JIEMS Journal of Industrial Engineering and Management Studies

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1 JIEMS Jounal of Industial Engineeing and Management Studies Vol. 4, No. 2, 2017, pp DOI: /JIEMS Pefomance of CCC- contol chat with vaiable sampling intevals Mohammad Sabe Fallah Nezhad 1,*, Yousof Shamstaba 1, Mohammad Mahdi Vali Sia 2 Abstact The CCC- chat is developed based on cumulative count of a confoming (CCC) contol chat that consides the cumulative numbe of items inspected until obseving nonconfoming ones. Typically, the samples obtained fom the pocess ae analyzed though 100% inspection to exploit the CCC- chat. Howeve, consideing the inspection cost and time would limit its implementation. In this pape, we investigate the pefomance of CCC- chat with vaiable sampling inteval (CCC- VSI chat). The efficiency of CCC- VSI chat is compaed with fixed sampling inteval (FSI) scheme of CCC- chat (CCC- FSI chat) and CCC VSI chat. The compaison esults show that CCC- VSI chat is moe efficient than the CCC VSI chat in educing the aveage time to signal (ATS) and also CCC- VSI chat pefoms bette than CCC- FSI chat. In addition, some sensitivity analyses ae pefomed to illustate the effect of the input paametes on the pefomance of CCC- VSI chat. Keywods: Attibute contol chat; vaiable sampling inteval; CCC- chat; aveage time to signal Received: July Revised: Octobe Accepted: Decembe Intoduction Attibute contol chat is a tool fo statistical pocess contol (SPC) that is applied fo inceasing stability and impoving quality though vaiability eduction in the poduction pocess. The cumulative count of confoming (CCC) contol chat is a type of attibute contol chat that is based on the cumulative numbe of confoming items between two consecutive nonconfoming items and follows the geometic distibution (Calvin, 1983; Goh, 1987). It is useful fo high-quality pocess like automated manufactuing systems. The extended state of CCC contol chat is called CCC- contol chat which consides the cumulative count of confoming items until obseving non-confoming ones, and it is based on the negative binomial distibution (Ohta et al,2001; Kudo et al,2004). CCC chat is based * Coesponding Autho; Fallahnezhad@yazd.ac.i 1 Depatment of Industial Engineeing, Yazd Univesity, Yazd, Ian. 2 Depatment of Industial Engineeing, K.N. Toosi Univesity of Technology, Tehan, Ian.

