RETRACTED ARTICLE: Monitoring the temperature through moving average control under uncertainty environment

Abstract

The existing moving average control charts can be only applied when all observations in the data are determined, precise, and certain. But, in practice, the data from the weather monitoring is not exact and express in the interval. In this situation, the available monitoring plans cannot be applied for the monitoring of weather data. A new moving average control chart for the normal distribution is offered under the neutrosophic statistics. The parameters of the offered chart are determined through simulation under neutrosophic statistics. The comparison study shows the superiority of the proposed chart over the moving average control chart under classical statistics. A real example from the weather is chosen to present the implementation of the chart. From the simulation study and real data, the proposed chart is found to be effective to be applied for temperature monitoring than the existing control chart.

Introduction

The monitoring of the process is done to reduce the percentage of non-conforming items during the process. The control charts are very important tools and used to watch the effect of extraneous factors on the process. The timely alert about the shift in the process is helpful to identify the reasons for this shift. The Shewhart control charts are simple to apply in the industry but detect only the larger shift in the process. Therefore, the moving average (MA), cumulative sum (CUSUM), and exponentially weighted moving average (EWMA) are the alternatives of the Shewhart control chart. These charts are sensitive to detect a small shift in the process. A little attention has been paid to design MA control charts for various situations. Chen and Yang¹ developed an economic model for the MA chart. Wong et al.² discussed the sensitivity power of the MA chart. Khoo and Wong³ and Areepong⁴ proposed double MA control charts. Mohsin et al.⁵ worked on a weighted MA chart using loss function. Alghamdi et al.⁶ designed the MA chart for the Weibull distribution. Ye et al.⁷ applied quality control methods for air data. Su et al.⁸ used the machine learning technique for metrology data.

The control charts under classical statistics can be only applied when all observations from the industrial process are determined. In complex systems such as monitoring the level of water in a river, the process data may not be determined. “Fuzzy data exist ubiquitously in the modern manufacturing process”⁹. Therefore, an alternative to the existing chart is the use of a control chart using the fuzzy approach. A piece of detailed information about the fuzzy control charts can be seen in references^{10,11,12,13,14,15,16}.

The fuzzy logic that provides information about the measure of truth and falseness is the special case of neutrosophic logic, see Smarandache¹⁷. The neutrosophic logic considers an additional measure is known as the measure of indeterminacy. A detailed discussion on neutrosophic logic can be seen in references^{18,19,20,21,22,23,24,25,26,27,28}. Smarandache²⁹ introduced the neutrosophic statistics (NS0 using the idea of neutrosophic logic. The NS is a generalization of the classical statistics and deals with the data having Neutrosophy. Chen et al.^30,31 introduced the neutrosophic numbers with applications. The np and X-bar control chart using NS were developed by Aslam et al.^32,33. Aslam et al.^34,35 worked on EWMA charts under NS.

A rich literature is available on control charts under classical statistics and fuzzy approach. By exploring the literature and best of our knowledge, there is no work on moving average control chart under NS. In this paper, we will introduce this neutrosophic moving average (NMA) chart and evaluate its performance over the existing MA charts under classical statistics. We will present a real example from the weather-monitoring department.

Materials and Methods

Suppose that $X_{N} = X_{L} + b_{N} I_{N} ;I_{N} \epsilon \left[ {I_{L} ,I_{U} } \right]$ be a neutrosophic random variable consists of the variable under classical statistics $X_{L}$ and indeterminate part $b_{N} I_{N} ;I_{N} \epsilon \left[ {I_{L} ,I_{U} } \right]$. The neutrosophic $X_{N} \epsilon \left[ {X_{L} ,X_{U} } \right]$; $I_{N} \epsilon \left[ {I_{L} ,I_{U} } \right]$ reduces to $X$ when $I_{L}$ = 0. Suppose $n_{N} \epsilon \left[ {n_{L} ,n_{U} } \right]$ presents the neutrosophic group size. Suppose that $\overline{X}_{iN} \epsilon \left[ {\overline{X}_{iL} ,\overline{X}_{iU} } \right]$ denotes the neutrosophic sample average for $ith$ subgroup. Here we assume that $X_{ijN}$ follows the neutrosophic normal distribution with a population mean $\mu_{N} \epsilon \left[ {\mu_{L} ,\mu_{L} } \right]$ and variance $\sigma_{N}^{2} \epsilon \left[ {\sigma_{L}^{2} ,\sigma_{U}^{2} } \right]$, for $i = 1,2, \ldots ,$ and $j = 1,2, \ldots ,n$. By following Montgomery³⁶, NMA statistic is defined as

$$MA_{iN} = \frac{{\overline{X}_{\left( i \right)N} + \overline{X}_{{\left( {i - 1} \right)N}} + \ldots . + \overline{X}_{{\left( {i - w + 1} \right)N}} }}{{w_{N} }};w_{N} \epsilon \left[ {w_{L} ,w_{U} } \right]$$

(1)

where $w_{N} \epsilon \left[ {w_{L} ,w_{U} } \right]$ shows the span at a time $i$. Note here that the statistic is given in Eq. (1) is similar to the exponentially weighted moving average (EWMA) statistic. The main difference between MA statistic and the EWMA statistic is the sensitivity that each statistic shows in the calculation of the data. The EWMA gives the higher weights to the current values while the MA gives equal weight to all values in the data, see Li et al.³⁷.

