Bilevel-Programming Based Time-Censored Ramp-Stress ALTSP with Warranty Cost

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1 International Journal of Performability Engineering Vol., No.3, May, 24, pp RAMS Consultants Printed in India Bilevel-Programming Based Time-Censored Ramp-Stress ALTSP with Warranty Cost PREETI WANTI SRIVASTAVA * and DEEPMALA SHARMA Department of Operational Research, University of Delhi, Delhi 7, INDIA (Received on October 9, 23, revised on December, 23 and March 25, 24) Abstract : Life test sampling plans are used to determine the acceptability of a product with respect to its lifetimes. Introducing acceleration in life testing experiments helps in obtaining estimates of reliability measures of high reliability products quickly. An accelerated life test (ALT), with linearly increasing stress is a ramp-stress test. This paper deals with a design of optimum time-censored ramp-stress accelerated life test sampling plan (ALTSP) where the product life is assumed to follow log-logistic distribution. The operating characteristic function of the uniformly most powerful (UMP) life test is deduced. The decision criterion is based on UMP test and the estimated median life time. The optimal plan consists in finding optimum sample size, sample proportions allocated to each stress, and stress rate factor by minimizing the expected total cost per lot. Bilevelprogramming approach is used for this purpose. The method developed has been illustrated using a numerical example.. Introduction Keywords: reliability, quality control, ramp-stress, warranty cost, bilevel-programming. Quality-control methods are commonly used to determine the acceptability of a product with regards to its usefulness at the time it is put into service. It is essential from the consumer s point of view that a product performs the required function without failure for the desired period of time. The lifetime of the product is therefore the most important quality characteristic In life testing, a fixed number of items are often tested simultaneously and the test continues for a fixed period of time (time-censoring or Type- I censoring) or until some fixed number of items on test fail (failure-censoring or Type-II censoring). Many modern high reliability products are designed to operate without failure for a very long time. Introducing acceleration in life testing experiments helps in obtaining reasonable failure information about the product faster. Acceleration of life test is applied in materials, products and degradation mechanisms such as insulation life, conductive particle-filled adhesives that have been widely used for flex-to-rigid board interconnections in calculators, multi-layer ceramic capacitors and power capacitors. Life tests under accelerated environmental conditions may be fully accelerated or partially accelerated. In fully accelerated life testing all the test units are run at accelerated conditions, while in partially accelerated life testing they are run at both normal and accelerated conditions. The t erm fully accelerated life test has been coined by [], and the term partially accelerated life test is due to [5]. A fully accelerated life test is commonly referred to as accelerated life test in the literature, and therefore the two terms can be used interchangeably.[5] has stressed the need for introducing ALT to the future plans of Military Standard: Reliability Testing For Engineering Development, Qualification, And Production, MIL-STD-78. This standard specifies the general requirements and specific tasks for reliability testing during the development, qualification, and production of systems and equipment. [9] has discussed the statistical models, test plans and methods of data analyses for the ALTs. *Corresponding author s preetisrivastava.saxena@gmail.com 28

2 282 Preeti Wanti Srivastava and Deepmala Sharma Stress under accelerated condition can be applied using constant-stress, step-stress, progressive-stress, cyclic-stress, random-stress, or combinations of such loadings. The choice of a stress loading depends on how the product or unit is used in service and other practical and theoretical limitations ([9]). Progressive stress ALTs have been used on various products. [4] and [3] used the ramp test on capacitors whereas [] used the ramp test on insulations. The ramped stress breakdown test have been applied by [4] to estimate lifetime of tunneling magnitoresistance (TMR) heads for the intrinsic and extrinsic failure mode. Optimum design of life test sampling plans under fully accelerated environmental conditions using constant-stress and step-stress loadings have been studied extensively by several authors such as [3], [], [2]. In designing accelerated life test sampling plans for product sold under warranty, it is advisable to consider the costs associated with the warranty policies. Warranties are contracts that product providers offer customers with guarantees of repairing products for free within the warranty terms. This paper deals with formulation of optimum time censored ramp-stress ALTSP for log-logistic distribution which has not been studied in the literature so far. Bilevelprogramming approach has been used for obtaining optimum plan. The method developed has been explained using an example. In Section 2, the model for the proposed plan has been formulated followed by Section 3 in which asymptotic variance of median loglifetime is derived. The operating characteristic function of the uniformly most powerful (UMP) life test is deduced in Section 4. The decision criterion is based on the UMP test and the MLE of median log-lifetime. Section 5 deals with the cost structure incorporating rejection, acceptance, testing, and warranty costs that have to be minimized. The design of optimal sampling plan using Bilevel-programming has been explained in Section 6. The method developed has been explained using an example in Section 7. Finally some concluding remarks are made in Section 8. Notation ^, k i Implies a MLE, Stress rate at level i, k i > * φφφ,, Proportion of sample allocated to stress rate k and k 2, respectively; φ= φ, Optimum proportion of sample allocated to stress rate k γ, γ Parameters of the inverse power law, γ >, < γ < n φ,nφ Items allocated to low and high stress rates, respectively α, λ Shape and scale parameters of log-logistic life distribution s(x), Stress at failure time x, s Stress level under normal operating conditions or design stress μ, σ Location, scale parameter of the logistic distribution C s, Cost of sampling and sending n items to the testing ground, Testing cost per unit time Φ( ), H( ) Cdf of standard normal distribution, Cdf of logistic distribution C re, Rejection cost per item when the whole lot has been rejected,warranty cost incurred within c f N, n, τ Product lot size, Total number of test items in a ALT, Censoring time ξ i Stress-rate factor; ξ < ξ 2 = ; ξ i = k i/k 2, i =, 2. u (ln(y) μ(ξ i))/σ ; Standardized log failure time at stress rate factor ξ i, i =, 2 c τ α, β Producer s risk and consumer s risk; < α <, < β < Lot acceptability constant i (ln(τ) μ(ξ ))/σ ; Standardized log-censoring time at stress rate factor ξ i, i =, 2

