A group acceptance sampling plan truncated life test for alpha power transformation inverted perks distribution based on quality control reliability

Abstract

In the present study, a group-acceptance sampling plan was presented when the lifetime of an item followed an alpha power transformation-inverted perks distribution. The median is used as a quality index to attain design parameters, such as the acceptance number and minimum group size, which are determined when the test termination and consumer risk are specified. The operating characteristic values are presented both graphically and in the form of tables. Two real-life datasets are used as examples to illustrate the tables.

Keywords:

• Operating characteristic
• sampling plan
• APTIP
• quality

1. Introduction

Group acceptance sampling plans (GASP) are a statistical method used in quality control in order to determine whether to accept or reject a batch of products or services based on the results of a sample taken from that batch. There is, however, a difference between a GASP and an acceptance sampling plan (ASP) in terms of the way samples are taken and analyzed (Al-Omari et al., 2019; Banihashemi et al., 2021). ASP involves selecting a random sample from a batch and inspecting it for defects or other quality issues. The sampling plan is typically based on a statistical model that considers batch size, sample size, and acceptable quality level (AQL). If the number of defects or issues in the sample is within the AQL, the batch is accepted; otherwise, it is rejected (Stern, 2004). On the other hand, GASP involves dividing the batch into subgroups or groups and then randomly selecting a sample from each group. The samples are then evaluated together, and if the number of defects or issues in the combined sample is within the AQL, the entire batch is accepted. If not, the entire batch is rejected (Moussawi-Haidar et al., 2013). GASP is used when it is difficult or costly to inspect each unit individually or when there is a risk of lot-to-lot variation. It can also be used when the product or service is complex and different parts of the batch have different levels of quality (Bouslah et al., 2013). In a nutshell, ASP is used to evaluate a batch based on a random sample, whereas GASP divides the batch into subgroups and evaluates the combined sample from each group. These methods are useful for quality control, and the choice of method depends on the specific circumstances and requirements of the product or service being evaluated (Mobley, 2002).

Advancements in technology have accelerated the production process, and as far as quality is concerned, 100% inspection cannot be highlighted in a statistical experiment owing to time and monetary limitations (Mohamed et al., 2015). Numerous statistical and sampling distribution tests have been conducted to ensure effective product quality (Ryan, 2011). In the area of quality control, ASP have gained attention for the life testing of different distributions that will help in the acceptance or rejection of lots taken from samples (Balakrishnan et al., 2007). ASP is a method that typically specifies the lot disposition criteria and the necessary sample size. It also determines the number of units used for screening, such as n, as well as the acceptance number c. Hence, if there are no more than c failures out of n items, the lot is accepted; otherwise, it is rejected. Consumer and producer risks are the possibility that a poor lot will be authorized and a good lot will be disallowed (Wang et al., 2021). Every sampling plan included information on consumer and producer risks (Wu et al., 2017).

Potential motivation for using the GASP is to increase the efficiency and reduce the costs of the quality control process. Instead of checking each item in a batch, which can be time-consuming and costly, GASP allows a representative sample of items to be tested. This provides a good idea of the overall quality of the batch. Another motivation provided by (Al-Nasser et al., 2018) for using GASP is to reduce the risk of accepting a batch with a high proportion of defective items. Batches with many flaws can be identified and rejected by testing a small number of items. This can help avoid expensive recalls, returns, or customer complaints (Aslam, et al., 2013). In addition, GASP can help increase confidence in the quality of the products being produced, which can be important for maintaining customer satisfaction and loyalty. By implementing a quality control process that includes GASP, organizations can demonstrate their commitment to producing high-quality products, which can help build trust and confidence in customers (Montgomery, 2020).

Researchers have been interested in the parameter induction strategy, which uses the composite approach and baseline distribution to introduce new distributions. Evans et al. (2011) provided a detailed description of the generation of new distributions utilising additional factors. Elbatal et al. (2022) presented an inverted perks distribution, which is the most important perspective in this research. To date, many scholars have also introduced many other transformed distributions, such as the alpha power transformation Lindley distribution (Dey et al., 2019), the APT Lomax distribution (ZeinEldin et al., 2020), the APT Weibull distribution (Ramadan & Magdy, 2018), the APT exponential distribution (Ijaz et al., 2021), the APT top-leone distribution (Hozaien et al., 2022), the APT log logistic distribution (Aldahlan, 2020), the APT-exponentiated T-X distribution (Klakattawi & Aljuhani, 2021), the APT Pareto distribution (Kinaci et al., 2019), and the APT Aradhana distribution (Mohiuddin & Kannan, 2021), Marshall–Olkin alpha power inverse exponential distribution (Basheer, 2022), generalization of Lindley distribution (Ahsan-Ul-Haq, Choudhary, et al., 2022), and power Lomax distribution (Al-Naseer & Haq, 2021). Many developments have been made by researchers on APT-inverted distributions; however, no work has been performed using the APTIP distribution. To date, information on performing the GASP technique to maintain the quality of products and improve the selection of products with minimum consumer and producer risk has been provided in this research article, along with some important features of the APTIP distribution are given;

