# Treating Measurement Errors in the Run Rule Schemes Integrated with ShewhartX¯ Chart

Authors: TingTing Shan; WeiDong Huang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In modern quality control applications, there often exist significant measurement errors because observations are measured quickly in time order. As a result, the errors influence the power of a control chart to detect a given change in the process parameter(s) of a quality characteristic. In this paper, by using a covariate error model, the properties of the ShewhartX¯ chart integrated with run rules are investigated when errors exist in the measurement of quality characteristic. Two metrics, the average run length and 95% quantile of the run length, are adopted to evaluate the chart’s performance for different mean shifts and sample sizes. Numerous simulations are conducted, and the results indicate that the errors in the measurement significantly affect the performance of the run rule X¯ chart, especially when the errors are large. To reduce this negative effect on the run rule X¯ chart, measuring more times of each item in each subgroup and increasing the coefficient in the covariate error model are shown to be good choices for practitioners.


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## Body

## 1. Introduction

As it is known in statistical process control (SPC), the Shewhart chart is effective in detecting large shifts in the process but is insensitive to small or moderate shifts. Even though the Shewhart-type charts still received much attention for the simple representation and implementation, as an intermediate solution, Western Electric [1] suggested the run rule scheme integrated with Shewhart-type charts to improve the chart’s performance. Then, run rules combined with control charts were studied by many researchers, such as Divoky and Taylor [2]; Champ and Woodall [3]; Klein [4]; Fu et al. [5]; and so on. More recent works on control charts integrated with run rule schemes can be found in Hu and Castagliola [6]; Chew et al. [7]; Shongwe and Graham [8]; Shongwe and Graham [8]; Shongwe et al. [9]; Khaw and Chew [10]; Chew et al. [11]; Shongwe [12]; Malela-Majika and Rapoo [13]; Malela-Majika et al. [14]; Hu et al. [15]; and Ali et al. [16]. All these existing research studies focused on how to improve the chart’s performance by incorporating run rules. They all assumed that observations are measured with no errors in the quality characteristic.However, in the actual measuring equipment, an exact measurement of a quality characteristic is a rare phenomenon, causing measurement errors in practice. The properties of a control chart for monitoring observations with measurement errors differ from the chart for monitoring observations with precise measurements. Most existing works have only studied the measurement errors’ effect on a specific monitoring scheme. For example, by using a linear covariate model, Linna and Woodall [17] investigated the measurement errors’ effect on the Shewhart mean (X¯) and variance (S2) charts. By extending the covariate model to the multivariate case, Linna et al. [18] studied the performance of multivariate control charts when measurement errors exist. Using the same linear model introduced in Linna and Woodall [17], the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts’ performance for monitoring observations with measurement errors was studied in Maravelakis et al. [19] and Maravelakis [20]. More recently, considering the autocorrelation and measurement errors, Costa and Castagliola [21] studied the Shewhart X¯ chart’s performance and suggested a new skipping sampling technique to reduce the negative effect of autocorrelation. Hu et al. [22, 23] investigated the overall performance of the synthetic and VSS X¯ chart when measurement errors exist. For a full account of publications on measurement errors prior to 2017, readers are referred to Maleki et al. [24]. For the most recent works on control charts with measurement errors, we suggest the research in Cheng and Wang [25]; Shongwe et al. [26]; Nguyen et al. [27]; Nguyen et al. [28]; Thanwane et al. [29]; Thanwane et al. [30]; and so on. Even the above research works are on the control charts with measurement errors. To the best of our knowledge, no research on the run rule schemes integrated with Shewhart X¯ chart with measurement errors can be found.In this paper, we focus on the properties of 2-of-3 run rules integrated with Shewhart-type charts in the presence of measurement errors. The run rule schemes are set by using warning limits, and it gives a signal when the selected run rule pattern occurs. This research can be extended to other run rule schemes, which are not presented here. It is noted that this paper is mainly to investigate the run ruleX¯ chart’s performance when measurement errors exist, not to increase the chart’s performance.This paper is organized as follows. In Section2, the linearly covariate error model used to represent the true value and the measurement value of observations is introduced. Then, the 2-of-3 run rule X¯ chart under this measurement error model is presented in Section 3. The detailed effect of measurement errors on the run rule X¯ chart is presented in Section 4. Finally, some conclusions and future research directions are given in Section 6.


## 2. The Linearly Covariate Measurement Error Model

In a process,n≥1 consecutive items Yi,1,Yi,2,…,Yi,n of the quality characteristic Y are collected at each sampling point i=1,2,…. These items are assumed to be independent normal random variables with mean μ0 and standard deviation σ0, i.e., Yi,j∼Nμ=μ0+δσ0,σ0. δ is the mean shift parameter in the process. Due to the measurement errors, the true value of quality characteristic Yi,j is observed from m≥1 measurements Xi,j,1,Xi,j,2,…,Xi,j,m, with each Xi,j,k being equal to the following equation:(1)Xi,j,k=A+BYi,j+εi,j,k,where A and B are two constants from the analysis of measurement device and εi,j,k∼N0,σM is an independent random error term.In practice, if there are errors in the measurements of observations, it is suggested to measure each item several times and average these measurements for each item. By doing this, a smaller variance in the measurement error component than for a single measurement can be obtained. This can be simplified to the following equation:(2)X¯i=1mn∑j=1n∑k=1mXi,j,k=A+1nB∑j=1nYi,j+1m∑j=1n∑k=1mεi,j,k,where m is the number of measurements per item. It can be derived that the expection EX¯i and the variance VX¯i of X¯i,(3)EX¯i=A+Bμ0+δσ0,(4)VX¯i=1nB2σ02+σM2m.Moreover, considering that the measurement error varianceσM2 sometimes linearly depends on the process mean μ, ε is normally distributed with mean 0 and variance σM2=C+Dμ, with two known constants C and D. EX¯i and VX¯i are derived to be(5)EX¯i=A+Bμ0+δσ0,(6)VX¯i=1nB2σ02+C+Dμm.


