Design of S-control chart for neutrosophic data: An application to manufacturing industry

Abstract

The Shewart S-control chart is commonly used as one of the statistical tools for monitoring the process variability. The existing design of S-control is based on the assumption that inspected quality of the observed process is, an exact and clearly specified quantitative quantity. If however, measured data involve some vague and imprecise observations, the conventional approach of the S-control, cannot be practiced. Designing of a generalized neutrosophic S-control chart which could support the indeterminate values in the processing data is originally developed in this article. The associated properties of proposed design under neutrosophic environment have been established in this study. The proposed chart represents a general design of the existence structure of the S-chart. Using neutrosophic average run length (ARL_n) as a performance measure, a comparative study of the proposed chart with the conventional approach of S-control under vague parameter values is evaluated. Findings both from analytical and simulation studies indicate that proposed design of S-control leads to efficient and more flexible approach over the traditional S-control. A real data example has been provided for demonstrating the implementation procedure of the proposed design.

Keywords

Quality control fuzzy charts neutrosophic measures

1 Introduction

The most of manufactured products are assumed to be come from a stable and capable process. Typically, stability and capability of a production process is monitored by control charts, as one of the most sophisticated statistical methods. The manufactured goods that meet the needs of marketplace are specifically defined as quality products [1]. Whereas, in statistical terms, quality is usually referred as degree of variation in what is being measured as a quality characteristic of the service or product. This variability always exists in any manufactured products, no matter how carefully and well-designed process is adopted in production [2]. The quality related characteristics of manufactured products are measured on some scales and provide based for implementation of control charts for quality monitoring. The control charts are implemented on observed quality characteristics with eventual goal of identifying the existence of assignable causes. Control charts are not only able to identify the different types of assignable causes but, also facilitate to find the process capability which is based on estimation results about parameters of the process [3]. If the observed quality characteristic remains within the specified control limits and essentially showing some random pattern, the process is assumed to be in statistical control [4]. According to different types of measurement scales, variety of control charts has been developed. Variable control charts are employed to deal with quality characteristics that can measure as numerical quantities whereas, attributes control charts are establish to handle with categorical data [5]. The manufacturing industries ensure the quality of their products by monitoring the quality of their products time by time in order to satisfy their customers and to survive in this competitive world [6].

$ɛ_{m, n} (t + 1) = ɛ_{m, n} (t) + β_{m, n} (t + 1), m = 1, \dots, 4, n = 1, \dots, l_{1}$ (1) $β_{m, n} (t + 1) = α_{m} (t) β_{m, n} (t) + s_{m, n} (t + 1), m = 1, \dots, 4, n = 1, \dots, l_{1}$ (2) $s_{m, n} (t + 1) = δ_{m} (t) [c_{m} φ_{1} (v_{m, n}^{b} - ɛ_{m, n} (t)) + d_{m} φ_{2} (v_{m, n}^{g} - ɛ_{m, n} (t))]$ (3) $α_{m} (t) = {\begin{matrix} α_{m 1} + \frac{φ_{3} (α_{m 1} - α_{m 2}) (σ_{m} - σ_{m 1})}{(σ_{m 3} - σ_{m 1})}, & if σ_{m} ⩽ σ_{m 1}, m = 1, \dots, 4 \\ α_{m 2}, & if σ_{m} > σ_{m 3}, m = 1, \dots, 4 \end{matrix}$ (4) $δ_{m} (t) = {\begin{matrix} \frac{δ_{m 0}}{\frac{(c_{m} + d_{m})}{2} - 1 + \sqrt{\frac{(c_{m} + d_{m})^{2}}{4} - (c_{m} + d_{m})}} &, (c_{m} + d_{m}) ⩾ 4, m = 1, \dots, 4 \\ δ_{m 0} &, 0 < (c_{m} + d_{m}) < 4, m = 1, \dots, 4 \end{matrix}$ (5)

The design of classical control charts is based on notion of accurate data values that is not always accessible [7]. The vagueness in control chart’s parameters or imprecision in observed quality characteristic may be resulted in imprecise data in many real situations. In such applications, dealing with classical control charts are not recommended however, it become necessary for SPC professionals to construct fuzzy structure for the control charts to accommodate vagueness in the processing data. The increased interest of numerous researchers has been reported in literature on profound designing of certain control charts that are based on vague parameter values. The major advantage of these fuzzified charts is their sensitivity over their conventional counterparts [8]. In fact, designs of these charts are enormously flexible and allow users to handle the vague data obtained during the measurement process. The merging concept of fuzzy theory in construction of control chart for products conformity to specified limits, was first time used by Bradshaw [9]. The suggested approach emphasizes that the graded criteria of a product in fuzzy concept is more realistic rather than its twofold classification. The amplification of fuzzy viewpoint in designing control charts for linguistic variables has further attempted in [10 –12]. The idea of fuzzification in construction of traditional CUSUM and EWMA control charts is developed [12 –14]. To handle with imprecise observations in data without using any defuzzification transformation method and developing the control charts for categorical variables are described by Murat et al. [15]. Designing of variable multivariate control charts in fuzzy mode is described in the work done by Moheb et al. [16]. Another approach by avoiding the defuzzification approach and constructing the variables control chart is explicitly portrayed in the work done by Alireza and Shapiro [17]. In addition, neural network schemes are also proposed by some researchers to identify the state of the process [18].

