Literature DB >> 35994483

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

Muhammad Arslan1, Syed Masroor Anwar2, Showkat Ahmad Lone3, Zahid Rasheed4, Majid Khan5, Saddam Akbar Abbasi6.   

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 (IAEWMAAIB) 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 IAEWMAAIB 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.

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Year:  2022        PMID： 35994483      PMCID： PMC9394848          DOI： 10.1371/journal.pone.0272584

Source DB:  PubMed          Journal:  PLoS One        ISSN： 1932-6203            Impact factor:   3.752


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 (AEWMAAIB) 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 (HEWMAAIB) 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 EWMAAIB 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-EWMAAIB and AIB mixed control charts, respectively, for improved process location monitoring. Recently, Haq et al. [22] suggested AIB adaptive Crosier CUSUM (ACCAIB) under fixed and variable sampling intervals and AIB adaptive EWMA (AEAIB) 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 AEWMAAIB (symbolized as IAEWMAAIB) control chart for enhanced monitoring of the process location. The proposed IAEWMAAIB 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 IAEWMAAIB control chart. To evaluate the performance of the proposed IAEWMAAIB 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 (CUSUMAIB), EWMAAIB, HEWMA, AEWMAAIB, AIB improved adaptive Crosier CUSUM (IACCUSUMAIB), ACCAIB, AEAIB, and mixed HWMA-CUSUM (MHC) control charts are considered for comparison. Moreover, to demonstrate the utility of the proposed IAEWMAAIB 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 IAEWMAAIB control chart, and also special cases of the proposed IAEWMAAIB 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 IAEWMAAIB control chart against CUSUMAIB, EWMAAIB, HEWMA, AEWMAAIB, and IACCUSUMAIB control charts. Additionally, the real-life applications of the proposed IAEWMAAIB 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, μ, σ, σ, ρ), where μ represents the mean and σ represents the standard deviation of X. Also, the ρ is the correlation coefficient corresponding to X and Y. Let (Y, X), 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 where , Here, follows a normal distribution with mean μ and the variance , that is, . Following Haq [29], we assume , then T~N(δ*, 1), where and known as the process mean shift. When a process is IC then T~N(0,1), that is, δ* = 0. Let δ = |μ−μ|/σ be the standardized shift in the σ units, where μ 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} the plotting statistic of the classical EWMA control chart is designed as follows: where, E0 = 0 and λ1 (0<λ1≤1) is the smoothing constant. If μ and σ are known, then the upper control limit (UCL) and lower control limit (LCL) are defined as: for large t, the control limits can be written as: where L denotes the control chart coefficient for a predefined false alarm rate. The EWMA statistic Z is plotted along UCL and LCL, the process is considered to be OOC when ZUCL; 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: Here, the HE is called HEWMA statistic at time t. The starting values of HE and E0 are assumed to be μ0, that is HE0 = E0 = μ0. Based on the statistic HE the LCL and UCL for HEWMA control chart are given by The control limits of the HEWMA control chart when t is very large. 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 HEUCL; otherwise, it is IC.

3: Proposed methods

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

3.1: Proposed IAEWMAAIB control chart

Here, we extend the work of Haq [29] and propose IAEWMAAIB control chart for the improved monitoring of process location. The proposed IAEWMAAIB 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} for IC process and is another sequance defined on , then the estimator of using the HEWMA statistic is defined as: 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 where . Following Haq [29], we assumed . Based on a sequence of IID random variable {T}, the plotting-statistic of the proposed IAEWMAAIB control chart is defined as: where AHE0 = 0 and is defined by Here is a smoothing multiplier of the proposed IAEWMAAIB 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 IAEWMAAIB control chart detects an OOC signal when |AHE| exceeds than threshold says .

