Optimal Designs of the Median Run Length Based Double Sampling X̄ Chart for Minimizing the Average Sample Size

Abstract

Designs of the double sampling (DS) chart are traditionally based on the average run length (ARL) criterion. However, the shape of the run length distribution changes with the process mean shifts, ranging from highly skewed when the process is in-control to almost symmetric when the mean shift is large. Therefore, we show that the ARL is a complicated performance measure and that the median run length (MRL) is a more meaningful measure to depend on. This is because the MRL provides an intuitive and a fair representation of the central tendency, especially for the rightly skewed run length distribution. Since the DS chart can effectively reduce the sample size without reducing the statistical efficiency, this paper proposes two optimal designs of the MRL-based DS chart, for minimizing (i) the in-control average sample size (ASS) and (ii) both the in-control and out-of-control ASSs. Comparisons with the optimal MRL-based EWMA and Shewhart charts demonstrate the superiority of the proposed optimal MRL-based DS chart, as the latter requires a smaller sample size on the average while maintaining the same detection speed as the two former charts. An example involving the added potassium sorbate in a yoghurt manufacturing process is used to illustrate the effectiveness of the proposed MRL-based DS chart in reducing the sample size needed.

Introduction

Statistical process control (SPC) is a powerful collection of statistical tools for achieving process stability. SPC is based on sound underlying principles, which is easy to use and can be applied in the manufacturing and service processes, such as in the food industries, automobile industries, as well as health-care and public-health surveillance [1]. A control chart is one of the valuable quality improvement techniques in SPC that can be used to attain process stability and reduce process variability over time. Since the double sampling (DS) chart was introduced by Croasdale [2] in 1974, the DS scheme has been studied extensively among researchers. By applying the concept of double sampling plans, Daudin [3] suggested an improved DS chart which incorporates both the ideas of variable sampling interval (VSI) and variable sample size (VSS). Unlike the VSI procedure, two successive samples are taken in the DS procedure without any intervening time; where, both the first and second samples of the DS chart are taken from the same population.

Recently, considerable efforts have been undertaken on the research of various DS type charts, which can be categorized into the DS type, DS S type and other DS type control charts. Costa and Machado [4], Khoo et al. [5] and Torng et al. [6] investigated the DS type charts for monitoring the process mean. Works on the DS S type charts for monitoring the process variance were discussed by He and Grigoryan [7], [8] and Lee et al. [9], [10]. Other DS type charts are the joint DS and S chart, proposed by He and Grigoryan [11], for a simultaneous monitoring of the process mean and variance, as well as the DS np chart for attributes, suggested by Rodrigues et al. [12].

It is known that the DS chart not only maintains the simplicity of the Shewhart chart, but the former also improves the statistical efficiency of the latter in detecting process mean shifts, besides reducing the sample size [13]. Compared to the Shewhart chart, He et al. [14] claimed that the sample size of the DS chart dramatically decreases to nearly 50% when the process is in-control. In addition, the DS chart has an advantage of having a lower total sample size when the incoming quality is either very excellent or very poor [15]. This is because only the first sample is required to sentence the process as either in-control or out-of-control. Therefore, the DS scheme is an appropriate choice for process monitoring with destructive testing and high inspection costs [16]. In view of these advantages, many researchers (see [2], [3], [7], [8], [14]) focused on proposing the DS chart for minimizing the in-control average sample size (). Hsu [17], [18] claimed that the conclusion made by He et al. [14] and He and Grigoryan [7] is questionable since the out-of-control average sample size () is disregarded when comparing the various charts' performances. Accordingly, Lee et al. [10] modified the design model of He and Grigoryan [8] to propose the DS S chart which minimizes both the and .

The average run length (ARL) has been traditionally used as a sole measure of a control chart's performance. The sole reliance on the ARL has been widely criticized by Das [19], Gan [20] and Golosnoy and Schmid [21]. This criticism comes from two concerns [1]. First, the value of the standard deviation of the run length (SDRL) is quite large. Second, the run length distribution is highly skewed. Furthermore, Thaga [22] stated that only a fraction of a chart's behavior is reflected by the size of the ARL. Therefore, misleading conclusion is drawn based on the ARL as it is not necessarily a typical run length. On the other hand, the median run length (MRL) is a more credible measure of a chart's performance since it is less affected by the skewness of the run length distribution [20], [23]. The MRL is the 50^th percentile of the run length distribution, representing “half of the time” [24]. For example, when the in-control MRL () is 250, a practitioner can claim that a false alarm will occur by the 250^th sample in half of the time; while an out-of-control MRL () of 10 means that for this particular shift, there is a 50% chance that an out-of-control signal will be produced in not later than the 10^th sample. For ease of interpretation and a better understanding of a chart's performance, Gan [20], Golosnoy and Schmid [21], Maravelakis et al. [23], Khoo et al. [25] and Low et al. [26] have all advocated using MRL as an alternative measure to evaluate a chart's performance.

Similar to other charts, the ARL is widely used in the literature as a performance and design criteria of the DS chart. However, when the run length distribution is highly skewed to the right, especially for an in-control process or when the shift is small, we show that the ARL is a peculiar measure of a typical chart's performance and that the MRL is a more meaningful quantity to rely on. Keeping this in mind, two new optimal design procedures for the MRL-based DS chart by minimizing the (i) and (ii) are developed in this paper. In this paper, the average sample size (ASS) is chosen as the objective function of the optimal design models. This is because these optimal design models are applicable to small enterprises which have a low production volume or are useful for monitoring destructive testing processes. The reason for minimizing the is due to the fact that the process will operate in the in-control state for most of the time [1], [3], [8], [14]. Hsu [17], [18] stated that the ASS for both the in-control and out-of-control situations should be taken into consideration when designing a control chart. Therefore, the second optimal design, i.e. minimizing the and , is proposed in accordance with the argument of Hsu [17], [18]. Consequently, a smaller sample size is used and this leads to a substantial reduction of inspection and sampling costs.

