New extended distribution-free homogenously weighted monitoring schemes for monitoring abrupt shifts in the location parameter

Tokelo Irene Letshedi; Jean-Claude Malela-Majika; Sandile Charles Shongwe

doi:10.1371/journal.pone.0261217

Abstract

A homogeneously weighted moving average (HWMA) monitoring scheme is a recently proposed memory-type scheme that gained its popularity because of its simplicity and superiority over the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) schemes in detecting small disturbances in the process. Most of the existing HWMA schemes are designed based on the assumption of normality. It is well-known that the performance of such monitoring schemes degrades significantly when this assumption is violated. Therefore, in this paper, three distribution-free monitoring schemes are developed based on the Wilcoxon rank-sum W statistic. First, the HWMA W scheme is introduced. Secondly, the double HWMA (DHWMA) W scheme is proposed to improve the ability of the HWMA W scheme in detecting very small disturbances in the location parameter and at last, the hybrid HWMA (HHWMA) W scheme is also proposed because of its flexibility and better performance in detecting shifts of different sizes. The zero-state performances of the proposed schemes are investigated using the characteristics of the run-length distribution. The proposed schemes outperform their existing competitors, i.e. EWMA, CUSUM and DEWMA W schemes, in many situations, and particularly the HHWMA W scheme is superior to these competitors regardless of the size of the shift in the location parameter. Real-life data are used to illustrate the implementation and application of the new monitoring schemes.

Citation: Letshedi TI, Malela-Majika J-C, Shongwe SC (2022) New extended distribution-free homogenously weighted monitoring schemes for monitoring abrupt shifts in the location parameter. PLoS ONE 17(1): e0261217. https://doi.org/10.1371/journal.pone.0261217

Editor: Valerie L. Shalin, Wright State University, UNITED STATES

Received: June 2, 2021; Accepted: November 27, 2021; Published: January 21, 2022

Copyright: © 2022 Letshedi et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: The data used for the illustration example are available from Mukherjee et al. (2019) (10.1016/j.cie.2019.106059).

Funding: The author(s) received no specific funding for this work.

Competing interests: The authors have declared that no competing interests exist.

1. Introduction

There are numerous monitoring schemes documented in the statistical process monitoring (SPM) literature. Over the years, a fact has been proven that SPM tools are crucial tools to monitor most industrial processes to enhance the quality of products and services. Monitoring schemes are intended to monitor and identify unnatural (i.e., assignable) causes of variation as soon as they occur. In-control (IC) processes are favourable as they consist of common (or natural) causes of variations that cannot disturb the process to the extent of ruining the outputs. Walter A. Shewhart was the first to design a modern monitoring scheme in the 1920s, see the introductory chapter of [1]. Shewhart schemes are widely known to be responsive to large shifts in a process. However, they are relatively slow in detecting small to moderate shifts in a process. When it comes to a quick detection of small to moderate shifts, the SPM literature recommends the use of memory-type monitoring schemes such as the exponentially weighted moving average (EWMA) by [2] and the cumulative sum (CUSUM) by [3]. Memory-type schemes have their disadvantages of being slow in detecting large shifts. More contributions and modifications of the memory-type schemes have been recorded over the years (see, [4–13]; just to cite a few). More recently, [14] introduced a new memory-type scheme called the homogeneously weighted moving average (HWMA). The charting statistic of the HWMA scheme assigns a weight λ (i.e., a smoothing parameter, where 0 < λ < 1) to the current sample and a weight (1 − λ) is homogeneously (or equally) distributed to all the previous samples. Some recent publications on HWMA schemes can be found in [15–21].

Researchers have also introduced different combinations of control charts for better results in improving the quality of products and services for end users. The double EWMA (DEWMA) scheme was introduced by [4] to increase the sensitivity of the EWMA scheme in monitoring small to moderate shifts; see also, [22, 23]. Furthermore, the hybrid EWMA (HEWMA) scheme was first introduced by [24, 25] to also increase the sensitivity of EWMA scheme and it was shown that the DEWMA scheme is a special case of the HEWMA scheme. Similarly, the double HWMA (DHWMA) and hybrid HWMA (HHWMA) schemes were discussed in [26–29].

Firstly, it is important to mention that a monitoring scheme is defined as distribution-free if the IC run-length distribution is the same for every continuous distribution. Next, the abovementioned HWMA-type monitoring schemes are based on normally distributed data. However, there are some distribution-free HWMA-type schemes that have been discussed in the SPM literature. That is, [30] proposed the HWMA schemes based on the sign and signed rank statistics and more recently, [31] and [32] provided different designs of the DHWMA scheme based on the sign statistic. The sign and signed-rank monitoring schemes are applicable in situations where the true process location value is known. Note though, in most practical situations, the true process location value may not be known; which means that the use of simple sign and signed-rank statistics are invalid; hence, more complex nonparametric methods are applicable. These can be nonparametric methods based on the precedence, exceedance, Wilcoxon rank-sum, Mann-Whitney, etc.; see [33] for more details. The focus of this paper is on the Wilcoxon-rank-sum (W) test by [34]. In essence, the W test statistic compares the outcomes between two independently and identically distributed (i.i.d.) samples (or groups) and tests whether the two samples are likely derived from the same population distribution, i.e., the two samples are collected from a population with the same shape. For some discussions on monitoring schemes based on the W statistic, see [5, 35–40]. The limitations of the existing research works that motivated this paper are that the SPM literature publications are mostly populated with parametric monitoring schemes. However, in most real-life applications, data are not always normally distributed. Therefore, there is a need of robust monitoring schemes that do not rely on the parametric settings or assumption. Moreover, there is a need of more efficient memory-type monitoring schemes that will perform better regardless of the size of the shift and/or the smoothing parameter.

Note that there is a limited number of HWMA-type monitoring schemes in the literature. Table 1 present the summary and gaps of the HWMA-type schemes in the SPM literature. As it can be noticed, there are only two dozen of articles related to the HWMA scheme. These include parametric and nonparametric variable and attribute HWMA-related schemes. For attribute schemes, there is only one article by [41] and only four nonparametric HWMA schemes based on the sign, signed-rank and Lepage statistics (see [31, 32] and [42]). Thus, there is an increasing need in the design of attribute and parametric monitoring schemes to cater for processes that are based on discrete and non-normal distribution, respectively. For more recent articles on HWMA scheme, readers are referred to the articles by [41–48]. A summary and more details of the existing HWMA-related articles are presented in Table 1.

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Table 1. Summary of the literature on HWMA-related monitoring schemes.

https://doi.org/10.1371/journal.pone.0261217.t001

Therefore, the objective of this paper is to propose three new distribution-free HWMA-type monitoring schemes based on the two-sample W statistic. These are the basic HWMA W scheme, DHWMA W scheme and the HHWMA W scheme. Moreover, the run-length properties of the latter schemes are derived and evaluated using Monte Carlo simulations. The run-length performance of the HWMA, DHWMA and HHWMA W schemes are compared to each other and thereafter, the corresponding existing competitors (i.e., EWMA and CUSUM W schemes by [5]; DEWMA W scheme by [49]) are also compared to the newly proposed monitoring schemes. Thus, the contribution of this paper can be summarised as follows:

The design of robust and efficient monitoring schemes under the violation of the normality assumption.
The introduction of new HWMA, DHWMA and HHWMA schemes based on the WRS statistic.
The investigation of both specific and overall performances of the proposed HWMA-type monitoring scheme since most of the existing ones are investigated using specific measures of performance.

The rest of this paper is organised as follows: Section 2 introduces the W statistic and the fundamental concepts of the proposed HWMA, DHWMA and HHWMA W schemes. The operation and implementation steps of the proposed schemes are presented in Section 3. Section 4 presents the performance analysis of the proposed schemes. Moreover, the HWMA, DHWMA and HHWMA W schemes are further compared to the existing CUSUM, EWMA and DEWMA W schemes. An illustrative example using real-life data is provided in Section 5, and finally, Section 6 presents the concluding remarks, recommendations and future research works.

2. The proposed HWMA W monitoring scheme and its extended versions

2.1 The Wilcoxon rank sum W statistic

Assuming that X = {x_i: i = 1,2, …, m} is IC reference sample (i.e., phase I) of size m and Y = {y_tj: t = 1,2, …; j = 1,2,…,n} a test sample (i.e., phase II) of size n. The samples X and Y are independent from one another and from the samples’ observations with unknown cumulative distribution function (c.d.f.), denoted as F(x) and G(y), respectively. The test and reference samples use the same distribution, but they differ in the location shift (δ). Thus, when δ = 0, the two distributions are the same (i.e., F = G) and the process is considered to be IC. Otherwise, the process is considered to be out-of-control (OOC).