A fuzzy set does not deal with indeterminate situations and represents a special case of the neutrosophic set where, indeterminacy is taken as independent component [19]. Interested readers may see further details on Neutrosophic set in [20 –22]. The idea of neutrosophic logic is further emerged to deal with indeterminacies in variety of fields. For example, Samandachi [23], first time develop literature on conventional statistical methods in order to model indeterminacies in real problems.In recent years, this grounding immensely helped many researchers to develop statistical methods for analyzing the broad class of data that may include imprecise and indeterminate observations. Chen et al. [24, 25], have proposed the neutrosophic extension of the classical statistical method in order to utilize the more information about key parameters in rock study. The designs for variable control charts such as X bar and chart, S² using neutrosophic consideration have been developed by Aslam et al. [26, 27].

In this work, a new generalized design of the S-control chart has been developed. This newly proposed design can handle the indeterminate values in the processing data. Theoretical framework for S-control chart under neutrosophic environment has been originally derived in this study. The conventional design of conventional S-control is appropriate only to monitor the variability of the process when employed statistics are clearly specified and processing data do not on include imprecise information. To best of our literature search, such design is not developed by utilizing the neutrosophic logic in order to handle indeterminacies that might exist in the measurement process.

The rest of this study is deliberately established in sections as follows. Section 2, briefly describes the designing of neutrosophic S-control chart. The performance of proposed design with the existing counterpart is readily examined in Section 3. The implementation procedure of neutrosophic S-control is discussed in Section 4 by utilizing real data from a manufacturing industry. This work is eventually concluded in Section 5.

2 Designing of neutrosophic S-control chart

Typical structure of the S-control is based on notion of independent and identically distributed (iid) crisp observations on certain quality characteristic. Analogue to classical approach, it has been assumed that an iid sample of the neutrosophic values ${\tilde{X}}_{i} \in [{\tilde{X}}_{l}, {\tilde{X}}_{u}], i = 1, 2 \dots . \tilde{n}$ is taken from a neutrosophic normal distribution with parameters which are denoted as: $\tilde{μ} = \frac{\sum_{j = 1}^{\tilde{N}} {\tilde{X}}_{j}}{\tilde{N}}, \tilde{μ} \in [{\tilde{μ}}_{l}, {\tilde{μ}}_{u}]$

${\tilde{σ}}^{2} = \sum_{j = 1}^{\tilde{N}} \frac{{({\tilde{X}}_{j} - \tilde{μ})}^{2}}{\tilde{N}}, \tilde{σ} \in [{\tilde{σ}}_{l}, {\tilde{σ}}_{u}], \tilde{N} \in [N_{l}, N_{u}] .$

The corresponding neutrosophic unbiased estimators of $\tilde{μ}$ and ${\tilde{σ}}^{2}$ are defined as: $\tilde{\bar{X}} = \frac{\sum_{i = 1}^{\tilde{n}} {\tilde{X}}_{i}}{\tilde{n}} and {\tilde{S}}^{2} = \sum_{i = 1}^{\tilde{n}} \frac{{({\tilde{X}}_{i} - \tilde{\bar{X}})}^{2}}{\tilde{n} - 1}$ where $\tilde{\bar{X}} \in [{\tilde{\bar{X}}}_{l}, {\tilde{\bar{X}}}_{u}]$ and ${\tilde{S}}^{2} \in [{\tilde{S}}_{l}^{2}, {\tilde{S}}_{u}^{2}]$ are representing neutrosophic measures whose values are not specified accurately. Setting up the control chart based on neutrosophic sample standard deviation $(\tilde{S})$ is used to monitor the variability of the process. In the $\tilde{S}$ -control chart, $\tilde{S}$ for jth subgroup is defined as:

${\tilde{S}}_{j} = \sqrt{\sum_{i = 1}^{\tilde{n}} \frac{{({\tilde{X}}_{ij} - {\tilde{\bar{X}}}_{j})}^{2}}{\tilde{n} - 1}}, j \geq 1$ where ${{\tilde{X}}_{1 j}, {\tilde{X}}_{2 j}, {\tilde{X}}_{3 j} \dots {\tilde{X}}_{\tilde{nj}}}$ represents the subgroup of size $\tilde{n} \in [n_{l}, n_{u}]$ with mean ${\tilde{\bar{X}}}_{j}$ . All the observations between and within subgroups are assumed from the neutrosophic normal distribution. Details on the neutrosophic normal distribution can be seen in [28]. More generally, for the larger subgroup sizes the distribution of ${\tilde{S}}_{j}$ is followed to approximate neutrosophic normal distribution based on the fact of normal approximation to chi-square distribution [28]. Therefore, three sigma control limits of the proposed $\tilde{S}$ -control chart are easier to derive as follows:

$\tilde{LCL} = {\tilde{σ}}_{0} [{\tilde{c}}_{4} - 3 \sqrt{1 - {\tilde{c}}_{4}^{2}}] = \tilde{a} {\tilde{σ}}_{0}$ $\tilde{UCL} = {\tilde{σ}}_{0} [{\tilde{c}}_{4} + 3 \sqrt{1 - {\tilde{c}}_{4}^{2}}] = \tilde{b} {\tilde{σ}}_{0}$ (6) where ${\tilde{c}}_{4} = {(2 / - (\tilde{n} - 1))}^{\frac{1}{2}} ⌈ (\frac{\tilde{n}}{2}) ⌈ (\frac{\tilde{n} - 1}{2})$ , $\tilde{a} \in [{\tilde{a}}_{l}, {\tilde{a}}_{u}]$ , $\tilde{b} \in [{\tilde{b}}_{l}, b_{u}]$ and ${\tilde{σ}}_{0}$ in control benchmark value of the process variability.

