Life truncated multiple dependent state plan for imprecise Weibull distributed data

This paper aims to provide a multiple dependent state (MDS) sampling technique for light-emitting diode luminous intensities under indeterminacy by employing time truncated sampling schemes and the Weibull distribution. This indicates that ASN is significantly impacted by the indeterminacy parameter. Furthermore, a comparison is shown between the existing, indeterminate sampling plans and the recommended sample designs. The projected sampling technique is illustrated by calculating the luminous intensities of LEDs using the Weibull distribution. Based on the findings and practical example, we conclude that the suggested strategy needs a smaller sample size than SSP and the current MDS sampling plan.

The present work aims at testing light-emitting diode luminous intensities under indeterminacy by employing an MDS sampling approach for the Weibull distribution.It is expected that the proposed sampling design demonstrates a smaller ASN than the current sampling designs, hence testing the luminous intensities of lightemitting diodes.
We demonstrated the MDS sampling plan for the Weibull distribution under indeterminacy in Section "Methodologies".Section "Comparative studies" presented a comparison of existing indeterminacy sampling strategies and existing classical sampling plans.A real-world scenario with the luminous intensities of light-emitting diodes is used in Section "LED manufacturing process data illustration" to illustrate the proposed sampling plan for the indeterminacy.The conclusions and upcoming research projects are covered in Section "Concluding remarks".

Methodologies
Aslam 44 introduced neutrosophic Weibull distribution that will be recalled in this section.We will also provide the architecture of the sample plan for determining the mean luminosities of light-emitting diodes in unclear conditions.
Consider the neutrosophic probability density function (NPDF) f (x N ) = f (x L ) + f (x U )I N ; I N ǫ[I L , I U ] which has a determinate part f (x L ) , an indeterminate part f (x U )I N and indeterminacy interval I N ǫ[I L , I U ] .It should be noted that the neutrosophic random variable x N ǫ[x L , x U ] follows the NPDF.The generalization of the PDF under classical statistics is the NPDF.When I L =0 the classical statistics of the proposed neutrosophic form of f (x N )ǫ f (x L ), f (x U ) simplifies to PDF.The Weibull distribution's NPDF is defined as follows using this information.
where α and β are scale and shape parameters, accordingly.Here, it should be noted that the Weibull distribution's proposed NPDF is a generalization of its PDF in terms of classical statistics.When I L =0 the neutrosophic Weibull distribution's NPDF simplifies to the Weibull distribution.The Weibull distribution's neutrosophic cumulative distribution function (NCDF) is given by The Weibull distribution's neutrosophic mean is given by The neutrosophic Weibull distribution's median life is given by Balamurali et al. 22 provide the following well-designed methodology for MDS sampling design, and under neutrosophic statistics proposed by 45 .
The following are the alternative and null hypotheses for the average luminous intensities of light-emitting diodes: Where µ 0 denotes the desired average wind speed and µ represents the actual average wind speed.According to these data, the suggested sample approach is presented as follows: Step 1: Pick a sample from the batch that is size n.These samples were put through a life test for a set amount of time t N0 .Mention the average µ 0N and the amount of indeterminacy Step 2: The test H 0 : µ N = µ 0N could be accepted if the average daily number of cases for c 1 days are greater or equal to µ 0 (i.e.,µ 0N ≤ c 1 ).If average daily number of cases in c 2 days are less than to µ 0 (i.e., µ 0 >c 2 ) then test H 0 : µ N = µ 0N could be rejected and come to an end the test, where c 1 ≤ c 2 .
Step 3: When c 1 < µ 0N ≤ c 2 then accept the current lot if m preceding lots, the mean number of cases must be less than or equal to c 1 before the test termination time t N0 .
The proposed plan has four values, namely, n, c 1 , c 2 and m where n is the sample size, and c 1 is the maximum number of allowable items that failed for unconditional acceptance c 1 , c 2 is the maximum number of additional items that failed for conditional acceptance c 1 ≤ c 2 , and m is the number of subsequent lots (prior) required to reach a conclusion.The characteristics of the MDS sampling plan converge to m → ∞ and/or c 1 = c 2 = c (say), and MDS oversimplifies SSP.The OC function can be used to determine the concert of any sampling design.
Using the binomial chance law, the OC function for an MDS sample design based on WD is expressed as follows: Suppose that t 0 = aµ N0 be the time in days, where a is the termination ratio.The probability of accepting where p N is the probability of rejecting H 0 : µ N = µ N0 and obtained using Eqs.( 2) and (3) as p N = F(t N ≤ t N0 ) and defined by where µ N /µ N0 is the difference between the specified average luminous intensities of light-emitting diodes and the actual average luminous intensities of light-emitting diodes.Assume that γ and δ be type-I and type-II errors, respectively.The proposed plan for testing H 0 : µ N = µ N0 N0 is one that the meteorologists are interested in using because it ensures that the probability of accepting H 0 : µ N = µ N0 when it is true should be greater than 1 − γ at µ N /µ N0 and the probability of accepting H 0 : µ N = µ N0 when it is incorrect should be lower than δ at µ N /µ N0 = 1 .The following two inequalities will be satisfied by the plan parameters for testing H 0 : µ N = µ N0 .
where p 1N and p 2N are defined by and The average sample size (ASN) has often been decreased by the use of on-hand sampling techniques.Any sample strategy's main objective is usually to lower the ASN, which also helps to lower the amount of time and money needed for the inspection.Accordingly, the goal of the suggested MDS sample design is to lower the ASN for WD in the suggested scenario.The non-linear programming approach yields the optimal quantities, which are stated as follows: Minimize ASN(p 1N ) = n Subject to P a (p 1N ) ≥ 1 − γ wherep 1N and p 2N are the likelihood of failure at producer's and consumer's risks respectively.These acceptance probabilities might be calculated by using the following equations: and The proposed plan consists of parameters c 1 , c 2 , m and ASN that are obtained by solving the non-linear programming problem in Eq. ( 12) for δ = {0.25,0.10, 0.05} , γ = 0.10 , a = 0.5, 1.0 and known I N are placed in Tables 1, 2, 3, and 4. Tables 1 and 2 show the WD for β = 2 , while Tables 3 and 4 show the WD with β = 1 (exponential distribution).The following points can be drawn from the results in the tables.
(a) As the value of a increases from 0.5 to 1.0, the value of ASN decreases (b) When all other parameters are held constant, ASN decreases as the shape parameter increases from β = 1 to β = 2. (c) Furthermore, it is discovered that the indeterminacy parameter I N has a significant influence on minimizing ASN values. (