The $MA_{iN} \epsilon \left[ {MA_{iL} ,MA_{iU} } \right]$ statistic in neutrosophic form can be written as

$$MA_{iN} = MA + c_{N} I_{N} ;I_{N} \epsilon \left[ {I_{L} ,I_{U} } \right]$$

(2)

where $MA$ is the determined part and $c_{N} I_{N}$; $I_{N} \epsilon \left[ {I_{L} ,I_{U} } \right]$ is indeterminate parts of $MA_{iN} \epsilon \left[ {MA_{iL} ,MA_{iU} } \right]$. The MA statistic mentioned by Montgomery³⁶ is a special case of the proposed $MA_{iN} \epsilon \left[ {MA_{iL} ,MA_{iU} } \right]$. The proposed $MA_{iN} \epsilon \left[ {MA_{iL} ,MA_{iU} } \right]$ becomes MA statistic if $I_{L}$ = 0. The neutrosophic mean and variance of $MA_{iN} \epsilon \left[ {MA_{iL} ,MA_{iU} } \right]$ when the process is an in-control $i \ge w_{N}$; $w_{N} \epsilon \left[ {w_{L} ,w_{U} } \right]$ are given as follows

$$E_{N} \left( {MA_{iN} } \right) = \mu_{0N} ;MA_{iN} \epsilon \left[ {MA_{iL} ,MA_{iU} } \right],\mu_{0N} \epsilon \left[ {\mu_{0L} ,\mu_{0U} } \right]$$

(3)

and

$$V_{N} \left( {MA_{iN} } \right) = \frac{{\sigma_{N}^{2} }}{{n_{N} w_{N} }};n_{N} \epsilon \left[ {n_{L} ,n_{U} } \right],w_{N} \epsilon \left[ {w_{L} ,w_{U} } \right],\sigma_{N}^{2} \epsilon \left[ {\sigma_{L}^{2} ,\sigma_{U}^{2} } \right]$$

(4)

Based on the given information, the proposed control chart is mentioned as below.

• Step 1: Select a neutrosophic sample of size $n_{N} \epsilon \left[ {n_{L} ,n_{U} } \right]$ from the production and compute $MA_{iN} \epsilon \left[ {MA_{iL} ,MA_{iU} } \right]$ for specified $w_{N} \epsilon \left[ {w_{L} ,w_{U} } \right]$.

• Step 2: Declare the process in-control if $LCL_{N} \le MA_{iN} \le UCL_{N}$.

• The operational process of the proposed control chart depends on two neutrosophic control limits, which are given as

  $$LCL_{N} = \mu_{0N} - k_{N} \sqrt {\frac{{\sigma_{N}^{2} }}{{n_{N} w_{N} }}} ;w_{N} \epsilon \left[ {w_{L} ,w_{U} } \right], n_{N} \epsilon \left[ {n_{L} ,n_{U} } \right]$$

  (5)
  $$UCL_{N} = \mu_{0N} + k_{N} \sqrt {\frac{{\sigma_{N}^{2} }}{{n_{N} w_{N} }}} ;w_{N} \epsilon \left[ {w_{L} ,w_{U} } \right], n_{N} \epsilon \left[ {n_{L} ,n_{U} } \right]$$

  (6)

  where $k_{N} \epsilon \left[ {k_{L} ,k_{U} } \right]$ is neutrosophic control limits.

Neutrosophic Monte Carlo simulation

In this section, we introduce the neutrosophic Monte Carlo (NMC) simulation for the proposed NMA control chart. Suppose that be neutrosophic mean for in-control process and $\mu_{1N} = \mu_{0N} + c\sigma_{N}$;$\mu_{1N} \epsilon \left[ {\mu_{1L} ,\mu_{1U} } \right]$, where $c$ is a shift constant. Let $r_{0N} \epsilon \left[ {r_{0L} ,r_{0U} } \right]$ be the specified neutrosophic average run length (NARL) when the process is in-control state, for more details, the reader may refer to Li et al.³⁸. The NMC simulation is stated as follows.