3.2: Special cases of IAEWMAAIB control chart

The AEWMA and AEWMAAIB control charts are special cases of the proposed IAEWMAAIB 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 IAEWMAAIB control chart tends to the AEWMA control chart. Proof: When ρ = 0, then the difference estimator reduces to The transformed T can be written as: . Now substitute the resulting in to obtain Also, put λ2 = 1 in Eq (8), then the reduced as follows: which is the estimated shift estimator of the existing AEWMA control chart. Based on Eq (10), the IAEWMAAIB chart can be expressed as: 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 IAEWMAAIB control chart becomes the statistic of AEWMA when ρ = 0 and λ2 = 1. Case 2: When λ2 = 1, the IAEWMAAIB control chart tends to the AEWMAAIB control chart. Proof: When λ2 = 1, the shift estimator in Eq (8) reduced as follows: Then the plotting statistic of the proposed control chart can be written as 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 AEWMAAIB control chart, which indicates that the proposed IAEWMAAIB control chart reduced to the AEWMAAIB 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 IAEWMAAIB 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 IAEWMAAIB 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, X), 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 IAEWMAAIB control chart is described as follows: Generate a random sample from BND. Calculate the estimator from Eq (1) and T. Calculate the statistic from Eq (8) using T estimator. Use the statistic as an input in . Estimate from Eq (9) and estimate using Eq (11). Estimate the IAEWMAAIB statistic AHE from Eq (10). Compute threshold for desired in-control ARL denoted as ARL0. Plot the |AHE| statistic against the threshold . If , record sequence order known as run-length (RL). Record RLs 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 and δ 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 IAEWMAAIB 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 IAEWMAAIB chart.
Table 1

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

ρ λ1 = 0.10, λ2 = 0.10, hIAEWMAAIB = 0.5172
δ 0.000.100.200.250.300.400.500.751.001.502.002.503.003.504.005.006.00
0.25ARL498.44215.2871.8444.7330.0016.6611.195.152.921.541.181.051.021.001.001.001.00
SDRL826.51348.65114.6368.9046.5524.9515.156.613.391.230.530.240.120.060.030.000.00
0.50ARL502.08156.5145.9228.6419.4210.767.263.231.961.221.051.011.001.001.001.001.00
SDRL820.70251.8171.1743.8729.0414.919.733.901.900.620.250.110.040.010.000.000.00
0.75ARL501.4860.6115.139.816.713.932.631.421.131.011.001.001.001.001.001.001.00
SDRL817.8394.5821.5013.289.004.732.961.000.440.090.000.000.000.000.000.000.00
0.95ARL504.293.211.221.081.021.001.001.001.001.001.001.001.001.001.001.001.00
 SDRL814.303.830.620.310.160.030.000.000.000.000.000.000.000.000.000.000.00
λ1 = 0.10, λ2 = 0.20, hIAEWMAAIB=0.5704
δ 0.000.100.200.250.300.400.500.751.001.502.002.503.003.504.005.006.00
0.25ARL498.60233.4381.4951.2735.7019.3412.135.273.051.541.191.051.011.001.001.001.00
SDRL775.38369.70126.7977.0254.6527.7216.326.873.681.200.540.250.120.060.030.000.00
0.50ARL501.33171.8551.9533.9022.8311.967.673.392.041.241.061.011.001.001.001.001.00
SDRL781.87273.1879.5950.3432.5316.3910.284.152.040.670.270.110.050.000.000.000.00
0.75ARL502.2771.4916.2710.757.344.072.701.491.121.011.001.001.001.001.001.001.00
SDRL777.55110.5223.2614.859.625.163.051.110.420.100.010.000.000.000.000.000.00
0.95ARL498.603.291.201.061.021.001.001.001.001.001.001.001.001.001.001.001.00
SDRL775.383.980.560.280.160.020.000.000.000.000.000.000.000.000.000.000.00
λ1 = 0.10, λ2 = 0.50, hIAEWMAAIB=0.6339
δ 0.000.100.200.250.300.400.500.751.001.502.002.503.003.504.005.006.00
0.25ARL497.33219.8381.3553.4939.5622.1313.515.743.171.601.201.071.021.011.001.001.00
SDRL770.60330.92119.4078.3255.6230.3017.807.403.711.230.530.280.150.070.020.000.00
0.50ARL498.68173.0654.9135.5424.4513.818.523.482.131.271.071.011.001.001.001.001.00
SDRL753.22252.8379.5951.0635.1318.9811.244.122.020.640.270.120.040.000.010.000.00
0.75ARL498.5875.5618.9611.557.924.242.761.471.161.011.001.001.001.001.001.001.00
SDRL745.77108.8226.3015.3710.525.233.020.990.470.110.020.000.000.000.000.000.00
0.95ARL497.313.471.241.081.031.001.001.001.001.001.001.001.001.001.001.001.00
SDRL740.134.150.620.320.180.050.000.000.000.000.000.000.000.000.000.000.00
λ1 = 0.10, λ2 = 0.75, hIAEWMAAIB=0.689
δ 0.000.100.200.250.300.400.500.751.001.502.002.503.003.504.005.006.00
0.25ARL498.38227.2885.6158.1940.7923.4614.105.853.231.701.231.081.031.011.001.001.00
SDRL730.60329.88119.8679.8955.5931.8019.347.553.631.300.570.300.170.090.040.000.00
0.50ARL500.98177.8359.3338.1626.9614.678.853.542.161.301.091.021.001.001.001.001.00
SDRL739.35255.2382.8952.3036.8919.8711.764.161.990.690.310.130.050.000.000.000.00
0.75ARL500.5376.7420.4712.238.064.312.901.521.181.011.001.001.001.001.001.001.00
SDRL740.81106.2527.0916.5111.045.223.091.040.460.120.010.000.000.000.000.000.00
0.95ARL502.983.431.281.091.031.001.001.001.001.001.001.001.001.001.001.001.00
SDRL725.943.980.650.310.180.060.000.000.000.000.000.000.000.000.000.000.00
Table 3

