Monitoring the Multivariate Coefficient of Variation using Run Rules Type Control Charts

01/03/2020 ∙ by P. H. Tran, et al. ∙ 0

In practice, there are processes where the in-control mean and standard deviation of a quality characteristic is not stable. In such cases, the coefficient of variation (CV) is a more appropriate measure for assessing process stability. In this paper, we consider the statistical design of Run Rules based control charts for monitoring the CV of multivariate data. A Markov chain approach is used to evaluate the statistical performance of the proposed charts. The computational results show that the Run Rules based charts outperform significantly the standard Shewhart control chart. Moreover, by choosing an appropriate scheme, the Run Rules based charts perform better than the Rum Sum control chart for monitoring the multivariate CV. An example in a spring manufacturing process is given to illustrate the implementation of the proposed charts.

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1 Introduction

Representing the ratio of the standard deviation to the mean, the coefficient of variation (CV) is a useful measure of relative dispersion of a random variable. It has the meaning that the higher the CV, the greater the level of dispersion around the mean. The CV is widely used in a large number of areas such as clinical chemistry, materials engineering, agricultural experiments and medicine, see, for example

Castagliola et al. 9. In laboratory medicine for comparing the reproducibility of assay techniques, the lower CV leads to the better analytical precision. In many manufacturing processes, keeping the CV in-control means ensuring the product quality. Monitoring the CV is then an important task attracted the interest of many authors and the literature of monitoring the CV is abundant. Kang et al. 15 are the first to use the Shewhart chart to monitor and detect changes in the CV. Their work is developed with a number of advantage-type control charts or adaptive strategies such as synthetic control chart (Calzada and Scariano 6), EWMA control chart (Castagliola et al. 9), CUSUM control chart (Tran and Tran 26), Run Rules based control chart (Castagliola et al. 7), the variable sampling interval (VSI) control chart (Castagliola et al. 8), the side-sensitive group runs (SSGR) control chart (You et al. 32), and the run sum control chart (Teoh et al. 24). The EWMA chart designed by Castagliola et al. 9 was further improved by Zhang et al. 33 (based on a modified EWMA charting statistic), Yeong et al. 31 (which integrates the VSI feature into the EWMA chart) and Zhang et al. 34 (by applying the resetting technique in Shu and Jiang 23).

The aforementioned studies focus only on the univariate CV. In fact, there are various situations where the multivariate coefficient of variation (MCV) is the main concern. For example, in biometry and genetics, it is quite often to measure multiple characteristics on individuals from several populations and the problem is to assess the relative variability of each population. The single calculation of the univariate CV of each characteristic is obviously insufficient because it does not consider the correlation between these features, see Albert and Zhang 2. However, a literature search reveals that not much attention has been paid to the CV for multivariate data despite its potential importance. Very recently, Yeong et al. 29 have suggested a Shewhart control chart for monitoring the MCV (denoted as ShewhartMCV chart in this paper). It is well-known that Shewhart-type charts are only efficient in detecting large and sudden process shifts. For moderate or small shifts, they take longer time to detect them. To enhance the performance of the Shewhart chart monitoring the MCV in the case of small shifts, Lim et al. 20 suggested to use a Rum Sum control chart. In this paper, we propose to apply supplementary Run Rules with the Shewhart control chart.

In the literature, several Run Rules have been studied and suggested by a number of authors. Champ and Woodall 11 were the first to obtain the exact formula evaluating the run length distribution and then calculated its average value, i.e. the average run length (ARL) which is the expected number of points plotted on chart until an out-of-control signal is given. They have also showed a disadvantage of the use of supplementary runs rules in the Shewhart chart that it reduces the in-control (denoted as ). In order to overcome this downside, alternative methods for the statistical design of control charts with Run Rules have been proposed: Klein 17 considered the 2-of-2 and 2-of-3 schemes while Khoo 16 considered additionally the 2-of-4, 3-of-3 and 3-of-4 schemes. A modified version of -out-of- control chart was studied in Antzoulakos and Rakitzis 4. An overview of control charts with supplementary runs rules until 2006 is presented in Koutras et al. 18. Recently, the Run Rules control chart are applied to monitor the coefficient of variation, the ratio of two normal variables as well as other non-normal processes; see, for example Acosta and Pignatiello 1; Amdouni et al. 3; Castagliola et al. 7; Faraz et al. 13, Tran et al. 25. However, up to now, the use of Run Rules control chart for tracking the MCV has not been investigated. Therefore, the goal of this study is to present the statistical design and the evaluation of the performance of one-sided Shewart-MCV charts with Run Rules. Numerical simulations shows that our proposed charts are efficient in detecting the process shifts. Moreover, the implementation of Run Rules control charts is also less complex compared to the Run Sum chart for MCV.

The paper is organized as follows. In Section 2, we provide a brief review of distribution of the sample MCV as discovered by Yeong et al. 29. In Section 3, we present the design and implementation of the Run Rules control charts for monitoring the MCV. Section 4 is devoted to assessing the performance of the proposed charts. An example is illustrated in Section 5 and some concluding remarks are given in Section 6.

2 A brief review of distribution of the sample multivariate coefficient of variation

We present in this Section a brief review of the distribution of the sample multivariate coefficient of variation (abbr. MCV). From the literature, there is a number different point of views about the definitions of the MCV. A formal definition for the multivariate coefficient of variation has firstly proposed by Reyment 22. Van Valen 27 has pointed out a few shortcomings of this definition and suggested a more appropriate definition. Another definition of the MCV was given by Nikulin and Voinov 21 based on the Mahalanobis distance. Recently, Albert and Zhang 2 have proposed a novel approach of defining the MCV to overcome the limitations of previous definitions. We use in this paper the definition of the MCV suggested by Nikulin and Voinov 21 which is considered as a natural generalization for the CV. This definition was also adopted by Yeong et al. 29 to monitor the MCV. Let

denote a random vector from a

-variate normal distribution with mean vector

and covariance matrix . The MCV is defined as

(1)

Suppose that a random sample of size , say , is taken from this distribution, i.e., , . Let and be the sample mean and the sample covariance matrix of , i.e.,

and

The sample multivariate coefficient of variation is then defined as

(2)

The

(cumulative distribution function) and the inverse

of are given in Yeong et al. 29 as

(3)

and

(4)

where and are the non-central distribution and the inverse of the non-central distribution with and degrees of freedom; the non-centrality parameter is .

3 Implementation of the RR-MCV control charts

Similar to the one-sided Run Rules control charts presented in Tran et al. 25 and Castagliola et al. 7, we suggest to define two one-sided Run Rules control charts for monitoring the MCV as follows:

  • A lower-sided -out-of- Run Rules control chart (denoted as RR MCV) to detect a decrease in with a lower control limit and a corresponding upper control limit .

  • An upper-sided -out-of- Run Rules control chart (denoted as RR MCV) to detect a decrease in with a lower control limit and a corresponding lower control limit .

Given the value of the control limits for each chart, an out-of-control signal is given at time if -out-of- consecutive values are plotted outside the control interval, i.e. in the lower-sided chart and in the upper-sided chart. The control chart designed above is called Run Rules type chart. Compared to Run Rules type charts which require both control and warning limits, these type charts are more simple to implement and interpret, see Klein 17. In this study, we only consider the 2-out-of-3, 3-out-of-4 and 4-out-of-5 Run Rules charts. More complex Run Rules schemes with larger values of are possible to design in a similar manner. However, their efficiency should be taken into consideration in terms of increased complexity of implementation.