The rest of this paper is organized as follows: The DS chart's procedure and its run length properties are briefly introduced in Section 2. Section 3 examines the performance of the DS chart, in terms of the percentiles of the run length distribution, ARL and ASS. Two optimal designs of the MRL-based DS chart, for minimizing the (i) and (ii) are proposed in Section 4. Besides providing the optimal chart parameters for the MRL-based DS chart, Section 5 compares the sample-size performance of the optimal MRL-based DS , EWMA and Shewhart charts. An illustrative example on the construction of the optimal MRL-based DS chart is given in Section 6. Conclusions are drawn in Section 7.

The DS Control Chart for Monitoring the Process Mean

Assume that the observations of the quality characteristic X are independent and follow an identical normal distribution with the in-control mean and variance . We further assume that and are known. By referring to Figure 1, let 0 and be the warning and control limits of the first-sample stage, respectively; while 0 is the control limit of the combined-sample stage. The regions of the DS chart can be divided into , , and . The charting procedure of the DS chart is as follows:

Figure 1. Schematic representation of the DS chart's operation.

The DS chart consists of two stages, i.e. the first-sample stage and the combined-sample stage.

1. Determine the limits , and .

2. Take a first sample of size and calculate the first sample mean . Here, for , is the j ^th observation at the i ^th sampling time of the first sample.

3. Declare the process as in-control if . Then the control flow returns to Step (2).

4. Declare the process as out-of-control if and then proceed to Step (8).

5. Take a second sample of size from the same population as the first sample if . Then compute the second sample mean . Here, for , is the j ^th observation at the i ^th sampling time of the second sample.

6. Calculate the combined-sample mean at the i ^th sampling time.

7. Declare the process as in-control if ; otherwise, declare the process as out-of-control and advance to Step (8).

8. Issue an out-of-control signal at the i ^th sampling time to indicate a process mean shift.

9. Investigate and remove assignable cause(s) and then return to Step (2).

Note that the i ^th sampling time refers to the i ^th time when either only the first sample of size or both the first and second samples of size , are collected.

Let be the size of a standardized mean shift, where is the out-of-control mean. If 0, the process is considered as in-control; otherwise, it is deemed as out-of-control. Let and represent the probabilities that the process remains in-control “by the first sample” and “after taking the second sample”, respectively. Then, is the probability that the process is regarded as in-control, where and are given as [3]

(1)

and

(2)

respectively, where and are the standard normal cumulative distribution function (cdf) and standard normal probability density function (pdf), respectively. In Equation (2), , and .

Let RL represents the run length which is the number of samples collected until the first out-of-control signal is detected. The RL distribution of a Shewhart chart follows a geometric distribution when the chart's control limits are known constants and the plotted statistics are independently and identically distributed random variables [1]. Since the DS chart is a two-stage Shewhart chart, all the RL properties of the DS chart can be characterized by those of the geometric distribution. Hence, the cdf of the RL for the DS chart, defined for {1, 2, 3, …}, is calculated as

(3)

It follows that the MRL of the DS chart is equal to [20]

(4)

while the other percentiles of the RL distribution are computed as the value , such that

(5)

where is in the range of .

Daudin [3] also showed that the ARL of the DS chart is

(6)

while the ASS at each sampling time is defined as

(7)

where .

Performance of the DS Chart Based on the Percentiles of the Run Length Distribution, ARL and ASS

Palm [24] claimed that a practitioner is more interested in the percentiles of the RL distribution as they provide additional and detailed information regarding the expected behavior of the RL. Therefore, we investigate the performance of the optimal ARL-based DS chart for minimizing , in terms of ARL, ASS and the percentiles of the RL distribution. Table 1 summarizes these performance measures for the DS chart when the in-control ARL, 370.0 and the out-of-control ARL, . Here, is the desired value corresponding to a shift . The optimization procedure given by Daudin [3] is applied here. The value is specified as the value of the optimal EWMA chart, where the value and the sample size () of this EWMA chart are set as 370.0 and (3, 5), respectively. In Table 1, the optimal (, , , , ) combinations of the ARL-based DS chart for minimizing are obtained such that 370.0 and . Here, the values in Table 1 are selected so that when 0.5, {11.9, 8.1} for {3, 5}; and when 1.0, {4.2, 2.8} for {3, 5}. These optimal chart parameters are used to calculate the ARL, ASS and the percentiles of the RL distribution based on the formulae shown in Section 2. Note that the in Table 1 represents the sample size of the ARL-based Shewhart chart, matching approximately a similar design of the ARL-based DS chart.

Table 1. ARLs, ASSs and percentiles of the run length distribution for the DS chart when 370.0 and .

