Testing of Grouped Product for the Weibull Distribution Using Neutrosophic Statistics

: Parts manufacturers use sudden death testing to reduce the testing time of experiments. The sudden death testing plan in the literature can only be applied when all observations of failure time/parameters are crisp. In practice however, it is noted that not all measurements of continuous variables are precise. Therefore, the existing sudden death test plan can be applied if failure data/or parameters are imprecise, incomplete, and fuzzy. The classical statistics have the special case of neutrosophic statistics when there are no fuzzy observations/parameters. The neutrosophic fuzzy statistics can be applied for the testing of manufacturing parts when observations are imprecise, incomplete and fuzzy. In this paper, we will design an original neutrosophic fuzzy sudden death testing plan for the inspection/testing of the electronic product or parts manufacturing. We will assume that the lifetime of the product follows the neutrosophic fuzzy Weibull distribution. The neutrosophic fuzzy operating function will be given and used to determine the neutrosophic fuzzy plan parameters through a neutrosophic fuzzy optimization problem. The results of the proposed neutrosophic fuzzy death testing plan will be implemented with the aid of an example.


Introduction
In life testing experiments, a random sample of items is usually selected, and a single item is installed on a single tester for the lot sentencing.This type of testing is costly as the number of testers is equal to the number of items that are selected for the testing purpose.Alternatively, the group sampling scheme is applied for the testing of more than one item on the single tester.Sudden death testing is implemented in groups to reduce the testing/inspection cost of the product.In sudden death testing, a random sample of size n is equally distributed to g groups having r items in each of the g groups.Reference [1] proposed the sudden test for the Weibull distribution.According to Reference [1] "The specimens in each group are tested identically & simultaneously on different testers.The 1st group of specimens is run until the 1st failure occurs.At this point, the surviving specimens are suspended & removed from testing.An equal set of new specimens numbering is next tested until the 1st failure.This process is repeated until one failure is generated from each of the groups".The sampling plan for sudden death testing has been considered by several authors in the literature, see for example References [1][2][3].
In the modern era, products are manufactured using advanced technology, which result in high quality and reliability.For the testing/inspection of a highly reliable product, it may not be possible to wait for the failures of the product for the lot sentencing.The two types of censoring widely applied to reduce the testing cost for a highly reliable product, are known as type-I censoring and type-II censoring.The testing is said to be type-I censoring if the time of experiment is fixed, and type-II if the number of failures is specified for the testing/inspection of the product.The acceptance sampling plans for type-I and type-II censoring when the lifetime follows the Weibull distribution is designed by a number of researchers, including, for example, References [4][5][6][7][8][9][10].
Parts manufacturers use sudden death testing to reduce the testing time of the experiment.The sudden death testing plan in the literature can only be applied when all observations of failure time/parameters are crisp.According to Reference [27], "all observations and measurements of continuous variables are not precise numbers but more or less non-precise.This imprecision is different from variability and errors.Therefore, lifetime data are also not precise numbers but more or less fuzzy.The best up-to-date mathematical model for this imprecision is so-called non-precise numbers".Therefore, the existing sudden death test plan can be applied if failure data or parameters are imprecise, incomplete, and fuzzy.The classical statistics have the special case of the neutrosophic statistics when there are no fuzzy observations/parameters.Thus, the neutrosophic fuzzy statistics can be applied for the testing of manufacturing parts when observations are imprecise, incomplete, and fuzzy.Recently, the authors of Reference [28] introduced the neutrosophic statistics in the area of acceptance sampling plan.
By exploring the literature, and to the best of the author's knowledge, there is no work on the design of neutrosophic fuzzy sudden death testing plan using the Weibull distribution.In this paper, we will design an original neutrosophic fuzzy sudden death testing plan for the inspection/testing of the electronic product or parts manufacturing.We will assume that the lifetime of the product follows the neutrosophic fuzzy Weibull distribution.The neutrosophic fuzzy operating function will be given and used to determine the neutrosophic fuzzy plan parameters through a neutrosophic fuzzy optimization problem.The results of the proposed neutrosophic fuzzy death testing plan will be implemented with the aid of an example.

