DesignofMultipleDependent State SamplingPlanApplication for COVID-19 Data Using Exponentiated Weibull Distribution

'e proposed sampling plan in this article is referred to asmultiple dependent state (MDS) sampling plans, for rejecting a lot based on properties of the current and preceding lot sampled. 'e median life of the product for the proposed sampling plan is assured based on a time-truncated life test, when a lifetime of the product follows exponentiated Weibull distribution (EWD). For the proposed plan, optimal parameters such as the number of preceding lots required for deciding whether to accept or reject the current lot, sample size, and rejection and acceptance numbers are obtained by the approach of two points on the operating characteristic curve (OC curve). Tables are constructed for various combinations of consumer and producer’s risks for various shape parameters. 'e proposed MDS sampling plan for EWD is demonstrated using the coronavirus (COVID-19) outbreak in China.'e performance of the proposed sampling plan is compared with the existing single-sampling plan (SSP) when the quality of the product follows EWD.


Introduction
e demand for high-quality products by consumers has always put pressure on the manufacturing industry, which has consequently resulted in time to time stretching its efficiency of production processes, to cater for consumer demand. e quality of the product management is based on the continuous manufacturing processes, and the final product could be tested for quality characteristics. For the continuous production processes, management is always done by using quality control charts, while sampling inspection is used for final product inspection by both consumers and producers, which, in contemporary literature, is referred to as acceptance sampling plan (ASP). e theme of sampling plans (SPs) is to decide to reject or accept a lot at the minimum inspection costs, while satisfying both consumers and producers' risks simultaneously. To arrive at this decision, a random sample was taken from a randomly selected lot, from produced lots by the industry. e two risks are considered simultaneously to have a more reliable decision on the disposition of the lot coming for inspection, at the minimum cost in terms of finances and time consumed to inspect all products. is reduces the possibility of the cost incurred of acceptance of bad lots by a consumer (consumer's risk β) and rejection of a good lot by the producers (producers' risk α).
e levels of quality connected to consumer and producer's risks have been referred to by quality control expertise as limiting quality level (LQL) and acceptable quality level (AQL), respectively. e field of ASP has experienced several improvements so far depending on the nature of the interest of the author and researcher. is paper intends to concentrate on time-truncated life that has proven to be more effective and gained popularity among acceptance sampling techniques. Normally, life test is used in the inspection of a lifetime of a particular product and sets the limit of time, which ultimately aids in the provision of life assurance of the product to both consumer and producer of the product [1]. e ASPs using time-truncated life tests have been used to study manufacturing industries product's reliability. Several SPs have been proposed in the literature using life tests for various distributions (for quality characteristics) under different situations. Aslam et al. [2] discussed extensively a tightened-normal-tightened group acceptance sampling plan with the assumption that the percentile life of the product is taken as product's quality. Aslam et al. [3] established a group sampling plan centred on truncated life tests considering an inverse Gaussian distribution. In the literature, the impact of misspecification of the model parameters for group sampling plan has also been explored. In addition, Gui [4] worked on ASP considering truncated life tests for half exponential power life distribution. AL-Omari [5] worked on an ASP for truncated life tests based on three parameter-kappa distributions. A single-and double-sampling plan for variable sampling inspection was proposed [6] under the Weibull distribution based on sudden death testing. Balamurali et al. [7] dealt with Weibull distributed lifetime assuring mean life for optimal design of multiple deferred state sampling plans. In addition, Balamurali et al. [7] took more effort to design multiple deferred state sampling plans for the generalized inverted exponential distribution. Based on the literature that we have gone through, there is no work that has been done for multiple deferred state sampling plans based on exponentiated Weibull distribution (EWD), hence the interest of this article.