## 3. The ShewhartX¯ Chart Integrated with Run Rules under Linearly Covariate Measurement Errors

Considering the constant measurement error variance in equations (3) and (4), by setting δ=0 in the in-control state, two warning limits, LWL (the lower warning limit) and UWL (the upper warning limit), are given as follows:(7)LWL=A+Bμ0−W1nB2σ02+σM2m,UWL=A+Bμ0+W1nB2σ02+σM2m.Second, considering the linearly increasing measurement error variance in equations (5) and (6), the warning limits of the chart are given as(8)LWL=A+Bμ0−W1nB2σ02+C+Dμm,UWL=A+Bμ0+W1nB2σ02+C+Dμm.In the implementation of the ShewhartX¯ chart integrated with 2-of-3 run rules, if two out of the three successive points fall above UWL or fall below LWL, an out-of-control signal is triggered. By modeling the 2-of-3 run rule chart with a Markov chain, Figure 1 shows all the transient states corresponding to the 2-of-3 run rules.Figure 1
The transient states of 2-of-3 run rules. (a) (1). (b) (2). (c) (3). (d) (4). (e) (5). (f) (6). (g) (7).
(a)(b)(c)(d)(e)(f)(g)When a new sample is collected and the corresponding sample mean (point) falls into one of the above seven transient states or into the 8th state, i.e., the absorbing state, where two out of the three successive points fall outside [LWL, UWL], then the run length properties of the Shewhart X¯ chart integrated with 2-of-3 run rules can be obtained by using the Markov chain matrix P:(9)P=Qr0T1=000pCpL00pU000000pCpL+pUpCpL00000pU00pUpCpL00000000pUpCpLpC000000pL+pU00pUpC000pL00000001,with the probabilities pL=PXi¯<LWL, pC=PLWL≤Xi¯≤UWL, and pU=PXi¯>UWL. Q refers to the transient probabilities between 7 states in the figure and the vector r=1−Q1, with 1=1,1,1,1,1,1,1T. For more details of the run rule charts, readers can refer to the related research works introduced in Section 1.For the case in equations (3) and (4), the transient probabilities are equal to(10)pL=Φ−W−δn1+γ2/B2m,(11)pC=ΦW−δn1+γ2/B2m−Φ−W−δn1+γ2/B2m,(12)pU=Φ−W+δn1+γ2/B2m,where γ2=σM2/σ02 is the measurement error ratio and Φ⋅ is the cumulative distribution function (c.d.f.) of the standard normal distribution.For the case in equations (5) and (6), the transient probabilities are equal to(13)pL=Φ−WB2σ02+C+Dμ0/m−Bδσ0nB2σ02+C+Dμ/m,(14)pC=ΦWB2σ02+C+Dμ0/m−Bδσ0nB2σ02+C+Dμ/m−Φ−WB2σ02+C+Dμ0/m−Bδσ0nB2σ02+C+Dμ/m,(15)pU=Φ−WB2σ02+C+Dμ0/m+Bδσ0nB2σ02+C+Dμ/m.From equations (10)–(15), it is shown that the parameter A has no influence on the chart, while in practice, as it has been explained in Linna and Woodall [17], this parameter should be taken into account for the process location or the process capability.Using the transition matrixQ in the Markov chain, the probability mass function (p.m.f.) fLℓ|Q,q and the c.d.f. FLℓ|Q,q of the run length distribution of the chart with measurement errors are given as(16)fLℓ|Q,q=qTQℓ−1r,FLℓ|Q,q=1−qTQℓ1,with q=0,0,0,1,0,0,0T being the initial probabilities of transient states.Furthermore, theARL of the chart can be derived as(17)ARL=qTI−Q−11,and the α∈0,1 quantile RLα of the Shewhart X¯ chart with 2-of-3 run rules under linearly covariate measurement error model can be obtained as (see Gan [31])(18)FLRLα−1|Q,q≤α,(19)FLRLα|Q,q>α.With equations (18) and (19), any quantile of the run length distribution can be computed. By setting α=0.95, the 95% quantile RL0.95 of the run length distribution can be obtained.


