# Time-Truncated Group Plan under a Weibull Distribution based on Neutrosophic Statistics
## 2 Design of the Proposed Plan using Neutrosophic Statistics
The operating procedure of the proposed GASP is represented by a flow chart and is shown in Figure 1.  
In general, an operating characteristic (OC) function helps to investigate the performance of the sampling plan. The OC function of the proposed GASP under a Weibull model based on time-truncated test is given by  
\[P_{AN}(p_{N})=\left[\sum_{d_{N}=0}^{c_{N}}\left( \begin{array}{c}r_{N}\\ d_{N}\end{array} \right)p_{N}^{d_{N}}\left(1-p_{N}\right)^{n_{N}-d_{N}}\right]^ {g_{N}}. \tag{5}\]  
Generally, each producer wishes that the sampling plan should provide a chance greater than \((1-\alpha)\) to accept the product when the product quality is at AQL, where \(\alpha\) is the producer's risk, whereas the consumer wants that the chance to accept the lot to be less than \(\beta\) when quality of the product is at LQL, where \(\beta\) is the consumer's risk. Obviously, the sampling plan that involves the minimum risks to both producer and consumer will be favorable. The design of the sampling plan by considering AQL and LQL, along with producer and consumer risks is known as two points on the OC curve approach and this approach is considered as the most important among others. Similarly, the sampling plan that makes its decision on the submitted lot using minimum sample size or average sample number (ASN) will be attractive. Therefore, in this study, we design GASP with the intention  
Figure 1: Operating procedure of the proposed group acceptance sampling plan (GASP) under a truncated life test. of assuring a Weibull-distributed mean life of the products with minimum sample size and minimum cost using two points on the OC curve approach. It should be mentioned that the ASN of the proposed plan is the product of the number of groups and group size (i.e., \(n_{N}=g_{N}r_{N}\)). For determining the optimal parameters, we use the following optimization problem. \[ \begin{array}{c}\text{Minimize }g_{N}\\ \text{Subject to }P_{a}(p_{N1})\geq 1-\alpha,\\ \\ P_{a}(p_{N2})\leq\beta,\\ g_{N}\geq 1,r_{N}>1,\ c_{N}\geq 0,\end{array}  \tag{6}\]  
where \(p_{N1}\) and \(p_{N2}\) are the failure probabilities obtained from the following equations  
\[p_{N1}=1-\exp\left(-a^{\delta_{N1}}\left(\frac{\mu_{N0}}{\mu_{N}}\right)^{ \delta_{N1}}\left(\frac{\Gamma\left(\frac{1}{\delta_{N1}}\right)}{\delta_{N1}} \right)^{\delta_{N1}}\right)\delta_{N1}\in\{\delta_{L1},\delta_{U1}\}, \tag{7}\]  
\[p_{N2}=1-\exp\left(-a^{\delta_{N2}}\left(\frac{\mu_{N0}}{\mu_{N}}\right)^{ \delta_{N2}}\left(\frac{\Gamma\left(\frac{1}{\delta_{N2}}\right)}{\delta_{N2} }\right)^{\delta_{N2}}\right)\delta_{N2}\in\{\delta_{L2},\delta_{U2}\}, \tag{8}\]  
\[P_{aN}(p_{N1})=\left[\sum_{d_{N}=0}^{c_{N}}\left( \begin{array}{c}r_{N}\\ d_{N}\end{array} \right)p_{N1}^{d_{N}}\left(1-p_{N1}\right)^{r_{N}-d_{N}}\right] ^{\delta_{N}}, \tag{9}\]  
\[P_{aN}(p_{N2})=\left[\sum_{d_{N}=0}^{c_{N}}\left( \begin{array}{c}r_{N}\\ d_{N}\end{array} \right)p_{N2}^{d_{N}}\left(1-p_{N2}\right)^{r_{N}-d_{N}}\right] ^{\delta_{N}}. \tag{10}\]  
In this designing, we define AQL as the failure probability corresponding to the mean ratios \(\mu_{N}/\mu_{N0}=2\), 4, 6, 8, 10. Similarly, the LQL is defined as the failure probability corresponding to the mean ratio \(\mu_{N}/\mu_{N0}=1\). The optimal parameters of the proposed GASP are determined for various combinations of group size, shape parameter, and producer's risk. We used the grid search method under neutrosophic statistics to find the optimal values of parameters [\(g_{L}\), \(g_{U}\)] and [\(c_{L}\), \(c_{U1}\)]. We selected those values of parameters from several combinations of parameters that satisfy the given conditions where the range between \(g_{L}\) and \(g_{U}\) is at a minimum. For this determination, we considered two sets of group sizes, such as \(r_{N}=\{10,\,12\}\) and \(r_{N}=\{4,\,6\}\), and two sets of shape parameters, such as \(\delta_{N}=\{0.9,\,1.1\}\) and \(\delta_{N}=\{1.9,\,2.1\}\).