A Reliability Sampling Plan to ensure Percentiles through Weibull Poisson Distribution

Volume 117 No. 13 2017, 155-163 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu A Reliability Sampling Plan to ensure Percentiles through Weibull Poisson Distribution V.Kaviayarasu 1 and P.Fawaz 2 1,2 Department of Statistics, Bharathiar University Coimbatore-641046, India. 1 kaviyarasu@buc.edu.ins 2 fawaku@gmail.com Abstract A new Reliability Sampling Plan Procedure has proposed to carried out to design the Double Acceptance Sampling Plan (DASP) based on Percentiles under Weibull Poisson Distribution (WPD). This distribution is widely used in real time data through life test experiments to ensure a specified lifetime of an item. The design parameters are carried out through its sample size and acceptance numbers for both the samples. The minimum sample size and their ratio dq for specified life time percentile were calculated for the producer risk. The Operating Characteristic curve values are obtained for the plan and their corresponding table values are generated through simulated data. AMS Subject Classification:62pxx. Key Words and Phrases:Weibull Poisson Distribution, Double Acceptance Sampling Plan, percentiles, operating characteristic function. 1 Introduction Statistical Quality Control plays a significant role for finding the success or failure of an item in manufacturing industry. Acceptance sampling plan takes an important role in Statistical Quality Control to determine the product quality which are concerned with inspection and decision making through accepting or rejecting a lot in large size of products. If the quality characteristics of product follows a lifetime of an item, then it is called as acceptance sampling procedure for life test experiments or Reliability Sampling Plan. Most circumstance in real time data follows a life testing models but not to determine the lot quality which is one of the major issues in determining the smallest sample size which ensure a specified life percentile. It gives more information about the life of the product than the mean lifetime of the product. Traditionally, the mean value divides the distribution into two parts and the median value divides into two equal parts or the 50 th Percentile in the case of 155

skewed life time distribution. Most of the sampling procedures are not concerned with deciles and percentiles which may divide the distribution into ten and hundred equal parts. Here an attempt is made for the WPD with percentiles to determine the disposition of the lot for life time product. Weibull Poisson Distribution is one of the widely used continuous probability distribution to determine the probability models for life time data. It is one of the popularly used model for modeling the data in reliability analysis, applied statistics, engineering techniques and biological studies. In this paper a life time of the product is assumed to follow Weibull-Poisson Distribution (WPD) which was introduced by Lu and Shi [15] and the plan parameters were designed and developed for the attribute double acceptance sampling. Many authors considered the design of acceptance sampling plans based on life test. Epstein [5]considered truncated life test for exponential distribution, Gupta and Groll [6] developed Gamma distribution in acceptance sampling based on life tests, Kantam and Rosaiah [7] introduced half logistic distribution for life test and developed sampling plan. Wu and Tsai[14] considered the problem of acceptance sampling plan based on truncated life tests when the lifetime of product follows the generalized Rayleigh distribution for known shape parameter. Balakrishnan et al[4] proposed acceptance sampling plans from truncated life tests based on the generalized Birnbaum-Saunders distribution, Srinivasa Rao et al [12]developed acceptance sampling plans from truncated life tests based on log-logistic distribution for percentiles. Aslam et al[2] acceptance sampling plans for Burr type XII distribution percentiles under the truncated life test. All these authors developed acceptance sampling plans based on the mean life time under a truncated life test. Later, Lio et al[10] developed acceptance sampling plans for median and percentiles using truncated life tests and assuming Birnbaum-Saunders distribution. Kaviyarasu and Fawaz[8]Proposed Acceptance Sampling Plans for Percentiles Based on the Modified Weibull Distribution Kaviyarasu and Fawaz [9] developed Single Sampling Plans for Percentiles Based on the Weibull Poisson Distribution. Literatures gives the generalization of acceptance sampling plans based on the life of products. Aslam and Jun [1] designed a double acceptance sampling plan for generalized log-logistic distributions with known shape parameters. Rao[11] proposed double acceptance sampling plans based on truncated life tests for the Marshall-Olkin extended exponential distribution. Aslam et al [3] developed double acceptance sampling plans for Burr type XII distribution percentiles under the truncated life test. In this paper Double Acceptance Sampling Plan (DASP) is developed for the life time of the product assumed to follow Weibull Poisson Distribution based on percentiles. Smaller sample size, Operating Characteristic (OC) and producers risk ratio were developed and relevant tables are given. Few real time examples are given based on life time data sets are provided through suitable illustration. 2 DASP for Percentiles of WPD Weibull-Poisson distribution is a compound distribution, it contain three Parameters which is used for life testing as the shape of the failure rate is flexible, it can be decreasing, increasing, upside-down bathtub-shaped or unimodel. According to Lu 156

