New Acceptance Sampling Plans Based on Truncated Life Tests for Akash Distribution With an Application to Electric Carts Data

In this paper, we develop new acceptance sampling plans based on truncated life tests for the Akash distribution. With various values of the Akash distribution parameter, the minimum sample sizes required to assert the specified mean life are obtained, also the operating characteristic function values and producer’s risk of the proposed sampling plan are presented. The results are illustrated by different examples for different values of the sampling plans. A real data of 20 small electric carts is used to illustrate the power of the new sampling plans. The results revealed that the suggested acceptance sampling plan is useful for researchers and engineering in producing lots.


I. INTRODUCTION
In statistical process control, even used most effective statistical techniques, defective or not satisfying some standard requirements products are inevitable. For this reason, it is required to check the lot after production and producers have to prevent them from reaching consumer. Acceptance sampling plans have been widely used to see if the lot is acceptable or not by inspecting sample. The acceptance of a lot is decided when the number of failures exceed acceptance number and one can terminate the test. Now, the process started by obtaining the minimum sample size that is necessary to emphasize a certain average life when the life test is terminated at a predetermined time. Such tests are called truncated lifetime tests.
An acceptance sampling plan based on truncated life tests consists of the following quantities: 1) The number of units (n) on test.
2) An acceptance number (c), where if c or less failures happened within the test time (t), the lot is accepted.
3) The maximum test duration time, t. 4) The ratio d = t/µ 0 , where µ 0 is the specified average life.
The associate editor coordinating the review of this manuscript and approving it for publication was Jenny Mahoney.
The acceptance sampling plan based on truncated life tests is studied by many authors for a variety of distribution. We can list some key studies in acceptance sampling literature considered special distributions as follows: exponential distribution by Sobel and Tischendrof [31], log-logistic distribution Kantam et al. [21] generalized Rayleigh distribution by Tsai and Wu [32], generalized exponential distribution by Aslam et al. [14], Maxwell distribution by Lu [23], inverse Rayleigh distribution by Rao et al. [26]. Govindaraju and Kissling [17] proposed sampling plans for beta distributed compositional fractions.
Braimah et al. [16] investigated single truncated acceptance sampling plans for Weibull product life distributions. Malathi and Muthulakshmi [2] proposed an economic design of acceptance sampling plans for truncated life test using Fréchet distribution. Gogah and Al-Nasser [18] developed a ranked acceptance sampling plan by attribute for exponential distribution. Mahdy et al. [24] considered skew-generalized inverse Weibull distribution in acceptance sampling. Al-Omari et al. [10] and [11] proposed acceptance sampling plans based on truncated life tests for Rama distribution and three-parameter Lindley distribution. Al-Omari and Al-Nasser [9] introduced new acceptance sampling plans for two parameter quasi Lindley distribution.
In the recent years about acceptance sampling plans see for exponentiated Fréchet distribution Al-Nasser and Al-Omari [1], for transmuted inverse Rayleigh distribution Al-Omari [2], for exponentiated inverse Rayleigh distribution Sriramachandran and Palanivel [30], for generalized inverted exponential distribution Al-Omari [3], for generalized inverse Weibull distribution Al-Omari [4], Al-Omari and Alhadrami [8] for extended exponential distribution, and for weighted exponential distribution Gui and Aslam [19] proposed acceptance sampling plans based on truncated lifetime tests.
Also, in newly published studies, Al-Omari [5] has studied the Garima distribution, transmuted generalized inverse Weibull distribution, and Sushila distribution respectively, Al-Omari et al. [6] used Marshall-Olkin Esscher transformed Laplace distribution and Al-Omari et al. [7] considered new Weibull-Pareto distribution in acceptance sampling plans based on truncated life tests.
The rest of this paper is organized as follows: Section 2 provides the Akash distribution as well as some statistical properties. In Section 3, we illustrated the new sampling plans based on the Akash distribution and its properties as the minimum sample size, the operating characteristic function and the producer's risk. The important tables and illustrated examples are given in Section 4. An application of real data is given in Section 5. Finally, the paper is concluded in Section 6. Shanker (2015) suggested Akash distribution probability density function (PDF) defined as (1) and cumulative distribution function (CDF) given by

II. AKASH DISTRIBUTION
The following three figures are the PDF, CDF, and reliability and Hazard functions of the Akash distribution, respectively. The hazard and reliability functions of the Akash distribution, respectively are given in Figure 3 and Figure 4, respectively.   The r th moment of the AD distribution is given by and the mean of the AD distribution is E (X ) = δ 2 +6 δ(δ 2 +2) . The coefficient of variation (CV) and coefficient of skewness (Sk), respectively, are The hazard rate function and mean residual life function are and Minimum sample sizes to be tested for a time t to assert with probability P * and acceptance number c that µ ≥ µ 0 for δ = 2 in the Akash distribution.