1. APTIP distribution has a wide range of applications in the analysis of lifespan data.

2. In the APTIP model, pdf and hrf shapes are more feasible.

3. The APTIP model outperformed its competitors in terms of fitting.

GASP has already been studied and worked on by many authors and has developed numerous lifetime distributions: Marshall–Olkin extended to the Lomax distribution (Rao, 2009a), generalized exponential distribution (Rao, 2009b), gamma distribution (Gupta & Gupta, 1961), log logistic distribution (Aslam et al., 2009), inverse Rayleigh (Ramaswamy & Anburajan, 2012), Weibull distribution (Aslam & Jun, 2009), generalized inverted exponential (S. Singh et al., 2015), inverse power Lomax (Ashraf et al., 2022), exponentiated inverted Weibull (Suseela & Rao, 2021), skew-generalized inverse Weibull (N. Singh et al., 2022), and Odd generalized exponential log-logistic distribution (Sivakumar et al., 2019), Exponential Lifetime Distribution (Mughal, 2011), Marshall–Olkin Kumaraswamy Exponential (MOKw-E) Distribution by Almarashi et al. (2021), Generalized Inverted Kumaraswamy Distribution by Alsultan and Khogeer (2022), Burr Type X Percentiles (Aslam, Wu, et al., 2013), and Generalized Transmuted‐Exponential Distribution (Fayomi & Khan, 2022).

The structure of this research paper is as follows: We introduced GASP in section 1. The cumulative distribution function (cdf), probability distribution function (pdf), and quantile function (qf) are presented in section 2. We describe GASP under the APTIP model in section 3. In Section 4 we describe the tables. In Sections 5 (Application) and 6, a discussion is presented. In Section 7, conclusions are presented.

2. Alpha power transformed inverted perks distribution

The random variable Y has the inverted perks distribution with two parameters β and λ, and its cdf is given by

(1) $\mathrm{F}\left(\mathrm{Y}\right)=1-\left(\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{\mathit{\lambda }{\mathrm{y}}^{-1}}}\right),\mathrm{y}\ge 0,\mathit{\beta },\mathit{\lambda }>0$(1)

pdf is given by

(2) $\mathrm{f}\left(\mathrm{Y}\right)=\frac{\mathit{\beta }\left(1+\mathit{\beta }\right)\mathit{\lambda }{\mathrm{e}}^{\mathit{\lambda }{\mathrm{y}}^{-1}}}{{\left(\mathrm{x}+\mathit{\beta }\mathrm{y}{\mathrm{e}}^{\mathit{\lambda }{\mathrm{y}}^{-1}}\right)}^2}$(2)

The cdf of the APT is given below

(3) $\mathrm{FAPT}\left(\mathrm{y};\mathit{\alpha }\right)=\left\{\begin{array}{c}\frac{{\mathit{\alpha }}^{\mathit{F}\left(\mathit{y}\right)}-1}{\mathit{\alpha }-1},\mathit{a}>0,\mathit{a}\ne 1\\ \mathit{F}\left(\mathit{y}\right)\mathit{if}\mathit{a}=1\end{array}\right.$(3)

The corresponding to their pdf is given as

(4) ${\mathrm{f}}_{\mathrm{APT}}\left(\mathrm{y};\mathit{\alpha }\right)=\left\{\begin{array}{c}\frac{\mathit{log\alpha }}{\mathit{\alpha }-1}\mathit{f}\left(\mathit{y}\right){\alpha }^{\mathit{F}\left(\mathit{y}\right)},\mathit{a}>0,\mathit{a}\ne 1\\ \mathit{F}\left(\mathit{y}\right)\mathit{if}\mathit{a}=1\end{array}\right.$(4)

Based on the alpha power transformation, many new distributions are liked APT by Mahdavi and Kundu (2017), and cdf of APTIP is given, where α > 0, β > 0,

(5) ${\mathrm{F}}_{\mathrm{APTIP}}\left(\mathrm{x}\right)=\left\{\begin{array}{c}\frac{{\mathit{\alpha }}^{1-\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{\mathit{\lambda }{\mathrm{x}}^{-1}}}}-1}{\mathit{\alpha }-1},\mathit{\alpha }>0,\mathit{\alpha }\ne 1\\ \left(1-\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{\mathit{\lambda }{\mathrm{x}}^{-1}}}\right)\mathit{if}\mathit{\alpha }=1\end{array}\right.$(5)

The corresponding pdf of APTIP distribution is

$\mathit{fAPTIP}\left(\mathit{x}\right)=\left[\left(\frac{(log\left(\mathit{\alpha }\right)}{\mathit{\alpha }-1}\right)\left(\frac{\left(\mathit{\beta }\left(1+\mathit{\beta }\right)\lambda {\mathit{e}}^{\mathit{\lambda }{\mathit{x}}^{-1}}\right)}{{\left(\mathit{x}+\mathit{\beta x}{\mathit{e}}^{\mathit{\lambda }{\mathit{x}}^{-1}}\right)}^2}\right){\alpha }^{\left(\frac{1+\mathit{\beta }}{1+\mathit{\beta \lambda }{\mathit{e}}^{\mathit{\lambda }{\mathit{x}}^{-1}}}\right)},\mathit{\alpha }>0,\mathit{\alpha }\ne 1\mathit{and}\left(\frac{\left(\mathit{\beta }\left(1+\mathit{\beta }\right)\lambda {\mathit{e}}^{\mathit{\lambda }{\mathit{x}}^{-1}}\right)}{{\left(\mathit{x}+\mathit{\beta x}{\mathit{e}}^{\mathit{\lambda }{\mathit{x}}^{-1}}\right)}^2}\right)\mathit{if}\mathit{a}=1\right]\left(6\right)$

