Optimal designs of the double sampling X¯ chart with estimated parameters

Int. J. Production Economics 144 (2013) 345–357

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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Optimal designs of the double sampling X¯ chart with estimated parameters Michael B.C. Khoo a,n, W.L. Teoh a, Philippe Castagliola b, M.H. Lee c a b c

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia Department of Quality and Logistics, LUNAM Universite´, Universite´ de Nantes & IRCCyN UMR CNRS 6597, Carquefou, France School of Engineering, Computing and Science, Swinburne University of Technology (Sarawak Campus), Sarawak, Malaysia

a r t i c l e i n f o

abstract

Article history: Received 24 April 2012 Accepted 20 February 2013 Available online 7 March 2013

The double sampling (DS) X chart detects small and moderate mean shifts quickly. Furthermore, this chart can reduce the sample size. The DS X chart is usually investigated assuming that the process parameters are known. Nevertheless, the process parameters are usually unknown and are estimated from an in-control Phase-I dataset. This paper (i) evaluates the performances of the DS X chart when process parameters are estimated by means of a new proposed theoretical method, (ii) shows that performances with estimated parameters are different from that with known parameters, and (iii) proposes three optimal design procedures: the first design minimizes the out-of-control average run length, the second design minimizes the in-control average sample size and the third design minimizes the average extra quadratic loss, by considering the number of Phase-I samples in these three designs. Additionally, for ease of implementation, this paper provides the new optimal parameters specially computed for the DS X chart with estimated parameters, based on the number of Phase-I samples used in practice. These findings will lead to a more economically feasible process monitoring situation, especially when the process parameters are unknown. & 2013 Elsevier B.V. All rights reserved.

Keywords: Double sampling (DS) X chart Average run length Average sample size Standard deviation of the run length Average extra quadratic loss Optimization design

1. Introduction Statistical Process Control (SPC) is a collection of powerful problem-solving tools to improve process capability and achieve process stability (Montgomery, 2009). A control chart is one of the most useful techniques in SPC. Recently, many researchers have contributed to the area of control charts, such as Wu et al. (2011), Ou et al. (2012b), Sun et al. (2012), Ahmad et al. (2013) and Du and Lv (2013), to name a few. The Shewhart X chart is extensively used to monitor large process mean shifts in the manufacturing and service sectors. The main limitation of this chart is its poor sensitivity toward small and moderate process mean shifts. To overcome this problem, various adaptive charts are developed and widely investigated in the literature. An adaptive control chart is a chart in which the chart parameters, such as the sample size and sampling interval are allowed to vary, depending on the value of the sample statistic (Montgomery, 2009). Recent works on adaptive charts were made by Nenes (2011), Zhou and Lian (2011) and Ou et al. (2012a). The double sampling (DS) type chart, which has attracted a great amount of attention among researchers, is one of the adaptive charts. The X chart with the DS

n

Corresponding author. Tel.: þ60 4 6533941; fax: þ 60 4 6570910. E-mail addresses: [email protected] (M.B.C. Khoo), [email protected] (W.L. Teoh). [email protected] (P. Castagliola). [email protected] (M.H. Lee). 0925-5273/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2013.02.022

procedure was introduced by Croasdale (1974), where declaring an out-of-control signal is prohibited at the first-sample stage. The first sample provides information as to whether the second sample needs to be taken but the process is then sentenced based on only the information provided by the second sample. The DS X chart suggested by Daudin (1992) is a two-stage Shewhart X chart which uses the information from both samples to make an out-ofcontrol decision at the combined-sample stage. Daudin’s procedure has a higher detection sensitivity than that of Croasdale’s (Irianto and Shinozaki, 1998). Daudin’s procedure minimizes the in-control average sample size (ASS0 ), while Irianto and Shinozaki (1998) modified Daudin’s DS method to minimize the out-ofcontrol average run length ðARL1 Þ. Works on the DS method continue to generate interest among researchers as some of the properties of the DS chart are superior to those of the Shewhart, EWMA, CUSUM, variable sampling interval (VSI) and variable sample size (VSS) charts (Daudin 1992; Costa, 1994). When the process is in-control, the sample size of the DS X chart decreases to nearly 50% compared to that of the Shewhart X chart (He et al., 2002). Daudin (1992) found that the DS X chart is quicker at detecting large mean shifts than both the EWMA and CUSUM charts. The CUSUM and EWMA charts have a good performance in detecting small shifts, but the control procedures of these two charts are not as easy as that of the DS X chart (Torng et al., 2010). Gupta and Walker (2007) pointed out that when the incoming quality is either very excellent or very poor, the DS X chart requires a lower total

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sample size because the process is either proclaimed as in-control or out-of-control, based on the decision made at the first sample. This leads to higher quality and uniformity in the output with fewer defects to rework and less scrap cost. In view of these remarkable properties of the DS chart, the optimal designs of the DS X chart with estimated parameters, considered in this paper are justified. Research works on the DS method can be grouped into the DS charts for monitoring the process mean and variance, i.e. the DS X type and DS S type charts, respectively. Among the DS X type charts proposed in the literature are as follows: He et al. (2002) solved the optimization problems in the designs of the DS and triple sampling (TS) X charts using genetic algorithm. Hsu (2004) commented on the conclusions made by He et al. (2002) as the latter only considered the average sample size (ASS) when the process is in-control without considering the situations when the process is out-of-control and yet not detected. Carot et al. (2002) studied a combined DS and VSI (or DSVSI) X chart. Owing to the efficiency of this DSVSI X chart in detecting small mean shifts, the economic design of this chart was studied by Lee et al. (2012b). Costa and Claro (2008) considered the DS X chart to monitor a first-order autoregressive moving average (ARMA (1,1)) process model. To reduce the total costs involved, Torng et al. (2009a) presented an economic statistical design of the DS X chart. An economic design of the DS X chart for correlated data using genetic algorithms was studied by Torng et al. (2009b). The performances of the variable parameters (VP) X chart and the DS X chart in the presence of correlation were compared by Costa and Machado (2011). In the VP X chart, all the design parameters, i.e. the sample size n, the sampling interval h and the factor k, used in determining the width of the chart’s action limits, are variables (Costa, 1996). Khoo et al. (2011) suggested a synthetic DS X chart and showed that it outperforms the standard synthetic X and DS X charts, for all levels of shifts and the EWMA chart for moderate and large shifts. The DS S type charts found in the literature include that suggested by He and Grigoryan (2002, 2003), where the optimization problems were solved using genetic algorithm. Hsu (2007) pointed out the setback in He and Grigoryan’s (2002) work which only considered the ASS when the process is in-control but not the average number of samples required to signal a process standard deviation shift. The effectiveness of the DS S chart in reducing the sample size of destructive testing was studied by Lee et al. (2010). Lee et al. (2012a) proposed a combined DS chart and VSI S chart, called the DSVSI S chart. A fundamental assumption in the construction of control charts, for most of the existing works in the literature, is that the process parameters are known. In practice, the process parameters are usually unknown and are estimated from an incontrol historical or Phase-I dataset. However, the fact is that when process parameters are estimated, the performance of a control chart is different from the case with known parameters because of the variability of the estimators. Numerous works have studied the effects of the estimation of process parameters on the performances of different types of control charts. A thorough review on the early research in this topic was given in Jensen et al. (2006). The more recent works that were not reported therein will be reviewed here. Most of these works deal with the X type charts, for example see Bischak and Trietsch (2007), Chakraborti (2007), Zhang et al. (2011, 2012) and Castagliola et al. (2012). Works that involve the variance type charts were made by Castagliola et al. (2009) and Schoonhoven et al. (2011). Authors that considered the EWMA and CUSUM type charts include Maravelakis and Castagliola (2009), Capizzi and Masarotto (2010), and Castagliola and Maravelakis (2011). Research on parameter estimation for attributes charts were reported in Testik et al. (2006) and Testik (2007).

A primary issue in SPC for variables is to effectively detect process mean shifts (Stoumbos and Reynolds, 1997; Jearkpaporn et al., 2007; Wu et al., 2011). Wu et al. (2011) stated that in most of today’s industrial SPC applications, the control chart is used to monitor only the process mean, rather than both the mean and variance. Similarly, most of the works in the existing literature on the DS type charts considered the process monitoring of only mean shifts (Carot et al., 2002; Torng et al., 2009a; Costa and Machado, 2011). Thus, in this paper, we only focus on the DS X chart that monitors the process mean shifts. Like most control charts, all the existing works in the literature on the DS type charts are designed based on the assumption that the process parameters are known. This paper investigates the performance (i.e. average run length (ARL), standard deviation of the run length (SDRL), ASS and average extra quadratic loss (AEQL)) of the DS X chart with estimated parameters, compares them with the case involving known parameters and proposes three optimal design procedures with estimated parameters, for (i) minimizing the ARL1 , (ii) minimizing the ASS0 and (iii) minimizing the AEQL. By employing the optimal design procedure, based on ASS0 , a smaller sample size should be used when the process is in-control and thus leading to the reduction of inspection and sampling costs (Daudin, 1992; He et al., 2002; He and Grigoryan, 2003). Spiring and Yeung (1998) stated that the loss function, which is used to measure the cost due to poor quality, is broadly applied in industry. The AEQL, which is based on the quadratic loss function (Taguchi and Wu, 1980), is widely used by many researchers (see Reynolds and Stoumbos, 2004; Wu et al., 2009, 2011; Ou et al., 2012a, 2012b) to design a control chart. Therefore, to evaluate the overall effectiveness of the DS X chart with estimated parameters, the AEQL is proposed as the third optimal design criterion in this paper. The remainder of the paper is organized as follows: Section 2 discusses the DS X chart of Daudin (1992). Section 3 presents formulae taken from Daudin (1992) so that an evaluation of the performance of the DS X chart when the process parameters are known can be made. This section also describes the derivation of equations to evaluate the performance of the DS X chart when process parameters are estimated. Section 4 compares the ARL, SDRL, ASS and AEQL performances of the DS X chart, for cases with known and estimated parameters. Three optimal designs of the DS X chart with estimated parameters for minimizing the (i) ARL1 , (ii) ASS0 and (iii) AEQL, are presented in Section 5. Optimal chart parameter combinations for the DS X chart with estimated parameters, based on the number of Phase-I samples and sample size used in practice are given in this section. Section 6 illustrates the application of the DS X chart with estimated parameters by means of an example. Conclusions are drawn in Section 7.