The W statistic was proposed by [34] to compare two independent samples in order to check whether their populations mean ranks differ. The W statistic for a phase I sample of size m and a phase II sample of size n is given by (1) where the indicator I_Z = 1 when the z^th observation of the ordered N (with N = m + n) observations obtained after combining the phase I and phase II samples is from the phase I sample and I_Z = 0 when the z^th observation is from the phase II sample.

Assuming that no ties are observed, then the mean and the variance of the W statistic are given by (see [5]) (2) respectively.

2.2 The HWMA W scheme

The proposed HWMA W monitoring scheme is designed following [14]’s idea. Thus, the HWMA W statistic is defined by (3) where λ ∈ (0,1] is the smoothing parameter and is the mean of the lagged W_t statistics (i.e., ) and is set to be equal to μ_W.

Thus, the mean and variance of H_t statistic are defined as follows: (4) and (5) where is defined in Eq (2). For more details on the derivation of the mean and variance of the H_t statistic, readers are referred to S1 Appendix of this paper. Using the above properties, the time-varying upper and lower control limits (UCL and LCL) of the HWMA W scheme are then given as (6) where L_H(L_H > 0) is the control limit constant of the HWMA W scheme which is determined such that the attained IC average run-length (ARL) is much closer or equal to the prespecified nominal IC ARL (ARL₀) value. The center line is equal to μ_W (i.e., CL = μ_W). Hence, the HWMA W scheme gives an OOC signal if the H_t statistic plots beyond the control limits defined in Eq (6).

2.3 The DHWMA W scheme

The DHWMA scheme for monitoring the process mean was proposed by [28]. In this paper, the DHWMA scheme is designed using the W statistic, and it is denoted as DHWMA W scheme. The charting statistic of the DHWMA W statistic is defined by (7) where H_t is defined in Eq (3), and is defined as (8) Thus, at a sampling time t, the mean and variance of the DH_t statistic are given by (9) and (10) respectively. The expressions of the mean and variance of the DHWMA W statistic are derived in S2 Appendix.

Thus, the control limits of the DHWMA W scheme are defined by (11) where L_DH (L_DH > 0) is the is the control limit constant of the DHWMA W scheme. The center line is equal to μ_W (i.e., CL = μ_W). Hence, the DHWMA W scheme gives an OOC signal if the DH_t statistic plots beyond the control limits defined in Eq (11).

2.4 The HHWMA W scheme

This paper also proposes the HHWMA monitoring scheme using the W statistic and it is denoted as HHWMA W scheme. The charting statistic of the HHWMA W statistic is defined by (12) where and is defined in Eq (8) and μ_W as explained earlier. Note that this scheme has two different smoothing parameters, i.e., λ₁ and λ₂, which are defined in the interval (0,1]. When λ₁ = λ₂, then the HHWMA W scheme reduces to the DHWMA W scheme.

Thus, at sampling time t, the mean and variance of HH_t statistic are defined by (13) and (14) respectively. The expressions of the mean and variance of HH_t statistic are derived in S3 Appendix of this paper.

Using the above properties, the control limits for HHWMA W scheme are defined by (15) where L_HH (L_HH > 0) is the control limits constant of the HHWMA W scheme. The center line is equal to μ_W (i.e., CL = μ_W). Thus, the HHWMA W scheme gives an OOC signal if the HH_t statistic plots beyond the control limits defined in Eq (15).

3. The operational procedure of the proposed HWMA, DHWMA and HHWMA W schemes

Most of the nonparametric monitoring schemes do not have specific or exact expressions of the run-length distribution. Hence, exact formulas as well as the Markov chain approach cannot be used in such a scenario. Therefore, Monte Carlo simulation approach is recommended since it can solve any complex problem through organised computer programs coding. In this paper, the proposed HWMA, DHWMA and HHWMA W schemes are constructed using the following steps:

Step 1: Generate a reference sample, X, of size m from an IC process, say N(0,1) distribution.
Step 2: Generate a test sample, Y, of size n independently from the reference sample. For the IC case, the distributions of the reference and test samples are identical (we say that δ = 0, thus Y ~ N (0,1)). For the OOC case, the distribution for the test sample is taken to be the same form as that for the reference sample, but with a shift in the location parameter in units of the population standard deviation (δ ≠ 0), in our example, Y ~ N(δ,1).
Step 3: The W statistic is calculated using Eq (1) as explained in Section 2.
Step 4: Calculate the mean and variance of the W statistic when the process is IC using Eq (2). These characteristics are used in Steps 5 and 6.
Step 5: (a) To construct the HWMA W scheme, we use the statistic defined in Eq (3).
(b) To construct the DHWMA W scheme, we use the statistic defined in Eq (7).
(c) To construct the HHWMA W scheme, we use the statistic defined in Eq (12).
Step 6: The control limits constants and the process design parameters are chosen such that the attained ARL₀ is equal (or closer) to the nominal value of 500.
(a) The time varying control limits of the HWMA W scheme are computed using Eq (6). The HWMA W scheme gives a signal if the charting statistic computed in Step 5(a) plots beyond the control limits calculated using Eq (6).
(b) The control limits of the DHWMA W sheme are computed using Eq (11). The DHWMA W scheme gives a signal if the charting statistic computed in Step 5(b) plots beyond the control limits calculated using Eq (11).
(c) The control limits of the HHWMA W sheme are computed using Eq (15). The HHWMA W scheme gives a signal if the charting statistic computed in Step 5(c) plots beyond the control limits calculated using Eq (15).

Note that other memory-type schemes based on the W statistic can also be constructed in a similar way.

4. IC and OOC performance analyses

4.1 Performance metrics

The characteristics of the run-length (RL) such as the average RL (ARL), the standard deviation of the RL (SDRL) as well as the expected ARL (EARL) and the expected SDRL (ESDRL) are the most popular metrics used to evaluate the performance of a monitoring scheme. The RL is the number of rational subgroups to be plotted on a scheme before it signals an OOC situation for the first time. Since the RL distribution is highly skewed, researchers advocate that the percentiles of the RL distribution provide more and meaningful information than the ARL value; see for example, [50]. It is very important to know that the ARL values measure the performance of a scheme for specific shifts. However, in practice, it is more crucial to investigate the performance of a monitoring scheme for a range of shifts, i.e., the overall performance for certain range of shifts. In this case, the SPM literature recommends the use of EARL or other expected characteristics of the RL to assess the overall performance of a scheme; see [51].

This paper uses Monte Carlo simulations in SAS ®9.4/IML15.4 with 20000 replications to assess the HWMA W, DHWMA W and HHWMA W schemes in terms of the above-mentioned metrics using different distributions. The distribution considered in this paper are the standard normal distribution (denoted as N(0,1)), Student t distribution with degrees of freedom 5 (denoted as t(5)) and gamma distribution with a shape parameter of 3 and a scale parameter of 1 (denoted as GAM(3,1)).

The EARL and ESDRL are mathematically defined as follows (16) and (17) where ARL(δ) and SDRL(δ) are the values of the ARL and SDRL for a specific shift δ in standard deviation unit and Δ is the number of increments between the lower and upper bound shifts (i.e., δ_min and δ_max). In this paper, we use increments equal to 0.1.

4.2 Performance analysis

In this section, we study the sensitivity (or performance) of the proposed HWMA, DHWMA and HHWMA W schemes under the N(0,1), t(5) and GAM(3,1) distributions.

4.2.1 Performance of the proposed HWMA W scheme.

Table 2 presents the IC and OOC ARL profiles of the proposed HWMA W scheme under N(0,1), t(5) and GAM(3,1) distributions when n = 5, m = 100, λ ∈{0.05,0.25,0.5} for a nominal IC ARL (ARL₀) of 500. From Table 2, it can be seen that when λ = 0.05, the proposed HWMA W scheme yields the attained ARL₀ values of 502.47, 499.37 and 504.89 under the N(0,1), t(5) and GAM(3,1) distributions, respectively. It can be observed that these attained ARL₀ values are much closer to the nominal ARL₀ value of 500, i.e. within the 10% of the desired nominal ARL₀ value as suggested by the SPM literature; see for example [33]. Other values of the attained ARL₀ values are also much closer to the nominal ARL₀ value of 500. Therefore, it can be concluded that the HWMA W scheme is IC robust which means that the characteristics of the IC RL are much closer to 500 across all continuous probability distributions.

