An Economic Reliability Test Plan: Log-Logistic Distribution

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1 Journal of Applied Statistics Vol. 33, No. 3, , April 2006 An Economic Reliability Test Plan: Log-Logistic Distribution R. R. L. KANTAM, G. SRINIVASA RAO & B. SRIRAM Department of Statistics, Acharya Nagarjuna University, India and Department of Statistics, J.K.C. College, Guntur , India ABSTRACT Sampling plans in which items that are put to test, to collect the life of the items in order to decide upon accepting or rejecting a submitted lot, are called reliability test plans. The basic probability model of the life of the product is specified as the well-known log-logistic distribution with a known shape parameter. For a given producer s risk, sample size, termination number, and waiting time to terminate the test plan are computed. The preferability of the test plan over similar plans existing in the literature is established with respect to cost and time of the experiment. KEY WORDS: number Log-logistic distribution, reliability test plan, producer s risk, acceptance sample Introduction Acceptance sampling plans in Statistical Quality Control are concerned with accepting or rejecting a submitted lot of a large size of products on the basis of the quality of the products inspected in a sample taken from the lot. If the quality of the product is the lifetime of the product, then a sample of such products, considered to be representive of the observed life times of the products are put forward for testing. If a decision to accept or reject the lot, subject to the risks associated with the two types of errors (rejecting a good lot/ accepting a bad lot), is possible then such a procedure is termed Acceptance sampling based on life test or Reliability test plans. It requires specifications of a probability model governing the life of the products, since an exponential distribution is a central distribution in reliability studies. The log-logistic distribution has been studied in detail by Shah & Dave (1963) and Tadikamalla & Johnson (1982). Survival models based on logistic and log-logistic distributions haven been given by O Quigley & Struthers (1982). The loglogistic distribution has been considered by Ragab and Green (1984), Balakrishnan & Malik (1987) with reference to moments of order statistics and linear unbiased estimation of location and scale parameters. Truncated life tests of this type are developed by Sobel & Tischendrof (1959) for exponential distribution. Gupta & Groll (1961) constructed similar sampling plans based on gamma life test sample data. Goode & Kao (1961) constructed sampling plans based on the Weibull distribution. Kantam & Rosaiah (1998) suggested Correspondence Address: G. Srinivasa Rao, Department of Statistics, J.K.C. College, Guntur , India. gaddesrao@yahoo.com Print= Online=06= # 2006 Taylor & Francis DOI: =

2 292 R. R. L. Kantam et al. acceptance sampling plans based on life tests when the failure density model of the products is half the logistic distribution. Kantam et al. (2001) studied acceptance sampling based on life tests when the failure density model of the products is a log-logistic distribution. In this paper we construct sampling plans for a log-logistic model in a different approach from that of Kantam et al. (2001) and make a relative comparison of the two approaches. The necessary theory is presented along with the developed plans in the next section, and a comparative study is presented in the section after. The Sampling Test Plan Let us assume that the distribution of the life of a product is the scaled log-logistic distribution whose probability density function and cumulative distribution function are given respectively by f (x) ¼ b (x=s) b 1 s ½1 þ (x=s) b ; x. 0, s. 0, b. 1 (1) 2 Š F(x) ¼ (x=s)b ½1 þ (x=s) b ; x. 0, s. 0, b. 1 (2) Š where b is know shape parameter (b ¼ 2). Let a lot of products of infinitely large size be submitted for sampling inspection, in order for a decision to be made on acceptance or rejection on the basis of this sampling inspection. Let a be the probability of rejecting the submitted lot, which is, in fact, actually a good lot in some sense this is known as producer s risk. Naturally, a should be as small as possible. We can think of this decision making in two ways. Let s 0 be a specified value of s so that s 0 would represent the specified average life of the product. Let t 0 be a pre-assigned time at which the life testing experiment, aimed to collect the lifetimes of the sample products, is terminated. Hence, t 0 may be called the terminating time. The research findings of Gupta & Groll (1961), Kantam & Rosaiah (1998), Kantam et al. (2001) suggest the minimum sample size n required and the corresponding acceptance number c such that if c or fewer failures occur out of n, before the time t 0, the lot would be accepted with a probability of 1 a. This approach is basically counting the number of failures out of n within the terminating time t 0, and hence the life testing experiment would be stopped as soon as the time t 0 is reached or (c þ 1)st failure is realized, whichever is earlier. A typical table of Kantam et al. (2001) is reproduced here in order to illustrate by an example (Table 1). Suppose an experimenter is interested in establishing that the true unknown average life is at least 15707(i.e s 0 ) with a confidence of It is desired to stop the experiment at 10,000 hours. Then, for an acceptance number c ¼ 2, the required sample size n is an entry in the above table corresponding to t 0 /s 0 ¼ 0.628, c ¼ 2. This number is 20. Thus, 20 items have to be put on test. If, during 10,000 hours, not more than two failures out of 20 are observed then the experimenter can assert with a confidence level of 0.95 that the average life is at least hours. The lot shall be rejected if three or more failures within 10,000 hours are noticed. Alternatively, we can think of another approach for a reliability test plan. Let n stand for the number of sampled items to be inspected. Let r be a natural number such that if r failures out of n are realized before the terminated time t 0 the lot would be rejected if the experiment is stopped as soon as the rth failure is reached or termination time t 0 is

