Improved adaptive EWMA control chart for process location with applications in groundwater physicochemical parameters and glass manufacturing industry

Muhammad Arslan; Syed Masroor Anwar; Showkat Ahmad Lone; Zahid Rasheed; Majid Khan; Saddam Akbar Abbasi

doi:10.1371/journal.pone.0272584

Abstract

The adaptive exponentially weighted moving average (AEWMA) control charts are the advanced form of classical memory control charts used for efficiently monitoring small-to-large shifts in the process parameters (location and/or dispersion). These AEWMA control charts estimate the unknown shifts using exponentially weighted moving average (EWMA) or cumulative sum (CUSUM) control charts statistics. The hybrid EWMA (HEWMA) control chart is preferred over classical memory control charts to detect early shifts in process parameters. So, this study presents a new auxiliary information-based (AIB) AEWMA (IAEWMA_AIB) control chart for process location that estimates the unknown location shift using HEWMA statistic. The objective is to develop an unbiased location shift estimator using HEWMA statistic and then adaptively update the smoothing constant. The shift estimation using HEWMA statistic instead of EWMA or CUSUM statistics boosts the performance of the proposed IAEWMA_AIB control chart. The Monte Carlo simulation technique is used to get the numerical results. Famous performance evaluation measures like average run length, extra quadratic loss, relative average run length, and performance comparison index are used to evaluate the performance of the proposed chart with existing counterparts. The comparison reveals the superiority of the proposed control chart. Finally, two real-life applications from the glass manufacturing industry and physicochemical parameters of groundwater are considered to show the proposed control chart’s implementation procedure and dominance.

Citation: Arslan M, Anwar SM, Lone SA, Rasheed Z, Khan M, Abbasi SA (2022) Improved adaptive EWMA control chart for process location with applications in groundwater physicochemical parameters and glass manufacturing industry. PLoS ONE 17(8): e0272584. https://doi.org/10.1371/journal.pone.0272584

Editor: Inés P. Mariño, Universidad Rey Juan Carlos, SPAIN

Received: October 22, 2021; Accepted: July 21, 2022; Published: August 22, 2022

Copyright: © 2022 Arslan 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: All relevant data are within the manuscript and its Supporting Information files.

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

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

1: Introduction

Ever since Shewhart [1] presented a conventional Shewhart control chart, it has become a typical practice to utilize these control charts for monitoring the assignable cause variations (shifts) in different manufacturing/production processes [2]. The Shewhart control charts are referred to as memoryless control charts; these charts do not carry previous information. The key limitation of these control charts is that they generally monitor large shift sizes in the process parameters (location and/or dispersion). On the other hand, the memory control chart, like the exponentially weighted moving average (EWMA) control chart proposed by Roberts [3], is incredibly beneficial for monitoring small to moderate process shifts.

Recently, different processes demand balanced and sufficient safety against the unknown shifts of large and small sizes in industries. The adaptive control charts are often suggested to catch both Shewhart and EWMA control charts’ desirable properties and simultaneously disclose large and small shifts. In this regard, Capizzi and Masarotto [4] introduced an adaptive EWMA (AEWMA) control chart based on score (Huber’s and Tukey’s bi-square) functions, which are very effective for monitoring small and large shifts simultaneously. Likewise, Zaman et al. [5] extended the existing structure of the AEWMA control chart with CUSUM accumulation error using score functions that signal out shift more precisely than the AEWMA control chart. Similarly, Haq et al. [6] introduced the AEWMA control chart by utilizing the unbiased estimator of the process location shift through EWMA statistic, and then the EWMA control chart’s smoothing constant is adaptively updated. Also, Haq [7] introduced the auxiliary information-based (AIB) estimator AEWMA (AEWMA_AIB) control chart for the process location by estimating the unknown process location shift.

Mixing or combining the features of various control charts improves the conventional control charts’ performance. For instance, Haq [8] presented a hybrid EWMA (HEWMA) control chart by combining two EWMA statistics. The one EWMA plotting statistic serves as an input for the other EWMA control chart. The HEWMA control chart is more efficient than the classical EWMA control chart in monitoring small-to-moderate shifts in the process location. Later on, different researchers examined the HEWMA control chart with some additional features. For example, Azam et al. [9] provided a HEWMA control chart for process location under the assumption of repetitive sampling. Similarly, Aslam et al. [10] presented the HEWMA chart for Com-Poisson distribution. Likewise, Noor-ul-Amin et al. [11] demonstrated an AIB HEWMA (HEWMA_AIB) chart under the assumption of phase-II process location monitoring. Similarly, Aslam et al. [12] suggested a HEWMA-CUSUM chart by combining the HEWMA and CUSUM statistics. Recently, Anwar et al. [13] introduced AIB double homogeneously weighted moving average (AIB DHWMA) control chart to monitor process location. Also, Rasheed et al. [14] and Rasheed et al. [15] suggested mixed memory and nonparametric triple EWMA control charts for improved process location monitoring.