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Table 2. The ARL profile of the HWMA W scheme when (m,n) = (100,5) and λ ∈ {0.05, 0.25, 0.5} for a nominal ARL₀ = 500 under N(0,1), t(5) and GAM(3,1) distributions.

https://doi.org/10.1371/journal.pone.0261217.t002

In terms of the OOC ARL profile, as the value of λ increases, the width of the control limits decreases; however, the sensitivity of the HWMA W decreases. For small shifts in the process mean, the sensitivity of the HWMA W scheme decreases as λ increases. However, the converse is true for moderate shifts. For instance, under the N(0,1) distribution, when λ = 0.05, 0.25 and 0.5, the EARL_(0.7,1.5] is found to be equal to 3.26, 2.79 and 2.62, respectively. This shows that for moderate shifts in the process mean, there is an increase in the sensitivity of the HWMA W scheme as λ increases. For small-to-moderate as well as from small-to-large shifts, the smaller the value of λ, the higher the sensitivity of the proposed HWMA W scheme. From Table 2, it can also be seen that for small values of λ, i.e., 0 < λ < 0.25, and small shifts in the process mean, the proposed HWMA W scheme performs better under heavy-tailed distributions followed by skewed distributions. For instance, when λ = 0.05, the HWMA W scheme yields EARL_(0,0.7] values of 76.2, 58.49 and 72.07 under the N(0,1), t(5) and GAM(3,1) distributions, respectively. However, when 0.25 ≤ λ < 1, the proposed HWMA W scheme performs better under heavy-tailed distributions followed by symmetric distributions. For instance, when λ = 0.5, the HWMA scheme yields EARL_(0,0.7] values of 93.31, 79.26 and 104.84 under the N(0,1), t(5) and GAM(3,1) distributions, respectively. This conclusion is also true in terms of the overall performance when δ ∈ (0,1.5]; see Table 2 performance in terms of the EARL_(0,1.5]. For moderate shifts in the process mean, regardless of the magnitude of λ, the proposed HWMA W scheme performs better under heavy-tailed distributions followed by skewed distributions. For instance, when λ = (0.05, 0.25, 0.5), the HWMA W scheme yields EARL_(0.7,1.5] values of (3.26, 2.79, 2.62), (2.85, 2.43, 2.21) and (3.00, 2.51, 2.31) under the N(0,1), t(5) and GAM(3,1) distributions, respectively.

Fig 1 investigates the effect of the test sample (i.e. phase II) on the overall performance of the HWMA W scheme in terms of the EARL_(0,1.5] and ESDRL_(0,1.5] when m = 100, n ∈{3,5,10} and λ ∈{0.05, 0.25} for a nominal ARL₀ value of 500. Thus, it can be observed that the larger the test sample, the more sensitive the proposed scheme is. For instance, when λ = 0.05, under the N(0,1) distribution, the HWMA W scheme yields EARL values of 43.22, 37.41 and 25.67 when n = 3, 5 and 10, respectively (see Fig 1A). The findings remain the same for other values of λ across all continuous probability distributions; see also Fig 1B). The same conclusion is also deduced in terms of the ESDRL profile (see Fig 1C and 1D).

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Fig 1. Effect of the phase II sample size on the overall performance of the HWMA W scheme when m = 100, λ ∈ {0.05, 0.25} and n ∈ {3, 5, 10} under N(0,1), t(5) and GAM(3,1) distributions.

https://doi.org/10.1371/journal.pone.0261217.g001

Fig 2 investigates the effect of the reference sample (i.e. phase I) on the overall performance of the HWMA W scheme in terms of the EARL_(0,1.5] and ESDRL_(0,1.5] when m ∈ {50,100,500}, n = 5 and λ ∈{0.05, 0.25} for a nominal ARL₀ value of 500. Thus, it can be observed that the larger the phase I sample size, the more sensitive the HWMA W scheme is. The smaller the phase I sample size, the higher the variability in the RL distribution. For instance, when λ = 0.25, under the t(5) distribution, the HWMA W scheme yields ESDRL values of 129.99, 83.43 and 56.11 when m = 50, 100 and 500, respectively (see Fig 2B. The findings remain the same for other values of λ.

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Fig 2. Effect of the phase I sample size on the performance of HWMA W scheme when n = 5, λ ∈ {0.05, 0.25} and m ∈ {50, 100,500} under N(0,1), t(5) and GAM(3,1) distributions.

https://doi.org/10.1371/journal.pone.0261217.g002

4.2.2 Performance of the proposed DHWMA W scheme.

Table 3 displays the IC and OOC ARL profiles of the proposed DHWMA W scheme under N(0,1), t(5) and GAM(3,1) distributions when n = 5, m = 100, λ ∈{0.05,0.25,0.5} for a nominal ARL₀ value of 500. From Table 3, it can be seen that when λ = 0.05, 0.25 and 0.5, the proposed DHWMA W scheme yields attained ARL₀ values of (499.39, 502.63, 503.37), (495.43, 512.80, 512.27) and (508.70, 499.08, 501.13) under the N(0,1), t(5) and GAM(3,1) distributions, respectively. It can be observed these attained ARL₀ values are much closer to the nominal ARL₀ value of 500. Therefore, it can be concluded that the DHWMA W scheme is IC robust.

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Table 3. The ARL profile of the DHWMA W scheme when (m,n) = (100,5) and λ ∈ {0.05, 0.25, 0.5} for a nominal ARL₀ = 500 under N(0,1), t(5) and GAM(3,1) distributions.

https://doi.org/10.1371/journal.pone.0261217.t003

In terms of the OOC ARL profile, as the value of λ increases, the width of the control limits increases as well; however, the sensitivity of the DHWMA W decreases. For small shifts in the process mean, the sensitivity of the DHWMA W scheme decreases as λ increases. However, the converse is true for moderate shifts under symmetric and heavy-tailed distribution. For instance, under the t(5) distribution, when λ = 0.05, 0.25 and 0.5, the EARL_(0.7,1.5] is found to be equal to 2.41, 2.65 and 2.46, respectively. For small-to-moderate as well as from small-to-large shifts, the smaller the value of λ, the higher the sensitivity of the proposed DHWMA W scheme. From Table 3, it can also be seen that for small shifts in the process mean, the proposed DHWMA W scheme performs better under heavy-tailed distributions followed by symmetric distributions. For instance, when λ = 0.05, the DHWMA scheme yields EARL_(0,0.7] values of 57.57, 51.16 and 74.58 under the N(0,1), t(5) and GAM(3,1) distributions, respectively. However, for moderate shifts in the process mean, regardless of the magnitude of λ, the proposed DHWMA W scheme performs better under heavy-tailed distributions followed by skewed distributions. For instance, when λ = 0.25, the DHWMA scheme yields EARL_(0.7,1.5] values of 3.08, 2.65 and 2.51 under the N(0,1), t(5) and GAM(3,1) distributions, respectively. This conclusion is also true in terms of the overall performance when δ ∈ (0,1.5]; see Table 3 in terms of the EARL_(0,1.5] values.

Fig 3 shows that as the phase II sample size n increases, the better the proposed DHWMA W scheme becomes in detecting shifts in the process mean. Note that as the value of λ increases, the proposed scheme becomes less sensitive. This is revealed by larger EARL and ESDRL values for large values of λ and small EARL and ESDRL values for small values of λ. Fig 4 shows that the larger the phase I sample size m, the more sensitive the DHWMA W scheme becomes–except for cases where λ is small under skewed distributions, see Fig 4A. As λ increases, Fig 4 also reveals that the DHWMA W scheme becomes less sensitive since the EARL and ESDRL values get larger. Note that while increasing m and/or n helps to increase the sensitivity of the proposed scheme, it also increases the costs of inspection and implementation; moreover, large samples are not easy to find. Therefore, a more reasonable sample size must be chosen in order to strike a balance between the sensitivity of the proposed scheme and the cost related to its implementation.

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Fig 3. Effect of the phase II sample size on the performance of DHWMA W scheme when m = 100, λ ∈ {0.05, 0.25} and n ∈ {3, 5, 10} under N(0,1), t(5) and GAM(3,1) distributions.

https://doi.org/10.1371/journal.pone.0261217.g003

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Fig 4. Effect of the phase I sample size on the performance of DHWMA W scheme when n = 5, λ ∈ {0.05, 0.25} and m ∈ {50, 100,500} under N(0,1), t(5) and GAM(3,1) distributions.

https://doi.org/10.1371/journal.pone.0261217.g004

4.2.3 Performance of the proposed HHWMA W scheme.

Table 4 displays the IC and OOC ARL profiles of the proposed HHWMA W scheme under N(0,1), t(5) and GAM(3,1) distributions when n = 5, m = 100, λ₁ ∈{0.1, 0.2} and λ₂ ∈{0.05,0.25,0.5} for a nominal ARL₀ = 500. From Table 4, it can be seen that when λ₁ = 0.1 and λ₂ = 0.05, 0.25 and 0.5, the proposed HHWMA W scheme yields attained ARL₀ values of (498.09, 502.89, 498.16), (494.78, 497.08, 508.29) and (509.48, 498.9, 496.86) under the N(0,1), t(5) and GAM(3,1) distributions, respectively. Also, it can be seen that when λ₁ = 0.2 and λ₂ = 0.05, 0.25 and 0.5, the proposed HHWMA W scheme yields attained ARL₀ values of (503.16, 506.08, 499.80), (502.44, 493.90, 499.70) and (502.86, 502.90, 505.40) under the N(0,1), t(5) and GAM(3,1) distributions, respectively. Thus, it can be observed that in both cases the attained ARL₀ values are much closer to the nominal ARL₀ value of 500 across different continuous probability distributions. Therefore, it can be concluded that the HHWMA W scheme is IC robust.