## 4. Influence of Linearly Covariate Measurement Errors on the Run Rule ShewhartX¯ Chart

Some characteristics of run length are usually used to show the control chart’s performance. Two metrics,ARL and RL0.95, are selected to evaluate the performance of the run rule X¯ chart. As it has been stated in Khoo et al. [32], by computing a higher percentile of the run length distribution, say RL0.95 in this paper, a practitioner can state with 95% confidence that when a process shift δ=0.5 occurs (see the case γ2=0, n=5 in Table 1), a signal will be triggered by the 50th sample. The information from the percentiles of the run length acts as practical guidance to study the behavior of control charts. The subscripts “0” and “1” of ARL represent the corresponding in-control and out-of-control performance of the chart, respectively. Without loos of generality, it is assumed that ARL0=370.4. Consequently, the parameter W of the run rule Shewhart X¯ chart with linearly covariate measurement errors satisfies the following constraint:(20)ARLm,n,B,γ2,W,δ=0=ARL0.Table 1
Out-of-controlARL,RL0.95 when ARL0=370.4, m=1, and B=1 for γ2∈0,0.1,0.2,0.3,0.5,0.7,1, δ∈0.1,0.3,0.5,0.7,1,1.5,2, and n∈1,3,5,7.
γ200.10.20.30.50.71δn=10.1(339.52, 1013)(342.12, 1021)(344.32, 1028)(346.20, 1033)(349.25, 1042)(351.62, 1050)(354.31, 1058)0.3(198.43, 591)(207.70, 618)(216.03, 643)(223.55, 666)(236.60, 705)(247.52, 738)(260.93, 778)0.5(100.86, 299)(109.07, 323)(116.82, 346)(124.15, 368)(137.66, 409)(149.80, 445)(165.86, 493)0.7(53.16, 156)(58.74, 172)(64.17, 189)(69.47, 205)(79.62, 235)(89.21, 264)(102.57, 304)1(23.30, 66)(26.19, 75)(29.10, 84)(32.01, 93)(37.82, 110)(43.58, 127)(52.03, 152)1.5(8.38, 22)(9.44, 25)(10.53, 28)(11.65, 32)(13.98, 39)(16.39, 46)(20.12, 57)2(4.33, 10)(4.78, 11)(5.26, 13)(5.76, 14)(6.81, 17)(7.93, 21)(9.71, 26)δn=30.1(290.13, 865)(296.08, 883)(301.21, 899)(305.67, 912)(313.07, 934)(318.95, 952)(325.81, 972)0.3(94.46, 279)(102.41, 303)(109.95, 326)(117.10, 347)(130.34, 387)(153.17, 455)(158.23, 470)0.5(32.98, 95)(36.86, 107)(40.71, 119)(44.53, 130)(52.03, 152)(59.32, 174)(69.82, 206)0.7(14.34, 40)(16.19, 45)(18.08, 51)(19.99, 57)(23.88, 68)(27.83, 80)(33.77, 98)1(5.93, 15)(6.63, 17)(7.36, 19)(8.12, 21)(9.71, 26)(11.37, 31)(13.98, 39)1.5(2.76, 5)(2.96, 6)(3.16, 6)(3.38, 7)(3.85, 9)(4.36, 10)(5.19, 13)2(2.14, 3)(2.19, 3)(2.26, 3)(2.33, 3)(2.50, 4)(2.68, 5)(3.01, 6)δn=50.1(252.34, 752)(260.12, 775)(266.95, 796)(272.98, 814)(283.15, 844)(291.40, 869)(301.21, 899)0.3(58.13, 171)(64.06, 188)(69.82, 206)(75.41, 222)(86.08, 254)(96.09, 284)(109.95, 326)0.5(17.62, 50)(19.87, 56)(22.15, 63)(24.50, 70)(29.10, 84)(33.76, 98)(40.71, 119)0.7(7.55, 20)(8.49, 22)(9.46, 25)(10.46, 28)(12.54, 34)(14.71, 41)(18.08, 51)1(3.49, 7)(3.81, 8)(4.15, 9)(4.51, 11)(5.26, 13)(6.07, 15)(7.36, 19)1.5(2.17, 3)(2.24, 3)(2.31, 3)(2.40, 4)(2.59, 5)(2.80, 5)(3.16, 1)2(2.01, 2)(2.02, 2)(2.03, 2)(2.05, 2)(2.09, 3)(2.15, 3)(2.26, 3)δn=70.1(222.52, 663)(231.30, 689)(239.10, 713)(246.07, 733)(258.01, 769)(267.85, 799)(279.77, 834)0.3(40.35, 117)(44.89, 131)(49.38, 144)(53.79, 158)(62.38, 183)(70.63, 208)(82.34, 243)0.5(11.49, 31)(12.97, 36)(14.49, 40)(16.04, 45)(19.22, 54)(22.47, 64)(27.43, 79)0.7(5.10, 12)(5.67, 14)(6.28, 16)(6.90, 18)(8.21, 22)(9.60, 26)(11.79, 32)1(2.70, 5)(2.88, 6)(3.07, 6)(3.28, 7)(3.72, 8)(4.20, 10)(4.99, 12)1.5(2.04, 2)(2.07, 3)(2.10, 3)(2.13, 3)(2.22, 3)(2.33, 3)(2.52, 4)2(2.00, 2)(2.00, 2)(2.00, 2)(2.01, 2)(2.02, 2)(2.03, 2)(2.07, 3)In this case, assumingδ=0 in equations (10)–(15), we can compute the only value of W using the “vert solvevert” function in MATLAB, irrespective of m, n, B, and γ2.