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

ρ λ1 = 0.50, λ2 = 0.20, hIAEWMAAIB=0.7771575
δ 0.000.100.200.250.300.400.500.751.001.502.002.503.003.504.005.006.00
0.25ARL502.04293.58124.8786.4861.5434.3521.438.804.822.261.491.171.051.011.001.001.00
SDRL514.23297.67125.2486.2660.5033.5220.468.044.631.840.940.510.240.120.060.000.00
0.50ARL505.48177.8359.3338.1626.9614.678.853.542.161.301.091.021.001.001.001.001.00
SDRL511.33181.2359.8937.8026.8914.578.263.161.990.690.310.130.050.000.000.000.00
0.75ARL504.48113.2528.5318.4812.266.664.212.071.351.031.001.001.001.001.001.001.00
SDRL509.01114.2628.2017.6212.086.553.881.590.760.190.030.000.000.000.000.000.00
0.95ARL505.325.291.551.211.061.001.001.001.001.001.001.001.001.001.001.001.00
 SDRL510.474.961.010.560.270.070.000.000.000.000.000.000.000.000.000.000.00
λ1 = 0.50, λ2 = 0.50, hIAEWMAAIB=1.072987
δ 0.000.100.200.250.300.400.500.751.001.502.002.503.003.504.005.006.00
0.25ARL500.54318.67147.50101.8374.0143.1627.3612.396.692.801.691.261.101.031.011.001.00
SDRL503.61317.10142.67100.9173.3442.6725.9311.325.862.361.140.580.320.190.100.020.00
0.50ARL499.76259.65102.0368.9149.4727.7717.537.484.051.841.281.071.021.001.001.001.00
SDRL500.34258.98101.7767.2847.2625.6516.106.653.551.320.610.270.120.050.010.000.00
0.75ARL500.09129.8936.8224.0716.799.045.742.491.501.061.001.001.001.001.001.001.00
SDRL501.30128.0034.9622.3815.047.935.042.010.900.250.050.010.000.000.000.000.00
0.95ARL498.457.221.801.311.121.011.001.001.001.001.001.001.001.001.001.001.00
SDRL500.836.501.250.660.360.110.020.000.000.000.000.000.000.000.000.000.00
λ1 = 0.50, λ2 = 0.75, hIAEWMAAIB=1.40871
δ 0.000.100.200.250.300.400.500.751.001.502.002.503.003.504.005.006.00
0.25ARL503.19398.92236.18175.69133.7878.4748.9018.769.553.702.171.541.251.101.031.001.00
SDRL499.54399.97236.13175.59133.4878.4547.7818.188.572.771.330.770.490.310.180.040.00
0.50ARL500.36351.12172.43119.4188.9548.8729.9411.285.572.361.541.201.061.011.001.001.00
SDRL500.05352.87171.25118.8086.4847.3528.1510.264.691.530.760.440.240.110.050.000.00
0.75ARL503.56218.5268.0842.5628.2313.828.003.211.941.171.011.001.001.001.001.001.00
SDRL498.86220.4068.0641.3326.6812.657.152.321.080.400.110.030.000.000.000.000.00
0.95ARL498.8010.622.271.631.291.051.001.001.001.001.001.001.001.001.001.001.00
SDRL499.449.671.390.850.530.220.060.000.000.000.000.000.000.000.000.000.00
λ1 = 0.50, λ2 = 0.90, hIAEWMAAIB=1.607892
δ 0.000.100.200.250.300.400.500.751.001.502.002.503.003.504.005.006.00
0.25ARL497.19369.01212.60158.58120.4675.4248.1020.0310.574.222.401.651.291.111.041.001.00
SDRL500.04368.17212.36155.69118.5872.7245.5017.728.873.051.500.880.570.330.190.040.00
0.50ARL500.64333.70161.27115.6684.4048.1829.7112.256.392.721.681.231.071.021.001.001.00
SDRL496.49333.62157.22115.3081.3545.2226.9210.274.971.760.900.490.260.130.040.000.00
0.75ARL498.62203.4267.7042.6129.1215.199.123.712.141.201.021.001.001.001.001.001.00
SDRL496.60200.5765.0240.4526.2812.997.552.621.320.450.140.020.000.000.000.000.00
0.95ARL497.6711.392.571.761.351.061.001.001.001.001.001.001.001.001.001.001.00
SDRL498.039.471.711.000.620.230.050.000.000.000.000.000.000.000.000.000.00