2. The DS X chart Assume that the quality characteristic Y, in a Phase-II process, follows a normal Nðm0 , s20 Þ distribution, where m0 and s20 are the incontrol population mean and variance, respectively. Let L 40 and L1 ZL be the warning and control limits in the first-sample stage of the DS X chart, respectively, and let L2 4 0 be the control limit in the combined-sample stage of the DS X chart (see Fig. 1). Note that L, L1 and L2 refer to the standardized limits. Let I1 ¼ ½L,L, I2 ¼ ½L1 ,LÞ [ ðL,L1 , I3 ¼ ð1,L1 Þ [ ðL1 , þ 1Þ and I4 ¼ ½L2 ,L2 . By referring to the graphical view of the DS X chart in Fig. 1, the operation of the chart is as follows: (1) Set the limits L, L1 and L2. (2) Take a first sample of size n1 at sampling time t and compute the 1 sample mean Y 1t ¼ Snj ¼ Y =n1 , where Y 1t,j , for j ¼ 1,2,. . .,n1 , 1 1t,j

M.B.C. Khoo et al. / Int. J. Production Economics 144 (2013) 345–357

In Eq. (3), r ¼ (Daudin, 1992)

out-of-control (I3) L1

out-of-control take a second sample (I2)

L2

ARL ¼

L in-control (I1) take a second sample (I2)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 þ n2 and c ¼ ðn1 þ n2 Þ=n2 . It follows that

1 1P a

ð4Þ

and

in-control (I4)

-L SDRL ¼

-L2

-L1

347

out-of-control

pffiffiffiffiffi Pa : 1Pa

ð5Þ

The average sample size (ASS) at each sampling time is;

out-of-control (I3)

ASS ¼ n1 þn2 P2 ,

First sample

Combined samples

ð6Þ

where the probability of taking the second sample P 2 is equal to P2 ¼ PrðZ 1t A I2 Þ pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi ¼ FðL1 þ d n1 ÞFðL þ d n1 Þ þ FðL þ d n1 ÞFðL1 þ d n1 Þ:

Fig. 1. Graphical view of the DS X chart’s operation.

ð7Þ

(3) (4) (5)

(6) (7)

(8)

are the Phase-II observations in the first sample at sampling time t. pffiffiffiffiffi If Z 1t ¼ ½ðY 1t m0 Þ n1 =s0 A I1 , the process is declared as incontrol. Then, the control flow returns to Step (2). If Z 1t A I3 , the process is declared as out-of-control and the control flow proceeds to Step (8). If Z 1t A I2 , take at sampling time t, a second sample of size n2 from the same population as for the first sample. Compute the 2 second sample mean Y 2t ¼ Snj ¼ Y =n2 , where Y 2t,j , for 1 2t,j j ¼ 1,2,. . .,n2 , are the Phase-II observations in the second sample at sampling time t. Compute the combined-sample mean Y t ¼ ðn1 Y 1t þ n2 Y 2t Þ= ðn1 þn2 Þ, for sampling time t. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi If Z t ¼ ½ðY t m0 Þ n1 þ n2 =s0 A I4 , the process is declared as incontrol. Otherwise, the process is declared as out-of-control and the control flow advances to Step (8). An out-of-control is signaled at sampling time t and corrective actions are taken to investigate and remove the assignable cause(s). Then return to Step (2).

Note that each sampling time may either contain the observations of only the first sample or the observations of both first and second samples. The AEQL for the DS X chart with known parameters can be calculated by (Wu et al., 2009) Z dmax 1 AEQL ¼ d2 ARLðdÞdd; ð8Þ dmax dmin dmin where dmax and dmin are the upper bound and lower bound of the mean shift, respectively. Here, ARLðdÞ, which can be computed from Eq. (4), is the ARL value for a mean shift d. 3.2. The DS X chart with estimated parameters When both the in-control process mean m0 and standard deviation s0 are unknown, they are estimated from an in-control Phase-I dataset comprising m samples, each having n observations {Xt,1, Xt,2, y, Xt,n}, for t¼1, 2, y, m. Assume that X t,j  Nðm0 , s20 Þ and there is independence within and between samples. A commonly used ^ 0 of m0 is the grand mean, i.e. estimator m

m^ 0 ¼ 3. The properties of the DS X chart 3.1. The DS X chart with known parameters Let P a1 and P a2 represent the probabilities that the process is declared as in-control ‘‘by the first sample’’ and ‘‘after taking the second sample’’, respectively. Then P a ¼ P a1 þ P a2 is the probability that the process is declared as in-control. Daudin (1992) showed that pffiffiffiffiffi pffiffiffiffiffi P a1 ¼ PrðZ 1t A I1 Þ ¼ FðL þ d n1 ÞFðLþ d n1 Þ, ð1Þ where d ¼ 9m1 m0 9=s0 (magnitude of the standardized mean shift) and Fð:Þ is the standard normal cumulative distribution function (cdf). Daudin (1992) also showed that Z P 4 fðzÞdz, ð2Þ P a2 ¼ PrðZ t A I4 and Z 1t A I2 Þ ¼ z A In2

and

The detailed derivations of the conditional probabilities P^ a1 and P^ a2 for the DS X chart with estimated parameters are shown in Appendix A. When parameters are estimated, P^ a ¼ P^ a1 þ P^ a2 is the probability that the process is deemed as in-control, where  rffiffiffiffiffiffiffi   rffiffiffiffiffiffiffi  pffiffiffiffiffi pffiffiffiffiffi n1 n1 P^ a1 ¼ F U þ VLd n1 F U VLd n1 ð11Þ mn mn and P^ a2 ¼

Z

^ 0,s ^ 0 Þdz, P^ 4 f Z^ 1t ðz9m

with  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi  rffiffiffiffiffiffiffi  pffiffiffiffiffi n2 L2 n1 þ n2 z n1 P^ 4 ¼ F U þV pffiffiffiffiffi d n2 mn n2  rffiffiffiffiffiffiffi   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi n2 L2 n1 þ n2 þ z n1 V d n2 F U pffiffiffiffiffi mn n2 and

ð3Þ

ð9Þ

where X t is the sample mean of {Xt,1, Xt,2, y, Xt,n}, i.e. ^ 0 of s0 is the X t ¼ Snj¼ 1 X t,j =n and a commonly used estimator s pooled estimator (Jensen et al., 2006), i.e. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u m X n X u 1 s^ 0 ¼ t ðX X t Þ2 : ð10Þ mðn1Þ t ¼ 1 j ¼ 1 t,j

z A I2

where fð:Þ is the standard normal probability density function (pdf); pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi In2 ¼ ½L1 þ d n1 ,Lþ d n1 Þ [ ðL þ d n1 ,L1 þ d n1 

P 4 ¼ PrðZ t A I4 9Z 1t ¼ zÞ rffiffiffiffiffi  rffiffiffiffiffi    n1 n1 z F cL2 þrcd z : ¼ F cL2 þrcd n2 n2

m 1 X Xt , mt¼1

 rffiffiffiffiffiffiffi  pffiffiffiffiffi n1 ^ 0,s ^ 0Þ ¼ V f U þ Vzd n1 : f Z^ 1t ðz9m mn

ð12Þ

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Here, U and V are the random variables defined as pffiffiffiffiffiffiffi mn ^ 0 m0 Þ U ¼ ðm

s0

ð13Þ

and V¼

s^ 0 , s0

ð14Þ

respectively. Note that the random variable U follows a standard ^ 0  N½m0 , s0 2 =ðmnÞ. Then the pdf of U is normal distribution as m f U ðuÞ ¼ fðuÞ:

ð15Þ

^ 20 =s20 Þ  gððmðn1Þ=2Þ, According to Zhang et al. (2011), V ¼ ðs ð2=mðn1ÞÞÞ, i.e. a gamma distribution with parameters ½mðn1Þ=2 and 2=½mðn1Þ. Then the pdf of V is deduced as    mðn1Þ 2 f V ðvÞ ¼ 2vf g v2  , , ð16Þ 2 mðn1Þ 2

where f g ð:Þ is the pdf of the gamma distribution with parameters ½mðn1Þ=2 and 2=½mðn1Þ. Eq. (4) gives the ARL with known parameters. Consequently, the ARL of the DS X chart with estimated parameters is Z þ1 Z þ1 1 ARL ¼ f U ðuÞf V ðvÞdvdu, ð17Þ 1P^ a 1 0 with P^ a ¼ P^ a1 þ P^ a2 , where P^ a1 and P^ a2 are shown in Eqs. (11) and (12), respectively. Also, the pdf f U ðuÞ and f V ðvÞ are presented in Eqs. (15) and (16), respectively. The SDRL of the DS X chart with estimated parameters is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SDRL ¼ E½ðRLÞ2 ARL2 , ð18Þ where h i Z E ðRLÞ2 ¼

þ1

1

Z 0

þ1

1 þ P^ a f U ðuÞf V ðvÞdvdu: ð1P^ a Þ2

ð19Þ

Note that when the process parameters are known, E½ðRLÞ2  ¼ ð1 þ Pa Þ=ð1Pa Þ2 . The ASS at each sampling time when the process parameters are estimated is Z þ1 Z þ1 ASS ¼ ðn1 þn2 P^ 2 Þf U ðuÞf V ðvÞdvdu, ð20Þ 1

0

where the conditional probability of taking the second sample, P^ 2 is equal to ^ 0,s ^ 0Þ P^ 2 ¼ PrðZ^ 1t A I2 9m  rffiffiffiffiffiffiffi   rffiffiffiffiffiffiffi  pffiffiffiffiffi pffiffiffiffiffi n1 n1 VLd n1 F U VL1 d n1 ¼F U mn mn  rffiffiffiffiffiffiffi   rffiffiffiffiffiffiffi  pffiffiffiffiffi pffiffiffiffiffi n1 n1 þ VL1 d n1 F U þ VLd n1 : þF U mn mn

ð21Þ

When parameters are estimated, the AEQL can be obtained as   Z dmax Z þ 1 Z þ 1 1 1 AEQL ¼ d2 v2 f U ðuÞf V ðvÞdvdudd: dmax dmin dmin 1 0 1P^ a ð22Þ Note that the detailed derivations of the AEQL for the DS X chart with estimated parameters can be found in Appendix B.