III. DESIGN OF THE ACCEPTANCE SAMPLING PLAN
Assume that life time of the product follow the Akash distribution defined in Equation (1 During the experiment the researchers assume that the lot size is infinitely large so that the theory of binomial VOLUME 8, 2020 TABLE 2. Operating characteristic function values for the sampling plan n, c = 2, t /µ 0 with a given probability P * for δ = 2 in the Akash distribution. distribution can be applied. Assume that the consumer's risk (the probability of acceptance a bad lot) is determined to be at most 1 − P * , i.e., the probability that the real mean life µ is less than µ 0 , not exceeds 1 − P * . Our problem is to get the smallest sample size n necessary to satisfy the inequality where c is the acceptance number for given values of P * ∈ (0, 1), where p = F(t; µ 0 ) is the probability of a failure observed within the time t which depends only on the and µ 0 = . If the number of observed failures within the time t is at most c, then from (7) we can confirm with probability P that   [21], and Gupta and Groll [20].
The operating characteristic function of the sampling plan (n, c, t/µ 0 ) is the probability of accepting the lot. Indeed, it can be can be considered as a source for choosing the minimum sample size, n, or the acceptance number, c. The operating characteristic function of the suggested acceptance sampling plan is defined as OC(p) = P (Accepting a lot |µ < µ 0 ) where p = F (t 0 ; µ). The producer's risk (PR) is the probability of rejection of the lot when it is good, i.e., µ > µ 0 . It is defined as For the suggested sampling plan and a given value for the producer's risk, , the experimenter is interesting in knowing the value of µ > µ 0 that will assert the PR to be at most .
µ is a function of µ/µ 0 , then µ/µ 0 is the smallest positive number for which p satisfies the inequality given by  Minimum sample sizes to be tested for a time t to assert with probability P * and acceptance number c that µ ≥ µ 0 for δ = 5 in the Akash distribution. where For a given value of the producer's risk, say λ, under this sampling plan, one may be interested in knowing what is the smallest value of the ratio µ/µ 0 that will assert the producer's risk is at most λ. This value is the minimum positive number for which p = F t µ 0 µ 0 µ satisfies the inequality TABLE 5. Operating characteristic function values for the sampling plan n, c = 2, t /µ 0 with a given probability P * for δ = 5 in the Akash distribution.
For a given acceptance sampling plan (n, c, t/µ 0 ) based on the AD at a specified confidence level P * , the smallest values of µ/µ 0 satisfying Inequality (13) are given in Table (3) for δ = 2.

IV. ILLUSTRATION OF TABLES AND EXAMPLES
In this section, we studied the performance of the proposed sampling plans in terms of the minimum sample sizes, operating characteristic function and minimum ratio. Various values of the Akash distribution parameter values δ = 2, 5. The results for δ = 2 are presented in Tables (1-3) and for δ = 5 are given in Tables (4-6).