						Percentiles of the run length distribution
δ *	n EWMA	nX -bar	δ	ARL	ASS	5th	10th	20th	30th	40th	50th	60th	70th	80th	90th	95th
0.5	3	11	n 1 = 2, n 2 = 18, L 1 = 1.847, L = 5.885, L 2 = 2.368
			0.00	370.0	3.165	19	39	83	132	189	257	339	445	595	851	1107
			0.25	57.8	3.467	3	7	13	21	30	40	53	70	93	133	172
			0.50	11.9	4.384	1	2	3	5	6	8	11	14	19	27	35
			1.00	3.0	7.995	1	1	1	1	2	2	3	3	5	6	8
			1.50	1.6	12.943	1	1	1	1	1	1	1	2	2	3	4
			2.00	1.2	17.041	1	1	1	1	1	1	1	1	1	2	2
			3.00	1.0	18.946	1	1	1	1	1	1	1	1	1	1	1
	5	14	n 1 = 3, n 2 = 17, L 1 = 1.647, L = 5.796, L 2 = 2.599
			0.00	370.0	4.691	19	39	83	132	189	257	339	445	595	851	1107
			0.25	46.7	5.228	3	5	11	17	24	33	43	56	75	107	139
			0.50	8.1	6.796	1	1	2	3	4	6	7	10	13	18	23
			1.00	1.9	12.080	1	1	1	1	1	1	2	2	3	4	4
			1.50	1.2	17.084	1	1	1	1	1	1	1	1	1	2	2
			2.00	1.0	19.244	1	1	1	1	1	1	1	1	1	1	1
			3.00	1.0	15.334	1	1	1	1	1	1	1	1	1	1	1
1.0	3	5	n 1 = 1, n 2 = 8, L 1 = 1.525, L = 5.977, L 2 = 2.591
			0.00	370.0	2.018	19	39	83	132	189	257	339	445	595	851	1107
			0.25	103.2	2.112	6	11	23	37	53	72	95	124	166	237	308
			0.50	23.7	2.392	2	3	6	9	12	17	22	28	38	54	70
			1.00	4.2	3.444	1	1	1	2	2	3	4	5	6	9	11
			1.50	2.1	4.929	1	1	1	1	1	2	2	2	3	4	5
			2.00	1.5	6.462	1	1	1	1	1	1	1	2	2	3	3
			3.00	1.1	8.427	1	1	1	1	1	1	1	1	1	1	2
	5	7	n 1 = 2, n 2 = 9, L 1 = 1.659, L = 5.899, L 2 = 2.646
			0.00	370.0	2.875	19	39	83	132	189	257	339	445	595	851	1107
			0.25	82.7	3.063	5	9	19	30	42	57	76	99	133	190	247
			0.50	16.7	3.617	1	2	4	6	9	12	15	20	27	38	49
			1.00	2.8	5.641	1	1	1	1	2	2	3	3	4	6	7
			1.50	1.5	8.104	1	1	1	1	1	1	1	2	2	3	3
			2.00	1.1	9.901	1	1	1	1	1	1	1	1	1	2	2
			3.00	1.0	10.517	1	1	1	1	1	1	1	1	1	1	1

From Table 1, we observe that the difference between the values of ARL and MRL is large when 0 and it diminishes as increases. This indicates that the shape and the skewness of the RL distribution change with the magnitude of the process mean shift . Also, the ARLs shown in Table 1 are all larger than the MRLs (i.e. 50^th percentile of the RL distribution) when 2.0. This is due to the fact that in a right-skewed RL distribution, the value of the average of the RL is greater than the median of the RL. Thus, the MRL is a better representation of the central tendency compared to the ARL. Note that the ARL only measures the expected run length and does not indicate the likelihood of getting a signal by a certain probability. For example, when 1.0, 3 and 0.25 are considered, there could exist a risk where a practitioner falsely interprets that an out-of-control is detected by the 103^rd sampling time (103.2) in 50% of the time, but in actual fact, this event occurs noticeably earlier, i.e. by the 72^nd sampling time (72).

An advantage of computing the lower percentiles (e.g. 5^th, 10^th and 20^th percentiles) of the RL distribution for 0 is that it allows the probability analysis of early false signals to be carried out. From Table 1, we notice that even when the value of is large, the lower percentiles are remarkably shorter. This suggests that even when the false alarm rate (FAR = 0.0027) is low, a relatively large percent of false signals occur very early in the process monitoring. The computation of the higher percentiles (e.g. 80^th, 90^th and 95^th percentiles) of the RL distribution also provides some useful information to a practitioner. For instance, when 0.5, 5 and 1.0 are considered, a practitioner can state with a 90% confidence that a shift with magnitude 1.0 is signaled by the fourth sampling time.

Table 1 provides clear evidence that the in-control RL distribution is highly skewed and that the skewness of the RL distribution changes with . Therefore, interpretation based on the average of the RL (or ARL) with respect to a highly skewed RL distribution is certainly misleading compared to the case if the RL distribution is symmetric. When the associated RL distribution has different levels of skewness as changes, the MRL provides a more meaningful performance measure for the DS chart. Along this line, we are motivated to propose two optimal designs (see Section 4) of the MRL-based DS chart.

Optimal Designs of the MRL-Based DS Chart

The optimal designs of the MRL-based DS chart having the smallest (i) and (ii) both the and , are proposed in Sections 4.1 and 4.2, respectively. The optimization programs are written using the ScicosLab software (www.scicoslab.org). It is not easy to optimally determine the five charting parameters, i.e. , , , and of the DS chart. Therefore, these optimal chart parameters are searched through the implementation of the Nelder Mead's nonlinear optimization algorithm [27]. Since the sample sizes and are parameters to be optimized, we need to limit the allowable upper bound, i.e. . Thus, 20 is fixed in this paper because it is a common practice in industries to use small and moderate sample sizes.

4.1 Minimizing the in-control ASS

The proposed optimal design of the MRL-based DS chart for minimizing the is illustrated as follows:

(8)

subject to

1. (9)

   where is the desired in-control MRL.
2. (10)

   where is the desired out-of-control MRL corresponding to a shift .
3. (11)

   where is the sample size of the MRL-based Shewhart chat, matching approximately a similar design of the MRL-based DS chart.

By applying the optimization model (8)–(11), the steps for obtaining the optimal MRL-based DS chart's parameters (, , , , ) are demonstrated as follows:

1. Specify the desired values of , , , and .

2. Search the parameters , and for all the (, ) pairs selected based on constraint (11). A nonlinear equation solver is used to determine these three parameters. Note that for any given value of , the values of and are adjusted simultaneously to satisfy both the constraints (9) () and (10) (). At the end of this step, all the possible (, , , , ) combinations fulfilling constraints (9)–(11) are obtained.

3. Identify the optimal (, , , , ) combination which has the smallest value of from all the chart-parameter combinations found in Step (2).

