A New Dispersion Control Chart for Handling the Neutrosophic Data

The control chart based on mean deviation (MD) is customary used as a robust alternative to the existing Shewhart control charts for observing changes in dispersion parameter of the process. The existing model of MD control chart is rooted under the assumption that indeterminate observations are not included in measured quality characteristic. If, inspected quality data entail some indeterminate and vague information, typical design of the MD control chart could not be effectively employed. This study originally presents an appropriate generalized design namely neutrosophic mean deviation (<inline-formula> <tex-math notation="LaTeX">$\widetilde {\text {MD}}$ </tex-math></inline-formula>) control chart that could accommodate imprecise observations in collected quality characteristic variables. Under the neutrosophic situation, the related properties of this newly <inline-formula> <tex-math notation="LaTeX">$\widetilde {\text {MD}}$ </tex-math></inline-formula> design have been derived. Using simulated data, performance of the <inline-formula> <tex-math notation="LaTeX">$\widetilde {\text {MD}}$ </tex-math></inline-formula> control chart in terms of neutrosophic average run length (<inline-formula> <tex-math notation="LaTeX">$\widetilde {\text {ARL}}$ </tex-math></inline-formula>) is investigated. The performance of proposed <inline-formula> <tex-math notation="LaTeX">$\widetilde {\text {MD}}$ </tex-math></inline-formula> control chart relative to existing competitor designs has been evaluated. The study reveals that proposed design of <inline-formula> <tex-math notation="LaTeX">$\widetilde {\text {MD}}$ </tex-math></inline-formula> chart outperforms as to existing counterparts in terms of statistical power. To illustrate the efficacy of this new design, real data from a manufacturing company has been used to describe the control procedure of the proposed <inline-formula> <tex-math notation="LaTeX">$\widetilde {\text {MD}}$ </tex-math></inline-formula> control chart.


I. INTRODUCTION
The existence of normal variation in almost every manufactured item is the most likely phenomenon due to different inherent factors. In statistical procedures, control chart approach is used as one of primary tools to study different variability components so that abnormal variation, resulting from different causing factors can be pinpointed [1]. Currently, control charts are extensively employed as a statistical procedure, particularly in industries to identify the contributing factors of normal and abnormal variation in manufactured products in view of some quality characteristics. In quality control frameworks, measured quality characteristic variables are assumed to be stable both in the location and scale parameters because unfavorable changes significantly The associate editor coordinating the review of this manuscript and approving it for publication was Martin Reisslein . effect on these distribution parameters of quality characteristic measurements [2]. The process average and variability are therefore needed to be controlled by using variety of charts available in literature. In back, 1920s, Shewhart proposed a variety of control charts for observing stability in mean and process variability [3]. The R chart is based on simplest statistical measure namely range (R) and S chart rooted on a relatively good measure of variability called standard deviation (S). Both primary charts are generally suggested for observing the process variability [4]. Additionally, R chart is inferior in view of losing efficiency for smaller sample size and wasting data information in its calculation. In contrast, the underlying statistic of S control chart is not a robust measure of variability and becomes greatly influenced by existence of outliers in data [5]. In many realistic situations where data with tiny errors are processed, the MD statistic which is another dispersion measure has advantage over the traditional S statistic as mentioned in [6]. The mean deviation has a variety of versions in which deviation from median is not potentially influenced by outliers in the processing data [7]. The MD measure is superior to S in contaminated situations in the processing data [8].
Dealing with empirical data, while employing MD statistic occasionally works better than S [9]. The MD is not consistently efficient than S statistic under ideal assumption of perfect normal distribution [10]. Many realistic situations requite assumption of normality that is not being met [11]. Due to this apparent superiority of MD statistic, Sun et al. [6] presented MD control chart for dispersion monitoring. Based on other related measures many researchers proposed various control charts for monitoring the dispersion parameter of the process details on which can be seen in [12]. The design of Shewharttype control charts is found on notion of crisp data that are not always available in some real situations [13]. The imprecision in distribution parameters or vagueness in measured quality characteristics may result in uncertainty environment. In such scenarios, dealing with traditional control chart approach is not applicable however; it becomes necessary for control chart practitioners to construct fuzzy structure for the control charts in order to accommodate indeterminacy in data. Huang et al. [14] presented extensive overview of the fuzzy control charts.
