Study on special type double sampling plan based on truncated Life-test for assuring percentile lifetime using exponentiated generalized FRECHET distribution

Introduction

In modern technological world, customers are more expertise in the reliable quality products. Assume that a producer sends a consumer a shipment of components known as a lot. The inspection samples have been taken from the lot, and the number of defective items will be counted. The selection of the sample size 'n' and acceptance number 'c' are the basic criterion of acceptance sampling plan. If number of defectives less than or equal to acceptance number, the entire lot will be accepted. If the number of defectives greater than acceptance number, the lot will be rejected. In some situations, the experimenters might take a time while before recording the necessary number of failures. As a result, sampling plan use time-truncated life tests to make decisions quickly. Time-truncated life tests have a defined number of failures and experiment duration. This approach is excellent for reducing the cost and time of a submitted lot's inspection. In acceptance sampling based on truncated life tests have the following assumptions: (i) the units are destructible or are degraded after the life test, and (ii) there are several distributions that model the product life reasonably well. The main problem in acceptance sampling plan is finding the minimum sample size, Operating Characteristic function and producer's risk exceeds the specified and true lifetime of product. In this paper focuses Special Type Double sampling plan based on truncated life test using Exponentiated Generalized Frechet Distribution. Special Type Double sampling plan is a special cases of Double sampling plan. Schilling and Neubauer proposed a concept of Single Sampling Plan with a zero acceptance number is extremely helpful in the field of compliance and safetyrelated testing. A lot by lot special type double sampling strategy is employed as an alternative to the Single Sampling Plan in order to avoid the problems with smaller acceptance numbers. Special type double sampling plan based on truncated lifetest approach will be more efficient and save time and cost.

In Reliability Acceptance Sampling Plan based on truncated life test using various distribution of review of literature given below: Epstein (1954) [5] has developed truncate life test in the Exponential cases. Gupta and Groll (1961) [6] has developed Sampling Plans Based on the Gamma Distribution. Rao et al. (2010) [10] has designed the life test sampling plans for log-logistic Distributions. Kantam and Rosaiah (2005) [11] has progressed an acceptance sampling based on the Inverse Rayleigh Distribution. Rosaiah. et al. (2006) [12] has described the source of reliability of test plans for Exponentiated Log-Logistic Distribution. Balakrishnan et al. (2007) [3] has designed an acceptance sampling plan from truncated life tests based on the Generalized Birnbaum-Saunders Distribution. Aslam, et al. (2010) [2] has designed acceptance sampling plan using generalized exponential Distribution. Pradeepa Veerakumari, et al. (2016) [9] has developed for Exponentiated Rayleigh Distribution. Kaviyarasu et.al (2017) [8] has developed for Weibull-Poisson Distribution. Dodge and Romig (1959) [4] have studied the use of DSP (0,1) plan to product characteristics involving costly and destructive testing. Sudamani Ramaswamy and Sowmya (2016) [13] discussed time truncated special purpose double sampling DSP (0, 1) for compound Rayleigh distribution. Jayalakshmi, et al. (2022) [7] has developed a Special Type Double Sampling Plan for Life Tests Based on Percentiles Using Gompertz Frechet distribution. Abd-Elfattah (2016) [1] has developed Exponentiated Generalized Frechet Distribution. The Exponentiated Generalized family of densities, denoted as f(x), offers enhanced adaptability in its tail regions and finds widespread applications across engineering and biological fields.

Exponentiated Generalized Frechet Distribution

The Cumulative Distribution Function of Exponentiated Generalized Frechet distribution is

The Probability density function of Exponentiated Generalized Frechet distribution is

Where, 𝛼, 𝛽, λ are Shape parameters and 𝜎 be a Scale parameter.

Survival Function of Exponentiated Generalized Frechet distribution is

Hazard Function of Exponentiated Generalized Frechet distribution is

Quantile Function of Exponentiated Generalized Frechet distribution is

Percentile Estimator of Exponentiated Generalized Frechet Distribution

The q th percentile function of the any distribution is given below, Pr(T≤ 𝑡 𝑞 )=q

Replacing the scale parameter 𝜎 in equation ( 1) then we get Cumulative Distribution function for Exponentiated Generalized Frechet distribution is

Taking a first derivative of partial differentiation with respect to 𝛿, we get ). The proposed sampling plan the probability of acceptance is given by,

The operating procedure of special double sampling plan for the truncated life test consists of the following steps:  Draw a random sample of size 𝑛 1 from the lot and put on test for pre-assigned experimental time to and observe the number of defectives 𝑑 1 . If 𝑑 1 ≥ 1 reject the lot.



If 𝑑 1 =0, draw a second random sample of size 𝑛 2 and put on test for time 𝑡 0 and observe the number of defectives𝑑 2 .  If 𝑑 2 ≤1 accept the lot, otherwise reject the lot.

Minimum Sample Size

The life test terminates at the predetermined period 𝑡 0 , and the number of failures that occurred between [0, t] is noted. If and only if the total number of failures at the decision about the termination time does not exceed the acceptance numbers𝑐 1 𝑎𝑛𝑑 𝑐 2 , the lot is decided to be accepted. The theory of binomial distribution can be used because the lot size is supposed to be infinitely large. The probability of acceptance must be satisfy the conditions,

Where, 𝑛 = 𝑛 1 + 𝑛 2 , p is the probability that items fails before𝑡 0 .