4. A comparison of the performances of the DS X chart with estimated and known parameters It is important to note that the goal of this paper is to investigate the impact of Phase-I parameter estimation on the performance of the DS X chart, and not to show the superiority of the DS X chart to other control charts. Table 1 provides the ARL, SDRL and ASS values for the cases of estimated parameters

(m A {10, 20, 40, 80}) and known parameters (m ¼ þ 1), as well as different (n1 , n2 , L, L1 , L2 ) parameter combinations and shifts d. When d ¼ 0, a set of six randomly selected (n1 , n2 , L, L1 , L2 ) parameter combinations having an in-control ARL, ARL0 ¼ 370.4 and in-control ASS, ASS0 ¼ n A f3,5g, for the chart with known parameters ðm ¼ þ 1Þ are listed in the upper part of Table 1. Meanwhile, the optimal (n1 , n2 , L, L1 , L2 ) combination that gives the lowest ARL1 value for the DS X chart with known parameters ðm ¼ þ1Þ, for the specific shift d is given in the lower part of the table. For the known-parameter case, the optimization procedure in Irianto and Shinozaki (1998), based on the formulae in Section 3.1 is employed. Note that the ARL, SDRL and ASS values for mA {10, 20, 40, 80} are computed for the specific (n1 , n2 , L, L1 , L2 ) combination in columns two to six using numerical integration, based on Eqs. (17), (18) and (20), respectively. The ScicosLab software, which is available at www.scicoslab.org, is used. The accuracy of all the results in Table 1 has been verified with simulation. For example, if n ¼ 5, d ¼ 0.5, the optimal parameter combination is (n1 ¼ 3, n2 ¼ 12, L ¼ 1.383, L1 ¼ 5.280, L2 ¼ 2.781) corresponding to the known-parameter case ðm ¼ þ1Þ. With these optimal parameters, the out-of-control ARL (ARL1 ), SDRL (SDRL1 ) and ASS (ASS1 ) are 19.9, 93.8 and 6.939, respectively, for m ¼ 10. Table 1 provides clear evidence that the actual ARL and SDRL values deviate significantly from the known-parameter (m ¼ þ 1) case, especially when d, m and n are small. There is also some deviation in the actual ASS value from the case with known parameters. Adopting estimates in place of known parameters increase the ARL and SDRL values, where the increase is more pronounced for small d, m and n values. Table 1 shows that as m increases, for fixed n and d, the difference between the ARL, SDRL and ASS values associated with the cases with estimated and known parameters decreases. It is obvious that for large shifts ðd Z 1:5Þ, the difference becomes negligible, even for small m and n values. For small shifts ðd r 0:5Þ, at least m ¼ 80 Phase-I samples are required for the chart with estimated parameters to have a performance that does not vary much from the chart with known parameters. The performance comparison of the AEQL values between the chart with estimated parameters and known parameters is shown in Table 2. For the known-parameter case (m ¼ þ 1), the optimization procedure demonstrated in Irianto and Shinozaki (1998) is modified and applied here to obtain the optimal chart parameters (n1 , n2 , L, L1 , L2 ). The modifications are as follows: (i) the objective function of minimizing AEQL is used instead of ARL1 and (ii) a mean shift domain ðd A ½dmin , dmax Þ is adopted here rather than a specific mean shift. Note that Eqs. (8) and (22), for the AEQL formulae of the known-parameter and estimated-parameter cases, respectively, require the use of a shift domain. Similar to Table 1, the two constraints, i.e. ARL0 ¼ 370.4 and ASS0 ¼ n A f3,5g are used for all the known-parameter cases considered in Table 2. The optimal chart parameters (n1 , n2 , L, L1 , L2 ) corresponding to the known-parameter case ðm ¼ þ1Þ are displayed in columns 4–8 of Table 2. The (ARL0 , ASS0 , AEQL) values for the estimatedparameter case (m A {10, 20, 40, 80}) are computed using these optimal chart parameters. We notice that from Table 2, the (ARL0 , ASS0 , AEQL) values are very different between the estimated-parameter and knownparameter cases. The differences are obvious, especially when n, m and dmin are small. For the fixed values of n, dmin and dmax , the ASS0 and AEQL values decrease and converge to those of the known parameters as m increases. The higher values of AEQL for the estimated-parameter case imply that the loss in quality cost due to the out-of-control conditions is higher than that of the known-parameter case. Table 2 also shows that more than 80 Phase-I samples are required to minimize the difference of the

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349

Table 1 ARL, SDRL and ASS of the DS X chart when n A {3, 5}, mA {10, 20, 40, 80, þ 1} and d A {0, 0.1, 0.2, 0.3, 0.5, 0.7, 1.0, 1.5, 2.0} with the (n1 , n2 , L, L1 , L2 ) combination corresponding to the known parameters case.

d

n1 n2

L

L1

L2

m ¼ 10

m ¼ 20

m ¼ 40

m ¼ 80

m ¼ þ 1 (known-parameter case)

(ARL, SDRL, ASS)

(ARL, SDRL, ASS)

(ARL, SDRL, ASS)

(ARL, SDRL, ASS)

(ARL, SDRL, ASS)

n ¼3 0.0 1 0.0 1 0.0 1 0.0 2 0.0 2 0.0 2 0.1 1 0.2 1 0.3 1 0.5 2 0.7 2 1.0 2 1.5 2 2.0 2

3 8 14 2 7 13 14 14 14 13 13 9 6 4

0.430 1.150 1.460 0.674 1.464 1.757 1.465 1.465 1.465 1.769 1.769 1.593 1.383 1.150

3.344 3.627 3.203 3.834 3.638 3.089 5.230 5.230 5.230 5.345 5.345 5.491 5.280 5.044

3.079 2.789 2.785 3.005 2.813 2.997 2.531 2.531 2.531 2.499 2.499 2.677 2.831 2.933

(1740.9, 45,259.7, 3.020) (768.5, 9427.2, 3.156) (1091.2, 22,914.1, 3.258) (1276.8, 24,807.5, 3.040) (763.6, 9691.3, 3.187) (1412.4, 38,132.8, 3.264) (492.9, 4761.4, 3.336) (362.3, 3887.8, 3.417) (221.1, 2790.0, 3.552) (54.4, 904.5, 4.289) (13.0, 187.5, 5.177) (3.7, 13.3, 5.977) (1.5, 1.1, 6.527) (1.1, 0.4, 5.595)

(668.3, 2898.8, 3.011) (460.7, 1403.9, 3.081) (529.6, 2100.8, 3.135) (583.8, 2135.5, 3.021) (455.6, 1397.0, 3.096) (585.7, 2520.4, 3.137) (325.0, 924.4, 3.183) (207.1, 662.9, 3.266) (106.4, 383.1, 3.402) (23.9, 75.8, 4.128) (7.9, 14.2, 5.030) (3.1, 3.4, 5.928) (1.4, 0.8, 6.577) (1.1, 0.3, 5.682)

(478.3, 947.6, 3.006) (393.8, 679.0, 3.041) (420.0, 810.6, 3.069) (447.4, 820.2, 3.011) (390.4, 672.1, 3.049) (442.3, 867.6, 3.070) (279.9, 486.4, 3.106) (153.8, 299.0, 3.189) (71.0, 141.5, 3.327) (17.5, 26.8, 4.045) (6.7, 7.9, 4.952) (2.9, 2.6, 5.901) (1.4, 0.8, 6.598) (1.1, 0.3, 5.722)

(416.8, 586.1, 3.003) (375.3, 494.7, 3.021) (387.5, 539.2, 3.035) (402.8, 544.6, 3.005) (373.2, 490.8, 3.025) (398.8, 556.5, 3.035) (260.4, 355.9, 3.067) (127.9, 187.4, 3.150) (57.5, 81.0, 3.288) (15.3, 18.2, 4.003) (6.3, 6.4, 4.912) (2.8, 2.3, 5.887) (1.4, 0.7, 6.607) (1.1, 0.3, 5.742)

(370.4, 369.9, 3.000) (370.4, 369.9, 3.000) (370.4, 369.9, 3.000) (370.4, 369.9, 3.000) (370.4, 369.9, 3.000) (370.4, 369.9, 3.000) (234.5, 234.0, 3.028) (102.6, 102.1, 3.112) (47.1, 46.6, 3.250) (13.7, 13.2, 3.961) (5.9, 5.4, 4.872) (2.7, 2.1, 5.872) (1.4, 0.7, 6.616) (1.1, 0.3, 5.760)

n ¼5 0.0 1 0.0 1 0.0 2 0.0 3 0.0 4 0.0 4 0.1 3 0.2 3 0.3 3 0.5 3 0.7 3 1.0 3 1.5 4 2.0 4

5 14 8 7 2 11 12 12 12 12 12 12 7 5

0.251 1.068 0.886 1.064 0.674 1.681 1.383 1.383 1.383 1.383 1.383 1.383 1.465 1.282

3.126 4.229 3.345 3.139 3.528 3.118 5.280 5.280 5.280 5.280 5.280 5.280 5.230 5.104

3.283 2.721 3.014 3.213 3.023 3.112 2.781 2.781 2.781 2.781 2.781 2.781 2.895 2.959

(692.6, 3356.0, 5.007) (363.9, 1093.4, 5.154) (501.1, 2049.2, 5.106) (578.7, 2783.4, 5.125) (534.5, 2100.5, 5.038) (516.9, 2363.4, 5.199) (305.8, 971.9, 5.314) (198.0, 720.9, 5.530) (101.5, 438.1, 5.882) (19.9, 93.8, 6.939) (5.3, 13.5, 8.333) (1.9, 1.8, 10.613) (1.1, 0.4, 10.277) (1.0, 0.1, 8.063)

(482.2, 996.4, 5.004) (335.5, 578.5, 5.078) (402.4, 785.3, 5.054) (432.3, 888.1, 5.065) (420.2, 799.5, 5.020) (404.7, 817.6, 5.101) (263.7, 486.6, 5.196) (143.3, 307.1, 5.417) (62.5, 146.3, 5.777) (12.7, 23.1, 6.860) (4.2, 5.2, 8.289) (1.8, 1.3, 10.631) (1.1, 0.3, 10.378) (1.0, 0.1, 8.178)

(417.4, 597.8, 5.002) (338.3, 446.2, 5.039) (376.6, 526.6, 5.027) (391.5, 558.5, 5.033) (387.1, 533.5, 5.010) (376.6, 533.5, 5.051) (246.2, 351.5, 5.136) (115.1, 183.7, 5.360) (47.4, 74.2, 5.724) (10.7, 13.4, 6.819) (3.9, 3.9, 8.267) (1.7, 1.2, 10.639) (1.1, 0.3, 10.427) (1.0, 0.1, 8.241)

(391.9, 469.6, 5.001) (348.0, 399.3, 5.020) (370.2, 438.1, 5.014) (378.1, 451.6, 5.017) (376.3, 442.0, 5.005) (369.9, 440.0, 5.026) (235.3, 290.5, 5.106) (100.5, 130.5, 5.331) (41.4, 51.3, 5.697) (9.9, 10.6, 6.799) (3.7, 3.4, 8.255) (1.7, 1.1, 10.643) (1.1, 0.3, 10.450) (1.0, 0.1, 8.275)

(370.4, 369.9, 5.000) (370.4, 369.9, 5.000) (370.4, 369.9, 5.000) (370.4, 369.9, 5.000) (370.4, 369.9, 5.000) (370.4, 369.9, 5.000) (218.0, 217.5, 5.076) (86.4, 85.9, 5.302) (36.3, 35.8, 5.670) (9.2, 8.7, 6.778) (3.6, 3.1, 8.244) (1.7, 1.1, 10.646) (1.1, 0.3, 10.473) (1.0, 0.1, 8.310)

Table 2 ARL0 , ASS0 and AEQL of the DS X chart when n A {3, 5}, m A {10, 20, 40, 80, þ 1}, various combinations of (dmin , dmax ) and (n1 , n2 , L, L1 , L2 ) combinations corresponding to the known parameters case. n

dmin

dmax n1 n2

L

L1

L2

m¼ 10

m¼ 20

m ¼40

m¼ 80

m¼ þN (known-parameter case)

(ARL0, ASS0, AEQL)

(ARL0, ASS0, AEQL)

(ARL0, ASS0, AEQL)