Thus, quantile function of APTIP as given,

(7) ${\mathrm{Q}}_{\mathrm{APTIP}}\left(\mathrm{p}\right)=\mathit{\lambda }\left[\frac{1}{\log \left\{\frac{1}{\mathit{\beta }}\left(1+\mathit{\beta }\right)\left(\frac{\mathrm{log\alpha }}{\mathrm{log\alpha }-\log \left({\mathrm{p}}_{\mathit{\alpha }-\mathrm{P}+1}\right)}\right)-1\right\}}\right]$(7)

3. GASP described under the APTIP model

In this study, we used the median as the quality index. As stated by (Shafiq et al., 2022), medians perform better for skewed types of distributions. However, the primary goal of this study is to provide a GASP when the lifetime of an item follows APTIP distributions with the specified parameters α, β, λ, and cdf in Equation 5(5) ${\mathrm{F}}_{\mathrm{APTIP}}\left(\mathrm{x}\right)=\left\{\begin{array}{c}\frac{{\mathit{\alpha }}^{1-\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{\mathit{\lambda }{\mathrm{x}}^{-1}}}}-1}{\mathit{\alpha }-1},\mathit{\alpha }>0,\mathit{\alpha }\ne 1\\ \left(1-\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{\mathit{\lambda }{\mathrm{x}}^{-1}}}\right)\mathit{if}\mathit{\alpha }=1\end{array}\right.$(5) . GASP provides valuable information about the lot by employing statistical techniques chosen at random to effectively test the overall performance of the product. Numerous steps were used to obtain the design parameters in GASP; the following steps are as follows:

1. The original group sampling inspection is carried out by randomly selecting n items and evenly distributing them into g groups, with r items and n = r × g in each group.

2. Determine the duration of the experiment, t₀, and the acceptance number, c.

3. The lot is accepted if the total number of failures across all g groups is less than c within t₀; otherwise, it is rejected.

4. The experiment is terminated when the total number of failures from all groups exceeds c or when the experiment time runs out, whichever comes first.

GASP is defined for the given hypothesized value of r under the two design parameters (g, c) for the APTIP model. In addition, the cdf defined in Equation 5(5) ${\mathrm{F}}_{\mathrm{APTIP}}\left(\mathrm{x}\right)=\left\{\begin{array}{c}\frac{{\mathit{\alpha }}^{1-\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{\mathit{\lambda }{\mathrm{x}}^{-1}}}}-1}{\mathit{\alpha }-1},\mathit{\alpha }>0,\mathit{\alpha }\ne 1\\ \left(1-\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{\mathit{\lambda }{\mathrm{x}}^{-1}}}\right)\mathit{if}\mathit{\alpha }=1\end{array}\right.$(5) is dependent on the parameters α, β, and λ. The median lifetime of the truncated life test is defined by Equation 6(7) ${\mathrm{Q}}_{\mathrm{APTIP}}\left(\mathrm{p}\right)=\mathit{\lambda }\left[\frac{1}{\log \left\{\frac{1}{\mathit{\beta }}\left(1+\mathit{\beta }\right)\left(\frac{\mathrm{log\alpha }}{\mathrm{log\alpha }-\log \left({\mathrm{p}}_{\mathit{\alpha }-\mathrm{P}+1}\right)}\right)-1\right\}}\right]$(7) , and the acceptance of the lot probability is expressed as:

(8) ${\mathrm{P}}_{\mathrm{a}}\left(\mathrm{p}\right)={\left[{\sum }_{\mathrm{i}=0}^{\mathrm{c}}\left(\begin{array}{c}\mathrm{r}\\ \mathrm{i}\end{array}\right){\mathrm{p}}^{\mathrm{i}}{\left(1-\mathrm{p}\right)}^{\mathrm{r}-1}\right]}^{\mathrm{g}},$(8)

Therefore, p represents the likelihood of an item in a group that has failed previously t_o. To insert Equation 6(7) ${\mathrm{Q}}_{\mathrm{APTIP}}\left(\mathrm{p}\right)=\mathit{\lambda }\left[\frac{1}{\log \left\{\frac{1}{\mathit{\beta }}\left(1+\mathit{\beta }\right)\left(\frac{\mathrm{log\alpha }}{\mathrm{log\alpha }-\log \left({\mathrm{p}}_{\mathit{\alpha }-\mathrm{P}+1}\right)}\right)-1\right\}}\right]$(7) into Equation 5(5) ${\mathrm{F}}_{\mathrm{APTIP}}\left(\mathrm{x}\right)=\left\{\begin{array}{c}\frac{{\mathit{\alpha }}^{1-\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{\mathit{\lambda }{\mathrm{x}}^{-1}}}}-1}{\mathit{\alpha }-1},\mathit{\alpha }>0,\mathit{\alpha }\ne 1\\ \left(1-\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{\mathit{\lambda }{\mathrm{x}}^{-1}}}\right)\mathit{if}\mathit{\alpha }=1\end{array}\right.$(5) . Based on Equation 6(7) ${\mathrm{Q}}_{\mathrm{APTIP}}\left(\mathrm{p}\right)=\mathit{\lambda }\left[\frac{1}{\log \left\{\frac{1}{\mathit{\beta }}\left(1+\mathit{\beta }\right)\left(\frac{\mathrm{log\alpha }}{\mathrm{log\alpha }-\log \left({\mathrm{p}}_{\mathit{\alpha }-\mathrm{P}+1}\right)}\right)-1\right\}}\right]$(7) , we set,

