https://orcid.org/0000-0001-7590-298X
Figures
Abstract
Quality control (QC) is a systematic approach to ensuring that products and services meet customer requirements. It is an essential part of manufacturing and industry, as it helps to improve product quality, customer satisfaction, and profitability. Quality practitioners generally apply control charts to monitor the industrial process, among many other statistical process control tools, and to detect changes. New developments in control charting schemes for high-quality monitoring are the need of the hour. In this paper, we have enhanced the performance of the mixed homogeneously weighted moving average (HWMA)-cumulative sum (CUSUM) control chart by using the auxiliary information-based (AIB) regression estimator and named it MHCAIB. The proposed MHCAIB chart provided an unbiased and more efficient estimator of the process location. The various measures of the run length are used to judge the performance of the proposed MHCAIB and to compare it with existing AIB charts like CUSUMAIB, EWMAAIB, MECAIB (mixed AIB EWMA-CUSUM), and HWMAAIB. The Run length (RL) based performance comparisons indicate that the MHCAIB chart performs relatively better in monitoring small to moderate shifts over its competitor’s charts. It is shown that the chart’s performance improves with the increase in correlation between the study variable and the auxiliary variable. An illustrative application of the proposed MHCAIB chart is also provided to show its implementation in practical situations.
Citation: Zubair F, Ahmad Khan Sherwani R, Abid M (2023) Enhanced performance of mixed HWMA-CUSUM charts using auxiliary information. PLoS ONE 18(9): e0290727. https://doi.org/10.1371/journal.pone.0290727
Editor: Robin Haunschild, Max Planck Institute for Solid State Research, GERMANY
Received: January 8, 2023; Accepted: August 15, 2023; Published: September 15, 2023
Copyright: © 2023 Zubair 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
Statistical process control and monitoring (SPCM) consists of several statistical tools, and control charts are considered the most efficient. The control charts resolve irregular deviations from the required standards in manufacturing and industrial processes. The memory-less and memory types are the two core divisions of the control charts (cf. Montgomery [1]). Shewhart [2] proposed memory-less control charts, which use only current sample information for process monitoring. The memory type charting procedures, for instance, the cumulative sum (CUSUM), the exponentially weighted moving average (EWMA), the progressive mean (PM), and the homogeneously weighted moving average (HWMA) were developed by Page [3], Roberts [4], Abbas et al. [5] and Abbas [6] respectively and the monitoring statistics of these charts grasp earlier sample information along with the recent information.
On control charts, various types of extensions have been introduced in the literature on the SPCM. Combining two control charts also improved the efficiency of the control charts. Lucas [7] and Lucas and Saccucci [8] suggested the combined design structure of the Shewhart-EWMA and Shewhart-CUSUM charts, respectively. Shamma and Shamma [9] proposed a double EWMA chart. Mixed design structures of EWMA-CUSUM (MEC) and CUSUM-EWMA (MCE) charts were suggested by Abbas et al. [10] and Zaman et al. [11] respectively. Motivated by the study of Shamma and Shamma [8], double PM and HWMA charts were suggested by Abbas et al. [12] and Abid et al. [13] respectively. A mixture of PM and EWMA charts was proposed by Abbas et al. [14]. Taking inspiration from Abbas et al. [9] and Abid et al. [15] developed a mixed HWMA-CUSUM (MHC) chart in which statistic of the CUSUM chart runs as the output and MHC chart outperforms against the EWMA, HWMA, and MEC charts.
In sample surveys, the precision of the estimates of the population parameters can be increased by using auxiliary information. The auxiliary variable is a variable known for all units of the population but not a variable under study. The auxiliary information-based (AIB) charts are usually developed using regression and ratio estimators to monitor the process variable effectively. In the SPCM literature, much work has been done related to the AIB charts. Riaz [16]and Riaz [17] proposed a regression estimator-based Shewhart (AIB) chart for monitoring process location and dispersion, respectively. The regression EWMAAIB chart was proposed by Abbas et al. [18] and the EWMAAIB performed well against the usual EWMA chart without the AIB information. Abbas [19] suggested the CUSUMAIB chart performed relatively better than the usual CUSUM chart. Ahmad et al. [20] suggested some AIB charts for the autocorrelated processes. Adegoke et al. [21] designed a regression HWMAAIB chart when the process variable is investigated under normal and non-normal environments and revealed that the HWMAAIB chart is more powerful than the EWMAAIB and CUSUMAIB charts. Sanusi et al. [22] suggested various ratio estimators based on EWMA charts. The regression PMAIB chart was suggested by Abbas et al. [12] under zero-state and steady-state processes. Interested readers can see the work of Ahmad et al. [20], Haq and Khoo [23], Abbasi and Haq [24], Noor-ul-Amin et al. [25], and Hussain et al. [26] on AIB charts. Dirbaz et al. [27] suggested two new AIB-based control charts, AIB-MEWMA and AIB-DMEWMA charts, to detect shifts in model parameters. Arslan et al. [28] designed a sensitive homogeneously weighted moving average chart using two supplementary variables (hereafter, TAHWMA), which is an efficient and unbiased estimator for the process mean if the two supplementary variables correlate with the study variable.