### 4.1. A Constant Measurement Error Variance

It can be seen from equations (17)–(19) that ARL1 and RL0.95 are functions of n, m, B, γ2, W, and δ. Table 1 presents ARL and RL0.95 of the run rule X¯ chart with linearly covariate error model for γ2∈0,0.1,0.2,0.3,0.5,0.7,1, δ∈0.1,0.3,0.5,0.7,1,1.5,2, and n∈1,3,5,7, when m=1 and B=1. The case γ2=0 corresponds to the “no measurement error” case. For example, when n=1, γ2=0, and δ=0, the chart parameter W=1.9293. With this value, we have ARL=198.43 and RL0.95=591 at δ=0.3. As it can be seen from Table 1, for a specified δ, ARL and RL0.95 increase when the measurement error ratio γ2 increases. This fact demonstrates the negative effect of the measurement errors on the run rule X¯ chart. For instance, for the example above but with γ2=1, we have ARL=260.93 and RL0.95=778. We can see that with the measurement error ratio increasing, larger ARL and RL0.95 are needed to detect a shift.The conclusions drawn above can also be seen from the four graphics in Figure2 where ARL and RL0.95 are plotted for δ∈0.5,1.5, n∈1,3,5,7 and γ2 when m=1 and B=1. This figure confirms the conclusion that larger ARL and RL0.95 are observed with increasing measurement errors.Figure 2
The influence ofγ2 on the run rule Shewhart X¯ chart when m=1, B=1, δ∈0.5,1.5, and n∈1,3,5,7.
(a)(b)(c)(d)Table2 presents ARL and RL0.95 of the chart for B∈1,2,3,4,5, δ∈0.1,0.3,0.5,0.7,1,1.5,2, and n∈1,3,5,7, when m=1 and γ2=1. It can be seen that for specified δ and n, the negative effect of measurement errors on the run rule X¯ chart decreases as B increases. Moreover, through the comparison between Tables 1 and 2, it can be seen that when B=5, the difference between ARL and SDRL corresponding to γ2=0 (no measurement error) and γ2=1 is negligible. For instance, when δ=0.3, m=1, γ2=1, n=1, and B=5 (Table 2), ARL=202.26 and RL0.95=602, while in Table 1, for the same values of m and δ, ARL=198.43 and RL0.95=591 when γ2=0.Table 2
Out-of-controlARL,RL0.95 when ARL0=370.4, m=1, and γ2=1 for n∈1,3,5,7, δ∈0.1,0.3,0.5,0.7,1,1.5,2, and B∈1,2,3,4,5.
B12345δn=10.1(354.31, 1058)(345.30, 1031)(342.39, 1022)(341.20, 1018)(340.61, 1017)0.3(260.93, 778)(219.88, 655)(208.67, 621)(204.34, 608)(202.26, 602)0.5(165.86, 493)(120.54, 357)(109.95, 326)(106.04, 314)(104.20, 309)0.7(102.57, 304)(66.84, 197)(59.35, 174)(56.66, 166)(55.41, 162)1.0(52.03, 152)(30.55, 88)(26.51, 76)(25.10, 72)(24.45, 70)1.5(20.12, 57)(11.09, 30)(9.56, 26)(9.04, 24)(8.80, 23)2.0(9.71, 26)(5.51, 14)(4.84, 12)(4.61, 11)(4.51, 11)δn=30.1(325.81, 972)(303.51, 905)(296.69, 885)(293.95, 877)(292.62, 873)0.3(158.23, 470)(113.57, 337)(103.27, 306)(99.48, 294)(99.69, 289)0.5(69.82, 206)(42.63, 124)(37.29, 108)(35.41, 103)(34.53, 100)0.7(33.77, 98)(19.03, 54)(16.40, 46)(15.49, 43)(15.08, 42)1.0(13.98, 39)(7.74, 20)(6.71, 17)(6.37, 16)(6.21, 16)1.5(5.19, 13)(3.27, 7)(2.98, 6)(2.88, 6)(2.84, 6)2.0(3.01, 6)(2.29, 3)(2.20, 3)(2.17, 3)(2.16, 3)δn=50.1(301.21, 899)(270.05, 805)(260.93, 778)(257.33, 767)(255.58, 762)0.3(109.95, 326)(72.63, 214)(64.70, 190)(61.85, 182)(60.52, 178)0.5(40.71, 119)(23.30, 66)(20.12, 57)(19.02, 54)(18.51, 52)0.7(18.08, 51)(9.96, 27)(8.59, 23)(8.13, 21)(7.92, 21)1.0(7.36, 19)(4.33, 10)(3.85, 9)(3.69, 8)(3.62, 8)1.5(3.16, 6)(2.36, 4)(2.25, 3)(2.21, 3)(2.20, 3)2.0(2.26, 3)(2.04, 2)(2.02, 2)(2.02, 2)(2.01, 2)δn=70.1(279.77, 834)(242.68, 723)(232.21, 692)(228.13, 680)(226.16, 674)0.3(82.34, 243)(51.59, 151)(45.39, 133)(43.19, 126)(42.17, 123)0.5(27.43, 79)(15.26, 43)(13.14, 36)(12.41, 34)(12.08, 33)0.7(11.79, 32)(6.59, 17)(5.74, 14)(5.46, 13)(5.33, 13)1.0(4.99, 12)(3.17, 6)(2.90, 6)(2.81, 5)(2.77, 5)1.5(2.52, 4)(2.11, 3)(2.07, 3)(2.06, 3)(2.05, 2)2.0(2.07, 3)(2.01, 2)(2.00, 2)(2.00, 2)(2.00, 2)In Figure3, ARL and RL0.95 of the run rule X¯ chart in the presence of measurement errors are plotted for δ∈0.5,1.5, n∈1,3,5,7, and B when m=1 and γ2=1. This figure also confirms the conclusion that with the value of B increasing, both ARL and RL0.95 decrease.Figure 3
The influence ofB on the run rule Shewhart X¯ chart when m=1, γ2=1, δ∈0.5,1.5, and n∈1,3,5,7.