The 3 sigma limits as given above are often employed at phase II stage for monitored quality characteristics. Such limits however, are avoided sometimes for observing the process variability and hence probability limits are essentially recommended [30].

It has been supposed throughout in this study that process variability is ${\tilde{σ}}_{1} = k {\tilde{σ}}_{0}$ , where k = 1, 0 < k < 1 and k > 1 indicating whether the process benchmark value is in control, has decreased and increased respectively. Therefore, the neutrosophic average run length $(\tilde{ARL})$ is a function of parameters that involved in upper and lower neutrosophic control limits and shift constant k. The average run length (ARL) is an important indicator for measuring the performance of control charts. Likewise, the neutrosophic structure of ARL has been used in many recent studies for assessing the performance of proposed control charts [31]. The $\tilde{ARL}$ for proposed $\tilde{S}$ -chart can be determined as follows:

According to classical defined of ARL [32], $\tilde{ARL}$ for in control process can be defined as: ${\tilde{ARL}}_{0} = \frac{1}{1 - \Pr [\tilde{LCL} {\tilde{\leq S}}_{i} \tilde{\leq UCL}]}$ (7) where $1 - \Pr [\tilde{LCL} {\tilde{\leq S}}_{i} \tilde{\leq UCL}]$ is the probability of exceeding. ${\tilde{S}}_{i}$ from the above neutrosophic control limits.

Using the relevant fact of sampling distribution of ${\tilde{S}}_{i}^{2}$ , the ratio $(\tilde{n} - 1) {\tilde{S}}_{i}^{2} / - σ^{2}$ under the assumptions of normality and independence has the neutrosophic ${\tilde{Δ}}_{\tilde{n} - 1}^{2}$ distribution.

We have $\Pr [\tilde{LCL} {\tilde{\leq S}}_{i} \tilde{\leq UCL}] = \Pr (\tilde{a} {\tilde{σ}}_{0} < {\tilde{S}}_{i} < \tilde{b} {\tilde{σ}}_{0})$ $= \Pr [{\tilde{a}}^{2} (\tilde{n} - 1) {\tilde{σ}}_{0}^{2} < (\tilde{n} - 1) {\tilde{S}}_{i}^{2} < {\tilde{b}}^{2} (\tilde{n} - 1) {\tilde{σ}}_{0}^{2}]$ $= \Pr [{\tilde{a}}^{2} (\tilde{n} - 1) {\tilde{σ}}_{0}^{2} \leq (\tilde{n} - 1) {\tilde{S}}_{i}^{2} \leq {\tilde{b}}^{2} (\tilde{n} - 1) {\tilde{σ}}_{0}^{2}]$ $= \Pr [{\tilde{a}}^{2} (\tilde{n} - 1) \leq \frac{(\tilde{n} - 1) {\tilde{S}}_{i}^{2}}{{\tilde{σ}}_{0}^{2}} \leq {\tilde{b}}^{2} (\tilde{n} - 1)]$ (8)

Simplifying (3) implies: $\Pr (\tilde{a} {\tilde{σ}}_{0} \leq {\tilde{S}}_{i} \leq \tilde{b} {\tilde{σ}}_{0}) = \tilde{F} [{\tilde{b}}^{2} (\tilde{n} - 1)] - \tilde{F} [{\tilde{a}}^{2} (\tilde{n} - 1)]$ where the $\tilde{F} \in [F_{l}, F_{u}]$ is the probability distribution of neutrosophic ${\tilde{Δ}}_{\tilde{n} - 1}^{2}$ distribution.

Hence equation (1) becomes ${\tilde{ARL}}_{0} = \frac{1}{1 - \tilde{F} [{\tilde{b}}^{2} (\tilde{n} - 1)] + \tilde{F} [{\tilde{a}}^{2} (\tilde{n} - 1)]}$ (9)

Similarly, when the process is shifted to out of control with shift constant k > 1 or 0 < k < 1, then ${\tilde{ARL}}_{1} = \frac{1}{\Pr [{\tilde{S}}_{i} \geq \tilde{UCL} | {\tilde{σ}}_{1}] + \Pr [{\tilde{S}}_{i} \leq \tilde{LCL} | {\tilde{σ}}_{1}]}$ (10) where $\Pr [{\tilde{S}}_{i} \geq \tilde{UCL} | {\tilde{σ}}_{1}] + \Pr [{\tilde{S}}_{i} \leq \tilde{LCL} | {\tilde{σ}}_{1}]$ is probability under the alternative hypothesis that process has been shifted to new standard deviation ${\tilde{σ}}_{1}$ .

Now

$\begin{matrix} \Pr [{\tilde{S}}_{i} \geq \tilde{UCL} | {\tilde{σ}}_{1}] \\ = \Pr [\frac{(\tilde{n} - 1) {\tilde{S}}_{i}^{2}}{{\tilde{σ}}_{1}^{2}} \geq \frac{(\tilde{n} - 1) {\tilde{UCL}}^{2}}{{\tilde{σ}}_{1}^{2}}] \end{matrix}$ (11)

Simplifying (6) implies that $\Pr [{\tilde{S}}_{i} \geq \tilde{UCL} | {\tilde{σ}}_{1}] = 1 - \tilde{F} [\frac{{\tilde{b}}^{2} (\tilde{n} - 1)}{k^{2}}]$ (12)