(ARL0, ASS0, AEQL)

(ARL0, ASS0, AEQL)

3

0.25 0.25 0.75 0.75

2.00 3.00 2.00 3.00

2 2 2 2

11 10 7 7

1.691 1.645 1.465 1.465

5.127 5.125 5.230 5.230

2.584 2.629 2.778 2.778

(567.4, (586.7, (679.9, (679.9,

3.289, 3.269, 3.199, 3.199,

9.197) 8.498) 4.502) 5.413)

(388.7, (396.2, (428.9, (428.9,

3.145, 3.135, 3.101, 3.101,

4.894) 5.564) 3.621) 4.907)

(355.8, (360.1, (377.7, (377.7,

3.073, 3.068, 3.051, 3.051,

3.986) 4.950) 3.381) 4.767)

(353.5, (356.2, (366.5, (366.5,

3.036, 3.034, 3.025, 3.025,

3.678) 4.741) 3.286) 4.710)

(370.4, (370.4, (370.4, (370.4,

3.000, 3.000, 3.000, 3.000,

3.437) 4.576) 3.202) 4.660)

5

0.25 0.25 0.75 0.75

2.00 3.00 2.00 3.00

3 4 4 4

12 11 9 9

1.383 1.691 1.593 1.593

5.280 5.127 5.491 5.491

2.781 2.745 2.821 2.821

(355.5, (355.7, (374.0, (374.0,

5.241, 5.228, 5.198, 5.198,

3.724) 4.690) 2.625) 4.278)

(329.6, (330.2, (340.7, (340.7,

5.122, 5.114, 5.100, 5.100,

2.886) 4.150) 2.506) 4.210)

(334.4, (334.9, (341.6, (341.6,

5.061, 5.057, 5.050, 5.050,

2.627) 3.982) 2.460) 4.185)

(345.4, (345.8, (350.0, (350.0,

5.031, 5.028, 5.025, 5.025,

2.526) 3.916) 2.440) 4.173)

(370.4, (370.4, (370.4, (370.4,

5.000, 5.000, 5.000, 5.000,

2.441) 3.860) 2.421) 4.162)

chart’s performance between the estimated-parameter and known-parameter cases. Sun et al. (2012) described today’s business as turbulence markets, which have the characteristics of short product life cycles, uncertain product types and fluctuating production volumes. Therefore, using a large number of Phase-I samples is infeasible and economically impractical. Furthermore, collecting a large number of Phase-I samples is time consuming and undetected parameter shifts may occur during this critical time. This study shows how different the ARL, SDRL, AEQL and to a certain extent the ASS performances can be in the known-parameter versus the estimated-parameter cases, hence making the use of the optimal parameter combination (n1 , n2 , L, L1 , L2 ) associated with the known-parameter case in the estimated-parameter case

inappropriate. For this reason, the optimization methods to obtain the new optimal chart parameters (n1 , n2 , L, L1 , L2 ), for the DS X chart with estimated parameters are proposed.

5. Optimal designs of the DS X chart with estimated parameters This section presents three optimal designs of the DS X chart with estimated parameters, for (i) sensitizing the detection of a specific shift in the mean, (ii) having the smallest in-control average sample size and (iii) having the lowest loss in quality cost or damage incurred in the out-of-control cases. Therefore, three optimization algorithms for minimizing (i) ARL1 ðdopt Þ,

350

M.B.C. Khoo et al. / Int. J. Production Economics 144 (2013) 345–357

(ii) ASS0 and (iii) AEQL are suggested. Here, ARL1 ðdopt Þ denotes the out-of-control ARL for a desired size of a standardized mean shift dopt (in multiples of standard deviation units), where a quick detection is needed. The optimization programs to compute the optimal chart parameters for the DS X chart with estimated parameters are written using the ScicosLab software. Because of the complexities of the mathematics and computations involved, the Nelder Mead’s nonlinear optimization algorithm (Nelder and Mead, 1965) is used here to obtain the five optimal chart parameters (n1 , n2 , L, L1 , L2 ). Castagliola et al. (2012) asserted that small and moderate sample sizes are common practice in an industrial context, hence the upper bound for n1 þ n2 ¼ nmax is set as nmax r15. Note that the ARL, ASS and AEQL formulae in Eqs. (17), (20) and (22), respectively, are considered. 5.1. Computation of optimal chart parameters for minimizing ARL1 ðdopt Þ The proposed optimal design of the DS X chart with estimated parameters to minimize ARL1 ðdopt Þ, modeled as a nonlinear minimization problem is as follows: Specifications: (i) Set a desired value of the in-control ARL ðARL0 Þ: t (ii) Set a desired value of the in-control ASS ðASS0 Þ which coincides with the Phase-I sample size: n (iii) Set the number of Phase-I samples: m (iv) Set the size of a standardized mean shift, for which a quick detection is needed: dopt The value of t is selected, based on the cost associated with the false alarm rate. The number of Phase-I samples, m and sample size, n are chosen according to the available resources and the rate of production. The value of dopt is specified based on customer’s requirements and knowledge about the process. Then find

 sample sizes, for both the first and second samples: n1 and n2 ;  warning and control limits for the DS X chart: L, L1 and L2 such that the objective function is minimized, which is mathematically expressed as Min

n1 ,n2 ,L,L1 ,L2

ARL1 ðdopt Þ

ð23Þ

subject to constraints ARL0 ¼ t

ð24Þ

ASS0 ¼ n

ð25Þ

1 rn1 o n on1 þ n2 r nmax

ð26Þ

Using the above optimization model (23)–(26), a heuristic search is performed to obtain the optimal (n1 , n2 , L, L1 , L2 ) combination, for the DS X chart with estimated parameters as follows: (a) For all (n1 , n2 ) pairs selected based on constraint (26), the parameters L, L1 and L2 are searched using a nonlinear equations solver. Here, for any fixed value of L, the values of L1 and then L2 are uniquely determined from constraint (25) ðASS0 ¼ nÞ and constraint (24) ðARL0 ¼ tÞ, respectively.

Therefore, in this step, all possible (n1 , n2 , L, L1 , L2 ) combinations that satisfy constraints (24)–(26), when d ¼ 0 can be obtained. (b) For any out-of-control condition ðdopt a 0Þ, the optimal (n1 , n2 , L, L1 , L2 ) combination which minimizes ARL1 ðdopt Þ is identified from all the parameter combinations found in Step (a). Table 3 presents the (ARL1 , SDRL1 , ASS1 ) values, for the out-ofcontrol cases together with the optimal (n1 , n2 , L, L1 , L2 ) combinations, for different combinations of ASS0 ¼ n, number of Phase-I samples m and shift dopt . The results for the case with known parameters (m ¼ þ 1) are also presented in the last column. The optimal design procedure mentioned above in this section is used. Here, it is ensured that ARL0 ¼ t ¼ 370.4 (constraint (24)) and ASS0 ¼ n A f3,5g (constraint (25)) are attained, for each (n1 , n2 , L, L1 , L2 ) combination in Table 3. The accuracy of the results in Table 3 has been verified with simulation. For each ðn,m, dopt Þ combination, the first row of each cell gives the optimal (n1 , n2 , L, L1 , L2 ) combinations while the second row of each cell gives the corresponding (ARL1 , SDRL1 , ASS1 ) values. For example, when n ¼ 5, m ¼ 40 and dopt ¼ 0.7, the optimal chart parameters are (n1 ¼ 3, n2 ¼ 12, L ¼ 1.400, L1 ¼ 5.248, L2 ¼ 2.813) and these optimal chart parameters give ARL1 ¼ 4.0, SDRL1 ¼ 4.1 and ASS1 ¼ 8.186. Here, ARL1 ¼ 4.0 is the smallest possible out-ofcontrol ARL value for a shift of dopt ¼ 0.7 when n ¼ 5 and m ¼ 40, for the DS X chart with estimated parameters. Concerning the optimal pair of sample sizes (n1 , n2 ), the difference between the cases with estimated and known parameters is generally negligible. Therefore, the optimal (n1 , n2 ) pair for the known-parameter case can generally be used in the case of estimated parameters. However, there is no specific trend for the limits L, L1 and L2 , as m or dopt increases. The ARL1 and SDRL1 values increase while the ASS1 value decreases as m decreases, for a fixed dopt . Thus, as m reduces, the chart’s sensitivity in detecting process changes decreases. Note that as m increases, the ARL1 , SDRL1 and ASS1 values converge to the corresponding values with known parameters. For large shifts ðdopt Z1:5Þ, the optimal ARL1 and SDRL1 values in the case with estimated parameters are almost similar to the corresponding values for the knownparameter case. However, for small shifts, the ARL1 and SDRL1 values of the estimated-parameter case are greatly different from that of the known-parameter case, except when m is large ðm 4 80Þ. 5.2. Computation of optimal chart parameters for minimizing ASS0 The proposed optimal design of the DS X chart with estimated parameters to minimize ASS0 , which is modeled as a nonlinear minimization problem is as follows: Specifications: (i) Set a desired value of the in-control ARL ðARL0 Þ: t (ii) Set a desired value of the out-of-control ARL ðARL1 Þ corren sponding to a shift d : e (iii) Set the number of Phase-I samples: m (iv) Set the Phase-I sample size: nX The values of the specifications in (i), (iii) and (iv) are similarly determined as in Section 5.1. The value of e is decided based on the cost of investigating an out-of-control signal. Then find

 sample sizes for both the first and second samples: n1 and n2 ;  warning and control limits for the DS X chart: L, L1 and L2

M.B.C. Khoo et al. / Int. J. Production Economics 144 (2013) 345–357

351

Table 3 (n1 , n2 , L, L1 , L2 ) combination (first row of each cell) and ðARL1 , SDRL1 , ASS1 Þ values (second row of each cell) of the optimal DS X chart with estimated parameters when ARL0 ¼ 370.4, ASS0 ¼ nA {3, 5}, m A {10, 20, 40, 80, þ 1} and dopt A {0.1, 0.2, 0.3, 0.5, 0.7, 1.0, 1.5, 2.0}.

dopt n ¼3 0.1 0.2 0.3 0.5 0.7 1.0 1.5 2.0 n ¼5 0.1 0.2 0.3 0.5 0.7 1.0 1.5 2.0

m ¼ 10

m ¼ 20

m ¼ 40

m ¼ 80

m ¼ þ 1 (known-parameter case)

(1, 14, 1.547, 4.050, 2.370) (335.1, 2501.0, 3.025) (1, 14, 1.547, 4.041, 2.371) (249.6, 2034.0, 3.101) (1, 14, 1.547, 3.988, 2.371) (156.1, 1439.2, 3.225) (1, 14, 1.546, 3.907, 2.372) (43.1, 484.3, 3.615) (1, 14, 1.544, 3.684, 2.377) (12.2, 114.0, 4.163) (2, 9, 1.698, 3.279, 2.512) (3.7, 9.1, 5.135) (2, 5, 1.358, 3.434, 2.696) (1.5, 1.1, 5.165) (2, 4, 1.213, 3.374, 2.738) (1.1, 0.3, 4.427)