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Table 4. The ARL profile of the HHWMA W scheme when (m,n) = (100,5), λ₁ ∈ {0.1,0.2} and λ₂ ∈ {0.05,0.25,0.5} for a nominal ARL₀ = 500 under N(0,1), t(5) and GAM(3,1) distributions.

https://doi.org/10.1371/journal.pone.0261217.t004

Since the proposed HHWMA W scheme is confirmed to be IC robust, it is now fair enough to evaluate its performance under different distributions. When λ₁ is kept constant, as λ₂ increases, the width of the control limits get wider; however, the sensitivity of the HHWMA W scheme in terms of the ARL values decreases. For small shifts in the process mean, the sensitivity of the HHWMA W scheme decreases as λ₂ increases. However, the converse is true for moderate shifts. For instance, under the t(5) distribution, when λ₁ = 0.1 and λ₂ = 0.05, 0.25 and 0.5, for small shifts, the EARL_(0,0.7] is equal to 52.51, 54.65 and 64.24, respectively; whereas, for moderate shifts, the EARL_(0.7,1.5] is equal to 2.63, 2.11 and 1.39, respectively. This shows that for moderate shifts in the process mean, there is an increase in the sensitivity of the HHWMA W scheme as λ₂ increases. For small-to-moderate as well as from small-to-large shifts, the smaller the value of λ₂, the higher the sensitivity of the proposed HHWMA W scheme. From Table 4, it can also be seen that for small shifts in the process mean, the proposed HHWMA W scheme performs better under heavy-tailed distributions followed by skewed distributions. For instance, when λ₁ = 0.1 and λ₂ = 0.05, the HHWMA scheme yields EARL_(0,0.7] values of 69.68, 52.51 and 58.86 under the N(0,1), t(5) and GAM(3,1) distributions, respectively. However, for moderate shifts in the process mean, regardless of the magnitude of λ₂, the proposed HHWMA W scheme performs better under skewed distributions followed by heavy-tailed distributions when λ₁ is kept small. For instance, when λ₁ = 0.1 and λ₂ = 0.25, the HHWMA W scheme yields EARL_(0.7,1.5] values of 2.56, 2.11 and 1.73 under the N(0,1), t(5) and GAM(3,1) distributions, respectively. However, as λ₁ increases, the HHWMA W scheme performs better under heavy-tailed distributions followed by symmetric ones. This conclusion is also true in terms of the overall performance when δ ∈ (0,1.5] regardless of the values of λ₁ and λ₂; see Table 4 performance in terms of the EARL_(0,1.5] values.

Fig 5 shows that as the phase II sample size increases, the more sensitive the proposed HHWMA W scheme becomes in detecting shifts in the process mean. For a given value of λ₁, as the value of λ₂ increases, the proposed scheme becomes less sensitive. This is revealed by larger EARL and ESDRL values for large values of λ₂ and smaller EARL and ESDRL values for small values of λ₂. Fig 6 shows that the larger the phase I sample size m, the more sensitive the HHWMA W scheme becomes. Fig 6 also reveals that the HHWMA W scheme becomes less sensitive as λ₂ increases. For instance, under the N(0,1) distribution, when m = 500, λ₁ = 0.1, λ₂ = 0.05 and 0.25, the EARL_(0,1.5] = 57.42 and 63.37, respectively. A similar pattern is observed in terms of the ESDRL values.

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Fig 5. Effect of the phase II sample size on the performance of HHWMA W scheme when m = 100, λ₁ = 0.1, λ₂ ∈ {0.05, 0.25} and n ∈ {3, 5, 10} under N(0,1), t(5) and GAM(3,1) distributions.

https://doi.org/10.1371/journal.pone.0261217.g005

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Fig 6. Effect of the phase I sample size on the performance of HHWMA W scheme when n = 5, λ₁ = 0.1, λ₂ ∈ {0.05, 0.25} and m ∈ {50, 100,500} under N(0,1), t(5) and GAM(3,1) distributions.

https://doi.org/10.1371/journal.pone.0261217.g006

4.3 Performance comparison study

In this section, the proposed HWMA, DHWMA and HHWMA W schemes are compared to some existing memory-type monitoring schemes competitors, that is, the CUSUM and EWMA W schemes of [5] and the DEWMA W scheme of [49]. The comparison is made when m = 100, n = 5 and λ = 0.5 and for HHWMA W (λ₁,₂) = (0.1, 0.5) with shifts increasing by 0.1 from 0.1 to 1.5 under N(0,1), t(5) and GAM(3,1) distributions. Tables 5 to 7 present the OOC ARL performances of the competing schemes for specific shifts and overall performances along with their corresponding control limits constants. To study the overall performance of the competing charts for small, moderate and large shifts, the EARL values were calculated using Eq (16) where EARL_(0,0.7], EARL_(0.7,1.5] and EARL_(0,1.5] values correspond to small, moderate and small-to-moderate shifts, respectively. In Tables 5–7, the EARL results of the most effective scheme are boldfaced.

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Table 5. The OOC ARL profiles of the proposed HWMA, DHWMA and HHWMA W schemes against the CUSUM, EWMA and DEWMA W schemes when (m,n,λ) = (100,5,0.5) for a nominal ARL₀ = 500 under N(0,1) distribution.

https://doi.org/10.1371/journal.pone.0261217.t005

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Table 6. The OOC ARL profiles of the proposed HWMA, DHWMA and HHWMA W schemes against the CUSUM, EWMA and DEWMA W schemes when (m,n,λ) = (100,5,0.5) for a nominal ARL₀ = 500 under t(5) distribution.

https://doi.org/10.1371/journal.pone.0261217.t006

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Table 7. The OOC ARL profiles of the proposed HWMA, DHWMA and HHWMA W schemes against the CUSUM, EWMA and DEWMA W schemes when (m,n,λ) = (100,5,0.5) for a nominal ARL₀ = 500 under GAM(3,1) distribution.

https://doi.org/10.1371/journal.pone.0261217.t007

Table 5 shows that under the N(0,1) distribution, the EARL_(0,0.7] = 100.89, 96.98, 94.90, 93.31, 86.92 and 80.19 for the CUSUM, EWMA, DEWMA, HWMA, DHWMA and HHWMA W schemes, respectively. This reveals that the HHWMA W scheme performs better compared to other competing schemes for small shifts. For the moderate shifts, it can be seen that the HHWMA W scheme outperforms the competing schemes and it is followed by the HWMA W scheme. For instance, the EARL_(0.7,1.5] are 1.64 and 2.62 for the HHWMA and HWMA W schemes, respectively, while the ones for other competing schemes are greater than 2.70. For small-to-moderate shifts, the EARL_(0,1.5] = 48.54, 46.90, 45.80, 44.94, 42.07 and 38.30 for the CUSUM, EWMA, DEWMA, HWMA, DHWMA and HHWMA W schemes, respectively, which reveals that the HHWMA W scheme is superior over all competing schemes and it is followed by the DHWMA and HWMA W schemes in that order. Table 6 presents the OOC performance comparison of the competing schemes under the t(5) distribution. Similar findings to the ones found for the comparison under the N(0,1) distribution is also observed under the t(5) distribution. In a nutshell, the HHWMA W scheme outperforms all competing schemes considered in this paper regardless of the magnitude (or size) of shifts, i.e. small, moderate, large (not shown here to conserve space) and small-to-moderate. For small shifts as well as for small-to-moderate shifts, the HHWMA W scheme performs better followed by the DHWMA W scheme. For moderate shifts, the HHWMA W scheme is superior over the competing schemes followed by the HWMA W scheme. Table 7 displays the OOC performance comparison of the competing schemes under the GAM(3,1) distribution. From Table 7, it can be observed that the HHWMA W scheme outperforms all competing schemes considered in this paper regardless of the magnitude of the shift in the process mean followed by the HWMA W scheme.