The performance of the proposed one-sided RRMCV control charts is measured by the out-of-control , to denote as . We utilize a Markov chain method, similar to the one initially proposed by Brook and Evans 5, to calculate the value. Further details on this method can be found in Fu et al. 14, Castagliola et al. 7 and Li et al. 19. Let us now suppose that the occurrence of an unexpected condition shifts the in-control MCV value to the out-of-control value , where is the shift size. Values of correspond to a decrease of the , while values of correspond to an increase of the in-control MCV. It is worth mentioning that a decrease (resp. increase) in

is related to process improvement (resp. deterioration). The probability

of the event that a sample drops into control interval is equal to:

  • for the RRMCV chart:

    (5)
  • for the RRMCV chart:

    (6)

where .

The Transition Probability Matrix (TPM) matrix of the embedded Markov chain for the two RRMCV control charts is

(7)

where is the matrix of transient probabilities, is the vector satisfied with and . The corresponding vector of initial probabilities associated with the transient states is equal to , i.e. the third state is the initial state.

Extended to Run Rules charts with larger values, the matrix of transient probabilities for the two RRMCV control charts is given by

(8)

In this case, the seventh state in the vector is the initial state. The (15,15) matrix of transient probabilities for the two RR-MCV control charts is not presented here due to space economy, but it can be seen in, for example, Tran, Castagliola, and Celano 25. Once matrix and vector have been determined, the and (standard deviation of run length) are given by

(9)
(10)

with

(11)
(12)
(13)

It is customary that a control chart is considered to be better than its competitors if it gives smaller value of the while the is the same. Therefore, the control limit of RRMCV control charts should be found out as a solution of the following equations:

  • for the RRMCV chart:

    (14)
  • for the RRMCV chart:

    (15)

where is the predetermined in-control value.

4 The Performance of the RR-MCV control charts

In this Section, we investigate the performance of RR MCV control charts. The desired in-control value, say is set at 370.4, for all the considered IC cases. The control limit of lower-sided chart and of upper-sided chart, which are the solution of equations (14) and (15), are shown in Table 1 for different combinations of , and , , , , .

RR-MCV chart RR-MCV chart RR-MCV chart
0.10 (0.027, 0.146) (0.053, 0.135) (0.063, 0.129) (0.039, 0.124) (0.063, 0.121) (0.071, 0.119) (0.048, 0.111) (0.070, 0.113) (0.077, 0.112)
0.20 (0.053, 0.296) (0.104, 0.274) (0.125, 0.261) (0.077, 0.251) (0.125, 0.245) (0.142, 0.239) (0.095, 0.223) (0.139, 0.227) (0.154, 0.224)
0.30 (0.079, 0.457) (0.155, 0.419) (0.185, 0.399) (0.114, 0.382) (0.185, 0.372) (0.211, 0.362) (0.142, 0.337) (0.206, 0.343) (0.229, 0.339)
0.40 (0.104, 0.633) (0.203, 0.574) (0.242, 0.544) (0.150, 0.520) (0.243, 0.504) (0.277, 0.490) (0.186, 0.455) (0.272, 0.462) (0.302, 0.456)
0.50 (0.127, 0.831) (0.248, 0.744) (0.297, 0.700) (0.184, 0.667) (0.299, 0.643) (0.342, 0.623) (0.229, 0.576) (0.335, 0.584) (0.373, 0.577)
0.10 (0.014, 0.128) (0.047, 0.129) (0.059, 0.125) (0.024, 0.106) (0.057, 0.115) (0.067, 0.115) (0.031, 0.093) (0.063, 0.106) (0.073, 0.108)
0.20 (0.028, 0.259) (0.092, 0.260) (0.117, 0.253) (0.047, 0.213) (0.112, 0.231) (0.134, 0.231) (0.062, 0.185) (0.126, 0.213) (0.146, 0.216)
0.30 (0.041, 0.395) (0.137, 0.397) (0.173, 0.385) (0.069, 0.322) (0.166, 0.350) (0.199, 0.349) (0.092, 0.279) (0.187, 0.321) (0.217, 0.326)
0.40 (0.054, 0.539) (0.179, 0.541) (0.227, 0.524) (0.090, 0.434) (0.218, 0.472) (0.262, 0.470) (0.121, 0.372) (0.246, 0.431) (0.286, 0.438)
0.50 (0.065, 0.692) (0.219, 0.695) (0.278, 0.670) (0.110, 0.547) (0.267, 0.599) (0.322, 0.596) (0.148, 0.465) (0.303, 0.543) (0.352, 0.551)
0.10 (0.002, 0.104) (0.040, 0.122) (0.055, 0.121) (0.007, 0.081) (0.050, 0.108) (0.063, 0.111) (0.011, 0.067) (0.057, 0.099) (0.069, 0.104)
0.20 (0.005, 0.208) (0.080, 0.246) (0.109, 0.245) (0.013, 0.162) (0.099, 0.217) (0.126, 0.222) (0.023, 0.133) (0.113, 0.199) (0.138, 0.208)
0.30 (0.007, 0.314) (0.118, 0.373) (0.161, 0.371) (0.019, 0.242) (0.146, 0.327) (0.187, 0.335) (0.033, 0.199) (0.167, 0.299) (0.204, 0.313)
0.40 (0.009, 0.421) (0.154, 0.506) (0.211, 0.503) (0.025, 0.321) (0.192, 0.440) (0.245, 0.451) (0.044, 0.262) (0.219, 0.399) (0.269, 0.418)
0.50 (0.011, 0.529) (0.188, 0.645) (0.259, 0.641) (0.031, 0.399) (0.234, 0.553) (0.301, 0.569) (0.053, 0.324) (0.268, 0.500) (0.331, 0.525)
Table 1: The control limits of RR-MCV control charts, on left side and on right side, for different values of and .

It can be seen from Table 1 that given and the values of and depend (in general) on . In particular, small values of lead to small values of and , as well. For example, in RRMCV chart with and , we have and when , while and when . Also, given and , the values of and depend on . The larger the value of , the smaller the value of and . For example, in RRMCV chart with and , we have and when , while and when .

Using the control limits in Table 1, the values of and (out-of-control) of the RRMCV control charts are provided in Tables 2-4. We set different combinations of , , , and , , , . Some simple conclusions can be drawn from these tables as follows:

RR-MCV chart RR-MCV chart RR-MCV chart
0.50 (14.2, 12.6) (2.8, 1.3) (2.1, 0.4) (8.5, 6.1) (3.3, 0.7) (3.0, 0.2) (7.1, 3.9) (4.1, 0.4) (4.0, 0.1)
0.75 (84.0, 82.2) (21.6, 20.0) (10.4, 8.9) (55.6, 52.9) (14.8, 12.4) (8.0, 5.7) (42.2, 38.9) (12.4, 9.3) (7.5, 4.4)
0.90 (211.8, 210.0) (116.2, 114.4) (77.7, 75.9) (177.0, 174.2) (90.3, 87.6) (58.9, 56.3) (154.8, 151.1) (76.6, 73.1) (49.8, 46.4)
1.10 (109.6, 107.7) (67.0, 65.2) (48.9, 47.1) (109.3, 106.5) (63.4, 60.8) (45.2, 42.6) (111.1, 107.5) (62.6, 59.1) (44.1, 40.7)
1.25 (32.5, 30.8) (14.7, 13.0) (9.4, 7.9) (33.5, 30.9) (14.7, 12.3) (9.6, 7.2) (35.3, 32.0) (15.3, 12.2) (10.2, 7.1)
1.50 (10.5, 8.9) (4.6, 3.1) (3.3, 1.7) (11.7, 9.3) (5.4, 3.0) (4.0, 1.6) (13.1, 10.0) (6.3, 3.1) (4.9, 1.6)
0.50 (14.6, 13.0) (2.9, 1.3) (2.1, 0.4) (8.7, 6.3) (3.3, 0.7) (3.0, 0.2) (7.2, 4.0) (4.1, 0.4) (4.0, 0.1)
0.75 (85.6, 83.8) (22.5, 20.9) (10.9, 9.3) (56.8, 54.2) (15.4, 12.9) (8.3, 6.0) (43.3, 39.9) (12.9, 9.8) (7.8, 4.7)
0.90 (213.7, 211.8) (118.9, 117.1) (80.2, 78.4) (179.0, 176.2) (92.8, 90.0) (61.0, 58.4) (156.9, 153.2) (78.8, 75.3) (51.7, 48.3)
1.10 (111.9, 110.1) (69.5, 67.8) (51.1, 49.4) (111.6, 108.8) (65.8, 63.1) (47.3, 44.7) (113.3, 109.7) (64.9, 61.4) (46.1, 42.7)
1.25 (33.8, 32.1) (15.6, 13.9) (10.1, 8.5) (34.8, 32.2) (15.5, 13.1) (10.1, 7.8) (36.6, 33.3) (16.1, 13.0) (10.7, 7.6)
1.50 (11.1, 9.5) (4.9, 3.4) (3.4, 1.9) (12.3, 9.9) (5.7, 3.3) (4.2, 1.8) (13.7, 10.5) (6.5, 3.4) (5.1, 1.8)
0.50 (15.2, 13.6) (3.0, 1.5) (2.2, 0.5) (9.0, 6.6) (3.4, 0.8) (3.0, 0.2) (7.4, 4.3) (4.1, 0.5) (4.0, 0.1)
0.75 (88.2, 86.4) (24.0, 22.3) (11.7, 10.2) (58.9, 56.3) (16.3, 13.9) (8.9, 6.5) (45.0, 41.6) (13.7, 10.5) (8.2, 5.1)
0.90 (216.6, 214.8) (123.1, 121.3) (84.1, 82.3) (182.3, 179.5) (96.6, 93.9) (64.5, 61.8) (160.2, 156.5) (82.4, 78.9) (54.7, 51.3)
1.10 (116.0, 114.2) (73.8, 72.0) (54.9, 53.2) (115.5, 112.7) (69.8, 67.1) (50.7, 48.1) (117.2, 113.5) (68.7, 65.2) (49.4, 46.0)
1.25 (36.2, 34.4) (17.1, 15.5) (11.1, 9.6) (37.0, 34.4) (16.9, 14.5) (11.1, 8.7) (38.9, 35.5) (17.5, 14.3) (11.6, 8.5)
1.50 (12.1, 10.5) (5.4, 3.9) (3.7, 2.2) (13.3, 10.9) (6.1, 3.8) (4.5, 2.1) (14.7, 11.6) (7.0, 3.8) (5.3, 2.1)
0.50 (16.0, 14.4) (3.1, 1.6) (2.2, 0.5) (9.4, 7.1) (3.4, 0.9) (3.1, 0.2) (7.7, 4.6) (4.2, 0.5) (4.0, 0.1)
0.75 (91.7, 89.9) (26.0, 24.3) (12.9, 11.3) (61.8, 59.2) (17.7, 15.3) (9.6, 7.3) (47.4, 44.0) (14.7, 11.6) (8.8, 5.7)
0.90 (220.5, 218.7) (128.5, 126.7) (89.3, 87.5) (186.6, 183.8) (101.8, 99.0) (69.0, 66.3) (164.6, 160.9) (87.2, 83.7) (58.8, 55.3)
1.10 (121.9, 120.1) (79.6, 77.8) (60.1, 58.4) (121.0, 118.2) (75.2, 72.5) (55.4, 52.8) (122.5, 118.9) (73.9, 70.4) (53.9, 50.4)
1.25 (39.7, 38.0) (19.4, 17.8) (12.7, 11.2) (40.3, 37.7) (18.9, 16.5) (12.5, 10.1) (42.1, 38.7) (19.4, 16.2) (12.9, 9.8)
1.50 (13.7, 12.1) (6.2, 4.7) (4.2, 2.7) (14.8, 12.4) (6.8, 4.5) (4.9, 2.5) (16.2, 13.1) (7.6, 4.5) (5.7, 2.5)
0.50 (17.1, 15.5) (3.3, 1.8) (2.3, 0.6) (10.0, 7.7) (3.5, 1.0) (3.1, 0.3) (8.1, 5.0) (4.2, 0.6) (4.0, 0.2)
0.75 (96.1, 94.3) (28.6, 26.9) (14.4, 12.8) (65.5, 62.8) (19.5, 17.0) (10.6, 8.3) (50.5, 47.1) (16.1, 12.9) (9.6, 6.5)
0.90 (225.1, 223.3) (134.9, 133.1) (95.5, 93.7) (191.7, 188.9) (107.8, 105.1) (74.5, 71.8) (170.0, 166.3) (93.0, 89.4) (63.8, 60.3)
1.10 (129.7, 127.9) (87.0, 85.2) (66.7, 64.9) (128.1, 125.3) (81.9, 79.2) (61.3, 58.7) (129.2, 125.6) (80.3, 76.8) (59.4, 55.9)
1.25 (44.7, 43.0) (22.5, 20.9) (14.9, 13.3) (44.8, 42.2) (21.6, 19.2) (14.3, 11.9) (46.4, 43.0) (22.0, 18.7) (14.7, 11.5)
1.50 (16.0, 14.4) (7.3, 5.7) (4.8, 3.3) (16.9, 14.5) (7.8, 5.4) (5.4, 3.1) (18.3, 15.1) (8.5, 5.4) (6.2, 3.0)
Table 2: The values of and using the RRMCV control charts for and different values of and .
RR-MCV chart RR-MCV chart RR-MCV chart
0.50 (32.7, 31.0) (3.3, 1.8) (2.2, 0.5) (17.4, 15.0) (3.5, 0.9) (3.0, 0.2) (12.4, 9.2) (4.2, 0.6) (4.0, 0.1)
0.75 (129.4, 127.6) (26.5, 24.8) (11.7, 10.2) (90.5, 87.8) (17.7, 15.3) (8.8, 6.5) (69.5, 66.0) (14.6, 11.4) (8.1, 5.0)
0.90 (250.7, 248.9) (128.2, 126.4) (83.4, 81.6) (217.2, 214.4) (100.5, 97.8) (63.5, 60.8) (193.7, 190.0) (85.5, 81.9) (53.7, 50.2)
1.10 (128.4, 126.6) (72.4, 70.7) (51.7, 49.9) (130.8, 128.0) (69.1, 66.4) (48.0, 45.3) (134.6, 130.9) (68.4, 64.9) (46.9, 43.5)
1.25 (43.1, 41.3) (16.5, 14.9) (10.1, 8.6) (45.6, 43.0) (16.5, 14.1) (10.3, 7.9) (48.8, 45.4) (17.2, 14.0) (10.9, 7.8)
1.50 (14.8, 13.2) (5.1, 3.6) (3.4, 1.9) (16.7, 14.3) (5.9, 3.6) (4.2, 1.8) (18.8, 15.6) (6.8, 3.7) (5.1, 1.8)
0.50 (33.4, 31.7) (3.4, 1.8) (2.2, 0.5) (17.8, 15.4) (3.5, 1.0) (3.0, 0.2) (12.6, 9.5) (4.2, 0.6) (4.0, 0.1)
0.75 (131.2, 129.4) (27.5, 25.8) (12.3, 10.7) (92.1, 89.4) (18.4, 15.9) (9.2, 6.8) (70.9, 67.4) (15.1, 11.9) (8.4, 5.3)
0.90 (252.3, 250.4) (131.0, 129.1) (86.0, 84.2) (219.0, 216.2) (103.0, 100.3) (65.7, 63.0) (195.7, 192.0) (87.8, 84.3) (55.6, 52.2)
1.10 (130.9, 129.0) (75.1, 73.3) (54.0, 52.3) (133.2, 130.4) (71.6, 68.9) (50.1, 47.5) (137.0, 133.3) (70.8, 67.3) (49.0, 45.6)
1.25 (44.7, 43.0) (17.5, 15.9) (10.8, 9.3) (47.2, 44.6) (17.4, 15.0) (10.9, 8.5) (50.5, 47.0) (18.1, 14.9) (11.5, 8.3)
1.50 (15.6, 14.0) (5.5, 4.0) (3.6, 2.1) (17.5, 15.1) (6.2, 3.9) (4.4, 2.0) (19.7, 16.5) (7.1, 4.0) (5.3, 2.0)
0.50 (34.5, 32.8) (3.5, 2.0) (2.2, 0.6) (18.5, 16.1) (3.6, 1.1) (3.1, 0.3) (13.1, 9.9) (4.3, 0.7) (4.0, 0.1)
0.75 (134.2, 132.4) (29.2, 27.5) (13.2, 11.6) (94.8, 92.1) (19.6, 17.1) (9.8, 7.4) (73.3, 69.8) (16.0, 12.8) (8.9, 5.8)
0.90 (254.8, 252.9) (135.3, 133.4) (90.1, 88.3) (222.0, 219.1) (107.1, 104.3) (69.3, 66.6) (198.9, 195.2) (91.6, 88.1) (58.8, 55.4)
1.10 (135.1, 133.2) (79.5, 77.7) (57.9, 56.2) (137.2, 134.4) (75.7, 73.0) (53.7, 51.1) (141.0, 137.3) (74.9, 71.3) (52.4, 49.0)
1.25 (47.6, 45.8) (19.2, 17.6) (12.0, 10.4) (50.1, 47.4) (19.0, 16.6) (11.9, 9.5) (53.3, 49.9) (19.7, 16.4) (12.5, 9.3)
1.50 (17.0, 15.4) (6.1, 4.6) (4.0, 2.5) (19.0, 16.5) (6.8, 4.4) (4.7, 2.3) (21.2, 18.0) (7.7, 4.5) (5.5, 2.3)
