Mixed Tukey EWMA-CUSUM control chart and its ...

Quality Technology & Quantitative Management

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Mixed Tukey EWMA-CUSUM control chart and its applications Muhammad Riaz, Qurat-Ul-Ain Khaliq & Shahla Gul To cite this article: Muhammad Riaz, Qurat-Ul-Ain Khaliq & Shahla Gul (2017): Mixed Tukey EWMA-CUSUM control chart and its applications, Quality Technology & Quantitative Management, DOI: 10.1080/16843703.2017.1304034 To link to this article: http://dx.doi.org/10.1080/16843703.2017.1304034

Published online: 17 Apr 2017.

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Date: 22 April 2017, At: 20:30

Quality Technology & Quantitative Management, 2017 http://dx.doi.org/10.1080/16843703.2017.1304034

Mixed Tukey EWMA-CUSUM control chart and its applications Muhammad Riaza, Qurat-Ul-Ain Khaliqb and Shahla Gulc a

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia; Department of Statistics, Allama Iqbal Open University, Islamabad, Pakistan; cDepartment of Education, University of Arid Agriculture, Rawalpindi, Pakistan b

ABSTRACT

Tukey control chart (TCC) is a popular choice for robust monitoring of process parameters. With the advancement in technology, we develop refined techniques that incorporate multiple aspects in a single structure. This article is a similar effort to design an improved charting structure in the form of mixed Tukey EWMA-CUSUM chart (namely MEC-TCC). We have investigated the performance of the proposed chart using different run length properties. We have observed that the proposed MEC-TCC design serves the dual objectives, namely the efficient detection of shifts and robustness against non-normality. The comparative analysis has revealed that the proposed scheme is an effective competitor to the existing counterparts, including classical Shewhart, EWMA, CUSUM, Tukey and some other variants such as mixed EWMA-CUSUM, Tukey EWMA, Tukey CUSUM. Moreover, the proposed design presents some of the aforementioned charts as special cases. For real life considerations, we have implemented the proposed and existing charts to two real life data sets, one related to air quality and the other one concerning smartphone accelerometer.

ARTICLE HISTORY

Accepted 6 March 2017 KEYWORDS

Control charts; run length; Tukey chart; mixed charts; non-normality

1. Introduction Statistical Process Control (SPC) refers to a set of several useful methods used for an effective process monitoring. Control charts are efficient monitoring tools and they appear as graphical displays with specific lower and upper bounds to monitor the quality of ongoing processes. There are a variety of control charts available for each exclusive type of data. Control charts have application in a different fields like health care, dairy production, chemical industries, aircraft manufacturing, software engineering, monitoring atmospheric changes, packaging and manufacturing industries, among many others. The fundamental structures of these charting designs require certain assumptions such as normality and independence. However, in many real situations, these assumptions may not always be fulfilled, for example, in measurements of semiconductor and chemical processes, quantification of cutting tool wear processes, lifetimes of accelerated life test samples. These types of characteristics are often of the skewed nature (cf. Bai & Choi, 1995; Choobineh & Branting, 1986; Nelson, 1979). Figueiredo and Gomes (2013) designed a method for observing the performance of control chart when data follow skewed Gaussian distribution. For individual observations based data, some fundamental charts include Individual/Moving Range (X/MR chart), Cumulative Sum (CUSUM) chart and Exponentially Weighted Moving Average CONTACT Muhammad Riaz

[email protected]

