs as well as ther correlaton matrx and knowng the jont dstrbuton of qualty characterstcs s not requred. Ths approach s based on the transformaton of Normal random vectors nto our desrable random vectors. In ths paper, we use the normal to anythng (NORTA) method whch s a smulaton-based method to generate the multvarate-attrbute qualty characterstc. In order to generate the vector X n a multvarate-attrbute process where F s the margnal dstrbuton of th; =,..., p + q qualty characterstc the followng steps should be appled. Suppose the correlaton matrx between the qualty characterstcs s known accordng to Equaton (7): Fg. : The structure of the proposed neural network for montorng the process mean

6 48 M. R. Malek and A. Amr. Smultaneous Montorng of Multvarate-Attrbute Process ρ L ρ,( p+ ρ L ρ,( p+ R X =. (7) M M O M ρ,( p+ ρ,( p+ K. The random vector of Z = ( z,..., z + ) form a multvarate standardzed Normal dstrbuton wth followng correlaton matrx: ρ L ρ,( p+ ρ L ρ,( p+ R Z =. M M O M ρ,( p+ ρ,( p+ K. The vector X s calculated by usng the followng equaton: p q Fx ( Φ( z )) X = M, (9) Fx ( Φ( z )) p+ q p+ q where Φ (.) s the cumulatve dstrbuton functon of a standard Normal dstrbuton. Note that, the correlaton matrx of R z n step depends on the orgnal correlaton matrx of R X. Fndng the elements of the R z matrx to acheve the orgnal correlaton matrx between qualty characterstcs are usually done through smulaton or Newton methods whch are very tme consumng. Ths ssue s mportant especally n stuatons that the number of qualty characterstcs ncreases because the elements of correlaton matrx ncreases. In ths paper, the Gaussan copula functon n MATLAB software s used to fnd the best values of Φ(z ) whch lead to obtanng the orgnal correlaton matrx of qualty characterstcs n both proposed ANN-based method as well as the extended T -MEWMS AS control chart. Ths facltates usng the NORTA method n generatng multvarate-attrbute qualty characterstcs. For more nformaton about the Gaussan copula method, refer to Cherubn et al. (004). In the tranng process of neural network A, frst for each state that the multvarate-attrbute process mean s out-ofcontrol, we generate 00 random samples of sze n. After that, the n-control random samples of sze n are prepared as equal as the total generated out-of-control random samples. It should be added that, the ncreasng the number of data sets n tranng process do not have a sgnfcant effect on the neural network performance. On the other hand, n the stuatons that the number of tranng data sets s not adequate, learnng performance of the neural network wll not be satsfactory. After generatng all n-control and out-of-control data sets, the sample mean value of each qualty characterstc n the generated dataset s computed and used as the nput value of the desgned neural network A. Fnally, the neural network A for recognzng the process mean state s traned va the generated nput vectors as well as ther correspondng target values. Note that target value for n-control and all out-of-control random samples are consdered equal to zero and one, respectvely. In each teraton of tranng process, the mean square error (MSE) that s based on the dfference between observed values of output layer and the target value s propagated backward from output towards the nput layer. Then the weghts assocated to the connectons are modfed. Ths terated process s contnued untl the MSE crteron decreases adequately. Note that all the smulatons ncludng generatng data sets as well as tranng the neural networks are done n MATLAB computer package. B. Detectng varance shfts n multvarate-attrbute process In order to detect dfferent varance shfts n multvarate-attrbute processes, a three layer perceptron neural network wth back-propagaton tranng algorthm s suggested. The structure of the neural network B that s desgned for montorng the process varablty ncludng the number of the nodes n the nput and output layer, number of hdden layers as well as the transfer functon s smlar to the neural network A that s presented for montorng the process mean. The number of nodes n the hdden layer s also determned based on tral an error procedure. We use the column vector of S j =[S j,s j,,s p+qj ] T as the nput vector of the neural network B, where S j,=,,,p+q s the standard devaton of th qualty characterstcs n jth subgroup. The structure of neural network B s represented n Fgure (): (8)

7 JQEPO Vol., No., PP , Fg. : The structure of the proposed neural network for montorng the process varablty In order to tran the neural network B for recognzng varance shfts, frst we prepare 00 random samples of sze n for each state that covarance matrx of qualty characterstcs s out-of-control. Then as equal as the number of out-ofcontrol random samples, the n-control random samples each of sze n are generated. After that, the sample standard devaton of each qualty characterstc n the generated dataset are calculated and used as the nput vectors of the desgned neural network. Smlar to neural network A, the target value for n-control and out-of-control states are consdered equal to zero and one, respectvely. After generatng all nput vectors as well ther correspondng target values, the neural network B s traned usng back-propagaton