Moving average control chart under neutrosophic statistics

1.   Introduction

A control chart, a primary instrument for process monitoring, consists of a set of systematic procedures for observing and refining the quality of items produced through a process. The notion of a control chart was floated by Walter A. Shewhart, a physicist and statistician at Bell Laboratories, during 1920s. Since its inception, several techniques have been introduced by researchers for different conditions. A fast and quick indication by the chart refers to the efficient monitoring chart. Two types of variations are commonly threatened to the running process, first cause is negligible and no harmful to the production process whereas is known as the assignable cause which is harmful to the process ^[1], reporting the process as out-of-control and required to be identified quickly and fast corrective actions are crucial to rectify and streamline the process ^[2]. Control charts are constructed using three lines i.e., the lower control line (LCL), the upper control line (UCL) and a central line (CL).

In general, the control charts are presented for any monitoring phenomena consisting of a single observation but particular situations like interval values cannot be monitored using these techniques. For example, the data on the depth of water in a river cannot be expressed using the existing control chart techniques. To monitor the temperature variations in which the observations are collected in the form of intervals, Aslam ^[3] developed F-test to study interval data under discrete distribution.

The traditional Shewhart chart is relatively good for spotting big process shifts and is less popular in literature due to the incompetency to detect small process disturbances. In order to detect slight process shifts the researchers have proposed the exponentially weighted moving average (EWMA), cumulative sum (CUSUM) and moving average (MA) charts which are significantly delicate for smaller process shifts in the parameters of interest and achieved significant attention of the quality control researchers ^[4]. Due to the inertia problem, the detection capability of the EWMA chart is abhorrent among the quality control personals ^[4]. The CUSUM charts are tougher to apply and use than other control charts for detecting smaller shifts in the parameters. The MA charts are easy to understand and apply because it is created using simple averages but are slightly less efficient as compared to the EWMA and CUSUM charts whereas its collective working is relatively the same ^[5]. Chen and Yang ^[6] developed an MA chart for the continuous process failure mechanism. Wong et al. ^[5] suggested a simple MA chart as well as the joint Shewhart-MA chart. Khoo ^[7] proposed the MA chart for monitoring the number of non-conforming products. Khoo and Wong ^[8] suggested a double MA chart by measuring double moving averages and showed its efficiency over the simple MA on the basis of the average run lengths. Lin, Chou ^[9] designed an economic control chart using MA for auto-correlated observations. Maghsoodloo, Barnes ^[10] discussed the MA control chart and discussed their conditional average run length. More details on MA control charts can be seen in ^{[11,12,13,14]}. Knoth et al. ^[15] criticized modifications made to standard exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) control charts. Abbas et al. ^[16] discussed some concerns raised on the auxiliary information-based control charts.

In recent years, neutrosophic statistics is being used very commonly by quality control researchers due to the nature of collected data which cannot be treated using traditional statistics ^[17]. The idea of neutrosophic statistics is based upon the fussy logic which considers the percentages of truth and falseness ^[18]. Several generalizations of fuzzy logic have been presented by ^[19,20,21]. Albassam, Khan ^[22] presented the application of D'Agostino test, a method used for evaluating the data, for neutrosophic statistics. In the real world, there are many situations in which exact, determined, certain, crisp or clear observations are not possible. For example, the depth of water in any river cannot be described in an exact figure but only the intervals of different areas can be stated. More references on neutrosophic can be appreciated in ^{[17,19,23,24,25,26,27,28,29,30]}. Woodall et al. ^[31] suggested that in the designing of the control charts under neutrosophic statistics, the sample size should not be imprecise.

However, little attention has been given to developing a control chart using the moving average scheme for the neutrosophic data which may attract the attention of several quality control experts. This paper attempts to introduce a charting scheme for dealing with neutrosophic statistic. The basic idea of this paper is to monitor the smaller variations in the uncertain data. The paper is arranged under different sections, appended below.

2.   Materials and methods

Let the neutrosophic random variable is defined as ${X}_{N} = {X}_{L}+{X}_{U}{I}_{N}; {I}_{N}ϵ\left[{I}_{L}, {I}_{U}\right]$ representing two parts ${X}_{L}$ under traditional statistics and ${X}_{U}{I}_{N}; {I}_{N}ϵ\left[{I}_{L}, {I}_{U}\right]$ representing the indeterminate part. Then neutrosophic random variable ${X}_{N}ϵ\left[{X}_{L}, {X}_{U}\right]$ reduces to ${X}_{L}$ when ${I}_{L}$ = 0. Let $n$ presents the sample size. The mean of the neutrosophic $ith$ subgroup can be described as ${\stackrel{-}{X}}_{iN}ϵ\left[{\stackrel{-}{X}}_{iL}, {\stackrel{-}{X}}_{iU}\right]$. Now we suppose that ${X}_{ijN}$ belongs to the neutrosophic normal distribution with mean ${\mu }_{N}ϵ\left[{\mu }_{L}, {\mu }_{L}\right]$ and variance ${\sigma }_{N}^{2}ϵ\left[{\sigma }_{L}^{2}, {\sigma }_{U}^{2}\right]$ respectively, for $i = \mathrm{1, 2}, \dots,$ and $j = \mathrm{1, 2}, \dots, n$. With reference to ^[2], NMA statistic is presented as