• Step-1: Generate a random sample of size $n_{N} \epsilon \left[ {n_{L} ,n_{U} } \right]$ from the neutrosophic standard normal distribution with $\mu_{0N} \epsilon \left[ {\mu_{0L} ,\mu_{0U} } \right]$ and variance $\sigma_{N}^{2} \epsilon \left[ {\sigma_{L}^{2} ,\sigma_{U}^{2} } \right]$. Compute $\overline{X}_{iN} \epsilon \left[ {\overline{X}_{iL} ,\overline{X}_{iU} } \right]$ for $ith$ subgroup.

• Step-2: Compute the statistic $MA_{iN} \epsilon \left[ {MA_{iL} ,MA_{iU} } \right]$ and plot it on $LCL_{N} \epsilon \left[ {LCL_{L} ,LCL_{U} } \right]$ and $UCL_{N} \epsilon \left[ {UCL_{L} ,UCL_{U} } \right]$. Note the first out-of-control value, which is called the run length.

• Step-3: Repeat the process 10,000 times and compute the neutrosophic average run length (NARL) and neutrosophic standard deviation (NSD). Choose $k_{N} \epsilon \left[ {k_{L} ,k_{U} } \right]$ for which NARL for in control process, say $ARL_{0N} \ge r_{0N}$; $ARL_{0N} \epsilon \left[ {ARL_{0L} ,ARL_{0U} } \right]$.

• Step-4: Generate a random sample of size $n_{N} \epsilon \left[ {n_{L} ,n_{U} } \right]$ from the neutrosophic standard normal distribution with $\mu_{1N} \epsilon \left[ {\mu_{1L} ,\mu_{1U} } \right]$ and variance $\sigma_{N}^{2} \epsilon \left[ {\sigma_{L}^{2} ,\sigma_{U}^{2} } \right]$. Compute $\overline{X}_{iN} \epsilon \left[ {\overline{X}_{iL} ,\overline{X}_{iU} } \right]$ for $ith$ subgroup.

• Step-5: Compute the statistic $MA_{iN} \epsilon \left[ {MA_{iL} ,MA_{iU} } \right]$ and plot it on $LCL_{N} \epsilon \left[ {LCL_{L} ,LCL_{U} } \right]$ and $UCL_{N} \epsilon \left[ {UCL_{L} ,UCL_{U} } \right]$. Note the first out-of-control value, which is called the run length for the shifted process.

• Step-3: Repeat the process 10,000 times and compute the NARL and NSD, say $ARL_{1N} \epsilon \left[ {ARL_{0L} ,ARL_{0U} } \right]$ at $\mu_{1N} \epsilon \left[ {\mu_{1L} ,\mu_{1U} } \right]$ for various values of $c$.

Using the NMC simulation process, $ARL_{1N} \epsilon \left[ {ARL_{1L} ,ARL_{1U} } \right]$ and NSD for various $c$, $n_{N} \epsilon \left[ {n_{L} ,n_{U} } \right]$ and $w_{N} \epsilon \left[ {w_{L} ,w_{U} } \right]$ are determined and placed in Tables 1, 2, 3 and 4. From Tables 1, 2, 3 and 4, the following trends can be noted in the values of $ARL_{1N} \epsilon \left[ {ARL_{1L} ,ARL_{1U} } \right]$.

• 1 For the fixed values of $w_{N} \epsilon \left[ {w_{L} ,w_{U} } \right]$, the values of $ARL_{1N} \epsilon \left[ {ARL_{1L} ,ARL_{1U} } \right]$ decreases as $n_{N} \epsilon \left[ {n_{L} ,n_{U} } \right]$ increases.

• 2 The values of $ARL_{1N} \epsilon \left[ {ARL_{1L} ,ARL_{1U} } \right]$ increases as $r_{0N}$ increases from 300 to 370.

Table 1 The values of NARL when n $\epsilon$ [4, 6] and w $\epsilon$ [3, 5].

Table 2 The values of NARL when n $\epsilon$ [6, 8] and w $\epsilon$ [3, 5].

Table 3 The values of NARL when n $\epsilon$ [8, 10] and w $\epsilon$ [3, 5].

Table 4 The values of NARL when n $\epsilon$ [18, 22] and w $\epsilon$ [3, 5].

Results and discussion

In this section, we will compare the performance of the proposed NMA chart over the existing MA chart under classical statistics in terms of $ARL_{1N} \epsilon \left[ {ARL_{1L} ,ARL_{1U} } \right]$ and by a simulation study. For this comparison, the same values of $r_{0N} \epsilon \left[ {r_{0L} ,r_{0U} } \right]$, $c$, $w_{N} \epsilon \left[ {w_{L} ,w_{U} } \right]$ and $n_{N} \epsilon \left[ {n_{L} ,n_{U} } \right]$ are considered.