For example, when , , , and , the output listing and the optimal (, , , , ) combination (see the last row of the output listing) are obtained as

  n1 n2 L1    L   L2  MRL0 MRL1 ASS0 ASS1

  1 6 0.869930 5.027832 2.857255 250 2 3.306028 4.494778

  1 7 1.142700 5.592773 2.763021 250 2 2.772141 4.215309

  1 8 1.283047 5.327637 2.692279 250 2 2.595803 4.198212

  .

  .

  .

  5 13 2.774352 3.005859 2.985578 250 2 5.037477 5.968226

  5 14 2.779625 3.035400 2.489431 250 2 5.042560 6.138543

  5 15 2.780030 3.035889 2.454888 250 2 5.045557 6.219900

  2 7 1.787 5.133  2.633  250 2 2.517 4.486

The output listing is not shown completely here as there are 66 (, ) pairs with the corresponding smallest value (see the 8^th column of each row in the output listing), for each (, ) pair.

4.2 Minimizing both the in-control and out-of-control ASSs

Hsu [17], [18] indicated that the optimal design of a control chart should take into consideration both the in-control and out-of-control situations. Therefore, in order to provide the best performance of the MRL-based DS chart for a specified mean shift , two objective functions, i.e. minimizing and are proposed in this section. The weighting average method suggested by Zadeh [28] is used to integrate these two objective functions. This weighting average method allows us to assign a weight to each objective function and then combine them into a single objective function. Since the performance of both the in-control and out-of-control cases are equally important, we let the weights of the and equal to each other. Hence, the integrated objective function of this proposed optimal design model is simplified to the minimization of .

The proposed optimal design of the MRL-based DS chart to minimize both the and , which is modeled as a nonlinear minimization problem, is mathematically expressed as follows:

(12)

subject to

1. (13)
2. (14)
3. (15)

   The design procedure of the optimization model (12)–(15) is similar to that presented in Step (1) to Step (3) of Section 4.1. The only difference is that we are minimizing instead of .

Comparative Studies

The performance of the optimal MRL-based DS chart is now compared with the Shewhart and optimal EWMA charts. The {250, 500} and various values of corresponding to {0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00, 2.50, 3.00} are considered. Thus, the three charts are compared based on their sample-size performance. Note that only moderate and large are considered in this paper because in many real industrial applications, small shifts in the process are usually not desirable to be detected in order to avoid too frequent process interruptions [5], [29].

For the Shewhart chart, the upper control limit , lower control limit and center line are computed as [1]

(16a)

and

(16b)

respectively, where is a multiplier controlling the width of both the and .

For the EWMA chart, the plotting statistics is expressed as [1]

(17a)

and

(17b)

where is the sample mean at the i ^th sampling time and . Then the upper and lower control limits, i.e. and , respectively, as well as the center line are defined as follows [1]:

(18a)

and

(18b)

respectively, where with the multiplier to be ascertained.

In this study, {3, 5, 7} are considered. The optimization procedure shown in Khoo et al. [25] is used to optimally design the MRL-based EWMA chart for minimizing the .

5.1 Study 1: The DS chart for minimizing the

In Study 1, we compare the sample size performance of the optimal MRL-based EWMA , Shewhart and DS charts. Tables 2 and 3 present the optimal chart parameters for these three charts, together with their corresponding values of and sample size (, or ASS). For the EWMA chart, the optimal parameters (, ) and the corresponding (, ) values are shown in the first and second rows of each cell, respectively. Meanwhile, the charting constant and the corresponding (, ) values of the Shewhart chart are listed in the first and second rows of each cell, respectively. For the DS chart, the optimal combination (, , , , ) is presented in the first and second rows of each cell, while the corresponding (, , ) values are presented in the third row of each cell.

Table 2. Optimal chart parameters for the EWMA , Shewhart and DS charts, together with their corresponding values of and sample size when 250 and is minimized.

