# A SkSP-R Plan under the Assumption of Gompertz Distribution
## 3 Design of the SkSP-R Plan for GmzD  
This section defines and discusses the SkSP-R for the GmzD distribution considered in this study. SkSP-R is introduced by [31] and showed its advantages over some other popular ASIPs. The SkSP-R plan parameters are \((n,c,i,f,k,\) and \(m)\). The procedure of time truncated SkSP-R can be described as follows:  
1. Begin with the normal inspection using the reference plan, and then place \(n\) items on test for prefixed time \(t_{0}\). Notice and count the number of sample items that failed before the experiment duration, say, \(d\). If \(d\leq c\), then accept the lot and reject it if \(d>c\).
2. Stop the normal inspection and utilize the skipping inspection (SI) if \(i\) successive units are accepted under normal inspection based on time truncated life tests.
3. Within SI, inspect only a fraction \(f\) of lots that is randomly selected. SI is continued until a sampled lot is rejected.
4. If a lot is not accepted after \(k\) consecutively sampled lots have been accepted, then the resampling procedure is employed for the immediate next lot as given below (Step 5).
5. Within the resampling technique, conduct the inspection based on the reference plan and continue SI if the lot is accepted. If the lot is not accepted, resampling is performed \(m\) times and the lot is rejected if it has not accepted on \((m-1)\)st resubmission.
6. If a lot is not accepted based on resampling scheme, then directly revert to the normal inspection (Step 1).
7. Remove or correct all the nonconforming units found with conforming units in the rejected lots.  
The average sample number (ASN) of the SkSP-R is as follows:  
\[ASN=\frac{nfQP^{i+k}-nfP^{k}(1-P^{i})(1-Q^{m})+nf}{P^{i}(1+fQP^{k})+f(1-P^{i} )[1-P^{k}(1-Q^{m})]}, \tag{4}\]  
and the OC function probability od acceptance of the proposed SkSP-R is as follows:  
\[P_{a}=\frac{(1-f)P^{i}+fp^{k}(P^{i}-P)(1-Q^{m})+fP}{P^{i}(1+fQP^{k})+f(1-P^{i} )[1-P^{k}(1-Q^{m})]}, \tag{5}\]  
where \(P=\sum_{j=0}^{c}{n\choose j}p^{j}(1-p)^{n-j}\), and \(p=F(t_{0})\) is the CDF of GmzD, which can be modified in terms of termination ratio and quality ratio as follows.  
\[p=1-e^{-\theta[e^{[t_{0}/n_{0}]\times(\theta^{2}\Gamma(\theta)/n/p_{0})}-1]}. \tag{6}\]  
Figure 2: Plots of hazard rate (**a**) and reliability (**b**) functions of the GmzD for various values of parameters.  
Now, we employed a two-point strategy to determine plan parameters; in this approach, the OC curve passes through both AQL and LQL. As a result, the optimization problem for determining plan parameters using the two-point technique AQL and LQL, as well as the optimization problem, is the following:  
\[ASN=\frac{nfQP^{i+k}-nfP^{k}(1-P^{i})(1-Q^{m})+nf}{P^{i}(1+fQ^{pk})+f(1-P^{i})[1 -P^{k}(1-Q^{m})]}, \tag{7}\]  
\[\frac{fP_{0}+(1-f)P_{0}^{i}+fP_{0}^{k}(P_{0}^{i}-P_{0})(1-Q_{0}^{m})}{f(1-P_{0} ^{i})[1-P_{0}^{k}(1-Q_{0}^{m})]+P_{0}^{i}(1+fQ_{0}P_{0}^{k})}\geq 1-\alpha_{p}, \tag{8}\]  
\[\frac{fP_{1}+(1-f)P_{1}^{i}+fP_{1}^{k}(P_{1}^{i}-P_{1})(1-Q_{1}^{m})}{f(1-P_{1} ^{i})[1-P_{1}^{k}(1-Q_{1}^{m})]+P_{1}^{i}(1+fQ_{1}P_{1}^{k})}\leq\beta, \tag{9}\]  
where \(P_{0}\) and \(P_{1}\) are probabilities at AQL and LQL, with \(Q_{0}\) and \(Q_{1}\) being \(1-P_{0}\) and \(1-P_{1}\), respectively. Our aim is to minimize the ASN of proposed SkSP-R where ASN depends on the sample size \(n\). Therefore, we minimize the sample size by using the above mentioned optimization problem (Equations (6)-(8)).