The classical set logic is further extended to instituted and subsequently neutrosophic set by incorporating indeterminacy as an independent membership component [15]. In practical applications, assumption of exact values available for computing quality characteristics is not likely to be met [16]. Zhao et al. [17] initially interpreted an idea of neutrosophic data which is an extension of classical data. As a result, many researchers working in the field of quality control also introduced generalized design structures of many existing control charts which allow dealing vague data information on the studied variables. First time, Asalm introduced the applications of the neutrosophic statistics in the field of statistical quality control [18]. Recently, Aslam and Khan designed the neutrosophic version of X-bar chart for monitoring the indeterminate mean parameter of the process [19]. The neutrosophic S 2 control chart for observing the scale parameter is given in the work [20]. Under neutrosophic and repetitive sampling scenarios attribute control chart has been established in [21]. There are many other neutrosophic version of well documented control charts and related areas, details on which can be found in references [22]- [27].
This study develops a new variability control chart namely a MD under the assumption that imprecise observations are encountered in the processing data and exact values are not readily available. The propose design of MD represents a generalized structure and coincides with the existing design of the MD control chart under zero indeterminacy in processing data and design parameters.
The remainder of paper is outlined in sections as follows. The proposed design of MD control chart is presented in Section II. Section III, illustrates the monitoring process of MD control chart and covers the related concept of the ARL using simulated data. The usefulness of proposed chart with other competitor designs is provided in Section IV. Section V, demonstrates the set-up procedure of MD control chart on real data case in details. Eventually, Section VI concludes the findings of the study.

II. PROPOSED DESIGN
The proposed design of MD control chart is relying on the assumption that an indeterminate random sample n ∈ [n L , n U ] is taken from the neutrosophic normal distribution with imprecise parameter values µ ∈ [µ L , µ U ] and σ ∈ [σ L , σ U ]. The neutrosophic distribution is an extended form of the classical normal distribution [26]. For instant, neutrosophic normal density curves for various imprecise parameters are portrayed in Figure 1. Figure 1 (a) shows that distribution is stable at precise shape parameter which is equal to 1 but, the mean is VOLUME 8, 2020 indeterminate and varies in range [1, 2.5] whereas Figure 1(b) shows distribution with indeterminate standard deviation that varies in range [1, 1.5]. In each graph, shaded area between the curves represents indeterminate region of the neutrosophic normal distribution. This example illustrates the idea behind the neutrosophic normal distribution. Similarly, distribution with both neutrosophic parameters can be established.
The proposed chart utilizes the MD which is defined for neutrosophic random sample of size n as: wherem is sample neutrosophic median.
Equation (1) represents the neutrosophic mean deviation based on neutrosophic random sample and is a robust measure of process variability.
Let the quality characteristic say W follows the neutrosophic normal distribution N µ, σ 2 . To derive the proposed chart, it is required to define the relationship between MD and unknown but true value of neutrosophic dispersion parameter σ . Let a neutrosophic random sample x 1 , x 2 , . . . . x n is taken from N µ, σ 2 and neutrosophic random variable T represents the required relationship as: The distributional curve of the neutrosophic test statistic T exclusively depends on the neutrosophic sample size n. For smaller sample size; the distribution behavior of T statistic is not symmetric even for the normally distributed quality characteristic. In order to meet the general model of the Shewhart control chart for the neutrosophic data, it is required to derivate the mean and standard deviation of the statistic defined in (2). Taking expectation, (2) provides For in-control process, E MD can be replaced safely by an estimate MD, the mean of all neutrosophic samples for the quality characteristic under study.