0.50 (36.2, 34.5) (3.7, 2.2) (2.3, 0.6) (19.5, 17.0) (3.7, 1.2) (3.1, 0.3) (13.7, 10.6) (4.3, 0.7) (4.0, 0.2)
0.75 (138.2, 136.4) (31.6, 29.9) (14.5, 12.9) (98.5, 95.8) (21.2, 18.7) (10.6, 8.3) (76.6, 73.0) (17.3, 14.1) (9.5, 6.4)
0.90 (258.1, 256.2) (140.9, 139.0) (95.5, 93.7) (225.9, 223.1) (112.5, 109.7) (74.0, 71.3) (203.1, 199.4) (96.7, 93.1) (63.1, 59.7)
1.10 (141.2, 139.3) (85.6, 83.8) (63.4, 61.6) (143.0, 140.2) (81.4, 78.7) (58.6, 56.0) (146.5, 142.9) (80.3, 76.8) (57.1, 53.6)
1.25 (51.9, 50.2) (21.8, 20.2) (13.7, 12.2) (54.2, 51.6) (21.3, 18.8) (13.4, 11.0) (57.5, 54.0) (21.9, 18.6) (13.9, 10.7)
1.50 (19.3, 17.6) (7.0, 5.4) (4.5, 3.0) (21.2, 18.7) (7.6, 5.3) (5.1, 2.8) (23.5, 20.2) (8.5, 5.3) (6.0, 2.8)
0.50 (38.3, 36.6) (4.0, 2.5) (2.4, 0.7) (20.8, 18.3) (3.8, 1.3) (3.1, 0.4) (14.6, 11.4) (4.4, 0.9) (4.0, 0.2)
0.75 (143.2, 141.4) (34.6, 32.9) (16.2, 14.6) (103.2, 100.5) (23.3, 20.8) (11.8, 9.4) (80.8, 77.2) (18.9, 15.7) (10.4, 7.3)
0.90 (262.1, 260.3) (147.4, 145.6) (101.9, 100.1) (230.7, 227.9) (118.8, 116.1) (79.8, 77.1) (208.4, 204.7) (102.8, 99.2) (68.4, 64.9)
1.10 (149.3, 147.5) (93.4, 91.6) (70.2, 68.4) (150.5, 147.7) (88.5, 85.8) (64.8, 62.1) (153.7, 150.0) (87.1, 83.6) (62.9, 59.4)
1.25 (58.0, 56.3) (25.3, 23.6) (16.1, 14.5) (59.9, 57.2) (24.4, 21.9) (15.5, 13.0) (63.0, 59.5) (24.8, 21.5) (15.8, 12.6)
1.50 (22.6, 21.0) (8.2, 6.7) (5.2, 3.7) (24.4, 21.9) (8.7, 6.4) (5.8, 3.4) (26.7, 23.4) (9.5, 6.4) (6.6, 3.4)
Table 3: The values of and using the RRMCV control charts for and different values of and .
RR-MCV chart RR-MCV chart RR-MCV chart
0.50 (101.5, 99.6) (4.0, 2.5) (2.3, 0.6) (61.8, 59.1) (3.8, 1.3) (3.1, 0.3) (41.7, 38.3) (4.4, 0.8) (4.0, 0.1)
0.75 (215.1, 213.3) (33.3, 31.6) (13.4, 11.8) (172.2, 169.3) (21.9, 19.4) (9.8, 7.4) (141.9, 138.2) (17.6, 14.4) (8.8, 5.7)
0.90 (303.4, 301.5) (142.7, 140.9) (90.0, 88.2) (278.9, 276.1) (113.0, 110.3) (68.7, 66.0) (258.8, 255.1) (96.5, 92.9) (58.1, 54.7)
1.10 (161.2, 159.4) (78.9, 77.1) (54.8, 53.1) (169.6, 166.8) (75.8, 73.1) (51.1, 48.5) (178.0, 174.3) (75.5, 71.9) (50.0, 46.6)
1.25 (65.9, 64.2) (18.8, 17.2) (11.0, 9.4) (73.5, 70.8) (18.9, 16.4) (11.1, 8.7) (81.1, 77.5) (19.7, 16.5) (11.7, 8.6)
1.50 (25.8, 24.1) (5.8, 4.3) (3.6, 2.1) (30.7, 28.1) (6.6, 4.3) (4.4, 2.0) (35.6, 32.2) (7.6, 4.4) (5.3, 2.0)
0.50 (102.7, 100.9) (4.1, 2.6) (2.3, 0.6) (62.8, 60.1) (3.8, 1.4) (3.1, 0.3) (42.5, 39.1) (4.4, 0.9) (4.0, 0.2)
0.75 (216.7, 214.9) (34.5, 32.8) (14.0, 12.4) (173.9, 171.1) (22.7, 20.2) (10.2, 7.8) (143.7, 140.0) (18.2, 15.0) (9.2, 6.0)
0.90 (304.3, 302.4) (145.5, 143.7) (92.6, 90.8) (280.2, 277.3) (115.7, 112.9) (71.1, 68.4) (260.3, 256.6) (98.9, 95.4) (60.2, 56.7)
1.10 (163.7, 161.8) (81.7, 79.9) (57.3, 55.5) (172.0, 169.2) (78.5, 75.8) (53.4, 50.7) (180.4, 176.7) (78.0, 74.5) (52.2, 48.8)
1.25 (68.0, 66.3) (20.0, 18.3) (11.7, 10.2) (75.6, 72.9) (19.9, 17.5) (11.7, 9.4) (83.3, 79.8) (20.7, 17.5) (12.3, 9.2)
1.50 (27.1, 25.4) (6.2, 4.7) (3.9, 2.4) (32.1, 29.5) (7.0, 4.7) (4.6, 2.2) (37.1, 33.7) (7.9, 4.8) (5.5, 2.2)
0.50 (104.8, 103.0) (4.3, 2.8) (2.3, 0.7) (64.5, 61.8) (3.9, 1.5) (3.1, 0.3) (43.8, 40.4) (4.5, 1.0) (4.0, 0.2)
0.75 (219.3, 217.5) (36.5, 34.8) (15.1, 13.4) (176.8, 174.0) (24.1, 21.6) (10.9, 8.5) (146.6, 143.0) (19.3, 16.1) (9.7, 6.6)
0.90 (305.9, 304.0) (149.8, 148.0) (96.9, 95.1) (282.3, 279.4) (119.9, 117.1) (74.8, 72.1) (262.7, 259.0) (102.9, 99.4) (63.6, 60.1)
1.10 (167.8, 166.0) (86.2, 84.5) (61.3, 59.6) (176.1, 173.2) (82.8, 80.1) (57.1, 54.4) (184.3, 180.6) (82.3, 78.7) (55.8, 52.3)
1.25 (71.7, 69.9) (21.9, 20.3) (13.0, 11.4) (79.4, 76.7) (21.8, 19.3) (12.9, 10.5) (87.1, 83.6) (22.5, 19.2) (13.4, 10.3)
1.50 (29.4, 27.7) (6.9, 5.4) (4.2, 2.7) (34.5, 32.0) (7.7, 5.3) (5.0, 2.6) (39.8, 36.4) (8.6, 5.5) (5.8, 2.6)
0.50 (107.7, 105.8) (4.6, 3.1) (2.4, 0.8) (66.9, 64.2) (4.1, 1.7) (3.1, 0.4) (45.7, 42.2) (4.5, 1.1) (4.0, 0.2)
0.75 (222.9, 221.0) (39.4, 37.6) (16.5, 14.9) (180.8, 178.0) (26.1, 23.6) (11.9, 9.5) (150.7, 147.1) (20.9, 17.6) (10.5, 7.3)
0.90 (308.0, 306.1) (155.5, 153.7) (102.5, 100.7) (285.0, 282.2) (125.4, 122.7) (79.8, 77.1) (266.0, 262.3) (108.3, 104.7) (68.1, 64.6)
1.10 (173.9, 172.0) (92.6, 90.8) (66.9, 65.2) (181.7, 178.9) (88.8, 86.1) (62.2, 59.6) (189.8, 186.1) (88.1, 84.5) (60.7, 57.2)
1.25 (77.2, 75.4) (24.8, 23.2) (14.9, 13.3) (84.8, 82.1) (24.4, 21.9) (14.6, 12.1) (92.6, 89.1) (25.0, 21.7) (15.0, 11.8)
1.50 (33.0, 31.3) (8.0, 6.5) (4.8, 3.3) (38.3, 35.7) (8.7, 6.3) (5.5, 3.1) (43.8, 40.4) (9.6, 6.4) (6.3, 3.1)
0.50 (111.3, 109.5) (5.0, 3.5) (2.5, 0.9) (69.9, 67.3) (4.3, 1.9) (3.2, 0.5) (48.1, 44.7) (4.7, 1.3) (4.1, 0.3)
0.75 (227.3, 225.4) (42.9, 41.2) (18.4, 16.8) (185.8, 183.0) (28.7, 26.1) (13.2, 10.8) (155.9, 152.3) (22.9, 19.6) (11.5, 8.3)
0.90 (310.6, 308.8) (162.2, 160.4) (109.1, 107.3) (288.5, 285.6) (132.1, 129.3) (85.8, 83.1) (270.0, 266.3) (114.7, 111.1) (73.6, 70.1)
1.10 (181.9, 180.0) (100.7, 98.9) (74.0, 72.3) (189.1, 186.3) (96.3, 93.6) (68.7, 66.0) (196.7, 193.0) (95.3, 91.7) (66.8, 63.3)
1.25 (84.8, 83.0) (28.8, 27.1) (17.5, 15.9) (92.1, 89.4) (27.9, 25.4) (16.8, 14.3) (99.9, 96.3) (28.4, 25.1) (17.1, 13.9)
1.50 (38.3, 36.5) (9.5, 8.0) (5.6, 4.1) (43.6, 41.0) (10.0, 7.7) (6.2, 3.9) (49.3, 45.9) (10.9, 7.8) (7.0, 3.8)
Table 4: The values of and using the RRMCV control charts for and different values of and .
  • The in-control value of MCV and the value of multivariate level have strong influence on the performance of RRMCV control charts. In particular, the increase of and results in the increase of the . For example, in Table 2 with and , we have when and when . Also, with the same and , we have when in Table 2 and when in Table 4. That is to say, the RRMCV charts are more efficient for processes with small values of in-control MCV and multivariate levels.