2 Pefomance of CCC- contol chat with vaiable sampling intevals on the cumulative numbe of confoming items, but it makes the chat insensitive to pocess shifts elatively (Chen, 2014). Kualmani et al. (2002) and Nooossana et al. (2007) poposed the conditional chat which detects the shifts of nonconfoming faction based on the cuent point and some of the pevious ones in ode to impove the pefomance of the CCC contol chat. Utilizing the scheme of vaiable sampling inteval (VSI) that vaies the next sampling inteval between successive samples based on of the cuent point is anothe method fo impoving the sensitivity of the CCC chat (Liu et al., 2006). The motivation of using vaiable sampling scheme is to decease the cost of contol chat implementation. Also, this chat detects the changes in nonconfoming faction moe quickly than CCC chat with fixed sampling intevals (FSI). The length of sampling inteval depends on the pocess condition in a VSI chat. If thee ae some alams which show a change in the pocess, then a shote sampling inteval should be used and if thee is no such signal, then a longe sampling inteval will be used. If the cuent value of the contol statistic is nea to the taget, then a longe sampling inteval should be applied next, and a shote sampling inteval should be applied when the cuent value of the contol statistic is nea to but not outside the contol limits. A substantial numbe of eseaches have consideed VSI contol chat in ode to impove the pefomance of chats. X contol chat with vaiable sampling inteval was studied by some eseaches (see fo example, Reynolds et al., 1988; Reynolds and Anold, 1989; Runge and Pignatiello J, 1991; Runge and Montgomey, 1993; Amin and Mille, 1993; Zhang et al., 2012; Lee et al., 2016). Apaisi and Hao (2001), Villalobos et al (2005) and Nadekhani and Makis (2016) consideed a VSI multivaiate Shewhat chat. Reynolds et al. (1990) and Luo and Wang, (2009) poposed the VSI CUSUM contol chat to monito the shift in the pocess mean. VSI EWMA chats have been investigated by some eseaches (see fo example, Shamma et al., 1991; Saccucci et al, 1992; Castagliolaet al, 2006; Castagliolaet et al, 2006). Vaughan(1992 ), Eppecht et al. (2003), Wu and Luo( 2004), Zhou et al. (2016) and Lee and Khoo (2017) studied the VSI np contol chat. Some studies consideed CCC contol chat with vaiable sampling inteval (CCCVSI). Liu et al. (2006) poposed cumulative count of confoming chat with vaiable sampling inteval. They found that CCCVSI chat is moe effective than CCCFSI chat. Chen et al. (2011) investigated CCC chat with vaiable sampling intevals and contol limits (CCCVSI/VCL chat) and they concluded that the CCCVSI/VCL chat is moe sensitive than both the CCCVSI and CCC chat in detecting the pocess shift. Chen (2013) studied CCC chat with vaiable sampling inteval fo coelated samples (GCCCVSI chat). Thei esults demonstated that using the GCCCVSI chat can enhance the speed of GCCC chat in detecting changes of the nonconfoming faction. Zhang et al. (2014) investigated the pefomance of CCCVSI chat with estimated contol limits. The pefomance of CCCVSI chat with estimated paametes was compaed with CCCFSI based on two pefomance metics, ATS and SDATS. They concluded that thei poposed design is moe efficient than CCCFSI chat. Lee and Khoo (2015) poposed a combination of uns ules and vaiable sampling inteval scheme fo CCC chat. In this pape, we study CCC-VSI chat and compae its pefomance with the pefomance of CCCFSI chat and CCCVSI poposed by Liu et al. (2006). The est of the pape is oganized as follows. In section 2, the CCC-VSI Chat has been descibed. In section 3, the compaison study between the CCC-VSI chat and the CCC-FSI chat and CCCVSI chat is conducted and the effect of paametes is elaboated using sensitivity analysis. In section 4, a pactical case study fo the implementation of the CCC-VSI chat is descibed. Finally, the pape is concluded. Jounal of Industial Engineeing and Management Studies (JIEMS), Vol.4, No.2 Page 20

3 M. S. Fallah Nezhad, Y. Shamstaba, M. M. Vali Sia 2. Desciption of the CCC-VSI Chat 2.1. Notations p 0 p 1 X i N d j d IL Li ARL ATS ATSV ATSF ATS ATS F I q j qj V the in-contol nonconfoming faction, the out-of-contol nonconfoming faction, the tue pobability of false alam, the cumulative count of items inspected between the (i-1)th nonconfoming item and the ith nonconfoming item (including the last nonconfoming item), the numbe of sampling inteval lengths fo the CCC-VSI chat, j =1,2,,n. Sampling inteval lengths of the CCC-VSI chat,, i.e., the time between inspections of two consecutive items (dn<dn-1< <d2<d1) the sampling inteval length of the CCC-FSI chat, the limits in the CCC-VSI chat which divide the egion between UCL and LCL into n sub-egions I1; I2;... ; I n (ILn-1<ILn-2< <IL1), the sampling inteval length which is used to obtain Xi, the aveage un length, the aveage time to signal, the in-contol ATS of the CCC-VSI chat, the in-contol ATS of the CCC-FSI chat, The out-of-contol ATS of the CCC-VSI chat, the out-of-contol ATS of the CCC-FSI chat, impovement facto, defined as I ATS / ATS the pobability that point Xi falls within the egion nonconfoming faction is p0, the pobability that point Xi falls within the egion nonconfoming faction is p 1, V F I j I j, when the pocess when the pocess 2.2. CCC-VSI contol chat As mentioned befoe, The CCC chats ae based on the cumulative count of confoming items until detecting nonconfoming item. Assume a pocess with nonconfoming faction of p and let X denote the cumulative count of items inspected until the detection of the th nonconfoming item, then X follows the negative binomial distibution with paametes (, p) and the pobability mass function of X as is following, x 1 x f ( x) (1 p) p, x, 1,..., 1 (1) Jounal of Industial Engineeing and Management Studies (JIEMS), Vol.4, No.2 Page 21