	n EWMA = 3			n EWMA = 5			n EWMA = 7
	EWMA X-bar	Shewhart X-bar	DS X-bar	EWMA X-bar	Shewhart X-bar	DS X-bar	EWMA X-bar	Shewhart X-bar	DS X-bar
	(λ, K EWMA)	KX -bar	(n 1, n 2,	(λ, K EWMA)	KX -bar	(n 1, n 2,	(λ, K EWMA)	KX -bar	(n 1, n 2,
	(MRL1, n EWMA)	(MRL1, nX -bar)	L 1, L, L 2)	(MRL1, n EWMA)	(MRL1, nX -bar)	L 1, L, L 2)	(MRL1, n EWMA)	(MRL1, nX -bar)	L 1, L, L 2)
δ *			(MRL1, ASS0, ASS1)			(MRL1, ASS0, ASS1)			(MRL1, ASS0, ASS1)
0.50	(0.175, 0.505)	2.992	(1, 19,	(0.300, 0.548)	2.992	(2, 18,	(0.300, 0.463)	2.992	(3, 17,
	(10, 3)	(10, 9)	1.787, 5.561, 2.261)	(7, 5)	(7, 12)	1.742, 5.171, 2.431)	(5, 7)	(5, 14)	1.594, 4.955, 2.617)
			(10, 2.405, 3.092)			(7, 3.467, 4.835)			(5, 4.885, 7.083)
0.75	(0.300, 0.707)	2.992	(1, 10,	(0.550, 0.820)	2.992	(1, 13,	(0.550, 0.693)	2.992	(2, 13,
	(6, 3)	(6, 6)	1.794, 5.939, 2.373)	(4, 5)	(4, 8)	1.583, 5.163, 2.463)	(3, 7)	(3, 9)	1.728, 5.248, 2.513)
			(6, 1.728, 2.537)			(4, 2.475, 3.760)			(3, 3.091, 5.312)
1.00	(0.550, 1.058)	2.992	(1, 6,	(0.550, 0.820)	2.992	(2, 7,	(0.550, 0.693)	2.992	(2, 7,
	(4, 3)	(4, 4)	1.818, 5.337, 2.477)	(2, 5)	(2, 6)	1.787, 5.133, 2.633)	(2, 7)	(2, 6)	1.787, 5.133, 2.633)
			(4, 1.415, 2.255)			(2, 2.517, 4.486)			(2, 2.517, 4.486)
1.25	(0.550, 1.058)	2.992	(1, 4,	(0.550, 0.820)	2.992	(1, 5,	(0.800, 0.923)	2.992	(2, 6,
	(3, 3)	(4, 3)	1.919, 5.145, 2.521)	(2, 5)	(2, 4)	1.615, 4.700, 2.634)	(1, 7)	(1, 6)	1.618, 4.992, 2.744)
			(3, 1.220, 2.010)			(2, 1.531, 2.796)			(1, 2.634, 5.356)
1.50	(0.550, 1.058)	2.992	(1, 3,	(0.550, 0.820)	2.992	(1, 5,	(0.550, 0.693)	2.992	(1, 5,
	(2, 3)	(2, 3)	1.893, 5.047, 2.618)	(1, 5)	(1, 4)	1.366, 5.110, 2.738)	(1, 7)	(1, 4)	1.366, 5.110, 2.738)
			(2, 1.175, 2.042)			(1, 1.859, 3.775)			(1, 1.859, 3.775)
1.75	(0.550, 1.058)	2.992	(1, 2,	(0.550, 0.820)	2.992	(1, 3,	(0.550, 0.693)	2.992	(1, 3,
	(2, 3)	(2, 2)	2.206, 4.603, 2.545)	(1, 5)	(1, 3)	1.581, 5.164, 2.758)	(1, 7)	(1, 3)	1.581, 5.164, 2.758)
			(2, 1.055, 1.644)			(1, 1.342, 2.702)			(1, 1.342, 2.702)
2.00	(0.550, 1.058)	2.992	(1, 3,	(0.550, 0.820)	2.992	(1, 3,	(0.550, 0.693)	2.992	(1, 3,
	(1, 3)	(1, 3)	1.970, 5.431, 2.573)	(1, 5)	(1, 3)	1.970, 5.431, 2.573)	(1, 7)	(1, 3)	1.970, 5.431, 2.573)
			(1, 1.146, 2.535)			(1, 1.146, 2.535)			(1, 1.146, 2.535)
2.50	(0.550, 1.058)	2.992	(1, 2,	(0.550, 0.820)	2.992	(1, 2,	(0.550, 0.693)	2.992	(1, 2,
	(1, 3)	(1, 2)	2.485, 3.061, 2.893)	(1, 5)	(1, 2)	2.485, 3.061, 2.893)	(1, 7)	(1, 2)	2.485, 3.061, 2.893)
			(1, 1.021, 1.437)			(1, 1.021, 1.437)			(1, 1.021, 1.437)
3.00	(0.550, 1.058)	2.992	(1, 2,	(0.550, 0.820)	2.992	(1, 2,	(0.550, 0.693)	2.992	(1, 2,
	(1, 3)	(1, 2)	2.992, 3.437, 0.000)	(1, 5)	(1, 2)	2.992, 3.437, 0.000)	(1, 7)	(1, 2)	2.992, 3.437, 0.000)
			(1, 1.004, 1.344)			(1, 1.004, 1.344)			(1, 1.004, 1.344)

Table 3. Optimal chart parameters for the EWMA , Shewhart and DS charts, together with their corresponding values of and sample size when 500 and is minimized.