Similarly $\Pr [{\tilde{S}}_{i} \leq \tilde{LCL} | {\tilde{σ}}_{1}] = \tilde{F} [\frac{{\tilde{a}}^{2} (\tilde{n} - 1)}{k^{2}}]$ (13)

Hence from (7) and (8) we have ${\tilde{ARL}}_{1} = \frac{1}{1 - \tilde{F} [\frac{{\tilde{b}}^{2} (\tilde{n} - 1)}{k^{2}}] + \tilde{F} [\frac{{\tilde{a}}^{2} (\tilde{n} - 1)}{k^{2}}]}$ (14)

Thus ${\tilde{ARL}}_{0}$ and ${\tilde{ARL}}_{1}$ are two performance metrics that provide the average number of neutrosophic samples until $\tilde{S}$ -control chart shows an out of control condition when actual process is in fact in control and out of control respectively. Essentially, the larger value for ${\tilde{ARL}}_{0}$ and smaller for ${\tilde{ARL}}_{1}$ are expected in the performance assessment. Note that when ${\tilde{σ}}_{0}$ appears to be unknown in real scenario, it could be obtained from the processing data with the following estimate: $\bar{\tilde{S}} = \frac{\sum_{i}^{T} {\tilde{S}}_{i}}{T}, \bar{\tilde{S}} \in [{\bar{\tilde{S}}}_{l}, {\bar{\tilde{S}}}_{u}]$ where T is number of available groups of preliminary samples each of size $\tilde{n}$ and ${\tilde{S}}_{i}$ is a neutrosophic standard deviation of ith group.

Therefore, neutrosophic control limits for the $\tilde{S}$ -control chart would be $\tilde{LCL} = \bar{\tilde{S}} - 3 \frac{\bar{\tilde{S}}}{{\tilde{c}}_{4}} \sqrt{1 - {\tilde{c}}_{4}^{2}}$ (15) $\tilde{UCL} = \bar{\tilde{S}} + 3 \frac{\bar{\tilde{S}}}{{\tilde{c}}_{4}} \sqrt{1 - {\tilde{c}}_{4}^{2}}$ (16) where $\frac{\bar{\tilde{S}}}{{\tilde{c}}_{4}}$ is an unbiased estimator of ${\tilde{σ}}_{0} .$

The overall procedure used to find the control limits of the proposed $\tilde{S}$ -control chart can be seen in Fig. 1.

Fig. 1

Flowchart of the proposed $\tilde{S}$ -control chart.

Flowchart as given in Fig. 1 may also use to reveal the step-by-step procedure involved to check state of the observed process by using the proposed control chart.

Now, we define all parameters given in (1) in order to obtain the ${\tilde{ARL}}_{1}$ for the proposed chart. To this end, let specify in control ${\tilde{ARL}}_{0} \in [370, 370]$ and $\tilde{n}$ belongs to different range of sample sizes says $\tilde{n} \in {2, 5}, \tilde{n} \in {6, 9}$ and $\tilde{n} \in {12, 15}$ and $\tilde{n} \in {18, 21}$ . For each neutrosophic sample, ${\tilde{ARL}}_{1}$ values for $\tilde{S}$ -control chart at different values of shift constant k are computed and displayed in Table 1, 2, 3 and 4.

Table 1

The values of ${\tilde{ARL}}_{1}$ at different values of shift constant k when $\tilde{n} \in {2, 5}$

k	${\tilde{ARL}}_{1}$
0.75	[1955.078 61449.709]
0.90	[264.2091 1300.5096]
1.00	[109.1626 257.1329]
1.10	[56.07937 79.63913]
1.25	[23.47833 26.96329]
1.50	[6.962926 12.146484]
2.00	[2.349276 5.192788]
2.50	[1.537859 3.364440]
3.00	[1.26886 2.597204]

Table 2

The values of ${\tilde{ARL}}_{1}$ at different values of shift constant k when $\tilde{n} \in {6, 9}$

k	${\tilde{ARL}}_{1}$
0.75	[1496.643 505085.160]
0.90	[1297.028 2883.263]
1.00	[275.6435 334.7412]
1.10	[73.85218 78.95839]
1.25	[16.45954 21.34014]
1.50	[4.155545 5.975013]
2.00	[1.500318 2.021200]
2.50	[1.135527 1.371128]
3.00	[1.044498 1.168392]

Table 3

The values of ${\tilde{ARL}}_{1}$ at different values of shift constant k when $\tilde{n} \in {12, 15}$

k	${\tilde{ARL}}_{1}$
0.75	[4.303111 12.124506]
0.90	[132.2549 321.8763]
1.00	[785.5926 1374.9074]
1.10	[332.6631 364.5042]
1.25	[61.53332 67.67421]
1.50	[10.77865 13.15563]
2.00	[2.566459 3.171874]
2.50	[1.148556 1.268066]
3.00	[1.019893 1.052068]

Table 4

The values of ${\tilde{ARL}}_{1}$ at different values of shift constant k when $\tilde{n} \in {18, 21}$

k	${\tilde{ARL}}_{1}$
0.75	[1.611968 2.330060]
0.90	[43.18100 70.29913]
1.00	[442.9519 599.0161]
1.10	[348.0952 368.2477]
1.25	[51.96204 56.39742]
1.50	[7.783937 9.069887]
2.00	[1.897794 2.171471]
2.50	[1.047174 1.083618]
3.00	[1.002749 1.007477]

Results in Table 1, 2, 3 and 4 show that if the process does not shift i.e., k = 1, the in control value of the average length never attend and it always remains less than 370. However, for larger neutrosophic sample say $\tilde{n} \geq 9$ , average run length became closer to advertised value of 370. It can be seen that if process quality is improved in terms of ${\tilde{σ}}_{0}$ i.e., k < 1, then more samples are required to detect the shift in case of small neutrosophic sample size. Whereas, smaller number of samples are required to detect such shift in case of larger neutrosophic sample size. In addition, if the quality of the process is dropped by any amount of shift constant i.e., k > 1, proposed chart is highly sensitve and immediately dectect this shift.