5. Illustrative example

In this section, the application of the proposed HWMA, DHWMA and HHWMA W monitoring schemes is illustrated using mining data to monitor the percentage of the silicon dioxide in iron ore used by [37]. High concentration of silica is a sign of contamination or impurity and as a result, it is undesirable. Consequently, it is crucial to incessantly monitor the silica concentration in iron ore on the flotation process in order to manage and resolve any peculiarities that may occur. On that note, this paper uses the proposed monitoring schemes to control the level of the percentage of silica in iron ore. The iron ore mining data used in this section consists of two subsets considered as phase I and phase II samples. In this example, it is assumed that a sample of size 5 is drawn every hour. The chi-square test of normality show that the iron ore data is not normally distributed at 5% level of significance (p-value = 0.00). Thus, nonparametric schemes are good choice to monitor these data. The phase I data consist of 104 samples of size 5 (m = 520) collected when the process was assumed to be IC. In phase II, 78 samples of size 5 each (i.e., n = 5) are monitored. The proposed HWMA and DHWMA W schemes are implemented using λ = 0.5 and it was found that L_H = 2.9069 and L_DH = 2.0095, respectively, for a nominal ARL₀ = 500. The HHWMA W scheme is implemented using λ₁ = 0.75 and λ₂ = 0.5 and it was found that L_HH = 2.1171 so that the attained ARL₀ is much closer to the nominal ARL₀ = 500. The plots of the proposed HWMA, DHWMA and HHWMA W monitoring schemes are shown in Fig 7A–7C. Table 8 presents the charting statistics and control limits as well as the signalling and non-signalling events of the proposed monitoring schemes when (m,n) = (520,5) for a nominal ARL₀ = 500. The control limits of the proposed HWMA, DHWMA and HHWMA W schemes are denoted as (LCL_t, LCL_t), (DLCL_t, DLCL_t) and (HLCL_t, HLCL_t), respectively, and the OOC signals are boldfaced in Table 8. The findings in both Fig 7 and Table 8 show that the HHWMA W scheme gives a signal for the first time on the 8^th subgroup in the prospective phase, while the DHWMA and HWMA W schemes give a signal for the first time on the 14^th subgroup in the prospective phase. This example confirms the superiority of the proposed HHWMA W scheme over the HWMA and DHWMA W schemes (see Fig 7 and Table 8).

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Fig 7. HWMA, DHWMA and HHWMA monitoring schemes of the iron ore date when m = 520, n = 5, λ = 0.5, λ₁ = 0.75 and λ₂ = 0.5 for and nominal ARL₀ value of 500.

https://doi.org/10.1371/journal.pone.0261217.g007

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Table 8. Illustration example of the proposed HWMA, DHWMA and HHWMA W charts using the mining iron ores data with (m,n,λ,λ₁,λ₂) = (520, 5, 0.5, 0.75, 0.5) for a nominal ARL₀ = 500.

https://doi.org/10.1371/journal.pone.0261217.t008

6. Conclusion

This paper introduced the HWMA, DHWMA and HHWMA monitoring schemes based on the Wilcoxon rank sum W statistic. The DHWMA W (HHWMA W) scheme is the extension of the HWMA W scheme where the same (different) smoothing parameter(s) are applied twice. The performances of the proposed schemes in terms of the RL characteristics were evaluated using the Monte Carlo simulations in SAS®9.4/IML15.4 with 20000 simulations. It was found that the proposed schemes are IC robust and present very interesting properties regardless of the nature of the underlying probability distribution. The HHWMA W scheme is preferred over the HWMA and DHWMA W schemes because of its flexibility and sensitivity in monitoring shifts of different sizes. In terms of the EARL and ESDRL profile, regardless of the value of smoothing parameters, the larger the phase I or/and sample size, the better the overall performance. Therefore, a balance must be strike in order to choose rational sample sizes to avoid high implementation and inspection costs. Compared to the existing CUSUM, EWMA and DEWMA W schemes in zero-state, it is observed that the proposed HHWMA scheme outperforms the existing schemes considered in this paper regardless of the magnitude of the shift in the process mean parameter.

From the results of this study, we recommend that quality engineers should put the proposed monitoring schemes into use when monitoring the location parameter regardless of the nature of the probability distribution. For future research, researchers who are interested in the design of efficient and robust schemes can look at the design of the HWMA, DHWMA and HHWMA W schemes with fast initial response features. Also, other complex extended HWMA monitoring schemes based on nonparametric statistics can be studied, such as those in [42]. Researchers are also advised to study more advanced schemes based on multi-objective cluster head selection using fitness averaged rider optimization algorithm for IoT networks in smart cities, see for instance [52]. Finally, given some concerns raised in [53], a steady-state mode performances of the proposed schemes need to be examined.

Supporting information

S1 Appendix. Properties of the HWMA W scheme.

https://doi.org/10.1371/journal.pone.0261217.s001

(DOCX)

S2 Appendix. Properties of the DHWMA W scheme.

https://doi.org/10.1371/journal.pone.0261217.s002

(DOCX)

S3 Appendix. Properties of the HHWMA W chart.

https://doi.org/10.1371/journal.pone.0261217.s003

(DOCX)