and Shi[15] the probability density function of WPD is given by, f(x; θ) = αβλxα 1 1 e λ e λ βxα +λexp( βxα), x > 0, (1) Where, θ = (α, β, λ),α(> 0) is the shape parameter and β(> 0) is the scale parameter of Weibull distribution and λ(> 0) is the Poisson parameter. It is noted that, WPD reduces to two parameter Weibull distribution as λ tends to 0 and the density function of WP distribution is monotonically decreasing if 0 < α 1. The mean and variance of WPD is as follows, and λ E(x, θ) = (e λ 1) 1 e λ [ β 1 (log(y) log(λ))] 1 α dy 0 λ V ar(x) = (e λ 1) 1 e y [ β 1 (log(y) log(λ))] 2 α dy [E(X)] 2 0 Further, they derived the properties of WPD and its cumulative distribution function of WPD is given by, Where θ = (α, β, λ). 2.1 Percentile Estimator F (x, θ) = (e λexp( βxα) )(1 e λ ) 1 (2) The 100q th percentile or the q th quantile of any distribution is given by, P r = (T t q ) = q t q = { β 1 log[λ 1 logq(1 e λ ) + e λ ]} 1 α Assuming when α = 1.Then t q becomes t q = { β 1 log[λ 1 logq(1 e λ ) + e λ ]} t q and q are directly proportional. Let, ψ = log[(λ 1 logq(1 e λ ) + e λ )] t q = 1 β ψ β = t q ψ Replacing the scale parameter (β) by (3), one can get the cumulative distribution function of WPD as, t λexp( F (t) = (e tq ψ) e λ )(1 e λ ) 1.t > 0, δ > 0 (3) 157

Letting δ = t t q F (t; δ) = (e λexp( δψ) e λ )(1 e λ ) 1.t > 0, δ > 0 (4) Assume that a life test is conducted and will be terminated at time t 0. A probability p to reject a bad lot is used to protect consumers. A bad lot means that the true 100q th percentile t q is below the supposed 100q th percentile that is, t q <. The lot is confirmed as a good one if the lifetime data hold the null hypothesis H 0 : t q against the alternative H 1 : t q <. The consumer s risk 1 P is used as the significance level for this hypothesis testing and P is the consumer s confidence level. According to Aslam et al[1]the procedure for WPD with a truncated censoring scheme is proposed as follows: (1) Draw the first random sample of size n 1 from the lot and put them on test. If c 1 or fewer failures are observed at the pre-determined time t 0, the lot is accepted. Otherwise, the life test is truncated to reject the lot before or at t 0 if (c 2 +1) failures are cumulated before or at t 0, where c 1 < c 2. (2) If the observed number of failures (d 1 ) by t 0 is between c 1 +1 and c 2 (c 2 included), then draw a second sample of size n 2 for life testing till a prescribed termination time t 0. The lot is accepted if the cumulated number of failures from two samples (d 2 ) is smaller or equal to c 2. Otherwise, the lot is rejected. Let us represent the Double Acceptance Sampling Plan as (n 1, n 2, c 1, c 2, δ 0 ) Here, n i and c i are the sample size and acceptance number associated with the i th sample respectively, i = 1, 2. For the proposed acceptance double sampling plan, the probability of acceptance of lot is given by, L(p) = Σ c 1 d 1 =0 ( n1 +Σ c 2 d 1 =c 1 +1 ( Σ c 2+d 1 n2 d 2 =0 d 2 ) P d 1 (1 P ) n 1 d 1 d 1 ( n1 d 1 ) P d 1 (1 P 0 ) n 1 d 1 ) P d 2 (1 P ) n 2 d 2 (5) Where, p is the failure probability before the time t, given a specified 100q th percentile lifetime, is obtained from P = F (t; δ 0 ). Where, δ 0 = t. We have F (t, δ) F (t, δ 0 ) t q t 0 The minimum sample size is obtained for the development of sampling plan by satisfying the condition L(p) 1 p. 3 EXAMPLE In this section consider the lifetime distribution as an Weibull Poisson Distribution (WPD) if the investigator is concerned in showing that the true unknown 10 th percentile life t 0.1 is at least 1000hrs. Let the consumer risk is set to 1 P = 0.05. It is desire to quit the experiment at time t = 1000hrs. When the acceptance numbers c1and c 2 as 0 and 1 respectively from Table: 1, the double acceptance sampling plan t (n 1, n 2, c 1, c 2, ) = (27, 61, 0, 1, 0.3).The operating characteristic curve for the plan t 0 0.1 obtained from Table-1 is given below; 158