Exponentiated Weibull Distribution (EWD)
e EWD originated from Weibull distribution, and the said distribution fits lifetime data, with exception of data with empirical hazard rates with non-monotone shapes, which are commonly encountered in survival analysis, and this makes the Weibull model useless in analysing data [8]. Generalization and extension of Weibull distribution was the result of such limitations, which therefore gave flexible alternatives in terms of modelling. For more details, please refer, Marshall and Olkin [9]; the extended Weibull distribution, the new extended Weibull distribution by Peng and Yan [10]; Lee et al. [11] the beta Weibull distribution, the modified Weibull distribution by Sarhan and Zaindin [12]. Others are the (P-A-L) extended Weibull distribution by Al-Zahrani et al. [13]; the additive Weibull distribution by [14], the generalized Weibull distribution by [15,16] proposed EWD, the Kumaraswamy Weibull distribution as proposed by Cordeiro, et al. [17] and the generalized modified Weibull distribution established by Carrasco et al. [18].
EWD was introduced by Gupta and Kapoor [19]; the distribution is characterized by shape and scale parameters which make it look like the gamma or Weibull family. e two-parameter gamma and Weibull distributions are the popular distributions used for analysing lifetime data. e gamma distribution is widely applied apart from survival analysis. Its survival function cannot be obtained in a closed form unless the shape parameter is an integer. On its part, the Weibull distribution's survival function and failure rate have simple forms. is characteristic has made the Weibull distribution much useful in analysing lifetime data. In this paper, we consider the exponentiated Weibull family that was introduced by Mudholkar and Srivastava [16], which has one scale parameter and two shape parameters. Let us assume that lifetime of quality characteristic follows EWD with two shape parameters. en, EWD has the following cumulative distribution function (CDF): where t > 0, δ > 0, c > 0, λ > 0. e probability density function for such a variable is given as follows: where t > 0, δ > 0, c > 0, λ > 0. For known shape parameters (δ, c), the CDF depends only on λt and the q th quantile (t q ) of the product; when its products' lifetime follows the EWD, it is given as But for median, quantile (q) � 0.5.
e failure probability of products before the experiment time t 0 under the EWD is given as follows: e termination time (t 0 ) can be expressed as t 0 q ), t 0 � at 0 q , where t 0 q is time truncation; also, the scale parameter can be written as For simplicity, λ 0 � (t 0 q /η). Hence, the failure probability of the product given in equation (3) can be expressed by substituting values for λ and t 0 as follows: en, in expanded form, it becomes When the quantile ratio (t q /t 0 q ) is greater than one, the above failure probability can be considered as AQL (p 1 ), and when it is equal to one, this probability is called the LQL (p 2 ). In this study, the assumption is that the shape parameters (δ, c) of the EWD are known. e shape parameters can be estimated from a previous production process, and generally, the manufacturer maintains the history of the product.

Multiple Dependent State Sampling Plans (MDSSPs)
e MDSSPs are regarded as one of the conditional SPs under special-purpose SP. Baker and Wortham introduced the MDS sampling concept. ese are the modifications of the chain-sampling plan proposed by Dodge as MDS-1 (c 1 , c 2 ) [20,21]. is can be applied for continuous production, and lots are submitted for serial inspection in the production order. e sample size can be reduced by implementing the MDSSP since the decision regarding the disposition of the current lot is made using the results of samples drawn from both current and successive lots. Several scholars have investigated MDSSP in various situations. Few to mention are Govindaraju and Subramani who discussed the selection of MDSSP for given AQL and LQL [22]. Balamurali and Jun [23] studied the MDSSP based on measurement data. Balamurali et al. [24] investigated the MDSSP using Bayesian methodology. For more details on MDSSP, one may refer to Vaerst [25]; Soundararajan and Vijayaraghavan [26]; Subramani and Haridoss [27]; and Aslam et al. [28]. Recently, the concept of MDS sampling has been used in control chart design. Aslam et al. [29,30] studied an attribute control chart for monitoring the manufacturing process based on an MDS sampling approach. In this work, we followed the methods of Rao et al. [31]. MDSSP is proposed to utilize the median life of the product based on a time-truncated life test, when the lifetime of the product (quality characteristics of the product) follows a EWD. e operating procedure of the MDSSP for EWD is given in the next section. e proposed plan and existing plans' performances are compared to reveal the best among them.

