A new nonparametric CUSUM mean chart - Wiley Online Library
Research Article Published online 29 December 2010 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/qre.1171
A New Non-parametric CUSUM Mean Chart Su-Fen Yanga ∗† and Smiley W. Chengb Not all data in practice came from a process with normal distribution. When the process distribution is non-normal or unknown, the commonly used Shewhart control charts are not suitable. In this paper, a new non-parametric CUSUM Mean Chart is proposed to monitor the possible small mean shifts in the process. The sampling properties of the new monitoring statistics are examined and the average run lengths of the proposed chart are examined. Two numerical examples are used to illustrate the proposed chart and compare with the two existing charts, assuming normality and Beta distribution, respectively. The CUSUM Mean Chart showed better detection ability than those two charts in monitoring and detecting small process mean shifts. Copyright © 2010 John Wiley & Sons, Ltd. Keywords: CUSUM chart; process mean; binomial distribution; beta distribution; ARL
1. Introduction
C
ontrol charts are commonly used tools to improve the quality of a process/service; such as the X-bar, R, EWMA, and CUSUM charts for variables data; p and c charts for attributes data. However, in order to properly construct the chart and study the chart’s behavior and assess its performance we need to know the sampling properties of the monitoring statistic. In most cases, a normal distribution or a distribution with a known form was assumed for variables data. Hence a question arises—What if the underlying process distribution is non-normal or unknown? How do we construct the proper chart(s) or evaluate their performances without normality or known distribution form? Hence we need to find an alternate way—using a non-parametric approach seems to be a good starting point. Only a few researches have been done in this area; see, for example, Ferrell1 , Bakir and Reynolds2 , Amin, et al.3 , Chakraborti et al. 4 , Altukife5, 6 ; Bakir7, 8 , Chakraborti and Eryilmaz9 , Chakraborti and Graham10 , Chakraborti and Van der Wiel11 , and Das and Bhattacharya12 . A major drawback of the Shewhart charts is its inability to detect small shifts; hence, a EWMA or CUSUM chart is used to rectify this deficiency. In this paper, we propose a new non-parametric version of the CUSUM chart for variables data to monitor the process mean, without assuming a process distribution. The paper is organized as follows: In Section 2, we discuss the construction of a newly proposed non-parametric CUSUM Mean Chart and its performance. In Section 3, we compare the performance of the CUSUM Mean Chart with two existing charts by numerical examples. In Section 4, we summarize the findings and give a recommendation.
2. The proposed non-parametric CUSUM Mean Chart Assume that a critical quality characteristic, X, has a mean . Let Y = X − and p = P(Y>0) = the ‘Process Proportion’. If the processes were in-control then p = p0 , or if the processes were out-of-control, that is had shifted, then p = p1 = p0 . If p0 is not given, it will be estimated using the preliminary data set (i.e. the Phase I of SPC). To monitor the process mean, a random sample of size n, X1 , X2 ,. . ., Xn , is taken from X. Define Yj = Xj −
Ij =
and
Let M be the total number of Yj >0, then M = in-control process.
1
if Yj >0
0
otherwise
for j = 1, 2,. . ., n.
n
j=1 Ij would follow a Binomial distribution with parameters (n, p0 ) for an
a Department
Copyright © 2010 John Wiley & Sons, Ltd.
Qual. Reliab. Engng. Int. 2011, 27 867--875
867
of Statistics, National Chengchi University, Muzha, Taipei 116, Taiwan of Statistics, University of Manitoba, Winnipeg, MB, Canada R3T 2N2 ∗ Correspondence to: Su-Fen Yang, Department of Statistics, National Chengchi University, Muzha, Taipei 116, Taiwan. † E-mail: [email protected] b Department
S.-F. YANG AND S. W. CHENG 2.1. The construction of the CUSUM Mean Chart Monitoring small process mean shifts is equivalent to monitoring small changes in process proportion, p. Let = |p0 −p1 |, >0, and we are interested in detecting the shift quickly. We propose a ‘Two-sided CUSUM Mean Chart’ as follows: First, we define two CUSUM monitoring statistics at time t + +Mt −(np0 +k1 )), Ct+ = max(0, Ct−1
t = 1, 2,. . .
(1)
− −(np0 −k1 )+Mt ), Ct− = min(0, Ct−1
t = 1, 2,. . .