(1, 14, 1.507, 5.689, 2.497) (316.6, 884.2, 3.027) (1, 14, 1.507, 5.689, 2.497) (202.8, 635.1, 3.107) (1, 14, 1.507, 5.532, 2.497) (105.0, 369.2, 3.240) (1, 14, 1.507, 5.211, 2.497) (24.4, 75.6, 3.658) (1, 14, 1.507, 4.710, 2.497) (8.4, 14.6, 4.265) (2, 9, 1.655, 4.050, 2.620) (3.2, 3.5, 5.636) (2, 6, 1.431, 3.968, 2.764) (1.4, 0.8, 6.213) (2, 4, 1.186, 4.248, 2.848) (1.1, 0.3, 5.358)

(1, 14, 1.486, 5.616, 2.540) (295.4, 517.6, 3.027) (1, 14, 1.486, 5.616, 2.540) (162.0, 317.5, 3.109) (1, 14, 1.486, 5.616, 2.540) (74.5, 149.8, 3.245) (1, 14, 1.486, 5.616, 2.540) (18.4, 28.3, 3.672) (2, 13, 1.807, 5.089, 2.499) (7.1, 8.3, 4.806) (2, 9, 1.624, 5.074, 2.668) (2.9, 2.7, 5.790) (2, 6, 1.407, 4.864, 2.810) (1.4, 0.8, 6.531) (2, 4, 1.168, 5.163, 2.900) (1.1, 0.3, 5.734)

(1, 14, 1.476, 5.312, 2.547) (275.0, 378.0, 3.028) (1, 14, 1.476, 5.312, 2.547) (134.3, 198.1, 3.110) (1, 14, 1.476, 5.312, 2.547) (60.0, 85.1, 3.247) (2, 13, 1.788, 5.532, 2.512) (15.9, 19.0, 3.944) (2, 13, 1.788, 5.532, 2.512) (6.5, 6.6, 4.839) (2, 9, 1.609, 5.513, 2.682) (2.8, 2.4, 5.832) (2, 6, 1.395, 5.365, 2.827) (1.4, 0.7, 6.587) (2, 4, 1.159, 5.235, 2.920) (1.1, 0.3, 5.763)

(1, 14, 1.465, 5.230, 2.531) (234.5, 234.0, 3.028) (1, 14, 1.465, 5.230, 2.531) (102.6, 102.1, 3.112) (1, 14, 1.465, 5.230, 2.531) (47.1, 46.6, 3.250) (2, 13, 1.769, 5.345, 2.499) (13.7, 13.2, 3.961) (2, 13, 1.769, 5.345, 2.499) (5.9, 5.4, 4.872) (2, 9, 1.593, 5.491, 2.677) ( 2.7, 2.1, 5.872) (2, 6, 1.383, 5.280, 2.831) (1.4, 0.7, 6.616) (2, 4, 1.150, 5.044, 2.933) (1.1, 0.3, 5.760)

(3, 12, 1.450, 4.819, 2.772) (318.7, 1019.0, 5.070) (3, 12, 1.450, 4.968, 2.772) (206.3, 751.6, 5.278) (3, 12, 1.450, 5.186, 2.772) (105.9, 455.5, 5.618) (3, 12, 1.450, 5.092, 2.772) (20.8, 97.6, 6.645) (3, 12, 1.450, 4.788, 2.772) (5.6, 14.3, 8.011) (3, 12, 1.450, 4.226, 2.773) (2.0, 1.9, 10.162) (4, 7, 1.550, 3.746, 2.862) (1.1, 0.4, 8.602) (4, 5, 1.353, 3.952, 2.909) (1.0, 0.1, 6.340)

(3, 12, 1.416, 5.735, 2.811) (295.6, 556.9, 5.073) (3, 12, 1.416, 5.735, 2.811) (159.5, 349.8, 5.290) (3, 12, 1.416, 5.735, 2.811) (68.9, 165.4, 5.643) (3, 12, 1.416, 5.735, 2.811) (13.7, 25.4, 6.711) (3, 12, 1.416, 5.735, 2.811) (4.5, 5.5, 8.131) (3, 12, 1.416, 5.148, 2.811) (1.8, 1.4, 10.478) (4, 7, 1.509, 4.464, 2.900) (1.1, 0.3, 9.866) (4, 5, 1.318, 4.649, 2.953) (1.0, 0.1, 7.581)

(3, 12, 1.400, 5.248, 2.813) (271.7, 392.3, 5.075) (3, 12, 1.400, 5.248, 2.813) (125.8, 203.5, 5.296) (3, 12, 1.400, 5.248, 2.813) (51.2, 81.2, 5.657) (3, 12, 1.400, 5.248, 2.813) (11.3, 14.3, 6.744) (3, 12, 1.400, 5.248, 2.813) (4.0, 4.1, 8.186) (3, 12, 1.400, 5.248, 2.813) (1.8, 1.2, 10.563) (4, 7, 1.487, 5.066, 2.909) (1.1, 0.3, 10.354) (4, 5, 1.300, 5.082, 2.965) (1.0, 0.1, 8.216)

(3, 12, 1.391, 5.337, 2.804) (251.3, 311.8, 5.075) (3, 12, 1.391, 5.337, 2.804) (106.4, 139.0, 5.299) (3, 12, 1.391, 5.337, 2.804) (43.4, 54.2, 5.663) (3, 12, 1.391, 5.337, 2.804) (10.2, 11.1, 6.761) (3, 12, 1.391, 5.337, 2.804) (3.8, 3.5, 8.215) (3, 12, 1.391, 5.337, 2.804) (1.7, 1.2, 10.606) (4, 7, 1.476, 5.396, 2.907) (1.1, 0.3, 10.476) (4, 5, 1.291, 5.088, 2.966) (1.0, 0.1, 8.257)

(3, 12, 1.383, 5.280, 2.781) (218.0, 217.5, 5.076) (3, 12, 1.383, 5.280, 2.781) (86.4, 85.9, 5.302) (3, 12, 1.383, 5.280, 2.781) (36.3, 35.8, 5.670) (3, 12, 1.383, 5.280, 2.781) (9.2, 8.7, 6.778) (3, 12, 1.383, 5.280, 2.781) (3.6, 3.1, 8.244) (3, 12, 1.383, 5.280, 2.781) (1.7, 1.1, 10.646) (4, 7, 1.465, 5.230, 2.895) (1.1, 0.3, 10.473) (4, 5, 1.282, 5.104, 2.959) (1.0, 0.1, 8.310)

such that the objective function is minimized, which is mathematically expressed as Min

n1 ,n2 ,L,L1 ,L2

ASS0

ð27Þ

subject to constraints ARL0 ¼ t

ð28Þ

ARL1 ¼ e

ð29Þ

1 rn1 o ðnX r nest Þ o n1 þ n2 rnmax

and

n1 r n2

ð30Þ

The nX and nest in constraint (30) are the sample sizes corresponding to the Shewhart X chart with known and estimated parameters, respectively, matching approximately a similar design of the DS X chart. Note that only nX is used to compute the ARL, SDRL and ASS for the DS X chart with estimated parameters. In contrast, nest only facilitates the selection of appropriate ðn1 ,n2 Þ pairs and is not directly used in the computations. The optimization model (27)–(30) is used to optimally determine the (n1 , n2 , L, L1 , L2 ) combination of the DS X chart with estimated parameters. The steps are described as follows: (a) For all (n1 , n2 ) pairs selected based on constraint (30), the parameters L, L1 and L2 are searched using a nonlinear equations solver. Here, for any given value of L1 , values of L and L2 are adjusted simultaneously to satisfy ARL0 ¼ t (see constraint (28)) and ARL1 ¼ e (see constraint (29)). Hence, in

this step, all possible (n1 , n2 , L, L1 , L2 ) combinations that satisfy constraints (28)–(30) are obtained. (b) The optimal (n1 , n2 , L, L1 , L2 ) combination which minimizes ASS0 is identified from all the parameter combinations found in Step (a). The optimization model (27)–(30) is employed to obtain the optimal (n1 , n2 , L, L1 , L2 ) combinations of the DS X chart with estimated parameters. For the chart with known parameters ðm ¼ þ1Þ, the optimization approach given in Daudin (1992) is used. Table 4 shows the optimal (n1 , n2 , L, L1 , L2 ) combinations in the first row of each cell and the corresponding ðASS0 ,ASS1 ,SDRL1 Þ values in the second row of each cell. The accuracy of all the entries in Table 4 has been verified with simulation. The optimal (n1 , n2 , L, L1 , L2 ) combination of the DS X chart with known and estimated parameters shown in Table 4, produces the same (ARL0 , ARL1 ) values as that of the optimal EWMA chart with known n parameters, for a specified shift d . This will ensure that the optimal DS X chart which gives the smallest ASS0 value, will also have reasonable (ARL0 , ARL1 ) values. The (n1 , n2 , L, L1 , L2 ) combinations for the DS X chart with estimated and known (m ¼ þ 1) parameters are selected so that the ARL1 , for both cases match an optimally designed EWMA chart with n known parameters, for a shift d , where the ARL0 of the two charts is fixed as 370.4. Here, the sample sizes of the EWMA chart considered are nEWMA A f1,3g. The optimal design of the EWMA chart is based on the procedure described in Lucas and Saccucci (1990). For example, n when nEWMA ¼ 3 and d ¼ 1.0, the optimal EWMA chart’s ARL1 is 4.16. The DS X chart with known (m ¼ þ 1) and estimated

352

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Table 4 (n1 , n2 , L, L1 , L2 ) combination (first row of each cell) and (ASS0 , ASS1 , SDRL1 ) values (second row of each cell) of the optimal DS X chart with estimated parameters when n

m A {10, 20, 40, 80, þ 1} and d A {0.5, 1.0, 1.5, 2.0, 2.5, 3.0}, matching approximately a similar design of the Shewhart X chart and the optimal EWMA chart (ARL0 ¼ 370.4, nEWMA A f1,3g) with known parameters.

dn

ARL1

Shewhart

DS X Chart

X chart nX

m ¼ 10

m ¼ 20

m ¼ 40

m ¼ 80

m ¼ þ 1 (knownparameter case)

(1, 14, 1.615, 5.446, 2.465) (2.618, 3.222, 70.8) (1, 5, 1.870, 4.299, 2.290) (1.402, 2.059, 24.1) (1, 2, 1.999, 2.836, 2.121) (1.119, 1.398, 17.7) (1, 2, 2.266, 2.599, 1.861) (1.044, 1.205, 7.3) (1, 2, 2.440, 2.499, 1.274) (1.007, 1.040, 4.2) –

(1, 14, 1.904, 5.479, 2.260) (1.851, 2.295, 37.4) (1, 5, 1.999, 3.733, 2.329) (1.269, 1.827, 13.3) (1, 2, 2.082, 2.971, 2.428) (1.090, 1.405, 8.1) (1, 2, 2.386, 2.972, 2.097) (1.041, 1.341, 4.2) (1, 2, 2.564, 2.713, 2.548) (1.011, 1.106, 2.6) (1, 2, 2.693, 2.725, 1.187) (1.002, 1.022, 1.5)