5: Evaluation and performance comparison

This section includes comprehensive comparisons of the proposed IAEWMAAIB control chart with CUSUMAIB [33], EWMAAIB [16], HEWMA [8], AEWMAAIB [29], IACCUSUMAIB [19], ACCAIB and AEAIB [22], MHC [34] control charts.

5.1: Proposed versus HEWMA control chart

The proposed IAEWMAAIB 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 IAEWMAAIB 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 IAEWMAAIB control chart is superior to the HEWMA chart. In terms of overall effectiveness (see Table 5), the proposed IAEWMAAIB 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.
Table 4

The run length profile EWMAAIB, AEMWAAIB, CUSUMAIB, IACCUSUMAIB and HEWMA control charts at ARL0 = 500.

EWMAAIBAEWMAAIBCUSUMAIBIACCUSUMAIBHEWMA
ρ 0.250.500.750.250.500.750.250.500.750.250.500.75
δ λ = 0.2λ = 0.2k = 0.5, h = 5.071λ = 0.2, δmin+=0.5λ1 = 0.1, λ2 = 0.2
0.00502500.34498.13499.56500.24499.96500500.88499.06500.19500.02501.14501.15
0.25140.59117.8871.7695.5778.6348.21138.23114.5268.9981.1671.2350.6991.64
0.5037.8229.6616.9926.321.0112.0536.3628.917.0724.8620.7113.6025.67
0.7515.9512.727.5211.439.065.2816.3213.388.6512.9710.756.9012.47
1.009.067.274.546.355.093.029.998.415.768.376.894.417.70
1.504.313.592.352.902.371.565.584.833.514.533.732.433.87
2.002.722.301.581.781.521.163.913.432.582.952.461.652.45
2.501.961.681.211.341.21.043.042.72.112.141.801.261.75
3.001.531.331.061.141.071.012.522.271.871.671.421.081.37
3.501.271.151.011.051.021.002.192.011.611.381.201.011.17
4.001.121.051.001.021.001.001.981.841.321.191.081.001.07
5.001.021.001.001.001.001.001.661.421.021.031.001.001.01
Fig 1

ARL profile of IAEWMAAIB, HEWMA, EWMAAIB, AEWMAAIB, IACCUSUMAIB, and CUSUMAIB control charts at ρ = 0.25, (λ, λ2) = 0.20, λ1 = 0.10 when ARL0 = 500.

Fig 2

ARL profile of IAEWMAAIB, HEWMA, EWMAAIB, AEWMAAIB, IACCUSUMAIB, and CUSUMAIB control charts at ρ = 0.75, (λ, λ2) = 0.20, λ1 = 0.10 when ARL0 = 500.

Table 5

Overall performance measures of the existing and proposed IAEWMAAIB control charts at different values of ρ.

ControlEQLPCIRARLEQLPCIRARLEQLPCIRARL
Chartρ = 0.25ρ = 0.50ρ = 0.75
EWMAAIB13.491.441.8210.001.131.3710.001.182.11
AEWMAAIB10.891.161.3510.151.141.449.181.081.61
CUSUMAIB20.672.212.5018.342.072.6013.621.612.71
IACCUSUMAIB13.721.471.7712.091.361.829.961.171.96
HEWMA12.241.311.5812.241.381.9212.241.443.04
IAEWMAAIB9.351.001.008.881.001.008.481.001.00

5.2: Proposed versus EWMAAIB control chart

In comparison to the EWMAAIB control chart, the proposed IAEWMAAIB 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 EWMAAIB control chart are larger than the ARL1 (41.68, 10.01, 4.46, 2.50, 1.42, 1.14) values of the proposed IAEWMAAIB control chart (see Tables 2 and 4). Likewise, Figs 1 and 2 displays the dominance of the proposed IAEWMAAIB control chart over EWMAAIB control chart. In addition, the EQL, RARL, and PCI values also show the dominance of the IAEWMAAIB control chart against the EWMAAIB control chart. For instance, at ρ = 0.25, the proposed IAEWMAAIB control chart has EQL = 9.35, PCI = 1.00, and RARL = 1.00, while the EWMAAIB control chart has EQL = 13.49, PCI = 1.44, RARL = 1.82 (see Table 5).
Table 2