4 Pefomance of CCC- contol chat with vaiable sampling intevals The cumulative distibution function of X as is follows, x i 1 i F( x) (1 p) p i 1 (2) If the acceptable isk of false alam is α, then the uppe contol limit, UCL the cental contol limit, CL and the lowe contol limit, LCL of the CCC- contol chats ae as following(xie et al, 2012): i 1 (1 ) 1 / 2 UCL 1 i p0 p0 i 1 (3) CL i i 1 (1 0 ) i p p (4) LCL i i 1 (1 0 ) i p p0 / 2 1 (5) When a point falls outside the contol limits, then the pocess is consideed to be out of contol. It is necessay to mention that if a plotted point of the CCC chat is above the uppe contol limit, then we have an impovement in the pocess. On the othe hand, a point below the lowe contol limit is a signal of pocess deteioation. The ARL (aveage un length) is the aveage numbe of points plotted until eceiving an outof-contol signal. Fo the CCC chats, ARL can be calculated fom Eq. (6), with specified values of nonconfoming faction p p0 and p=p0 fo out of contol and in contol situations espectively. 1 ARL UCLi 1 1 p (1 p) LCL 1 i (6) ANI is the aveage numbe of items inspected until a nonconfoming signal occus, that can be calculated fo CCC-FSI and CCC-VSI chat by using the following equation: ANI ARL (7) p When the CCC-VSI chat is applied, then a finite numbe of inteval lengths d1, d2,..., dn( d1 d2... dn) ae used. These inteval lengths should be detemined based on the pactical conditions of manufactuing pocesses. The inteval limits IL1, IL2,..., ILn ae designed in the CCC-VSI chat, so that the egion between UCL and LCL is divided into n sub-egions I 1, I 2,..., I n, coesponding to the n diffeent intevals. Thus following famewok is used fo sampling fom the pocess, Jounal of Industial Engineeing and Management Studies (JIEMS), Vol.4, No.2 Page 22

5 M. S. Fallah Nezhad, Y. Shamstaba, M. M. Vali Sia L i d1,x i1 I1 ( IL1, UCL) d2,x i1 I2 ( IL2, IL1 )... dn,x i1 In ( LCL, ILn ) (8) The sampling inteval length, Li that is used fo inspection between the (i-1) th nonconfoming item and the ith nonconfoming item depends on the value of Xi-1. The inteval limits IL1, IL2,..., ILn 1 can be computed as follows ( is invese function of the negative binomial distibution with paametes and p0): F 1 1 IL1 F (1 q1 ) 2 1 IL2 F (1 q 1 q2 ) ILn 1 F (1 q1 q2... qn ) 2 (9) This scheme continues until the IL values fall between UCL and LCL. It can be elaboated as follows: LCL IL IL... IL IL UCL n1 n IL1 F (1 q1 ) UCL, ILn 1 F (1 q1 q2... qn ) LCL 2 2 (10) (11) Fo example, when n=4; then fou diffeent intevals ae applied. Fo implementing the CCCVSI scheme, contol limits IL1, IL2 and IL3 ae designed between UCL and LCL, so that the egion (LCL, UCL) is divided into fou sub-egions. Jounal of Industial Engineeing and Management Studies (JIEMS), Vol.4, No.2 Page 23

6 Pefomance of CCC- contol chat with vaiable sampling intevals Theefoe, ATS can be calculated based on the following equations: ATSF ANI d ARL d (12) p d q d q... d q n ATSV ARL p q 1 q 2... q n n (13) 3. Pefomance compaisons between the CCC-VSI and the CCC-FSI chat In ode to evaluate the efficiency of the CCC-VSI chat, its pefomance is compaed with the CCC-FSI chat in this section. Note that the fixed values of nonconfoming faction p0 and acceptable isk of false alam ae assumed fo both the CCCFSI and the CCCVSI chat so that these chats have the same ANI value. In ode to compae thei ATS values, we opt appopiate design paametes fo the CCC-VSI and the CCC-FSI chat so that the equation ATSF=ATSV is satisfied at the in contol state. Theefoe the CCCVSI chat will be matched to the coesponding CCC-FSI chat when p=p0 and both of them have the same in-contol ATS. On the othe hand, when the pocess nonconfoming faction changes to, the values of ATS ATS F and ATS V p ( p ) 1 0 should be compaed. The contol chat with smalle out-of-contol can detect the shift of nonconfoming faction moe quickly. Let ATS F = ATS V, thus, d d q d q... d q d q d q... d q q q... q n n n n 1 2 n (14) Thus, the sampling inteval length of the CCC-FSI chat is adjusted to be equal 1(d =1), then by selecting appopiate value values of ( d1, d2,..., d n) and ( q1, q2,..., q n) that satisfy Eq. (14), we can obtain the matched CCC-VSI and CCC-FSI chat that have the same in-contol value of ATS. Then, when the nonconfoming faction shifts to p 1, the pefomance of the CCC-VSI chat can be analyzed by computing the value of I, that is the atio of out-of-contol ATS of the CCC-VSI and the CCC-FSI chat: I ATS d q d q... d q ATS q q q V n n F n (15) ATS When I is less than 1.00, the ATS V is less than F. It indicates that the CCC-VSI chat pefoms bette than the CCC-FSI chat. Thus the atio I is the impovement facto. The smalle values of the impovement facto denote an impovement in the pefomance of the CCC-VSI chat. The values of q j can be calculated using the following equations: Jounal of Industial Engineeing and Management Studies (JIEMS), Vol.4, No.2 Page 24

