Designing a control chart of extended EWMA statistic based on multiple dependent state sampling

Abstract

In this paper, we presented a memory type control chart (CC) based on multiple dependent state (MDS) sampling to pinpoint the slight variation in the process mean for the quality trait of normal distribution (ND). Two pairs of control limits denominated as internal and external control limits are derived using under control mean and variance. The essential steps are taken to get the value of average run length (ARL) for stable and disturb process. Various tables of ARLs are erected using different smoothing constants, shifts and MDS parameter. Comparisons are established to assess the effectiveness of initiated CC with the various existing CC in term of ARL. It has been ascertained that offered CC manifest the best performance in searching out the diminutive changes in the process mean. Two examples, one is based on simulation study and other is related to real-life data, have been discussed for its practical purpose.

1. Introduction

No doubt this is an era of quality attention, manufacturing industries in the world are challenging each other on the basis of the quality of their manufacturing items. The best quality of the item means the good reputation of that product in the market. Therefore, the main concern of the companies is to maintain their reputation in the market by attaining the high quality of their product. Statistical quality control (SCQ) is one of the potent instruments to attain this goal. Control charts (CCs) are the most influential type of SQC. The CCs are commanding instrument in the manufacturing industry to monitor the quality features of the products. CC gives the signal when the running process is disturbed due to some cause of variation (named as a natural cause of variation and assignable cause of variation), under this situation engineer uses some technique to bring the manufacturing operation in-control condition. The operation is said to be in-control when the plotted data of production process falls within the lower and upper control limits. There are two main categories of CCs, one is designated as memory less CCs and the second one is memory type CCs. The concept of memory less type CC was initiated by Shewhart [22]. The significant drawback of using memory less type CC is that it cannot detect minor variation efficiently. So Roberts [18] recommends the memory type CC named as Exponential Weighted Moving Average (EWMA) CC. The salient feature of this type of CC is that EWMA statistic gathers two types of information, i.e. information collected from a recent value of study variable as well as collect the information of past values of statistic. These two types of information empower the memory type CC to capture the smaller shifts very quickly. Various CCs have been constructed using EWMA statistic to search out the minor variation more efficiently like Borror et al. [9] suggested the EWMA CC using passion data and compute the value of ARL using Markov chain method. The finding showed the ameliorate performance in searching the smaller shifts as equating to the existing chart. Castagliola [10] suggested an EWMA CC for surveillance the sample variance of the production process. More work can be seen in [26,21,23,14,27,19]. Recently researchers have made some modification in memory type CC to enhance its capability in term of detecting the smaller shifts like [17,11,12,16].

The effectiveness of CC can also be increased in the form of early identification by using different sampling schemes like Repetitive sampling, Rank set sampling, Sequential sampling and Multiple Dependent State (MDS) sampling. MDS sampling was a revolutionary step in the field of Statistical process control (SPC) given by Wortham and Baker [24] in which judgments are made about the lot sentencing with the support of information gathered from recent sample and the information accumulated from previous lot. Actually this sampling technique does not demand for additional sampling but utilize previous information of sample data. After that Balamurali and Jun [8] used MDS-based plan and showed the outstanding performance by comparing it with a single sampling plan in term of smaller average sample number. More work using MDS can be seen in [2,6,25,7]. The development of CC using the concept of MDS sampling was proposed by Aslam et al. [3] to monitor the process parameter for exponential life data. The operational procedure of MDS-based CC is quite unique in its implementation. The authors used four control limits to monitor the process mean. If the value of statistic falls within the interior limits the running process is promulgated as in control. If the value of statistic falls outside the exterior limits the ongoing process is proclaimed as out of control. If the value of statistic falls in re-sampling state, then we stop our sampling, and decision made about the process in or out of control of that particular value of statistic is based on the previous rth information. While in RSS, we continue our sampling until we reached the decision of in or out of control process. The determination of the value of parameter $r$ is the mutual understanding between producer and consumer. If the quality history of producer is excellent and constantly meet the requirement of market, then the smaller value of $r$ is the better choice otherwise vice versa. The results are compared with antagonist CC proposed by Santiago and Smith [20] in the form of ARL which showed the outstanding performance of recommended idea in capturing the minor variations very quickly. After that Aslam et al. [4] suggested an Attribute CC using the concept of MDS technique in which current as well as previous information is used for enhancing the competency of offered chart. Results are matched in the form of ARL with competitor chart and found to be a notable performance of proposed idea in the form of lesser values of ARL. More work about the application of MDS in developing CCs can be viewed in [5,13,1].