(a)(b)(c)(d)Table3 presents ARL and RL0.95 for m∈1,2,3,4,5, δ∈0.1,0.3,0.5,0.7,1,1.5,2, and n∈1,3,5,7 when B=1 and γ2=1. It can be seen from Table 3 that as m increases, the negative effect of the measurement errors on the run rule X¯ chart decreases. For instance, for the case where δ=0.3, B=1, n=1, and γ2=1, ARL=260.93 and RL0.95=778 decrease down to ARL=216.03 and RL0.95=643, respectively, when m=1 increases up to m=5. So, it is concluded that if there is a relative high variability of the measurement error compared with the process variability, a large m is needed to reduce the negative effect of measurement errors on the run rule X¯ chart.Table 3
Out-of-controlARL,RL0.95 when ARL0=370.4, B=1, and γ2=1 for n∈1,3,5,7, δ∈0.1,0.3,0.5,0.7,1,1.5,2, and m∈1,2,3,4,5.
m12345δn=10.1(354.31, 1058)(349.25, 1042)(346.77, 1035)(345.30, 1031)(344.32, 1028)0.3(260.93, 778)(236.60, 705)(225.90, 673)(219.88, 655)(216.03, 643)0.5(165.86, 493)(137.66, 409)(126.51, 375)(120.54, 357)(116.82, 346)0.7(102.57, 304)(79.62, 235)(71.20, 210)(66.84, 197)(64.17, 189)1.0(52.03, 152)(37.82, 110)(32.98, 95)(30.55, 88)(29.10, 84)1.5(20.12, 57)(13.98, 39)(12.03, 33)(11.09, 30)(10.53, 28)2.0(9.71, 26)(6.81, 17)(5.93, 15)(5.51, 14)(5.26, 13)δn=30.1(325.81, 972)(313.07, 934)(307.03, 916)(303.51, 905)(301.21, 899)0.3(158.23, 470)(130.34, 387)(119.40, 354)(113.57, 337)(109.95, 326)0.5(69.82, 206)(52.03, 152)(45.79, 134)(42.63, 124)(40.71, 119)0.7(33.77, 98)(23.88, 68)(20.63, 59)(19.03, 54)(18.08, 51)1.0(13.98, 39)(9.71, 26)(8.38, 22)(7.74, 20)(7.36, 19)1.5(5.19, 13)(3.85, 9)(3.46, 7)(3.27, 7)(3.16, 6)2.0(3.01, 6)(2.50, 4)(2.36, 4)(2.29, 3)(2.26, 3)δn=50.1(301.21, 899)(283.15, 844)(274.83, 820)(270.05, 805)(266.95, 796)0.3(109.95, 326)(86.08, 254)(77.23, 228)(72.63, 214)(69.82, 206)0.5(40.71, 119)(29.10, 84)(25.22, 72)(23.30, 66)(22.15, 63)0.7(18.08, 51)(12.54, 34)(10.80, 29)(9.96, 27)(9.46, 25)1.0(7.36, 19)(5.26, 13)(4.63, 11)(4.33, 10)(4.15, 9)1.5(3.16, 6)(2.59, 5)(2.43, 4)(2.36, 4)(2.31, 3)2.0(2.26, 3)(2.09, 7)(2.05, 3)(2.04, 2)(2.03, 2)δn=70.1(279.77, 834)(258.01, 769)(248.23, 740)(242.68, 723)(239.10, 713)0.3(82.34, 243)(62.38, 183)(55.24, 162)(51.59, 151)(49.38, 144)0.5(27.43, 79)(19.22, 54)(16.57, 46)(15.26, 43)(14.49, 40)0.7(11.79, 32)(8.21, 22)(7.11, 18)(6.59, 17)(6.28, 16)1.0(4.99, 12)(3.72, 8)(3.35, 7)(3.17, 6)(3.07, 6)1.5(2.52, 4)(2.22, 3)(2.15, 3)(2.11, 3)(2.10, 3)2.0(2.07, 3)(2.02, 2)(2.01, 2)(2.01, 2)(2.00, 2)In Figure4, ARL and RL0.95 of the run rule X¯ chart in the presence of measurement errors are plotted for δ∈0.5,1.5, n∈1,3,5,7, and m when γ2=1 and B=1. This figure confirms the conclusion that by measuring the sample item several times, the negative effect of measurement errors on the run rule X¯ chart can be reduced to some extent.Figure 4
The effect ofm on the run rule Shewhart X¯ chart when B=1, γ2=1, δ∈0.5,1.5, and n∈1,3,5,7.
(a)(b)(c)(d)Since measuring the item in each sample several times can reduce the measurement errors’ negative effect on control charts, the run length performance of run ruleX¯ chart with linearly covariate error is presented in Figure 5, for the case of m=5 measurements per item, using γ2, δ∈0.5,1.5 and n∈1,3,5,7 when B=1. It can be noted that for fixed n, with m=5 measurements per item, ARL and RL0.95 only have a slightly increasing trend with γ2 increasing. Moreover, through the comparisons between the four graphics in Figure 2, the negative effect of measurement errors on the run rule X¯ chart is reduced by taking multiple measurements of m=5, especially for the cases when γ2≥0.5.Figure 5
The effect ofγ2 on the run rule Shewhart X¯ chart when m=5, B=1, δ∈0.5,1.5, and n∈1,3,5,7.
(a)(b)(c)(d)Finally, Figure6 presents the run length performance of the run rule X¯ chart for B, δ∈0.5,1.5, and n∈1,3,5,7 when γ2=1. From Figure 3, it is known that increasing B reduces the negative effect of measurement errors on the run rule X¯ chart. Meanwhile, through the comparison of Figures 3 and 6, it is noted that taking m=5 measurements and increasing B enable the negative effect of measurement errors on the run rule X¯ chart to be diminished faster compared to the cases in Figure 3.Figure 6
The effect ofB on the run rule Shewhart X¯ chart when m=5, γ2=1, δ∈0.5,1.5, and n∈1,3,5,7.
(a)(b)(c)(d)