7 M. S. Fallah Nezhad, Y. Shamstaba, M. M. Vali Sia UCL1 x 1 q 1 p1 (1 p1) xil1 1 1 x 1 q p (1 p ) 1 xil x IL x IL n2 x 1 q n1 p1 (1 p1) x IL n ILn 1 x 1 q n p1 (1 p1) xlcl1 1 x x (16) 3.1. Application of CCC- chats with vaiable sampling intevals We now study the behavio of CCC-VSI chat based on equal pobabilities fo each inteval: 1 q1 q2... qn (17) n By using Eq. (14) and substitutingd 1, we have, 1 1 d1q1 d2q2... dnqn ( d1 d2... dn) n Thus following equation is obtained, d1 d2... dn n (18) In this pape, we used data of Liu et.al (2006) fo compaison study. They have assumed , p and sampling inteval lengths ( d1, d2,..., d n) with the fixed value of d 1can be chosen as follows: n 2, d 1.9, d 0.1; n 3, d 1.9, d 1, d 0.1; n 4, d 1.9, d 1.2, d 0.8, d 0.1; n 5, d 1.9, d 1.5, d 1, d 0.5, d 0.1; Based on the equations 10 and 11 fo each value of the paamete, possible values fo the numbe of intevals (n) will be obtained. Fo example in the case CCCVSI (=1), when n=1538, the value of IL1537 is equal to LCL accoding to the equation 11. Thus, this scheme is only possible fo the cases n=2, 3,, This analysis is denoted in Table 1. Also, the possible value of n can be obtained fo othe values of based on constaint (10) and (11). Jounal of Industial Engineeing and Management Studies (JIEMS), Vol.4, No.2 Page 25

8 Pefomance of CCC- contol chat with vaiable sampling intevals Table 1. Possible numbe of intevals (n) fo implementing CCCVSI contol chat n IL1,ILn-1 esult LCL 3 IL 1385 UCL possible IL 2193 UCL, IL 812 LCL possible IL UCL, IL 4 LCL possible IL UCL, IL 3 LCL impossible IL UCL, IL 3 LCL impossible Impovement factos fo diffeent CCC- contol chat based on the numbe of intevals In this subsection, by fixing n and based on the assumed paametes in the pevious subsection and possible values of, it is obseved that when nonconfoming faction, p1 inceases then the impovement facto deceases which demonstates bette pefomance of CCC-VSI compaed to CCC-FSI. The esults ae denoted in Table 2. Also when the paamete inceases, then impovement facto deceases which denotes an impovement in the pefomance of CCC-VSI chat. As can be seen in Table 2, (fo all values of nonconfoming faction and the paamete ) the values of I ae less than 1, thus the pefomance of CCC-VSI chat is always bette than CCC-FSI chat. Table 2. Impovement factos with fixed numbe of intevals (n=3) n=3 impovement facto, I p1/p0 =1 =2 =3 = Jounal of Industial Engineeing and Management Studies (JIEMS), Vol.4, No.2 Page 26