3 An Economic Reliability Test Plan 293 Table 1. Minimum sample sizes necessary to assert the average life to exceed a given value, s 0, with probability 1 2 a and corresponding acceptance number, c, using Binomial probability (b ¼ 2) 1 2 a c t/s 0 ¼ reached, whichever is earlier. In this sense, r is called a termination number. The sample size naturally depends upon cost consideration and the expected waiting time to reach a decision. Larger sample sizes may decrease expected waiting times but increase the cost of experimentation. As a balance between these two aspects let us consider that the sample size be a multiple of the termination number. We know that the probability of r failures out of n tested items is given by n c r p r q n r where p ¼ F(x,s) given in equation (2) and q ¼ 1 2 p. Hence, the probability of accepting the lot in the present notation is X r 1 i¼0 n c r p i q n i (3) If a is denoted as the producer s risk, the above equation shows that P r 1 i¼0 n c r p i q n i ¼ 1 a. Because n is assumed to be a multiple of the termination number, it would be of the form kr. Given a, r, k we could solve the above equation for p using cumulative probabilities of binomial distribution. The values of p so obtained when used in equation (2) would give the corresponding value of t/s. This is useful in solving for the termination time t 0 for a specified population average life and hence a specified s. The values of t/s for r ¼ 1(1) 10, k ¼ 2(1) 10, a ¼ 0:05 and 0.01 are tabulated in Tables 2 and 3 respectively. As an example, suppose we have to construct a life test sampling plan with an acceptance probability of 0.95 for lots with an acceptable mean life of 15,707 hours and 10, 5 as, respectively the sample size and termination number. From Table 2, the entry

4 294 R. R. L. Kantam et al. Table 2. Reliability test plan for log-logistic distribution (a ¼ 0.05) r \ n 2r 3r 4r 5r 6r 7r 8r 9r 10r against r ¼ 5 under the column 2r is Since the acceptable mean life is given to be hours this shows that s 0 ¼ that is s 0 ¼ 15,707=1:5707 ¼ 10,000. The reading from Table 2 is , which can be treated as t/s 0. Using the specified value of s 0 we can get t 0 ¼ s 0 ( ) ¼ and this give the termination time. This test plan will be implemented as follows. Select 10 items from the submitted lot and put them to test. If the fifth failure is realized before hours we reject the lot otherwise accept the lot. In either case, terminate the experiment as soon as the fifth failure occurs or the th hour is reached, whichever is earlier. As a comparison of sampling plans for the log-logistic model under the two approaches, the common entries for n, r/c from Tables 2 and 3 and Table 1 of Kantam et al. (2001) are borrowed to comment about the savings in the cost of sampling and the expected waiting time of the two sampling plans. These are given in Table 4. Comparative Study The upper entry in each occupied cell of Table 4 corresponds to the proportion of termination time of the test plan suggested in the present paper. The lower entry corresponds to the similar quantity of the sampling plan of Kantam et al. (2001). These entries revel that the terminating time of this test plan is uniformly smaller than the corresponding time of Table 3. Reliability test plan for log-logistic distribution (a ¼ 0.01) r \ n 2r 3r 4r 5r 6r 7r 8r 9r 10r

5 An Economic Reliability Test Plan 295 Table 4. Comparison of proportion of termination time r\n 2r 3r 4r 5r 6r 10r (a ¼ 0.05) r\n 2r 3r 4r 5r 8r 9r (a ¼ 0.01) the plan in Kantam et al. (2001). Therefore, the present test plans would result in many savings in the experimental time in order to decide to accept or reject a submitted lot on the basis of the sample lifetimes taken from the lot. References Balakrishnan, N. & Malik, H.J. (1987) Best linear unbiased estimation of location and scale parameter of log-logistic distribution, Communication in Statistics Theory and Methods, 16, pp Goode, H. P. & Kao, J. H. K. (1961) Sampling plans based on the Weibull distribution, Proceedings of Seventh National Symposium on Reliability and Quality Control, Philadelphia, Pennsylvania, pp Gupta, S. S. & Groll, P. A. (1961) Gamma distribution in acceptance sampling based on life tests, Journal of the American Statistical Association, 56, pp Kantam, R.R. L. & Rosaiah, K. (1998) Half logistic distribution in acceptance sampling based on life tests, Indian Association for Productivity Quality & Reliability Transactions, 23(2), pp Kantam, R.R. L., Rosaiah, K. & Srinivasa Rao, G. (2001) Acceptance sampling based on life tests: log-logistic model, Journal of Applied Statistics, 28(1), pp

6 296 R. R. L. Kantam et al. O Quigley, J. & Struthers, L. (1982) Survival models based upon the logistic and log-logistic distribution, Computer Programs in Biomedicine, 15, pp Ragab, A. & Green, J. (1984) On order statistics from the log-logistic distribution and their properties, Communications in Statistics Theory and Methods., 13, pp Shah, B.K. & Dave, P. H. (1963) A note on log-logistic distribution, Journal of Mathematical Sciences of University of Baroda (Science number) 12, pp Sobel, M. & Tischendrof, J. A. (1959) Acceptance sampling with new life test objectives, Proceedings of Fifth National Symposium on Reliability and Quality Control, Philadelphia, Pennsylvania, pp Tadikamalla, P. R. & Johnson, N. L. (1982) Systems of frequency curves generated by the transformation of logistic variables, Biometrika, 69, pp