(9) $\mathit{\mu }=\left[\frac{\mathit{\lambda }}{\log \left\{\frac{1}{\mathit{\beta }}\left(1+\mathit{\beta }\right)\left(\frac{\mathrm{log\alpha }}{\mathrm{log\alpha }-\log \left({\mathrm{p}}_{\mathit{\alpha }-\mathrm{P}+1}\right)}\right)-1\right\}}\right]$(9)

and

(10) $\mathit{\eta }=\left[\frac{1}{\log \left\{\frac{1}{\mathit{\beta }}\left(1+\mathit{\beta }\right)\left(\frac{\mathrm{log\alpha }}{\mathrm{log\alpha }-\log \left({\mathrm{p}}_{\mathit{\alpha }-\mathrm{P}+1}\right)}\right)-1\right\}}\right]$(10)

To substitute η in Equation 8(8) ${\mathrm{P}}_{\mathrm{a}}\left(\mathrm{p}\right)={\left[{\sum }_{\mathrm{i}=0}^{\mathrm{c}}\left(\begin{array}{c}\mathrm{r}\\ \mathrm{i}\end{array}\right){\mathrm{p}}^{\mathrm{i}}{\left(1-\mathrm{p}\right)}^{\mathrm{r}-1}\right]}^{\mathrm{g}},$(8) , λ=$\frac{\mathit{\mu }}{\mathit{\eta }}$ is obtained. Let λ=$\frac{\mathit{\mu }}{\mathit{\eta }}$ and t_o =a₁${\mathit{\mu }}_{\mathrm{o}}$, where the product quality can be expressed as a ratio and the median of its median life as the prescribed lifespan is μ/μ_o. Now substituting λ=$\frac{\mathit{\mu }}{\mathit{\eta }}$ and to=a₁μ_o in Equation 5(5) ${\mathrm{F}}_{\mathrm{APTIP}}\left(\mathrm{x}\right)=\left\{\begin{array}{c}\frac{{\mathit{\alpha }}^{1-\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{\mathit{\lambda }{\mathrm{x}}^{-1}}}}-1}{\mathit{\alpha }-1},\mathit{\alpha }>0,\mathit{\alpha }\ne 1\\ \left(1-\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{\mathit{\lambda }{\mathrm{x}}^{-1}}}\right)\mathit{if}\mathit{\alpha }=1\end{array}\right.$(5) , probability of failure is expressed as

(11) $\mathrm{p}=\frac{{\mathit{\alpha }}^{1-\left(\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{\left(\frac{\mu }{\eta }\right){\left(a_{1{\mu }_o}\right)}^{-1}}}\right)}-1}{\mathit{\alpha }-1}$(11)

it can also be written as

(12) $\mathrm{p}=\frac{{\mathit{\alpha }}^{1-\left(\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{{({\mathit{a}}_{1\mathit{\eta }})}^{-1}}\left(\frac{\mu }{{\mu }_o}\right)}\right)}-1}{\mathit{\alpha }-1}$(12)

Equation 12(12) $\mathrm{p}=\frac{{\mathit{\alpha }}^{1-\left(\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{{({\mathit{a}}_{1\mathit{\eta }})}^{-1}}\left(\frac{\mu }{{\mu }_o}\right)}\right)}-1}{\mathit{\alpha }-1}$(12) can be used to calculate p for a specified value of α, β, when a₁ is given and r₂= μ/μ_o; however, we have two failure probabilities, p₁ and p₂, and used to represent the consumer and producer risk, where producer risk refers to the probability of rejecting a good lot and consumer risk refers to accepting the probability of a bad lot. To satisfy the following two Equations simultaneously for a given value of α, β, a₁, and r₂, we must evaluate the values of g and c

(13) ${\mathrm{P}}_{\mathrm{a}}\left({\mathrm{p}}_1|\mathit{\mu }/{\mathit{\mu }}_{\mathrm{o}}={\mathrm{r}}_1\right)={\left[{\sum }_{\mathrm{i}=0}^{\mathrm{c}}\left(\begin{array}{c}\mathrm{r}\\ \mathrm{i}\end{array}\right){\mathrm{p}}_1^{\mathrm{i}}{\left(1-{\mathrm{p}}_1\right)}^{\mathrm{r}-\mathrm{i}}\right]}^{\mathrm{g}}\le ,\mathit{\gamma }$(13)

and

(14) ${\mathrm{P}}_{\mathrm{a}}\left({\mathrm{p}}_2|\mathit{\mu }/{\mathit{\mu }}_{\mathrm{o}}={\mathrm{r}}_2\right)={\left[{\sum }_{\mathrm{i}=0}^{\mathrm{c}}\left(\begin{array}{c}\mathrm{r}\\ \mathrm{i}\end{array}\right){\mathrm{p}}_2^{\mathrm{i}}{\left(1-{\mathrm{p}}_2\right)}^{\mathrm{r}-\mathrm{i}}\right]}^{\mathrm{g}}\ge 1-\mathrm{\Im },$(14)

