Performance of a New Time-Truncated Control Chart for Weibull Distribution Under Uncertainty

Ali Hussein AL-Marshadi; Ambreen Shafqat; Muhammad Aslam; Abdullah Alharbey

doi:10.2991/ijcis.d.210331.001

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Volume 14, Issue 1, 2021, Pages 1256 - 1262

Performance of a New Time-Truncated Control Chart for Weibull Distribution Under Uncertainty

Authors

Ali Hussein AL-Marshadi¹, Ambreen Shafqat², Muhammad Aslam¹^{, *}^,, Abdullah Alharbey¹

¹Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah, 21551, Saudi Arabia

²Department of Statistics and Financial Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing, Jiangsu, P.R. China

^*Corresponding author. Email: aslam_ravian@hotmail.com

Corresponding Author

Muhammad Aslam

Received 22 January 2021, Accepted 23 March 2021, Available Online 6 April 2021.

DOI: 10.2991/ijcis.d.210331.001 How to use a DOI?
Keywords: Weibull distribution; Neutrosophic Weibull distribution; Attribute chart; Time truncated; Shift
Abstract: To detect indeterminacy effect in the manufacturing process, attribute control chart using neutrosophic Weibull distribution is proposed in this paper. To make the attribute control chart more efficient for persistent shifts in the industrial process, an attribute control chart using Weibull distribution has been proposed recently. In this study, a neutrosophic Weibull distribution-based attribute control chart develop for efficient monitoring of the process. The indeterminacy effect was studied with the control chart's performance using characteristics of run length. In addition, the proposed chart effectively detected shifts in uncertainty. The relative efficiency of the proposed structure is compared with the existing attribute control chart under the Weibull time-truncated life test. The relative analysis reveals that the proposed time-truncated control chart for Weibull distribution under uncertainty design performance more efficiently than the existing counterparts. From the comparison, the proposed chart provides smaller values for the out-of-control average run length as compared to the existing attribute control chart. An illustrative application related to automobile manufacturing is also incorporated to demonstrate the proposal.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

A leading tool in the manufacturing process is the control chart that is applied to watch manufacturing shifts. A slight change in the quality target may cause big losses for a company. The variable control charts are designed to monitor the process using measurement data. Nonconforming items can be monitored using an attribute control chart. According to [1], “the traditional Shewhart np control charts are the statistical control scheme most commonly used for monitoring the number of non-conforming items.” For more details, the reader may refer to [1–4].

Usually, control charts work when data is derived from a normal process. When data moves away from normality, the chart designed with normal distribution cannot monitor the process. Control charts designed with non-normal distributions present a good alternative. According to [5], “In most life testing, to reduce the test time of the experiment, a failure-censored (type-II) scheme, or time-censored (type-I) scheme is usually adopted.” Therefore, the control chart designed with a time-truncated control can be used to save monitoring time. In the operational procedure of time-truncated charts, the experiment and watching times are fixed in advance, where an item is labeled as defective if its failure/service time is less than the specified time. The noted numbers of defects are plotted on the control chart to decide the state of the process. Previous studies [6] have proposed an attribute chart for Weibull distribution. Specifically, some studies [7] designed control charts using censored data. Others [8] have studied time-truncated charts for exponentiated half logistic distribution and Dagum distribution, respectively. Further, [9–12] show further control chart applications for various statistical distributions.

The Shewhart control charts given in the literature are unable to apply if some observations in the production data are uncertain or fuzzy. Fuzzy-based control charts are applied when uncertainty is presented in observations and parameters. According to [13], “fuzzy control charts are more sensitive than traditional ones; hence, they provide better quality products.” Furthermore, Darestani et al. [14] proposed a u-chart using a fuzzy approach, whereas [15] proposed a chart using fuzzy logic and [16] proposed a c-chart using fuzzy logic [17]. More work can be seen in [17–21].

The major drawback of a fuzzy-based control chart is that it is less informative than other charts. As such, Smarandache [22] mentioned that the generalized form of fuzzy logic is known as neutrosophic logic. Moreover, Smarandache and Khalid [23] stated that neutrosophic logic is more efficient than fuzzy logic and interval-based analysis. Other researchers [24] have discussed neutrosophic logic applications. Neutrosophic statistics is a branch of statistics that analyzes neutrosophic data, and many studies [25] have presented methods to analyze the indeterminacy data. Other works [26] have introduced indeterminacy in an attribute chart and proposed attribute charts for neutrosophic statistics [27]. Moreover, Khan et al. [28] proposed an S-chart using neutrosophic statistics. Lastly, Aslam et al. [29] proposed an efficient attribute chart. More information in neutrosophic statistics can be seen in [30,31].

