# Time-Truncated Group Plan under a Weibull Distribution based on Neutrosophic Statistics
## 2 Design of the Proposed Plan using Neutrosophic Statistics  
The method to design the proposed GASP for providing quality assurance of the product in terms of mean life is discussed in this section. The ratio between the true mean life and the specified mean life of the product is considered as the quality of the product. A Weibull distribution is considered as an appropriate model to express the lifetime of the product because of its flexible nature. So, we assume that the lifetime of the product \(t_{N}\in\{t_{L},t_{U}\}\) under study follows a neutrosophic Weibull distribution, which has the shape parameter \(\delta_{N}\in\{\delta_{L},\delta_{U}\}\) and scale parameter \(\lambda_{N}\in\{\lambda_{L},\lambda_{U}\}\). Then, the cumulative distribution function (cdf) of the Weibull distribution is obtained as follows. \[F(t_{N};\lambda_{N},\delta_{N})=1-\exp\left(-\left(\frac{t_{N}}{\lambda_{N}} \right)^{\delta_{N}}\right)t_{N}\geq 0,\lambda_{N}>0,\delta_{N}>0. \tag{1}\]  
In this study, it is assumed that the scale parameter \(\lambda_{N}\) is unknown and the shape parameter \(\delta_{N}\) is known. It can be seen that the cdf depends only on \(t_{N}/\lambda_{N}\) since the shape parameter is known. One can estimate the shape parameter from the available history of the production process when it is unknown. The true mean life of the product under the neutrosophic Weibull distribution is calculated by the following equation  
\[\mu_{N}=\left(\frac{\lambda_{N}}{\delta_{N}}\right)\Gamma\left(\frac{1}{\delta _{N}}\right), \tag{2}\]  
where \(\Gamma(. )\) represents the complete gamma function. Then, the probability that the product will fail before it reaches the experiment time \(t_{N0}\),is denoted by \(p_{N}\) and is given as follows  
\[p_{N}=1-\exp\left(-\left(\frac{t_{N0}}{\lambda_{N}}\right)^{\delta_{N}}\right). \tag{3}\]  
As pointed out by [PERSON] and [PERSON] [8], we can write \(t_{N0}\) as a constant multiple of the specified mean life \(\mu_{N0}\), such as \(t_{N0}=t_{0}=\mathrm{a}\mathrm{\mu_{0}}a\mathrm{\mu_{N0}}\) where '\(a\)' is called the experiment termination ratio. Also, we can express the unknown scale parameter in terms of the true mean life and known shape parameters. After \(t_{N0}\), \(\lambda_{N}\) value substitution, and possible simplification, one can obtain the probability that the product will fail before attaining the experiment time \(t_{N0}\), using the following equation. \[p_{N} = 1-\exp\left(-a^{\delta_{N}}\left(\frac{\mu_{N0}}{\mu_{N}}\right)^{ \delta_{N}}\left(\frac{\Gamma\left(\frac{1}{\delta_{N}}\right)}{\delta_{N}} \right)^{\delta_{N}}\right), \tag{4}\]  
With respect to the ratios between the true mean life and the specified mean life, \(\mu_{N}/\mu_{N0}\), the acceptable quality level (AQL, i.e., \(p_{N1}\)) and limiting quality level (LQL, i.e., \(p_{N2}\)) are defined. That is, the failure probabilities obtained when the mean ratio values greater than one are taken as AQL and the same are obtained at a mean ratio equal to one are considered as LQL. The operating procedure of the proposed GASP for a time-truncated life test is described as follows:  
1. Take a sample of \(n_{N}\in\{n_{L},n_{U}\}\) items randomly from the submitted lot and distribute \(r_{N}\in\{r_{L},r_{U}\}\) items into \(g_{N}\in\{g_{L},g_{U}\}\) groups. Then, conduct the life test on the sample items for the specified time \(t_{N0}\). **Step 2. **: Observe the test and count number of sample items failed in each group before reaching experiment time \(t_{N0}\) and denote it as \(d_{N}\in\{d_{L},d_{U}\}\). **Step 3. **: If at most, \(c_{N}\) sample items found to be failed in each of all \(g_{N}\) groups, then accept the lot where \(c_{N}\in\{c_{L},c_{U}\}\). Otherwise, reject the lot. Two parameters used to characterize the proposed plan are number of groups \(g_{N}\) and the acceptance number \(c_{N}\). It is to be noted that \(r_{N}\in\{r_{L},r_{U}\}\) denotes the number of items in each group and is called the group size.