By scrutinizing the literature, CC based on MDS using Extended EWMA (EEWMA) statistic has not been suggested yet. Therefore, we suggested a MDS-based CC using Extended EWMA statistic in this paper. It is expected that the initiated CC which is the combination of improved sampling scheme as well as improved memory type estimator will be more powerful in term of quick identification of switched process in mean. The rest of the paper is organized as follows: the designing of the recommended CC is given in Section 2. In Section 3, the superiority of the suggested CC with the existing CC is discussed. A simulation study is conducted in Section 4. In Section 5, the utilization of the proposed idea in practical life data is considered. Finally concluding remarks are discussed in Section 6.

2. Designing of proposed CC

Suppose that manufacturing procedure is operating in stable condition and quality trait designated as $Y_i$ follows the normal distribution (ND) with mean $w$ and standard deviation $\sigma$, then the statistic initiated by Naveed et al. [16] named as EEWMA statistic is given as

$$Z_i={\theta }_1\overline{Y_i}-{\theta }_2{\overline{Y}}_{i-1}+(1-{\theta }_1+{\theta }_2)Z_{i-1},$$ (1)

where $\overline{Y_i}$ is the mean quality trait of the sample of size $n$ and ${\theta }_1\text{and}{\theta }_2$ are the smoothing constants (weights) such that $0<{\theta }_1\le 1\text{and}0\le {\theta }_2<{\theta }_1$ under the condition that the sum of weights is unity and ${\theta }_1>{\theta }_2$. The mean and variance of the statistic are given as

$$\begin{array}{rl}E(Z_i)& =w,\end{array}$$ (2)

$$\begin{array}{rl}var(Z_i)& =\frac{{\sigma }^2}{n}\left[({\theta }_1^2+{\theta }_2^2)\left\{\frac{1-{\gamma }^{2i}}{1-{\gamma }^2}\right\}-2\gamma {\theta }_1{\theta }_2\left\{\frac{1-{\gamma }^{2i-2}}{1-{\gamma }^2}\right\}\right],\end{array}$$ (3)

where $\gamma =(1-{\theta }_1+{\theta }_2)\text{and }w,{\sigma }^2$ indicates the target mean and variance of $Y_i$ respectively. For larger value of $i,var(Z_i)$ reduce to

$$var(Z_i)=\frac{{\sigma }^2}{n}\left(\frac{{\theta }_1^2+{\theta }_2^2-2\gamma {\theta }_1{\theta }_2}{1-{\gamma }^2}\right).$$ (4)

The functioning methodology of the recommended chart is as follows:

Step 1: Take a random sample of size $n$ from the quality feature $Y_i$ and compute $\overline{Y_i}$, then calculate the value of $Z_i$ using predefine smoothing constants.

Step 2: Process is proclaimed to be under control if the value of $Z_i$ lies within the internal limits. On the other hand, if the values of statistic $Z_i$ fall outside the external limits process is professed to be out of control.

Step 3: The functioning process is considered to be under control if $r$ prior subgroups fall within the internal limits otherwise declare the procedure as out of control.

The external pair of limits named as upper and lower external limits are given as

$$\begin{array}{rl}UC{L}_1& =w+L_1\sqrt{\frac{{\sigma }^2}{n}\left(\frac{{\theta }_1^2+{\theta }_2^2-2\gamma {\theta }_1{\theta }_2}{1-{\gamma }^2}\right)},\end{array}$$ (5)

$$\begin{array}{rl}LC{L}_1& =w-L_1\sqrt{\frac{{\sigma }^2}{n}\left(\frac{{\theta }_1^2+{\theta }_2^2-2\gamma {\theta }_1{\theta }_2}{1-{\gamma }^2}\right)}.\end{array}$$ (6)

The internal limits are given as

$$\begin{array}{rl}UC{L}_2& =w+L_2\sqrt{\frac{{\sigma }^2}{n}\left(\frac{{\theta }_1^2+{\theta }_2^2-2\gamma {\theta }_1{\theta }_2}{1-{\gamma }^2}\right)},\end{array}$$ (7)

$$\begin{array}{rl}LC{L}_2& =w-L_2\sqrt{\frac{{\sigma }^2}{n}\left(\frac{{\theta }_1^2+{\theta }_2^2-2\gamma {\theta }_1{\theta }_2}{1-{\gamma }^2}\right)}.\end{array}$$ (8)

The initiated chart comprises of two control constants $L_{1}and{L}_2$ as well as parameter $r.$ The intended CC is reduced to EWMA CC based on MDS when ${\theta }_2=0.$ Furthermore given idea is transformed to [16] if $L_1=L_2=L\text{and}r=0$.