References

1. Montgomery DC. Introduction to Statistical Quality Control (8th ed.), New Jersey, Hoboken: John Wiley & Sons, 2019.
2. Roberts SW. Control chart tests based on geometric moving averages. Technometrics. 1959; 1(3):239–250.
- View Article
- Google Scholar
3. Page ES. Continuous inspection schemes. Biometrika. 1954; 41(1/2):100–115.
- View Article
- Google Scholar
4. Shamma SE, Shamma AK. Development and evaluation of control charts using double exponentially weighted moving averages. International Journal of Quality & Reliability Management. 1992; 9(6):18–26.
- View Article
- Google Scholar
5. Li SY, Tang LC, Ng SH. Nonparametric CUSUM and EWMA control charts for detecting mean shifts. Journal of Quality Technology. 2010; 42(2):209–226.
- View Article
- Google Scholar
6. Hawkins DM, Wu Q. The CUSUM and the EWMA head-to-head. Quality Engineering. 2014; 26(2):215–222.
- View Article
- Google Scholar
7. Malela-Majika J-C, Rapoo E. Distribution-free cumulative sum and exponentially weighted moving average control charts based on the Wilcoxon rank-sum statistic using ranked set sampling for monitoring mean shifts. Journal of Statistical Computation and Simulation. 2016; 86(18):3715–3734.
- View Article
- Google Scholar
8. Riaz M, Abid M, Nazir HZ, Abbasi SA. An enhanced nonparametric EWMA sign control chart using sequential mechanism. PLoS ONE. 2019; 14(11):e0225330. pmid:31751403
- View Article
- PubMed/NCBI
- Google Scholar
9. Adeoti OA, Malela-Majika J-C. Double exponentially weighted moving average control chart with supplementary runs-rules. Quality Technology & Quantitative Management. 2020; 17(2):149–172.
- View Article
- Google Scholar
10. Lima-Filho LMA, Pereira TL, Souza TC, Bayer FM. Process monitoring using inflated beta regression control chart. PLoS ONE. 2020; 15(7):e0236756. pmid:32730316
- View Article
- PubMed/NCBI
- Google Scholar
11. Aslam M, Shafqat A, Albassam M, Malela-Majika J-C, Shongwe SC. A new CUSUM control chart under uncertainty with applications in petroleum and meteorology. PLoS ONE. 2021; 16(2):e0246185. pmid:33539442
- View Article
- PubMed/NCBI
- Google Scholar
12. Yeganeh A, Shadman A, Triantafyllou IS, Shongwe SC, Abbasi SA. Run rules-based EWMA charts for efficient monitoring of profile parameters. IEEE Access. 2021; 9:38503–38521.
- View Article
- Google Scholar
13. Inkelas M, Blair C, Furukawa D, Manuel VG, Malenfant JH, Martin E, et al. Using control charts to understand community variation in COVID-19. PLoS ONE. 2021; 16(4):e0248500. pmid:33930013
- View Article
- PubMed/NCBI
- Google Scholar
14. Abbas N. Homogeneously weighted moving average control chart with an application in substrate manufacturing process. Computers & Industrial Engineering. 2018; 120:460–470.
- View Article
- Google Scholar
15. Adegoke NA, Smith ANH, Anderson MJ, Sanusi RA, Pawley MDM. Efficient homogeneously weighted moving average chart for monitoring process mean using an auxiliary variable. IEEE Access. 2019; 7:94021–94032.
- View Article
- Google Scholar
16. Nawaz T, Han D. Monitoring the process location by using new ranked set sampling-based memory control charts. Quality Technology & Quantitative Management. 2020; 17(3):255–284.
- View Article
- Google Scholar
17. Riaz M, Abbasi SA, Abid M, Hamzat AK. A new HWMA dispersion control chart with an application to wind farm data. Mathematics. 2020; 8(12):2136.
- View Article
- Google Scholar
18. Dawod AB, Adegoke NA, Abbasi SA. (2020). Efficient linear profile schemes for monitoring bivariate correlated processes with applications in the pharmaceutical industry. Chemometrics and Intelligent Laboratory Systems. 2020, 206:104137.
- View Article
- Google Scholar
19. Thanwane M, Malela-Majika J-C, Castagliola P, Shongwe SC. The effect of measurement errors on the performance of the homogenously weighted moving average monitoring scheme. Transactions of the Institute of Measurement and Control. 2021; 43(3):728–745.
- View Article
- Google Scholar
20. Thanwane M, Abbasi SA, Malela-Majika J-C, Aslam M, Shongwe SC. The use of fast initial response features on the homogeneously weighted moving average chart with estimated parameters under the effect of measurement errors. Quality and Reliability Engineering International. 2021; 37(6):2568–2586.
- View Article
- Google Scholar
21. Abbasi SA, Nassa SH, Aldosari MM, Adeoti OA. Efficient homogeneously weighted dispersion control charts with an application to distillation process. Quality and Reliability Engineering International. 2021; 37(8):3221–3241.
- View Article
- Google Scholar
22. Zhang L, Chen G. An extended EWMA mean chart. Quality Technology & Quantitative Management. 2005; 2(1):39–52.
- View Article
- Google Scholar
23. Haq A, Ejaz S, Khoo MBC. A new double EWMA-t chart for process mean. Communications in Statistics-Simulation and Computation. 2020;
- View Article
- Google Scholar
24. Haq A. A new hybrid exponentially weighted moving average control chart for monitoring process mean. Quality and Reliability Engineering International. 2013; 29(7):1015–1025.
- View Article
- Google Scholar
25. Haq A. A new hybrid exponentially weighted moving average control chart for monitoring process mean: discussion. Quality and Reliability Engineering International. 2017; 33(7):1629–1631.
- View Article
- Google Scholar
26. Abid M, Shabbir A, Nazir HZ, Sherwani RAK, Riaz M. A double homogeneously weighted moving average control chart for monitoring of the process mean. Quality and Reliability Engineering International. 2020; 36(5):1513–1527.
- View Article
- Google Scholar
27. Adeoti OA, Koleoso SO. A hybrid homogeneously weighted moving average control chart for process monitoring. Quality and Reliability Engineering International. 2020; 36(6):2170–2186.
- View Article
- Google Scholar
28. Alevizakos V, Chatterjee K, Koukouvinos C. The extended homogeneously weighted moving average control chart. Quality and Reliability Engineering International. 2021; 37(5):2134–2155.
- View Article
- Google Scholar
29. Malela-Majika J-C, Shongwe SC, Adeoti OA. A hybrid homogeneously weighted moving average control chart for process monitoring: Discussion. Quality and Reliability Engineering International. 2021; 37(8):3314–3322.
- View Article
- Google Scholar
30. Raza MA, Nawaz T, Han D. On designing distribution-free homogeneously weighted moving average control charts. Journal of Testing and Evaluation. 2020; 48(4):3154–3171.
- View Article
- Google Scholar
31. Riaz M, Abid M, Shabbir A, Nazir HZ, Abbas Z, Abbasi SA. A non-parametric double homogeneously weighted moving average control chart under sign statistic. Quality and Reliability Engineering International. 2021; 37(4):1544–1560.
- View Article
- Google Scholar
32. Alevizakos V, Chatterjee K, Koukouvinos C. The extended nonparametric homogeneously weighted moving average sign control chart. Quality and Reliability Engineering International. 2021; 37(8):3395–3416.
- View Article
- Google Scholar
33. Chakraborti S, Graham MA. Nonparametric (distribution-free) control charts: An updated overview and some results. Quality Engineering. 2019; 31(4):523–544.
- View Article
- Google Scholar
34. Wilcoxon F. Individual comparisons by ranking methods. Biometry Bulletin. 1945; 1:80–83.
- View Article
- Google Scholar
35. Tercero-Gomez V, Aguilar-Lleyda V, Cordero-Franco A, Conover W. A distribution-free CUSUM chart for joint monitoring of location and scale based on the combination of Wilcoxon and Mood statistics. Quality and Reliability Engineering International. 2020; 36(4):1422–1453.
- View Article
- Google Scholar
36. Chong ZL, Mukherjee A, Khoo MBC. Some simplified Shewhart-type distribution-free joint monitoring schemes and its application in monitoring drinking water turbidity. Quality Engineering. 2020; 32(1):91–110.
- View Article
- Google Scholar
37. Mukherjee A, Chong ZL, Khoo MBC. Comparisons of some distribution-free CUSUM and EWMA schemes and their applications in monitoring impurity in mining process flotation. Computers & Industrial Engineering. 2019; 137:106059.
- View Article
- Google Scholar
38. Triantafyllou IS, Ram M. Nonparametric EWMA-type control charts for monitoring industrial processes: An Overview. International Journal of Mathematical, Engineering and Management Sciences. 2021; 6(3):708–751.
- View Article
- Google Scholar
39. Mabude K, Malela-Majika J-C, Shongwe SC. A new distribution-free generally weighted moving average monitoring scheme for detecting unknown shifts in the process location. International Journal of Industrial Engineering Computations. 2020; 11(2):235–254.
- View Article
- Google Scholar
40. Letshedi TI, Malela-Majika J-C, Castagliola P, Shongwe SC. Distribution-free triple EWMA control chart for monitoring the process location using the Wilcoxon rank-sum statistic with fast initial response feature. Quality and Reliability Engineering International. 2021; 37(5):1996–2013.
- View Article
- Google Scholar
41. Adeoti OA, Malela-Majika JC, Shongwe SC, Aslam M. A homogeneously weighted moving average control chart for Conway–Maxwell Poisson distribution, Journal of Applied Statistics. 2021;
- View Article
- Google Scholar
42. Chan KM, Mukherjee A, Chong ZL, Lee HC. Distribution-free double exponentially and homogeneously weighted moving average Lepage schemes with an application in monitoring exit rate. Computers & Industrial Engineering. 2021; 161: 107370.
- View Article
- Google Scholar
43. Abbas N, Riaz M, Ahmad S, Abid M, Zaman B. On the efficient monitoring of multivariate processes with unknown parameters. Mathematics. 2020; 8(5), 823, https://doi.org/10.3390/math8050823.
- View Article
- Google Scholar
44. Abid M, Mei S, Nazir HZ, Riaz M, Hussain S. A mixed HWMA-CUSUM mean chart with an application to manufacturing process. Quality and Reliability Engineering International. 2020; 37(2), 618–631.
- View Article
- Google Scholar
45. Adegoke NA, Riaz M, Ganiyu KO, Abbasi SA. One-Sided and Two One-Sided Multivariate Homogeneously Weighted Moving Charts for Monitoring Process Mean; IEEE Access. 2021; 9, 80388–80404, https//doi.org/10.1109/ACCESS.2021.3085349.
- View Article
- Google Scholar
46. Thanwane M, Shongwe SC, Aslam M, Malela-Majika J-C, Albassam M. A homogenously weighted moving average scheme for observations under the effect of serial dependence and measurement inaccuracy. International Journal of Industrial Engineering Computations. 2021; 12(4), 401–414.
- View Article
- Google Scholar
47. Thanwane M, Malela-Majika JC, Castagliola P, Shongwe S. The effect of measurement errors on the performance of the homogenously weighted moving average monitoring scheme with estimated parameters. Journal of Statistical Computation and Simulation. 2021; 91(7):1306–1330.
- View Article
- Google Scholar
48. Thanwane M, Shongwe SC, Malela-Majika JC, Aslam M. Parameter estimation effect of the HWMA scheme to monitor the process mean under the combined effect of autocorrelation and measurement errors. IEEE Access. 2020; 8:221352–221366.
- View Article
- Google Scholar
49. Malela-Majika J-C. New distribution-free memory-type control charts based on the Wilcoxon rank-sum statistic. Quality Technology & Quantitative Management, 2021; 18(2):135–155.
- View Article
- Google Scholar
50. Teoh WL, Khoo MBC, Castagliola P, Lee MH. The exact run length distribution and design of the Shewhart chart with estimated parameters based on median run length. Communications in Statistics-Simulation and Computation. 2016; 45(6):2081–2103.
- View Article
- Google Scholar
51. Ou Y, Wu Z, Tsung F. A comparison study of effectiveness and robustness of control charts for monitoring process mean. International Journal of Production Economics. 2012; 135(1):479–490.
- View Article
- Google Scholar
52. Alazab M, Lakshmanna K, Reddy GT, Pham Q-V, Maddikunta PKR. Multi-objective cluster head selection using fitness averaged rider optimization algorithm for IoT networks in smart cities. Sustainable Energy Technologies and Assessments. 2021; 43: 100973.
- View Article
- Google Scholar
53. Knoth S, Tercero-Gómez VG, Khakifirooz M, Woodall WH. The impracticality of homogeneously weighted moving average and progressive mean control chart approaches. Quality Reliability Engineering International. 2021; 37(8):3779–3794.
- View Article
- Google Scholar

[ref1] 1. Montgomery DC. Introduction to Statistical Quality Control (8th ed.), New Jersey, Hoboken: John Wiley & Sons, 2019.