Design of Proposed Plan
Suppose that T Ni {T L , T U } = i =1, 2, 3, . . ., n N being a random sample from the neutrosophic fuzzy Weibull distribution with neutrosophic fuzzy shape parameter m N and neutrosophic fuzzy scale parameter λ N .The neutrosophic fuzzy Weibull distribution is defined by: Some more details on fuzzy based Weibull distribution can be seen in References [29,30].It is assumed that items are tested identically and simultaneously in each group.We propose following a neutrosophic fuzzy sudden death testing plan.The quality characteristic beyond the lower specification limit L is defined as defective.Thus, the probability of defectiveness is given by: For the given p N , the corresponding λ N L from Equation ( 2) is given by: Step 1. Select a random sample n N {n L , n U } and distribute r items into g N {g L , g U } groups.
Step 2. Record first failure from ith group (i = 1, 2, . . .g N ) and calculate neutrosophic fuzzy statistic The proposed plan has two neutrosophic fuzzy parameters g N {g L , g U } and k N {k aL , k aU }.The proposed sampling plan is the extension of the plan proposed in Reference [1].The proposed sampling plan reduces to the plan in Reference [1] when g L = g U = g and k aL = k aU = k.The operational process of the proposed plan is also shown in Figure 1.sampling plan reduces to the plan in Reference [1] when   =   =  and   =   = .The operational process of the proposed plan is also shown in Figure 1.
where    is the neutrosophic fuzzy cumulative distribution function of the neutrosophic fuzzy gamma with a neutrosophic degree of freedom   .

Neutrosophic Fuzzy Non-Linear Optimization
Let  and  respectively be the producer's risk and consumer's risk.The plan parameters   {  ,   } and   {  ,   } will be determined (  ) ≥ 1 −  at acceptable quality level (AQL =  1 ) and (  ) ≤  at limiting quality level (LQL =  2 ).Therefore, the neutrosophic fuzzy plan parameters of the proposed plan will be determined by following the neutrosophic fuzzy non-linear optimization problem: minimize   {  ,   } (5) The plan parameters   {  ,   } and   {  ,   } will be determined through Equations ( 5) -( 7) using the grid search method.The following algorithm is used to determine the plan parameters.
Step 1. Specify the value of r.
Step 2. Determine the values of   and   using the search grid method through Equations ( 5)- (7).
where G δ N is the neutrosophic fuzzy cumulative distribution function of the neutrosophic fuzzy gamma with a neutrosophic degree of freedom δ N .

Neutrosophic Fuzzy Non-Linear Optimization
Let α and β respectively be the producer's risk and consumer's risk.The plan parameters g N {g L , g U } and k N {k aL , k aU } will be determined L C AQL ≥ 1 − α at acceptable quality level (AQL = p 1 ) and L C LQL ≤ β at limiting quality level (LQL = p 2 ).Therefore, the neutrosophic fuzzy plan parameters of the proposed plan will be determined by following the neutrosophic fuzzy non-linear optimization problem: minimize The plan parameters g N {g L , g U } and k N {k aL , k aU } will be determined through Equations ( 5)-( 7) using the grid search method.The following algorithm is used to determine the plan parameters.
Step 1. Specify the value of r.
Step 2. Determine the values of g N and k N using the search grid method through Equations ( 5)- (7).
Step 3. Choose the parameters for the plan where indeterminacy interval in g N is minimum.
The combinations that have smaller values of n N {n L , n U } are selected and reported in Tables 1  and 2. Table 1 reports for g N {g L , g U } and k N {k aL , k aU } when r = 5 and Table 2 reports for g N {g L , g U } and k N {k aL , k aU } when r = 10.From Tables 1 and 2, we noted the following trend in g N {g L , g U } and k N {k aL , k aU }.

1.
For the fixed values of neutrosophic parameters, g N {g L , g U } and k N {k aL , k aU } decrease as r increases from 5 to 10.