### 4.2. Linearly Increasing Measurement Error Variance

Considering the linearly increasing variance case, it can be seen that parameterC in equations (5) and (6) behaves like σM2 in equation (4). If there is a shift in the process mean, then it is amplified by parameter D, which also changes the variance of the measurement error. So, the influence of D on the run rule X¯ chart is investigated in Figure 7. In this figure, for specified values of n∈1,3,5,7, B=1, C=5, m=1, μ0=10, and σ02=10 are set for illustration. Through this figure, it is seen that increasing D can significantly increase the values of ARL and RL0.95. This fact indicates the negative effect of linearly increasing variance on the run rule X¯ chart. For instance, if n=3 and δ=0.5, when D=1 increases up to D=3, ARL=74.80 and RL0.95=221 increase up to ARL=104.61 and RL0.95=310, respectively.Figure 7
The effect ofD on the run rule Shewhart X¯ chart when C=5, m=1, μ0=10, σ02=10, δ∈0.5,1.5, and n∈1,3,5,7.
(a)(b)(c)(d)


## 4.1. A Constant Measurement Error Variance

It can be seen from equations (17)–(19) that ARL1 and RL0.95 are functions of n, m, B, γ2, W, and δ. Table 1 presents ARL and RL0.95 of the run rule X¯ chart with linearly covariate error model for γ2∈0,0.1,0.2,0.3,0.5,0.7,1, δ∈0.1,0.3,0.5,0.7,1,1.5,2, and n∈1,3,5,7, when m=1 and B=1. The case γ2=0 corresponds to the “no measurement error” case. For example, when n=1, γ2=0, and δ=0, the chart parameter W=1.9293. With this value, we have ARL=198.43 and RL0.95=591 at δ=0.3. As it can be seen from Table 1, for a specified δ, ARL and RL0.95 increase when the measurement error ratio γ2 increases. This fact demonstrates the negative effect of the measurement errors on the run rule X¯ chart. For instance, for the example above but with γ2=1, we have ARL=260.93 and RL0.95=778. We can see that with the measurement error ratio increasing, larger ARL and RL0.95 are needed to detect a shift.The conclusions drawn above can also be seen from the four graphics in Figure2 where ARL and RL0.95 are plotted for δ∈0.5,1.5, n∈1,3,5,7 and γ2 when m=1 and B=1. This figure confirms the conclusion that larger ARL and RL0.95 are observed with increasing measurement errors.Figure 2
The influence ofγ2 on the run rule Shewhart X¯ chart when m=1, B=1, δ∈0.5,1.5, and n∈1,3,5,7.
(a)(b)(c)(d)Table2 presents ARL and RL0.95 of the chart for B∈1,2,3,4,5, δ∈0.1,0.3,0.5,0.7,1,1.5,2, and n∈1,3,5,7, when m=1 and γ2=1. It can be seen that for specified δ and n, the negative effect of measurement errors on the run rule X¯ chart decreases as B increases. Moreover, through the comparison between Tables 1 and 2, it can be seen that when B=5, the difference between ARL and SDRL corresponding to γ2=0 (no measurement error) and γ2=1 is negligible. For instance, when δ=0.3, m=1, γ2=1, n=1, and B=5 (Table 2), ARL=202.26 and RL0.95=602, while in Table 1, for the same values of m and δ, ARL=198.43 and RL0.95=591 when γ2=0.Table 2
Out-of-controlARL,RL0.95 when ARL0=370.4, m=1, and γ2=1 for n∈1,3,5,7, δ∈0.1,0.3,0.5,0.7,1,1.5,2, and B∈1,2,3,4,5.
B12345δn=10.1(354.31, 1058)(345.30, 1031)(342.39, 1022)(341.20, 1018)(340.61, 1017)0.3(260.93, 778)(219.88, 655)(208.67, 621)(204.34, 608)(202.26, 602)0.5(165.86, 493)(120.54, 357)(109.95, 326)(106.04, 314)(104.20, 309)0.7(102.57, 304)(66.84, 197)(59.35, 174)(56.66, 166)(55.41, 162)1.0(52.03, 152)(30.55, 88)(26.51, 76)(25.10, 72)(24.45, 70)1.5(20.12, 57)(11.09, 30)(9.56, 26)(9.04, 24)(8.80, 23)2.0(9.71, 26)(5.51, 14)(4.84, 12)(4.61, 11)(4.51, 11)δn=30.1(325.81, 972)(303.51, 905)(296.69, 885)(293.95, 877)(292.62, 873)0.3(158.23, 470)(113.57, 337)(103.27, 306)(99.48, 294)(99.69, 289)0.5(69.82, 206)(42.63, 124)(37.29, 108)(35.41, 103)(34.53, 100)0.7(33.77, 98)(19.03, 54)(16.40, 46)(15.49, 43)(15.08, 42)1.0(13.98, 39)(7.74, 20)(6.71, 17)(6.37, 16)(6.21, 16)1.5(5.19, 13)(3.27, 7)(2.98, 6)(2.88, 6)(2.84, 6)2.0(3.01, 6)(2.29, 3)(2.20, 3)(2.17, 3)(2.16, 3)δn=50.1(301.21, 899)(270.05, 805)(260.93, 778)(257.33, 767)(255.58, 762)0.3(109.95, 326)(72.63, 214)(64.70, 190)(61.85, 182)(60.52, 178)0.5(40.71, 119)(23.30, 66)(20.12, 57)(19.02, 54)(18.51, 52)0.7(18.08, 51)(9.96, 27)(8.59, 23)(8.13, 21)(7.92, 21)1.0(7.36, 19)(4.33, 10)(3.85, 9)(3.69, 8)(3.62, 8)1.5(3.16, 6)(2.36, 4)(2.25, 3)(2.21, 3)(2.20, 3)2.0(2.26, 3)(2.04, 2)(2.02, 2)(2.02, 2)(2.01, 2)δn=70.1(279.77, 834)(242.68, 723)(232.21, 692)(228.13, 680)(226.16, 674)0.3(82.34, 243)(51.59, 151)(45.39, 133)(43.19, 126)(42.17, 123)0.5(27.43, 79)(15.26, 43)(13.14, 36)(12.41, 34)(12.08, 33)0.7(11.79, 32)(6.59, 17)(5.74, 14)(5.46, 13)(5.33, 13)1.0(4.99, 12)(3.17, 6)(2.90, 6)(2.81, 5)(2.77, 5)1.5(2.52, 4)(2.11, 3)(2.07, 3)(2.06, 3)(2.05, 2)2.0(2.07, 3)(2.01, 2)(2.00, 2)(2.00, 2)(2.00, 2)In Figure3, ARL and RL0.95 of the run rule X¯ chart in the presence of measurement errors are plotted for δ∈0.5,1.5, n∈1,3,5,7, and B when m=1 and γ2=1. This figure also confirms the conclusion that with the value of B increasing, both ARL and RL0.95 decrease.Figure 3