Over the last few years, auxiliary information has been highly esteemed for process monitoring. Many authors have studied various features of the AIB EWMA type control charts. For example, Abbas et al. [16] designed an AIB EWMA control chart. Similarly, Adegoke et al. [17] introduced an AIB EWMA control chart using different sampling schemes. Also, Haq [18] proposed nonparametric EWMA_AIB sign control chart. Later, Abbasi and Haq [19] recommended an auxiliary-based optimal and adaptive CUSUM control chart for process location. Likewise, Anwar et al. [20] and Anwar et al. [21] introduced modified-EWMA_AIB and AIB mixed control charts, respectively, for improved process location monitoring. Recently, Haq et al. [22] suggested AIB adaptive Crosier CUSUM (ACC_AIB) under fixed and variable sampling intervals and AIB adaptive EWMA (AE_AIB) control charts for the monitoring of process mean. For more details about adaptive EWMA, hybrid EWMA, and AIB memory control charts, see [23–27] and reference therein.

As mentioned before, Zaman et al. [5] suggested the enhanced AEWMA control chart that estimates the unknown location shift using CUSUM statistic instead of EWMA statistic. Inspired by the innovation in AEWMA structure of [5], we intend to present an improved AEWMA_AIB (symbolized as IAEWMA_AIB) control chart for enhanced monitoring of the process location. The proposed IAEWMA_AIB control chart uses the HEWMA statistic and an unbiased shift estimator to estimate the process location shift. Then, depending on the magnitude of the shift, it determines an appropriate value for the smoothing constant. The shift estimation with the HEWMA statistic rather than the CUSUM or EWMA statistic boosts the detection ability of the proposed IAEWMA_AIB control chart. To evaluate the performance of the proposed IAEWMA_AIB control chart against other control charts, performance evaluation measures such as average run length (ARL), extra quadratic loss (EQL), performance comparison index (PCI), and relative ARL (RARL) measures are considered. Besides, the Monte Carlo simulation method is used to calculate the performance evaluation measures. Existing control chart such as AIB CUSUM (CUSUM_AIB), EWMA_AIB, HEWMA, AEWMA_AIB, AIB improved adaptive Crosier CUSUM (IACCUSUM_AIB), ACC_AIB, AE_AIB, and mixed HWMA-CUSUM (MHC) control charts are considered for comparison. Moreover, to demonstrate the utility of the proposed IAEWMA_AIB control chart for practical importance, two real-life applications are also provided.

The remainder of the article is arranged as follows: The existing memory control charts are described in Section 2. Similarly, Section 3 illustrates the construction of the proposed IAEWMA_AIB control chart, and also special cases of the proposed IAEWMA_AIB control chart. Additionally, the performance evaluation measures and parameter choices are available in Section 4. Similarly, Section 5 provides the performance comparison of the proposed IAEWMA_AIB control chart against CUSUM_AIB, EWMA_AIB, HEWMA, AEWMA_AIB, and IACCUSUM_AIB control charts. Additionally, the real-life applications of the proposed IAEWMA_AIB control chart are provided in section 6. The last section presents the overall summary, conclusions, and recommendations.

2: Existing methods

This section provides insight into the variable of interest and AIB estimator in Subsection 2.1. Likewise, the methodologies of the classical EWMA and HEWMA control charts are presented in Subsections 2.2 and 2.3, respectively.

2.1: Variable of interest and transformation

Let be the variable of interest from a normal distribution. Let be the sample mean and be the sample standard deviation of Y. So, for the in-control (IC) situation, δ = 0, and . Let X be an auxiliary variable of Y. The variables X and Y follow a bivariate normal distribution (BND) (i.e., (Y, X)~N(μ_Y, μ_X, σ_Y, σ_X, ρ), where μ_X represents the mean and σ_X represents the standard deviation of X. Also, the ρ is the correlation coefficient corresponding to X and Y. Let (Y_it, X_it), i = 1,2,…n be a random sample of size n at time t, for t≥1. Following Haq and Khoo [28] and Haq [29], the AIB difference estimator for monitoring the process location is (1) where , Here, follows a normal distribution with mean μ_Y and the variance , that is, . Following Haq [29], we assume , then T_t~N(δ*, 1), where and known as the process mean shift. When a process is IC then T_t~N(0,1), that is, δ* = 0. Let δ = |μ_Y−μ_Y,1|/σ_Y be the standardized shift in the σ_Y units, where μ_Y,1 is OOC process mean. The process is IC for δ = 0, else, out-of-control (OOC).