© 2017 International Chinese Association of Quantitative Management

2

M. RIAZ ET AL.

(EWMA) chart (cf. Page, 1951; Roberts, 1959; Shewhart, 1931). For subgroups based data, several authors worked on the performance of the Shewhart, CUSUM and EWMA charts (cf. Graham, Mukherjee, & Chakraborti, 2012; Mukherjee & Sen, 2015). Several authors have proposed CUSUM type non-parametric charts, which showed more sensitivity to the shifts in process location or dispersion (cf. Graham, Chakraborti, & Mukherjee, 2014; Mukherjee & Marozzi, 2016; Mukherjee, Graham, & Chakraborti, 2013). For more details Qin(2013) and Montogomery (2009) may be considered. The structures of these charts, along with other charts, have been investigated in the literature and their properties have been reported in different contexts. We provide here a brief review of some selective charts for our study purposes. Alemi (2004) used Tukey control chart (TCC) as a good alternative to X/MR chart for individual process monitoring, when data follow skewed distribution and/or have some outliers. Borckardt et al. (2005, 2006) operated the Tukey chart with serially dependent data. Several authors investigated the performance of Tukey chart under conditions (cf. Lee, 2011; Lee, Kuo, & Lin, 2013; Torng & Lee, 2008; Torng, Lee, & Tseng, 2009; Torng, Liao, Lee, & Wu, 2009; Sukparungsee, 2012, 2013; Saithanu, Nojit, & Mekparyup, 2015). Several authors introduced different designs of design of Tukey chart (cf. Khaliq & Riaz, 2015; Khaliq, Riaz, & Ahmad, 2016; Mekparyup & Saithanu, 2015; Mekparyup, Kornpetpanee, & Saithanu, 2014, 2014; Tercero-Gomez, Ramirez-Galindo, Cordero-Franco, Smith, & Beruvides, 2012; Tercero-Gómez, Cordero-Franco, Alvarado-Puente, & del Carmen Temblador-Pérez, 2014; Torng et al., 2009). Lee and Torng (2015) introduced some modifications to MTCC in order to enhance the insensitivity of signalling mean shifts for highly skewed populations. Khaliq, Riaz, and Alemi (2015) carried out a comparative analysis to judge the performance of Tukey chart vs. X/MR charts under several probability models and found Tukey chart as an attractive choice. One such extended option of EWMA and CUSUM is combined mixed structure based on these basic charts, such as combined Shewhart-EWMA, combined Shewhart-CUSUM. Westgard, Groth, Aronsson, and De Verdier (1977), Lucas (1982), Ncube and Woodall (1984), Gibbons (1999), Klein (1996, 1997), Capizzi and Masarotto (2010) and the references therein may be seen in this regard. Abbas, Riaz, and Does (2013a, 2013b, 2014) and Zaman, Riaz, Abbas, and Does (2015) proposed the mixed version of EWMA and CUSUM designs (MEC) for location/dispersion parameters. They claimed superior performance of the Mixed chart than classical CUSUM and EWMA designs. In this study, we intend to propose a mixed design of Tukey-EWMA and Tukey-CUSUM charts, namely MEC-TCC, for an efficient and robust monitoring of location parameter. The remaining part of the article is organized as: Section 2 includes the details regarding different control designs. The proposed MEC-TCC chart is given in Section 3. Section 4 presents performance evaluation techniques. Section 5 provides performance and comparative analysis of the proposed chart with other competing charts. Section 6 offers real applications based on two real data sets. Section 7 gives the concluding remarks of this study. We have also provided a complete list of nomenclature used in this study at the end of the manuscript.

2. Design structures of control charts Let y1, y2, … ,yn be individual and independent observations having mean μ0 and standard deviation σ0. Based on past data in an in-control state, the fitted probability density function (pdf) denoted by f, along with the cumulative density function (CDF) denoted by F are given as: x

g = F(x) =

∫−∞

f (y)dx ⇒ x = F −1 (g)

(1)

where g is probability of given y and it may used to work out the values of different quantiles, such as .25, .5 and .75 for the first, second and third quartiles.