2.
For the fixed values of neutrosophic parameters, g N {g L , g U } and k N {k aL , k aU } decrease as LQL increases.

Application of Proposed Plan
The application of the proposed plan will be given on the ball bearing quality assurance, as the quality characteristic is measurable and as there is a chance that some observations may be fuzzy.Suppose that the ball bearing manufacturer is interested in applying the proposed sampling plan, but is not certain how to select suitable plan parameters for the testing of his product.Suppose he decides to install five items on the single tester.Let AQL = AQL = 0.001, LQL = 0.010, α = 0.05 and β = 0.10.From Table 1, we have g N {3, 5} and k N {106.0, 160.8}.Hence, he can select g between 3 and 5. Suppose experimenter decides to select a random sample size n = 25 and g = 5.
Step 1. Select a random sample 25 and distribute five items into five groups.
Step 2. Perform sudden death testing and note down the first failure from each of the five groups so reject the lot of ball bearing product.

Comparison Study
In this section, we will compare the efficiency of the proposed plan using the neutrosophic statistics and the sudden death sampling plan proposed in Reference [2] using the classical (crisp) statistics in terms of plan parameter g.For fair comparison, we will consider the same plan parameter values.By comparing Table 1 of the proposed plan with Table 1 of Reference [2] when r = 5, it can be seen that the proposed plan provides smaller values of g as compared to the plan in Reference [2].For easy reference, we present the comparison in g N between the proposed plan and Reference [2] in Table 3.For example, when AQL = 0.001 and LQL = 0.002 in Table 3, the proposed plan has a minimum value of g = 17 and a maximum value of g = 21, while the plan in [2] provides a crisp value of g = 231.Therefore, the proposed plan needs a sample size of n = r*g between 85 and 105.By comparison, the existing plan needs a sample size of 1155 for the testing of the same lot of the product.From Table 3, we note that the proposed plan has a smaller g N for all combinations of AQL and LQL.Therefore, the proposed plan required a smaller sample size for the inspection of the same product.Hence, less inspection cost is needed when the proposed sampling plan is implemented.

Concluding Remarks
In this paper, we have designed a neutrosophic fuzzy sudden death testing plan for the inspection/testing of the electronic product or parts manufacturing.The NFOC and neutrosophic fuzzy non-linear optimization problem are used to evaluate the proposed sampling plan.The proposed plan can be applied for the testing of parts when some observations or parameters are fuzzy.The proposed plan is the extension of a sudden death plan based on classical statistics.However, the proposed plan is more flexible than the plan based on classical statistics.Some tables are given and discussed with the help of a ball bearing example.The proposed plan can be applied for testing products in the automobile, aerospace and electronics industries.The proposed plan has the limitation that it can only be applied when the failure time follows the neutrosophic fuzzy Weibull distribution.The proposed plan for some other distribution can be considered as future research.The proposed sampling plan using big data can also be considered as future research.

Figure 1 .
Figure 1.Operational procedure of proposed plan.By following Reference [1] that Y m iN in v N = ∑ g N i=1 Y m iN follows the i.i.d.neutrosophic exponential distribution with neutrosophic parameter λ m N N r N , so v N follows the neutrosophic fuzzy gamma distribution with neutrosophic parameters g N , λ m N N r N ; g N {g L , g U }.The neutrosophic fuzzy operating function (NFOC) will be derived as follows: (i = 1, 2, . . ., 5).The number of first failures from the five groups are Y 1 = [220, 230], Y 2 = [300, 320], Y 3 = 285, Y 4 = [155, 165] and Y 5 = [365, 375].The lifetime of ball bearing follows the neutrosophic Weibull distribution with parameter m N {2, 2} and lower specification limit L = 200.The statistic v N is calculated as:

Table 1 .
The neutrosophic plan optimal parameters when r = 5.

Table 2 .
The neutrosophic plan optimal parameters when r = 10.

Table 3 .
[2] comparison of proposed plan with the plan in Reference[2].