2.2: Classical EWMA control chart

Roberts [3] suggested the classic EWMA control chart for monitoring small-to-moderate shifts in the process location. Based on {Y_t} the plotting statistic of the classical EWMA control chart is designed as follows: (2) where, E₀ = 0 and λ₁ (0<λ₁≤1) is the smoothing constant. If μ_Y and σ_Y are known, then the upper control limit (UCL) and lower control limit (LCL) are defined as: (3) for large t, the control limits can be written as: (4) where L denotes the control chart coefficient for a predefined false alarm rate. The EWMA statistic Z_t is plotted along UCL_t and LCL_t, the process is considered to be OOC when Z_t<LCL_t or Z_t>UCL_t; otherwise, it is IC.

2.3: HEWMA control chart

Haq [8] introduced the HEWMA control chart for process location. This control chart is constructed by using one EWMA statistic as an input for another EWMA statistic given as: (5) Here, the HE_t is called HEWMA statistic at time t. The starting values of HE_t and E₀ are assumed to be μ₀, that is HE₀ = E₀ = μ₀. Based on the statistic HE_t the LCL and UCL for HEWMA control chart are given by (6)

The control limits of the HEWMA control chart when t is very large. (7) where the L is the control chart coefficient, and it is decided in such a way that the IC ARL (ARL₀) of the HEWMA control chart is determined at a pre-specified desired level. The process is considered to be OOC when HE_t<LCL_t or HE_t>UCL_t; otherwise, it is IC.

3: Proposed methods

This section contains the methodology of the proposed IAEWMA_AIB control chart for monitoring the process location. Subsection 3.1 covers the design structure of the proposed IAEWMA_AIB control chart. Besides, the special cases of the IAEWMA_AIB control chart are given in Subsection 3.2.

3.1: Proposed IAEWMA_AIB control chart

Here, we extend the work of Haq [29] and propose IAEWMA_AIB control chart for the improved monitoring of process location. The proposed IAEWMA_AIB control chart first estimates the shift estimator δ* using HEWMA statistic and based on the estimated value of the shift estimator, appropriate values of smoothing parameters are selected to construct the proposed chart. Let is a sequence based on {T_t} for IC process and is another sequance defined on , then the estimator of using the HEWMA statistic is defined as: (8) where λ₁ and λ₂ are the smoothing constants for the HEWMA statistic. The initial values of and are zero, i.e., . The shift estimator is unbiased for the IC process and biased for the OOC process. The unbiased estimator of , even if the process is either IC or OOC, it is given by (9) where . Following Haq [29], we assumed . Based on a sequence of IID random variable {T_t}, the plotting-statistic of the proposed IAEWMA_AIB control chart is defined as: (10) where AHE₀ = 0 and is defined by (11)

Here is a smoothing multiplier of the proposed IAEWMA_AIB control chart, which is a function of the shift estimator . The control chart chooses smaller values of when is small and larger values of when is large. The proposed IAEWMA_AIB control chart detects an OOC signal when |AHE_t| exceeds than threshold says .

3.2: Special cases of IAEWMA_AIB control chart

The AEWMA and AEWMA_AIB control charts are special cases of the proposed IAEWMA_AIB control chart by considering special values of parameters. The special cases with their proofs are provided here.

Case 1: When ρ = 0 and λ₂ = 1, the proposed IAEWMA_AIB control chart tends to the AEWMA control chart.

Proof: When ρ = 0, then the difference estimator reduces to (12)

The transformed T can be written as: .

Now substitute the resulting in to obtain (13)

Also, put λ₂ = 1 in Eq (8), then the reduced as follows: (14) which is the estimated shift estimator of the existing AEWMA control chart. Based on Eq (10), the IAEWMA_AIB chart can be expressed as: (15) where can be obtained in similar ways as described in Eq (11). The statistic and shift estimator are similar to the plotting statistic and shift estimator of AEWMA proposed by Haq et al. [6] except for their notations. Hence, the statistic of the proposed IAEWMA_AIB control chart becomes the statistic of AEWMA when ρ = 0 and λ₂ = 1.

Case 2: When λ₂ = 1, the IAEWMA_AIB control chart tends to the AEWMA_AIB control chart.