A. TABLES FOR δ = 2
For an acceptance number c, the smallest sample sizes necessary to assert that the mean life exceeds µ 0 with probability greater than or equal P * for δ = 2 in Akash distribution are presented in Table (1). VOLUME 8, 2020 TABLE 6. Minimum ratio of µ/µ 0 for the acceptability of a lot with producer's risk of 0.05 for δ = 5 in the Akash distribution.
Assume that the life time of the products follows the Akash distribution with parameter δ = 2, and that the researcher like to establish that the mean life is greater than or equal to at least µ 0 = 1000 hours with probability P * = 0.99. Also, assume that the life test will be terminated at t 0 = 1257 hours. Since Table (1) provides the smallest sample size, then when P * = 0.99, d = t 0 /µ 0 = 1.257, and c = 2, the corresponding Table (1) entry is n = 9 units. Now, these 9 units should be tested and if out of the 9 items if no more than two items are fail within 1257 hours, the researcher can confirm that the true mean life µ of the items is at least 1000 hours with confidence level of 0.99.
The operating characteristic function values for the suggested sampling plan based on the Akash distribution adopted in Table (1) for various values of P * and d = t 0 /µ 0 = 1.257 with acceptance number c = 2 are presented in Table (2). For illustration, when P * = 0.99, c = 2, t 0 /µ 0 = 1.257, µ/µ 0 = 4, the corresponding table entry is 0.525701, it implies, based on the above acceptance sampling plan, that is the lot is accepted if out of 9 items, less than or equal 2 items fail before time point t 0 = 1257 hours, then if µ ≥ 4 × t 0 /1.257 = 3.1822t 0 = 4000 hours, then the product will be accepted with probability of at least 0.525701.
The minimum ratio of the true mean lifetime to the specified one for the proposed acceptance plan of a lot with producer's risk = 0.05 are given in Table (3). For illustration, when P * = 0.99 (consumer's risk is 0.01), c = 2, d = t/µ 0 = 1.257, the corresponding table entry µ/µ 0 = 13.277, which implies that if µ ≥ 13.277 × t 0 /1.257 = 10.5625 t 0 = 13277 hours, then with c = 2 and sample size n = 9, the lot will be rejected with probability less than or equal to 0.05. That is, the product is accepted with probability of at least 0.95.
Minimum sample sizes to be tested for a time t to assert with probability P * and an acceptance number c that µ ≥ µ 0 for δ = 5 in the Akash distribution are summarized in Table 4. The operating characteristic function values for the new sampling plan and the minimum ratio of µ/µ 0 for the acceptability of a lot with producer's risk of 0.05 for δ = 5 in the Akash distribution are provided in Tables 4 and 5, respectively.
When are compared the minimum sample sizes presented in Tables (1) and (4) to investigate the effect of the life time distribution parameter, it is found that the minimum sample sizes calculated when δ = 5 are less than their counterparts of δ = 2 for fixed P * and t/µ 0 .
Also, when we compare the results obtained based on the suggested acceptance sampling plans for the Akash distribution with their competitive in Al-Nasser and Al-Omari [1], Baklizi and El Masri [15], and Kantam et al. [21], it turns out that the samples size obtained in this paper are smaller than their counterparts.

V. AN APPLICATION OF ELECTRIC CARTS DATA
A real data set is considered in this section to investigate the performance of the suggested acceptance sampling plans. These data was already considered by Zimmer et al. (1998), Gui and Aslam (2017), and Lio et al. (2010). The data consists of the lifetime (in months) to first failure of 20 small electric carts used for internal transportation and delivery in a large manufacturing facility. The data are 0.9, 1.5, 2.3, 3.2, 3.9, 5, 6.2, 7. 5, 8.3, 10.4, 11.1, 12.6, 15, 16.3, 19.3, 22.6, 24.8, 31.5, 38.1, 53. The analysis of the data is given below in Table 7.  Table (7) where AIC = −2MLL + 2w, CAIC = −2MLL + 2w n n − w − 1 , where w is the number of parameters and n is the sample size. The results are presented in Table 8. Let the specified mean lifetime and the testing time are µ 0 = 14.6535 and t = 9.202 months, respectively. Therefore, for P * = 0.75 and d = t/µ 0 = 0.628, the acceptance number and the corresponding minimum sample sizes are given in Table 9, which is found to be c = 4. Hence, if the number of failures before t = 9.202 months, is less than or equal to 4, we can accept the lot with the assured mean lifetime 14.6535 months with probability 0.70. Since the number of failures before t = 9.202 months is 9, then the lot is rejected. The values of OC for the ASP (n = 20, c = 4, t/µ 0 = 0.628) and the corresponding producer's risk are presented in Table 10, while the minimum ratios for this example are given in Table 11.

VI. CONCLUSION
In this paper, new acceptance sampling plans based on truncated life tests for the Akash distribution are proposed. The necessary tables are presented for the minimum sample size needed to guarantee a certain mean life of the test units. The operating characteristic function values as well as the associated producer's risks are also provided. The suggested sampling plans are applied for real data set. The outcomes of this paper can be used to develop other kinds of acceptance sampling plans such as group and double acceptance sampling plans for Akash and other distributions.
NURSEL KOYUNCU received the B.Sc., M.Sc., and Ph.D. degrees from the Department of Statistics, Hacettepe University, Turkey. She has studied for one year at the Erasmus MC University Medical Center, Department of Biostatistics, The Netherlands. She is currently working as an Associate Professor with the Department of Statistics, Hacettepe University. Her research interests include sampling, quality control, and calibration and simulation techniques. He is currently an Assistant Professor of statistics with Majmaah University, Majmaah, Saudi Arabia. His research interests include Bayesian statistics, data analysis, regression analysis, statistics in phylogenetics, biostatistics, sampling in general, distribution theory, and the analysis of stock market time series data. VOLUME 8, 2020