(1, 14, 2.004, 5.225, 2.166) (1.657, 2.044, 30.5) (1, 5, 2.065, 4.365, 2.321) (1.215, 1.748, 10.8) (1, 2, 2.154, 3.364, 2.479) (1.073, 1.451, 6.0) (1, 2, 2.451, 3.998, 2.130) (1.038, 1.604, 3.4) (1, 2, 2.628, 2.864, 2.750) (1.011, 1.172, 2.1) (1, 2, 2.775, 2.850, 2.692) (1.003, 1.055, 1.3)

(1, 13, 2.012, 5.350, 2.164) (1.587, 1.940, 28.1) (1, 5, 2.098, 4.913, 2.316) (1.189, 1.700, 9.9) (1, 2, 2.195, 5.333, 2.488) (1.063, 1.499, 5.2) (1, 2, 2.464, 3.044, 2.592) (1.026, 1.339, 3.1) (1, 2, 2.665, 2.975, 2.564) (1.011, 1.227, 2.0) (1, 2, 2.810, 2.947, 2.438) (1.004, 1.105, 1.3)

(1, 14, 2.090, 5.935, 2.058) (1.513, 1.851, 26.0) (1, 4, 2.041, 6.421, 2.449) (1.165, 1.600, 9.1) (1, 2, 2.227, 5.062, 2.541) (1.052, 1.467, 4.6) (1, 2, 2.514, 3.533, 2.319) (1.023, 1.482, 2.8) (1, 2, 2.702, 3.375, 2.081) (1.012, 1.459, 1.8) (1, 2, 2.843, 3.004, 3.373) (1.004, 1.128, 1.2)

(4, 11, 1.287, 3.938, 2.923) (6.300, 8.409, 20.9) (1, 9, 1.432, 5.585, 2.593) (2.475, 4.135, 5.0) (1, 5, 1.450, 3.659, 2.624) (1.806, 3.490, 2.0) (1, 3, 1.227, 3.452, 2.663) (1.716, 3.002, 0.9) (1, 3, 1.035, 3.362, 2.703) (1.952, 3.062, 0.3) (2, 2, 1.869, 2.938, 2.744) (2.151, 2.210, 0.1)

(3, 12, 1.560, 5.785, 2.757) (4.474, 6.057, 14.4) (1, 9, 1.507, 5.176, 2.588) (2.238, 3.847, 4.1) (1, 5, 1.494, 4.839, 2.663) (1.714, 3.527, 1.8) (1, 3, 1.255, 4.375, 2.766) (1.661, 3.259, 0.8) (1, 3, 1.063, 4.760, 2.804) (1.894, 3.694, 0.3) (2, 2, 1.903, 2.920, 3.150) (2.124, 2.186, 0.1)

(2, 13, 1.442, 5.278, 2.696) (3.965, 5.232, 12.7) (1, 9, 1.543, 4.265, 2.576) (2.130, 3.703, 3.8) (1, 5, 1.516, 5.190, 2.678) (1.667, 3.480, 1.7) (1, 3, 1.273, 5.412, 2.816) (1.626, 3.296, 0.8) (1, 3, 1.079, 5.540, 2.856) (1.858, 3.756, 0.3) (2, 2, 1.982, 3.121, 2.981) (2.100, 2.249, 0.1)

(2, 13, 1.506, 4.124, 2.660) (3.726, 4.936, 12.0) (1, 9, 1.563, 5.460, 2.560) (2.076, 3.638, 3.7) (1, 5, 1.530, 5.309, 2.682) (1.637, 3.448, 1.7) (1, 3, 1.283, 5.249, 2.837) (1.607, 3.287, 0.8) (1, 3, 1.088, 5.103, 2.878) (1.837, 3.743, 0.3) (2, 2, 2.001, 3.126, 3.049) (2.092, 2.245, 0.1)

(2, 13, 1.580, 6.325, 2.605) (3.483, 4.631, 11.4) (1, 8, 1.525, 5.688, 2.592) (2.018, 3.444, 3.6) (1, 5, 1.541, 5.864, 2.678) (1.617, 3.424, 1.7) (1, 3, 1.296, 5.957, 2.853) (1.585, 3.279, 0.8) (1, 3, 1.098, 5.499, 2.896) (1.817, 3.755, 0.3) (2, 2, 2.035, 3.439, 2.898) (2.083, 2.394, 0.1)

nEWMA ¼ 1 0.5 26.47

6

1.0

9.58

3

1.5

5.18

2

2.0

3.35

2

2.5

2.38

2

3.0

1.78

2

nEWMA ¼ 3 0.5 11.89 11 1.0

4.16

5

1.5

2.24

4

2.0

1.42

3

2.5

1.10

3

3.0

1.01

3

n

parameters must attain this ARL1 value, for d ¼ 1.0 so that the two charts have equal sensitivity for this shift. At the same time, the DS X chart with estimated and known (m ¼ þ 1) parameters is designed to match the two ARL points of the Shewhart X chart with known parameters. The ARL points are ARL0 ¼ t ¼ 370.4 and an appropriate ARL1 value, where a suitable sample size of the Shewhart X chart, nX (see the third column in Table 4) is chosen such that the X chart’s ARL1 is as close as possible to that of the EWMA chart (the ARL1 value in the second column of Table 4), for the n same shift d . The ‘‘ 73’’ standard deviation width is used for the Shewhart X chart. For practical implementation, suppose that ARL0 ¼ t ¼ 370.4, n ARL1 ¼ e ¼ 2.24, d ¼ 1.5 and m ¼ 10 are considered, then a practitioner needs to collect 10 samples, each having 4 ðn ¼ nX ¼ 4Þ measurements to estimate m0 and s0 , for subsequent use in Phase-II process monitoring (see Table 4). The optimal parameter combination (n1 ¼ 1, n2 ¼ 5, L ¼ 1.450, L1 ¼ 3.659, L2 ¼ 2.624) n which minimizes ASS0 , for d ¼ 1.5, is used to optimally design the DS X chart with estimated parameters in the Phase-II analysis. These optimal chart parameters give ASS0 ¼ 1.806, ASS1 ¼ 3.490 and SDRL1 ¼ 2.0. Concerning the optimal (n1 , n2 ) pair, the difference between the cases with estimated and known parameters is minimal, except for n the cases ðm, d Þ A {(10, 0.5), (20, 0.5)} when nEWMA ¼ 3 (see Table 4). n However, no trend exists for L, L1 and L2 as m or d changes. It is obvious that the ðASS0 ,ASS1 ,SDRL1 Þ values approach the corresponding values with known parameters when m increases. An interesting n observation is that for moderate and large shifts ðd Z1:0Þ, the difference between ðASS0 ,ASS1 Þ values corresponding to the estimated and known-parameter cases is minimal. In contrast, for small dn ð r0:5Þ and mð r 20Þ values, the ðASS0 ,ASS1 Þ values corresponding

to the estimated-parameter case are larger than those of the knownparameter case (see Table 4). n The cell for ðm ¼ 10, d ¼ 3Þ when nEWMA ¼ 1 (see Table 4) is empty because all the permissible (n1 , n2 , L, L1 , L2 ) combinations cannot satisfy constraints (28) and (29) simultaneously, as all the ARL1 values obtained are lower than 1.78 when ARL0 is fixed as 370.4. This phenomenon is expected as the DS X chart is more sensitive towards a large shift than the EWMA chart.

5.3. Computation of optimal chart parameters for minimizing AEQL The proposed optimal design of the DS X chart with estimated parameters to minimize AEQL is conducted using the following model: Specifications: (i) Set a desired value of the in-control ARL ðARL0 Þ: t (ii) Set a desired value of the in-control ASS ðASS0 Þ which coincides with the Phase-I sample size: n (iii) Set the number of Phase-I samples: m (iv) Set a desired lower bound of the mean shift: dmin (v) Set a desired upper bound of the mean shift: dmax The value of dmin (dmin 4 0) is selected such that process shifts smaller than dmin can be ignored in order to avoid over-correction that may introduce extra variability into the process (Woodall, 1985). The value of dmax is decided based on the quality practitioner’s knowledge about the process, i.e. the maximum possible mean shift in the process. Then find  sample sizes, for both the first and second samples: n1 and n2

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 warning and control limits for the DS X chart: L, L1 and L2 such that the objective function is minimized, which is mathematically expressed as Min

n1 ,n2 ,L,L1 ,L2

AEQL

ð31Þ

subject to constraints ARL0 ¼ t

ð32Þ

ASS0 ¼ n

ð33Þ

1 rn1 o n on1 þ n2 r nmax

ð34Þ

By using the above optimization model (31)–(34), the optimal (n1 , n2 , L, L1 , L2 ) combination for the DS X chart with estimated parameters is obtained as follows:

353

n2 ¼ 12, L ¼ 1.416, L1 ¼ 5.735, L2 ¼ 2.811) combination is used to design the chart in the Phase-II process monitoring. These optimal chart parameters give AEQL ¼ 2.582 for the interval of mean shifts 0.5 r d r2.0. From Table 5, we observe that the difference for the optimal (n1 , n2 ) pair between the estimated-parameter case and the known-parameter case is small, except for the cases ðn,m, dmin , dmax Þ A {(5, 10, 0.75, 2.00), (5, 10, 0.75, 3.00)}. It is apparent from Table 5 that the AEQL value for the smaller shift domain is generally lower than that for the larger shift domain. For fixed n, dmin and dmax , the AEQL value of the estimatedparameter case is higher and approaches that of the knownparameter case as m increases. Also, the difference in the AEQL values between the cases of known and estimated parameters decreases as n or m increases.