tranng algorthm. Note that as smlar to neural network A, the mean square error (MSE) crteron s used for evaluaton of tranng the neural network B. V. PROPOSED NN-BASED METHOD FOR SIMULTANEOUS MONITORING OF PROCESS MEAN AND VARIABILITY After desgnng the structure of neural networks A and B as well as ther tranng steps, the proposed neural network-based control scheme can be mplemented for smultaneous montorng of mean vector as well as covarance matrx of multvarate-attrbute qualty characterstcs. For ths purpose, we determne the threshold values for the output neurons of both neural networks A and B. The process state s determned based on the smultaneous comparson between the outputs of two proposed neural networks and ther correspondng threshold values. The threshold values of the neural networks A and B are calculated such that:. When the neural networks A and B are appled separately for detectng mean and varance shfts, respectvely; the n-control average run length (ARL 0 ) obtaned by them should be approxmately equal.. When the neural networks A and B are appled smultaneously, the ARL 0 value obtaned by them should be approxmately equal to ARL 0 obtaned by extended combnatory control charts. The procedure of calculatng the threshold value for the output neuron of both neural networks A and B that are traned for detectng mean and varance shfts s smlar. In order to calculate the threshold value for the output neuron of neural network, whch s desgned for montorng the mean vector (covarance matrx) of the multvarate-attrbute processes, the followng steps should be appled:. Generatng 0000 random samples each of sze n form a multvarate-attrbute process whose mean vector (covarance matrx) s n-control.. Calculatng the sample mean (sample standard devaton) value of qualty characterstcs n each sample taken and enterng the nput vector that s comprsed of p+q elements to the neural network A (B) whch s proposed for detectng mean (varance) shfts. 3. Sortng the observed values of output neuron of neural network A (neural network B) n ascendng order and savng them n a vector lke c (d ). 4. Determnng an element of the vector c (d ) as the threshold value of output neuron of neural network A (neural network B) such that the ARL 0 obtaned by the neural network A (B) becomes equal to a predetermned value. In order to dentfy the process state we set two rules as follows:

8 50 M. R. Malek and A. Amr. Smultaneous Montorng of Multvarate-Attrbute Process. If the observed values of output neurons of both desgned neural networks are equal or less than ther correspondng thresholds, the process s ntroduced as an n-control state.. Otherwse, f the observed values of at least one output neuron of any neural networks A or B are greater than ther correspondng thresholds, the process wll be out-of-control. We also extend a combnatory control charts and compare the results of the proposed neural network-based methodology wth t. For ths purpose, based on NORTA Inverse technque we extend multvarate T and MEWMS AS control charts and apply them for montorng mean vector and covarance matrx of multvarate-attrbute processes where the qualty characterstcs are correlated. Then, we apply the extended T -MEWMS AS control charts for smultaneous montorng of mean vector and covarance matrx of multvarate-attrbute processes. VI. NUMERICAL EXAMPLE In ths secton a numercal example based on smulaton s presented to llustrate the hgh performance of the proposed neural network-based method and then the results are compared wth developed multvarate-attrbute T - MEWMS AS control chart. In the numercal example the qualty of product s consdered to be expressed by the combnaton of a Posson attrbute and a Normal varable. The parameters of qualty characterstcs are known based on Phase I analyss. Accordngly, the parameter of Posson attrbute (x ) s equal to 4 and the mean and varance of Normal varable (x ) are equal to 3 and 4, respectvely. The coeffcent constant between qualty characterstcs s consdered equal to and the random samples of sze 0 are used for montorng the process. In order to provde a control scheme for smultaneous montorng of the mean vector and covarance matrx of ths process, two three-layer perceptron neural networks should be desgned. Accordng to subsectons A and B, both neural networks have two (number of qualty characterstcs) and one nodes n ther nput and output layers, respectvely. Based on tral and error experments, the desgned neural networks A and B that are desgned for montorng the mean vector and covarance matrx of the process have one hdden layer wth and 0 nodes, respectvely. Accordng to subsecton A, n order to tran the neural network A for detectng mean shfts, frst for each out-ofcontrol states of the process mean we generate 00 random samples of sze n=0. Because, there are three out-ofcontrol states, we prepare totally 600 out-of-control random samples. After that, 600 n-control random samples of sze n=0 are also generated. Then, the sample mean values of both qualty characterstcs n all 00 generated data sets are calculated and used as the nput vectors of the neural network A. The nput vectors of the proposed neural network A T for detectng mean shfts s the column vector of X = [ x j, x j ], j =,,...,00, whch x j nd x j are the mean j values of Posson and Normal qualty characterstcs n the jth tranng nput vector. It should be mentoned that n order to generate out-of-control random samples, the shft of µ = µ + σ s used n whch µ and µ are the mean value of th qualty characterstc before and after shft, respectvely and σ s the standard devaton of th qualty characterstc under n-control state. Table () represents the requred nformaton n tranng process of the frst neural network: The frst neural network desgned for montorng the multvarate-attrbute process mean s traned wth the generated nput vectors as well as ther correspondng target values usng back-propagaton algorthm. Fnally, the MSE value that s obtaned from tranng step s obtaned equal to TABLE I. DETAILS OF TRAINING THE DESIGNED NN FOR MONITORING THE PROCESS MEAN Shfted qualty Mean process state Number of data sets Mean value of x Mean value of x Target value characterstc - n-control 600 λ = 4 µ = 3 0 x out-of-control 00 λ = 8 µ = 3 x out-of-control 00 λ = 4 µ = 7 x and x out-of-control 00 λ = 8 µ = 7

9 JQEPO Vol., No., PP , 05 5 TABLE II. DETAILS OF TRAINING THE DESIGNED NN FOR MONITORING THE PROCESS VARIABILITY Shfted qualty characterstc Mean process state Number of data sets Varance of of x Varance value of x Target value - n-control 600 λ = 4 σ = 4 0 x out-of-control 00 λ = 6 σ = 4 x out-of-control 00 λ = 4 x and x out-of-control 00 λ = 6 σ = 6 σ = 6 The procedure and the number of data sets requred for tranng second neural network that s desgned for detectng varance shfts s almost dentcal to the frst neural network. The only dfference s that the nput vectors of the second neural network s the column vector of S j =[S j,s j ] T, j=,,,00, whch S j and S j are the sample standard devatons of Posson and Normal qualty characterstcs n the jth tranng data sets. In order to generate out-of-control data sets n tranng process of the second neural network, the shft of σ = σ s used whch σ and σ are the standard devaton of th qualty characterstc before and after shft n ts varablty. The nformaton requred for tranng the second neural network s summarzed n Table (). The second neural network for varance shfts s traned wth the all 00 nput vectors as well as ther correspondng target values usng back-propagaton algorthm. Fnally, the MSE value of the tranng process s obtaned equal to In order to compare the performance of the proposed neural network-based methodology wth the combned T - MEWMS AS control chart, we set the threshold values of the desgned neural networks as well as the parameters of the extended control charts such that the ARL 0 obtaned by both methods become roughly equal to 00. The ARL 0 obtaned by smultaneous applcaton of the desgned neural networks wll be equal to 00 f the ARL 0 obtaned by separate applcaton of each neural network become equal to 400. For ths purpose, accordng to secton 3 the output thresholds of the frst and second neural networks are consdered equal to and 0.900, respectvely. Consequently, based on 0000 replcates the ARL 0 obtaned by the frst and second neural networks n montorng the process mean and the process varablty are to and , respectvely. Then, the overall ARL 0 based on the smultaneous usng of the desgned neural networks s calculated equal to 94.. After calculatng the threshold values of both neural networks, the process state can be determned by smultaneous comparson between outputs values and ther threshold values accordng to secton 3. Note that, the mean and the varance of the Posson qualty characterstc are dependent. Hence, the proposed NNs wll be dependent. It s clear that the dependency between the proposed NNs mproves the detecton performance of the proposed NN-based method. Because t can ncrease the probablty of detectng shfts, even very small shfts n the parameter of Posson qualty characterstc by at least one of the desgned NNs. The results of the proposed neural network-based method n smultaneous montorng of mean vector and covarance matrx of the process n terms of out-of-control average run length (ARL ) crteron are presented n Table (3) and the results are compared wth the combned T -MEWMS AS multvarate-attrbute control chart. For each smultaneous shft, the standard devatons of run lengths are also presented n bracket. The results of both combned methods are obtaned based on 0000 replcates. The results of Table (3) show hgh detecton performance of the proposed neural network-based approach n detectng dfferent smultaneous shfts n the multvarate-attrbute process. The frst column (frst sx smultaneous shfts) of Table (3) are shfts n the mean of x and the varance of x, whle the others are shfts n the varance of x and the mean of x. We can conclude that n the frst sx smultaneous shfts, the proposed methodology outperforms the extended combned T -MEWMS AS control chart, however n the last sx shfts; the detecton performance of both methods s almost the same. The results of Table (3) show that the standard devatons of run lengths for both NN-based approach and the extended control chart are small. Consequently, the performance of both methods n detectng smultaneous shfts n the mean vector and covarance matrx of the process n terms of ARL crteron are vald.