The attribute control charts under neutrosophic statistics are presented in the literature. No previous work has designed a time-truncated attribute control chart for neutrosophic Weibull distribution using neutrosophic statistics. In this paper, we designed a time-truncated attribute control chart using neutrosophic statistics. The effect of the indeterminacy parameter was studied on the average run length (ARL). The advantages of the proposed chart are also discussed. In next section, we describe the design of the proposed chart. In Section 3, the comparison of the proposed chart with the counterpart chart and simulation study is described. The Illustrative example is described in Section 4. In the final section, the conclusion and future recommendations are displayed.

2. DESIGN OF THE PROPOSED CHART

First, we introduce the neutrosophic Weibull distribution. Then, we present the design of the current chart.

Suppose that xNϵxL,xU be a neutrosophic random variable having neutrosophic Weibull distribution, where xL and xU are the lower and upper values of neutrosophic random variable. The Weibull distribution neutrosophic probability density function (NPDF) and neutrosophic cumulative distribution function (NCDF) are defined as follows

fxN=βαxNαβ−1e−xNαβ+βαxNαβ−1e−xNαβIN; INϵIL,IU,xNϵxL,xU(1)

FxN=1−e−xNαβ1+IN +IN;INϵIL,IU,xNϵxL,xU(2)

where β is the shape parameter, α is the scale parameter, and IN shows the neutrosophic interval. The neutrosophic mean, say μN of the neutrosophic Weibull distribution is given by

μN=αΓ1+1/β1+IN(3)

where Γ. denotes the gamma function. Let μN0 target the product's lifetime and xN0 be a truncated time. The probability pN that an item fails by time xN0 is given as

pN=1−e−xNαβ1+IN+IN;INϵIL,IU,xNϵxL,xU(4)

Let xN0=aμN0 be the termination time, where a is constant and we express the unknown scale parameter α in term of μN using Eq. (3). Then, Eq. (4) can be rewritten as

pN=1−e−aβμNμN0−βΓ1+1/β1+INβ1+IN +IN;INϵIL,IU,xNϵxL,xU(5)

When μN=μN0, then the probability in Eq. (5) reduces to

pN0=1−e−aβΓ1+1/β1+INβ1+IN +IN;INϵIL,IU,xNϵxL,xU(6)

The np attribute control chart having lower control limit (LCL) and upper control limit (UCL) for the neutrosophic Weibull distribution is explained in the following steps:

Step 1. Count defective items of the products (D) from the selected sample.

Step 2. If D>UCL or D<LCL, the process is considered out of control. The process is considered in control if LCL≤D≤UCL.

Note that D follows a neutrosophic binomial distribution with parameters n and in-control probability pN0. Therefore, the control limits of the proposed np control chart are as follows:

UCL=npN0+knpN01−pN0(7a)

LCL=max0,npN0−knpN01−pN0(7b)

where k is the coefficient or control constant of the control limits to be resolute. The pN0 values are normally unknown; therefore, the averages of failed items (D = defective items) such as D¯ are taken. Thus, the neutrosophic control limits for practical use are

UCLN=D¯+kD¯1−D¯/n(8a)

LCLN=max0,D¯−kD¯1−D¯n(8b)

The probability for the in-control process (i.e., PNin0) is given by

PNin0=PLCL≤D≤UCL|pN0 =∑d=LCL+1UCLndpN0d1−pN0n−d(9)

If LCL=0, d should be 0 when applied. Let μN1 be the shifted average and the probability at the new mean is evaluated by

pN1=1−e−aβμN1μN0−βΓ1+1/β1+INβ1+IN +IN;INϵIL,IU,xNϵxL,xU(10)

If the shifted mean is μN1=fμN0 for a constant f, then Eq. (10) can be rewritten as

pN1=1−e−aβf−βΓ1+1/β1+INβ1+IN +IN;INϵIL,IU,xNϵxL,xU(11)

The in-process probability for the shifted mean can be written as

PNin1=PLCL≤D≤UCL|pN1 =∑d=LCL+1UCLndpN1d1−pN1n−d(12)

The in-control ARL can be calculated as

ARL0=11−PNin0(13)

The ARL for the shifted process can be calculated as

ARL1=11−PNin1(14)

ARL values for various values (e.g., IN, f, and β) are shown in Tables 1–3, where the following behavior can be noted.