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

ρ λ1 = 0.20, λ2 = 0.10, hIAEWMAAIB=0.55061
δ 0.000.100.200.250.300.400.500.751.001.502.002.503.003.504.005.006.00
0.25ARL503.56243.8086.6257.9139.3022.9314.386.243.611.821.281.101.031.011.001.001.00
SDRL540.72253.9886.6454.1938.9922.7213.396.073.531.430.640.350.180.100.050.000.00
0.50ARL502.54181.4758.5136.6525.8013.979.274.072.441.381.101.021.001.001.001.001.00
SDRL535.21186.4158.4936.5425.1413.439.294.012.280.810.340.140.060.020.000.000.00
0.75ARL502.1177.3619.2712.778.294.883.181.641.211.021.001.001.001.001.001.001.00
SDRL533.2081.8119.6911.538.274.503.141.190.540.150.010.000.000.000.000.000.00
0.95ARL503.423.861.341.121.051.001.001.001.001.001.001.001.001.001.001.001.00
 SDRL530.204.050.730.370.230.050.000.000.000.000.000.000.000.000.000.000.00
λ1 = 0.20, λ2 = 0.20, hIAEWMAAIB=0.5989394
δ 0.000.100.200.250.300.400.500.751.001.502.002.503.003.504.005.006.00
0.25ARL503.26250.6794.6863.9646.2925.3116.177.023.851.891.331.131.041.011.001.001.00
SDRL531.50263.6995.6463.4745.2724.6915.947.073.021.460.690.380.210.110.060.000.00
0.50ARL502.63193.3563.1941.6829.5816.2810.014.462.501.421.141.031.011.001.001.001.00
SDRL529.12196.1163.2440.0328.8816.719.284.362.380.810.390.180.070.010.000.000.00
0.75ARL501.1887.1821.9314.109.735.233.341.731.251.021.001.001.001.001.001.001.00
SDRL525.0490.3021.3714.078.925.063.231.250.580.150.040.000.000.000.000.000.00
0.95ARL498.204.201.401.151.051.011.001.001.001.001.001.001.001.001.001.001.00
SDRL527.404.420.800.410.220.070.000.000.000.000.000.000.000.000.000.000.00
λ1 = 0.20, λ2 = 0.50, hIAEWMAAIB=0.7795878
δ 0.000.100.200.250.300.400.500.751.001.502.002.503.003.504.005.006.00
0.25ARL497.86284.87119.9883.6458.8532.5920.638.224.552.101.411.151.051.011.001.001.00
SDRL514.75295.73119.4682.6857.3931.2918.248.064.541.700.810.430.240.120.050.000.00
0.50ARL498.49231.0383.7156.1638.1720.0712.125.242.921.551.151.041.011.001.001.001.00
SDRL517.14235.6082.8555.3137.4119.6811.144.382.651.000.430.200.090.030.000.000.00
0.75ARL498.37108.9928.8117.5011.836.273.981.871.301.031.001.001.001.001.001.001.00
SDRL514.44109.6227.8116.7710.826.153.921.400.650.170.050.000.000.000.000.000.00
0.95ARL500.604.991.471.191.061.001.001.001.001.001.001.001.001.001.001.001.00
SDRL511.685.160.920.490.260.060.000.000.000.000.000.000.000.000.000.000.00
λ1 = 0.20, λ2 = 0.75, hIAEWMAAIB=0.9002
δ 0.000.100.200.250.300.400.500.751.001.502.002.503.003.504.005.006.00
0.25ARL501.10301.04123.4384.4862.8435.0522.429.024.932.291.431.171.041.011.001.001.00
SDRL504.66300.22121.8482.8160.2033.3021.969.014.861.910.890.480.220.120.060.000.00
0.50ARL498.87232.6585.5054.8538.8721.5813.405.663.181.611.171.041.011.001.001.001.00
SDRL505.32233.7484.2253.6337.5220.8713.375.622.941.110.480.200.090.030.000.000.00
0.75ARL501.58110.5029.5918.7712.806.844.422.041.331.031.001.001.001.001.001.001.00
SDRL501.12108.3128.5618.6611.476.684.181.620.750.190.040.000.000.000.000.000.00
0.95ARL503.505.291.541.191.071.001.001.001.001.001.001.001.001.001.001.001.00
SDRL503.315.261.010.530.290.060.000.000.000.000.000.000.000.000.000.000.00