The probability that the running process is operating under a control condition when genuinely it is working in stable condition is designated by $P_{in}^0$ using MDS sampling is given as

$$\begin{array}{rl}{P}_{in}^0& =P(LC{L}_2\le Z_i\le UC{L}_2)\\ & +\left\{\begin{array}{c}P(UC{L}_2<Z_i<UC{L}_1)+\\ P(LC{L}_1<Z_i<LC{L}_2)\end{array}\right\}\ast \{P(LC{L}_2\le Z_i\le UC{L}_2){\}}^r.\end{array}$$ (9)

By solving Equation 9, we have

$$P_{in}^0=(2\varphi (L_2)-1)+2\{\varphi (L_1)-\varphi (L_2)\}\ast \{(2\varphi (L_2)-1){\}}^r.$$ (10)

The effectiveness of the intended CC is evaluated with the support of ARL. ARL reveals the average number of subgroups before the ongoing procedure is professed to be out of control. The ARL when the functioning process is operating under control condition denominated as in-control ARL designated by $AR{L}_{0}or{r}_0$ and calculated as

$$AR{L}_0=\frac{1}{1-P_{in}^0}.$$ (11)

Now we presume that operating procedure is changed owing to some exterior variation and switch to the new process mean that is $w\text{to}{w}_1$, where $w_1=w+d\sigma$.

Now the probability that the manufacturing process is proclaimed as in control when assured it is switched to the new process mean is labelled as $P_{in}^1$ and calculated as

$$\begin{array}{rl}{P}_{in}^1& =\varphi \left(L_2-\frac{d\sqrt{n}}{\sqrt{\left({\displaystyle \frac{{\theta }_1^2+{\theta }_2^2-2\gamma {\theta }_1{\theta }_2}{1-{\gamma }^2}}\right)}}\right)-\varphi \left(-L_2-\frac{d\sqrt{n}}{\sqrt{\left({\displaystyle \frac{{\theta }_1^2+{\theta }_2^2-2\gamma {\theta }_1{\theta }_2}{1-{\gamma }^2}}\right)}}\right)\\ & +\{\varphi \left(L_1-\frac{d\sqrt{n}}{\sqrt{\left({\displaystyle \frac{{\theta }_1^2+{\theta }_2^2-2\gamma {\theta }_1{\theta }_2}{1-{\gamma }^2}}\right)}}\right)-\varphi \left(L_2-\frac{d\sqrt{n}}{\sqrt{\left({\displaystyle \frac{{\theta }_1^2+{\theta }_2^2-2\gamma {\theta }_1{\theta }_2}{1-{\gamma }^2}}\right)}}\right)\\ & +\varphi \left(-L_2-\frac{d\sqrt{n}}{\sqrt{\left({\displaystyle \frac{{\theta }_1^2+{\theta }_2^2-2\gamma {\theta }_1{\theta }_2}{1-{\gamma }^2}}\right)}}\right)-\varphi \left(-L_1-\frac{d\sqrt{n}}{\sqrt{\left({\displaystyle \frac{{\theta }_1^2+{\theta }_2^2-2\gamma {\theta }_1{\theta }_2}{1-{\gamma }^2}}\right)}}\right)\}\\ & \text{* }{\left\{\varphi \left(L_2-\frac{d\sqrt{n}}{\sqrt{\left({\displaystyle \frac{{\theta }_1^2+{\theta }_2^2-2\gamma {\theta }_1{\theta }_2}{1-{\gamma }^2}}\right)}}\right)-\varphi \left(-L_2-\frac{d\sqrt{n}}{\sqrt{\left({\displaystyle \frac{{\theta }_1^2+{\theta }_2^2-2\gamma {\theta }_1{\theta }_2}{1-{\gamma }^2}}\right)}}\right)\right\}}^r.\end{array}$$ (12)

The ARL for alter process is denoted by $AR{L}_1$ and calculated as

$$AR{L}_1=\frac{1}{1-P_{in}^1}.$$ (13)

Now we construct various tables of ARL using proposed concept by setting predefined value of ARL, different values of MDS parameter $r$. The subsequent tendency is viewed in $AR{L}_1$ values of recommended chart provided by Tables 1–3.