6. An illustrative example (a) This step is similar to Step (a) presented in Section 5.1. (b) For any desired out-of-control shift range ð0 o dmin o dmax Þ, the optimal (n1 , n2 , L, L1 , L2 ) combination which minimizes AEQL is identified from the set of parameter combinations found in Step (a). Table 5 displays the optimal (n1 , n2 , L, L1 , L2 ) combinations listed in the first row of each cell, together with their corresponding AEQL value listed in the second row of each cell, for various combinations of (n, m, dmin , dmax ). Note that ARL0 ¼ t ¼ 370.4 (constraint (32)) and ASS0 ¼ n A {3, 5} (constraint (33)) are attained for all the cases considered in Table 5. Besides providing the optimal chart parameters for the DS X chart, Table 5 also enlightens a practitioner about the loss in quality cost arose from the out-of-control conditions and the overall effectiveness of the chart within a mean shift domain. Additionally, the chart having a small AEQL value implies that its ARL1 value at each d point is generally small. Suppose that ARL0 ¼ 370.4, ASS0 ¼ n ¼ 5, m ¼ 20, dmin ¼ 0.5 and dmax ¼ 2.0 are considered, then the optimal (n1 ¼ 3,

An example is provided to illustrate the construction of the optimal DS X chart with estimated parameters. Three designs are considered. The first design minimizes ARL1 ðdopt Þ, the second design minimizes ASS0 and the third design minimizes AEQL. By means of the Statistical Analysis System (SAS) program, the data are generated from a normal distribution with mean m0 ¼ 100 and standard deviation s0 ¼ 5. Various summary statistics for the Phase-I and Phase-II data of these three designs by minimizing ARL1 ðdopt Þ, ASS0 and AEQL, are provided in Table 6. The following steps explain the construction of the optimal DS X chart with estimated parameters for minimizing ARL1 ðdopt Þ:

(1) Assume that ARL0 ¼ 370.4 and dopt ¼ 1.0 are desired. (2) Also, assume that in the Phase-I process, m ¼ 20 samples, each of size n ¼ 3 are used to estimate process parameters. (3) An analysis to ensure that the Phase-I data are in-control is made by using the Bonferroni-type adjustment (Ryan, 2000) which is a viable alternative to Phase-I control charts. The charts’ limits of

Table 5 A (n1 , n2 , L, L1 , L2 ) combination (first row of each cell) and AEQL value (second row of each cell) of the optimal DS X chart with estimated parameters when ARL0 ¼ 370.4, ASS0 ¼ n A {3, 5}, m A {10, 20, 40, 80, þ 1} and various combinations of (dmin , dmax ). n dmin

dmax m ¼ 10

3 0.25 2.00 0.25 3.00 0.50 2.00 0.50 3.00 0.75 2.00 0.75 3.00

5 0.25 2.00 0.25 3.00 0.50 2.00 0.50 3.00 0.75 2.00 0.75 3.00

m ¼ 20

m ¼ 40

m ¼ 80

m ¼ þ 1 (known-parameter case)

(2, 11, 1.761, 4.488, 2.526) 5.014 (2, 10, 1.711, 4.547, 2.571) 5.614 (2, 9, 1.656, 4.362, 2.617) 4.077 (2, 8, 1.592, 4.421, 2.665) 5.090 (2, 7, 1.519, 4.360, 2.713) 3.647 (2, 7, 1.519, 4.242, 2.714) 4.932

(2, 11, 1.725, 5.312, 2.580) 4.108 (2, 10, 1.678, 5.602, 2.623) 5.022 (2, 9, 1.624, 5.381, 2.668) 3.625 (2, 8, 1.563, 5.331, 2.714) 4.802 (2, 8, 1.563, 4.980, 2.714) 3.412 (2, 7, 1.492, 5.147, 2.762) 4.789

(2, 11, 1.708, 5.535, 2.593) 3.749 (2, 10, 1.661, 5.443, 2.636) 4.785 (2, 9, 1.609, 5.513, 2.682) 3.439 (2, 8, 1.549, 5.815, 2.729) 4.683 (2, 8, 1.549, 5.815, 2.729) 3.306 (2, 7, 1.479, 5.337, 2.777) 4.724

(2, 11, 1.691, 5.127, 2.584) 3.437 (2, 10, 1.645, 5.125, 2.629) 4.576 (2, 9, 1.593, 5.491, 2.677) 3.267 (2, 8, 1.534, 5.196, 2.727) 4.571 (2, 7, 1.465, 5.230, 2.778) 3.202 (2, 7, 1.465, 5.230, 2.778) 4.660

1.450, 5.294, 2.772) (3, 12, 1.416, 5.735, 2.811) 3.000 1.450, 5.186, 2.772) (3, 12, 1.416, 5.735, 2.811) 4.221 1.450, 4.704, 2.772) (3, 12, 1.416, 5.735, 2.811) 2.582 1.450, 4.664, 2.772) (3, 12, 1.416, 5.735, 2.811) 4.093 1.398, 4.314, 2.798) (4, 9, 1.643, 4.772, 2.834) 2.541 1.398, 4.263, 2.798) (4, 9, 1.643, 4.772, 2.834) 4.231

(3, 12, 1.400, 5.248, 2.813) 2.692 (3, 12, 1.400, 5.248, 2.813) 4.024 (4, 11, 1.718, 5.355, 2.772) 2.469 (4, 11, 1.718, 5.355, 2.772) 4.019 (4, 9, 1.618, 5.484, 2.842) 2.480 (4, 9, 1.618, 5.484, 2.842) 4.196

(3, 12, 1.391, 5.337, 2.804) 2.562 (3, 12, 1.391, 5.337, 2.804) 3.941 (4, 11, 1.704, 5.242, 2.766) 2.412 (4, 11, 1.704, 5.242, 2.766) 3.984 (4, 9, 1.606, 5.196, 2.838) 2.450 (4, 9, 1.606, 5.196, 2.838) 4.179

(3, 12, 1.383, 5.280, 2.781) 2.441 (4, 11, 1.691, 5.127, 2.745) 3.860 (4, 11, 1.691, 5.127, 2.745) 2.357 (4, 11, 1.691, 5.127, 2.745) 3.951 (4, 9, 1.593, 5.491, 2.821) 2.421 (4, 9, 1.593, 5.491, 2.821) 4.162

(2, 10, 1.773, 3.784, 2.426) 7.739 (2, 9, 1.715, 3.847, 2.478) 7.385 (2, 9, 1.709, 3.562, 2.487) 5.538 (2, 8, 1.644, 3.621, 2.537) 5.999 (2, 7, 1.563, 3.502, 2.594) 4.272 (2, 7, 1.561, 3.431, 2.598) 5.310 (3, 12, 3.851 (3, 12, 4.764 (3, 12, 2.893 (3, 12, 4.282 (3, 11, 2.674 (3, 11, 4.313

354

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Table 6 Summary statistics of the Phase-I and Phase-II dataset for an illustrative example. Phase I

Phase II The first design (minimizes ARL1 )

Sampling time, t X t

St

Sampling time, t Y 1t

Z^ 1t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2.634 7.086 2.708 3.290 3.681 7.046 5.241 9.530 5.365 7.713 1.105 0.917 3.932 6.115 7.356 5.646 4.253 3.848 5.565 5.492

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

0.558 1.350 0.474 0.528  0.755  0.050 0.747  0.089  0.871  0.402 1.535 2.276 1.595 1.887 2.522 2.024

99.954 105.719 98.766 95.421 99.733 100.480 98.497 97.483 103.274 96.051 98.285 101.177 101.413 94.987 103.673 99.216 100.308 95.256 99.217 100.214

101.583 104.595 101.262 101.467 96.582 99.267 102.300 99.116 96.141 97.926 105.300 108.124 105.531 106.641 109.061 107.162

Yt

Z^ t

106.065 4.070 105.954 4.002 105.334 3.620 104.922 3.366

The second design (minimizes ASS0 ) Y 1t

Z^ 1t

99.684 103.481 98.214 110.977 101.248 98.511 100.022 103.456 101.144 98.475 104.757 98.035 104.248 103.196 102.655 110.730 104.026 102.245

0.042 0.747  0.231 2.139 0.333  0.175 0.105 0.743 0.313  0.182 0.984  0.264 0.890 0.695 0.594 2.094 0.849 0.518

Z^ t

Yt

101.392

107.840

0.881

The third design (minimizes AEQL) Y 1t

Z^ 1t

101.583 104.595 101.262 101.467 96.582 99.267 102.300 99.116 96.141 97.926 105.300 103.077 108.328 107.524

0.558 1.350 0.474 0.528 – 0.755 – 0.050 0.747 – 0.089 – 0.871 -0.402 1.535 0.951 2.330 2.119

Yt

Z^ t

106.056

3.677

105.929 107.155

3.606 4.289

3.814

Remarks: The boldfaced values denote the out-of-control cases.

X Chart

Xt

S Chart

St

110

UCLX = 108.439

105

14

UCLS = 12.134

12 10

100

8

95

6

LCLX = 90.473

90 85

4 2 0

1

3

5

7

9

11

13

15

17

19

Sampling time, t

1

3

5

7

9

11

13

15

17

19

LCLS = 0

Sampling time, t

Fig. 2. An analysis of the Bonferroni-adjusted (a) X and (b) S charts, in evaluating the Phase-I data.

the retrospective Bonferroni-adjusted X and S charts (see Fig. 2) pffiffiffi are calculated using UCLX =LCLX ¼ X 7Z FAP=ð2mÞ ðS=c4 Þ= n and qffiffiffiffiffiffiffiffiffiffiffiffi UCLS =LCLS ¼ S 7 Z FAP=ð2mÞ 1c24 ðS=c4 Þ, respectively, where X ¼ 99.456 is the sample grand average, S ¼ 4.926 is the average of the m sample standard deviations, Z B is the ð1BÞ100th percentage point of the standard normal distribution, c4 is an unbiased constant and FAP is the false alarm probability. Here, FAP is set as 0.1025, implying that each chart gives an overall probability of false alarm of at most 0.1025. In this example, 0.0027 is the probability of a false alarm at each sampling time, thus, FAP ¼ 1ð10:0027Þ40 ¼ 0:1025 is the probability of at least one false alarm in 40 samples. (4) Fig. 2(a) and (b) shows that the Phase-I data are in-control. ^ 0 ¼ 99.456 and s ^ 0 ¼ 5.385, which are Thus, we deduce that m computed using Eqs. (9) and (10), respectively. (5) From Table 3, for m ¼ 20, n ¼ 3 and dopt ¼ 1.0, the optimal chart parameters for the DS X chart with estimated parameters which give ARL0 ¼ 370.4 and ASS0 ¼ 3 are

ðn1 ,n2 ,L,L1 ,L2 Þ ¼ (2, 9, 1.655, 4.050, 2.620). These optimal chart parameters are used for process monitoring in Phase-II. A similar procedure is adopted for the second and third designs, i.e. by minimizing ASS0 and minimizing AEQL, respectively. The only difference is in Step (5). For the second design, n n assume that ARL0 ¼ 370.4, ARL1 ðd Þ ¼ 9.58 and d ¼ 1.0 are intended, then Table 4 suggests the use of optimal parameters ðn1 ,n2 ,L,L1 ,L2 Þ ¼ (1, 5, 1.999, 3.733, 2.329) as m ¼ 20 and n ¼ nX ¼ 3. For the third design, since d ¼ 1.0 is within all the shift domains considered in Table 5, the optimal chart parameters having the smallest AEQL will be selected for this example. Therefore, when ARL0 ¼ 370.4, m ¼ 20, n ¼ 3 and the shift domain 0.75r d r2.00, the (n1 ¼ 2, n2 ¼ 7, L ¼ 1.519, L1 ¼ 4.360, L2 ¼ 2.713) combination is used to optimally design the DS X chart with estimated parameters (see Table 5). The randomly generated Phase-II data consist of 16 samples of size (n1 ¼ 2, n2 ¼ 9), 18 samples of size (n1 ¼ 1, n2 ¼ 5) and 14 samples of size (n1 ¼ 2, n2 ¼ 7), for the first, second and third designs, respectively. For all the three designs, measurements for the first 10 sampling times (t ¼ 21 to 30) are

M.B.C. Khoo et al. / Int. J. Production Economics 144 (2013) 345–357

generated based on an in-control condition, while measurements for the subsequent sampling times (t ¼ 31 onwards) are generated with d ¼ 1.0. The DS X chart with estimated parameters for minimizing ARL1 ðdopt Þ (first design), minimizing ASS0 (second design) and minimizing AEQL (third design) are plotted in Fig. 3(a)–(c), respectively, where the solid dots denote Z^ 1t while the hollow dots represent Z^ t . The DS X chart for the first design detects the first out-of-control signal at sampling time t ¼ 32 as Z^ 32 ¼ 4.070 4L2 ¼ 2.620, while the second design signals at sampling time t ¼ 36 as Z^ 36 ¼ 3.814 4L2 ¼ 2.329. The detection speed in the first design (t ¼ 32) is faster than the second design (t ¼ 36) as the first design minimizes ARL1 ðdopt Þ. Concerning the number of observations sampled (t ¼ 21 onwards), the second design requires less observations (only 26 observations) compared to the first design (33 observations) as the former minimizes ASS0 . For the third design, it is observed that sample point 31 overshoots the control limit L2 (i.e. Z^ 31 ¼ 3.677 4L2 ¼ 2.713), confirming an out-of-control condition. Then immediate actions should be taken to find and remove any special causes.