[ref2] 2. Roberts SW. Control chart tests based on geometric moving averages. Technometrics. 1959; 1(3):239–250.
View Article
Google Scholar

[3] View Article

[4] Google Scholar

[ref3] 3. Page ES. Continuous inspection schemes. Biometrika. 1954; 41(1/2):100–115.
View Article
Google Scholar

[6] View Article

[7] Google Scholar

[ref4] 4. Shamma SE, Shamma AK. Development and evaluation of control charts using double exponentially weighted moving averages. International Journal of Quality & Reliability Management. 1992; 9(6):18–26.
View Article
Google Scholar

[9] View Article

[10] Google Scholar

[ref5] 5. Li SY, Tang LC, Ng SH. Nonparametric CUSUM and EWMA control charts for detecting mean shifts. Journal of Quality Technology. 2010; 42(2):209–226.
View Article
Google Scholar

[12] View Article

[13] Google Scholar

[ref6] 6. Hawkins DM, Wu Q. The CUSUM and the EWMA head-to-head. Quality Engineering. 2014; 26(2):215–222.
View Article
Google Scholar

[15] View Article

[16] Google Scholar

[ref7] 7. Malela-Majika J-C, Rapoo E. Distribution-free cumulative sum and exponentially weighted moving average control charts based on the Wilcoxon rank-sum statistic using ranked set sampling for monitoring mean shifts. Journal of Statistical Computation and Simulation. 2016; 86(18):3715–3734.
View Article
Google Scholar

[18] View Article

[19] Google Scholar

[ref8] 8. Riaz M, Abid M, Nazir HZ, Abbasi SA. An enhanced nonparametric EWMA sign control chart using sequential mechanism. PLoS ONE. 2019; 14(11):e0225330. pmid:31751403
View Article
PubMed/NCBI
Google Scholar

[21] View Article

[22] PubMed/NCBI

[23] Google Scholar

[ref9] 9. Adeoti OA, Malela-Majika J-C. Double exponentially weighted moving average control chart with supplementary runs-rules. Quality Technology & Quantitative Management. 2020; 17(2):149–172.
View Article
Google Scholar

[25] View Article

[26] Google Scholar

[ref10] 10. Lima-Filho LMA, Pereira TL, Souza TC, Bayer FM. Process monitoring using inflated beta regression control chart. PLoS ONE. 2020; 15(7):e0236756. pmid:32730316
View Article
PubMed/NCBI
Google Scholar

[28] View Article

[29] PubMed/NCBI

[30] Google Scholar

[ref11] 11. Aslam M, Shafqat A, Albassam M, Malela-Majika J-C, Shongwe SC. A new CUSUM control chart under uncertainty with applications in petroleum and meteorology. PLoS ONE. 2021; 16(2):e0246185. pmid:33539442
View Article
PubMed/NCBI
Google Scholar

[32] View Article

[33] PubMed/NCBI

[34] Google Scholar

[ref12] 12. Yeganeh A, Shadman A, Triantafyllou IS, Shongwe SC, Abbasi SA. Run rules-based EWMA charts for efficient monitoring of profile parameters. IEEE Access. 2021; 9:38503–38521.
View Article
Google Scholar

[36] View Article

[37] Google Scholar

[ref13] 13. Inkelas M, Blair C, Furukawa D, Manuel VG, Malenfant JH, Martin E, et al. Using control charts to understand community variation in COVID-19. PLoS ONE. 2021; 16(4):e0248500. pmid:33930013
View Article
PubMed/NCBI
Google Scholar

[39] View Article

[40] PubMed/NCBI

[41] Google Scholar

[ref14] 14. Abbas N. Homogeneously weighted moving average control chart with an application in substrate manufacturing process. Computers & Industrial Engineering. 2018; 120:460–470.
View Article
Google Scholar

[43] View Article

[44] Google Scholar

[ref15] 15. Adegoke NA, Smith ANH, Anderson MJ, Sanusi RA, Pawley MDM. Efficient homogeneously weighted moving average chart for monitoring process mean using an auxiliary variable. IEEE Access. 2019; 7:94021–94032.
View Article
Google Scholar

[46] View Article

[47] Google Scholar

[ref16] 16. Nawaz T, Han D. Monitoring the process location by using new ranked set sampling-based memory control charts. Quality Technology & Quantitative Management. 2020; 17(3):255–284.
View Article
Google Scholar

[49] View Article

[50] Google Scholar

[ref17] 17. Riaz M, Abbasi SA, Abid M, Hamzat AK. A new HWMA dispersion control chart with an application to wind farm data. Mathematics. 2020; 8(12):2136.
View Article
Google Scholar

[52] View Article

[53] Google Scholar

[ref18] 18. Dawod AB, Adegoke NA, Abbasi SA. (2020). Efficient linear profile schemes for monitoring bivariate correlated processes with applications in the pharmaceutical industry. Chemometrics and Intelligent Laboratory Systems. 2020, 206:104137.
View Article
Google Scholar

[55] View Article

[56] Google Scholar

[ref19] 19. Thanwane M, Malela-Majika J-C, Castagliola P, Shongwe SC. The effect of measurement errors on the performance of the homogenously weighted moving average monitoring scheme. Transactions of the Institute of Measurement and Control. 2021; 43(3):728–745.
View Article
Google Scholar

[58] View Article

[59] Google Scholar

[ref20] 20. Thanwane M, Abbasi SA, Malela-Majika J-C, Aslam M, Shongwe SC. The use of fast initial response features on the homogeneously weighted moving average chart with estimated parameters under the effect of measurement errors. Quality and Reliability Engineering International. 2021; 37(6):2568–2586.
View Article
Google Scholar

[61] View Article

[62] Google Scholar

[ref21] 21. Abbasi SA, Nassa SH, Aldosari MM, Adeoti OA. Efficient homogeneously weighted dispersion control charts with an application to distillation process. Quality and Reliability Engineering International. 2021; 37(8):3221–3241.
View Article
Google Scholar

[64] View Article

[65] Google Scholar

[ref22] 22. Zhang L, Chen G. An extended EWMA mean chart. Quality Technology & Quantitative Management. 2005; 2(1):39–52.
View Article
Google Scholar

[67] View Article

[68] Google Scholar

[ref23] 23. Haq A, Ejaz S, Khoo MBC. A new double EWMA-t chart for process mean. Communications in Statistics-Simulation and Computation. 2020;
View Article
Google Scholar

[70] View Article

[71] Google Scholar

[ref24] 24. Haq A. A new hybrid exponentially weighted moving average control chart for monitoring process mean. Quality and Reliability Engineering International. 2013; 29(7):1015–1025.
View Article
Google Scholar

[73] View Article

[74] Google Scholar

[ref25] 25. Haq A. A new hybrid exponentially weighted moving average control chart for monitoring process mean: discussion. Quality and Reliability Engineering International. 2017; 33(7):1629–1631.
View Article
Google Scholar

[76] View Article

[77] Google Scholar

[ref26] 26. Abid M, Shabbir A, Nazir HZ, Sherwani RAK, Riaz M. A double homogeneously weighted moving average control chart for monitoring of the process mean. Quality and Reliability Engineering International. 2020; 36(5):1513–1527.
View Article
Google Scholar

[79] View Article

[80] Google Scholar

[ref27] 27. Adeoti OA, Koleoso SO. A hybrid homogeneously weighted moving average control chart for process monitoring. Quality and Reliability Engineering International. 2020; 36(6):2170–2186.
View Article
Google Scholar

[82] View Article

[83] Google Scholar

[ref28] 28. Alevizakos V, Chatterjee K, Koukouvinos C. The extended homogeneously weighted moving average control chart. Quality and Reliability Engineering International. 2021; 37(5):2134–2155.
View Article
Google Scholar