Proof: When λ₂ = 1, the shift estimator in Eq (8) reduced as follows: (16)

Then the plotting statistic of the proposed control chart can be written as (17) where can be obtained in similar ways as described in Eq (11). The shift estimator and plotting statistic in Eqs (16) and (17) are similar to the shift estimator and plotting statistic of AEWMA_AIB control chart, which indicates that the proposed IAEWMA_AIB control chart reduced to the AEWMA_AIB control chart for λ₂ = 1.

4: Performance evaluation measures

This section introduces the performance evaluation measures to analyze the control charts’ performance. The Monte Carlo simulation detail is given in Subsection 4.1. Likewise, the description of the ARL is enlisted in Subsection 4.2. Similarly, the overall performance evaluation measures are defined in Subsection 4.3. The choices of parameters of the proposed IAEWMA_AIB control chart is given in Subsection 4.4.

4.1: Monte Carlo simulation

The Monte Carlo simulation procedure is regarded as a computational technique for obtaining numerical results for evaluating the performance of the proposed IAEWMA_AIB control chart. Monte Carlo simulation with 10⁵ iterations is conducted for each displacement of δ using R software to obtain the ARL and standard deviation of RL (SDRL) of the proposed control chart. The sample values of (Y_t, X_t), for t>1 are generated from BND. The shift reflected in the process location is considered here δ = 0.10, 0.20, 0.25, 0.30, 0.40, 0.50, 0.75, 1.00, 1.50, 2.00, 2.5, 3, 3.5, 4, 5, 6. The simulation algorithm of the proposed IAEWMA_AIB control chart is described as follows:

Generate a random sample from BND.
Calculate the estimator from Eq (1) and T_t.
Calculate the statistic from Eq (8) using T_t estimator.
Use the statistic as an input in .
Estimate from Eq (9) and estimate using Eq (11).
Estimate the IAEWMA_AIB statistic AHE_t from Eq (10).
Compute threshold for desired in-control ARL denoted as ARL₀.
Plot the |AHE_t| statistic against the threshold .
If , record sequence order known as run-length (RL).
Record RL_s after repeating steps (i)-(ix) 10⁵ times.
Determine the average of 10⁵ RL, which is ARL₀.
For out-of-control ARL values, considered and repeat from steps (ii)-(x).

4.2: The ARL measure

The average run length (ARL) is commonly used to evaluate a control chart’s performance at a shift. The ARL is listed as IC ARL (ARL₀) and OOC ARL (ARL₁). If a process is functioning in an in-control state, the ARL₀ is chosen to be sufficiently large to eliminate the effect of the false alarm rate. On the other hand, the ARL₁ should be small enough to detect a shift quickly. A control chart is preferred over the other competing charts if it should have a smaller ARL₁ value at predefined ARL₀ [30].

4.3: Overall performance measures

The EQL, RARL, and PCI performance evaluation measures are used to evaluate a control chart’s overall effectiveness. The EQL is the weighted average ARL over the domain of shifts with δ² as a weight [31]. It is defined as where ARL(δ) is the ARL at specific δ; δ_min and δ_max are the smallest and largest shift values of the domain, respectively. The lower the EQL value signifies, the better the performance of the control chart [20].

The RARL, like EQL, is also used to assess the efficiency of a control chart. It can be defined as follows:

The ARL(δ) is ARL of the competing control chart. The ARL*(δ) is ARL of benchmark control chart at a δ. A control chart is decided as a benchmark control chart for a smaller ARL at specific δ [21]. The RARL value of the benchmark control chart is assumed to be one. If the competing control chart has RARL>1, the benchmark control chart is more efficient than the competing one.

The PCI corresponds to the EQL ratio of the specific chart to theEQL of the benchmark chart. Here EQL* represents the EQL of the benchmark chart, whereas the EQL represents the EQL of the competing control chart. According to the [32], the PCI is presented as:

The PCI of the benchmark control chart should be one. If the PCI>1, the benchmark chart is superior to the competing control chart [20].

4.4: Effect of parameters choices

The parameters (λ₁, λ₂, and ρ) has their effects on the performance of the proposed IAEWMA_AIB control chart. Various combinations of these parameters are chosen; hence, corresponding ARL and SDRL are computed. The parameter λ₁ is set as 0.10, 0.20, and 0.50, whereas the parameter λ₂ is set as 0.10, 0.20,0.50, 0.75,0.90 to obtain ARL₀ = 500. Similarly, the values of ρ are assumed as 0.25, 0.50, 0.75, and 0.95. Tables 1–3 present the numerical results of the proposed IAEWMA_AIB chart.