When the indeterminacy parameter IN increases, the decreasing trend in ARL1 is noted.
For the same values, the values of ARL1 decreases as β increases.

f	β = 1, n = 30
	k = 3.26994, I_N = 0.1, a = 0.1545	k = 3.12394, I_N = 0.2, a = 0.1289	k = 3.23313, I_N = 0.4, a = 0.1082	k = 3.22172, I_N = 0.5, a = 0.09385
	ARL
1.0	374.93	373.97	373.41	370.43
0.9	165.38	163.88	154.50	152.48
0.8	70.06	68.93	61.44	60.33
0.7	28.65	27.97	23.68	23.13
0.6	11.47	11.11	9.03	8.79
0.5	4.65	4.48	3.59	3.48
0.4	2.07	1.99	1.65	1.61
0.3	1.18	1.15	1.06	1.05
0.2	1.00	1.00	1.00	1.00
0.1	1.00	1.00	1.00	1.00

Table 1

ARLs of the proposed chart when r0=370.

f	β = 1.1, n = 30
	k = 3.13684, I_N = 0.1, a = 0.1883	k = 3.19848, I_N = 0.2, a = 0.1584	k = 3.1863, I_N = 0.4, a = 0.1332	k = 3.21308, I_N = 0.5, a = 0.1310
	ARL
1.0	371.62	371.68	371.37	374.92
0.9	151.50	150.41	141.13	131.00
0.8	59.51	58.61	51.77	46.61
0.7	22.75	22.22	18.61	16.04
0.6	8.67	8.40	6.78	5.68
0.5	3.47	3.35	2.69	2.27
0.4	1.64	1.50	1.34	1.21
0.3	1.07	1.00	1.01	1.00
0.2	1.00	1.00	1.00	1.00
0.1	1.00	1.00	1.00	1.00

Table 2

ARLs of the proposed chart when r0=370.

f	β = 2, n = 30
	k = 3.58595, I_N = 0.1, a = 0.2167	k = 3.58132 I_N = 0.2, a = 0.2224	k = 2.89913, I_N = 0.4, a = 0.2472	k = 3.3222, I_N = 0.5, a = 0.2428
	ARL
1.0	372.83	370.03	371.83	370.58
0.9	136.70	120.64	88.27	79.58
0.8	47.76	37.90	21.24	17.70
0.7	16.20	11.85	5.64	4.53
0.6	5.58	3.96	1.94	1.62
0.5	2.17	1.64	1.09	1.03
0.4	1.17	1.05	1.00	1.00
0.3	1.00	1.00	1.00	1.001
0.2	1.00	1.00	1.00	1.00
0.1	1.00	1.00	1.00	1.00

Table 3

ARLs of the proposed chart when r0=370.

The values of k, ARL0, and ARL1 can be obtained through the following steps:

Step 1: Fix the values of β, n, and IN, thus determining the values of k via the grid search method.

Step 2: Several values of k are noted during simulation, where ARL0≥r0 is a specified ARL value.

Step 3: Choose the k value when ARL0 is close to r0.

Step 4: Determine ARL1 values for various values of f.

3. COMPARATIVE STUDIES

Our proposed control chart becomes the same chart proposed in [6] when IN=0. Table 4 presents both control charts when IN=0.1 and IN=0.5. Table 4 shows that the proposed control chart provides smaller ARLs values when compared to the chart proposed by [6]. For example, when f=0.9, β=1.1, n=30, and IN=0.1, the proposed control chart detects the first out-of-control value at the 151st sample, whereas the existing chart proposed by [6] detects the first out-of-control value at the 177th sample. Similarly, when f=0.9, β=1.1, n=30, and IN=0.5, the proposed control chart detects the first out-of-control value at the 131st sample, whereas and the existing chart proposed by [6] detects the first out-of-control value at the 177th sample. From this study, it can be seen that the proposed chart provides a quick indication about the shift in the process when compared to the control chart proposed by [6]. This study thus proved the efficiency of the proposed chart.

f	[6] chart (I_N = 0)		The proposed chart
	β = 1	β = 1.1	I_N = 0.1		I_N = 0.5
	k = 3.40009, a = 0.1344	k = 3.33227, a = 0.1394	k = 3.26994, I_N = 0.1, a = 0.1545	k = 3.13684, β = 1.1, a = 0.1883	k = 3.22172, β = 1, a = 0.09385	k = 3.21308, β = 1.1, a = 0.1310
	ARL
1.0	370.32	370.80	374.93	371.62	370.43	374.92
0.9	180.12	177.71	165.38	151.50	152.48	131.00
0.8	83.88	81.21	70.06	59.51	60.33	46.61
0.7	37.40	35.50	28.65	22.75	23.13	16.04
0.6	16.05	14.94	11.47	8.67	8.79	5.68
0.5	6.74	6.18	4.65	3.47	3.48	2.27
0.4	2.91	2.66	2.07	1.64	1.61	1.21
0.3	1.45	1.35	1.18	1.07	1.05	1.00
0.2	1.03	1.01	1.00	1.00	1.00	1.00
0.1	1.00	1.00	1.00	1.00	1.00	1.00

Table 4

Comparison of the proposed chart with the existing chart in ARLs.