	n EWMA = 3			n EWMA = 5			n EWMA = 7
	EWMA X-bar	Shewhart X-bar	DS X-bar	EWMA X-bar	Shewhart X-bar	DS X-bar	EWMA X-bar	Shewhart X-bar	DS X-bar
	(λ, K EWMA)	KX -bar	(n 1, n 2,	(λ, K EWMA)	KX -bar	(n 1, n 2,	(λ, K EWMA)	KX -bar	(n 1, n 2,
	(MRL1, n EWMA)	(MRL1, nX -bar)	L 1, L, L 2)	(MRL1, n EWMA)	(MRL1, nX -bar)	L 1, L, L 2)	(MRL1, n EWMA)	(MRL1, nX -bar)	L 1, L, L 2)
δ *			(MRL1, ASS0, ASS1)			(MRL1, ASS0, ASS1)			(MRL1, ASS0, ASS1)
0.50	(0.175, 0.548)	3.198	(1, 19,	(0.175, 0.424)	3.198	(2, 18,	(0.300, 0.499)	3.198	(3, 17,
	(12, 3)	(13, 10)	1.767, 5.024, 2.558)	(8, 5)	(8, 14)	1.650, 5.147, 2.760)	(6, 7)	(6, 16)	1.506, 5.586, 2.915)
			(12, 2.466, 3.170)			(8, 3.781, 5.278)			(6, 5.246, 7.591)
0.75	(0.238, 0.659)	3.198	(1, 14,	(0.300, 0.590)	3.198	(2, 14,	(0.550, 0.721)	3.198	(2, 16,
	(6, 3)	(6, 7)	1.842, 5.116, 2.560)	(4, 5)	(4, 9)	1.916, 5.084, 2.664)	(3, 7)	(3, 11)	1.729, 5.945, 2.743)
			(6, 1.917, 2.991)			(4, 2.775, 4.765)			(3, 3.342, 6.075)
1.00	(0.550, 1.133)	3.198	(1, 8,	(0.550, 0.878)	3.198	(1, 9,	(0.550, 0.721)	3.198	(2, 9,
	(4, 3)	(4, 5)	1.844, 5.038, 2.677)	(3, 5)	(3, 6)	1.651, 5.409, 2.760)	(2, 7)	(2, 8)	1.805, 5.292, 2.836)
			(4, 1.521, 2.612)			(3, 1.889, 3.354)			(2, 2.640, 5.137)
1.25	(0.550, 1.133)	3.198	(1, 5,	(0.550, 0.878)	3.198	(1, 7,	(0.800, 0.959)	3.198	(2, 8,
	(3, 3)	(3, 4)	1.910, 5.297, 2.755)	(2, 5)	(2, 5)	1.686, 5.758, 2.794)	(1, 7)	(1, 7)	1.670, 5.274, 2.921)
			(3, 1.281, 2.277)			(2, 1.643, 3.333)			(1, 2.760, 6.313)
1.50	(0.550, 1.133)	3.198	(1, 4,	(0.800, 1.167)	3.198	(1, 6,	(0.550, 0.721)	3.198	(1, 6,
	(2, 3)	(2, 4)	1.918, 5.294, 2.809)	(1, 5)	(1, 5)	1.381, 5.582, 2.949)	(1, 7)	(1, 5)	1.381, 5.582, 2.949)
			(2, 1.220, 2.352)			(1, 2.003, 4.295)			(1, 2.003, 4.295)
1.75	(0.550, 1.133)	3.198	(1, 3,	(0.550, 0.878)	3.198	(1, 4,	(0.550, 0.721)	3.198	(1, 4,
	(2, 3)	(2, 3)	2.236, 5.253, 2.704)	(1, 5)	(1, 4)	1.639, 4.736, 2.934)	(1, 7)	(1, 4)	1.639, 4.736, 2.934)
			(2, 1.076, 1.940)			(1, 1.405, 3.172)			(1, 1.405, 3.172)
2.00	(0.800, 1.507)	3.198	(1, 3,	(0.550, 0.878)	3.198	(1, 3,	(0.550, 0.721)	3.198	(1, 3,
	(1, 3)	(1, 3)	1.934, 5.785, 2.877)	(1, 5)	(1, 3)	1.934, 5.785, 2.877)	(1, 7)	(1, 3)	1.934, 5.785, 2.877)
			(1, 1.159, 2.579)			(1, 1.159, 2.579)			(1, 1.159, 2.579)
2.50	(0.550, 1.133)	3.198	(1, 2,	(0.550, 0.878)	3.198	(1, 2,	(0.550, 0.721)	3.198	(1, 2,
	(1, 3)	(1, 2)	2.482, 3.399, 2.899)	(1, 5)	(1, 2)	2.482, 3.399, 2.899)	(1, 7)	(1, 2)	2.482, 3.399, 2.899)
			(1, 1.025, 1.646)			(1, 1.025, 1.646)			(1, 1.025, 1.646)
3.00	(0.550, 1.133)	3.198	(1, 2,	(0.550, 0.878)	3.198	(1, 2,	(0.550, 0.721)	3.198	(1, 2,
	(1, 3)	(1, 2)	3.000, 3.259, 2.578)	(1, 5)	(1, 2)	3.000, 3.259, 2.578)	(1, 7)	(1, 2)	3.000, 3.259, 2.578)
			(1, 1.003, 1.205)			(1, 1.003, 1.205)			(1, 1.003, 1.205)

In this study, all the three charts are designed to have a similar sensitivity for a particular , i.e. by having a similar value as that of the optimal MRL-based EWMA chart when {250, 500}. In particular, the optimal combination (, , , , ) of the DS chart is obtained such that {250, 500} (constraint (9)) and (constraint (10)) for a . Here, the value is specified as the value (see the values in the second, fifth and eighth columns of Tables 2 and 3) of the optimal MRL-based EWMA chart. Therefore, with the implementation of the optimization model (8)–(11) (see Section 4.1), the optimal DS chart having the smallest value, will also possess a reasonable value which is similar to that of the optimal EWMA chart for the specified . For the Shewhart chart, it is designed to match the two MRL points of the EWMA chart. Note that the two MRL points are the {250, 500} and a suitable value, which is chosen such that the value of the Shewhart chart, with an appropriate , is as close as possible to that of the optimal EWMA chart. For example, when 250, 7 and  = 0.5, the optimal value for EWMA chart is five. Thus, both the DS and Shewhart charts must have 5 for 0.5.

From these tables, it is obvious that the optimal MRL-based DS chart generally outperforms the optimal EWMA and Shewhart charts, in terms of the average sample size. Precisely, the and values of the optimal MRL-based DS chart when 0.50 and 0.75, respectively, are lower than the corresponding values of and . When compared with the optimal MRL-based EWMA chart, the decrease in of the optimal DS chart is around 36–85% when 0.75; while the decrease in is around 13–82% when 1.00. Tables 2 and 3 also reveal that there are substantial improvements in the and values of the optimal MRL-based DS chart, in comparison to the of the MRL-based Shewhart chart, where reductions of around 50–75% and 33–68%, respectively, exist, for 0.50. It is clear that from these two tables, the reduction of the out-of-control ASS is not as high as that of the in-control ASS. Also, by using the optimal MRL-based DS chart, we need a smaller sample size to detect moderate to large and a larger sample size to detect small . Generally, the optimal MRL-based DS chart requires much smaller sample sizes on the average when the process is in-control and out-of-control and thus, using the chart reduces costs.

5.2 Study 2: The DS chart for minimizing the

Table 4 summarizes the optimal (, , , , ) combination (listed in the first and second rows of each cell) of the DS chart for minimizing the , together with their respective (, , ) values (listed in the third row of each cell). The optimization model (12)–(15) in Section 4.2 is employed here. Therefore, it is ensured that all the optimal (, , , , ) combinations in Table 4 attain {250, 500} (constraint (13)) and (constraint (14)) for a . Here, the value is specified as the value of the optimal MRL-based EWMA chart. In other words, both the optimal MRL-based DS charts for minimizing the (see Study 1 of Section 5.1) and (see Study 2 of Section 5.2) have the same and values.