QUALITY TECHNOLOGY & QUANTITATIVE MANAGEMENT

3

Using the above information, the lower control limit (LCL), centre line (CL) and upper control limit (UCL) of different charts are given below. The design structures of the usual Shewhart, EWMA and CUSUM design structures may be seen in Montgomery (2009). 2.1. Tukey control chart The control limits for TCC are defined as:

LCL = Q1 − L(IQR),

CL = Q2 ,

UCL = Q3 + L(IQR)

(2)

where L denotes the control limits coefficient that may be fixed at pre specified ARL0 and Q1, Q2 and Q3 are first, second and third quartiles respectively. 2.2. EWMA-TCC The plotting statistic for EWMA–TCC is (cf. Khaliq et al., 2016) given as: Gi = 𝜆xi + (1 − 𝜆)Gi−1. The variance of EWMA-TCC statistic is as follows: ⌢

IQR(𝜆(1 − (1 − 𝜆)2i )) Var(Gi ) = 2−𝜆

(3)

where Gi is initially set to zero, G0 = μ0 and 𝜆 is the smoothing parameter. The choice of 𝜆 = 1 produces the usual TCC. The time varying control limits in this case are given as: √ √ ⌢ ⌢ 𝜆(1 − (1 − 𝜆)2i ) 𝜆(1 − (1 − 𝜆)2i ) LCL = Q1 − Lt (IQR) , CL = Q2 , UCL = Q3 + Lt (IQR) 2−𝜆 2−𝜆 (4) 2.3. CUSUM-TCC The CUSUM chart plots the cumulative sums of the deviations of sample values from a target value. The cumulative sum, up to and including the ith sample, is denoted by Ci and is given in the form of two statistics Ci+ and Ci− as: + Ci+ = max(0, xi − Q3s − k + Ci−1 ),

where Q1s and Q3s are

Q3s

− Ci− = max(0, Q1s − xi − k + Ci−1 )

(5)

adjusted Q1 and Q3 respectively defined as:

= (Q3 − .75Q3 ) + .5Q2 + .25Q1 ,

Q1s = (Q1 − .075Q1 ) + .5Q2 + .25Q3

(6)

where Ci+ and Ci− are known as upper and lower Tukey-CUSUM statistic which to zero Ci+ = Ci− = 0. Q3s and Q1s are adjusted quartile values. H = h(IQR) is the

are initially adjusted decision interval for CUSUM statistic, IQR is the Interquartile range of the sample values used in forming CUSUM . The constant k is a reference value and is usually chosen as k = .5δ, (relative to the size of shift (δ)), where δ is shift in standard deviation (σ) units. Once k is selected, h is adjusted accordingly to obtain desired ARL0. (cf. Khaliq & Riaz, 2015). 2.4. MEC chart The design structure of MEC chart depends on two statistics Si+ and Si− (cf. Abbas et al., 2013a) as given below. + Si+ = max(0, (Ri − 𝜇0 ) − qi + Si−1 ),

− Si− = max(0, −(Ri − 𝜇0 ) − qi + Si−1 )

(7)

where and are recognized as the upper and lower CUSUM statistics, respectively. Initially, these values are S0+ = S0− = 0 and these values are based on the EWMA statistic Ri, that is deﬁned as:

Si+

Si−

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M. RIAZ ET AL.

Ri = xi 𝜆 + (1 − 𝜆)Ri−1, where 𝜆 is the time varying constant. The Initial value of Ri is set equal to mean, R0 = μ0. The mean and variance of Ri are given as: 𝜎x2 (𝜆(1 − (1 − 𝜆)2i )) . 2−𝜆 Moreover, qi is a time-varying slack value and is defined as: � √ 𝜆(1 − (1 − 𝜆)2i ) qi = a∗ Var(Ri ) = a∗ 𝜎0 2−𝜆 Mean(Ri ) = 𝜇0 ,

Var(Ri ) =

The statistics are plotted against the limit ‘bi’ defined as: � √ 𝜆(1 − (1 − 𝜆)2i ) bi = b∗ Var(Ri ) = b∗ 𝜎0 2−𝜆

(8)

(9)