In the SPCM literature, very little work is available on AIB mixed memory control charts. Recently, Anwar et al. [29] designed a regression estimator based MECAIB and MCEAIB charts for prompt detection of persistent changes, and the MECAIB chart is more effective than the MCEAIB chart and as well as the EWMAAIB and CUSUMAIB charts. Adegoke et al. [21] revealed that the performance of the regression estimator is relatively better than the ratio estimator. The core focus of this study is to propose an efficient mixed memory chart under the scenario of the regression estimator. So, this study proposes a new regression estimator based MHC chart labeled as MHCAIB for detecting persistent deviations in the process location. The MHCAIB chart is a mixture of the HWMAAIB and the usual CUSUM chart. In the recent age of development, improvements in quality assurance techniques are the need of the hour. In context, we have developed a new chart showing visible improvement in detecting shifts. Even a minute change and deviation in the quality can be a big hurdle in many industrial processes like lifesaving drugs, substrate manufacturing, missile equipment, etc. In some industrial and manufacturing processes, the auxiliary information is also recorded along with the under-study variables for different tasks. This information can be used to improve the control chart design without imparting any additional financial burden to the entrepreneur. We can use this information for the improvement of design. Our proposed chart is shown to have improved results compared to its counterparts and can be used for high-quality monitoring in different industrial applications.
The rest of the article is outlined as follows: the next section offers the structure of the MHC and the MHCAIB charts, along with the RL evaluation of the proposed MHCAIB chart. The performance evaluation and RL comparisons of the MHCAIB chart against the competitor’s charts are delivered in Section 3. A numerical example of the MHCAIB and existing charts are offered in Section 4, Section 5 gives the limitation of the study, and the article ends with a conclusion and recommendations.
2. The Mixed HWMA-CUSUM (MHC) and the proposed Mixed HWMA-CUSUM with auxiliary information (MHCAIB) control charts
This section includes a description of the construction of the existing Mixed HWMA-CUSUM (MHC) and the proposed Mixed HWMA-CUSUM with additional information MHCAIB charts:
2.1. The MHC chart
Let zij is the variable of interest follows the normal distribution, i.e., where μz and
is the in-control (IC) mean and variance of the process variable, respectively, i = 1, 2, 3, … and j = 1, 2, 3, …, n. Abbas [6] suggested the statistic of the HWMA chart as follows:
(1)
Where θ is the smoothing parameter (θ ∈ (0, 1]),
, and
. The IC mean and variance for Hi are as follows (cf. Abbas [6]):
Abid et al. [15] proposed the MHC chart by placing the statistic given in (1) with the CUSUM statistic, and the plotting statistics of the MHC chart are given as:
(2)
Where Hi is given in (1) and
. The K and H are defined as (cf. Abid et al. [15])
(3)
(4)
And these are the parameters of the MHC chart. The process is considered to be out-of-control (OOC) if any value of or
go beyond H; otherwise, it is considered to be in control.
2.2. The proposed MHCAIB chart
In most situations, there exists a positive/negative association between the study/process variable (zi) and the auxiliary variable(xi). Let us assume that xi is strongly correlated with zi and the strength of the correlation between xi and zi is represented by ρzx. The pair of observations xi and zi follows the bivariate normal distribution, i.e., (zi, xi) ~ BVN(μz + δσz, μx, σz, σx, ρzx), where δ is mathematically written as , where μ1 is the shifted mean. The regression estimator suggested by Cochran [30] is as follows:
(5)
where
is the regression coefficient,
and
is the sample mean of the zi and xi, respectively, and μx is the population mean of the xi. The mean and variance of Ri are given below (cf. Appendix A in S1 File):
(6)
The plotting statistic of the proposed MHCAIB using (2) is defined as:
(7)
Where
(8)
θ is already defined above, and
. The MHCAIB chart depends on two parameters, i.e., K and H and which are mathematically written as: (cf. Abid et al. [15]):
(9)
(10)
and
are plotted against the value of H given in (10). If
or
, the process is assumed to be out of control (OOC); otherwise, it is in control (IC).