Note that $\mathrm{w}$ presents span at time $i$. It can be observed that the statistic given in Eq (1) is parallel to (EWMA) statistic. The core modification is made for MA and EWMA schemes due to their sensitivity. Thus EWMA generates larger loads to the current observations whereas the MA provides an equal load to all observations, see ^[32].

2.1.   Mathematical proofs

In this subsection, we will provide the mathematical proofs of the neutrosophic form of MA statistic, mean and variance of MA statistic, and control limits under neutrosophic statistics.

Suppose that ${X}_{iN}ϵ\left[{X}_{iL}, {X}_{iU}\right]$ $\left(i = \mathrm{1, 2}, \dots \right)$ be a neutrosophic random variable having neutrosophic normal distribution with mean ${\mu }_{N}ϵ\left[{\mu }_{L}, {\mu }_{L}\right]$ and variance ${\sigma }_{N}^{2}ϵ\left[{\sigma }_{L}^{2}, {\sigma }_{U}^{2}\right]$. Suppose that ${\stackrel{-}{X}}_{iN}ϵ\left[{\stackrel{-}{X}}_{iL}, {\stackrel{-}{X}}_{iU}\right]$ be the neutrosophic average of $ith$ subgroup follows the neutrosophic normal distribution with mean ${\mu }_{0N}ϵ\left[{\mu }_{0L}, {\mu }_{0L}\right]$ and variance ${\sigma }_{N}^{2}/nϵ\left[{\sigma }_{L}^{2}/n, {\sigma }_{U}^{2}/n\right]$, where $n$ is a subgroup size. The neutrosophic form of ${X}_{iN}ϵ\left[{X}_{iL}, {X}_{iU}\right]$ is expressed as

The neutrosophic form of ${\stackrel{-}{X}}_{iN}ϵ\left[{\stackrel{-}{X}}_{iL}, {\stackrel{-}{X}}_{iU}\right]$ is expressed as

where the first values denote the determinate part, the second values denote indeterminate part and ${I}_{N}ϵ\left[{I}_{L}, {I}_{U}\right]$ is a measure of indeterminacy.

The MA statistic under neutrosophic statistics when for $w = 3$ and $i = 4$ is expressed as

or

or

In general, the MA statistic, say $M{A}_{iN}ϵ\left[M{A}_{iL}, M{A}_{iU}\right]$ under neutrosophic can be expressed as follows:

Using the above-mentioned information, the neutrosophic form of the statistic $M{A}_{iN}ϵ\left[M{A}_{iL}, M{A}_{iU}\right]$ can be expressed as

Hence proved.

The mean of MA statistic under neutrosophic statistics is derived as follows:

We know

Applying expectation on both sides

Hence proved.

The variance of $\mathrm{M}{\mathrm{A}}_{\mathrm{i}\mathrm{N}}$ is derived as follows:

We know

Taking variance on both sides

Hence proved.

By definition, the lower control limit is defined by

or

By definition, the upper control limit is defined by

or

where $k$ is the coefficient of control limits.

2.2.   Design of the proposed chart

The neutrosophic form of $M{A}_{iN}ϵ\left[M{A}_{iL}, M{A}_{iU}\right]$ is given by

The neutrosophic form of $M{A}_{iN}ϵ\left[M{A}_{iL}, M{A}_{iU}\right]$ has two parts. The first part $M{A}_{iL}$ presents the determinate part and $M{A}_{iU}{I}_{NMA}$ denote the indeterminate part and ${I}_{NMA}ϵ\left[{I}_{LMA}, {I}_{UMA}\right]$ shows the measure of indeterminacy associated with $M{A}_{iN}ϵ\left[M{A}_{iL}, M{A}_{iU}\right]$. Montgomery ^[2] presented the $\mathrm{M}\mathrm{A}$ statistics which is the special case of $M{A}_{iN}ϵ\left[M{A}_{iL}, M{A}_{iU}\right]$ and this neutrosophic form tends to $M{A}_{iL}$ statistic if ${I}_{LMA}$ = 0. Then the mean and variance of the neutrosophic statistics $M{A}_{iN}ϵ\left[M{A}_{iL}, M{A}_{iU}\right]$ for the in-control process $\mathrm{i}\ge \mathrm{w}$; are given as

and

Under the above-stated process, the steps of suggested chart may be written as:

Step 1: Choose a sample having size $n$ and $w$ and calculate $M{A}_{iN}ϵ\left[M{A}_{iL}, M{A}_{iU}\right]$.