(10)

where a* and b* are the constants (used in CUSUM chart, i.e. k & h). The time-varying reference values qi and bi are due to the standard deviation of EWMA statistic. The value of a* is generally taken .5 and the desired ARL0 may be achieved by adjusting the aforementioned coefficients. The decision criterion is: as far as Si+ and Si− remain inside the decision interval bi, the process is declared to be in-control and otherwise out-of-control. 2.5. The proposed MEC-TCC chart This section provides the complete design structure of the proposed MEC-TCC chart. It depends on the EWMA statistic Zi = 𝜆 xi + (1 − 𝜆) Zi−1as an input value and the two plotting statistics are given as:

( ) + Gi+ = max 0, Zi − Q3∗ − ti + Gi−1 ,

( ) − Gi− = max 0, Q1∗ − Zi − ti + Gi−1

(11)

where Gi+ and Gi− are known as the upper and lower plotting statistic of MEC-TCC design and are initially set to be zero, G0+ = G0− = 0. The quantities Q1∗ and Q3∗ are defined as:

Q1∗ = Q1 − .75Q1 + .5Q2 + .25Q3 ,

Q3∗ = Q3 − .75Q3 + .5Q2 + .25Q

Moreover, ti is time varying slack value and is defined as… √ ⌢ 2i/ ∗ ti = t IQR 𝜆(1 − (1 − 𝜆) 2 − 𝜆 s

⌢

(12)

(13)

⌢

where IQR = (Q3 − Q1 )∕2 (for symmetric class) and IQR = (Q3 − Q1 )∕1.34898 (for asymmetric class) s s (cf. Hubert & Vandervieren, 2008). The time varying decision limit of the mixed design for symmetric and skewed (asymmetric) classes (cf. Figure 1) may be given as:

√ ( ) 2i/ hi = h IQR 𝜆(1 − (1 − 𝜆) (2 − 𝜆). symmetric class such as Normal , s ⌢

∗

HUi =

h∗U

⌢

IQR

√ 𝜆 (1−(1−𝜆)2i ) 2−𝜆

as

HLi =

h∗L

⌢

IQR as

(14)

,

√ 𝜆 (1−(1−𝜆)2i ) 2−𝜆

(asymmetric class, such as Rayleigh (2) and Lognormal (0, .25)).

(15)

QUALITY TECHNOLOGY & QUANTITATIVE MANAGEMENT Symmetrical Distribution

f

0.0

0.00

0.05

0.1

0.10

0.2

0.15

0.20

0.3

0.25

0.4

0.30

Skewed Distribution

f

5

0

2

4

6

8

-4

x

-2

0

2

4

x

Figure 1. Shape of symmetrical and skewed distribution.

Here HUi and HLi are upper and lower decision intervals of the proposed chart and h∗U and h∗L are upper and lower decision intervals’ constants. Moreover, the quantities t* and h* are the constants relying on the time-varying reference values ti and hi. For this study, t* is taken .5 (like k). The desired ARL0 is be achieved by adjusting these aforementioned coefficients. The plotting statistic Gi+ and Gi− are tested against the decision intervals and the process is deemed in-control if points fall within the interval, otherwise out-of-control.

3. Performance measures The performance measures that are used to assess the performance of a control chart may be classified into two main types based on specific shifts and overall shifts. We have used location shifts in this format: μ1 = μ0 + δσ0 where δ refers to the amount of shift, μ1 is the shifted mean, μ0 is the in-control mean and σ0 is the controlled value of process standard deviation. We briefly define here some commonly used measures. A run is defined as a sequence of points that are plotted on a chart until a point indicates an outof-control signal. The number of such points in a run is known as run length (RL). There are two main states of RL named in-control and out-of-control. It is expected to have a higher in-control RL and a smaller out-of-control RL in general. We briefly define some common measures of a control chart based on RL. 3.1. Specific shifts based measures These measures are dependent upon the specific δ values to detect a certain amount of shift. The most popular of these measures is ARL, We evaluate the performance of a control chart generally in terms of average run length (ARL). The ARL of a control chart is defined as the average number of points that must be plotted until a point indicates an out-of-control condition. The ARL can be classified by ARL0 (when the process is in-control) and ARL1 (when the process is out-of-control). While some other supporting measures include median run length (MDRL), standard deviation of run length (SDRL) and percentiles of run length (PRL). In addition, RL curves are also used as an attractive supportive measure. These measures are mathematically defined as:

∑ ARL =

m

(RL)m , m

(16)

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M. RIAZ ET AL.

√ SDRL =

MDRL = Median(RL),

(17)

PRL = F −1 (q),

(18)

( ) √ { } E(RL)2 − (E(RL))2 E(RL)2 − ARL2 SDRL =

(19)

where q refers to a probability value, F−1 stands for the inverse cumulative distribution function of RL, E stands for expected value, m refers to the number of RL values produced to capture process behaviour. It is generally desirable to have smaller values for these measures. 3.2. Overall performance measures Besides the above mentioned individual performance measures, there are some commonly used overall performance measures, including Extra Quadratic Loss (EQL), Performance Comparison Index (PCI) and Relative Average Run Length (RARL). These assess the overall performance of a chart. These are defined as: 𝛿

EQL =

max 1 𝛿 2 ARL(𝛿) d𝛿, 𝛿max − 𝛿min ∫𝛿min

PCI =

RARL =

EQL , EQLbmk

𝛿max ARL(𝛿) 1 d𝛿, 𝛿max − 𝛿min ∫𝛿min ARLbmk (𝛿)

(20)

(21)

(22)

where δmin and δmax refer to minimum and maximum values of δ respectively, bmk is the benchmark chart. For a benchmark chart, we generally consider a chart having minimum EQL. Sometimes a standard existing chart may also be considered as a benchmark chart. RARL is measured relative to the benchmark chart and it observes how close a specific chart performs relative to bmk chart. PCI defines the ratio between EQL of a specific chart relative to that of a bmk chart. It is generally expected to have smaller outcomes for these measures for more effective chart(s).

4. Performance analysis and comparisons In this section we evaluate the performance of different control charts using the aforementioned measures. We also provide a comparative analysis of the proposed MEC-TCC chart and some other charts, including the Shewhart, TCC, the EWMA, EWMA-TCC, the CUSUM, CUSUM-TCC and MEC charts. For performance evaluation, we have used Monte Carlo simulations (105 repetitions) to get the RL properties and we have followed the procedure of Khaliq et al. (2016), at different combinations of λ and δ and evaluated RL properties of the different charts of this study We have covered different probability distributions to study and compare the performance of different types of charts. These probability distributions include Lognormal (0, .25), Logistic (6, 2), Laplace (0, 1), Maxwell (2), Normal(0, 1), Rayleigh (2), Student t (10 degrees of freedom) at λ = 1, .5, .25.


0.95

0.95 0.85

0.75 0.65 0.55 TCC

0.45 EWMA-TCC( =.25)

0.35

Commulative Probability


0.85

CUSUM-TCC

0.75 0.65 0.55 TCC

0.45

EWMA-TCC(λ=.25) CUSUM-TCC

0.35

MEC-TCC(λ=.25)

MEC-TCC( =.25)

0.25

75

175

275

375

0.25 75

475

175

Run Length

375

1

0.9

0.7 0.6 0.5

TCC

EWMA-TCC(λ=.25)

0.4


0.9

0.8

0.8 0.7 0.6 0.5

TCC EWMA-TCC(λ=.25)

0.4

CUSUM-TCC

0.3

CUSUM-TCC

0.3

MEC-TCC(λ=.25)

MEC-TCC(λ=.25)