[85] View Article

[86] Google Scholar

[ref29] 29. Malela-Majika J-C, Shongwe SC, Adeoti OA. A hybrid homogeneously weighted moving average control chart for process monitoring: Discussion. Quality and Reliability Engineering International. 2021; 37(8):3314–3322.
View Article
Google Scholar

[88] View Article

[89] Google Scholar

[ref30] 30. Raza MA, Nawaz T, Han D. On designing distribution-free homogeneously weighted moving average control charts. Journal of Testing and Evaluation. 2020; 48(4):3154–3171.
View Article
Google Scholar

[91] View Article

[92] Google Scholar

[ref31] 31. Riaz M, Abid M, Shabbir A, Nazir HZ, Abbas Z, Abbasi SA. A non-parametric double homogeneously weighted moving average control chart under sign statistic. Quality and Reliability Engineering International. 2021; 37(4):1544–1560.
View Article
Google Scholar

[94] View Article

[95] Google Scholar

[ref32] 32. Alevizakos V, Chatterjee K, Koukouvinos C. The extended nonparametric homogeneously weighted moving average sign control chart. Quality and Reliability Engineering International. 2021; 37(8):3395–3416.
View Article
Google Scholar

[97] View Article

[98] Google Scholar

[ref33] 33. Chakraborti S, Graham MA. Nonparametric (distribution-free) control charts: An updated overview and some results. Quality Engineering. 2019; 31(4):523–544.
View Article
Google Scholar

[100] View Article

[101] Google Scholar

[ref34] 34. Wilcoxon F. Individual comparisons by ranking methods. Biometry Bulletin. 1945; 1:80–83.
View Article
Google Scholar

[103] View Article

[104] Google Scholar

[ref35] 35. Tercero-Gomez V, Aguilar-Lleyda V, Cordero-Franco A, Conover W. A distribution-free CUSUM chart for joint monitoring of location and scale based on the combination of Wilcoxon and Mood statistics. Quality and Reliability Engineering International. 2020; 36(4):1422–1453.
View Article
Google Scholar

[106] View Article

[107] Google Scholar

[ref36] 36. Chong ZL, Mukherjee A, Khoo MBC. Some simplified Shewhart-type distribution-free joint monitoring schemes and its application in monitoring drinking water turbidity. Quality Engineering. 2020; 32(1):91–110.
View Article
Google Scholar

[109] View Article

[110] Google Scholar

[ref37] 37. Mukherjee A, Chong ZL, Khoo MBC. Comparisons of some distribution-free CUSUM and EWMA schemes and their applications in monitoring impurity in mining process flotation. Computers & Industrial Engineering. 2019; 137:106059.
View Article
Google Scholar

[112] View Article

[113] Google Scholar

[ref38] 38. Triantafyllou IS, Ram M. Nonparametric EWMA-type control charts for monitoring industrial processes: An Overview. International Journal of Mathematical, Engineering and Management Sciences. 2021; 6(3):708–751.
View Article
Google Scholar

[115] View Article

[116] Google Scholar

[ref39] 39. Mabude K, Malela-Majika J-C, Shongwe SC. A new distribution-free generally weighted moving average monitoring scheme for detecting unknown shifts in the process location. International Journal of Industrial Engineering Computations. 2020; 11(2):235–254.
View Article
Google Scholar

[118] View Article

[119] Google Scholar

[ref40] 40. Letshedi TI, Malela-Majika J-C, Castagliola P, Shongwe SC. Distribution-free triple EWMA control chart for monitoring the process location using the Wilcoxon rank-sum statistic with fast initial response feature. Quality and Reliability Engineering International. 2021; 37(5):1996–2013.
View Article
Google Scholar

[121] View Article

[122] Google Scholar

[ref41] 41. Adeoti OA, Malela-Majika JC, Shongwe SC, Aslam M. A homogeneously weighted moving average control chart for Conway–Maxwell Poisson distribution, Journal of Applied Statistics. 2021;
View Article
Google Scholar

[124] View Article

[125] Google Scholar

[ref42] 42. Chan KM, Mukherjee A, Chong ZL, Lee HC. Distribution-free double exponentially and homogeneously weighted moving average Lepage schemes with an application in monitoring exit rate. Computers & Industrial Engineering. 2021; 161: 107370.
View Article
Google Scholar

[127] View Article

[128] Google Scholar

[ref43] 43. Abbas N, Riaz M, Ahmad S, Abid M, Zaman B. On the efficient monitoring of multivariate processes with unknown parameters. Mathematics. 2020; 8(5), 823, https://doi.org/10.3390/math8050823.
View Article
Google Scholar

[130] View Article

[131] Google Scholar

[ref44] 44. Abid M, Mei S, Nazir HZ, Riaz M, Hussain S. A mixed HWMA-CUSUM mean chart with an application to manufacturing process. Quality and Reliability Engineering International. 2020; 37(2), 618–631.
View Article
Google Scholar

[133] View Article

[134] Google Scholar

[ref45] 45. Adegoke NA, Riaz M, Ganiyu KO, Abbasi SA. One-Sided and Two One-Sided Multivariate Homogeneously Weighted Moving Charts for Monitoring Process Mean; IEEE Access. 2021; 9, 80388–80404, https//doi.org/10.1109/ACCESS.2021.3085349.
View Article
Google Scholar

[136] View Article

[137] Google Scholar

[ref46] 46. Thanwane M, Shongwe SC, Aslam M, Malela-Majika J-C, Albassam M. A homogenously weighted moving average scheme for observations under the effect of serial dependence and measurement inaccuracy. International Journal of Industrial Engineering Computations. 2021; 12(4), 401–414.
View Article
Google Scholar

[139] View Article

[140] Google Scholar

[ref47] 47. Thanwane M, Malela-Majika JC, Castagliola P, Shongwe S. The effect of measurement errors on the performance of the homogenously weighted moving average monitoring scheme with estimated parameters. Journal of Statistical Computation and Simulation. 2021; 91(7):1306–1330.
View Article
Google Scholar

[142] View Article

[143] Google Scholar

[ref48] 48. Thanwane M, Shongwe SC, Malela-Majika JC, Aslam M. Parameter estimation effect of the HWMA scheme to monitor the process mean under the combined effect of autocorrelation and measurement errors. IEEE Access. 2020; 8:221352–221366.
View Article
Google Scholar

[145] View Article

[146] Google Scholar

[ref49] 49. Malela-Majika J-C. New distribution-free memory-type control charts based on the Wilcoxon rank-sum statistic. Quality Technology & Quantitative Management, 2021; 18(2):135–155.
View Article
Google Scholar

[148] View Article

[149] Google Scholar

[ref50] 50. Teoh WL, Khoo MBC, Castagliola P, Lee MH. The exact run length distribution and design of the Shewhart chart with estimated parameters based on median run length. Communications in Statistics-Simulation and Computation. 2016; 45(6):2081–2103.
View Article
Google Scholar

[151] View Article

[152] Google Scholar

[ref51] 51. Ou Y, Wu Z, Tsung F. A comparison study of effectiveness and robustness of control charts for monitoring process mean. International Journal of Production Economics. 2012; 135(1):479–490.
View Article
Google Scholar

[154] View Article

[155] Google Scholar

[ref52] 52. Alazab M, Lakshmanna K, Reddy GT, Pham Q-V, Maddikunta PKR. Multi-objective cluster head selection using fitness averaged rider optimization algorithm for IoT networks in smart cities. Sustainable Energy Technologies and Assessments. 2021; 43: 100973.
View Article
Google Scholar

[157] View Article

[158] Google Scholar

[ref53] 53. Knoth S, Tercero-Gómez VG, Khakifirooz M, Woodall WH. The impracticality of homogeneously weighted moving average and progressive mean control chart approaches. Quality Reliability Engineering International. 2021; 37(8):3779–3794.
View Article
Google Scholar

[160] View Article

[161] Google Scholar

Figures

Abstract

1. Introduction

2. The proposed HWMA W monitoring scheme and its extended versions

2.1 The Wilcoxon rank sum W statistic

2.2 The HWMA W scheme

2.3 The DHWMA W scheme

2.4 The HHWMA W scheme

3. The operational procedure of the proposed HWMA, DHWMA and HHWMA W schemes

4. IC and OOC performance analyses

4.1 Performance metrics

4.2 Performance analysis

4.2.1 Performance of the proposed HWMA W scheme.

4.2.2 Performance of the proposed DHWMA W scheme.

4.2.3 Performance of the proposed HHWMA W scheme.

4.3 Performance comparison study

5. Illustrative example

6. Conclusion

Supporting information

S1 Appendix. Properties of the HWMA W scheme.

S2 Appendix. Properties of the DHWMA W scheme.

S3 Appendix. Properties of the HHWMA W chart.

References