The influence ofB on the run rule Shewhart X¯ chart when m=1, γ2=1, δ∈0.5,1.5, and n∈1,3,5,7.
(a)(b)(c)(d)Table3 presents ARL and RL0.95 for m∈1,2,3,4,5, δ∈0.1,0.3,0.5,0.7,1,1.5,2, and n∈1,3,5,7 when B=1 and γ2=1. It can be seen from Table 3 that as m increases, the negative effect of the measurement errors on the run rule X¯ chart decreases. For instance, for the case where δ=0.3, B=1, n=1, and γ2=1, ARL=260.93 and RL0.95=778 decrease down to ARL=216.03 and RL0.95=643, respectively, when m=1 increases up to m=5. So, it is concluded that if there is a relative high variability of the measurement error compared with the process variability, a large m is needed to reduce the negative effect of measurement errors on the run rule X¯ chart.Table 3
Out-of-controlARL,RL0.95 when ARL0=370.4, B=1, and γ2=1 for n∈1,3,5,7, δ∈0.1,0.3,0.5,0.7,1,1.5,2, and m∈1,2,3,4,5.
m12345δn=10.1(354.31, 1058)(349.25, 1042)(346.77, 1035)(345.30, 1031)(344.32, 1028)0.3(260.93, 778)(236.60, 705)(225.90, 673)(219.88, 655)(216.03, 643)0.5(165.86, 493)(137.66, 409)(126.51, 375)(120.54, 357)(116.82, 346)0.7(102.57, 304)(79.62, 235)(71.20, 210)(66.84, 197)(64.17, 189)1.0(52.03, 152)(37.82, 110)(32.98, 95)(30.55, 88)(29.10, 84)1.5(20.12, 57)(13.98, 39)(12.03, 33)(11.09, 30)(10.53, 28)2.0(9.71, 26)(6.81, 17)(5.93, 15)(5.51, 14)(5.26, 13)δn=30.1(325.81, 972)(313.07, 934)(307.03, 916)(303.51, 905)(301.21, 899)0.3(158.23, 470)(130.34, 387)(119.40, 354)(113.57, 337)(109.95, 326)0.5(69.82, 206)(52.03, 152)(45.79, 134)(42.63, 124)(40.71, 119)0.7(33.77, 98)(23.88, 68)(20.63, 59)(19.03, 54)(18.08, 51)1.0(13.98, 39)(9.71, 26)(8.38, 22)(7.74, 20)(7.36, 19)1.5(5.19, 13)(3.85, 9)(3.46, 7)(3.27, 7)(3.16, 6)2.0(3.01, 6)(2.50, 4)(2.36, 4)(2.29, 3)(2.26, 3)δn=50.1(301.21, 899)(283.15, 844)(274.83, 820)(270.05, 805)(266.95, 796)0.3(109.95, 326)(86.08, 254)(77.23, 228)(72.63, 214)(69.82, 206)0.5(40.71, 119)(29.10, 84)(25.22, 72)(23.30, 66)(22.15, 63)0.7(18.08, 51)(12.54, 34)(10.80, 29)(9.96, 27)(9.46, 25)1.0(7.36, 19)(5.26, 13)(4.63, 11)(4.33, 10)(4.15, 9)1.5(3.16, 6)(2.59, 5)(2.43, 4)(2.36, 4)(2.31, 3)2.0(2.26, 3)(2.09, 7)(2.05, 3)(2.04, 2)(2.03, 2)δn=70.1(279.77, 834)(258.01, 769)(248.23, 740)(242.68, 723)(239.10, 713)0.3(82.34, 243)(62.38, 183)(55.24, 162)(51.59, 151)(49.38, 144)0.5(27.43, 79)(19.22, 54)(16.57, 46)(15.26, 43)(14.49, 40)0.7(11.79, 32)(8.21, 22)(7.11, 18)(6.59, 17)(6.28, 16)1.0(4.99, 12)(3.72, 8)(3.35, 7)(3.17, 6)(3.07, 6)1.5(2.52, 4)(2.22, 3)(2.15, 3)(2.11, 3)(2.10, 3)2.0(2.07, 3)(2.02, 2)(2.01, 2)(2.01, 2)(2.00, 2)In Figure4, ARL and RL0.95 of the run rule X¯ chart in the presence of measurement errors are plotted for δ∈0.5,1.5, n∈1,3,5,7, and m when γ2=1 and B=1. This figure confirms the conclusion that by measuring the sample item several times, the negative effect of measurement errors on the run rule X¯ chart can be reduced to some extent.Figure 4
The effect ofm on the run rule Shewhart X¯ chart when B=1, γ2=1, δ∈0.5,1.5, and n∈1,3,5,7.
(a)(b)(c)(d)Since measuring the item in each sample several times can reduce the measurement errors’ negative effect on control charts, the run length performance of run ruleX¯ chart with linearly covariate error is presented in Figure 5, for the case of m=5 measurements per item, using γ2, δ∈0.5,1.5 and n∈1,3,5,7 when B=1. It can be noted that for fixed n, with m=5 measurements per item, ARL and RL0.95 only have a slightly increasing trend with γ2 increasing. Moreover, through the comparisons between the four graphics in Figure 2, the negative effect of measurement errors on the run rule X¯ chart is reduced by taking multiple measurements of m=5, especially for the cases when γ2≥0.5.Figure 5
The effect ofγ2 on the run rule Shewhart X¯ chart when m=5, B=1, δ∈0.5,1.5, and n∈1,3,5,7.
(a)(b)(c)(d)Finally, Figure6 presents the run length performance of the run rule X¯ chart for B, δ∈0.5,1.5, and n∈1,3,5,7 when γ2=1. From Figure 3, it is known that increasing B reduces the negative effect of measurement errors on the run rule X¯ chart. Meanwhile, through the comparison of Figures 3 and 6, it is noted that taking m=5 measurements and increasing B enable the negative effect of measurement errors on the run rule X¯ chart to be diminished faster compared to the cases in Figure 3.Figure 6
The effect ofB on the run rule Shewhart X¯ chart when m=5, γ2=1, δ∈0.5,1.5, and n∈1,3,5,7.
(a)(b)(c)(d)