5.3: Proposed versus AEWMAAIB control chart

The proposed IAEWMAAIB control chart is superior to the AEWMAAIB control chart. For example, if (λ, λ1, λ2) = 0.20, ρ = 0.50, and δ = 0.25, the ARL1 values of the AEWMAAIB and IAEWMAAIB 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 IAEWMAAIB control chart reveals the edge over the AEWMAAIB control chart. As an illustration, at ρ = 0.50, the EQL, RARL, and PCI values of the AEWMAAIB and IAEWMAAIB 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 CUSUMAIB control chart

The proposed IAEWMAAIB control chart provides superior performance to the CUSUMAIB control chart. For example, at ρ = 0.75, δ = 0.25,0.50,1.00, the proposed IAEWMAAIB control chart (λ1 = 0.50, λ2 = 0.90) produces ARL1 = (42.61, 9.12, 2.14), whereas, the CUSUMAIB 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 IAEWMAAIB control chart over CUSUMAIB chart. Similarly, at a specific range of shifts, the proposed IAEWMAAIB control chart has smaller EQL, PCI, and RARL values than the CUSUMAIB 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 IAEWMAAIB and CUSUMAIB control charts, respectively (see Table 5).

5.5: Proposed versus IACCUSUMAIB control chart

In comparison with IACCUSUMAIB, the proposed IAEWMAAIB 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 IACCUSUMAIB and IAEWMAAIB 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 IAEWMAAIB control chart against the IACCUSUMAIB control chart. Similarly, at ρ = 0.75, the IACCUSUMAIB 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 IAEWMAAIB control charts. (see Table 5).

5.6: Proposed versus ACCAIB and AEAIB control charts

The proposed IAEWMAAIB control chart is compared with the ACCAIB and AEAIB control charts. The IAEWMAAIB control chart is superior as compared to the ACCAIB and AEAIB 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 ACCAIB, AEAIB and IAEWMAAIB 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.
Table 6

The run length profile ACCAIB, AEAIB, and MHC control charts at ARL0 = 500.

ACCAIB (δmin = 0.50)
δ 00.250.50.7511.52345
λ = 0.20, ρ = 0.25499.5280.7724.8813.028.324.532.961.671.191.03
λ = 0.20, ρ = 0.50499.5471.1520.8210.756.883.732.451.431.081.00
λ = 0.20, ρ = 0.75499.9350.7313.626.924.412.441.651.081.001.00
AEAIB
δ 00.250.50.7511.52345
λ = 0.20, ρ = 0.25499.7293.7226.0511.406.362.901.781.131.011.00
λ = 0.20, ρ = 0.50500.4076.2320.779.105.142.411.511.051.001.00
λ = 0.20, ρ = 0.75499.9344.6514.116.223.561.821.261.011.001.00
MHC
δ 00.10.150.20.250.50.7511.52
λ = 0.20501.0062.0041.0032.0025.0013.009.007.005.004.00
λ = 0.50501.00222.00128.0079.0052.0016.009.006.004.003.00

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 IAEWMAAIB control chart with MHC chart, it is clear that the proposed IAEWMAAIB 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 IAEWMAAIB control chart (ρ = 0.95) are 3.21 and 1.22, respectively (see Tables 1 and 6). So, the proposed IAEWMAAIB 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 IAEWMAAIB control chart: The use of HEWMA statistic with an adaptive scheme certainly boosts the detection ability of the proposed IAEWMAAIB control chart. The auxiliary information improves the performance of the proposed IAEWMAAIB control chart (see. Tables 1–3 and Fig 3)
Fig 3

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

The ARL1 values of the proposed IAEWMAAIB control chart are smaller than the competing control charts (i.e. CUSUMAIB, HEWMA, EWMAAIB, AEWMAAIB, and IACCUSUMAIB) (see Tables 1–3 versus Table 4). The overall performance evaluation measures show the dominance of the IAEWMAAIB control chart against other control charts (see Subsections 5.1–5.5). The proposed IAEWMAAIB control chart provides the best performance for larger values of ρ (see Fig 3). The ARL1 performance of the proposed IAEWMAAIB control chart is increased for smaller λ1 and λ2 (see Tables 1–3). In short, the proposed IAEWMAAIB control chart is superior for larger ρ and smaller λ1 and λ2 values.