Table 4. (, , , , ) combination (first and second rows of each cell) and (, , ) values (third row of each cell) of the DS charts when {250, 500} and is minimized.

	MRL0 = 250			MRL0 = 500
δ *	n EWMA = 3	n EWMA = 5	n EWMA = 7	n EWMA = 3	n EWMA = 5	n EWMA = 7
0.50	(1, 19,	(1, 19,	(3, 17,	(1, 19,	(2, 18,	(3, 17,
	1.787, 5.561, 2.261)	1.473, 5.321, 2.477)	1.594, 4.955, 2.617)	1.767, 5.024, 2.558)	1.650, 5.147, 2.760)	1.506, 5.586, 2.915)
	(10, 2.405, 3.092)	(7, 3.675, 4.601)	(5, 4.885, 7.083)	(12, 2.466, 3.170)	(8, 3.781, 5.278)	(6, 5.246, 7.591)
0.75	(1, 9,	(1, 12,	(2, 11,	(1, 12,	(1, 15,	(2, 14,
	1.736, 5.334, 2.435)	1.538, 5.010, 2.503)	1.634, 4.400, 2.605)	1.764, 5.098, 2.640)	1.553, 5.455, 2.724)	1.650, 5.554, 2.811)
	(6, 1.742, 2.516)	(4, 2.488, 3.716)	(3, 3.125, 5.151)	(6, 1.934, 2.936)	(4, 2.806, 4.324)	(3, 3.384, 5.935)
1.00	(1, 5,	(1, 8,	(1, 8,	(1, 7,	(1, 8,	(2, 8,
	1.706, 5.029, 2.588)	1.283, 5.328, 2.692)	1.283, 5.328, 2.692)	1.770, 5.282, 2.750)	1.576, 4.997, 2.820)	1.731, 4.411, 2.899)
	(4, 1.440, 2.218)	(2, 2.596, 4.198)	(2, 2.596, 4.198)	(4, 1.537, 2.563)	(3, 1.920, 3.299)	(2, 2.668, 5.002)
1.25	(1, 3,	(1, 5,	(2, 5,	(1, 4,	(1, 6,	(2, 6,
	1.732, 4.216, 2.700)	1.611, 3.937, 2.646)	1.472, 4.272, 2.829)	1.760, 4.764, 2.885)	1.605, 4.281, 2.867)	1.452, 4.074, 3.054)
	(3, 1.249, 1.944)	(2, 1.535, 2.787)	(1, 2.705, 5.054)	(3, 1.314, 2.225)	(2, 1.650, 3.172)	(1, 2.878, 5.683)
1.50	(1, 3,	(1, 5,	(1, 5,	(1, 4,	(1, 5,	(1, 5,
	1.874, 3.541, 2.680)	1.351, 3.571, 2.789)	1.351, 3.571, 2.789)	1.907, 3.781, 2.855)	1.248, 4.535, 3.020)	1.248, 4.535, 3.020)
	(2, 1.182, 2.002)	(1, 1.882, 3.711)	(1, 1.882, 3.711)	(2, 1.226, 2.325)	(1, 2.060, 4.006)	(1, 2.060, 4.006)
1.75	(1, 2,	(1, 3,	(1, 3,	(1, 3,	(1, 4,	(1, 4,
	2.205, 4.297, 2.547)	1.578, 4.003, 2.766)	1.578, 4.003, 2.766)	2.201, 3.434, 2.926)	1.615, 3.605, 3.024)	1.615, 3.605, 3.024)
	(2, 1.055, 1.638)	(1, 1.343, 2.669)	(1, 1.343, 2.669)	(2, 1.081, 1.840)	(1, 1.424, 3.090)	(1, 1.424, 3.090)
2.00	(1, 3,	(1, 3,	(1, 3,	(1, 3,	(1, 3,	(1, 3,
	1.970, 4.016, 2.580)	1.970, 4.016, 2.580)	1.970, 4.016, 2.580)	1.890, 3.416, 3.097)	1.890, 3.416, 3.097)	1.890, 3.416, 3.097)
	(1, 1.146, 2.470)	(1, 1.146, 2.470)	(1, 1.146, 2.470)	(1, 1.174, 2.396)	(1, 1.174, 2.396)	(1, 1.174, 2.396)
2.50	(1, 2,	(1, 2,	(1, 2,	(1, 2,	(1, 2,	(1, 2,
	2.468, 3.021, 3.184)	2.468, 3.021, 3.184)	2.468, 3.021, 3.184)	2.451, 3.254, 3.281)	2.451, 3.254, 3.281)	2.451, 3.254, 3.281)
	(1, 1.022, 1.423)	(1, 1.022, 1.423)	(1, 1.022, 1.423)	(1, 1.026, 1.589)	(1, 1.026, 1.589)	(1, 1.026, 1.589)
3.00	(1, 2,	(1, 2,	(1, 2,	(1, 2,	(1, 2,	(1, 2,
	2.992, 3.437, 0.000)	2.992, 3.437, 0.000)	2.992, 3.437, 0.000)	3.000, 3.259, 2.578)	3.000, 3.259, 2.578)	3.000, 3.259, 2.578)
	(1, 1.004, 1.344)	(1, 1.004, 1.344)	(1, 1.004, 1.344)	(1, 1.003, 1.205)	(1, 1.003, 1.205)	(1, 1.003, 1.205)

Note that similar conclusions regarding the comparative performance of the in-control and out-of-control sample sizes among the three charts which are discussed for Tables 2 and 3, are obtained for Table 4. Thus, we compare the chart settings between the optimal MRL-based DS chart for minimizing the (i) (see Tables 2 and 3 of Study 1) and (ii) (see Table 4) in this Study 2. In Table 4, it is noticeable that some of the optimal (, , , , ) combinations are different from those shown in Tables 2 and 3. In addition, we found that the and values in Studies 1 and 2 are fairly close to each other. We observe that there are some increments in the value and some decrements in the value for Study 2 as compared to that in Study 1. This is expected as we are minimizing both the and in Study 2. Note that the accuracies of all the results shown in Tables 1–4 have been verified with simulation.