Step 2: State the process in-control if $LC{L}_{N}\le M{A}_{iN}\le UC{L}_{N}$.

The functioning of the proposed control chart relay on the following pair of two neutrosophic control limits, given as

3.   Monte Carlo simulation of the proposed neutrosophic statistic

In this section, the neutrosophic Monte Carlo (NMC) simulation of the neutrosophic MA chart has been presented. Let the mean of the in-control process be ${\mu }_{1N} = {\mu }_{0N}+c{\sigma }_{N}$; ${\mu }_{1N}ϵ\left[{\mu }_{1L}, {\mu }_{1U}\right]$, here $c$ is the introduced shift. The pre-specified average run length (ARL) for the smooth process is ${r}_{0N}$, further literature can be seen in ^[33]. The NMC simulation may be listed in the following steps, as:

Step-1: Generate neutrosophic observations of size $n$ from the neutrosophic standard normal distribution with mean ${\mu }_{0N}ϵ\left[{\mu }_{0L}, {\mu }_{0U}\right]$ and variance ${\sigma }_{N}^{2}ϵ\left[{\sigma }_{L}^{2}, {\sigma }_{U}^{2}\right]$. Compute ${\stackrel{-}{X}}_{iN}ϵ\left[{\stackrel{-}{X}}_{iL}, {\stackrel{-}{X}}_{iU}\right]$ for $ith$ subgroup.

Step-2: Calculate the statistic $M{A}_{iN}ϵ\left[M{A}_{iL}, M{A}_{iU}\right]$ and show it on $LC{L}_{N}ϵ\left[LC{L}_{L}, LC{L}_{U}\right]$ and $UC{L}_{N}ϵ\left[UC{L}_{L}, UC{L}_{U}\right]$. Observe the first out-of-control number and note it as the run length.

Step-3: Run these steps 10,000 times and calculate ARL, say ${ARL}_{0N}$ and neutrosophic standard deviation (NSD) of run-length. Select $k$ with the condition that ${ARL}_{0N}\ge {r}_{0N}$.

Step-4: Generate neutrosophic observations of size $n$ from the neutrosophic standard normal distribution with mean ${\mu }_{0N}ϵ\left[{\mu }_{0L}, {\mu }_{0U}\right]$ and variance ${\sigma }_{N}^{2}ϵ\left[{\sigma }_{L}^{2}, {\sigma }_{U}^{2}\right]$ at a specified shift $c$. Compute ${\stackrel{-}{X}}_{iN}ϵ\left[{\stackrel{-}{X}}_{iL}, {\stackrel{-}{X}}_{iU}\right]$ for $ith$ subgroup.

Step-5: Calculate the value $M{A}_{iN}ϵ\left[M{A}_{iL}, M{A}_{iU}\right]$ and show it on $LC{L}_{N}ϵ\left[LC{L}_{L}, LC{L}_{U}\right]$ and $UC{L}_{N}ϵ\left[UC{L}_{L}, UC{L}_{U}\right]$. Observe the first out-of-control value which is known as the run length for the out-of-control process.

Step-6: Run the steps 10,000 spells and calculate ${ARL}_{1N}$ and NSD for various values of $c$. Using the simulation process, ${ARL}_{1N}$ and NSD for different values of $c$, $n$, ${I}_{N}$ and $w$ are computed and given in Tables 1–6. From Tables 1–6, the following findings related to ${ARL}_{1N}$ are observed.

(1) The values of ${ARL}_{1N}$ decreases as $n$ increases when $w$ is the same.

(2) The values of ${ARL}_{1N}$ increases as ${r}_{0N}$ increases from 300 to 370.

4.   Results and discussion

In this section, ARL evaluation of the suggested NMA chart has been given with the parallel prevailing chart discussed by ^[2] under the classical statistics using the same parameters in terms of ${ARL}_{1N}$. For the fair comparison, we used similar values of ${r}_{0N}$, $c$, $w$ and $n$.