3. Performance evaluation of the MHCAIB chart
The help of a well-known measure assesses the performance of the proposed MHCAIB and its competitor charts called the average run length (ARL). The Average Run length can be defined as “The number of sample points before a control chart gives alarm is called run length (RL) and an average value of RL distribution is called ARL.” The ARLIC and ARLOOC designated the in-control (IC) and out-of-control (OOC) ARL. A chart having a lesser value of ARLOOC for a particular shift is designated to be better over the competitor chart at a certain shift in the process parameter(s) for a fixed value of ARLIC. We have also included some other performance measures associated with RL, like (the standard deviation RL (SDRL) and the median RL (MDRL) (cf. Abid et al. [15])) and these measures are calculated through the Monte Carlo simulations approach. The computational algorithms are developed in the R programming language, and the computational algorithms’ flow chart is presented in Fig 1.
The MHCAIB chart has the design parameters n, θ, ρzx, h and k. the ARLOOC values of the proposed MHCAIB chart against numerous choices of θ = 0.1, 0.25, 0.5, 0.75 and ρzx = 0.25, 0.5, 0.75, 0.95 when the value of k = 0.5 are given in Tables 1–4. The proposed MHCAIB chart shows better performance against smaller values of θ(for instance δ = 0.1, ρzx = 0.25 when θ = 0.1, ARLOOC = 103.54 against θ = 0.75, ARLOOC = 300.25 (cf. Table 1 vs. Table 4)). An increase in ρzx enhanced the efficiency of the MHCAIB chart (for instance δ = 0.05, θ = 0.1 when ρzx = 0.25, ARLOOC = 117.43 against ρzx = 0.95, ARLOOC = 39.74 (cf. Table 1)). There is a decrease in the OOC SDRL and MDRL values with the increase in ρzx (for instance δ = 0.125, θ = 0.1 when ρzx = 0.25, OOC SDRL = 71.08, OOC MDRL = 22 against ρzx = 0.95, OOC SDRL = 11.66, OOC MDRL = 12 (cf. Table 1)).
The performance assessment of the MHCAIB chart in the form of a line graph is given in Fig 2A–2D against various choices of θ and ρzx. A decrease in θ enhanced the efficacy of the MHCAIB chart and vice versa (cf. Fig 2A vs. Fig 2D). Also, an increase in the value of ρzx improved the sensitivity of the MHCAIB chart and vice versa. It means that a higher correlation coefficient value increases the suggested chart’s efficiency. (cf. Fig 2A–2D).
(A). when θ = 0.1, (B). when θ = 0.25, (C). when θ = 0.5 and (D) when θ = 0.75.
3.1. Comparisons
This section offered the OOC performance assessment of the proposed MHCAIB with the CUSUMAIB, EWMAAIB, HWMAAIB, and MECAIB suggested by Abbas et al. [18], Abbas [19], Sanusi et al. [22], and Anwar et al. [27] respectively. Moreover, we have also expressed these comparisons as a percentage decrease in ARL. (ARLPD) and mathematically ARLPD is defined as . The chart with the highest ARLPD value is labeled an efficient chart for that specific shift.
3.1.1. MHCAIB versus CUSUMAIB.
Abbas et al. [18] suggested the CUSUMAIB chart and the ARLOOC results of the CUSUMAIB chart are provided in Tables 5 and 6 against various choices of θ, ρzx and δ. The proposed MHCAIB chart compromises enhanced performance over the CUSUMAIB chart for all selections of θ, ρzx, and δ (for instance when δ = (0.05, 0.1, 0.2), θ = 0.1 and ρzx = 0.5, the MHCAIB ARLOOC = (108, 54, 27) and the CUSUMAIB ARLOOC = (374, 220, 92) (cf. Table 5). Also, at δ = 0.05, the ARLPD in CUSUMAIB and MHCAIB charts are 25.2%and 78.4%, when θ = 0.1 and ρzx = 0.5, respectively.
3.1.2. MHCAIB versus EWMAAIB.
Abbas [19] recommended the EWMAAIB chart and the results of ARLOOC values of the EWMAAIB chart are specified in Tables 5 and 6. The suggested MHCAIB chart displays comparatively better performance over the EWMAAIB chart when δ ≤ 0.75 (for instance, at δ = (0.05, 0.1, 0.2), θ = 0.1, and ρzx = 0.5, the MHCAIB ARLOOC = (108, 54, 27) and the EWMAAIB ARLOOC = (421, 282, 118) (cf. Table 5) and when θ = 0.25, the MHCAIB ARLOOC = (229, 93, 34) and the EWMAAIB ARLOOC = (455, 361, 190) (cf. Table 6). Furthermore, at δ = 0.1, the ARLPD in AIB-EWMA and AIB-MHC charts are 43.6% and 89.2% when θ = 0.1 and ρzx = 0.5, respectively.