9 M. S. Fallah Nezhad, Y. Shamstaba, M. M. Vali Sia 3.3. Impovement factos fo diffeent numbes of sampling intevals As shown in Table 3, when n=2 and the atio p p 1 0 equals 2, then I = It indicates that the CCC-VSI chat has the best pefomance in the case n=2 among the othe values. Also, p with inceasing the atio 1 p0 the value of I deceases. Table 3. Impovement factos I fo diffeent values of numbe of intevals =3 Impovement facto, I p1/p0 n=2 n=3 n=4 n= Impovement factos based on diffeent lengths of sampling inteval In this subsection, the numbe of sampling intevals has been fixed (n=2) and the length of sampling intevals has been changed in ode to analyze how the pefomance of CCCVSI chats vaies. Fou diffeent cases of )d1, d2( ae chosen and without changing the othe paametes, thei coesponding impovement factos, I ae detemined. As can be seen in Table 4, when the atio p p 1 0 inceases, then the impovement factos, I deceases and the pefomance of CCC-VSI chats impoves compaed to the CCC-FSI chats. Besides, it demonstates that applying lage values fo the diffeences of inteval lengths, (d1 - d2) leads to smalle values of the impovement facto and consequently bette pefomance of CCC-VSI chats. Jounal of Industial Engineeing and Management Studies (JIEMS), Vol.4, No.2 Page 27

10 Pefomance of CCC- contol chat with vaiable sampling intevals Table 4. Impovement factos I with diffeent values of sampling inteval lengths (d1, d2) fo n= impovement facto, I (d1,d2) p1/p0 FSI(1,1) (1.9,0.1) (1.7,0.3) (1.5,0.5) (1.2,0.8) Impovement factos fo diffeent pobability allocations The oveall pefomance of CCCVSI is analyzed based on the equal in-contol pobability allocations. Thus, in ode to investigate the oveall pefomance of CCCVSI chat when the condition q1=q2 = =qn is not satisfied, we fix n=2 and d1=1.9 and only change the values of in contol pobability q1 as suggested by Liu et al. (2006). It should be noted that: q1 q2 1. Thus, d2 can be calculated as follows based on Eq. (19). Jounal of Industial Engineeing and Management Studies (JIEMS), Vol.4, No.2 Page 28

11 M. S. Fallah Nezhad, Y. Shamstaba, M. M. Vali Sia d 1 dq (19) q2 Thus, q1 should satisfy the inequality q 1 (1 ) / d 1 fo fixed value of d1. The esults in Table 5, indicate that when (q2-q1) deceases, then the impovement facto, I deceases and the pefomance of CCC-VSI chats impoves. Table 5. Impovement factos I with diffeent pobability allocation fo CCC- chat with n=2 Impovement factos I with diffeent pobability allocation (q1,q2) (0.10,0.8973) (0.2,0.7973) (0.3,0.6973) (0.4,0.5973) ( , ) p1/p0 (d1,d2) (1.9,0.8997) (1.9,0.7742) (1.9,0.6128) (1.9,0.3973) (1.9, 0.1) Design of a CCC-VSI chat In designing a CCC-VSI chat, the nonconfoming faction p0, acceptable false alam pobability, paamete and the numbe of intevals and thei lengths in the CCC-VSI chat Jounal of Industial Engineeing and Management Studies (JIEMS), Vol.4, No.2 Page 29

12 Pefomance of CCC- contol chat with vaiable sampling intevals should be detemined. Accoding to the methodologies elaboated so fo, the design pocedue is suggested as follows. Step 1: Detemine the contol limits based on fixed false alam pobability. Step 2: Select the numbe of sampling intevals. Step 3: Select the method of pobability allocation. Step 4: Detemine the length of sampling intevals (d1, d2,, dn). Step 5: Evaluate the efficiency of CCC-VSI chats: All of the design paametes of the CCCVSI chat can be detemined by implementing the above fou steps. When the shifted nonconfoming faction is, then the pobability allocation can be calculated using Eq. (16), then the impovement facto I can be computed using Eq. (15), that can be used to evaluate the efficiency of the CCC-VSI chat. p 1 ( q, q ) To illustate the design pocedue of a CCCVSI chat, we suppose that p , , n=2, (d1,d2)=(1.9,0.1). The data of Table 6 show the input defect sequence. As can be seen in Fig. 1, CCC-VSI chats ae dawn fo diffeent values of paamete. It is concluded that no signal is obseved. 1 2 Table 6. A set of data follow geometic distibution with nonconfoming ate p0 = Defect CCC Defect CCC Defect CCC Defect CCC Defect CCC sequence sequence sequence sequence sequence Jounal of Industial Engineeing and Management Studies (JIEMS), Vol.4, No.2 Page 30