Comparative studies
This section discusses the suggested plan's effectiveness in terms of ASN.The lower the sample size, the less expensive it is to test the luminous intensity hypothesis for average LEDs.The suggested sample plan is the expansion of the plan under classical statistics if there is no uncertainty or indeterminacy in the recording of the average LED's luminous intensity.When I N =0, the suggested sampling plan lowers to the current sampling plan.The plan parameters under the classical statistics are shown in the first column of Tables 1, 2, 3, and 4. Tables 1, 2, 3, and 4 show that when the indeterminacy parameter I N rises, lower values of the ASN are needed to test H 0 : µ N = µ N0 .For instance, it can be observed that ASN = 54 from the plan under classical statistics and ASN = 46 for the suggested sample plan when I L =0.05 from Table 1 when µ N /µ N0 =1.5, δ =0.10, γ = 0.10 , a = 0.5 and β =2.According to the study, the existing sample plan under classical statistics is less effective in ASN than the proposed plan under indeterminacy.Additionally, as the WD transforms into an exponential distribution  www.nature.com/scientificreports/when β =1 for comparison purposes, we created Tables 3 and 4. Tables 1, 2, 3, and 4 illustrate that the WD shows fewer samples than the exponential distribution.For instance, Table 3 shows that the ASN is 39 when the recommended plan values are ASN = 25 for β =2 and when µ N /µ N0 =1.5, δ =0.25, γ = 0.10 , a = 0.5 and I L =0.04.The study's findings indicate that the existing sampling strategy under traditional statistics is less effective in terms of sample size than the expected sampling plan under indeterminacy.Figure 1 shows the operating characteristic (OC) curve of the WD plan for the conditions when γ = 0.10; δ = 0.10, β = 2.0 and a = 0.50 .Therefore, in order to test the null hypothesis, the proposed plan needs a lower ASN than the existing plan H 0 : µ N = µ N0 (Fig. 2).When there is uncertainty, the industrialist can implement the suggested plan faster and with less effort.The ASN performance of MDS design values for β = {2, 1.0} and δ =0.05 are displayed in Figs. 3 and 4. In these figures the ratio µ N /µ N0 is displayed on horizontal axis and ASN is given on vertical axis.These figures indicate   that the existing sampling strategy under traditional statistics is less effective in terms of ASN as compared with the proposed sampling plan under indeterminacy.The likelihood of MDS acceptance at different levels of indeterminacy is depicted in Fig. 1.The Fig. 2 shows that probability of acceptance is higher using MDS than the existing sampling plan.

