Volume 109 No 9 2016, 125 133 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://wwwijpameu ijpameu Design of Optimal Bayesian Reliability Test Plans for a Series System P N Bajeel 1 and M Kumar 2 Department of Mathematics, National Institute of Technology Calicut, India 1 bajeelpn@gmailcom, 2 mahesh@nitcacin Corresponding author July 18, 2016 Abstract Consider a series system with n independent components Assume that lifetime of i th component follows exponential distribution with unknown parameter, 1 i n We assume each, 1 i n, is distinct and the priori information can be modeled by quasi-density function given by g 1 λ k, k 0, u i, where u i is a predefined upper bound A Bayesian estimator for, 1 i n, i based on data obtained through type-i censoring is used to get an estimate of system reliability Optimal reliability test plan is designed, and an optimization problem is formulated satisfying usual probability requirements Several numerical examples are considered to illustrate the Bayesian approach of obtaining optimal reliability test plan for a series system AMS Subject Classification: 62N05 Key Words and Phrases: Exponential distribution, Reliability, Life testing, Type-I censoring, Delta method, Type-I error, Type-II error, Error loss function ijpameu 125 2017
1 Introduction When a manufacturer produces a new product, he should test the reliability of the product for a particular duration of time to make sure that the product will perform to the best as expected by the consumer Fixing of the testing time based on prior information is one of the difficult tasks The incorporation of priori information about the failure rate and its upper bounds, take a significant role for fixing the duration of testing time in the design of reliability test plans Some of the references related to optimal reliability test plans for exponentially distributed lifetimes of components, with constant unknown failure rates are [1, 2, 3, 8, 9, 7, 6] The optimal test times or an optimal number of components to be tested, reported under various situations in these papers are supposed to be used under normal working conditions regardless of the environment in which component testing is to be carried out In [4], the reliability test plan for a series system is constructed by assuming constant failure rate that depends upon the mission performed For a parallel system consisting of n components, [5] designed reliability test plans with the assumption that the components have lifetimes that are exponentially distributed The parameters, 1 i n, of the exponential distributions depend upon k covariates through exponential relationships But, there is no literature available in Bayesian reliability test plans In this paper, we propose a Bayesian reliability test plan for a series system consisting of n components with the assumption that components have lifetimes that are exponentially distributed with non-informative quasi-prior; here, some preliminaries are given and then the problem is formulated as a reliability optimization problem The system is accepted if the reliability estimate ˆR d, where d (0, 1) and the reliability estimate is obtained using the Bayesian estimator of failure rate A system is considered at time t 1 to be satisfactory if R, the probability that the system survives for one unit time is greater than or equal to R 1 and it is considered to be unsatisfactory if R R 0, where R 0 and R 1 are constant such that 0 < R 0 < R 1 < 1 Then the solution procedure of the formulated optimization problem is explained and type-i censoring scheme is employed to obtain the data Finally, the developed optimal test plan is illustrated through examples ijpameu 126 2017
2 Some Preliminaries and Description of the Problem In this paper, we are considering the problem of testing the reliability of a series system with n independent components under type-i censoring, where the i th component has exponential lifetime with unknown parameter, u i i 1, 2,, n, and u i is the predefined upper bound of the failure rate Then the series system reliability for unit time period is given by R n e We consider the quasi-prior g 1, k 0, a simple prior as the λ k i non-informative quasi-prior for We test the i th component in (0, ] As soon as the component fails, it will be replaced by an identical component, so that the testing continue till the fixed time, 1 i n Posterior Distribution Based on Type-I Censoring: Since T ij, the lifetime of j th component of type i, follows Exponential distribution with failure rate, the pdf of T ij is given by f(t ij ) e t ij Then the likelihood function based on type-i censoring is given by L λi e t δj ij e t 1 δj ij λ X i i e, j N where δ i 1 if t ij, δ i 0 if t ij >, X i δ j and j N t ij j N Let f(t ij, ), g( t ij ) and f(t ij ) be the joint density of t ij and, the conditional density of, given t ij, and the conditional density of t ij, given, respectively Under the assumption that the marginal densities m(t ij ) and g of t ij and, respectively, satisfy the conditions required for the existence of conditional densities, we have g( t ij ) f(t ij, ) and f(t ij ) f(t ij, ) m(t ij ) g Thus the posterior distribution of is given by g( t ij ) g()f(t ij ) m(t ij ) λ X i i e 1 λ k i 0 g()f(t ij )d gf(t ij ) 0 g()f(t ij )d L X i k+1 i Γ (X i k + 1) λx i k i e ijpameu 127 2017