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Table 1. The run-length profile of proposed IAEWMA_AIB control chart under various choices of λ₂ when λ₁ is small and ARL₀ = 500.

https://doi.org/10.1371/journal.pone.0272584.t001

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Table 2. The run-length profile of proposed IAEWMA_AIB control chart under various choices of λ₂ when λ₁ is moderate and ARL₀ = 500.

https://doi.org/10.1371/journal.pone.0272584.t002

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Table 3. The run-length profile of proposed IAEWMA_AIB control chart under various choices of λ₂ when λ₁ is large and ARL₀ = 500.

https://doi.org/10.1371/journal.pone.0272584.t003

5: Evaluation and performance comparison

This section includes comprehensive comparisons of the proposed IAEWMA_AIB control chart with CUSUM_AIB [33], EWMA_AIB [16], HEWMA [8], AEWMA_AIB [29], IACCUSUM_AIB [19], ACC_AIB and AE_AIB [22], MHC [34] control charts.

5.1: Proposed versus HEWMA control chart

The proposed IAEWMA_AIB control chart provides better performance against the HEWMA control chart for different values of ρ. For example, at λ₁ = 0.10, λ₂ = 0.20, and δ = 0.25, 0.50, the proposed IAEWMA_AIB control chart (ρ = 0.50) has the ARL₁ values 33.90, 7.67, while the HEWMA control chart has 91.64, 25.67 (see Tables 1 and 4). Furthermore, Figs 1 and 2 also demonstrate that the proposed IAEWMA_AIB control chart is superior to the HEWMA chart. In terms of overall effectiveness (see Table 5), the proposed IAEWMA_AIB control chart has smaller EQL, RARL, and PCI (i.e., 8.88, 1.00, 1.00) values against the HEWMA control chart EQL, RARL, and PCI (i.e., 12.24, 1.38, 1.92) values.

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Fig 1. ARL profile of IAEWMA_AIB, HEWMA, EWMA_AIB, AEWMA_AIB, IACCUSUM_AIB, and CUSUM_AIB control charts at ρ = 0.25, (λ, λ₂) = 0.20, λ₁ = 0.10 when ARL₀ = 500.

https://doi.org/10.1371/journal.pone.0272584.g001

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Fig 2. ARL profile of IAEWMA_AIB, HEWMA, EWMA_AIB, AEWMA_AIB, IACCUSUM_AIB, and CUSUM_AIB control charts at ρ = 0.75, (λ, λ₂) = 0.20, λ₁ = 0.10 when ARL₀ = 500.

https://doi.org/10.1371/journal.pone.0272584.g002

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Table 4. The run length profile EWMA_AIB, AEMWA_AIB, CUSUM_AIB, IACCUSUM_AIB and HEWMA control charts at ARL₀ = 500.

https://doi.org/10.1371/journal.pone.0272584.t004

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Table 5. Overall performance measures of the existing and proposed IAEWMA_AIB control charts at different values of ρ.

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5.2: Proposed versus EWMA_AIB control chart

In comparison to the EWMA_AIB control chart, the proposed IAEWMA_AIB control chart exhibits superior performance. As an illustration, at (λ, λ₁, λ₂) = 0.20, ρ = 0.5, and δ ∈ (0.25,0.5,0.75,1,1.5,2), the ARL₁ (117.88, 29.66, 12.72, 7.27, 3.59, 2.30) values of the EWMA_AIB control chart are larger than the ARL₁ (41.68, 10.01, 4.46, 2.50, 1.42, 1.14) values of the proposed IAEWMA_AIB control chart (see Tables 2 and 4). Likewise, Figs 1 and 2 displays the dominance of the proposed IAEWMA_AIB control chart over EWMA_AIB control chart. In addition, the EQL, RARL, and PCI values also show the dominance of the IAEWMA_AIB control chart against the EWMA_AIB control chart. For instance, at ρ = 0.25, the proposed IAEWMA_AIB control chart has EQL = 9.35, PCI = 1.00, and RARL = 1.00, while the EWMA_AIB control chart has EQL = 13.49, PCI = 1.44, RARL = 1.82 (see Table 5).