3.1.3. MHCAIB versus HWMAAIB.
Sanusi et al. [22] introduced the HWMAAIB chart and the results of ARLOOC values of HWMAAIB chart are provided in Tables 5 and 6 for several choices of θ, ρzx, and δ. The suggested MHCAIB chart shows a better ARLOOC performance against the HWMAAIB chart for all choices of θ and ρzx when δ ≤ 0.75 (for instance at δ = (0.05, 0.1, 0.2), θ = 0.1, and ρzx = 0.75, the MHCAIB ARLOOC = (82, 41, 21) and the HWMAAIB ARLOOC = (315, 160, 62) (cf. Table 5) and when θ = 0.25, the MHCAIB ARLOOC = (169, 63, 23) and the HWMAAIB ARLOOC = (385, 224, 83), (cf. Table 6). Moreover, at δ = 0.1, the ARLPD in HWMAAIB and MHCAIB charts are 55.2% and 87.4% when θ = 0.25 and ρzx = 0.75, respectively.
3.1.4. MHCAIB versus MECAIB.
Anwar et al. [29] suggested the MECAIB chart and the results of ARLOOC values of MECAIB chart are specified in Tables 5 and 6. The suggested MHCAIB chart proposes reasonably superior performance against the MECAIB chart for all choices of θ, ρzx and δ (for instance at δ = 0.05, 0.1, 0.2, θ = 0.1 and ρzx = 0.75, the MHCAIB ARLOOC = (82, 41, 21) and the MECAIB ARLOOC = (303, 153, 62) (cf. Table 5) and when θ = 0.25, the MHCAIB ARLOOC = (169, 63, 23) and the MECAIB ARLOOC = (337, 173, 64) (cf. Table 6). Also, at δ = 0.1, the ARLPD in AIB-MEC and AIB-MHC charts are 65.4% and 87.4% when θ = 0.25 and ρzx = 0.75, respectively.
3.2. Graphical comparisons based on ARLOOC
The ARLOOC based graphical comparisons of the proposed MHCAIB, EWMAAIB, HWMAAIB, and MECAIB charts are presented in Fig 3A–3C for various choices ρzx. The proposed MHCAIB chart offers inferior performance for the selected choices of θ and ρzx (cf. Fig 3A–3C). Additionally, the proposed MHCAIB chart displays superiority over the EWMAAIB, HWMAAIB, and MECAIB charts as the value of ρzx is increased (cf. Fig 3A vs. Fig 3D).
(A) when ρzx = 0.5, (B) when ρzx = 0.75, (C) when when ρzx = 0.95.
4. An illustrative example
Based on the simulated dataset, this section offers an illustrative example of the proposed MHCAIB, HWMAAIB, and MECAIB charts. This dataset consists of 20 pairs of observations which are obtained from the bivariate normal distribution, i.e., (zi, xi) ~ N(μz + δσz, μx, σz, σx, ρzx) by using the following values of μz = 0, δ = 0.50, μx = 0, σz = 1, σx = 1 and ρzx = 0.50 (cf. Abbas et al. [18]). This dataset is used to evaluate the shift-detecting capability of the proposed MHCAIB, HWMAAIB, and MECAIB charts. The selected parameters for the practical implementation of the proposed MHCAIB, HWMAAIB, and MECAIB charts are as follows: for the proposed MHCAIB chart θ = 0.1, k = 0.5, and h = 8.575; for the MECAIB chart θ = 0.1, k = 0.5, and h = 37.35; for HWMAAIB chart θ = 0.1, and C = 2.936 when ARLIC ≈ 500. The control limits and the plotting statistics of the MHCAIB, HWMAAIB, and MECAIB charts are given in Table 7.
The HWMAAIB chart signals only one OOC point at the 18th sample (cf. Fig 4). The MECAIB chart cannot produce any OOC signal (cf. Fig 5). Moreover, the proposed MHCAIB chart produces nine OOC signals from sample numbers 12 to 20 (cf. Fig 6), and this is a piece of evidence about the enhanced shift-detecting capability of the proposed MHCAIB chart against the HWMAAIB, and MECAIB charts.
5. Limitation
The proposed chart uses auxiliary information in its design, so it should be used only if there is a high correlation between the auxiliary variable and the study variable.
6. Conclusion and recommendations
A control chart is the most famous statistical process control and monitoring tool to detect irregular variations from ongoing processes. In this article, we have suggested a new regression estimator-based MHC chart labeled MHCAIB for monitoring persistent deviations in the process location. The ARLOOC performance of the suggested MHCAIB chart is compared with the CUSUMAIB, EWMAAIB, HWMAAIB, and MECAIB, and the suggested MHCAIB chart performs exceptionally well in detecting shifts over its competitor charts for all selected sets of θ and ρzx. Also, it is noticed that the choice of the larger value of ρzx and a smaller value of θ is effective in enhancing the performance of the MHCAIB chart. An application based on simulated data has also identified the dominance of the MHCAIB chart against the HWMAAIB and MECAIB charts.
This study can also be extended for dual auxiliary information-based regression estimator for detecting deviations in the process location and dispersion.
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