1. As $r$ increases, $AR{L}_1$ values perceived a declining tendency, as well as we observed the asymptotic results of proposed chart for larger value of $r$ that is as we increase the value of $r$ the difference between ARLs is very close and still in diminishing trend.

2. As $d$ decreases, $AR{L}_1$ exhibits a speedy decreasing pattern.

3. For smaller values of smoothing constant, we exhibit a hasty diminishing trend.

Table 2. ARLs of proposed CC when $r_0=370,n=5$.

		EEWMA for MDS sampling
	EEWMA for single sampling	$r=1$	$r=2$	$r=5$	$r=20$	$r=30$
	$L=3.0$	$L_1=3.0186$	$L_1=3.979$	$L_1=3.444$	$L_1=3.64$	$L_1=4.37$
		$L_2=2.4544$	$L_2=2.090$	$L_2=2.301$	$L_2=2.52$	$L_2=2.57$
	${\theta }_1=0.20$	${\theta }_1=0.20$	${\theta }_1=0.20$	${\theta }_1=0.20$	${\theta }_1=0.20$	${\theta }_1=0.20$
	${\theta }_2=0.07$	${\theta }_2=0.07$	${\theta }_2=0.07$	${\theta }_2=0.07$	${\theta }_2=0.07$	${\theta }_2=0.07$
$d$	$ARL$	$ARL$	$ARL$	$ARL$	$ARL$	$ARL$
0	370.03	370.01	370.01	370.02	370.00	370.01
0.03	291.54	289.30	285.92	275.83	266.31	263.98
0.04	248.62	245.32	238.64	225.58	213.15	210.09
0.05	207.59	203.39	193.09	178.93	165.34	162.03
0.07	140.35	135.08	119.51	107.12	95.23	92.56
0.09	94.09	88.57	71.90	62.96	54.74	53.19
0.10	77.27	71.84	55.83	48.49	42.01	40.96
0.12	52.73	47.72	34.17	29.37	25.67	25.37
0.15	30.79	26.67	17.33	14.87	13.66	13.62
0.20	13.89	11.20	6.74	5.96	5.88	5.80
0.25	7.1	5.42	3.32	3.07	3.03	3.00
0.30	4.09	3.05	2.02	1.94	1.90	1.87

Table 1. ARLs of proposed CC when $r_0=370,n=5$.

		EEWMA for MDS sampling
	EEWMA for single sampling	$r=1$	$r=2$	$r=5$	$r=20$	$r=30$
	$L=2.9996$	$L_1=3.0499$	$L_1=3.105$	$L_1=3.472$	$L_1=3.58$	$L_1=4.42$
		$L_2=2.2987$	$L_2=2.307$	$L_2=2.297$	$L_2=2.52$	$L_2=2.56$
	${\theta }_1=0.10$	${\theta }_1=0.10$	${\theta }_1=0.10$	${\theta }_1=0.10$	${\theta }_1=0.10$	${\theta }_1=0.10$
	${\theta }_2=0.03$	${\theta }_2=0.03$	${\theta }_2=0.03$	${\theta }_2=0.03$	${\theta }_2=0.03$	${\theta }_2=0.03$
$d$	$ARL$	$ARL$	$ARL$	$ARL$	$ARL$	$ARL$
0	370.02	370.02	370.03	370.07	370.01	370.01
0.03	232.00	225.31	219.67	206.45	193.14	190.17
0.04	175.99	167.35	160.31	144.18	130.32	127.79
0.05	131.53	121.93	114.47	98.11	86.13	84.46
0.07	73.55	64.23	57.76	45.26	38.66	38.42
0.09	42.42	34.72	30.01	22.10	19.27	19.20
0.10	32.72	25.93	22.05	15.94	14.27	14.18
0.12	21.11	15.00	12.75	8.91	8.60	8.55
0.15	10.51	7.30	5.98	4.46	4.35	4.28
0.20	4.39	2.92	2.49	2.05	2.01	1.99
0.25	2.34	1.64	1.50	1.34	1.26	1.20
0.30	1.54	1.21	1.17	1.13	1.10	1.02

Table 3. ARLs of proposed CC when $r_0=370,n=5$.