Equation (5) represents the mean of the neutrosophic test statistic T which is difficult to derive analytically due to unclosed form of the integral for the normal density function. However, simulation results as provided in [6] can be utilized for finding the value of t 2 for different sample size ranges. Thus from (4), it is obvious that The expression forˆ σ in (6) is similar to the estimate of σ used in R and S control charts for dispersion monitoring [28]. Moreover, for the required standard error it has been supposed that Again, with difficulty in analytically solution the simulation results can be set-up to find the values of t 3 for different values of neutrosophic sample size. Thus the variance of neutrosophic random variable T from (2) can be written as: where σ MAD represents the neutrosophic standard error of the distribution of the statistic MD. From (7) and (8), the estimate of σ MD can be written as: Thus, the neutrosophic parameters (i.e., LCL, CL and UCL) of the proposed control charts by using the Shewhart's general model can be expressed as: The validity of limits given in (10) is based on the assumption that quality characteristic under study follows the approximate neutrosophic normal distribution. However, the true distribution of T is not symmetric for smaller sample size. Therefore the probability limits are realistic choice in real applications where generated data is not necessary from the assumed model [29].
Using the classical structures of probability limits for R, S and G control charts, the probability limits of the proposed neutrosophic designed then leads to: where α = α L + α R , is the specified type-I erro, divided in right (α R ) and left (α L ) tail probability,ˆ σ is estimated from subgroups data using eq (6), T L and T R are percentage points of the T distribution at the specified value of n.
Thus (10) and (11) can be used to construct desired neutrosophic control limits of the MD control chart. However, the LCL in 3-sigma limits sometimes set zero for negative values of LCL at small values of n which is unrealistic meaning in case of dispersion measure. On other hand, probability limits given as defined in (11), always have a value for LCL, hence are much more effective than conventional 3-sigma 96008 VOLUME 8, 2020 limits. The average run length (ARL) for the proposed chart based on probability limits can be derived.
Likewise, the neutrosophic structure of ARL has been used in many recent studies for assessing the performance of proposed control charts [30]. The ARL ∈ [ARL L , ARL U ] for proposed MD-chart can be determined.
According to classical defined of ARL [31], ARL for in control process can be defined as: where α = 1 − Pr LCL ≤ MD i ≤ UCL is the probability of exceeding MD i from the constructed neutrosophic control limits.
Considering (11) and simplifying (3) implies: Hence equation (12) becomes When the process is shifted to out of control state i.e., σ 1 = σ 0 , with shift constant > 1 then where probability under the alternative hypothesis that process has been shifted to new standard deviation σ 1 .
The ARL is an important indicator for measuring the performance of control charts. It has been used in the Section III for assessing the performance of MD control chart.

III. PROCESS MONITORING USING SIMULATED DATA
To monitor the scale parameter σ using the proposed chart depends upon the construction of conventional 3-sigma limits UCL and LCL defined in equation (10) and in case of probability limits, control limits defined in equation (11) would be used. The process is assumed to be working in a control state with scale parameter σ 0 . Now it has been assumed that process is shifted to the new state with a scale parameter σ 1 = σ 0 where ≥ 1 is a shift constant. Note here, = 1 implies that process is stable at the value of monitoring parameter σ 0 . If the quality of the observed process has been declined by the amount i.e., > 1, how many expected number of samples immediately following this change are required to detect the shift. This would result in the sensitivity of the proposed chart against this shift and measure in terms of ARL 1 . Now, we define all parameters given in (14) in order to obtain the ARL 1 for the proposed chart. To this, let preliminary specify in control ARL 0 ∈ [500, 500] and n belongs to different range of sample sizes says n ∈ [2, 4, ] and n ∈ [8, 10]. Total 10000 simulated random samples each  of size n have been generated from the neutrosophic normal distribution with indeterminate parameter µ = {0, 0.5} and σ = {1, 1}. From each sample, MD i has been obtained. For each neutrosophic sample size n, quantiles points of T distribution are obtained for subsequent used in control limits that based on probability limits approach as defined in (11). The resulting ARL 1 values for MD-chart at different values of the shift constant are computed and displayed in Table 1 and Table 2. control value of the average length i.e., ARL 0 = 500, is always attended and remains within the computed interval. Note that if the process shifted by the amount 1.5 using neutrosophic sample [2,4], on average [57.57, 112.11] number of samples are required to detect this shift. In general, it is obvious from Table 1 that at fixed value of n, the ARL 1 value decreases as the value of shift constant increased which indicates that large shift is reasonably more quickly detected by the proposed chart. Moreover, ARL 1 value at larger sample size approaches to unity with the increased of shift constant . Therefore, the proposed chart is reasonably sensitive to larger and moderate shifts in the process.  Similar conclusion can be followed from the results reported in Table 2.
