Inspection plan for COVID-19 patients for Weibull distribution using repetitive sampling under indeterminacy

It is broadly established that a huge number of COVID-19 cases are unnoticed worldwide. A rudimentary measure of population occurrence is the small part of positive cases for a given date in any country. On the other hand, this is subject to largely found that bias since tests are normally only ordered from suggestive cases, whereas a large proportion of infected people might show little symptoms or sometimes no symptoms for more details see [1]. Most governments are applying the mechanism of test randomly selected individuals to estimate the true disease occurrence in inhabitants in a particular locality. Nevertheless, when the disease occurrence is low and difficult to acquire from each patient/person by tests, under such situations we may use an acceptance sampling plan under indeterminacy. The health practitioners are paying attention to estimate the average number of deaths or ratio of deaths to the total number of COVID-19 death cases on daily basis, for the coming days, next week or month, etc. Reference [2]. In such a case, the health practitioners are paying attention to test the null hypothesis that the average number of deaths or ratio of deaths to the total number of COVID-19 death cases on daily basis is equal to the specified average number of deaths due to COVID-19 against the alternative hypothesis that the average number of deaths due to COVID-19 varies significantly. In this situation for testing of the hypothesis, practically it is difficult to record the average number of death for the whole year, whereas it is easy to record the daily basis and the average number of deaths can be obtained from the randomly selected days. The null hypothesis may be rejected if the daily average number of deaths due to COVID-19, state acceptance number of days, is more than or equal to the specified average number of deaths due to COVID-19 throughout the given number of days.

Many researchers have done studies on the time truncated life test for various distributions. Some of them are [3] developed the acceptance sampling plan for life tests: log-logistic models. Reference [4] derived acceptance sampling based on truncated life tests for generalized Rayleigh distribution. Reference [5] developed the acceptance sampling plans based on the generalized Birnbaum-Saunders distribution. Reference [6, 7] constructed the acceptance sampling plans for Birnbaum-Saunders and Burr XII distributions. References [8, 9] constructed acceptance sampling plans for extended exponential and generalized inverted exponential distributions. The details about the acceptance sampling plans can be seen in [10, 11]. The generalization of a single acceptance sampling plan namely repetitive sampling plan, [12] derived the decision rule of the repetitive acceptance sampling plan. The method of repetitive group acceptance sampling plan (RGASP) was first proposed by [13] for an attribute. Reference [14, 15] constructed the RASP for inverse Gaussian distribution and Burr type XII. Reference [16] developed generalized inverted exponential distributions. References [17‐19] studied the repetitive sampling plan under different situations.

More details about the neutrosophic logic, their measure of determinacy, and indeterminacy are given by [20]. Numerous authors studied the neutrosophic logic for various real problems and showed its efficiency over fuzzy logic, for more details refer [21‐26]. The idea of neutrosophic statistics was given using the idea of neutrosophic logic, [27‐29]. The neutrosophic statistics give information about the measure of determinacy and measure of indeterminacy, see [30]. The neutrosophic statistics reduce to classical statistics if no information is recorded about the measure of indeterminacy. References [31‐33] proposed the acceptance sampling plans using neutrosophic statistics [34]. proposed the time-truncated group plans for the Weibull distribution. Reference [35] worked on neutrosophic Weibull and neutrosophic family of Weibull distribution.

The existing sampling plans based on classical statistics and fuzzy philosophies do not give information about the measure of indeterminacy. Reference [36] worked on the single sampling plan using a fuzzy approach. Reference [37] discussed the effect of sampling error on inspection using a fuzzy approach. Reference [38] proposed a single plan using fuzzy logic. Reference [39] proposed the improved sampling plan using fuzzy logic. For details, the reader may refer to [40, 41]. To the best of our knowledge, there is no work on a time-truncated sampling plan for Weibull distribution under indeterminacy. In this paper, a repetitive acceptance sampling plan for Weibull distribution under indeterminacy is developed to testing the daily average deaths. We are anticipated the proposed sampling plan shows a fewer sample size as compared with the existing sampling plans for testing the daily average deaths.

In Section 2, we present an introduction of a repetitive acceptance sampling plan for Weibull distribution under indeterminacy. In Section 3, the proposed repetitive acceptance sampling plan under indeterminacy is compared with the single sampling plan proposed by [42]. The proposed sampling plan is illustrated using COVID-19 data belong to Italy, which was recorded from 1 April to 20 July 2020 in Section 4. Finally, the conclusions and future research works are established in Section 5.

The repetitive acceptance sampling plan depends upon the truncated life test procedure is developed by [43‐45]. The operational steps of this test are given as follows:

The above procedure of repetitive acceptance sampling plan (RASP) mainly depends on four characteristics those are n, c₁, c₂ and I_N, where I_Nϵ[I_L, I_U] is considered as the specified parameter and set according to the uncertainty level. RASP is nothing but the generalization of an ordinary single sampling plan under uncertainty. If c₁ = c₂ in RASP, it ultimately reduces to a single sampling plan under uncertainty. Suppose that t₀ = aμ₀ be the time in days, where a is the termination ratio. The lot acceptance probability is to be determined with the help of operating characteristic (OC) function for details see [13] and it is given by

Here P_a(p) is the probability of accepting H₀ : μ_N = μ_0N and P_r(p) is the probability of rejecting H₀ : μ_N = μ_0N, which are given by and

where p is the probability of unreliability.