5.3: Proposed versus AEWMA_AIB control chart

The proposed IAEWMA_AIB control chart is superior to the AEWMA_AIB control chart. For example, if (λ, λ₁, λ₂) = 0.20, ρ = 0.50, and δ = 0.25, the ARL₁ values of the AEWMA_AIB and IAEWMA_AIB are 78.63 and 41.68, respectively (see Tables 2 and 4 & Figs 1 and 2). Similarly, the EQL, PCI, and RARL values of the proposed IAEWMA_AIB control chart reveals the edge over the AEWMA_AIB control chart. As an illustration, at ρ = 0.50, the EQL, RARL, and PCI values of the AEWMA_AIB and IAEWMA_AIB control charts are presented as (10.15, 1.14, and 1.44), and (8.88, 1.00, and 1.00), respectively (see Table 5).

5.4: Proposed versus CUSUM_AIB control chart

The proposed IAEWMA_AIB control chart provides superior performance to the CUSUM_AIB control chart. For example, at ρ = 0.75, δ = 0.25,0.50,1.00, the proposed IAEWMA_AIB control chart (λ₁ = 0.50, λ₂ = 0.90) produces ARL₁ = (42.61, 9.12, 2.14), whereas, the CUSUM_AIB control chart (k = 0.5) has ARL₁ equal to (68.99, 17.07, 8.65) (see Tables 3 and 4). Furthermore, Figs 1 and 2 highlights the superiority of the proposed IAEWMA_AIB control chart over CUSUM_AIB chart. Similarly, at a specific range of shifts, the proposed IAEWMA_AIB control chart has smaller EQL, PCI, and RARL values than the CUSUM_AIB control chart. For example, at ρ = 0.75, the EQL, PCI, and RARL values are (8.48, 1.00, 1.00) and (13.62, 1.61, 2.71) for IAEWMA_AIB and CUSUM_AIB control charts, respectively (see Table 5).

5.5: Proposed versus IACCUSUM_AIB control chart

In comparison with IACCUSUM_AIB, the proposed IAEWMA_AIB control chart detects an earlier shift. In more detail, if δ = 0.25, (λ, λ₁, λ₂) = 0.20, , and ρ = 0.25, 0.50, 0.75, the ARL₁ values of IACCUSUM_AIB and IAEWMA_AIB control chart are (81.16, 71.23, 50.69) and (63.96, 41.68, 14.10), respectively (see Tables 2 and 4). Figs 1 and 2 also demonstrate the edge of the proposed IAEWMA_AIB control chart against the IACCUSUM_AIB control chart. Similarly, at ρ = 0.75, the IACCUSUM_AIB control chart EQL, PCI, and RARL (i.e., 9.96, 1.17, and 1.96) values are larger than the EQL, PCI, and RARL (8.48, 1.00, and 1.00) values of the proposed IAEWMA_AIB control charts. (see Table 5).

5.6: Proposed versus ACC_AIB and AE_AIB control charts

The proposed IAEWMA_AIB control chart is compared with the ACC_AIB and AE_AIB control charts. The IAEWMA_AIB control chart is superior as compared to the ACC_AIB and AE_AIB control charts in terms of early shift detection. In more detail, if δ = 0.25, (λ, λ₁, λ₂) = 0.20, , and ρ = 0.50, the ARL₁ values of ACC_AIB, AE_AIB and IAEWMA_AIB control chart are 80.77,93.72, and 41.68, respectively (see Tables 2 and 6). Similarly, the other entries in Tables 2 and 6 can be compared.

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Table 6. The run length profile ACC_AIB, AE_AIB, and MHC control charts at ARL₀ = 500.

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5.7: Proposed versus MHC control chart

The MHC control chart is an improved control chart used for enhanced process monitoring. On comparing our proposed IAEWMA_AIB control chart with MHC chart, it is clear that the proposed IAEWMA_AIB control chart is superior to the MHC control chart for higher correlation coefficient values. For example, if δ = 0.10, 0.20, (λ, λ₁, λ₂) = 0.10, the ARL₁ values of the MHC control chart are 62 and 32, whereas the ARL₁ values of IAEWMA_AIB control chart (ρ = 0.95) are 3.21 and 1.22, respectively (see Tables 1 and 6). So, the proposed IAEWMA_AIB control chart is preferred for higher ρ, else the MHC control chart.

5.8: Main findings of the study

The following are the key findings of the proposed IAEWMA_AIB control chart:

The use of HEWMA statistic with an adaptive scheme certainly boosts the detection ability of the proposed IAEWMA_AIB control chart.
The auxiliary information improves the performance of the proposed IAEWMA_AIB control chart (see. Tables 1–3 and Fig 3)
The ARL₁ values of the proposed IAEWMA_AIB control chart are smaller than the competing control charts (i.e. CUSUM_AIB, HEWMA, EWMA_AIB, AEWMA_AIB, and IACCUSUM_AIB) (see Tables 1–3 versus Table 4).
The overall performance evaluation measures show the dominance of the IAEWMA_AIB control chart against other control charts (see Subsections 5.1–5.5).
The proposed IAEWMA_AIB control chart provides the best performance for larger values of ρ (see Fig 3).
The ARL₁ performance of the proposed IAEWMA_AIB control chart is increased for smaller λ₁ and λ₂ (see Tables 1–3).
In short, the proposed IAEWMA_AIB control chart is superior for larger ρ and smaller λ₁ and λ₂ values.