6: Real-life applications

This section provides two real-life applications of the proposed IAEWMAAIB 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 IAEWMAAIB and existing AEWMAAIB 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 IAEWMAAIB control chart is considered along with the existing AEWMAAIB control chart. The parameters of the proposed IAEWMAAIB control chart are taken as λ1 = 0.2, λ2 = 0.1, and ρ = 0.905 with and ARL0 = 500. Similarly, the parameters of AEWMAAIB control chart are taken as λ = 0 and ρ = 0.905 with and ARL0 = 500. The existing AEWMAAIB control chart displays the first OOC signal at the 27th sample, whereas the proposed IAEWMAAIB control chart shows at 17th sample (see Table 7 and Figs 4 and 5). Overall, the existing AEWMAAIB control chart detects a total of 5 OOC signals, while the proposed control chart detects 17 OOC signals. The comparison indicates that the proposed IAEWMAAIB control chart is superior to the existing AEWMAAIB control chart in terms of process location monitoring.
Table 7

Application of the proposed IAEWMAAIB versus AEWMAAIB control charts for glass manufacturing data.

1234567891011121314151617181920
y i1 5.6929.0910.35.838.3123.363.4319.6713.6224.5225.6219.144.8617.0519.518.3723.779.4138.235.85
y i2 17.9722.6520.1318.1718.6311.6812.4520.411.927.416.0519.855.1724.2716.1520.929.656.4124.3831.38
y i3 25.4118.0224.8819.698.1921.815.2613.7821.8212.7422.5319.4819.449.4525.1816.4316.4636.9517.0418.34
y i4 23.0822.764.4713.424.2314.8918.6428.3230.3131.3327.3111.6515.319.1226.817.5126.542.6624.6410.06
y i5 18.1324.2915.569.5415.047.1614.1626.1523.8717.297.2323.1217.1618.617.9818.7824.7824.0310.1420.37
x i1 0.272.220.750.280.611.740.121.540.981.912.061.500.201.311.541.461.770.673.260.37
x i2 1.381.691.571.431.490.850.911.610.862.151.241.550.201.861.241.612.310.431.892.42
x i3 2.011.401.971.550.611.641.191.021.650.941.691.541.520.681.981.251.262.571.311.45
x i4 1.711.700.190.971.861.121.502.192.332.392.140.841.200.662.121.352.120.041.920.74
x i5 1.411.871.200.721.140.491.052.11.791.340.541.741.331.470.551.501.941.840.741.60
AEWMAAIB0.1660.5260.4720.4310.3940.4100.3620.4210.5040.5840.5970.6040.5570.5280.5360.5290.6220.6720.6890.687
IAEWMAAIB0.0830.2350.2370.2200.2040.2390.2080.2820.3800.4710.4960.5130.4750.4540.4700.4700.5690.6230.6450.648
2122232425262728293031323334353637383940
y i1 11.236.0121.0618.8427.0914.8734.222.4111.2523.8819.7711.226.6810.8121.9113.538.05811.3110.4118.97
y i2 15.1813.317.128.5214.2716.514.612.7921.8829.279.1310.5114.2714.7919.128.5075.9433.93216.748.048
y i3 18.1731.6914.489.7228.6821.6628.7326.3916.9926.9422.2810.6731.14.30316.3717.1223.6212.4323.5323.23
y i4 17.857.220.0317.1513.6213.8713.9622.1429.2125.742611.626.02221.7812.3516.236.20110.6915.8312.27
y i5 19.1616.6211.7225.2723.4924.7314.95.3819.2211.4712.7711.79.98621.4511.883.3645.1476.8719.239.489
x i1 0.760.411.641.52.141.122.571.680.771.811.771.092.661.072.191.480.681.110.841.65
x i2 1.150.961.320.641.071.281.090.941.672.290.750.851.481.491.670.700.460.161.560.61
x i3 1.452.491.070.732.21.642.22.121.292.122.270.912.780.181.541.582.441.472.402.33
x i4 1.370.501.561.330.991.031.041.672.232.072.651.150.482.021.451.540.501.051.501.33
x i5 1.511.290.861.991.761.941.130.261.510.801.471.220.811.881.300.150.360.521.740.79
AEWMAAIB0.6720.6460.6450.6090.6930.7100.8760.9050.9661.0820.7330.4770.4470.4190.1290.0560.2120.3910.6440.849
IAEWMAAIB0.6360.6150.6170.5830.6700.6890.7740.7990.8400.9100.5790.3390.1480.0040.2520.3990.5210.5430.5600.576
Fig 4

AEWMAAIB chart with real-life data of glass manufacturing industry using ρ = 0.905, λ = 0.1, and ARL0 = 500.