Therefore eq. (1) becomes

The Weibull distribution under neutrosophic statistics is developed by [42] for the design of the sampling scheme plan for testing the average wind speed under an indeterminate environment.

Suppose that f(x_N) = f(x_L) + f(x_U)I_N; I_Nϵ[I_L, I_U] be a neutrosophic probability density function (npdf) having determinate part f(x_L), indeterminate part f(x_U)I_N and indeterminacy interval I_Nϵ[I_L, I_U]. Note that x_Nϵ[x_L, x_U] be a neutrosophic random variable follows the npdf. The npdf is the generalization of pdf under classical statistics. The proposed neutrosophic form of f(x_N)ϵ[f(x_L), f(x_U)] reduces to pdf under classical statistics when I_L =0. Based on this information, the npdf of the Weibull distribution is defined as follows.

where α and β are scale and shape parameters, respectively. Note here that the proposed npdf of the Weibull distribution is the generalization of pdf of the Weibull distribution under classical statistics. The neutrosophic form of the npdf of the Weibull distribution reduces to the Weibull distribution when I_L =0. The neutrosophic cumulative distribution function (ncdf) of the Weibull distribution is given by

The neutrosophic mean of the Weibull distribution is given by.

The null and alternative hypotheses for the daily average deaths are stated as follows:

Where μ_N is a true daily average death and μ_0N is the specified daily average deaths. Suppose that t_0N = aμ_0N be the time in days, where a is the termination ratio. The probability of the item will fail before it reaches the experiment time t_0N is defined as follows: where μ_N/μ_0N is the ratio of true average daily wind speed to specified average daily wind speed. Suppose that \(\tilde{\alpha }\) and \(\tilde{\beta }\) be type-I and type-II errors. The medical practitioners are interested to apply the proposed plan for testing H₀ : μ_N = μ_0N such that the probability of accepting H₀ : μ_N = μ_0N when it is true should be larger than \(1-\tilde{\alpha }\) at μ_N/μ_0N and the probability of accepting H₀ : μ_N = μ_0N when it is wrong should be smaller than \(\tilde{\beta }\) at μ_N/μ_0N = 1. In order to find the design parameters n, c₁, c₂ and I_N for the proposed RASP, we consider two points on the OC function. In our approach, the quality level mainly depends on the ratio μ_N/μ_0N. This ratio is helpful for the producer to improve the lot quality. From in producer point of view, the probability of acceptance should be at least \(1-\tilde{\alpha }\) at acceptable quality level (AQL), p_1N. So, the producer demands the lot should be accepted at various levels of μ_N/μ_0N. Similarly, from in consumer point of view the lot rejection probability should not be exceeded \(\tilde{\beta }\) at limiting quality level (LQL), p_2N. The design parameters are determined by satisfying the following two inequalities where p_1N and p_2N are defined by

The estimated designed parameters of the proposed plan should be minimizing the average sample number (ASN) at an acceptable quality level. The ASN for the proposed plan with fraction defective (p) is derived to be

Therefore, the design parameters for the proposed plan with minimum sample size will be obtained by solving the below optimization technique

The values of the designed parameters n, c₁ and c₂ for various values of \(\tilde{\beta }\) =0.25, 0.10, 0.05, 0.01; \(\tilde{\alpha }=0.10\); a = 0.5 and 1.0, μ_N/μ_0N =1.1, 1.2, 1.3, 1.4, 1.5, 1.8, 2.0 and I_N =0.0, 0.02, 0.04 and 0.05 when shape parameter β = 1, 2 and 3 are given in Tables 1, 2, 3, 4, 5 and 6. Tables 1 and 2 are shown for the exponential distribution case. For exponential distribution, it can be seen that the values of ASN decrease as the values of a increases from 0.5 to 1.0. On the other hand for other the same parameters, the values of n decreases as the values of β increases. Note here that the indeterminacy parameter I_N also plays a significant role in minimizing the sample size. As indeterminacy parameter I_N increases the ASN values are decreasing.

A comparative study is carried out between the proposed sampling plans with the existing sampling plans available in the literature with respect to the sample size in this section. We know the cost of the study is always directly proportional to the sample size, a plan is said to be economical if it requires a smaller number of samples for testing the hypothesis about the daily new deaths from COVID-19. The proposed repetitive sampling plan under uncertainty/indeterminacy for Weibull distribution is the generalization of the testing average wind speed using sampling plan for Weibull distribution under indeterminacy plan developed by [42]. The comparison for the proposed and the existing sampling plan for Weibull distribution under indeterminacy plan developed by [42] are displayed in Tables 7 and 8 for \(\tilde{\alpha }=0.10;\beta =2\) at a = 0.5 and 1.0. The developed sampling plan reduces to the existing sampling plan when c₁ = c₂ = c. From Tables 7 and 8, it is noticed that the values of the sample size required for testing H₀ : μ_N = μ_0N smaller for the proposed sampling plan as compared with the existing sampling plan developed by [42]. For example, when μ_N/μ_0N =1.1 and a =0.5 from Table 7, it can be seen that ASN = 491.58 from the plan proposed sampling plan whereas existing sampling plan sample size n = 617 when I_N =0.02, β = 2 and a = 0.5. Hence, the proposed sampling plan is more economical than the existing sampling plan.