75

275

475

675

0.2

875

75

275

1

0.9

0.9

0.8

0.8

0.7 0.6 0.5 TCC

0.4

EWMA-TCC(λ=.25) CUSUM-TCC

0.3

MEC-TCC(λ=.25)

475

Run Length

(v)

675

875



1

275

675

875

(iv)

(iii)

75

475

Run Length

Run Length

0.2

475

(ii)

1


275

Run Length

(i)

0.2

7

0.7 0.6 0.5 TCC

0.4

EWMA -TCC(λ=.25) CUSUM-TCC

0.3 0.2

MEC-TCC(λ=.25)

75

275

475

675

875

Run Length

(vi)

Figure 2. RL curves (uisng ARL0 = 370) for (i) Normal (0, 1) at δ = −.25; (ii) Normal (0, 1) at δ = .25; (iii) Logistic (6, 2) at δ = −.25; (iv) Logistic (6, 2) at δ = .25; (v) T(10) at δ = −.25; (vi) T(10) at δ = .25.

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M. RIAZ ET AL.

TCC

351

TCC

351

CUSUM-TCC

CUSUMTCC

EWMATCC( =.25)

301

EWMATCC( =.25)

301

MECTCCy( =.25)

MECTCC( =.25)

251

201

201

ARL

ARL

251

151

151

101

101

51

51

1 -1.5

-1

-0.5

0

0.5

1

1

1.5

-3

-2

-1

0

(i)

1

2

3

(ii)

451

TCC TCC CUSUMTCC EWMATCC( =.25) MECTCC( =.25)

401 351

401

301

ARL

251

ARL

CUSUMTCC EWMATCC( =.25) MECTCC( =.25)

501

301

201 201

151 101

101 51 1

1 -3

-2

-1

0

(iii)

1

2

3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(iv)

Figure 3. ARL curves (uisng ARL0 = 370) of different control charts under (i) Normal (0, 1); (ii) Logistic (6, 2); (iii) Lognormal (0, .25); (iv) Rayleigh(2).

These results are arranged in tabular and graphical displays (cf. Tables 1–5, Figures 2–4 and Appendix Tables A1–A2) using ARL0 = 370. Table 1 provides ARL values of MEC-TCC vs. TCC, CUSUMTCC and EWMA-TCC. Table 2 includes MDRL performance of MEC-TCC vs. TCC, CUSUM-TCC and EWMA-TCC. Table 3 covers SDRL performance of MEC-TCC vs. TCC, CUSUM-TCC and EWMA-TCC. Table 4 offers a comparative ARL performance of MEC-TCC vs. MEC control charts. Table 5 presents ARL performance of MEC-TCC vs. Shewhart, EWMA and CUSUM charts. Table A1-A2 provide 95th and 5th PRL performance respectively, of MEC-TCC vs. TCC, CUSUM-TCC and EWMA-TCC. Figure 2 presents RL curves and Figures 3 and 4 provide ARL curves of different charts under different probability models. Section 4.1 provides the comparative ARL performance of proposed MEC-TC design vs. TCC, EWMA-TCC and CUSUM-TCC charts. Section 4.2 provides the comparative ARL performance of


350

Shewhart

300

MEC-TCC

9

ARL

250 200 150

100 50 0

-3

-2

-1

0

1

2

3

(i-a) 350

Shewhart

300

MEC-TCC

ARL

250 200 150 100 50 0

-3

-2

-1

0

1

2

3

(i-b) 350

Shewhart

300

MEC-TCC

ARL

250 200 150 100 50 0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(i-c) Figure 4. ARL Curves (uisng ARL0 = 370) of different control charts (i) Shewhart vs. MEC-TCC (a. Normal (0, 1), b. Logistic (6, 2), c. t(10)); (ii) EWMA vs. MEC-TCC (a. Normal (0, 1), b. Logistic (6, 2), c. t(10)); (iii) CUSUM vs. MEC-TCC (a. Normal (0, 1), b. Logistic (6, 2), c. t(10)); (iv) Shewhart, EWMA, CUSUM vs. MEC-TCC (a. Lognormal (0, .25)-Increase in shifts, b. Lognormal (0, .25)-Decrease in shifts).