		EEWMA for MDS sampling
	EEWMA for single sampling	$r=1$	$r=2$	$r=5$	$r=20$	$r=30$
	$L=3.0$	$L_1=3.0174$	$L_1=3.154$	$L_1=3.613$	$L_1=3.62$	$L_1=3.73$
		$L_2=2.104$	$L_2=2.252$	$L_2=2.281$	$L_2=2.52$	$L_2=2.58$
	${\theta }_1=0.30$	${\theta }_1=0.30$	${\theta }_1=0.30$	${\theta }_1=0.30$	${\theta }_1=0.30$	${\theta }_1=0.30$
	${\theta }_2=0.15$	${\theta }_2=0.15$	${\theta }_2=0.15$	${\theta }_2=0.15$	${\theta }_2=0.15$	${\theta }_2=0.15$
$d$	$ARL$	$ARL$	$ARL$	$ARL$	$ARL$	$ARL$
0	370.05	370.03	370.00	370.01	370.00	370.00
0.03	315.27	311.15	309.05	304.26	296.93	295.08
0.04	281.83	275.20	272.13	264.53	254.15	251.57
0.05	247.14	237.97	234.13	223.89	211.53	208.53
0.07	183.36	170.11	165.54	151.86	139.01	136.10
0.09	133.15	117.85	113.39	99.21	88.70	86.60
0.10	113.26	97.66	93.45	79.83	70.85	69.20
0.12	82.20	67.09	63.49	51.90	45.80	44.98
0.15	51.71	38.82	36.182	28.08	25.17	25.16
0.20	25.39	16.84	15.389	11.50	11.18	11.16
0.25	13.54	8.22	7.441	5.66	5.60	5.51
0.30	7.83	4.54	4.127	3.33	3.23	3.20

3. Advantages of proposed CC

This section is sundered into three subdivisions, in Section 3.1 benefit of the suggested CC is examined on the basis of ARL by comparing it with Naveed et al. [16], Section 3.2 gives the comparability with Khan et al. [13], Section 3.3 gives collation with MDS-based EWMA CC.

3.1. Proposed MDS-EEWMA control chart versus Naveed et al. [16]

The existing CC proposed by Naveed et al. [16] is based on single sampling scheme. Whereas recommended CC consists of two pairs of limits. Naveed et al. [16] is the special case of suggested CC when $L_1=L_2=L\text{and}r=0$. The ARL values are presented in the first column of Tables 1–3 which express the far better evaluation of proposed chart with lesser ARLs as collate to the existing chart. For example when $r_0=370,n=5,d=0.03$, then $AR{L}_1=232$ for existing CC and for intended CC $AR{L}_1=225.321\text{for}r=1,AR{L}_1=219.67\text{for}r=2,AR{L}_1=206.45\text{for}r=5,AR{L}_1=193.14\text{for}r=20$ and for $r=30,AR{L}_1=190.17$. We observed the lesser values of $AR{L}_1$ for every value of $r$. We also notice that for the greater value of $r$, the execution of proposed chart for identifying the smaller shifts has been increased.

3.2. Proposed MDS-EEWMA CC versus Khan et al. [13]

Here we examined the benefits of presented CC by collating it with Khan et al. [13] with the assist of ARL. The ARL values using Khan et al. [13] are placed in Table 4. We notice the lesser values of $AR{L}_1$ for presented chart for every value of $d$ which manifest the remarkable execution of presented chart. For example, when $r_0=370,n=5,{\theta }_1=0.1,{\theta }_2=0.03,r=2,d=0.03,AR{L}_1=219.67$ for presented chart and for $AR{L}_1=244.82$ for [13] when $\theta =0.1$ and rest of the parameters are same. Thus the best evaluation of suggested CC is observed in the form of lesser ARL.

Table 4. Comparisons of ARLs when $r_0=370,n=5$.

	Proposed EEWMA for MDS sampling	Khan et al. [13]	EWMA for MDS sampling
	$r=2$	$r=2$	$r=2$
	$L_1=3.105$	$L_1=3.8230$	$L_1=3.609$
	$L_2=2.307$	$L_2=2.085$	$L_2=2.156$
	${\theta }_1=0.10$	$\theta =0.1$	$\theta =0.1$
	${\theta }_2=0.03$
$d$	$ARL$	$ARL$	$ARL$
0	370.03	370.03	370.00
0.03	219.67	244.82	254.97
0.04	160.31	186.10	201.96
0.05	114.47	136.74	156.69
0.07	57.76	70.97	92.57
0.09	30.01	37.20	55.14
0.10	22.05	27.38	42.92
0.12	12.75	15.49	26.55
0.15	5.98	7.43	13.69
0.20	2.49	2.99	5.41
0.25	1.50	1.71	2.71
0.30	1.17	1.26	1.71