In addition to ARL, neutrosophic characteristic curve OC can be occasionally established to describe the ability of control for detecting the shift in the process. In order to construct OC curve for the proposed chart we assume the following hypothesis is being testing at specific value of α for the observed process: H 0 : σ = σ 0 i.e., process is in control at scale parameter σ 0 . H 1 : σ = σ 1 = σ 0 i.e., process is shifted at scale parameter σ 1 . Now the construction of OC curve involves the probability of type-II error (β) which is defined for proposed chart in term of probability limits as: Equation (15) denotes the probability of accept the null false null hypothesis i.e., in fact process has been moved to new state with scale parameter σ 1 but we are committing the error by accept the null hypothesis that process is in control state with scale parameter σ 0 . Using the same preceding simulated data, the OC curves of the MD control chart are constructed for different neutrosophic samples are displayed in Figure 2 below.
The OC curves in Figure 2 show that as the process shifted from hypothesized value σ 0 , the probability of committing the Type-II error decreased at a specific value of α and n.This means that the proposed chart will detect the larger shift more effectively as compared to smaller ones for given value of α and n.As the neutrosophic sample size increases, the probability β decreases for given value of α and n as shown in Figure 2(b). Accordingly, for larger sample sizes, the proposed control chart may become more powerful for detecting the shift of specific amount. For example, detecting the shift of 3 in Figure 2, the required power (i.e., 1 − β) of MD control chart belongs to intervals [0.036, 0.153] and [0.404, 0.703] for the neutrosophic samples [4,6] and [8,10] respectively. The neutrosophic area is represented by the shaded region in Figure 2. Thus, the power curves are useful for visualizing the particular shift with specific probability and can be computed for comparing the performance of proposed chart with another counterpart.

IV. COMPARISON WITH EXISTING CHARTS
In this section, performance analysis of the proposed control chart is given with other existing neutrosophic charts. As the MD control chart belongs to the class of charts usually used for monitoring the process variability, therefore, the comparison of the MD control chart is given with other charts that are designed for variability monitoring under the neutrosophic environment. The neutrosophic design of the conventional R and S charts for monitoring the dispersion of the process exists in literature [19], [20]. Among the several statistical approaches, power curve is used as one of the effective tools for assessing the performance of different controls. The power of the proposed chart is defined as: Equation (16) provides the probability of rightly rejecting the null hypothesis H 0 . In our situation, if the plotted statistics MD i goes beyond the lower and upper probability limits for given value of false alarm probability α and neutrosophic sample size n, it would constitute the power of the MD control chart. Any control chart that provides greater probability of correctly rejecting H 0 , would be considered as an efficient approach for detecting the shift when in fact process has been shifted to new state with parameter σ 1 . Employing this approach, power of the test for proposed chart and its counterparts R and S charts have been computed for fixed value of α = 0.002. To achieve the power curves, simulated data likewise the previous section have been generated at various neutrosophic sample sizes using software R.3.2.5. For given value of α and n, power curves of proposed chart in comparisons with R and S charts are depicted in Figure 3 and Figure 4 respectively.    Figure 3 and Figure 4 indicate that when process is in control with shift = 1, power is considerably closer to theoretical probability α = 0.002. Whereas in case the process shifts to out of control state with > 1 at specific value of n, power of each control chart increases. However, clearly from Figure 3, that for given value of n, proposed MD control chart is more powerful than neutrosophic R chart for detecting shift in process parameter σ 0 . For example, power of proposed MD control chart for shift, = 5 at n ∈ [4,6] is [0.147, 0.621] which is greater than the corresponding power [0.012, 0.613] of the neutrosophic R chart. The same is also true for other values of shift constant as shown in Figure 3. In addition, power curves in Figure 3 are evaluated at small and large neutrosophic sample sizes which show that different behaviors of power curves between the proposed chart and neutrosophic R chart. Here the difference of power is larger at smaller neutrosophic sample in comparison to large sample size. Whereas, the performance of neutrosophic S control chart as shown in Figure 4, is slightly better than R chart and closer to that of proposed MD control chart. However, overall performance of proposed chart substantially seems better than S chart across the various neutrosophic sample sizes. In addition to power curves, the performance of proposed design has been compared with the existing design of VOLUME 8, 2020 Riaz and Saghir [6] in relation to ARL 1 properties. The proposed design is set up for ARL 0 ∈ [100, 100] in view to estimate the value of ARL 1 . The proposed control chart has been evaluated at assumed neutrosophic samples n ∈ [2,4] and n ∈ [6,8] while the existing MD control chart is constructed at average sample sizes 3 and 7. The estimated ARL 1 are presented in Table 3.