Reliability ( Estimate of the System at Unit Period: ( E ˆλ ) ) 2 i ( ˆλ ) 2 i f(λi )d Differentiate on ˆ and equate to zero implies we have, ˆ E Now E from posterior distribution is nothing but the mean of gamma random variable That is, ˆ E X i k + 1 The reliability estimate of the system reliability is obtained using the Bayesian estimator of failure rates ˆR n e ˆ Xi k+1 n e Acceptance Rule Based on Reliability Estimate: Accept the system if the estimate of the system reliability based on Bayesian Xi k+1 estimator of given by ˆR n e is greater than or equal to some number d, where d (0, 1) Then ˆR d Xi k+1 n e d n Xi k + 1 ln d A system is said to be satisfactory for unit time if R, the survival probability, is greater than or equal to R 1, the acceptable reliability level (ARL) and, it is said to be unsatisfactory if R is less than or equal to R 0, the unacceptable reliability level (URL), where R 0 and R 1 are constants such that 0 < R 0 < R 1 < 1 Then we have the following relations: R R 1 n e R 1 n λ 1 ln R 1, R R 0 n e R 0 n λ 1 ln R 0 Mean and Variance of the Test Statistic: Since the lifetime is exponentially distributed, the number of failures follows Poisson distribution with mean and variance Then, Xi k + 1 E Xi k + 1 V ar Li k + 1 V ar E Li k + 1, ( ( X i k + 1 ) X i k + 1 ) ijpameu 128 2017
3 Optimal Design of the Problem Let c i denote the cost of testing the i th component per unit time Then the aim is to find the time periods, 1 i n that minimize the total testing cost subjected to type-i and type-ii error constraints That is, the problem is to determine the optimum values of by formulating the following optimization problem: Minimize C n c i such that P (Accept the system System is good) 1 α, (1) P (Accept the system System is bad) β, (2) where 0 < β, 1 α < 1 Here the first constraint is usually referred to as producer s risk while the second is the consumer s risk Using the acceptance rule defined in the previous section, the constraints (1) and (2) can be written as min P ( max P X i k + 1 ln d ( The exact distribution of ) ln R 1 1 α, (3) X i k + 1 ln d ) ln R 0 β (4) X i k + 1 is not easy to obtain, and in order to obtain the tractable problem, we therefore need to X i k + 1 approximate the distribution of Recall that n X i k + 1 has mean n k + 1 and variance Then constraints (3) and (4) become min ln d n k + 1 ln R 1 Z 1 α, (5) ijpameu 129 2017
ln d n k + 1 max ln R 0 Z β (6) Z 1 α and Z β are strictly positive and negative respectively for all values of α, β < 05 Now consider the optimization problem in left hand side of the constraint (5) Clearly, this optimization problem will attain the optimum when n ln R 1, then this optimization problem can be rewritten as 1 ln d + ln R 1 + (k 1) min such that ln R 1 Since the numerator is a positive and independent of, to minimize the objective function, it is enough to maximize the denominator Now, using the priori information on upper bound of failure rate, this optimization problem can be rewritten as max such that ln R 1, u i i 1, 2,, n This is an optimization problem in Define ϑ1 i u i if i j and ϑ1 i ln R 1 i j u i if i j for j 1, 2,, n Then it is clear that by assuming feasibility, the optimum solution to above maximization problem will be at any one of these ϑ1 i s, let it be ϑ1 i Then the constraint (5) can be written as Z 1 α n ϑ1 i (k 1) 1 ln d + ln R 1 (7) Similarly by defining ϑ2 i u i if i j and ϑ2 i ln R 0 i j u i if i j for j 1, 2,, n, and by assuming feasibility, the optimum solution of the maximization problem corresponding to constraint ijpameu 130 2017
(6) will be at any one of these ϑ2 i s, let it be ϑ2 i, and then the constraint (6) can be rewritten as Z β n ϑ2 i (k 1) 1 ln d + ln R 0 (8) Now for d (0, 1), the optimal design is a convex programming problem to minimize C n c i subjected to constraints (7) and (8) It can be solved easily 4 Numerical Results The method presented in the previous section is illustrated below with the help of numerical computation Let the number of components in a series system be 3 and the cost vector c (1, 15, 2) and the upper bound vector of failure rate u (007, 005, 007) Then the following Table gives the optimal test times and total testing cost for different inputs Table 1 Numerical examples for a three component series system α β R 0 R 1 k L 1 L 2 L 3 C ln d 005 005 08 099 0 185457 10543 108033 559668 00100503 005 005 08 099 2 46789 326688 301852 156163 00100503 005 005 08 099 05 107909 135219 224117 758971 0121366 008 005 085 098 05 256508 287306 485869 165921 00952889 005 008 085 098 2 622929 473764 398413 23104 00202027 005 005 085 099 0 253276 117293 179093 787403 00100503 The results are generated by running Visual C++ and LINGO11 in tandem The programming is done in Visual C++, within which LINGO11 is called whenever an optimization required 5 Conclusions In this paper, the designing of an optimal reliability test plan for a series system with a failure rate as random variable having quasidensity is discussed in detail The data are obtained through type-i censoring scheme, and the reliability estimator is obtained by estimating a Bayesian estimator of failure rate Some numerical examples are also computed to illustrate the Bayesian approach of ijpameu 131 2017
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