## 4.2. Linearly Increasing Measurement Error Variance

Considering the linearly increasing variance case, it can be seen that parameterC in equations (5) and (6) behaves like σM2 in equation (4). If there is a shift in the process mean, then it is amplified by parameter D, which also changes the variance of the measurement error. So, the influence of D on the run rule X¯ chart is investigated in Figure 7. In this figure, for specified values of n∈1,3,5,7, B=1, C=5, m=1, μ0=10, and σ02=10 are set for illustration. Through this figure, it is seen that increasing D can significantly increase the values of ARL and RL0.95. This fact indicates the negative effect of linearly increasing variance on the run rule X¯ chart. For instance, if n=3 and δ=0.5, when D=1 increases up to D=3, ARL=74.80 and RL0.95=221 increase up to ARL=104.61 and RL0.95=310, respectively.Figure 7
The effect ofD on the run rule Shewhart X¯ chart when C=5, m=1, μ0=10, σ02=10, δ∈0.5,1.5, and n∈1,3,5,7.
(a)(b)(c)(d)


## 5. An Illustrative Example

Similar to the research work in Costa and Castagliola [21], a yogurt cup filling process is used here to illustrate the implementation of the X¯ chart integrated with run rules when measurement error exists. The critical quality characteristic in the filling process is the weight Y of each yogurt cup. From the historical database of yogurt cup weights (Phase I), the in-control mean μ0=124.9 and the standard deviation σ0=0.76 are estimated accurately. Moreover, an independent R&R (repeatability and reproducibility) study has showed an estimated measurement standard deviation σM=0.24, which leads to γ=0.24/0.76=0.316. By fixing the parameter settings A=0 and B=1 in the linearly covariate error model, the quality practitioner decides to use Shewhart X¯ chart integrated with 2-of-3 run rules to monitor the process. Taking n=5 yogurt cups and measuring each item m=2 times, in a sample, the parameter W of the run rule X¯ chart can be calculated as W = 1.9293 using equation (20). Then, the warning limits in equations (7) and (8) are given as follows:(21)LWL=124.9−1.9293×150.762+0.2422=124.23,UWL=124.9+1.9293×150.762+0.2422=125.57.In addition, 20 samples corresponding to a sequence of production (Phase II) are recorded in Table4. We have 10 values for each sample, i.e., the weight of n=5 yogurt cups weighed m=2 times. The mean X¯i of these samples is presented in the right side of Table 4. In Figure 8, these mean values X¯i with the warning limits LWL=124.23 and UWL=125.57 are also plotted. As it can be seen in the figure, the Shewhart X¯ chart with 2-of-3 run rules actually triggers an out-of-control signal from samples 11 and 12 (shown in the frame), suggesting a downward shift in the process mean. One possible reason is that the pipe used for filling the cups is clogged.Table 4
20 samples corresponding to a sequence of production.
iXi,1,kXi,2,kXi,3,kXi,4,kXi,5,kX¯i1125.24125.35125.12125.15123.38123.13124.75125.38126.18126.50125.022124.79124.93124.42124.61123.90124.21123.46123.68123.89124.63124.253124.96124.95124.40124.54124.93124.94125.54125.59124.96124.80124.964125.55124.96125.76125.44124.88124.53125.85126.35123.76124.20125.135124.92125.01123.86124.27124.16124.29124.77125.29125.93125.52124.806124.95125.27125.92125.95124.23124.15124.90125.61124.50124.28124.987124.16124.41122.80123.15125.76124.96126.34126.21125.04125.06124.798125.73126.04125.08124.87123.98124.28124.29124.18124.28124.55124.739125.75125.63124.58124.41125.38125.07123.71123.78124.86124.73124.7910123.33123.48124.22124.49125.13125.06126.51126.10124.99125.16124.8511124.29124.86124.67125.03124.78125.32122.70122.39123.22122.95124.0212123.86123.63123.34123.84122.87123.04123.73123.93123.87123.83123.5913125.32125.12123.68123.48123.41122.97124.77124.00123.75123.73124.0214124.02124.43124.05124.43122.86122.94124.65124.55124.16124.09124.0215123.93123.85123.93123.47124.47124.96124.45124.61122.82122.56123.9016125.83125.87123.89124.09124.40124.15124.55124.35123.44123.14124.3717125.33124.97124.57124.37123.97124.29124.90125.24124.38124.16124.6218123.02123.10124.50124.45124.96125.15125.66125.23124.88124.34124.5319122.93123.35125.18124.79124.12124.15124.18124.03125.04125.08124.2820124.59124.64121.48121.21122.63123.08123.68123.42123.79123.21123.17Figure 8
X¯ chart with 2-of-3 run rules corresponding to the data in Table 4.


## 6. Conclusion

The properties of ShewhartX¯ chart integrated with run rules are investigated when errors exist in the measurement of observations in a quality characteristic. By using a linearly covariate error model, ARL,RL0.95 of the chart is derived using a Markov chain approach. The simulations results show that ARL,RL0.95 of the run rule Shewhart X¯ chart is seriously affected by the measurement errors. With the measurement error increasing, the run rule Shewhart X¯ chart’s performance deteriorates. To reduce the negative effect of measurement errors, it is beneficial to measure each item in each subgroup more times and increase the coefficient B in the linearly covariate error model whenever possible. Both are good alternatives to implement the run rule Shewhart X¯ chart. It should be careful that measuring each item more times practically causes more cost.All the results are based on the covariate error model, and it would be interesting to extend our research to different control charts with some other measurement error models.



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*Source: 8405059-2021-08-04.xml*