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M. RIAZ ET AL.

350

EWMA

300

MEC-TCC

ARL

250 200 150 100 50 0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(ii-a) 350

EWMA

300

MEC-TCC

ARL

250 200 150 100 50

0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(ii-b) 350

EWMA

300

MEC-TCC

ARL

250 200 150 100 50 0

-2

-1.5

-1

-0.5

0

(ii-c) Figure 4. (Continued).

0.5

1

1.5

2


350

CUSUM

300

MEC-TCC

ARL

250 200 150 100 50 0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(iii-a) 350

CUSUM

300

MEC-TCC

ARL

250 200 150 100 50 0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(iii-b) 350

CUSUM

300

MEC-TCC

ARL

250 200 150 100

50 0

-2

-1.5

-1

-0.5

0

(iii-c) Figure 4. (Continued).

0.5

1

1.5

2

11

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M. RIAZ ET AL. 2500

Shewhart EWMA

2000

CUSUM MEC-TCC

ARL

1500

1000

500

0

-3

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-1

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0

(iv-a) 350 Shewhart 300

EWMA CUSUM

ARL

250

MEC-TCC

200

150 100 50 0

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

(iv-b) Figure 4. (Continued).

proposed MEC-TC design vs. classical Shewhart, EWMA and CUSUM charts. Section 4.3 provides the comparative ARL performance of the proposed MEC-TCC vs. MEC designs using probability models Laplace (0, 1), Logistic (6, 2), Normal (0, 1), Rayleigh (2) and Student t (10) at λ = 1, .5 and .25. Section 4.4 provides an overall view of the performance of different charts. The details of these results are discussed in the following subsections. 4.1. MEC-TCC vs. TCC, EWMA-TCC and CUSUM-TCC The performance of MEC-TCC, TCC CUSUM-TCC and EWMA-TCC charts are given in Table 1 (in the form of ARL values) at pre-specified ARL0 ≅ 370. It is to be noted that when λ = 1, CUSUM-TCC becomes the special case of MEC-TCC (using IQR/1.34898 in asymmetrical and SIQR in symmetrical distribution). Tables 1–3 and Appendix Tables A1–A2 advocate the following.


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Figure 5. Adverse effects of PM10 to human (https://www.ec.gc.ca/Air/default.asp?lang=En&n=0768F92F1&offset=1&toc=hide).

Figure 6. Cement-dust coating on apple leaves and fruit (http://www.omafra.gov.on.ca/english/crops/facts/01-015.htm).

(i) ARL1 values of MEC-TCC chart decrease as the value of λ decreases from 1 to .5 and further to .25 i.e. ARL performance of the proposed chart is improving as the value of λ decreases. The other performance measure like MDRL, SDRL and PRL support it (cf. Tables 2, 3, A1, A2 and Figures 2 and 3)). (ii) For the case of Normal (0, 1) and Student-t (10), MEC-TCC performs more efficiently for smaller shifts. For example, ARL1 is 283.14 for TCC, 122.4 for CUSUM-TCC; 194.5739 for EWMA-TCC (at λ = .5) and 136.6377 at λ = .25; whereas MEC-TCC has ARL1 96.0435, 78.6805 and 74.0473 at λ = 1, .5 and .25. It indicates that MEC-TCC provides smaller ARL1 (for all λ) values among all designs. The other performance measures support these findings. Tables 1–3, A1, A2 and Figures 2 and 3 may be seen for more details. (iii) In Logistic model; MEC-TCC design has smaller ARL1 than TCC and EWMA-TCC (cf. Table 1). Both CUSUM-TCC and MEC-TCC have similar ARL1 performance except δ = −.25. MEC has slightly high ARL1 than CUSUM-TCC at δ = ±.25 when λ = 1 and .5. For instance, ARL1 values of TCC, EWMA-TCC and MEC-TCC (at λ = 1, .5, .25) are 267.10, 205.6 and (94.92, 77.08 and 81.87) at δ = .25.