3.3. Proposed MDS-EEWMA control chart versus MDS-EWMA control chart

In this section, we made a comparison between offered chart and MDS-EWMA CC with the assist of ARL. The offered chart is converted to MDS-EWMA CC if ${\theta }_2=0.$ For comparison purpose, rest of the values of parameters are the same to view the clear picture of the present chart. It can be seen in Table 4 that the offered chart has a lesser value of ARL as collating it to MDS-EWMA CC. For example when $r_0=370,{\theta }_1=0.1,{\theta }_2=0.03,r=2,n=5,d=0.04,AR{L}_1=160.31$ for offered chart and for MDS-EWMA CC $AR{L}_1=201.96$ when $\theta =0.1$ and other values of the parameters are same. We observe the positive difference between ARL which reveals the remarkable performance of offered chart in the form of early identification of shift.

4. Simulation study

In this section, the working operation of the offered chart is elaborated for its practical use with the help of simulated data. We presume that the operating process is working in stable condition and follows the ND. First we originate 20 observations of size 5 with mean, $w=0\text{and}\sigma =1$ using MDS sampling$.$ Again we originate 20 observations of size 5 from the switched process with mean, $w_1=w+d\sigma$ where $d=0.12$, using MDS sampling. The simulated data and corresponding values of a statistic $Z_i$ using the parameter $L_1=3.105,L_2=2.307,r=2,{\theta }_1=0.1\text{and}{\theta }_2=0.03\text{and}AR{L}_0=370$ are plotted in Figure 1. We notice that process is disturbed after 20 + 13 = 33 samples (same value was recorded in Table 1). We also plot the simulated data using Naveed et al. [16] in Figure 2 using parameter ${\theta }_1=0.1,{\theta }_2=0.03\text{and}L=\mathrm{2.999.}$ From Figure 2, we ascertained that switched process is not identified by using these CCs. Therefore, we can say that the current CC is more proficient in searching out the early variation in the process mean.

Figure 1.

Simulated graph of proposed chart.

Figure 2.

Simulated graph of Naveed et al. [16].

5. Industrial application

Here the utilization of the offered chart is elaborated with the cooperation of real-life data set which is taken from Montgomery [15, p. 284]. The data set comprises of 20 observations each of size 5 are the mensuration of the dimension of finished product produced by the machine. It is avowed that the measurement data follows the ND. The control limits for initiated chart using Equation (5)–(8) with ${\theta }_1=0.1,{\theta }_2=0.03,L_1=3.105,L_2=2.307$ and $AR{L}_0=370$ are $UC{L}_1=136.48,UC{L}_2=135.04$ $LC{L}_1=125.26,LC{L}_2=126.70$ whereas the control limits for competitor chart (Shewhart Type Chart) are $UCL=136.26,LCL=125.45$ with CC parameters $L=3.0,{\theta }_1=0.1,{\theta }_2=0.03.$ The data and corresponding values of statistic $Z_i$ using MDS technique with ${\theta }_1=0.1,{\theta }_2=0.03,r=2$ are plotted in Figure 3. An MDS-based CC is considered to be controlled if the value of $Z_i$ falls within $LC{L}_2\le Z_i\le UC{L}_2$. Using MDS technique the in-control span for suggested CC is 8.34, whereas Shewhart-based competitor chart controlled span was 10.84 which reveals the notable performance of intended CC in finding the slighter variation more quickly. We notice that process is under control, however, samples 7 and 8 are nearer to internal limits which demands the attentiveness of industrial engineers.

Figure 3.

Industrial data of proposed CC.

6. Concluding remarks

In this paper, we offered a memory type CC based on MDS sampling to recognize the petite shifts in the process mean. The execution of the initiated CC has been observed in the form of ARL. Comparison of the presented chart has been made with Naveed et al. [16], MDS-EWMA CC, Khan et al. [13] based on MDS technique on the basis of ARL which exhibit a far better evaluation of the initiated CC in term of faster identification of shorter variation in process mean. A simulation-based example has been operated to judge the efficacy of recommended CC. In addition, utilization of the intended CC has been given through practical example for the elucidation of recommended methodology. The initiated CC is a remarkable contribution in the toolkit of the quality control area. The suggested CC can be extended for other sampling schemes like modified MDS sampling, modified repetitive sampling for further research.

Disclosure statement

No potential conflict of interest was reported by the authors.