However, r₁ and r₂ correspond to the median ratios of the consumer and producer risks, respectively. The probabilities in Equation 13(13) ${\mathrm{P}}_{\mathrm{a}}\left({\mathrm{p}}_1|\mathit{\mu }/{\mathit{\mu }}_{\mathrm{o}}={\mathrm{r}}_1\right)={\left[{\sum }_{\mathrm{i}=0}^{\mathrm{c}}\left(\begin{array}{c}\mathrm{r}\\ \mathrm{i}\end{array}\right){\mathrm{p}}_1^{\mathrm{i}}{\left(1-{\mathrm{p}}_1\right)}^{\mathrm{r}-\mathrm{i}}\right]}^{\mathrm{g}}\le ,\mathit{\gamma }$(13) and (Equation 14(14) ${\mathrm{P}}_{\mathrm{a}}\left({\mathrm{p}}_2|\mathit{\mu }/{\mathit{\mu }}_{\mathrm{o}}={\mathrm{r}}_2\right)={\left[{\sum }_{\mathrm{i}=0}^{\mathrm{c}}\left(\begin{array}{c}\mathrm{r}\\ \mathrm{i}\end{array}\right){\mathrm{p}}_2^{\mathrm{i}}{\left(1-{\mathrm{p}}_2\right)}^{\mathrm{r}-\mathrm{i}}\right]}^{\mathrm{g}}\ge 1-\mathrm{\Im },$(14) ) are given by

(15) $\mathrm{p}1=\frac{{\mathit{\alpha }}^{1-\left(\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{{\left({\mathit{a}}_{1\mathit{\eta }}\right)}^{-1}}}\right)}-1}{\mathit{\alpha }-1}$(15)

(16) $\mathrm{p}2\mathit{\hspace{0.17em}}=\frac{{\mathit{\alpha }}^{1-\left(\frac{1+\mathit{\beta }}{1+\mathit{\beta }{\mathrm{e}}^{{\left(a_1\mathit{\eta }\right)}^{-1}\left(r_2\right)}}\right)}-1}{\mathit{\alpha }-1}$(16)

$\begin{array}{ccc}{\mathrm{H}}_0:\mathit{\mu }={\mathrm{r}}_2\times {\mathit{\mu }}_0& \mathrm{versus}& {\mathrm{H}}_1:\mathit{\mu }={\mathrm{r}}_1\times {\mathit{\mu }}_0\end{array},$

Ensuring the constraints while minimizing g and c as min {g, c : (g, r, c)}, to justify the following Equations simultaneously for given values $\mathit{\alpha },\mathit{\beta }$, ${\mathrm{a}}_1$, we evaluate the respective g and c values.

(17) $\left\{\begin{array}{c}\mathit{subject}\mathit{to}:\mathit{minimize}\mathit{g}\mathit{and}\mathit{c}\\ {\mathrm{P}}_{\mathrm{a}}\left(p_1|\mathit{\mu }/{\mathit{\mu }}_{\mathrm{o}}={\mathrm{r}}_1\right)\le \mathit{\gamma }\\ {\mathrm{P}}_{\mathrm{a}}\left(p_2|\mathit{\mu }/{\mathit{\mu }}_{\mathrm{o}}={\mathrm{r}}_2\right)\ge 1-\mathrm{\Im }\\ \mathit{where}\mathit{g},\mathit{and}\mathit{r}\mathit{ϵ}\mathit{Z}+,0\le \mathit{c}<\mathit{r}\end{array}\right.$(17)

4. Description of tables with examples

For different design parameters under the GASP, Tables 1 and 2 are demonstrated, where the values of β (0.2487, 0.455), ℽ (0.25, 0.50, and 0.01), r2 (2, 4, 6, and 8), with a1 (0.5, 1), and r (5,10), are respectively, where r1 is equal to 1, r2 is the ratio (2, 4, 6, 8), and r is the number of items in each group (5 and 10). In our findings with the minimization of consumer risk (ℽ), the number of groups g increases with constant c, and the probability of accepting a lot Pa(p) s also increases. However, in Tables 1 and 2, when we have r = 5, a₁ = 0.5, ℽ = 0.25, and r₂ = 4, 138 (69 × 2), the units are required to put a life test on it. As shown in Table 2, when we have r = 10, a1 = 1, ℽ = 0.25, and r₂ = 2, 448 (112 × 4) units are required. According to the APTIP model, when the OC value increases, the number of groups decreases, the median life increases to account for GASP, and the median is utilised as a quality metric. As seen in Table 3, the true median values are, when r = 5 and a₁ = 1, which are taken from Table 2, the true median life was found to rise with decreasing g and c beyond a certain point for the fixed (g, c) and Pa(p). Figure 1 depicts this occurrence by plotting r2 values with respect to OC and g values from Tables 1 and 2, respectively. Figure 1 clearly indicates that the true median lifetime grows, g decreases and remains constant, and OC values rise instantly. As a result, the inspected lot is accepted at these sites under the APTIP model. Accepting the lot for r = 5, as shown in Figure 1, enables the lowest number of groups to be tested, saving time and money.

Figure 1. Illustration of OC curves with values taken from Tables 1 and 2.

Table 1. GASP under APTIP model, α = 0.125, β = 0.2487

r = 5, a1 = 0.5					r = 5, a1 = 1			r = 10, a1 = 0.5			r = 10, a1 = 1
ℽ	r2	g	C	Pa(p)	g	c	Pa(p)	g	c	Pa(p)	g	c	Pa(p)
0.25	2	-	-	-	941	4	0.9582	1,758	5	0.9735	47	5	0.9636
	4	69	2	0.9759	11	2	0.9703	9	2	0.9668	5	3	0.9847
	6	10	1	0.9539	11	2	0.9909	9	2	0.9895	2	2	0.9833
	8	10	1	0.9734	3	1	0.9690	3	1	0.9660	1	1	0.9580
0.1	2	-	-	-	-	-	-	2,919	5	0.9564	-	-	-
	4	114	2	0.9604	17	2	0.9545	64	3	0.9859	8	3	0.9756
	6	114	2	0.9880	17	2	0.9860	15	2	0.9826	3	2	0.9750
	8	16	2	0.9577	17	2	0.9940	15	2	0.9924	3	2	0.9888
0.05	2	-	-	-	-	-	-	-	-	-	-	-	-
	4	2,004	3	0.9879	140	3	0.9866	83	3	0.9817	10	3	0.9696
	6	149	2	0.9844	22	2	0.9819	20	2	0.9769	4	2	0.9668
	8	149	2	0.9933	22	2	0.9922	20	2	0.9898	4	2	0.9851
0.01	2	-	-	-	-	-	-	-	-	-	-	-	-
	4	3,080	3	0.9815	215	3	0.9795	127	3	0.9722	15	3	0.9548
	6	228	2	0.9762	34	2	0.9722	30	2	0.9656	6	2	0.9506
	8	228	2	0.9897	34	2	0.9880	30	2	0.9848	6	2	0.9777