3 Performance evaluation

In this section proposed $\tilde{S}$ -control chart has been compared with conventional designed of S-chart in terms of the ${\tilde{ARL}}_{1}$ .properties. In order to estimate the value of ${\tilde{ARL}}_{1}$ , process is set up at 3-sigma limits and value of the shift is assumed to be in direction of shift constant k ≥ 1. In order to find performance difference across different sample sizes, both designed are evaluated at various sample sizes. For implementation of $\tilde{S}$ -control chart, we assume that $\tilde{n} \in {6, 9}$ and $\tilde{n} \in {12, 15}$ while the existing S-control is evaluated at average sample size values i.e., 8 and 14. The values of ${\tilde{ARL}}_{1}$ are shown in the Table 4 and 5.

Table 5
${\tilde{ARL}}_{1}$ performance under the existing and proposed designs

k ${\tilde{ARL}}_{1}$

Proposed method at $\tilde{n} \in {69}$ Existing method at n = 8

0.75 [1496.643 505085.160] 3795.2430

0.90 [1297.028 2883.263] 2054.4880

1.00 [275.644 334.7412] 314.4167

1.10 [73.852 78.9583] 75.5681

1.25 [16.459 21.3401] 17.8268

1.50 [4.1555 5.9750] 4.6240

2.00 [1.5003 2.0212] 1.62482

2.50 [1.1355 1.3711] 1.1874

3.00 [1.0444 1.1683] 1.0689

k	${\tilde{ARL}}_{1}$
0.75	[1496.643	505085.160]	3795.2430
0.90	[1297.028	2883.263]	2054.4880
1.00	[275.644	334.7412]	314.4167
1.10	[73.852	78.9583]	75.5681
1.25	[16.459	21.3401]	17.8268
1.50	[4.1555	5.9750]	4.6240
2.00	[1.5003	2.0212]	1.62482
2.50	[1.1355	1.3711]	1.1874
3.00	[1.0444	1.1683]	1.0689

We see from results reported in Table 5 and 6 that conventional S-control provides values within the indeterminacy intervals obtained from the proposed designed of $\tilde{S}$ -control chart. The in control value average run length at k = 1 is expected be at least 370. However this advertised value remain less than 370 in both designed charts irrespective of sample size due to biased structure of average run length distribution as discussed in [30]. Thus indeterminate intervals constructed under proposed designed are more accurate and effective over the determined values obtained from conventional approach. It is also worth mentioned here that $\tilde{S}$ -control chart provides equivalent results to S-control chat when n_l = n_u = n and δ_l = σ_u = σ.

4 Simulation study

A simulation study has been conducted in this section in order to validate the analytical expressions derived for the $\tilde{S}$ -control chart in previous section. To generate the data having indeterminate values, it has been assumed that process is in control at mean level 0, 2 and target standard deviation 5, 5. For this, data have been generated from the neutrosophic normal distribution with parameters $\tilde{μ} = {0, 2}$ and $\tilde{δ} = {5, 5}$ . Based on the ${\tilde{ARL}}_{0}$ equal to 370, in control limits have been constructed according to (1). Now in order to detect a shift of amount k standard deviation a neutrosophic sample $\tilde{n} = {10, 10}$ has been selected from neutrosophic normal distribution with parameters $\tilde{μ} = {0, 2}$ and ${\tilde{δ}}_{1} = k \tilde{σ}$ where k ≥ 1 .‘To facilitate these simulations for different values of k, a code has been written in R software [33]. Total 5000 Monte Carlo simulated realizations were generated for each increment of shift constant k that was assumed to vary from 1 to 4. For each simulated sample from the neutrosophic normal distribution, $\tilde{S}$ was calculated. Then value of ${\tilde{ARL}}_{1}$ was achieved according to (5) by observing the number of samples taken prior an out of control signal was detected, was recorded for each loop. The average value of ${\tilde{ARL}}_{1}$ over 5000 realizations was calculated and simulated results together values obtained exact from theoretical expression are given in Table 7.

Table 6
${\tilde{ARL}}_{1}$ performance under the existing and proposed designs

k ${\tilde{ARL}}_{1}$

Proposed method at $\tilde{n} \in {1215}$ Existing method at n = 14

0.75 [4.3031 12.1245] 5.7142

0.90 [132.254 321.8763] 170.9017

1.00 [785.592 1374.9074] 940.3643

1.10 [332.663 364.5042] 346.9119

1.25 [61.5333 67.6742] 63.1244

1.50 [10.7786 13.1556] 11.4407

2.00 [2.5664 3.1718] 2.7337

2.50 [1.1485 1.2680] 1.1799

3.00 [1.0198 1.0520] 1.0273

k	${\tilde{ARL}}_{1}$
0.75	[4.3031	12.1245]	5.7142
0.90	[132.254	321.8763]	170.9017
1.00	[785.592	1374.9074]	940.3643
1.10	[332.663	364.5042]	346.9119
1.25	[61.5333	67.6742]	63.1244
1.50	[10.7786	13.1556]	11.4407
2.00	[2.5664	3.1718]	2.7337
2.50	[1.1485	1.2680]	1.1799
3.00	[1.0198	1.0520]	1.0273