13 CCC CCC CCC M. S. Fallah Nezhad, Y. Shamstaba, M. M. Vali Sia CCC VSI chat (a) Defects sequence CCC LCL IL UCL CCC-5 VSI chat (c) Defects Sequence CCC LCL IL1 UCL IL CCC-2 VSI chat (b) Defects Sequence CCC LCL IL1 UCL Figue 1. An example of the CCCVSI chat(a), CCC-2VSI chat(b) and CCC-5VSI chat(c) with n=2. Jounal of Industial Engineeing and Management Studies (JIEMS), Vol.4, No.2 Page 31

14 Pefomance of CCC- contol chat with vaiable sampling intevals 5. Conclusion In this pape, we have poposed the CCC-VSI contol chat fo high quality pocesses. The CCC- chat is an impoved fom of CCC chats. Also, we compaed CCC-VSI chat with CCCVSI chat that has been studied by Liu et al. (2006). The esults of this study have demonstated that always CCCVSI chat is moe efficient than CCCFSI chat and also CCC-VSI chats pefom bette than CCCVSI chat. It is denoted that the efficiency of CCC-VSI chat can be enhanced by inceasing the diffeence between inteval lengths. The esults show that when pobability allocation is equal, then the pefomance of CCC-VSI chat becomes bette. In CCC-VSI chat, if paamete inceases, then the impovement facto of CCC-VSI chat will decease. Thus, CCC-VSI chat can detect the nonconfoming faction shift faste than CCCVSI chat. Also, we compaed CCC-VSI chat with CCC-FSI chat and concluded that CCC-VSI chat is always moe efficient than CCC-FSI chat. Refeences Amin, R. W., and Mille, R. W., (1993). "A obustness study of X chats with vaiable sampling intevals", Jounal of Quality Technology, Vol. 25, pp Apaisi, F., and Hao, C. L., (2001). "Hotellings T2 contol chat with vaiable sampling intevals", Intenational Jounal of Poduction Reseach, Vol. 39, No. 14, pp Calvin, T., (1983). "Quality Contol Techniques fo" Zeo Defects", IEEE Tansactions on Components, Hybids, and Manufactuing Technology, Vol. 6, No. 3, pp Castagliola, P., Celano, G., and Fichea, S., (2006). "Evaluation of the statistical pefomance of a vaiable sampling inteval R EWMA contol chat", Quality Technology & Quantitative Management, Vol. 3, No. 3, pp Castagliola, P., Celano, G., Fichea, S., and Giuffida, F., (2006). "A vaiable sampling inteval S2- EWMA contol chat fo monitoing the pocess vaiance", Intenational Jounal of Technology Management, Vol. 37, pp Chen, Y.-K., (2013). "Cumulative confomance count chats with vaiable sampling intevals fo coelated samples", Computes & Industial Engineeing, Vol. 64, No. 1, pp Chen, Y. K., Chen, C. Y., & Chiou, K. C., (2011). "Cumulative confomance count chat with vaiable sampling intevals and contol limits", Applied stochastic models in business and industy, Vol. 27, No. 4, pp Eppecht, E. K., Costa, A. F., and Mendes, F. C., (2003). "Adaptive contol chats fo attibutes", IIE Tansactions, Vol. 35, No. 6, pp Goh, T., A contol chat fo vey high yield pocesses. Quality Assuance, 13(1), Kudo, K., Ohta, H., and Kusukawa, E., (2004). "Economic Design of A Dynamic CCC Chat fo High-Yield Pocesses", Economic Quality Contol, Vol. 19, No. 1, pp Kualmani, V., Xie, M., Goh, T., and Gan, F., (2002). "A conditional decision pocedue fo high yield pocesses", IIE Tansactions, Vol. 34, No. 12, pp Lee, M. H., and Khoo, M. B., (2015). "Vaiable sampling inteval cumulative count of confoming chat with uns ules", Communications in statistics-simulation and computation, Vol. 44, No. 9, pp Jounal of Industial Engineeing and Management Studies (JIEMS), Vol.4, No.2 Page 32

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