Designing MDSSPs for EWD
In this section, the operating procedure and designing methodology of the MDSSP are discussed for EWD.

Operating Procedure.
In this section, the operating procedure and designing methodology of the MDSSP are discussed for EWD. e following are the various steps in the operating procedure to obtain the parameters.
(1) From a current lot, draw a random sample of n units. e proposed MDSSP is characterized by four parameters, namely, c 1 , c 2 , n, and m, where n is the sample size, m is the number of previous lots needed to make a decision, c 2 is the maximum number of failed items for conditional acceptance, and c 1 is the maximum number of failed items for unconditional acceptance.
Note that the attribute MDSSP is the generalization to a SSP, and it also reduces to SSP when either m ⟶ ∞ c 1 � c 2 � c or. e operating characteristic (OC) function of the MDSSP for EWD for time-truncated life test is given by For the lot acceptance, probability at failure probability p considering a binomial distribution is obtained using the following equation:

Designing Methodology.
SSPs are designed to focus on minimizing the average sample number (ASN). A preferable SSP is the one with a minimum ASN. is is because for the minimum ASN, the corresponding inspection time and inspection cost will be reduced. In this article, we attempt to minimize the ASN of the proposed MDSSP for EWD under truncated life tests. is is achieved through optimization problem that minimizes the ASN to obtain optimal parameters for proposed design as follows: where failure probability is p 1 at the producer's risk and the failure probability at the consumer's risk is p 2 . e quality level is expressed as the ratio of its true lifetime quantile ratio given as t 0 /t 0 q . e lifetime quantile ratio concept helps the Complexity 3 producer to enhance product quality. e probabilities of acceptance of the lot at AQL and LQL under an MDS sampling plan are, respectively, obtained using the following equations: e producer wishes that his product should be accepted when it is of good quality. erefore, we consider the quantile ratio t 0 /t 0 q � 2, 4, 6, 8, 10 at the producer's risk. e consumer wants to reject the product if it is at a poor quality level. erefore, we consider the mean ratio t 0 /t 0 q � 1 at the consumer's risk.
e optimal parameters of the proposed MDS sampling plan for EWD with known shape parameters (δ, c) � (2, 2), (2, 1.5), (1.5, 1.5), and (1.5, 2) under truncated life tests are given in Tables 1-5 by assuming that the consumer's risk β � 0.25, 0.10, 0.05, and 0.01 and producer's risk α � 0.05 at 50 th percentile and 25 th percentile. Also, the optimal parameters of the proposed MDS sampling plan for EWD with estimated shape parameters c � 0.9525 and δ � 4.4859 for the real-time data related to the coronavirus outbreak in China are given in Table 6. e termination ratio is considered as a � 0.5, a � 0.7, and a � 1.0. From the results in Tables 1-6, we observed the following: (1) When other parametric combinations are fixed, we noticed that as the termination ratio increases from 0.5 to 1.0, the required sample size n decreases. (2) It is interesting to note that as consumer's risk decreases, the sample sizes increase when other parametric combinations are fixed. (3) Furthermore, it is observed that shape parameters also influence sample size in the proposed design. (4) In addition, we have noticed that sample size decreases when quantile value increases (i.e., from 25 th percentile to 50 th percentile) assuming that all other parametric combinations are fixed. (5) Results revealed that the increase in quantile ratio also increases the producer's probability of`lot acceptance when other parametric combinations are fixed.