3.1. Simulation Study

The performance of the proposed control chart was compared with the existing chart proposed by [6] using simulated data. The data was generated when a=0.1548, n=30, β=1, and IN=0.1_. Among 50 observations, the first 20 values were generated when the process was in-control (IC) state when μ0=1. Further, the next 30 observations were generated from the shifted process when f=0.8 by using the R software. The values of the number of defective items (D) under the proposed scheme are plotted in Figure 1 and the existing control chart displayed in Figure 2. Figure 1 indicates the shift at the 31st sample; the existing chart proposed by [6] indicates no shift. By comparing both figures, we can conclude that the proposed control chart indicates an issue in the existing control chart. Therefore, the application of the proposed control chart under uncertainty is helpful for minimizing the number of non-conforming items and got conforming products in less time.

4. ILLUSTRATIVE EXAMPLE

The proposed control chart was applied in an automobile manufacturing company located in South Korea. The company was interested in monitoring the service time in months for specific subsystems [6]. The same service data was applied by [6], who showed that the data followed the Weibull distribution with β=1. For this study, let a=0.1883, k=3.1368, n=30, p0=0.1722, and IN=0.1, which leads to the value of β=11+0.1=1.1 for the neutrosophic Weibull distribution. Let the value of μN0=60, which leads to time xN0=aμN0=0.1883∗60=11.29. From 400 observations, the statistic D is made by counting the observation as a failure if it is smaller than 11.29 months. The numbers of defectives D are shown in Figures 3 and 4, and also listed as follows:

3, 1, 4, 4, 3, 3, 3, 6, 4, 3, 5, 2, 3, 3, 4, 4, 4, 4, 6, 1, 4, 5, 5, 6, 7, 7, 2, 1, 7, 8, 3, 3, 3, 9, 4, 5, 10, 5, 5, 7.

The control limits for the proposed chart are UCL=8 and LCL=0 when the values of parameters values are used from Table 2. The proposed chart is displayed in Figure 3 and the existing chart is shown in Figures 4, respectively. Figure 3 shows the shift in service time means the shift values detect out of control at 31st sample. The existing chart indicates that no action is needed for service time because all values are in control. By comparing both charts, it can be seen that the proposed control chart indicates an issue in monthly service time for the specific subsystem of the car while the existing chart does not show any issue in monthly service time for the specific subsystem of the car.

5. CONCLUSIONS

The time-truncated attribute control chart was herein presented using the neutrosophic Weibull distribution. The measurement's indeterminacy effect was studied on the performance of the proposed control chart with the counterpart control chart. The proposed chart was the generalized version of the attribute control chart under a neutrosophic environment. From the results presented in Tables 1–4, we observed that the indeterminacy parameter significantly affected ARL values. The out-of-control ARL values reduced when indeterminacy increased. The comparative study proved the current chart's efficiency. The proposed control chart was proven to monitor service time in the automobile industry. The proposed control chart has limitations, i.e., it can only be applied when the life/service time follows the neutrosophic Weibull distribution. Secondly, the proposed control chart cannot be applied for variable data. The proposed control chart can be applied in the automobile industry, aircraft industry, and mobile industry for monitoring the defective items. In the future, the proposed control chart should be used for other neutrosophic statistical distributions. Following [7], the proposed control chart used an exponentially weighted moving average (EWMA) statistic and cumulative sum (CUSUM) statistic, both of which should be considered in future research.

CONFLICTS OF INTEREST

The authors declare no conflict of interest.

AUTHORS' CONTRIBUTIONS

A.H.A.M, A.S, M.A and A.A wrote the paper.

ACKNOWLEDGMENTS

The authors are deeply thankful to the editors and reviewers who offered invaluable suggestions for improving this manuscript. This article was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah under project No. (G-22-130-1441). The authors, therefore, gratefully acknowledge the DSR technical and financial support.

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 14 - 1
Pages: 1256 - 1262
Publication Date: 2021/04/06
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.d.210331.001 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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ris enw bib

TY  - JOUR
AU  - Ali Hussein AL-Marshadi
AU  - Ambreen Shafqat
AU  - Muhammad Aslam
AU  - Abdullah Alharbey
PY  - 2021
DA  - 2021/04/06
TI  - Performance of a New Time-Truncated Control Chart for Weibull Distribution Under Uncertainty
JO  - International Journal of Computational Intelligence Systems
SP  - 1256
EP  - 1262
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.210331.001
DO  - 10.2991/ijcis.d.210331.001
ID  - AL-Marshadi2021
ER  -

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