Note: A large sample length in required cells contains hyphens (-).

Table 2. GASP under APTIP model, α = 1.5, β = 0.455487

r = 5, a1 = 0.5					r = 5, a1 = 1			r = 10, a1 = 0.5			r = 10, a1 = 1
ℽ	r2	g	c	Pa(p)	g	c	Pa(p)	g	c	Pa(p)	g	c	Pa(p)
0.25	2	-	-	-	-	-	-	3,237	4	0.9569	112	4	0.9532
	4	2	2	0.9798	45	2	0.9798	41	2	0.9750	7	2	0.9673
	6	29	1	0.9613	7	1	0.9625	8	1	0.9539	7	2	0.9899
	8	29	1	0.9782	7	1	0.9788	8	1	0.9737	3	1	0.9616
0.1	2	-	-	-	-	-	-	-	-	-	1,253	5	0.9652
	4	617	2	0.9667	74	2	0.9671	68	2	0.9590	38	3	0.9884
	6	617	2	0.9901	74	2	0.9903	68	2	0.9875	11	2	0.9842
	8	47	1	0.9649	12	1	0.9639	13	1	0.9576	11	2	0.9932
0.05	2	-	-	-	-	-	-	-	-	-	1,629	5	0.9550
	4	802	2	0.9569	96	2	0.9575	657	3	0.9874	50	3	0.9848
	6	802	2	0.9871	96	2	0.9874	88	2	0.9838	14	2	0.9800
	8	61	1	0.9546	16	1	0.9522	88	2	0.9932	14	2	0.9914
0.01	2	-	-	-	-	-	-	-	-	-	-	-	-
	4	1,823	3	0.9800	1,686	3	0.9859	1,010	3	0.9807	76	3	0.9770
	6	1,233	2	0.9803	148	2	0.9807	135	2	0.9753	21	2	0.9701
	8	1,233	2	0.9917	148	2	0.9919	135	2	0.9895	21	2	0.9871

Note: A large sample length in required cells contains hyphens (-).

Table 3. Optimized values taken from Tables 1 and 2

r2	2	4	6	8
g	-	17	17	17
c	-	2	2	2
Pa(p)	-	0.9545	0.9860	0.9940
g	-	1,823	1,233	1,233
c	-	3	2	2
Pa(p)	-	0.9800	0.9803	0.9917

Note: A large sample length in required cells contains hyphens (-).

Now, we have illustrated Tables 1–3, with an example, and for the readers, provided additional details (Aslam et al., 2010). Let the lifespan of the ball bearing be tested following the APTIP model, with a value of β = 0.2487. However, the mean lifespan of the ball bearing is 2,000 cycles. The consumer and producer face 25% and 5% risks, respectively, when the mean life is 3,000 and 5,000 cycles, respectively. Therefore, if the investigator runs the experiment of 1,000 cycles with 10 units in each group to see that the ball bearing mean life is longer than the prescribed life, for this structure, we have μ0 = 3,000, β = 0.2487, α1 = 0.5, ℽ = 0.25, r = 10, r2 = 4, and producer risk 0.05. However, from Table 1, we have g = 8, with approval number c = 3, which implies that (8 × 3) = 24, three units will be required among the eight groups. If no more than one unit expires in all these groups before 1,000 cycles, the mean life of the ball bearing will be higher than the stated life. Therefore, if the investigator wants to check the hypothesis that ball bearings have a life span of 5,000 cycles, the true median lifetime is four times that, he can test eight groups of three units each; if higher than one unit fails in 1,000 cycles, as a1 = 0.5, and the mean life length is 1,000 cycles, the integrator will conclude that the life is higher than 5,000 cycles with a 95% level of certainty; consequently, the lot is accepted.

Figure 1 depicts the aforementioned (g,c), and OC (probability values) phenomena by plotting the optimised values from Tables 2 and 3. Figure 1 shows that as the median life increases, the value of g decreases and then remains constant, whereas the value of probability (OC values) increases rapidly. At these points, the under-investigation lot will be accepted under the APTIP model, saving both money and time.