Application of Proposed Sampling Plan for COVID-19 Data
In this section, real data of COVID- 19  To fit the EWD, the shape parameters are estimated using the maximum likelihood approach, and they are c � 0.9525 and δ � 4.4859. e Kolmogorov-Smirnov test statistic value is 0.0684, and p value is 0.6925. Figure 1 shows the empirical and theoretical PDFs, empirical and theoretical CDFs, Q-Q plots, and P-P plots to highlight the goodness of fit of EWD. Hence, EWD yields a good fit for COVID-19 mortality rate data.
Suppose that the experimenter would like to use the proposed multiple dependent state sampling plans to implement the median life percentile of the product where the product lifetime follows an EWD with the shape parameters c � 0.9525 and δ � 4.4859. If the medical practitioner assume that the median mortality rate of the person suffering from COVID-19 is 2.0, the medical practitioner expected that the median mortality rate would be 4.0. e consumer's risk is 0.05 if the actual median mortality rate is 2.0 and the producer's risk is 0.10 if the actual median mortality rate is 4.0. With these constraints, the optimal parameters selected from Table 6 are n � 29, c 1 � 1, c 2 � 3, and m � 2 with values of c � 0.9525 and δ � 4.4859, t 0 q � 1.5, α � 0.05, β � 0.10, and t q /t 0 q � 2 at a � 0.5. e multiple dependent state sampling plans are established as follows.
A sample of 29 mortality rates due to COVID-19 will be selected at random for the group of people, and their mortality rate is 2.0. If the mortality rate before 2.0 is less than or equal to 1 case, then the group of people will be accepted and the group of people will be rejected if it is greater than 3 cases. ere will be a disposition of the present group of people deferred until the 2 preceding groups of people will be tested in case of the number of cases between 1 and 3. In this real example, there are seven cases before the mortality rate of 2.0 due to COVID-19 in Mexico. Hence, we reject the present group of people. us, medical 4 Complexity     Table 4: Optimal parameters of the proposed MDSSP for EWD with δ � 1.5, c � 2 at 50 th percentile.

Comparative Study
A comparitive study is made between single and MDSSP when quality control follows EWD. e OC curve is used to show the efficiency of the plan. e curve has displayed the difference in probabilities of accepting a good lot as well as rejecting the bad lot. Table 7 reveals the efficiency of the proposed MDSSP over the SSP while assuming the underlying distribution of data to follow exponentiated Weibull distribution considering quantile ratio t 0 /t 0 q � 2, 4, 6, 8, 10 for each consumer's risk β � 0.25, 0.10, 0.05, 0.01 while keeping producer's risk at α � 0.05. e comparison is basically on the sample size n and probability of acceptance P a (p 1 ). e acceptance sample size for the proposed MDSSP is smaller than the existing single-sampling plan for several set parameters (see Table 7). For quantile ratio 2, the plan parameters for MDSP are n � 17, c 1 � 1, c 2 � 5, and m � 2, whereas for SSP, the plan parameters are n � 31 and c 1 � 3 with the corresponding probability of acceptance of 0.972 and 0.968. e acceptance sample size is smaller for MDSP and relatively larger for SSP, while the acceptance probability is larger for MDSP and relatively larger than that of SSP. As the quantile ratio increased, the acceptance sample size decreased for both sampling plans. Figure 2 depicts the operating characteristic (OC) curve for comparison of MDSP with plan parameters n � 17, c 1 � 1, c 2 � 5, and m � 2 and SSP with n � 31 and c 1 � 3. It is noticed that the MDS plan is reasonably and greatly efficient than SSP in terms of sample size.

Conclusions
In this article, multiple deferred state sampling plans were developed under the assumption that the lifetime of the product follows an exponentiated Weibull distribution when lifetime tests are truncated. e optimal parameters of the proposed sampling plan are obtained by satisfying the respective consumer and producer's risks simultaneously. A comparative study of the proposed MDSSP has been performed using OC curves along with a single-sampling plan. We conclude that the proposed MDS sampling plan is more effective than the existing single-sampling plans to secure the consumer and producer with less inspection.

Data Availability
e datasets used to support the findings of this study are included within the article.