t Table 1: OC Values for the DASP 1 2 4 8 12 16 20 24 28 32 OC 0.0485 0.2627 0.6368 0.8867 0.9498 0.9727 0.9831 0.9887 0.9919 0.99395 It is observed from the above table that, if the actual 10 th percentile is equal to the required 10 th percentile ( tq = 1) the producers risk is approximately (1 0.0485) = 0.9515 The producers risk is almost equal to 0.05 or less when the actual 10 th percentile is greater than 12 times the specified 10 th percentile. Figure: 1 explains, the experiment is done up to 1000hrs and the following decision is made 1)d = 0, 1 the lot is accepted. 2)d 2, the lot is rejected and the inspector should advice the management to concentrate on the production process for better quality products. 3)d = 2, the inspector is suggested to go for second sample. The following OC curve for the DASP represent the above table values: Figure 1: 1 OC curve for WPD 4 CONCLUSION In this paper, a new distribution is proposed for the Double Acceptance Sampling Plan for life testing experiments. The model parameters of WPD have used to determine the minimum sample size and OC Curve. By the implication of this distribution one can easily obtain the producer risk ratio at 5% level. The results of these three parameter distribution suited for the acceptance sampling plan through life test experiments for percentiles. Further various tq and were obtained for the DASP different experiment times assumed that the life test follows WPD. This distribution provides the high probability values for tq > 4. Few tables are developed for establishing this Plan. ACKNOWLEDGMENT The authors would like to thank the unknown referees for their valuable com- 159

ments and suggestion. Further we acknowledge university authorities for their necessary facilities in the department and support through URF for the research program. References [1] M. Aslam and C.H Jun, A double acceptance sampling plan for generalized loglogistic distributions with known shape parameters, Journal of Applied Statistics, 37: (2010), PP:405-414. [2] M.Aslam, Y. Mahmood, YL. Lio, TR. Tsai and MA. Khan, Acceptance sampling plans from truncated life tests based on the Burr type XII percentiles, Journal of the Chinese Institute of Industrial Engineers, 27, (2010), PP:270-280. [3] M. Aslam, Y. Mahmood, Y.L. Lio, T.R. Tsai and M.A. Khan, Double acceptance sampling plans for Burr type XII distribution percentiles under the truncated life test, Journal of the Operational Research Society, 63, (2012), PP:1010-1017. [4] N. Balakrishnan, V. Leiva and J. Lopez, Acceptance sampling plans from truncated life tests based on the generalized Birnbaum-Saunders distribution, Communications in Statistics: Simulation and Computation, 36, (2007), pp: 643656. [5] B. Epstein. Truncated life tests in the exponential case, Annals of Mathematical Statistics, 25(1954),pp:555-564. [6] S.S. Gupta and P.A. Groll, Gamma distribution in acceptance sampling based on life tests, Journal of the American Statistical Association, 56(1961), pp. 942-970. [7] R. R. L. Kantam and K. Rosaiah, Half Logistic distribution in acceptance sampling based on life tests, CIAPQR Transactions, 23, (1998), pp:117-125. [8] V. Kaviyarasu and P. Fawaz, Certain Studies on Acceptance Sampling Plans for Percentiles Based on the Modified Weibull Distribution, International Journal of Statistics and Systems, 12, (2017), pp:343-354. [9] V. Kaviyarasu and P. Fawaz, Design of acceptance sampling plan for life tests based on percentiles using Weibull-Poisson distribution, International Journal of Statistics and Applied Mathematics, 2, (2017), pp:51-57. [10] Y. L. Lio, T. R. Tsai. and S. J. Wu, Acceptance sampling plans from truncated life tests based on the birnbaum-saunders distribution for percentiles, Communications in Statistics: Simulation and Computation, 39, (2010), pp:119-136. [11] G.S. Rao, Double acceptance sampling plans based on truncated life tests for the Marshall-Olkin extended exponential distribution, Austrian Journal of Statistics, 40, (2011), pp:169-176. 160