5. Application

5.1. Dataset 1

Data reflects an aerial communication transceiver’s active repair time (hours) (Alyami et al., 2022). The information of data is as follows: 0.50, 0.60, 0.60, 0.70, 0.70, 0.70, 0.80, 0.80, 1.00, 1.00, 1.00, 1.00, 1.10, 1.30, 1.50, 1.50, 1.50, 1.50, 2.00, 2.00, 2.20, 2.50, 2.70, 3.00, 3.00, 3.30, 4.00, 4.00, 4.50, 4.70, 5.00, 5.40, 5.40, 7.00, 7.50, 8.80, 9.00, 10.20, 22.00, and 24.50

The maximum likelihood estimates (MLE) with their standard error for two parameters of APTIP are $\hat{\alpha }$ = 0.8132096 (1.376690), $\hat{\beta }$$\hat{\beta }$ = 1.7222587 (4.026261), and $\hat{\lambda }$ = 1.9866405 (0.484034). The maximum distance between the fitted APTIP model as well as the data, according to the Kolmogorov –Smirnov test (k-s), is 0.082276 with a p-value = 0.9494. Figure 2 depicts a data (a) P-PP–P Plot, (b) Q-–Q plot, (c) histogram with estimated pdf, (d) estimated cdf with empirical cdf, and (e) increased estimated hazard rate function (f) TTT plot.

Figure 2. Graphical representation of APTIP model for data-set I (a) P-–P plot, (b) Q-–Q plot, (c) estimated pdf, (d) empirical and theoretical cdf, (e) estimated hrf, and (f) TTT plot.

5.2. Dataset second

The second dataset was initially given by (Ahsan-Ul-Haq, et al., 2022), and it depicts the maximum northern yearly flood flowsflow of Saskatchewan 1,000 cubic feet per second river near Edmonton over a 47-a calendar year. The data information is as follows: 19.885, 20.940, 21.820, 23.700, 24.888, 25.460, 25.760, 26.720, 27.500, 28.100, 28.600, 30.200, 30.380, 31.500, 32.600, 32.680, 34.400, 35.347, 35.700, 38.100, 39.020, 39.200, 40.000, 40.400, 40.400, 42.250, 44.020, 44.730, 44.900, 46.300, 50.330, 51.442, 57.220, 58.700, 58.800, 61.200, 61.740, 65.440, 65.597, 66.000, 74.100, 75.800, 84.100, 106.600, 109.700, 121.970, 121.970, and 185.560.

The maximum likelihood estimates (MLE) with their standard error for three parameters of APTIP are $\hat{\alpha }$ = 8.5678509 (6.094041), $\hat{\beta }$ = 0.5080798 (0.435501), and $\hat{\lambda }$ = 31.9016069 (11.999767). The maximum distance between the fitted APTIP model and the data, as determined by the Kolmogorov –Smirnov test (k-s), is 0.21745, with a p-value of 0.02136.

Figure 3 depicts a data probability plot (pp plot), quantile plot (QQ plot), histogram with estimated pdf, estimated cdf with empirical cdf, and increased estimated hazard rate function, TTT plot all of which demonstrate a better fit of failure times of deep groove ball bearings under APTIP model. The true median lifetimes, n (sample size), c, gg, and OC values for the two datasets are showedshown in Table 6. Assuming dataset I as shown in Table 4, with $\hat{\alpha }$ = 0.8132096, $\hat{\beta }$ = 1.7222587, and $\hat{\lambda }$ = 1.9866405, r = 10, a1 = 1, ν = 0.050.05, and when c and g decreasedecrease, the probability increaseincreases, hence lot will be accepted. Similarly, consider dataset II as shown in Table 5, with $\hat{\alpha }$$\hat{\alpha }$= = 8.5678509, $\hat{\beta }$= $\hat{\beta }$ = 0.5080798, and $\hat{\lambda }$$\hat{\lambda }$= = 31.9016069, r = 10, a₁ = 1, ν = 0.05, as c and g decrease, the probability increases, and the lot is accepted. As can be seen from the above results for both data sets data setsdatasets I and II, lessfewer groups are tested at optimised cost and time.

Figure 3. Graphical representation of APTIP model for dataset I (a) P–P plot, (b) Q–Q plot, (c) estimated pdf, (d) empirical and theoretical cdf, (e) estimated hrf, and (f) TTT plot.

Table 4. GASP under APTIP model, $\hat{\mathit{\alpha }}$ = 0.8132096, $\hat{\mathit{\beta }}$ = 1.7222587, $\hat{\mathit{\lambda }}$ = 1.9866405

r = 5, a1 = 0.5					r = 5, a1 = 1			r = 10, a1 = 0.5			r = 10, a1 = 1
ℽ	r2	g	c	Pa(p)	g	c	Pa(p)	g	c	Pa(p)	g	c	Pa(p)
0.25	2	-	-	-	-	-	-	-	-	-	-	-	-
	4	13	3	0.9874	2	3	0.9764	2	3	0.9512	1	5	0.9863
	6	4	2	0.9808	1	2	0.9686	1	2	0.9576	1	4	0.9872
	8	4	2	0.9915	1	2	0.9853	1	2	0.9797	1	3	0.9753
0.1	2	-	-	-	-	-	-	-	-	-	-	-	-
	4	21	3	0.9798	4	3	0.9533	5	4	0.9808	2	5	0.9728
	6	6	2	0.9714	4	3	0.9888	3	3	0.9813	1	4	0.9872
	8	6	2	0.9872	2	2	0.9708	2	2	0.9598	1	3	0.9753
0.05	2	-	-	-	-	-	-	-	-	-	-	-	-
	4	27	3	0.9741	18	3	0.9879	6	4	0.9770	2	5	0.9728
	6	7	2	0.9667	5	3	0.9860	3	3	0.9813	2	4	0.9746
	8	7	2	0.9851	2	2	0.9708	2	2	0.9598	1	3	0.9753
0.01	2	-	-	-	-	-	-	-	-	-	-	-	-
	4	41	3	0.9609	27	3	0.9819	9	4	0.9657	3	5	0.9594
	6	41	2	0.9915	7	3	0.9805	5	3	0.9690	2	4	0.9746
	8	11	2	0.9767	3	2	0.9565	5	2	0.9888	2	3	0.9512

RemarksNote: A large sample length in required cells contains hyphens (-).