  • The sample size has positive impact on the power of proposed charts: the larger the sample size, the smaller the average number of samples needed to detect the out-of-control status. For instance, with and (Table 3), we have when but this value dropped significantly to when on RRMCV chart.

  • Larger values for do not necessarily deliver better performance for the Run Rules based control charts; it depends on the value of the sample size , the shift size and especially the type of control chart. In general, using larger values for results in better performance for the lower-sided chart but worse performance for upper-sided chart. For example, with in Table 4, the RRMCV charts results in for (lower-sided) and for (upper-sided) compared to for (lower-sided) and for (upper-sided) in the RRMCV charts.

To compare the performance of the RRMCV control charts with the Shewhart MCV control chart, we define the index as

(16)

In this definition, and represent the value of the Shewhart-MCV chart and RR chart, respectively. Values indicate that the RRMCV charts outperform the Shewhart-MCV chart; conversely, values indicate that the Shewhart-MCV chart outperforms the RR-MCV charts. Tables 5-7 present the rounded results (to the nearest integer) of . It can be seen from these tables that the RR-MCV charts outperforms the Shewhart-MCV chart in most cases.

The above conclusions can be seen more clearly in Figure 1, where we draw the profiles for both Shewhart chart (designed by Yeong et al. 29) and Run Rules charts for a number of different in-control scenarios. Generally, in the case of decreasing shifts the 4-of-5 rule clearly outperforms the Shewhart chart and the other Run Rules charts, especially when and . As increase, the improvement is not as much as in the first case but still, it is substantial. For increasing shifts, we have also an improvement with Run Rules charts but it is not as much as in the case of decreases. In addition, a part of the curve of the Shehwart chart corresponding to very large shifts (i.e., 1.50 or 0.50) is below curves of Run Rules charts. We deduce that the Shewhart chart becomes more efficient than the proposed Run Rule based charts in detecting very large shifts. This is also confirmed by the negative values of index with the large values of in the Tables 5-7.