[12] G. Srinivasa Rao and R. R. L. Kantam, Acceptance sampling plans from truncated life tests based on log-logistic distribution for percentiles, Economic Quality Control, 25, (2010), pp:153-167. [13] G. Srinivasa Rao and Ch. Ramesh Naidu, Acceptance Sampling Plans for Percentiles Based on the Exponentiated Half Logistic Distribution, Applications and Applied Mathematics: An International Journal, 9, (2014), pp:39-53. [14] T.-R Tsai and S.-J. Wu, Acceptance sampling based on truncated life tests for generalized Rayleigh distribution, Journal of Applied Statistics, 33, (2006), pp:595-600. [15] Wanbo Lu and Daimin Shi, A new compounding life distribution: the WeibullPoisson distribution, Journal of Applied Statistics, 39, (2012), pp:21-38. Appendix P n 1 n 2 t Table 2: Minimum Sample Sizes and OC values for double acceptance sampling plan (n 1, n 2, c 1, c 2, δ 0 ) when c 1 = 0, and c 2 = 1 for10 th percentile of Weibull Poisson Distribution. t q 1 2 4 8 12 16 20 24 28 32 22 874 0.07 0.2594 0.5077 0.712 0.8449 0.899 0.9301 0.95 0.9632 0.9723 0.9786 18 81 0.09 0.2474 0.5524 0.8324 0.9578 0.9831 0.9914 0.9949 0.9967 0.9977 0.9983 15 60 0.11 0.2453 0.5634 0.843 0.9613 0.9845 0.992 0.9953 0.9969 0.9979 0.9984 13 46 0.13 0.2429 0.5743 0.8522 0.964 0.9855 0.9925 0.9956 0.9971 0.998 0.9985 0.75 11 45 0.15 0.2467 0.5621 0.8415 0.9608 0.9843 0.992 0.9953 0.9969 0.9979 0.9984 10 36 0.17 0.242 0.5709 0.85 0.9634 0.9853 0.9924 0.9955 0.997 0.9979 0.9985 9 32 0.19 0.2416 0.5716 0.8506 0.9636 0.9854 0.9925 0.9955 0.9971 0.9979 0.9985 8 31 0.21 0.2453 0.5663 0.8455 0.9621 0.9849 0.9922 0.9954 0.997 0.9979 0.9985 8 22 0.23 0.2331 0.5872 0.8627 0.9667 0.9864 0.9929 0.9957 0.9971 0.998 0.9985 7 23 0.25 0.241 0.5792 0.8562 0.9652 0.986 0.9928 0.9957 0.9972 0.998 0.9985 38 118 0.07 0.0986 0.3558 0.707 0.9146 0.9636 0.9807 0.9883 0.9923 0.9946 0.996 30 82 0.09 0.0973 0.3673 0.7237 0.9213 0.9665 0.9821 0.9891 0.9928 0.9949 0.9962 24 98 0.11 0.1003 0.333 0.6633 0.8937 0.9539 0.9757 0.9854 0.9904 0.9933 0.9951 21 55 0.13 0.0967 0.3706 0.7281 0.923 0.9671 0.9825 0.9893 0.9929 0.9949 0.9962 0.90 17 478 0.15 0.1109 0.371 0.7151 0.9175 0.965 0.9816 0.9889 0.9927 0.9949 0.9962 16 45 0.17 0.0977 0.3637 0.7191 0.9196 0.9657 0.9818 0.9889 0.9927 0.9948 0.9962 14 63 0.19 0.1008 0.3261 0.6458 0.8838 0.949 0.973 0.9838 0.9894 0.9927 0.9946 13 37 0.21 0.0979 0.3622 0.7171 0.9188 0.9654 0.9817 0.9889 0.9926 0.9948 0.9961 12 32 0.23 0.097 0.3678 0.7249 0.9218 0.9667 0.9823 0.9892 0.9928 0.9949 0.9962 11 31 0.25 0.0976 0.3621 0.7174 0.919 0.9655 0.9817 0.9889 0.9926 0.9948 0.9961 49 131 0.07 0.05 0.2515 0.6131 0.8755 0.9447 0.9701 0.9816 0.9877 0.9913 0.9935 38 113 0.09 0.0506 0.2441 0.5948 0.8657 0.9401 0.9676 0.9802 0.9868 0.9907 0.9931 31 602 0.11 0.0512 0.2327 0.5563 0.8411 0.9276 0.9607 0.976 0.9842 0.9889 0.9918 27 61 0.13 0.0485 0.2627 0.6368 0.8867 0.9498 0.9727 0.9831 0.9887 0.9919 0.994 0.95 23 69 0.15 0.0504 0.2419 0.591 0.8637 0.9391 0.9671 0.9799 0.9866 0.9906 0.993 20 378 0.17 0.0528 0.2335 0.5512 0.8367 0.9252 0.9593 0.9752 0.9836 0.9885 0.9916 19 37 0.19 0.047 0.2729 0.654 0.8937 0.9527 0.9741 0.9839 0.9891 0.9922 0.9941 17 36 0.21 0.0482 0.2673 0.6446 0.89 0.9512 0.9734 0.9835 0.9889 0.9921 0.9941 16 31 0.23 0.045 0.2661 0.6472 0.8909 0.9514 0.9734 0.9834 0.9888 0.9919 0.9939 14 38 0.25 0.0504 0.2483 0.6074 0.8726 0.9434 0.9694 0.9813 0.9875 0.9912 0.9934 75 121 0.07 0.0102 0.1223 0.4677 0.8026 0.906 0.9468 0.9663 0.9769 0.9833 0.9874 59 93 0.09 0.0099 0.1246 0.4762 0.8074 0.9083 0.9479 0.9669 0.9773 0.9835 0.9875 49 68 0.11 0.0096 0.1311 0.4933 0.8164 0.9124 0.95 0.9681 0.978 0.9839 0.987 40 103 0.13 0.0109 0.1108 0.4179 0.7678 0.8879 0.9367 0.9602 0.973 0.9807 0.9855 0.99 36 51 0.15 0.0097 0.13 0.4903 0.8148 0.9116 0.9496 0.9679 0.9778 0.9839 0.9877 31 60 0.17 0.0106 0.1154 0.4432 0.7869 0.8981 0.9424 0.9637 0.9752 0.9822 0.9866 Continued on next page 161