Comparison in NARL

The proposed control chart is the extension of the X-bar control chart under neutrosophic statistics proposed by Aslam and Khan³⁹. The proposed chart reduces to Aslam and Khan³⁹ chart when $w_{N} \epsilon \left[ {1,1} \right]$. We presented the values of $ARL_{1N} \epsilon \left[ {ARL_{1L} ,ARL_{1U} } \right]$ and NSD of both charts when $r_{0N} \epsilon \left[ {370,370} \right]$ in Table 5.

Table 5 The comparison of the proposed chart and existing chart when NARL = 370.

From Table 5, it can be noted that the proposed chart has smaller indeterminacy intervals of $ARL_{1N} \epsilon \left[ {ARL_{1L} ,ARL_{1U} } \right]$ and NSD as compared to the chart proposed by Aslam and Khan³⁹. For example, when $c$ = 0.1, the values of $ARL_{1N} \epsilon \left[ {ARL_{1L} ,ARL_{1U} } \right]$ and NSD from the proposed charts are [248.96, 173.35] and [241.98, 171.21], respectively. On the other hand, the values of $ARL_{1N} \epsilon \left[ {ARL_{1L} ,ARL_{1U} } \right]$ and NSD from the existing control chart are [320.91, 294.43] and [313.19, 289.75], respectively. From this comparison, it is concluded that when $c$ = 0.1, the proposed chart indicates the shift in the process between 173rd and 248th sample while the existing chart is expected to detect the shift between 294th and the 320th sample. We note that the proposed chart is an efficient chart in detecting the quick shift in the process as compared to Aslam and Khan³⁹ chart.

Comparison by Simulation Data

Now we compare the performance of the proposed chart with the chart proposed by Aslam and Khan³⁹ and X-bar control chart under classical statistics using the simulated data. Let $n_{N} \epsilon \left[ {6,8} \right]$, $w_{N} \epsilon \left[ {3,5} \right]$ and $r_{0N} \epsilon \left[ {370,370} \right]$ The 20 observations are generated from the neutrosophic normal distribution with $\mu_{N} \epsilon \left[ {0,0} \right]$ and variance $\sigma_{N}^{2} \epsilon \left[ {1,1} \right]$. Later 20 observations are generated from the process with a shift of 0.30. The values of statistic $MA_{iN} \epsilon \left[ {MA_{iL} ,MA_{iU} } \right]$ and two existing charts are calculated and plotted in Fig. 1. From Fig. 1, it can be seen that the proposed chart detects shift at the 20th sample. On the other hand, the chart proposed by Aslam and Khan³⁹ detects a shift in the process at around 30th sample while the traditional Shewhart control chart under classical statistics does not show any shift in the process. We also note that the proposed chart shows several points in the indeterminacy interval of control limits.

Figure 1

The control charts using the simulation data. Left figure (proposed chart), middle figure for Aslam and Khan³⁹ and right figure (Shewhart chart). This is generated using R version 3.2.1 (https://www.r-project.org/).

Case study

A meteorologist is interested to apply the proposed control chart for the better prediction and analysis of the weather. To explain the implementation of the proposed control chart, we collected the weather data in Jeddah, Saudi Arabia. The temperature data of October 2019 at three different occasions is collected from https://www.timeanddate.com/weather/saudi-arabia/jeddah/historic. The temperature data at morning 6, afternoon and evening 6 are reported in Table 6. From Table 6, it can be seen that the weather data has neutrosophic numbers. Therefore, the existing control chart under classical statistics cannot be applied for this type of data. The use of the existing chart under classical statistics may mislead the meteorologist. The proposed control chart under neutrosophic statistics can be applied for the monitoring of the weather. The proposed control chart and existing control charts for the weather data are shown in Fig. 2.

Table 6 The weather data.

Figure 2

The control charts for real data. Left figure (proposed chart) and right figure (Shewhart chart). This figure is generated using R version 3.2.1 (https://www.r-project.org/).

From Fig. 2, we note that several temperature values are close to the upper value of the control limits. On the other hand, the existing chart under the classical statistic shows that the weather is in-control and the meteorologist needs no action. From this comparison, it is concluded that the proposed chart indicates some issue in weather and meteorologist should be alert about it.

Concluding remarks

A new moving average control chart for the normal distribution was offered under the neutrosophic statistics. The parameters of the offered chart were determined through simulation under neutrosophic statistics. The comparison study showed the superiority of the proposed chart over the moving average control chart under classical statistics. A real example from the weather was chosen to present the implementation of the chart. From the simulation study and real data, the proposed chart was found to be effective to be applied for temperature monitoring than the existing control chart. The proposed control chart can be applied in the weather department for the monitoring of the process. The proposed chart using EWMA statistics can be extended for future research.