The results given in Table 3 show that the conventional approach of the MD control chart provides ARL 1 within the imprecise intervals resulted from the proposed approach. The design will be considered more adequate and effective if it provides estimates in indeterminate range rather than exact value under the uncertainty situations as described in [32]. It is also worth mentioned, results of the proposed design of MD control chart coincide with the conventional MD control chart under zero indeterminacy i.e., µ L = µ U = µ, σ L = σ U = σ , ARL L = ARL U = ARL, and n L = n U = n.
Moreover, the neutrosophic ARL value calculated previously is a function of some assumed constant values of the design parameters and shift amount.
In real-life circumstances, these parameters and shift constant are not known in advance [33]. Therefore some researchers recommend to use the ARL as a function of estimated parameters instead of constant values [34]. The ARL based on estimated parameters becomes a random variable   a neutrosophic normal distribution lead to various neutrosophic control limits of the proposed control chart. This resulted in a sampling distribution of in-control ARL 0 values which is given in Figure 5. Figure 5, shows the density plot of in-control neutrosophic ARL 0 values, obtained from 10000 simulated MD control charts based on α = 0.01 and n ∈ [4,6] . At the given false alarm rate α = 0.01, the desired value of ARL 0 is [100, 100]. It is clearly shown in Figure 5, averaging over 10000 calculated values of the neutrosophic EARL values provide EARL ∈ [99.12, 100.87] which is suitably closer to the desired value of ARL 0 .

V. ILLUSTRATIVE EXAMPLE
In order to demonstrate the control procedure of the new method, data from an operating manufacturing operation was utilized. Utilizing authentic data assists in developing understanding of the implementation procedure of the proposed MD control chart. The manufacturing enterprise is Phase I database of the yogurt cup filling process [35]. The indeterminate measured data on quality characteristic of the filling process is taken in thirty groups each with five observations that are recorded in range and given in Table 4. The data from an original source are taken in exact observations at assumed target value equal to 125g. To support in understanding the preceding idea of the proposed chart, neutrosophic data are generated according procedure defined in [36]. The generated data on each studied variable are now deliberately in range instead of exact values. The proposed control chart utilizes the neutrosophic median deviation to monitor the process variability. Therefore, neutrosophic median deviation for each subgroup is computed and given in last column of the Table 4.

VOLUME 8, 2020
The 3-sigma control limits of the MD control chart for data given in Table 3 can be obtained as: Here, the LCL is set to zero because of resulted negative values for given data at specified n. Whereas the probability limits at α equal to 0.002, using (11)  Both approaches provide neutrosophic probability limits that differ greatly for given data. The proposed MD control chart for observed quality characteristic is schematically shown in Figure 6 The Figure 6 indicates that the plotted neutrosophic median deviations exhibit random pattern and inside the control limits. Consequently, process may conclude as a state of in-control. In Figure 7, the conventional 3-sigma MD control chart has been set up for original data set that does not contain any indeterminacy. From this approach, similar conclusion may be inferred about the existing state of the process.

VI. CONCLUSION
In this study, a modified design of the MD control that deals with neutrosophic data has been presented. To evaluate the performance of the proposed chart some related neutrosophic measures as neutrosophic run length ( ARL), operating characteristic curve ( OC) and power are developed. It has been demonstrated that how this newly design could be constructed using simulated and real-life data. Comparisons of the proposed MD control chart with other neutrosophic competitor designs of R and S charts are provided. By employing a simulation study, it has been observed that proposed MD control chart considerably outperforms in terms of detecting the shift in the process as to existing neutrosophic R and S control charts.