LED manufacturing process data illustration
Park et al. 46 and Jin et al. 47 mentioned that quantum dot light-emitting diodes have uncertainty and inaccurate measurements of device parameters.Let's consider a case study on the production of light emitting light-emitting diodes (LEDs) that focuses on the luminous intensities of LED sources in order to illustrate the use of the provided methodologies.The operational process of light-emitting diode with the help of images can be seen in (https:// eepow er.com/ indus try-artic les/ an-intro ducti on-to-light-emitt ing-diodes/#).The justification for the process distribution is done and evidence that it resembles the Weibull distribution quite closely.The luminous intensities of LED data has been taken from 48 and they showed that the process distribution to be fairly close to the Weibull distribution.The steady process is sampled using n = 30 sample size.Due to the inevitable degree of imprecision in the data provided by a specific LED's luminous intensity, the luminous intensities of light-emitting diodes are supplied as lower and upper bounds as well as a point estimate, which are as follows:   The maximum distance between the real-time data and the fitted of WD is found from the Kolmogorov-Smirnov test statistic as [0.12964, 0.14889] and the p-value is [0.6473, 0.4742].It is established that the luminous intensities of light-emitting diodes data drawn from the WD with shape parameter β N = [1.75456,2.47762] and scale parameter α = [4.09506,6.23939] .As a result, WD is a good fit for data on the luminous intensities of light-emitting diodes.Table 5 lists the plan parameters for this shape parameter.The form parameter for the suggested strategy is β N = [1.75456, 2.47762] when I L =0.2918.Assume that engineering administrators want to test H 0 : µ N = [3.6186,5.4998] using the suggested sample scheme when I L =0.2918,γ = 0.10 , µ N /µ 0N =1.5, a=0.5 and δ=0.10.From Table 5, it can be noted that c 1 = 6 , c 2 =10, m = 1 and ASN = 25.
The created MDS sampling plan could function as follows: if the average luminous intensities of the lightemitting diodes in 6 measurements are greater than or equal to 23.5255 luminous intensities of the light-emitting diodes, accept the null hypothesis H 0 : µ N = [3.6186,5.4998] .For the batch of light-emitting diodes, a sample of 25 light-emitting diodes with varying luminous intensities will be chosen at random, using null hypothesis H 0 : µ N = [3.6186,5.4998] .The lot of light-emitting diodes will be allowed if the average luminous intensities of the light-emitting diodes prior to [3.6186, 5.4998] are less than or equal to 6 measures, and the lot of light-emitting diodes will be denied if it is larger than 10 measurements.A property of the current batch of light-emitting diodes will be delayed until the testing of the previous lot of light-emitting diodes if the luminous intensities of the light-emitting diodes are between 6 and 10 measurements.The average luminous intensities of light-emitting diodes are more than equal to [3.6186, 5.4998] in more than 17 measurements, which means the assertion that they are H 0 : µ N = [3.6186,5.4998] might be disproven based on the evidence.Whereas, when compared with the existing MDS plan to test H 0 : µ N = [3.6186,5.4998] when I L = 0,γ = 0.10 , µ N /µ 0N =1.5, a= 0.5 and δ= 0.10 the plan parameters are obtained as c 1 = 7 , c 2 = 16, m = 2 and ASN = 53.Which means that the lot of light-emitting https://doi.org/10.1038/s41598-024-55694-2www.nature.com/scientificreports/

Figure 1 .
Figure 1.At various indeterminacy values, the OC curve plan.

Figure 2 .
Figure 2. At various indeterminacy values, the OC curve of SSP and MDS.

Table 1 .
The MDS design values for β = 2 and a = 0.5.

Table 2 .
The MDS design values for β = 2 and a = 1.0.

Table 3 .
The MDS design values for β = 1 and a = 0.50.

Table 4 .
The MDS design values for β = 1 and a = 1.0.