An Illustrative Example

In this section, we consider the example given by Carot et al. [30]. This example illustrates the implementation of the optimal MRL-based DS chart to monitor the amount of potassium sorbate to be added to a yoghurt manufacturing process. For the sake of comparison, the construction of the optimal MRL-based EWMA and Shewhart charts are also discussed in this section.

It is well known that potassium sorbate is a preservative, a bactericide and a fungicide. Hence, it is one of the basic ingredients to preserve a number of edible products. According to the public health institutions, the advisable amount of potassium sorbate to be added is 0.5–2.0 g per kg product. Thus, let 1.5 g and 0.008 g as the desired process parameters of potassium sorbate in this yoghurt manufacturing process [29]. We initially generate the measurements of the first ten sampling times (1 to 10) based on an in-control condition; whereas the measurements for the subsequent six sampling times (11 to 16) are generated with 0.75. Table 5 tabulates various summary statistics for the DS , Shewhart and EWMA charts.

Table 5. Summary statistics of the simulated data for the amount of potassium sorbate (in grams, g) added to a yoghurt manufacturing process.

	DS X-bar chart				Shewhart X-bar chart	EWMA X-bar chart
Sample sizes	n 1 = 1, n 2 = 13				nX -bar = 8	n EWMA = 5
Sampling time, i	X-bar (1, i)	Z 1i	X-bar (i)	Zi	X-bar(Shewhart) (i)	Zi (EWMA)
1	1.5093	1.1567			1.5002	1.5009
2	1.5162	2.0279	1.5003	0.1494	1.5003	1.5021
3	1.4903	−1.2187			1.4997	1.4999
4	1.4957	−0.5422			1.4997	1.4985
5	1.5084	1.0502			1.5001	1.4987
6	1.5021	0.2635			1.5027	1.5002
7	1.4904	−1.1958			1.4997	1.4987
8	1.5028	0.3538			1.5019	1.5009
9	1.5044	0.5489			1.4959	1.5027
10	1.5013	0.1596			1.4965	1.4995
11	1.4982	−0.2227			1.5079	1.5050
12	1.4995	−0.0655			1.5008	1.5044
13	1.5125	1.5597			1.5086	1.5058
14	1.5199	2.4822	1.5061	2.8433	1.5074	1.5028
15	1.5080	1.0041			1.5091	1.5068
16	1.5144	1.8056	1.5075	3.4898	1.5044	1.5024

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

Let us assume that 250 and are desired. By referring to Table 2, the optimal chart parameters for the DS , Shewhart and EWMA charts are (, , , , ) = (1, 13, 1.583, 5.163, 2.463), (, ) = (8, 2.992) and (, , ) = (5, 0.550, 0.820), respectively. Figures 2 to 4 display the optimal MRL-based DS , Shewhart and EWMA charts. The solid and hollow dots in Figure 2 represent and of the DS chart, respectively. Also, the values of the and in Figures 3 and 4 are computed from Equations (16a) and (18a), respectively. Note that only the optimal chart parameters for the optimal MRL-based DS chart for minimizing the is considered in this example as both the optimal designs, i.e minimizing the and , have almost similar and values.

Figure 2. The DS chart.

The chart is used to monitor the amount of potassium sorbate to be added to a yoghurt manufacturing process. It produces the first out-of-control signal at sampling time 14.

Figure 4. The EWMA chart.

The chart is used to monitor the amount of potassium sorbate to be added to a yoghurt manufacturing process. It produces the first out-of-control signal at sampling time 15.

Figure 3. The Shewhart chart.

The chart is used to monitor the amount of potassium sorbate to be added to a yoghurt manufacturing process. It produces the first out-of-control signal at sampling time 13.

From Figures 2 to 4, it is observed that the DS , Shewhart and EWMA charts produce the first out-of-control signal at sampling time 14 as 2.8433>2.463, 13 as 1.5086>1.5085 and 15 as 1.5068>1.5066, respectively. This indicates that all the three charts have almost similar sensitivity in detecting 0.75. Concerning the number of observations sampled (from 1 onwards; see Table 5), relatively less number of observations (40 observations) are required for the DS chart compared to the Shewhart (104 observations) and EWMA (75 observations) charts. It is apparent that the DS chart needs around 53% and 38% of the total sample size of the EWMA and Shewhart charts to detect the mean shift of 0.75.

Conclusions

A good understanding of a control chart is vital as it helps to increase the quality engineers' confidence. Therefore, the MRL is chosen as the design measure in this paper because it is more readily comprehensible by the shop floor personnel and practitioners than the ARL. For completeness, this paper proposes two optimal designs of the MRL-based DS chart for minimizing the (i) and (ii) , which are not yet available in the existing literature. Specific optimal chart parameters are provided in Tables 2 to 4 for these two optimal designs. These optimal chart parameters aid the practitioners to implement the optimal MRL-based DS chart instantaneously.

From the comparative studies, it is found that the optimal MRL-based DS chart generally requires a smaller sample size on the average than the optimal EWMA and Shewhart charts when the process is either in-control or out-of-control. The effectiveness of the optimal MRL-based DS chart in reducing the sampling and inspection costs, provides a practical advantage for the practitioners in using this chart. Since both the optimal designs of the MRL-based DS chart for minimizing the (i) and (ii) , produce fairly close and values, either one of these two optimal designs can be implemented in practice. The optimal MRL-based DS chart proposed in this paper provides an alternative to the SPC user and may stimulate more research interests in the area of the optimal MRL-based control charts.

Funding Statement

This research is supported by the KPI allocation of the School of Mathematical Sciences, Universiti Sains Malaysia (USM). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.