4.1.   Comparison of charts using ARL

Montgomery ^[2] discussed the MA chart under classical statistics. As mentioned earlier, the proposed control chart is the generalization of the existing MA chart. It is important to note that the proposed control chart reduces to the existing MA chart if no uncertainty is found in the observations. The ARLs are used for the evaluation and comparison of any proposed study ^[2]. It is worth mentioning here that a control chart having the least number of ARLs is considered the better or more capable chart as compared to a chart having larger ARLs. The values of ARL and NSD of the existing control chart mentioned by ^[2] when ${I}_{N} = 0$ are shown in the first column of Tables 1–6. From Tables 1–6, it can be seen that as the values of ${I}_{N}$ increases, there is a decreasing trend in the values of ARL and NSD. For example, when ${I}_{N} = 0$ and $c$ = 0.2, from Table 5, the values of ARL and NSD are 90 and 89 and when ${I}_{N} = 0.02$ and $c$ = 0.2, from Table 5, the values of ARL and NSD are 81 and 80, respectively. This comparison shows that the proposed control chart gives smaller values of ARL and NSD as compared to the MA chart under classical statistics. Based on this information, it is concluded that the proposed control chart is more efficient than the control chart proposed by ^[2] under classical statistics.

4.2.   Comparison using simulation data

Now, we compare the efficiency of the proposed chart with the existing control chart discussed by ^[2] under classical statistics using the simulated data. The data is generated from the neutrosophic normal distribution with ${\mu }_{N}ϵ\left[0, 0\right]$ and ${\sigma }_{N}^{2}ϵ\left[1, 1\right]$. Forty neutrosophic observations are generated such that the first 20 neutrosophic observations are generated from the in-control process and the remaining neutrosophic observations are generated by considering $c$ = 0.4 and ${I}_{N}$ = 0.1. The values of $M{A}_{iL}$ is computed and plotted on the control chart in Figure 1 (left) and the values $M{A}_{iU}$ are computed and plotted on the control chart in Figure 1 (right). By comparing both figures in Figure 1, it can be noted that the values of $M{A}_{iL}$ fall within the control limits that indicate that the process is in-control. On the other hand, the proposed control chart shows that the values of $M{A}_{iU}$ go out the upper control limit at the 34^th sample. Under an uncertain environment, the proposed control chart detects shift earlier than the control chart under classical statistics. The simulation study also shows the superiority of the proposed control chart over the existing control chart under classical statistics.

5.   Real world example

The study of the practical execution of the anticipated chart using the temperature data is included in the paper. For this purpose, the temperature data from the meteorological department of the Kingdom of Saudi Arabia have been collected and given in Table 7. The determinate (lower) values of temperature data of three times as morning, afternoon and evening of Jeddah city for the month of October 2019 have been collected from https://www.timeanddate.com/weather/saudi-arabia/jeddah/historic. The uncertainty/indeterminacy is always present in measuring the temperature data; see ^[34,35,36]. The upper value of the temperature data is calculated by supposing a measure of intermediacy, say ${I}_{N} = 0.1$. Under indeterminacy, the use of the existing control chart under classical statistics may mislead decision-makers. Therefore, the proposed control chart under indeterminacy should be designed for monitoring the temperature. The values of ARL and NAS for the real data are shown in Table 8. The values of neutrosophic statistic $M{A}_{iN}$ are computed and plotted on the control chart in Figure 2. The left- hand side figure shows the MA control chart under classical statistics and the right-hand side figure shows the plot of the proposed control chart. On noticing Figure 2, it can be seen that all values of the statistic $M{A}_{iL}$ falls within the control limit and indicate that the temperature is in-control. But the proposed control chart shows the shift in the temperature. We may conclude that the proposed chart displays some concerns about the temperature of the city as some points fall close to the upper control limit and the meteorologist should be alert whereas the classical chart shows fully in-control observations.

By comparing the control charts in Figures 1 and 2, it can be seen that the control chart using simulated data in Figure 1 shows the out-of-control at the 34^th sample and the control chart using real data in Figure 2 shows the out-of-control at the 8^th sample.

6.   Conclusions

In this paper, the neutrosophic moving average chart has been presented under the neutrosophic statistics. The coefficients of the charts have been computed for altered process situations. The average run length values have been calculated which show the better discovering ability of the suggested chart. The practical application of the suggested chart has been included by constructing the control chart under the proposed methodology for the temperature data. It has been concluded that the suggested chart is an effective chart using simulated and real data. The proposed control chart has the ability to detect shift earlier than the control chart using classical statistics. It has also been concluded that the suggested chart is a significant addition to the toolkit of the meteorologist. The suggested chart can be extended for the exponentially weighted moving averages using a resampling scheme for future research.

Acknowledgements

The authors are deeply thankful to the editor and reviewers for their valuable suggestions to improve the quality and presentation of the paper.

Conflict of interest

The authors declare no conflict of interest.