Fig 5

Proposed IAEWMAAIB control chart with real-life data of glass manufacturing industry using ρ = 0.905, λ1 = 0.2, λ2 = 0.1, and ARL0 = 500.

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 = 836.06, μ = 4.93, , and ρ = 0.50. We consider the proposed IAEWMAAIB control chart along with the existing AEWMAAIB control chart for the practical implementation. The parameters of the proposed IAEWMAAIB control chart are taken as λ1 = 0.2, λ2 = 0.75, ρ = 0.50 with and ARL0 = 500. Similarly, the parameters of AEWMAAIB 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 AEWMAAIB control chart detects seven OOC signals whereas the proposed IAEWMAAIB control chart displays 16 OOC signals. Hence, the proposed IAEWMAAIB control chart is more efficient for the monitoring the stability of groundwater physicochemical parameters.
Table 8

Application of the proposed IAEWMAAIB versus AEWMAAIB control charts for ground water data.

123456789101112131415
y i1 860846850806750744870888775868825792933910950
y i2 817830828835792720810825792885810811870909960
y i3 880845879780760790840860750885825816909888927
y i4 912890803790720775815880812900850845933933990
y i5 856897887757791782820895742860830870925860890
x i1 6.43.87.54.74.23.21.94.55.03.52.55.86.05.37.0
x i2 5.83.56.32.75.83.61.85.14.03.81.85.96.57.26.3
x i3 4.43.66.03.04.63.51.74.84.94.01.74.76.07.36.6
x i4 6.53.86.23.73.74.92.04.73.04.51.85.36.36.37.3
x i5 4.84.26.95.13.04.73.05.03.73.22.06.06.66.25.1
AEWMAAIB0.1510.4800.4040.3820.1070.4830.3200.2960.4550.4160.2650.2710.0220.3070.988
IAEWMAAIB0.1510.4800.4040.3820.1070.3650.2140.1920.3610.3230.3020.3070.0550.1130.886
161718192021222324252627282930
y i1 967920850760781870858745990872830750880880747
y i2 960909828725740845873773933907780790840918730
y i3 914867879790798835880730940914867860867915730
y i4 895890803795803828900780980830820860909890720
y i5 935945887750812773820732933856825810867840790
x i1 5.06.47.55.74.06.83.13.85.257.05.75.85.06.32.9
x i2 6.06.26.35.53.66.33.34.86.06.75.54.84.36.83.6
x i3 4.55.26.05.95.06.43.03.64.77.35.95.54.96.62.2
x i4 6.55.96.25.85.06.23.53.94.06.45.75.35.26.83.6
x i5 6.47.26.95.65.26.03.74.36.36.05.65.04.86.03.2
AEWMAAIB1.0312.1322.0551.4711.0240.9850.9571.0460.6741.0270.8110.7850.8440.8780.827
IAEWMAAIB2.0602.0191.4431.0000.9600.9330.9470.8931.2231.1640.9700.9410.9851.0050.952
Fig 6

AEWMAAIB chart with real-life data of ground water using ρ = 0.50, λ = 0.2, and ARL0 = 500.

Fig 7

Proposed IAEWMAAIB control chart with real-life data of ground water using ρ = 0.50, λ1 = 0.2, λ2 = 0.75, and ARL0 = 500.

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 (AEWMAAIB) control chart and propose an enhanced AIB AEWMA, symbolized as IAEWMAAIB control chart for efficient monitoring of process location shift. The proposed IAEWMAAIB 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 IAEWMAAIB 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 IAEWMAAIB control chart outperforms against AIB CUSUM, AIB EWMA, AEWMAAIB, HEWMA, and AIB improved adaptive Crosier CUSUM. Furthermore, it is vital to mention, that the proposed IAEWMAAIB control chart converges to AEWMA and AEWMAAIB 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} for IC process, then the shift estimator using EWMA statistic is The can be written as Let is another sequance defined on , then the estimator of using HEWMA statistic is Similarly, the can be written as Where (λ2, λ2)∈(0,1] are the smoothing constants. The initial values of and are zero, i.e., . Using Mathematica, we can get the following Taking expectation on the both side of (A-4), Now, using “FullSimplify” command in Mathematica, we can get the following Which is the unbiased estimator. (PDF) Click here for additional data file. (PDF) Click here for additional data file.
  1 in total

1.  An improved Bayesian Modified-EWMA location chart and its applications in mechanical and sport industry.

Authors:  Muhammad Aslam; Syed Masroor Anwar
Journal:  PLoS One       Date:  2020-02-26       Impact factor: 3.240
  1 in total

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