355

7. Conclusions This paper shows that the ARL, SDRL, ASS and AEQL properties of the DS X chart with estimated parameters are quite different compared to those of known parameters, except for the case with a large number of Phase-I samples (m Z 80). A large number of Phase-I samples (m Z80) is needed so that the DS X chart with estimated parameters behaves like the chart with known parameters. This paper also presents the methodologies and results for choosing the optimal chart parameters of the DS X chart with estimated parameters by minimizing (i) ARL1 ðdopt Þ, (ii) ASS0 and (iii) AEQL, for different combinations of the sample size, number of Phase-I samples and shift. The third design algorithm, i.e. minimizing AEQL, of the DS X chart with estimated parameters aims at optimizing the overall statistical performance and the crucial quality cost. It should be pointed out that the results in this paper require the assumptions of a normal underlying distribution and the independence of the data. The results need to be re-examined if these assumptions are not satisfied.

Acknowledgments Zˆ1t / Zˆt 5 4 3 2 1 0 -1 -2 -3 -4 -5

L1 L2 L

–L –L2 –L1 21

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Sampling time, t

Zˆ1t / Zˆt 5 4 3 2 1 0 -1 -2 -3 -4 -5

L1 L2 L

–L –L 2 – L1 21

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Sampling time, t

Zˆ1t / Zˆt 5 4 3 2 1 0 -1 -2 -3 -4 -5

L1 L2 L

–L –L2 –L1 21

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25

26

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Sampling time, t Fig. 3. The DS X chart with estimated parameters for minimizing (a) ARL1 ðdopt Þ, (b) ASS0 and (c) AEQL.

This research is conducted while the first author is on sabbatical leave at the School of Mathematical Sciences, Universiti Sains Malaysia (USM) and it is supported by the USM Fundamental Research Grant Scheme (FRGS), No. 203/PMATHS/ 6711232 and the USM Research University Postgraduate Research Grant Scheme (USM-RU-PRGS), No. 1001/PMATHS/844087.

Appendix A Derivation of the conditional probabilities P^ a1 and P^ a2 for the DS X chart with estimated parameters Define pffiffiffiffiffi ^ 0 Þ n1 ðY 1t m , ðA:1Þ Z^ 1t ¼ s^ 0 where Y 1t ¼ Snj 1¼ 1 Y 1t,j =n1 is the sample mean of the first sample at sampling time t, computed from the Phase-II process. Here, Y 1t,j denotes the jth observation in the first sample at sampling time t, where Y 1t,j  Nðm0 þ ds0 , s20 Þ. If d ¼ 0, the process is in-control, ^ 0,s ^ 0 Þ be the otherwise the process is out-of-control. Let F Z^ 1t ðz9m ^ 0 and s ^ 0 . Then conditional cdf of Z^ 1t , given m ^ 0,s ^ 0 Þ ¼ PrðZ^ 1t r z9m ^ 0,s ^ 0Þ F Z^ 1t ðz9m   s^ 0 ^ 0 þ pffiffiffiffiffi z : ¼ Pr Y 1t r m n1

ðA:2Þ

Since Y 1t,j  Nðm0 þ ds0 , s20 Þ, then Y 1t  Nðm0 þ ds0 , s20 =n1 Þ. Consequently,    pffiffiffiffiffi   n1 s ^0 pffiffiffiffiffi ^ 0,s ^ 0Þ ¼ F m ^ 0 m0 F Z^ 1t ðzm þ zd n1

s0

s0

 rffiffiffiffiffiffiffi  pffiffiffiffiffi n1 þ Vzd n1 , ¼F U mn

ðA:3Þ

where U and V are defined in Eqs. (13) and (14), respectively. ^ 0,s ^ 0 Þ is obtained as Thus, P^ a1 ¼ PrðZ^ 1t A I1 9m  rffiffiffiffiffiffiffi   rffiffiffiffiffiffiffi  pffiffiffiffiffi pffiffiffiffiffi n1 n1 ðA:4Þ þ VLd n1 F U VLd n1 : P^ a1 ¼ F U mn mn Next, define pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 0 Þ n1 þ n2 ðY t m , Z^ t ¼ s^ 0

ðA:5Þ

356

M.B.C. Khoo et al. / Int. J. Production Economics 144 (2013) 345–357

where Y t ¼ ðn1 Y 1t þ n2 Y 2t Þ=ðn1 þ n2 Þ and Y 2t ¼ Snj 2¼ 1 Y 2t,j =n2 is the sample mean of the second sample at sampling time t, computed ^ 0,s ^ 0Þ from the Phase-II process. Then P^ a2 ¼ PrðZ^ t A I4 and Z^ 1t A I2 9m is given by Z ^ 0,s ^ 0 Þdz, ðA:6Þ P^ a2 ¼ P^ 4 f Z^ 1t ðz9m

By taking all the different d within the shift domain into consideration, AEQL for the DS X chart with estimated parameters is obtained as Z dmax AEQL ¼ QLðdÞf d ðdÞdd dmin

z A I2

^ 0,s ^ 0 Þ and f Z^ ðz9m ^ 0,s ^ 0 Þ is the pdf of where P^ 4 ¼ PrðZ^ t A I4 9Z^ 1t ¼ z, m 1t ^ 0,s ^ 0 Þ in Eq. (A.3), i.e. Z^ 1t , derived from F Z^ 1t ðz9m  rffiffiffiffiffiffiffi  pffiffiffiffiffi n1 ^ 0,s ^ 0Þ ¼ V f U ðA:7Þ þ Vzd n1 : f Z^ 1t ðz9m mn To evaluate Eq. (A.6), P^ 4 is solved as ^ 0,s ^ 0 Þ: P^ 4 ¼ PrðL2 r Z^ t rL2 9Z^ 1t ¼ z, m

ðA:8Þ

By substituting Z^ t (see Eq. (A.5)) into Eq. (A.8) and using the relationship Y t ¼ ðn1 Y 1t þn2 Y 2t Þ=ðn1 þn2 Þ, followed by rearranging the inequality, we obtain   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi L2 n1 þn2 z n1 L2 n1 þn2 z n1  P^ 4 ¼ Pr m^ 0 , s^ 0 , r Z^ 2t r pffiffiffiffiffi pffiffiffiffiffi  n2 n2 ðA:9Þ where pffiffiffiffiffi ^ 0 Þ n2 ðY 2t m : Z^ 2t ¼ ^ s0

ðA:10Þ

By substituting Z^ 2t into Eq. (A.9) and due to the fact that Y 2t  Nðm0 þ ds0 , s20 =n2 Þ, Eq. (A.9) simplifies to  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi  rffiffiffiffiffiffiffi  pffiffiffiffiffi n2 L2 n1 þ n2 z n1 þV d n2 P^ 4 ¼ F U pffiffiffiffiffi mn n2  rffiffiffiffiffiffiffi   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi n2 L2 n1 þ n2 þ z n1 ðA:11Þ V d n2 , F U pffiffiffiffiffi mn n2 where U and V are defined in Eqs. (13) and (14), respectively.

Appendix B

ðB:1Þ

where ‘ðdÞ is the extra quadratic loss (the quadratic loss owing to the shift d minus the quadratic loss when the process is incontrol) per unit, i.e;

‘ðdÞ ¼ K c ½s^ 20 þ ðm1 m^ 0 Þ2 K c s^ 20 ¼ K c s^ 20 d2 :

ðB:2Þ

Here, K c is the constant cost depending on an individual process. ^ 20 ¼ s20 V 2 (see Eq. (14)), Eq. (B.2) becomes Due to the fact that s

‘ðdÞ ¼ K c s20 V 2 d2 :

ðB:3Þ

Also, NðdÞ is the average number of units produced in an outof-control condition, i.e NðdÞ ¼ gh  ARLðdÞ;

Z

dmax

dmin

Z

þ1

1

Z 0

þ1

d2 v2



1

1P^ a

 f U ðuÞf V ðvÞf d ðdÞdvdudd;

ðB:6Þ 2 0

where f d ðdÞ is the pdf of the mean shift. The product of ghK c s is a constant and has no influence on the optimal design algorithm or the performance comparison; hence, for simplicity, it will be omitted. Then Eq. (B.6) is simplified to   Z dmax Z þ 1 Z þ 1 1 AEQL ¼ d2 v2 f U ðuÞf V ðvÞf d ðdÞdvdudd: 1P^ a dmin 1 0 ðB:7Þ In practice, since the data of the out-of-control cases are sparse and become obsolete when the assignable causes are identified and eliminated, it is very difficult, if not impossible, to estimate the probability distribution of the shift size (Ou et al., 2012b). To account for this problem, many researchers (Reynolds and Stoumbos, 2004; Wu et al., 2009, 2011; Ou et al., 2012a, 2012b) assumed that all the process mean shifts occur with equal probability and have a uniform distribution Uðdmin , dmax Þ. Therefore, Eq. (B.7) is further simplified to   Z dmax Z þ 1 Z þ 1 1 1 AEQL ¼ d2 v2 f U ðuÞf V ðvÞdvdudd: dmax dmin dmin 1 0 1P^ a ðB:8Þ Note that in this paper, AEQL is computed by Eq. (B.8), which is based on the assumption of the uniform distribution of the mean shift. However, other pdfs of the mean shift can be considered according to the quality practitioner’s knowledge about the process being monitored.

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Derivation of formulae to calculate the AEQL of the DS X chart with estimated parameters Let QLðdÞ in Eq. (B.1) be the loss incurred during an out-ofcontrol condition because of a particular shift d (Wu et al., 2009). QLðdÞ ¼ ‘ðdÞ  NðdÞ;

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ðB:4Þ

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