Table 2 Continued from previous page... P t n 1 n 2 1 2 4 8 12 16 20 24 28 32 28 50 0.19 0.0103 0.1179 0.4548 0.7945 0.9019 0.9445 0.9649 0.976 0.9827 0.9869 25 57 0.21 0.0109 0.1098 0.415 0.7657 0.8867 0.936 0.9598 0.9727 0.9805 0.9854 23 44 0.23 0.0107 0.1164 0.4459 0.7886 0.8989 0.9429 0.964 0.9755 0.9823 0.9867 22 31 0.25 0.0095 0.1276 0.4857 0.8121 0.9101 0.9487 0.9673 0.9774 0.9835 0.9875 Table 3: Gives the ratio d 0.1 for accepting the lot with the producers risk of 0.05. t q P 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.75 19.8901 6.61135 6.24333 6.02391 6.40803 6.01332 6.00286 6.31563 5.55308 5.92529 0.90 8.73535 8.19825 10.2681 8.15386 8.68386 8.41088 11.0067 8.48649 8.26575 8.49208 0.95 10.3396 10.9597 12.4406 9.50764 11.102 12.8512 8.98178 9.26591 9.07363 10.6793 0.99 12.0826 11.7375 11.1224 14.2298 11.292 13.0676 12.5939 14.3477 12.9458 11.41 162

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