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Fig 3. ARL profile of proposed IAEWMA_AIB control chart under different ρ for (λ₁, λ₂) = 0.10 at ARL₀ = 500.

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6: Real-life applications

This section provides two real-life applications of the proposed IAEWMA_AIB control chart. More details are provided in the following subsections.

6.1: Application 1

Here, a real-life data set from the glass industry is considered to illustrate how the proposed IAEWMA_AIB and existing AEWMA_AIB control charts are practically implemented. Data from Asadzadeh and Kiadaliry [35] are considered here for the glass thickness (X) and its impact on stress strength (Y) of glass bottles. Anwar et al. [27] used this data set for the simultaneous monitoring of process parameters. This data set contains 40 samples, each of size 5, of stress strength (kg/cm²) and thickness (cm) (see Table 7). The X and Y follow BND with , , , and . For the comparison, the proposed IAEWMA_AIB control chart is considered along with the existing AEWMA_AIB control chart. The parameters of the proposed IAEWMA_AIB control chart are taken as λ₁ = 0.2, λ₂ = 0.1, and ρ = 0.905 with and ARL₀ = 500. Similarly, the parameters of AEWMA_AIB control chart are taken as λ = 0 and ρ = 0.905 with and ARL₀ = 500. The existing AEWMA_AIB control chart displays the first OOC signal at the 27^th sample, whereas the proposed IAEWMA_AIB control chart shows at 17^th sample (see Table 7 and Figs 4 and 5). Overall, the existing AEWMA_AIB control chart detects a total of 5 OOC signals, while the proposed control chart detects 17 OOC signals. The comparison indicates that the proposed IAEWMA_AIB control chart is superior to the existing AEWMA_AIB control chart in terms of process location monitoring.

Download:

Fig 4. AEWMA_AIB chart with real-life data of glass manufacturing industry using ρ = 0.905, λ = 0.1,

and ARL₀ = 500.

https://doi.org/10.1371/journal.pone.0272584.g004

Download:

Fig 5. Proposed IAEWMA_AIB control chart with real-life data of glass manufacturing industry using ρ = 0.905, λ₁ = 0.2, λ₂ = 0.1,

and ARL₀ = 500.

https://doi.org/10.1371/journal.pone.0272584.g005

Download:

Table 7. Application of the proposed IAEWMA_AIB versus AEWMA_AIB control charts for glass manufacturing data.

https://doi.org/10.1371/journal.pone.0272584.t007

6.2: Application 2

This subsection illustrates how the proposed control charts can be used in practice to monitor the stability of groundwater physicochemical parameters. The stability of the soil water parameters is always desirable to industrial processes, crop yields and drinking water, all of which ultimately impact industrial production, crop production and human health. In particular, cultivation yield is influenced by certain factors, such as colour, acidity, hardness, pH, sulphite and temperature. We consider two physio-chemical groundwater parameters, including total dissolved solids and the total water hardness, to show the applicability of the proposed control chart. More precisely, total dissolved solids is a study variable Y (measured in terms of electric conductivity (EC)). In contrast, total hardness of water is an auxiliary variable X (measured in terms of calcium magnesium carbonates). We consider groundwater (used for crop irrigation) in District Rahim Yar Khan, Pakistan, to demonstrate the significance of the proposed location control charts. The data is taken from [36], and it is based on 30 different locations from each location, a sample of size five is collected. The X and Y follow BND with μ_Y = 836.06, μ_X = 4.93, , and ρ = 0.50.

We consider the proposed IAEWMA_AIB control chart along with the existing AEWMA_AIB control chart for the practical implementation. The parameters of the proposed IAEWMA_AIB control chart are taken as λ₁ = 0.2, λ₂ = 0.75, ρ = 0.50 with and ARL₀ = 500. Similarly, the parameters of AEWMA_AIB control chart are taken as λ = 0.2 and ρ = 0.50 with and ARL₀ = 500. From Table 8 and Figs 6 and 7, it can be seen that the existing AEWMA_AIB control chart detects seven OOC signals whereas the proposed IAEWMA_AIB control chart displays 16 OOC signals. Hence, the proposed IAEWMA_AIB control chart is more efficient for the monitoring the stability of groundwater physicochemical parameters.