Table 7

${\tilde{ARL}}_{1}$ values from simulated data and exact analytical expression

Shift constant k	Simulated Values	Exact Values
1.0	332.7418	332.060
1.2	23.0742	23.4206
1.4	5.7502	5.8035
1.6	2.7604	2.7210
1.8	1.7786	1.7832
2.0	1.3970	1.4027
2.2	1.2132	1.2219
2.4	1.1312	1.1277
2.6	1.0766	1.0758
2.8	1.0488	1.0460
3.0	1.0252	1.0286
3.2	1.0190	1.0180
3.4	1.0110	1.0116
3.6	1.0070	1.0075
3.8	1.0052	1.0050
4.0	1.0024	1.0033

Results given in Table 7 indicate that the simulated values of ${\tilde{ARL}}_{1}$ agree with expected values obtained from the analytical expression given in (1). Thus, the simulation produced validate the analytical expression derived for S-control chart design developed under the imprecise environment.

5 Case study

An application of the proposed Neutrosophic S-control chart has been illustrated in this section. The proposed model has been implemented on manufactured parts obtained from an injection molding machine as mentioned in [34].

To assess the material behavior of the manufactured items, twenty samples each with five parts are collected and subjected to strength test. The compression strength for each part of all samples is recorded in a range and shown in Table 8. Thus, the compression strength for each manufactured part is assumed to be monitored in range instead of exact values. In order to monitor in control state of the manufacturing process, the Neutrosophic s-control chart has been established for strength crisp values. In this application size is taken as 5, i.e., $\tilde{n} \in {5, 5}$ and $\tilde{σ}$ is estimated by $\bar{\tilde{S}}$ which is the average of neutrosophic standard deviations of all twenty subgroups. The neutrosophic standard deviation for each sample is calculated and shown in the last column of Table 8.

Table 8
Real Data and Neutrosophic measures

Sample Number Sample Observations

${\tilde{X}}_{1}$ ${\tilde{X}}_{2}$ ${\tilde{X}}_{3}$ ${\tilde{X}}_{4}$ ${\tilde{X}}_{5}$ ${\tilde{\bar{X}}}_{j}$ ${\tilde{S}}_{j}$

1 [82.22,83.41] [81.13,81.35] [78.42,78.91] [74.94,75.71] [76.16,77.40] [15.71,15.87] [5.79, 7.77]

2 [88.33,89.53] [78.12,78.96] [78.63,79.03] [70.41,71.71] [84.00,85.09] [15.98,16.17] [3.68, 5.58]

3 [85.33,85.91] [75.10,76.19] [83.49,84.32] [75.19,76.20] [80.31,81.72] [15.98,16.17] [2.82, 4.92]

4 [80.23,81.45] [73.83,75.24] [82.12,82.63] [73.81,74.61] [74.78,75.91] [15.39,15.59] [1.64, 3.02]

5 [82.49,83.53] [78.23,78.55] [82.27,82.69] [77.92,78.69] [78.62,79.13] [15.98,16.10] [4.87, 6.65]

6 [75.10,75.57] [78.96,80.25] [86.70,87.54] [88.89,90.35] [81.70,81.94] [16.45,16.63] [1.99,4.52]

7 [73.60,74.89] [77.06,78.49] [80.20,81.59] [73.14,74.23] [79.00,80.18] [15.32,15.58] [3.77, 5.43]

8 [78.26,79.21] [84.27,84.55] [81.38,82.10] [85.28,86.48] [73.97,74.94] [16.13,16.29] [7.01, 9.50]

9 [79.84,80.88] [85.37,86.56] [75.91,77.11] [63.19,64.94] [79.39,81.17] [15.35,15.63] [3.07, 5.13]

10 [75.07,76.57] [74.73,76.16] [70.52,71.66] [81.15,82.61] [73.34,74.44] [14.99,15.26] [2.31, 3.42]

11 [79.94,80.34] [80.95,81.63] [77.77,79.16] [73.73,74.33] [77.99,79.05] [15.62,15.78] [2.58, 4.16]

12 [80.39,81.08] [81.25,81.81] [78.79,79.68] [73.05,74.37] [81.43,82.14] [15.79,15.96] [1.03, 2.38]

13 [82.52,83.30] [81.06,81.46] [78.59,79.47] [81.71,82.24] [79.01,79.56] [16.12,16.24] [0.85, 3.04]

14 [78.51,79.69] [74.14,75.71] [78.07,78.77] [77.60,78.58] [74.98,75.58] [15.33,15.53] [5.27, 7.29

15 85.12,85.69] [81.92,82.97] [82.24,83.25] [72.45,74.05] [71.14,71.73] [15.71,15.91] [0.71, 1.72]

16 [78.03,79.63] [79.19,80.11] [79.33,80.46] [78.68,79.58] [80.54,80.81] [15.83,16.02] [1.97, 4.18]

17 [81.60,82.77] [77.35,78.83] [74.67,75.84] [77.74,79.17] [81.90,82.59] [15.73,15.97] [1.98, 4.60]

18 [83.78,85.29] [75.92,77.74] [83.39,84.39] [80.23,81.66] [78.81,79.80] [16.09,16.36] [1.79,3.63]

19 [78.01,79.11] [77.57,78.08] [80.50,81.40] [83.82,85.02] [80.71,82.20] [16.02,16.23] [3.27, 5.21]