(2)
where Mt is the number of (Yj >0) at time t, k1 is a reference value with k1 = n / 2, and the starting values are normally chosen as zero, i.e. C0+ = 0 and C0− = 0. Let H1 be the upper control limit (UCL) for Ct+ and −H1 the lower control limit (LCL) for Ct− of the Two-sided CUSUM Mean Chart. The center line is at 0. Both Ct+ and Ct− are plotted on the chart. If any Ct+ ≥ H1 or Ct− ≤ −H1 , the process is deemed to be out-of-control. The two design parameters of the chart, k1 and H1 , are so chosen so that they would satisfy certain required average run length (ARL). Note that the in-control ARL, denoted ARL0 , of the Two-sided CUSUM Mean Chart depends on the values of n, p0 , , k1 and H1 , i.e. ARL0 = f (n, p0 , , k1 , H1 ); hence, H1 can be expressed as H1 = g(ARL0 , n, p0 , , k1 ), 0
(wileyonlinelibrary.com) DOI: 10.1002/qre.1171
A New Non-parametric CUSUM Mean Chart Su-Fen Yanga ∗† and Smiley W. Chengb Not all data in practice came from a process with normal distribution. When the process distribution is non-normal or unknown, the commonly used Shewhart control charts are not suitable. In this paper, a new non-parametric CUSUM Mean Chart is proposed to monitor the possible small mean shifts in the process. The sampling properties of the new monitoring statistics are examined and the average run lengths of the proposed chart are examined. Two numerical examples are used to illustrate the proposed chart and compare with the two existing charts, assuming normality and Beta distribution, respectively. The CUSUM Mean Chart showed better detection ability than those two charts in monitoring and detecting small process mean shifts. Copyright © 2010 John Wiley & Sons, Ltd. Keywords: CUSUM chart; process mean; binomial distribution; beta distribution; ARL
1. Introduction
C
ontrol charts are commonly used tools to improve the quality of a process/service; such as the X-bar, R, EWMA, and CUSUM charts for variables data; p and c charts for attributes data. However, in order to properly construct the chart and study the chart’s behavior and assess its performance we need to know the sampling properties of the monitoring statistic. In most cases, a normal distribution or a distribution with a known form was assumed for variables data. Hence a question arises—What if the underlying process distribution is non-normal or unknown? How do we construct the proper chart(s) or evaluate their performances without normality or known distribution form? Hence we need to find an alternate way—using a non-parametric approach seems to be a good starting point. Only a few researches have been done in this area; see, for example, Ferrell1 , Bakir and Reynolds2 , Amin, et al.3 , Chakraborti et al. 4 , Altukife5, 6 ; Bakir7, 8 , Chakraborti and Eryilmaz9 , Chakraborti and Graham10 , Chakraborti and Van der Wiel11 , and Das and Bhattacharya12 . A major drawback of the Shewhart charts is its inability to detect small shifts; hence, a EWMA or CUSUM chart is used to rectify this deficiency. In this paper, we propose a new non-parametric version of the CUSUM chart for variables data to monitor the process mean, without assuming a process distribution. The paper is organized as follows: In Section 2, we discuss the construction of a newly proposed non-parametric CUSUM Mean Chart and its performance. In Section 3, we compare the performance of the CUSUM Mean Chart with two existing charts by numerical examples. In Section 4, we summarize the findings and give a recommendation.
2. The proposed non-parametric CUSUM Mean Chart Assume that a critical quality characteristic, X, has a mean . Let Y = X − and p = P(Y>0) = the ‘Process Proportion’. If the processes were in-control then p = p0 , or if the processes were out-of-control, that is had shifted, then p = p1 = p0 . If p0 is not given, it will be estimated using the preliminary data set (i.e. the Phase I of SPC). To monitor the process mean, a random sample of size n, X1 , X2 ,. . ., Xn , is taken from X. Define Yj = Xj −
Ij =
and
Let M be the total number of Yj >0, then M = in-control process.
1
if Yj >0
0
otherwise
for j = 1, 2,. . ., n.
n
j=1 Ij would follow a Binomial distribution with parameters (n, p0 ) for an
a Department
Copyright © 2010 John Wiley & Sons, Ltd.
Qual. Reliab. Engng. Int. 2011, 27 867--875
867
of Statistics, National Chengchi University, Muzha, Taipei 116, Taiwan of Statistics, University of Manitoba, Winnipeg, MB, Canada R3T 2N2 ∗ Correspondence to: Su-Fen Yang, Department of Statistics, National Chengchi University, Muzha, Taipei 116, Taiwan. † E-mail: [email protected] b Department
S.-F. YANG AND S. W. CHENG 2.1. The construction of the CUSUM Mean Chart Monitoring small process mean shifts is equivalent to monitoring small changes in process proportion, p. Let = |p0 −p1 |, >0, and we are interested in detecting the shift quickly. We propose a ‘Two-sided CUSUM Mean Chart’ as follows: First, we define two CUSUM monitoring statistics at time t + +Mt −(np0 +k1 )), Ct+ = max(0, Ct−1
t = 1, 2,. . .
(1)
− −(np0 −k1 )+Mt ), Ct− = min(0, Ct−1
t = 1, 2,. . .
(2)
where Mt is the number of (Yj >0) at time t, k1 is a reference value with k1 = n / 2, and the starting values are normally chosen as zero, i.e. C0+ = 0 and C0− = 0. Let H1 be the upper control limit (UCL) for Ct+ and −H1 the lower control limit (LCL) for Ct− of the Two-sided CUSUM Mean Chart. The center line is at 0. Both Ct+ and Ct− are plotted on the chart. If any Ct+ ≥ H1 or Ct− ≤ −H1 , the process is deemed to be out-of-control. The two design parameters of the chart, k1 and H1 , are so chosen so that they would satisfy certain required average run length (ARL). Note that the in-control ARL, denoted ARL0 , of the Two-sided CUSUM Mean Chart depends on the values of n, p0 , , k1 and H1 , i.e. ARL0 = f (n, p0 , , k1 , H1 ); hence, H1 can be expressed as H1 = g(ARL0 , n, p0 , , k1 ), 0