The minimum sample sizes are satisfying the conditions of Search procedure. To determine the minimum sample size of Special Type Double Sampling plan reduces to

Operating Characteristic Function

The operating characteristic curve plots between the probability of acceptance and specified lifetime of product. The operating characteristic function of the Special Type Double sampling plan must satisfying the following equation,

Where, p=F(t,𝛿) and 𝛿 can be indicated as a function of 𝛿 = 𝑡 𝑡 𝑞 ⁄ 𝑝 = 𝐹 ( 𝑡 𝑡 𝑞 .

𝑑 𝑞 )

Where 𝑑 𝑞 =𝑡 𝑞 𝑡 𝑞 0 ⁄ In the minimum sample size and operating Characteristic values of 25 th percentile using Exponentiated Generalized Frechet Distribution given in Table 1.

Minimum ratio of Producer's risk

The producer's Risk is described as the probability of rejecting a lot 𝑡 𝑞 >𝑡 𝑞 0 . For a given value of the producer's risk, say 𝛼 and satisfying the following equation.

Numerical Illustration

Assume that an Inspector conducts an experiment on lifetime of juice mixer, when he interests with use to Exponentiated Generalized Frechet Distribution with fixed shape parameter 𝛼 = 3, 𝛽 = 2, 𝛾 = 3. Figure 1 represents the images of juice mixer.

The investigator wants to run the experiment for 320 hours, but the laboratory has testers to true percentile life time is 400 hours, 𝑐 1 =0,𝑐 2 =1, 𝛼= 0.05, 𝛽=0.10, then, 𝜑 0.25 = 0.1173 is computed from the equation get under the percentile estimator and the minimum ratio and the minimum sample size is 𝑛 1 = 5,𝑛 2 = 3 get the information from the table 1. The probability of acceptance is given by (𝑛 1 ,𝑛 2 , 𝑐 1 ,𝑐 2 t/𝑡 0.25 ) = (5,3,0,1,0.8) with p* = 0.95 under the Exponentiated Generalized Frechet distribution. Step 4: Find the minimum ratio for the acceptability of a lot with producer's risk must satisfy the condition is,

Conclusion

In this paper deals with the Special Type double sampling based on truncated life test using Exponentiated Generalized Frechet Distribution. Special Type Double Sampling Plan is also called as Zero-one Double Sampling plan. In this paper mainly concentrates minimize the producer's risk and also satisfy the consumer expectations. So, we used the truncated life test approach because it very useful to save inspection time and cost. The probability of acceptance is given by (𝑛 1 ,𝑛 2 ,𝑐 1 ,𝑐 2 t/𝑡 0.25 ) = (7,3,0,1,0.8) with p* = 0.95 under the Exponentiated Generalized Frechet distribution. The table values are used for suitable applications of the product. So, it is strongly recommended that industrial practitioners can use the proposed plan for the testing of electronic products.

References

1. Elfattah AAM, Assar SM, Abd-Elghaffar HI. Exponentiated generalized Frechet distribution. International Journal of Mathematical Analysis and Applications. 2016;3(5):39-48.
2. Aslam M, Jun CH. A double acceptance sampling plan for generalized log-logistic distributions with known shape parameters. Journal of Applied Statistics. 2010;37(3):405-414.
3. Balakrishnan N, Leiva V, Lopez J. Acceptance sampling plans from truncated life tests based on generalized Birnbaum Saunders distribution. Communications in Statistics-Simulation and Computation. 2007;36(3):643-656.
4. Dodge HF, Romig HG. Sampling inspection tables-single and double sampling. 2nd ed. New York: John Wiley & Sons, 1959.
5. Epstein B. Truncated life tests in the exponential case. Annals of Mathematical Statistics. 1954;25:555-564.
6. Gupta SS, Groll PA. Gamma distribution in acceptance sampling based on life tests. Journal of the American Statistical Association. 1961;56(296):942-970.
7. Jayalakshmi S, Vijilamery S. Designing of special type double sampling plan for truncated life test using Gompertz Frechet distribution. International Journal of Mathematics Trends and Technology-IJMTT. 2022;68:1-7.
8. Kaviayarasu V, Fawaz P. A reliability sampling plan to ensure percentiles through Weibull Poisson distribution. International Journal of Pure and Applied Mathematics. 2017;117(13):155-163.
9. Pradeepa Veerakumari K, Ponneeswari P. Designing of acceptance sampling plan for life tests based on percentiles of exponentiated Rayleigh distribution. International Journal of Current Engineering and Technology. 2016;6(4):1148-1153.
10. Rao GS, Kantam RR. Acceptance sampling plans from truncated life tests based on the log-logistic distribution for percentiles. Economic Quality Control. 2010;25(2):153-167.
11. Rosaiah K, Kantam RR. Acceptance sampling based on inverse Rayleigh distribution. Economic Quality Control. 2005;20:277-286.
12. Rosaiah K, Kantam RR, Kumar S. Reliability test plans for exponentiated log-logistic distribution. Economic Quality Control. 2006;21(1):29-41.
13. Sudamani Ramaswamy AR, Sowmya N. Time truncated special purpose double sampling DSP(0,1) for compound Rayleigh distribution. International Journal of Engineering Sciences and Management Research. 2016;3(5):69-80.