RR-MCV chart RR-MCV chart RR-MCV chart
0.50 71 46 -10 83 38 -56 85 22 -107
0.75 47 59 58 65 72 68 73 76 70
0.90 22 35 40 35 49 55 43 57 62
1.10 8 17 22 8 22 28 6 23 30
1.25 9 18 18 6 18 17 1 14 12
1.50 -1 -5 -17 -13 -22 -45 -26 -42 -77
0.50 71 47 -7 83 40 -51 85 25 -100
0.75 47 59 58 65 72 68 73 76 70
0.90 22 34 39 34 49 54 42 56 61
1.10 8 17 22 8 22 28 6 23 30
1.25 9 18 19 6 18 19 1 15 14
1.50 0 -2 -13 -11 -18 -38 -24 -36 -67
0.50 70 49 -2 82 43 -43 86 29 -88
0.75 46 58 58 64 71 68 72 76 71
0.90 21 33 39 34 48 53 42 55 60
1.10 8 17 22 8 22 28 7 23 30
1.25 10 19 21 7 20 21 3 17 17
1.50 2 2 -7 -7 -11 -28 -19 -27 -53
0.50 70 51 4 82 46 -32 86 35 -74
0.75 45 57 58 63 71 68 72 76 71
0.90 21 32 37 33 46 52 41 54 59
1.10 8 17 22 9 22 28 8 23 30
1.25 11 20 23 10 22 24 6 20 21
1.50 6 7 -1 -2 -3 -17 -12 -15 -36
0.50 69 53 11 82 50 -21 85 40 -58
0.75 44 56 57 62 70 68 71 75 71
0.90 20 31 36 32 45 50 39 53 57
1.10 9 17 21 10 22 28 10 24 30
1.25 14 22 25 14 25 27 11 24 26
1.50 12 12 7 7 6 -5 -1 -3 -20
Table 5: The values of index for and different values of and .
RR-MCV chart RR-MCV chart RR-MCV chart
0.50 65 55 2 81 52 -37 87 42 -80
0.75 38 58 59 57 72 69 67 77 71
0.90 17 33 39 28 47 54 36 55 61
1.10 4 16 21 2 20 27 -1 21 29
1.25 3 17 19 -2 17 18 -9 13 13
1.50 -4 -3 -14 -18 -18 -40 -33 -36 -69
0.50 65 56 5 81 54 -32 87 44 -73
0.75 38 58 58 56 72 69 66 77 72
0.90 16 33 39 27 47 53 35 55 60
1.10 4 16 21 2 20 27 -1 21 29
1.25 4 17 19 -2 18 19 -9 14 15
1.50 -4 -1 -10 -17 -14 -34 -31 -31 -60
0.50 64 57 9 81 56 -24 86 47 -63
0.75 37 57 58 56 71 69 66 77 72
0.90 16 32 38 27 46 52 34 54 59
1.10 4 16 21 2 20 27 -0 21 29
1.25 4 18 21 -1 19 21 -7 16 18
1.50 -2 3 -5 -14 -8 -24 -27 -22 -47
0.50 64 58 15 80 58 -15 86 51 -50
0.75 36 56 58 55 71 69 65 76 72
0.90 15 31 37 26 45 51 33 52 58
1.10 4 16 21 3 20 27 1 21 29
1.25 6 19 22 2 21 24 -4 19 22
1.50 1 8 1 -8 -1 -13 -20 -12 -31
0.50 63 59 20 80 61 -5 86 55 -36
0.75 35 55 57 53 70 69 63 76 73
0.90 15 30 35 25 43 49 32 51 57
1.10 6 16 21 5 20 27 3 21 29
1.25 8 21 24 6 24 27 1 23 26
1.50 7 13 8 -0 8 -2 -10 -1 -16
Table 6: The values of index for and different values of and .
RR-MCV chart RR-MCV chart RR-MCV chart
0.50 45 62 14 67 64 -17 78 59 -53
0.75 23 57 59 38 72 70 49 77 73
0.90 9 31 38 16 45 53 22 53 60
1.10 -2 14 21 -8 18 26 -13 18 27
1.25 -7 16 19 -19 16 18 -31 12 13
1.50 -15 -1 -11 -37 -15 -35 -59 -31 -62
0.50 45 62 17 66 65 -12 77 60 -47
0.75 22 57 59 38 72 70 49 77 73
0.90 9 31 38 16 45 52 22 53 60
1.10 -2 14 20 -8 18 26 -13 18 27
1.25 -7 16 19 -19 16 19 -31 13 15
1.50 -15 1 -8 -36 -12 -29 -57 -26 -53
0.50 45 63 20 66 66 -6 77 62 -38
0.75 22 56 58 37 71 70 48 77 73
0.90 9 30 37 16 44 51 22 52 59
1.10 -2 14 20 -7 18 26 -12 18 27
1.25 -6 17 20 -17 18 21 -29 15 18
1.50 -13 4 -3 -33 -6 -21 -53 -19 -41
0.50 44 64 25 65 68 3 76 64 -26
0.75 21 55 58 36 70 70 47 76 73
0.90 8 29 36 15 43 50 21 50 57
1.10 -1 14 20 -6 18 26 -11 18 27
1.25 -4 18 22 -15 20 24 -25 17 22
1.50 -10 8 3 -28 1 -10 -46 -10 -27
0.50 43 64 30 64 69 11 75 66 -14
0.75 21 54 57 35 69 70 46 75 73
0.90 8 28 34 15 41 48 20 49 56
1.10 -0 14 20 -4 18 26 -9 19 28
1.25 -2 20 24 -11 22 27 -20 21 26
1.50 -5 13 9 -20 9 0 -35 1 -12
Table 7: The values of index for and different values of and .
(a)
(b)
(c)
(d)
(e)
(f)
Figure 1: The profiles of Shewhart chart and Run Rules charts for various in-control settings; left side: lower-sided charts, right side: upper-sided charts

The analysis presented above is only for the case of specific shift size. In practice, however, it is hard for quality practitioners to predetermine a specific shift without any previous experience. Thus, they usually have an interest in detecting a range of shifts

rather than preference for any particular size of the process shift. The use of the uniform distribution has been proposed to account for the unknown shift size by some authors (for instance, see

Chen and Chen 12 and Celano et al. 10). The statistical performance of the corresponding chart can be evaluated through the (Expected Average Run Length) given by

(17)

where is the , as a function of shift , of the proposed Run Rules charts, with for . In the following section, we will consider a specific range of shift (decreasing case, denoted as (D)) for lower-sided RRMCV control chart and (increasing case, denoted as (I)) for upper-sided RRMCV control chart.

