Group acceptance sampling plan based on truncated life tests using extended odd Weibull exponential distribution with application to the mortality rate of COVID-19 patients

It is well-known that nanomaterials greatly influence our lives, and appropriate statistical methods have aided their development. Lu et al.¹ provided a thorough analysis of the practical use of statistical techniques at the nanoscale. When using detrimental tests to determine an item’s quality, sampling is the only option. Classical sampling methods, such as systematic sampling, random samples, stratified samples, and cluster samples, are used to select an accurate representation of the population. However, improved sampling strategies have been developed for nano-processes to acquire more precise predictions and spend less on sampling by effectively using smaller sample sizes and sampling plans. In manufacturing, sampling plans decide whether to accept or reject incoming or outgoing collections based on a predetermined quality, known as lot judgment. Developers must prioritize optimizing the sample size and study timeframe to achieve better results. Acceptance sampling strategies are the key to reaching this optimization. By implementing simple acceptance plans, developers can determine the minimum sample size required for testing. Testing one item at a time can save both time and money. The implementation of groups can save time and money on research. The Group Acceptance Sampling Plan (GASP) considers the entire set as a group when a tester verifies multiple things. When the GASP is combined with truncated life testing, the resulting product is called the GASP based on a truncated life test, which presumes that the product’s lifespan fits a specific probability distribution. Initially developed by Aslam and Jun² for the truncated life test, the points of the group acceptance sampling plan assumed that each item’s lifetime would pursue either an inverse Rayleigh or a log-logistic distribution with known shape parameter distribution. The determination of the number of groups and acceptance numbers in this plan is based on the values of the manufacturer and consumer risks.

The utilization of the GASP, as recommended by Al-Nasser et al.,³ can effectively prevent the acceptance of batches that exhibit significant defects by testing a small sample size. This approach can prevent costly recalls, refunds, and customer criticism while simultaneously upgrading consumer confidence in the quality of the goods being constructed. By incorporating the GASP into their quality control strategy, organizations can showcase their unwavering commitment to producing premium products and reinforce consumer trust and loyalty, as noted by Montgomery.⁴ 

The extensive research conducted on the GASP has resulted in various lifetime distributions. These include Marshall-Olkin distribution extended to Lomax distribution, generalized exponential Rao distribution,⁵ gamma distribution by Gupta and Gupta,⁶ inverse Rayleigh and log-logistic Ramaswamy and Anburajan distribution,⁷ Weibull distribution by Aslam and Jun,⁸ inverse power Lomax distribution by Ashraf et al.,⁹ exponentiated inverted Weibull distribution by Suseela and Rao,¹⁰ odd generalized exponential log-logistic distribution by Sivakumar et al.,¹¹ exponential lifetime Mughal distribution,¹² Marshall–Olkin–Kumaraswamy exponential distribution by Almarashi et al.,¹³ generalized inverted Kumaraswamy distribution Alsultan and Khogeer,¹⁴ Burr type X percentile distribution by Aslam et al.,¹⁵ generalized transmuted exponential distribution Fayomi and Khan,¹⁶ and alpha power transformation inverted perk distribution by Ameeq et al.¹⁷ 

Lifetime data analysis must avoid relying solely on the exponential distribution because of its simple form and insufficient memory properties. In recent years, numerous authors have introduced new variations on the standard exponential distribution, expanding the distribution’s range of applicability. These include the Kumaraswamy transmuted exponential by Afifya et al.,¹⁸ the exponentiated exponential by Gupta and Kundu,¹⁹ the Marshall–Olkin exponential formulated by Marshall and Olkin,²⁰ and the odd exponentiated half-logistic exponential by Afify et al.,²¹ the modified exponential by Rasekhi et al.,²² the Marshall–Olkin logistic exponential by Mansour et al.,²³ the odd log-logistic Lindley exponential by Alizadeh et al.,²⁴ Harris’s extended exponential by Pinho et al.,²⁵ and the Marshall–Olkin alpha power exponential by Nassar et al.²⁶ The EOWEx distribution boasts numerous desirable properties as a statistical model. With its ability to represent various hazard rates using only three parameters of bathtub failure rates, it proves to be a versatile tool applicable to different fields, such as medicine, engineering, and manufacturing quality assurance. Its cumulative distribution function (CDF) and hazard rate function (HRF) are straightforward and can be used to analyze censored datasets.