10 5 M. R. Malek and A. Amr. Smultaneous Montorng of Multvarate-Attrbute Process TABLE III. ARL VALUES UNDER DIFFERENT SIMULATNEOUS SHIFTS smultaneous shft ANN T - MEWMS AS ( µ = µ + σ, σ =. σ ).5 (0.95).56 (.30) ( µ = µ + σ, σ =. σ ).55 (0.95).57 (.8) ( µ = µ +.5 σ, σ =. σ ).8 (0.47).3 (0.6) ( µ = µ +.5 σ, σ =. σ ).9 (0.47).0 (0.55) ( µ = µ +.5 σ, σ =. σ ).05 (0.4).07 (0.30) ( µ = µ +.5 σ, σ =. σ ).05 (0.4).06 (0.7) smultaneous shft ANN T - MEWMS AS ( σ =.3 σ, µ = µ σ ).7 (0.46).0 (0.53) ( σ =.35 σ, µ = µ σ ).04 (0.).05 (0.6) ( σ =.4 σ, µ = µ σ ).0 (0.0).0 (0.) ( σ =.3 σ, µ = µ σ ). (0.56).7 (0.49) ( σ =.35 σ, µ = µ σ ).05 (0.30).05 (0.6) ( σ =.4 σ, µ = µ σ ).0 (0.3).0 (0.4) We also evaluate the performance of the proposed method n detectng separate mean and varance shfts and compare t wth T -MEWMS AS control chart. The results of detectng dfferent step shfts n the mean vector and the covarance matrx by both methods n terms of average run length as well as the standard devaton of run lengths crtera are gven n Tables (4) and (5), respectvely. The results of Tables (4) and (5) represent the satsfactory mplementaton of both methods n detectng mean and varance shfts, respectvely. It can be also concluded from Tables (4) and (5) that the standard devatons values of run lengths n NN-based and the extended control chart are adequately small. Obvously, the results of ARL values n detectng dfferent mean shfts as well as the varance shfts are vald. TABLE IV. ARL VALUES UNDER DIFFERENT MEAN SHIFTS Mean shft ANN T -MEWMS AS Mean shft ANN T - MEWMS AS ( µ = µ σ ).76 (.4) 3.47 (3.74) ( µ = µ +.5 σ ).9 (0.48).7 (0.65) ( µ = µ σ ) 9.8 (9.6) 3.36 (3.55) ( µ = µ +.5 σ ).50 (0.87).6 (0.45) ( µ σ, µ σ ) 5.44 (5.0).9 (.67) ( µ +.5 σ, µ +.5 σ ).09 (0.76).04 (0.) ( µ = µ + σ ).58 (0.98).78 (.4) ( µ = µ + σ ).90 (.46).63 (.6) ( µ = µ +.5 σ ).05 (0.4).09 (0.33) ( µ = µ +.5 σ ). (0.37).03 (0.7) ( µ + σ, µ + σ ).35 (.84).9 (0.54) ( µ +.5 σ, µ +.5 σ ).09 (0.3).0 (0.06) TABLE V. ARL VALUES UNDER DIFFERENT VARIANCE SHIFTS Varance shft ANN T - MEWMS AS ( σ =.5 σ ) 4.0 (3.44) 5.8 (6.0) ( σ =.5 σ ) 5.76 (5.43) 4.9 (4.93) ( σ =.5 σ, σ =.5 σ ).7 (.6).04 (.95) ( σ =. σ ).95 (.43).35 (.5) ( σ =.75 σ ).58 (.04).6 (.4) ( σ =. σ, σ =.75 σ ).44 (0.80).4 (0.69) Varance shft ANN T -MEWMS AS ( σ =.3 σ ).0 (0.33).5 (0.44) ( σ = σ ).66 (.07).5 (.4) ( σ =.3 σ, σ = σ ).04 (0.).06 (0.6) ( σ =.35 σ ).0 (0.6).05 (0.) ( σ =.5 σ ).37 (0.70).3 (0.70) ( σ =.35 σ, σ =.5 σ ).0 (0.09).0 (0.07)