RR-MCV chart RR-MCV chart RR-MCV chart
(D) (101.8, 100.1) (48.1, 46.5) (33.0, 31.3) (79.4,76.7) (38.5,36.0) (26.9,24.3) (67.8,64.4) (34.0,30.6) (24.2,20.7)
(I) (29.4, 27.8) (17.4, 15.8) (13.2, 11.5) (30.3,27.9) (17.4,15.0) (13.1,10.6) (31.7,28.5) (18.0,14.7) (13.7,10.2)
(D) (103.1, 101.3) (49.2, 47.5) (33.9, 32.2) (80.4,77.8) (39.4,36.9) (27.6,25.1) (68.8,65.4) (34.8,31.4) (24.9,21.4)
(I) (30.3, 28.6) (18.1, 16.5) (13.7, 12.1) (31.2,28.7) (18.1,15.6) (13.6,11.1) (32.6,29.3) (18.6,15.3) (14.1,10.7)
(D) (105.1, 103.3) (50.9, 49.2) (35.3, 33.7) (82.2,79.6) (40.9,38.4) (28.8,26.3) (70.4,67.0) (36.1,32.7) (26.0,22.5)
(I) (31.8, 30.1) (19.2, 17.7) (14.6, 13.0) (32.6,30.1) (19.1,16.7) (14.5,12.0) (34.0,30.8) (19.6,16.3) (14.9,11.5)
(D) (107.8, 106.1) (53.2, 51.5) (37.2, 35.6) (84.7,82.0) (42.9,40.3) (30.5,27.9) (72.6,69.2) (37.9,34.5) (27.4,24.0)
(I) (34.0, 32.4) (20.9, 19.3) (15.9, 14.3) (34.7,32.3) (20.6,18.1) (15.6,13.1) (36.1,32.9) (21.1,17.8) (16.0,12.6)
(D) (111.2, 109.4) (55.9, 54.3) (39.5, 37.9) (87.7,85.0) (45.2,42.7) (32.4,29.9) (75.4,72.0) (40.1,36.7) (29.2,25.8)
(I) (37.2, 35.5) (23.0, 21.4) (17.5, 16.0) (37.6,35.1) (22.5,20.1) (17.1,14.6) (38.9,35.6) (22.9,19.6) (17.4,14.1)
RR-MCV chart RR-MCV chart RR-MCV chart
(D) (136.9, 135.1) (53.3, 51.7) (35.1, 33.5) (107.0,104.3) (42.5,40.0) (28.5,26.0) (90.6,87.2) (37.3,34.0) (25.6,22.1)
(I) (36.0, 34.3) (18.8, 17.2) (13.8, 12.2) (37.9,35.4) (18.8,16.4) (13.8,11.2) (40.2,36.9) (19.5,16.2) (14.3,10.9)
(D) (138.3, 136.5) (54.5, 52.8) (36.1, 34.4) (108.3,105.6) (43.5,41.0) (29.3,26.8) (91.8,88.3) (38.2,34.8) (26.3,22.9)
(I) (37.0, 35.4) (19.5, 17.9) (14.4, 12.8) (38.9,36.4) (19.5,17.1) (14.3,11.8) (41.2,38.0) (20.1,16.8) (14.8,11.4)
(D) (140.6, 138.8) (56.3, 54.7) (37.6, 35.9) (110.3,107.6) (45.1,42.5) (30.6,28.0) (93.7,90.2) (39.6,36.3) (27.4,24.0)
(I) (38.8, 37.2) (20.7, 19.2) (15.3, 13.7) (40.7,38.2) (20.7,18.2) (15.2,12.7) (43.1,39.8) (21.2,18.0) (15.6,12.3)
(D) (143.6, 141.8) (58.8, 57.1) (39.6, 37.9) (113.2,110.5) (47.2,44.7) (32.3,29.8) (96.4,92.9) (41.5,38.2) (29.0,25.6)
(I) (41.6, 40.0) (22.5, 20.9) (16.7, 15.1) (43.4,40.9) (22.3,19.8) (16.4,13.9) (45.8,42.5) (22.8,19.6) (16.8,13.4)
(D) (147.5, 145.7) (61.8, 60.1) (42.0, 40.4) (116.8,114.1) (49.8,47.2) (34.4,31.8) (99.7,96.2) (43.9,40.5) (30.9,27.4)
(I) (45.6, 43.9) (24.8, 23.3) (18.4, 16.9) (47.2,44.6) (24.4,22.0) (18.0,15.5) (49.5,46.2) (24.8,21.6) (18.3,15.0)
RR-MCV chart RR-MCV chart RR-MCV chart
(D) (208.8, 206.9) (60.1, 58.4) (37.6, 35.9) (171.7,168.9) (47.6,45.1) (30.4,27.9) (147.1,143.5) (41.6,38.2) (27.2,23.8)
(I) (50.0, 48.3) (20.5, 18.9) (14.5, 12.9) (55.1,52.6) (20.6,18.1) (14.5,12.0) (60.3,57.0) (21.3,18.0) (15.0,11.6)
(D) (210.1, 208.2) (61.3, 59.6) (38.6, 36.9) (173.1,170.3) (48.7,46.2) (31.3,28.7) (148.5,144.9) (42.5,39.2) (27.9,24.5)
(I) (51.4, 49.7) (21.3, 19.7) (15.1, 13.5) (56.6,54.1) (21.3,18.9) (15.0,12.5) (61.9,58.5) (22.0,18.7) (15.6,12.2)
(D) (212.3, 210.4) (63.3, 61.6) (40.2, 38.5) (175.4,172.6) (50.4,47.9) (32.6,30.1) (150.8,147.2) (44.1,40.7) (29.2,25.7)
(I) (53.9, 52.2) (22.6, 21.0) (16.1, 14.5) (59.2,56.7) (22.6,20.2) (16.0,13.5) (64.7,61.2) (23.3,20.0) (16.4,13.1)
(D) (215.2, 213.4) (66.0, 64.3) (42.3, 40.6) (178.6,175.8) (52.7,50.2) (34.4,31.9) (154.0,150.4) (46.2,42.8) (30.8,27.4)
(I) (57.7, 56.0) (24.5, 22.9) (17.6, 16.0) (63.1,60.5) (24.4,22.0) (17.3,14.8) (68.7,65.3) (25.0,21.8) (17.7,14.4)
(D) (218.9, 217.1) (69.3, 67.6) (44.8, 43.2) (182.6,179.8) (55.5,53.0) (36.6,34.1) (158.0,154.4) (48.8,45.4) (32.8,29.4)
(I) (63.1, 61.4) (27.1, 25.5) (19.5, 17.9) (68.5,65.8) (26.8, 24.3) (19.0,16.5) (74.1,70.7) (27.3,24.0) (19.3,16.0)
Table 8: The values for MCV control chart when and for RRMCV control chart when with different values of and .

Table 8 presents the values of and (Expected Standard Deviation Run Length) for various combinations of , and . The same trends as the case of specific shift size are observed from this table. The value of in the upper-sided Run Rule control chart corresponding to smaller values of is significantly smaller than that corresponding to larger values of In the contrary, the values of decrease from smaller scheme to larger scheme of Run Rules for lower-sided chart. Therefore, the choice of using RRMCV, RRMCV or RRMCV control charts depends on the goal of practitioners: It they want to detect increasing shifts, they are advised to choose smaller scheme of Run Rules (say RRMCV in this paper); if they want to detect decreasing shifts, the larger scheme of Run Rules (say RRMCV in this paper) should be used.

Similar to the specific shift size case, a comparison between the performance of RR-MCV control charts and the performance of Shewhart-MCV control chart is provided in Table 9. It has been undertaken by defining the index

(18)

where and are the value for the Shewhart-MCV and RRMCV chart. If , the RRMCV charts give better performance than the Shewhart-MCV chart; if , the Shewhart-MCV chart is better. Once again, the obtained results show that the RRMCV charts outperforms the Shewhart-MCV chart in most cases.