For this analysis, the product quality is assessed using the median value. In the study by Fayomi and Khan,¹⁶ the median is the superior quality parameter when handling a skewed distribution. The percentile point is used as the quality metric since the EOWEx distribution has a non-normal distribution. Our primary objective is to provide a GASP based on a truncated life test, assuming that the product lifetimes comply with the EOWEx distribution defined by Afify and Mohamed²⁷ with specific shape parameters. Assistance is needed in locating any mention of the GASP regarding the EOWEx distribution in the literature. The GASP for the EOWEx distribution will be tailored to meet the safety requirements of both producers and consumers of the EOWEx distribution. The minimum necessary groups and approval numbers based on customer risk and test termination time for a particular group size are identified.

In this paper, we will adhere to the designated format. Section II will extensively examine the underlying principles of the EOWEx distribution, offering a complete overview of the computation of pdf, cdf, and quantile functions. In Sec. III, we will delve into the construction of the GASP, which considers the percentiles of a truncated lifespan. Section IV will demonstrate practical applications of the methodology using actual data. Finally, in Sec. V, we will present the outcomes and examine different possible scenarios, leaving no aspect unexplored.

It is crucial to acknowledge that X ∼ EOW-G(α, γ, ξ) if α and γ are positive shape parameters with PDF (2).

It is imperative to utilize medians as the quality indicator for skewed distributions, as recommended by Shafiq et al.²⁹ However, our approach further introduces a revolutionary GASP for products with EOWEx distributions. This method includes essential parameters such as α, γ, θ, and cdf in Eq. (4) and utilizes randomized statistical techniques to provide invaluable insights into a product’s functionality and performance. The GASP is the key to unlocking the full potential of a product. The GASP design parameters were obtained through a meticulous series of stages. Below, we provide a comprehensive explanation of these stages:

1. During a group sampling inspection, it is imperative to randomly select n items and distribute them into g groups, each containing r items. It is non-negotiable that n must be equal to r multiplied by g.

2. It is essential to establish both the duration of the experiment, denoted as t₀, and the acceptance threshold, represented by c.

3. The experiment is being carried out for G groups at the same time, and the number of failures for each group is being recorded.

4. The acceptance of the lot depends on the condition that there should be no more than c failures across all groups.

5. If any group experiences more than c failures, the experiment will be immediately stopped, and the entire batch will be rejected without exception.

To determine the value of p, we must specify the exact values of a₁ and r₂ in $mm0$⁠. Once we have obtained these values, we can calculate the ratio of the mean lifetime of the product to the specified lifetime $mm0$⁠, which will allow us to determine the quality level of the product decisively. To determine the design parameters (g and c) for a given quality level, along with specific consumer and producer risks, the following optimization problem needs to be solved:

Minimize Average Sample Number (ASN), n = r × g.

Tables I and II present the design parameters for the proposed GASP based on the EOWEx distribution with γ values of 0.2 and 1.5, respectively. The suggested GASP is specifically designed for two distinct choices of r, 5 and 10, with an extensive range of values for the average risk ratio of the producer (r₂ = 2, 4, 6, and 8), associated with various values for the consumer’s risk (β = 0.25, 0.10, 0.05, and 0.01), utilizing R codes for this study. Tables I and II display how the number of groups changes based on consumer risk and how the number of groups decreases as the proportion of at-risk producer quality represented by “r₂” increases. Despite an unchanging number of groups and acceptance, the probability of accepting a“P_a” increases as consumer levels decrease. Table I also shows the impact of the time-termination multiplier for two values (a₁ = 0.5 and a₁ = 1.0). When r = 10 and r₂ = 4, the number of groups decreases as “a₁” increases. However, when r₂ = 8, the number of groups decreases while the acceptance number “c” tends to increase. Researchers should adjust the group size to decrease to “n.” For example, if we have β = 0.25, a = 0.5, r₂ = 4, and r = 5, we need 12 groups, or n = g × r = 12 × 5 = 60 units, to conduct a life-expectancy test. With r = 10, only four groups are needed, using forty test units. A group of size 4 with r = 10 elements is recommended. Table II lists the parameter values for design with γ = 1.5. As the parameter’s value increases, the related plane’s group size decreases. For β = 0.25, a₁ = 0.5, r₂ = 4, and r = 5, there are seven groups, or 35 units, for the life test. However, for r = 10, only two groups or 20 test units are needed. Groups of size 2 with r = 10 items per group would be optimal. Within the EOWEx distribution, the median life is used as a quality parameter for the GASP, and the number of groups is reduced. Generally speaking, $OCPa$ values increase as the median life of a population increases.