20 [84.12,85.22] [72.66,73.77] [77.70,79.18] [77.74,79.09] [80.05,81.00] [15.69,15.93] [2.29, 3.92]

Sample Number		Sample Observations
1	[82.22,83.41]	[81.13,81.35]	[78.42,78.91]	[74.94,75.71]	[76.16,77.40]	[15.71,15.87]	[5.79, 7.77]
2	[88.33,89.53]	[78.12,78.96]	[78.63,79.03]	[70.41,71.71]	[84.00,85.09]	[15.98,16.17]	[3.68, 5.58]
3	[85.33,85.91]	[75.10,76.19]	[83.49,84.32]	[75.19,76.20]	[80.31,81.72]	[15.98,16.17]	[2.82, 4.92]
4	[80.23,81.45]	[73.83,75.24]	[82.12,82.63]	[73.81,74.61]	[74.78,75.91]	[15.39,15.59]	[1.64, 3.02]
5	[82.49,83.53]	[78.23,78.55]	[82.27,82.69]	[77.92,78.69]	[78.62,79.13]	[15.98,16.10]	[4.87, 6.65]
6	[75.10,75.57]	[78.96,80.25]	[86.70,87.54]	[88.89,90.35]	[81.70,81.94]	[16.45,16.63]	[1.99,4.52]
7	[73.60,74.89]	[77.06,78.49]	[80.20,81.59]	[73.14,74.23]	[79.00,80.18]	[15.32,15.58]	[3.77, 5.43]
8	[78.26,79.21]	[84.27,84.55]	[81.38,82.10]	[85.28,86.48]	[73.97,74.94]	[16.13,16.29]	[7.01, 9.50]
9	[79.84,80.88]	[85.37,86.56]	[75.91,77.11]	[63.19,64.94]	[79.39,81.17]	[15.35,15.63]	[3.07, 5.13]
10	[75.07,76.57]	[74.73,76.16]	[70.52,71.66]	[81.15,82.61]	[73.34,74.44]	[14.99,15.26]	[2.31, 3.42]
11	[79.94,80.34]	[80.95,81.63]	[77.77,79.16]	[73.73,74.33]	[77.99,79.05]	[15.62,15.78]	[2.58, 4.16]
12	[80.39,81.08]	[81.25,81.81]	[78.79,79.68]	[73.05,74.37]	[81.43,82.14]	[15.79,15.96]	[1.03, 2.38]
13	[82.52,83.30]	[81.06,81.46]	[78.59,79.47]	[81.71,82.24]	[79.01,79.56]	[16.12,16.24]	[0.85, 3.04]
14	[78.51,79.69]	[74.14,75.71]	[78.07,78.77]	[77.60,78.58]	[74.98,75.58]	[15.33,15.53]	[5.27, 7.29
15	85.12,85.69]	[81.92,82.97]	[82.24,83.25]	[72.45,74.05]	[71.14,71.73]	[15.71,15.91]	[0.71, 1.72]
16	[78.03,79.63]	[79.19,80.11]	[79.33,80.46]	[78.68,79.58]	[80.54,80.81]	[15.83,16.02]	[1.97, 4.18]
17	[81.60,82.77]	[77.35,78.83]	[74.67,75.84]	[77.74,79.17]	[81.90,82.59]	[15.73,15.97]	[1.98, 4.60]
18	[83.78,85.29]	[75.92,77.74]	[83.39,84.39]	[80.23,81.66]	[78.81,79.80]	[16.09,16.36]	[1.79,3.63]
19	[78.01,79.11]	[77.57,78.08]	[80.50,81.40]	[83.82,85.02]	[80.71,82.20]	[16.02,16.23]	[3.27, 5.21]
20	[84.12,85.22]	[72.66,73.77]	[77.70,79.18]	[77.74,79.09]	[80.05,81.00]	[15.69,15.93]	[2.29, 3.92]

The Neutrosophic control limits based on ${\tilde{ARL}}_{0}$ equal to 370, have been set up according to Equations (10) and (11) as follows:

$\tilde{LCL} = [0, 0]$ $\tilde{UCL} = [6.14, 10.04]$

The upper and lower values of $\tilde{S}$ have been plotted in Fig. 2 in order to determine the presence of any assignable causes.

Fig. 2

The $\tilde{S}$ -control chart for real example data.

In Fig. 2, visual investigation indicates that all points are inside the control limits except value at the 7th sample is slightly over the lower $\tilde{UCL}$ . Thus, the point exceeding the control limit needs attention for further action. However, overall process is stable and working at usual nonrandom pattern. In Fig. 3, we have also set up conventional control chart for original data set. Similar interpretation is inferred from the conventional approach and process is only suspicious at 7th sample point. It is worth to mention that implementation of the proposed $\tilde{S}$ -control chart is not limited to industrial processing data but can be applied in a variety of cases with the aim to monitor the variability of the process.

Fig. 3

The existing S-control chart for real example data.

6 Conclusions

The designing of neutrosophic S-control chart has been presented in this study. The proposed chart represents a generalized extension of existing structure of S-control chart that could deal with data involving indeterminate measurements. The performance of proposed $\tilde{S}$ -control chart has been evaluated by developing the necessary neutrosophic measures. It has been observed from comparative analysis that the built design of the $\tilde{S}$ -control chart provides more accurate and effective results under inaccurate and ambiguous scenarios. The simulation study has also been conducted that exhibits the analytical expressions developed for the proposed $\tilde{S}$ -control chart. A real application example from manufacturing industry has considered that implicates the implementation procedure of the proposed design for processing data involving uncertain observations. Further, there are many other approaches such as different neutral network schemes that may be adopted to recognize the pattern of plotted points in control charts. Such future study would strengthen and expand the results of this work.