Table 5. GASP under APTIP model, $\hat{\alpha }$ $\hat{\alpha }$ = 8.5678509, $\hat{\beta }$ $\hat{\beta }$ = 0.5080798, and $\hat{\lambda }$ $\hat{\lambda }$ = 31.9016069

r = 5, a1 = 0.5					r = 5, a1 = 1			r = 10, a1 = 0.5			r = 10, a1 = 1
ℽ	r2	g	c	Pa(p)	g	c	Pa(p)	g	c	Pa(p)	g	c	Pa(p)
0.25	2	-	-	-	-	-	-	-	-	-	-	-	-
	4	-	-	-	1,049	2	0.9864	2	2	0.9812	106	2	0.9841
	6	267	1	0.9680	56	1	0.9718	64	1	0.9660	15	1	0.9669
	8	267	1	0.9822	56	1	0.9874	64	1	0.9810	15	1	0.9810
0.1	2	-	-	-	-	-	-	-	-	-	-	-	-
	4	-	-	-	1,743	2	0.9774	1,715	2	0.9690	176	2	0.9738
	6	-	-	-	93	1	0.9536	1,715	2	0.9914	176	2	0.9924
	8	444	1	0.9706	93	1	0.9735	106	1	0.9687	24	1	0.9698
0.05	2	-	-	-	-	-	-	-	-	-	-	-	-
	4	-	-	-	2,267	2	0.9708	2,231	2	0.9599	229	2	0.9661
	6	-	-	-	2,267	2	0.9917	2,231	2	0.9888	229	2	0.9902
	8	577	1	0.9619	121	1	0.9657	138	1	0.9594	31	1	0.9612
0.01	2	-	-	-	-	-	-	-	-	-	-	-	-
	4	-	-	-	3,485	2	0.9554	-	-	-	3,781	3	0.9890
	6	-	-	-	3,485	2	0.9872	3,429	2	0.9828	352	2	0.9849
	8	-	-	-	3,485	2	0.9946	3,429	2	0.9929	352	2	0.9936

Note: A large sample length in required cells contains hyphens (-).

Table 6. The true median lifetimes, sample sizes, g, c, and OC values of the two datasets

Dataset I					Dataset II
r2	2	4	6	8	r2	2	4	6	8
g	-	2	1	1	g	-	3,485	3,485	3,485
c	-	3	2	2	c	-	2	2	2
Pa(p)	-	0.9512	0.9576	0.9797	Pa(p)	-	0.9554	0.9872	0.9946

From a practical perspective, we calculated descriptive statistics in Table 7; however, due to the right-skewed nature of data, using the median as a quality index is the best option. Therefore, Figure 4 depicts the box-plot of datasets I and II with maximum (X_m), minimum (X_o), mean ($\bar{X}$$\bar{X}$), median ($\tilde{X}$$\tilde{X}$), mode ($\hat{X}$$\hat{\mathrm{X}}$), lower quartile (Q₁), upper quartile (Q₃), variance (S²), and standard deviation (S), respectively.

Figure 4. Box-plot of datasets one and two.I and II.

Table 7. Descriptive statistics

	${\mathbf{X}}_{\mathbf{m}}$	${\mathbf{X}}_{\mathbf{o}}$	$\bar{\mathbf{X}}$	$\tilde{\mathbf{X}}$	$\hat{\mathbf{X}}$	Q1	Q3	s^2	s
Dataset I	24.50	0.50	4.0125	2.1	1.00, 1.50	1	4.85	26.68	5.165
Dataset II	185.560	19.885	51.49	40.4	40.400, 12.970	30.29	61.47	1048.25	32.37

6. Discussion

Group acceptance sampling plans are important quality control tools used in many industries. They offer several advantages over individual sampling plans including cost-effectiveness, efficiency, and ease of use. GASP provide a statistically reliable method for ensuring that the quality of many products meets the required standards, while also reducing the time and resources required for inspection. These plans are widely used in manufacturing, aerospace, and healthcare industries, where quality and reliability are critical. These plans have been developed and refined over many years, and are recognized as an effective method for ensuring product quality and reliability. However, there are some limitations to group acceptance sampling plans that need to be considered. For instance, GASP can sometimes result in more defects being accepted in many or more good products being rejected. In addition, sampling plans cannot guarantee that all products in a lot will be of the same quality. Therefore, GASP is an effective and widely used method to ensure product quality and reliability. They offer several advantages over individual sampling plans, and are based on statistical principles that provide a reliable estimate of the quality of a large number of products. While there are limitations to these plans, they remain valuable tools in the quality control process and are likely to continue to be used in many industries in the future.

7. Conclusion

In this study, we present the GASP, which occurs when the lifetime of an item follows an alpha power transformation-inverted perks distribution. The median is used as a quality index for different constraint parameters, namely acceptance number, group size, predefined consumer risk, and test termination. We also noticed that the optimum value graphs when g and c tended to drop rather than the OC values tended to grow in the table. We visually displayed these optimized values and concluded that it is beneficial to accept the lot at such optimized values with little g and c because it optimizes the price and time involved in examining the items. These findings align with the most recent research by (Algarni, 2022), showing that such a tester would save time and money during testing.

Disclosure statement

No potential conflict of interest was reported by the author(s).