Data from the United Kingdom on COVID-19 from April 15 to June 30, 2020, are presented in Table III (Amaal and Ehab³⁰). These numbers reflect the rate of death due to drought. The results of the Kolmogorov–Smirnov (KS) statistics and associated P-values are provided in Table IV for the models fitted based on actual datasets. This table also includes the estimation and standard errors (SEs) of the model parameter values under consideration.

Observe that Fig. 1 comprehensively presents the P–P plot, Q–Q plot, empirical CDF, histogram, and estimated PDF for your reference. In addition, Fig. 2 provides a detailed TTT plot, including pertinent information concerning both the underlying and predicted hazard rate functions.

Figure 1 demonstrates the correlation between the EOWEx distribution and the UK’s COVID- 19 data. On the other hand, Fig. 2 clearly shows a declining hazard rate in the estimated HRF for the given dataset, as depicted in the TTT plot. As such, it is evident that the EOWEx distribution is a far superior option. Table V displays plan parameters based on median lifetime using fitted parametric values for consumers’ risk at 0.25, 0.10, 0.05, and 0.01 for r₂ = 2, 4, 6, and 8. It’s worth noting that the behavior of the plan parameters shown in Tables I and II is consistent with that of Table V.

Experimenters shall employ the GASP approach to assess whether the median failure rate of COVID-19 patients surpasses a specific value. A stratified random sample of 100 COVID-19 deaths is selected, with 50 subjects assigned to each group. Assuming a median death rate of 2.0 for COVID-19 patients, the medical practitioner expects a median mortality rate of 4.0. The consumer and producer risks are 25% and 5%, respectively. The optimal parameters are determined from Table V, comprising $γ̂=8.832616$⁠, $α̂=1.916093$⁠, m = 2, a₁ = 0.5, β = 0.25, r = 10, and $r2=42=2$⁠. These parameters result in g and c values of 77 and 6, respectively. To determine the sample size, the fixed value r = 10 shall be multiplied by g = 77 to yield n = 770. In simpler terms, a sample size of 770 shall be drawn and divided into 77 groups, each consisting of 50 individuals. If no more than six cases fail in each group before completing 100 cases, it shall be statistically confirmed that the median failure life is greater than the specified value. If the opposite is true, the lot shall be rejected. Tables I and V suggest that maximum likelihood estimation can be effectively used instead of arbitrary values for plan parameters.

A comparative study is made between the group sampling plan and the ordinary sampling plan (OSP) and focuses only on the group sampling plans with r = 5 and the GASP for the EOWEx distribution with $γ̂=8.832 616$ and $α̂=1.916 093$ to save space. Table VI provides the design parameters for these group sampling plans for easy reference. Table VI shows that for a given value of r₂, the group sampling plan parameters are consistently lower than the corresponding plan parameters from the ordinary sampling plan. It has been observed that the GASP method is more efficient than the OSP in terms of sample size. All entries in the table were generated using the R programming language, as outlined in the  Appendix.

In this study, we have formed a GASP using the EOWEx distribution to determine the optimal group size “g” and acceptance number “c” for truncated life testing based on median product lifetime. It is important to note that the offered plan can only be used when testing multiple items simultaneously and cannot be applied to a lifetime with a normal distribution. The logic and application of the GASP are illustrated through tables and examples. Real-world data demonstrate the effectiveness of the suggested GASP. It should be noted that the GASP sample size reduces when there is variation among the units used. The proposed assessment method has a broad scope in various sectors, including electrical, aerospace, and missile testing.

Authors are highly indebted to the editor and the reviewers for their valuable suggestions in improving the manuscript.

The author has no conflicts to disclose.

Rehab Alsultan: Conceptualization (equal); Data curation (equal); Formal analysis (lead); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.

Relevant R Codes.