Footnotes

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under project number (RGP-2019-5).

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k	${\tilde{ARL}}_{1}$
	Proposed method at $\tilde{n} \in {69}$		Existing method at n = 8
0.75	[1496.643	505085.160]	3795.2430
0.90	[1297.028	2883.263]	2054.4880
1.00	[275.644	334.7412]	314.4167
1.10	[73.852	78.9583]	75.5681
1.25	[16.459	21.3401]	17.8268
1.50	[4.1555	5.9750]	4.6240
2.00	[1.5003	2.0212]	1.62482
2.50	[1.1355	1.3711]	1.1874
3.00	[1.0444	1.1683]	1.0689

k	${\tilde{ARL}}_{1}$
	Proposed method at $\tilde{n} \in {1215}$		Existing method at n = 14
0.75	[4.3031	12.1245]	5.7142
0.90	[132.254	321.8763]	170.9017
1.00	[785.592	1374.9074]	940.3643
1.10	[332.663	364.5042]	346.9119
1.25	[61.5333	67.6742]	63.1244
1.50	[10.7786	13.1556]	11.4407
2.00	[2.5664	3.1718]	2.7337
2.50	[1.1485	1.2680]	1.1799
3.00	[1.0198	1.0520]	1.0273

Sample Number		Sample Observations
	${\tilde{X}}_{1}$	${\tilde{X}}_{2}$	${\tilde{X}}_{3}$	${\tilde{X}}_{4}$	${\tilde{X}}_{5}$	${\tilde{\bar{X}}}_{j}$	${\tilde{S}}_{j}$
1	[82.22,83.41]	[81.13,81.35]	[78.42,78.91]	[74.94,75.71]	[76.16,77.40]	[15.71,15.87]	[5.79, 7.77]
2	[88.33,89.53]	[78.12,78.96]	[78.63,79.03]	[70.41,71.71]	[84.00,85.09]	[15.98,16.17]	[3.68, 5.58]
3	[85.33,85.91]	[75.10,76.19]	[83.49,84.32]	[75.19,76.20]	[80.31,81.72]	[15.98,16.17]	[2.82, 4.92]
4	[80.23,81.45]	[73.83,75.24]	[82.12,82.63]	[73.81,74.61]	[74.78,75.91]	[15.39,15.59]	[1.64, 3.02]
5	[82.49,83.53]	[78.23,78.55]	[82.27,82.69]	[77.92,78.69]	[78.62,79.13]	[15.98,16.10]	[4.87, 6.65]
6	[75.10,75.57]	[78.96,80.25]	[86.70,87.54]	[88.89,90.35]	[81.70,81.94]	[16.45,16.63]	[1.99,4.52]
7	[73.60,74.89]	[77.06,78.49]	[80.20,81.59]	[73.14,74.23]	[79.00,80.18]	[15.32,15.58]	[3.77, 5.43]
8	[78.26,79.21]	[84.27,84.55]	[81.38,82.10]	[85.28,86.48]	[73.97,74.94]	[16.13,16.29]	[7.01, 9.50]
9	[79.84,80.88]	[85.37,86.56]	[75.91,77.11]	[63.19,64.94]	[79.39,81.17]	[15.35,15.63]	[3.07, 5.13]
10	[75.07,76.57]	[74.73,76.16]	[70.52,71.66]	[81.15,82.61]	[73.34,74.44]	[14.99,15.26]	[2.31, 3.42]
11	[79.94,80.34]	[80.95,81.63]	[77.77,79.16]	[73.73,74.33]	[77.99,79.05]	[15.62,15.78]	[2.58, 4.16]
12	[80.39,81.08]	[81.25,81.81]	[78.79,79.68]	[73.05,74.37]	[81.43,82.14]	[15.79,15.96]	[1.03, 2.38]
13	[82.52,83.30]	[81.06,81.46]	[78.59,79.47]	[81.71,82.24]	[79.01,79.56]	[16.12,16.24]	[0.85, 3.04]
14	[78.51,79.69]	[74.14,75.71]	[78.07,78.77]	[77.60,78.58]	[74.98,75.58]	[15.33,15.53]	[5.27, 7.29
15	85.12,85.69]	[81.92,82.97]	[82.24,83.25]	[72.45,74.05]	[71.14,71.73]	[15.71,15.91]	[0.71, 1.72]
16	[78.03,79.63]	[79.19,80.11]	[79.33,80.46]	[78.68,79.58]	[80.54,80.81]	[15.83,16.02]	[1.97, 4.18]
17	[81.60,82.77]	[77.35,78.83]	[74.67,75.84]	[77.74,79.17]	[81.90,82.59]	[15.73,15.97]	[1.98, 4.60]
18	[83.78,85.29]	[75.92,77.74]	[83.39,84.39]	[80.23,81.66]	[78.81,79.80]	[16.09,16.36]	[1.79,3.63]
19	[78.01,79.11]	[77.57,78.08]	[80.50,81.40]	[83.82,85.02]	[80.71,82.20]	[16.02,16.23]	[3.27, 5.21]
20	[84.12,85.22]	[72.66,73.77]	[77.70,79.18]	[77.74,79.09]	[80.05,81.00]	[15.69,15.93]	[2.29, 3.92]