14

M. RIAZ ET AL.

Figure 7. Probability plot of real life data.

(iv) For highly skewed probability model like Lognormal (0, .25), MEC-TCC has minimum ARL1 at all λ values (for both kinds of shifts ± δσ). It shows more sensitivity to shifts in the case of highly skewed data to all competing designs. For example, MEC-TCC has ARL1 = 59.03 (cf. δ = −.25, λ = .25) while TCC has ARL1 = 343.8. (v) For Rayleigh (2) model, EWMA-TCC has smaller ARL1 among all design for δ 0 CUSUM-TCC and MEC-TCC (λ = . 25) perform equally well. EWMATCC displays a biased ARL feature of (cf. Table 1 and Figure 3(ii)). (vi) For Maxwell (2) probability model, EWMA–TCC has smaller ARL1 than TCC, CUSUMTCC. One may observe similar ARL1 behavior of MEC-TCC and EWMA-TCC charts (cf. Table 1). (vii) For Normal (0, 1) at δ = −.25, MDRL values of different designs are as follows: TCC is having 195; CUSUM-TCC has 87; EWMA-TCC has 136 & 94 (at λ = .5 and .25); MEC-TCC is having 69, 62 and 61 at λ = 1, .5 & .25. It indicates that MEC-TCC has smaller MDRL values than other competing designs. MDRL and PRL values from Normal (0, 1), Student-t (10) & Logistic(6, 2) also support these results. When data have skewed class (cf. Lognormal (0, .25), Rayleigh (2) & Maxwell (2)) MEC-TCC has smaller MDRL values than other competing designs except EWMA-TCC. This is due to the biased feature of RL. (cf. Tables A1 and A2). (viii) MEC-TCC shows smaller SDRL values than other competing designs. This design shows the least variation in many cases such as Normal (0, 1), Student-t (10), Logistic (6, 2), Maxwell (2) and Lognormal (0, .25) (cf. Table 3). 4.2. MEC-TCC vs. Shewhart, CUSUM and EWMA designs This section provides the comparative ARL performance of MEC-TCC design vs. Shewhart, CUSUM and EWMA charts under the several probability models including Normal (0, 1), Logistic (6, 2), Student-t (10) and Lognormal (0, .25). The ARL values are reported in Table 4 that provide some useful findings, as stated below.


180

15

PM10

160 140

PM10

120 100 80 60 40 20

0

0

200

400

600

800

Time

(i)

PM10 UCL

140

LCL

PM10

90

40

-10 0

100

200

300

-60

400

500

600

700

Sample No

(ii)

PM10 UCL LCL

155

PM10

105

55

5

-45

0

100

200

300

400

500

600

700

800

Sample No

(iii)

Figure 8. (i) Data Display; (ii) TCC chart; (iii) Shewhart chart; (iv) EWMA chart; (v) EWMA-TCC; (vi) CUSUM chart; (vii) CUSUM-TCC (ACL) chart; (viii) MECchart; (ix) MEC-TCC (ACL) chart.

16

M. RIAZ ET AL. 89 79

UCL

69

LCL

59

Di

49 39 29 19 9 -1

0

100

200

300

400

500

600

700

Sample No

(iv) UCL LCL Gi

90 70 50 30 10 -10 0

100

200

300

400

500

600

700

800

Sample No

(v) H

Cp

Cn

1001

Cp and Cn

801 601 401 201 1

0

100

200

300

400

500

600

700

800

Sample No

(vi) Figure 8. (Continued).

(i) In case of normal distribution, MEC-TCC has a smaller ARL1 than Shewhart chart for −1.5