11 JQEPO Vol., No., PP , VII. CONCLUSION AND FUTURE RESEARCH In ths paper, we proposed a neural network-based approach control scheme for smultaneous montorng of the multvarate-attrbute process mean and varablty. For ths purpose, frst we desgned two three-layer perceptron neural networks for detectng mean and varance shfts, respectvely. Then, n order to determne the multvarate-attrbute process state, we appled both neural networks smultaneously. Because of lackng any method n the lterature for comparson study, we also developed two multvarate control charts ncludng T and MEWMS AS that are proposed for montorng multvarate process mean and varablty, respectvely. Then we combned and appled them for smultaneous detectng mean and varance shfts n multvarate-attrbute processes. The results of comparson showed that the proposed neural network-based method outperforms the T -MEWMS AS control chart n most smultaneous shfts. The results also confrmed the satsfactory performance of both methods n detectng separate mean and varance shfts n the process. As a future research, dentfyng the magntude of the smultaneous shfts n multvarate-attrbute processes can be nvestgated usng statstcal methods as well as artfcal neural networks. ACKNOWLEDGEMENT The authors are grateful to the respectful referees for ther precous comments whch led to mprovement n the paper. REFERENCES. Ahmadzadeh, F. (0). Change pont detecton wth multvarate control charts by artfcal neural network. The Internatonal Journal of Advanced Manufacturng Technology, -, publshed onlne. DOI: 0.007/s Amr, A., Malek, M. R., & Doroudyan, M. H. (05). Montorng Varablty of multvarate-attrbute processes usng artfcal neural network. Producton and Operatons Management, 5 () Apars, F., Avendaño, G., & Sanz, J. (006). Technques to nterpret T control chart sgnals. IIE Transactons, 38(8) Apars, F., García Bustos, S., & Epprecht, E. K. (04). Optmum multple and multvarate Posson statstcal control charts. Qualty and Relablty Engneerng Internatonal, 30() Bersms, S., Psaraks, S., & Panaretos, J. (007). Multvarate statstcal process control charts: an overvew. Qualty and Relablty Engneerng Internatonal, 3(5) Bran Hwarng, H., & Wang, Y. (00). Shft detecton and source dentfcaton n multvarate autocorrelated processes. Internatonal Journal of Producton Research, 48(3) Cheng, C. S. (995). A mult-layer neural network model for detectng changes n the process mean. Computers & Industral Engneerng, 8() Cheng, C. S., & Cheng, H. P. (0). Usng neural networks to detect the bvarate process varance shfts pattern. Computers & Industral Engneerng, 60() Cherubn, U., Lucano, E., & Vecchato, W. (004). Copula methods n fnance. John Wley & Sons. 0. Doroudan, M. H., Amr, A., Root transformaton method for montorng correlated varable and attrbute qualty characterstcs. Proceedngs of th Islamc Countres Conference on Statstcal Scences (ICCS-), Lahore, Pakstan, December 9-, 0.. Doroudyan, M. H., & Amr, A. (03). Montorng multvarate attrbute processes based on transformaton technques. The Internatonal Journal of Advanced Manufacturng Technology, 69(9-) Golnab, S., & Houshmand, A. A. (999). Multvarate shewhart x-bar chart. Inter Stat, 4.

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