Download:

Fig 6. AEWMA_AIB chart with real-life data of ground water using ρ = 0.50, λ = 0.2,

and ARL₀ = 500.

https://doi.org/10.1371/journal.pone.0272584.g006

Download:

Fig 7. Proposed IAEWMA_AIB control chart with real-life data of ground water using ρ = 0.50, λ₁ = 0.2, λ₂ = 0.75,

and ARL₀ = 500.

https://doi.org/10.1371/journal.pone.0272584.g007

Download:

Table 8. Application of the proposed IAEWMA_AIB versus AEWMA_AIB control charts for ground water data.

https://doi.org/10.1371/journal.pone.0272584.t008

7: Summary, conclusions, and recommendations

The Adaptive exponentially weighted moving average (AEWMA) control chart is an advanced form of the classical EWMA control chart to track small to large shifts in the process. The objective of this study is to improve the detection ability of the existing auxiliary information-based (AIB) AEWMA (AEWMA_AIB) control chart and propose an enhanced AIB AEWMA, symbolized as IAEWMA_AIB control chart for efficient monitoring of process location shift. The proposed IAEWMA_AIB control chart is designed by estimating unknown process location shift with hybrid EWMA statistic, and then smoothing constant is adaptively updated. To evaluate the performance of the proposed IAEWMA_AIB control chart against other control charts, Monte Carlo simulations technique is used for numerical results. Performance comparison tools like average run length, extra quadratic loss, performance comparison index, and relative average run length are used. The analysis based on performance evaluation measures and visual presentation reveals the proposed IAEWMA_AIB control chart outperforms against AIB CUSUM, AIB EWMA, AEWMA_AIB, HEWMA, and AIB improved adaptive Crosier CUSUM. Furthermore, it is vital to mention, that the proposed IAEWMA_AIB control chart converges to AEWMA and AEWMA_AIB control charts at the specific values of parameters. Finally, two real-life applications for users and practitioners are also provided to show the proposed study from a practical perspective. This study can be extended for non-normal distribution and the multivariate case.

Appendix

Let is a sequence based on {T_t} for IC process, then the shift estimator using EWMA statistic is (A-1)

The can be written as

Let is another sequance defined on , then the estimator of using HEWMA statistic is (A-2)

Similarly, the can be written as

Where (λ₂, λ₂)∈(0,1] are the smoothing constants. The initial values of and are zero, i.e., .

(A-3)

Using Mathematica, we can get the following (A-4)

Taking expectation on the both side of (A-4), (A-5)

Now, using “FullSimplify” command in Mathematica, we can get the following (A-6) (A-7) (A-8)

Which is the unbiased estimator.

Supporting information

S1 Data.

https://doi.org/10.1371/journal.pone.0272584.s001

(PDF)

S2 Data.

https://doi.org/10.1371/journal.pone.0272584.s002

(PDF)

References

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View Article
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[32] Google Scholar

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Figures

Abstract

1: Introduction

2: Existing methods

2.1: Variable of interest and transformation

2.2: Classical EWMA control chart

2.3: HEWMA control chart

3: Proposed methods

3.1: Proposed IAEWMAAIB control chart

3.2: Special cases of IAEWMAAIB control chart

4: Performance evaluation measures

4.1: Monte Carlo simulation

4.2: The ARL measure

4.3: Overall performance measures

4.4: Effect of parameters choices

5: Evaluation and performance comparison

5.1: Proposed versus HEWMA control chart

5.2: Proposed versus EWMAAIB control chart

5.3: Proposed versus AEWMAAIB control chart

5.4: Proposed versus CUSUMAIB control chart

5.5: Proposed versus IACCUSUMAIB control chart

5.6: Proposed versus ACCAIB and AEAIB control charts

5.7: Proposed versus MHC control chart

5.8: Main findings of the study

6: Real-life applications

6.1: Application 1

6.2: Application 2

7: Summary, conclusions, and recommendations

Appendix

Supporting information

S1 Data.

S2 Data.

References

3.1: Proposed IAEWMA_AIB control chart

3.2: Special cases of IAEWMA_AIB control chart

5.2: Proposed versus EWMA_AIB control chart

5.3: Proposed versus AEWMA_AIB control chart

5.4: Proposed versus CUSUM_AIB control chart

5.5: Proposed versus IACCUSUM_AIB control chart

5.6: Proposed versus ACC_AIB and AE_AIB control charts