In comparison with the Run Sum MCV control chart suggested by Lim et al. 20, the Run Rules based charts also have some outstanding advantages. As discussed above, the Run Rules give the best performance when the lower–sided RRMCV chart is used for decreasing shift size and when the upper-sided RRMCV chart is used for increasing shift size. We see that the value of from these Run Rules control charts are smaller than the optimal values of from the Run Sum control chart. For example, with and , for the upward chart we have in the RRMCV chart (Table 8 in this paper) while in the Run Sum control chart (Table 1 in Lim et al. 20); for the downward we have in the RRMCV chart (Table 8 in this paper) while in the Run Sum control chart (Table 3 in Lim et al. 20). Note also that these charts have the same in-control , i.e. . Moreover, the use of the Run Sum control chart requires to optimize the score vectors on the range of shifts that it is difficult in practice to predetermine, while the Run Rules charts require the determination of a single control limit value, for all shift sizes. This make the Run Rules more easier to implement.

RR-MCV chart RR-MCV chart RR-MCV chart
(D) 38 41 41 53 54 53 61 60 58
(I) 5 12 14 2 12 13 -4 8 8
(D) 38 41 41 53 54 53 60 60 58
(I) 5 12 14 2 12 14 -3 9 9
(D) 38 41 41 52 54 53 60 60 58
(I) 6 13 15 3 13 15 -2 10 11
(D) 37 40 40 52 53 52 59 59 57
(I) 7 14 16 5 15 17 1 12 14
(D) 36 39 39 51 52 51 59 58 57
(I) 10 16 17 9 17 19 5 15 17
RR-MCV chart RR-MCV chart RR-MCV chart
(D) 34 41 41 49 55 53 58 61 59
(I) 1 11 14 -5 11 13 -12 7 8
(D) 34 41 41 49 54 53 57 60 58
(I) 1 12 14 -5 11 14 -11 8 9
(D) 33 41 41 48 54 53 57 60 58
(I) 2 12 15 -4 12 15 -10 9 11
(D) 33 40 40 48 53 52 56 59 58
(I) 3 13 16 -1 14 17 -8 11 14
(D) 32 39 39 47 52 52 55 59 57
(I) 6 15 17 2 16 19 -3 14 17
RR-MCV chart RR-MCV chart RR-MCV chart
(D) 23 41 41 37 55 54 46 61 59
(I) -8 11 14 -20 10 13 -32 6 8
(D) 23 41 41 37 54 53 46 61 59
(I) -7 11 14 -19 10 13 -31 6 9
(D) 22 40 41 36 54 53 45 60 58
(I) -7 11 15 -18 11 15 -30 8 11
(D) 22 40 40 35 53 52 45 60 58
(I) -5 12 16 -16 13 16 -27 10 14
(D) 21 39 39 35 52 52 44 59 57
(I) -3 14 17 -12 15 19 -22 13 17
Table 9: The index values for different values of and .

5 Illustrative example

An illustrative example of RRMCV control chart is given in this Section. Let us consider a sintering process in an Italian company that manufactures sintered mechanical parts, which is introduced in Lim et al. 20. The data are recorded from a spring manufacturing process, for which the quality characteristics are the spring inner diameter () and the spring elasticity (

). From Phase I, we have the estimated value of

, while (according to Lim et al. 20) the assumption that the MCV during Phase I is constant holds. The data collected during the Phase II process with sample size are shown in Table 10. Further details on the process can be found in Lim et al. 20. From the obtained results in Section 3, the control limits for the different control charts are as follows.

  • For the upper-sided ShewhartMCV chart,

  • For the upper-sided RR chart,

  • For the upper-sided RR chart,

  • For the upper-sided RR chart,

Sample number
1 7.781 1.592 1.164 0.734 0.35645 0.113710
2 7.385 1.804 1.006 1.667 0.96049 0.104890
3 7.988 2.260 0.762 0.359 0.17373 0.108870
4 8.189 2.100 1.885 0.470 0.13026 0.156790
5 7.436 2.061 1.404 0.519 0.08280 0.139290
6 6.746 2.289 0.846 0.811 0.43835 0.133240
7 7.356 1.917 0.197 2.587 0.01597 0.059996
8 8.492 1.845 1.460 1.746 1.42051 0.055093
9 7.272 1.580 1.353 0.345 0.27988 0.117710
10 7.585 1.568 1.098 0.788 0.41252 0.109610
11 7.734 1.709 0.952 0.228 0.11462 0.102440
12 8.160 1.498 1.598 1.178 1.00757 0.122950
13 7.102 2.661 1.508 0.945 0.73607 0.101260
14 8.392 1.883 0.536 0.706 0.23234 0.085637
15 7.592 2.531 0.256 0.563 0.24827 0.043489
16 8.141 2.093 0.394 0.603 0.25584 0.072202
17 7.883 2.490 1.321 1.179 0.65037 0.142430
18 7.886 2.877 0.883 1.431 0.22524 0.106680
19 7.830 1.008 0.878 0.558 0.14223 0.112090
20 8.196 1.482 0.791 0.220 0.13724 0.088460
Table 10: Illustrative example of Phase II dataset.

[5mm]

Figure 2: RRMCV control chart corresponding to Phase II data set in Table 10

[5mm]

Figure 3: RRMCV control chart corresponding to Phase II data set in Table 10

[5mm]

Figure 4: RRMCV control chart corresponding to Phase II data set in Table 10

The corresponding values are presented in the rightmost column of 10 and plotted in Figures 2-4, respectively. Each figure consists of the for the upper-sided Shewhart - MCV chart along with the of the MCV control chart, for . It is not difficult to see that the RRMCV control chart signals the occurrence of the out-of-control condition by two out of three successive plotting points #4 and #5 above the control limit , see Figure 2. The RRMCV control chart signals the occurrence of the out-of-control condition by three out of four successive plotting points #4, #5 and #6 above the control limit , see Figure 3. The RRMCV control chart signals the occurrence of the out-of-control condition by four out of five successive plotting points #1, #2, #3 and #4 above the control limit , see Figure 4. The figures 2-4 also show that all the points are plotted below the . That is to say, in this situation, the upper Shewhart - MCV control chart (designed in Yeong et al. 29) fails to detect the out-control condition detected by the RRMCV control charts. In addition, it is worth noting that similar conclusions have been reached by Lim et al. 20 with the Run Sum MCV control chart. Specifically, their Run Sum chart gives an out-of-control signal at sample 4. However, the design of this run sum control chart requires the a priori selection for a shift of interest so as to determine the optimal scores for the chart. This means that the run sum chart is designed so as to be the optimal one in the selection of a specific shift. Their choice, for illustrative purposes, was a 25% increase in the IC value of the MCV. Clearly, this is not required here where only the value of the is needed for the application of the run rule chart. Before closing this section we mention that the lower-sided MCV control charts can be constructed in a similar manner. The respective limits are 0.010029, 0.02403, 0.03464, 0.04275 . As in the case of the lower-sided run sum chart of Lim et al. 20 no out-of-control signal is given. However, due to space economy, we do not provide the respective figures.

6 Concluding remarks

In this paper, we have investigated the one-sided control charts with Run Rules for monitoring the coefficient of variation of multivariate data. Two one-sided charts were considered to detect separately both increases and decreases in the multivariate CV. The performance of proposed charts is evaluated through for deterministic shift size and for unknown shift size. The numerical results showed that the Run Rules control charts enhance significantly the performance of Shewhart control chart. For purpose of optimizing the performance of Run Rules charts, it is recommended to use the RRMCV for detecting decreasing process shifts and RRMCV for detecting increasing process shifts. Moreover, under certain conditions, this careful choice of the Run Rules charts also lead to an improved efficiency compared to the Run Sum control chart for MCV.

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