3 Bilevel-Programming Based Time-Censored Ramp-Stress ALTSP with Warranty Cos The Model This section deals with model formulation for design of the optimal ALTSP under rampstress loading. Basic Assumptions. The stress rates k and k 2 (k < k 2 ) are used in a ramp-test. 2. At any constant stress, s, the product life, X, has a log-logistic distribution with cdf α F(x;α,λ) = ( + (x/λ(s)) ), x, α >, λ >, () where λ(s) is a function of stress, s. 3. For the effect of changing stress, the linear cumulative exposure model holds (see [8]). 4. The inverse power law holds for λ(s) where λ is linear function of a γ (possibly transformed) stress, i.e., γ λ(s) = e (s / s), where parameters γ > and < γ < are the characteristics of the product. 5. The test units are statistically independent and identically distributed. The stress applied to test units is continuously increased with constant rate k from zero i.e., the stress at failure time x is s(x) = kx. Life Test Procedure. Out of total n ( n = nφ+ nφ) items, nφ items randomly chosen are allocated to low stress rate k and the remaining nφ ( φ= φ) items are allocated to high stress rate k The test is continued until all test units fail, or a prescribed censoring time τ is reached. Log-Logistic Distribution The log-logistic life distribution has been found appropriate for high reliability components [2]. The cdf of log-logistic distribution is given in (), and its pdf is given by: α α 2 f(x;α,λ) = (α/λ)(x/λ) ( + (x/λ) ), x, α >, λ >, (2) where α and λ are shape and scale parameters, respectively. Life Distribution under Ramp-stress Test From the linear cumulative exposure model and the inverse power law, the cdf of the lifetime Y of a unit tested under stress rate k is: G(x) = F(ε(x)), (3) where F( ) is the assumed cdf (see ()) with the scale parameter λ set equal to one, x ε(x) = λ(s(t))dt (4) is the cumulative exposure (damage) model. Hence, the cdf, and pdf, respectively, of loglogistic distribution reduces to x x G(x;, ) = ( [dt / (s(t))]) α α λ / + (( [dt / λ(s(t))]) ), αη

4 284 Preeti Wanti Srivastava and Deepmala Sharma α α = (x / η ) / ( + (x / η ) ), (using (), (3) and (4)) α 2 g(x;, ) ( / ) ((x/ ) α αη = α η η /( + (x/ η) ) ), where α =α ( +γ ), η = / ( + γ (exp( γ ) ( + γ ))(s / k)), G( ) is the log-logistic distribution with scale parameter η and shape parameter α. The median lifetime of G( ) is η. Instead of using the actual life time X, Y = ln(x) is used. Thus, Y follows logistic distribution with location parameter μ = ln(η) and scale parameter σ = (/α ), the lower specification limit on Y being L' = ln(l). The pdf and cdf for Y, respectively are (y µ ) σ (y µ ) σ 2 h(y; µσ, ) = e / σ ( + e ), (y µ ) σ (y µ ) σ H(y; µσ, ) = e / ( + e ). Log-Likelihood The location parameter of lifetime distribution of a unit tested under stress rate factor ξ i is μ(ξ ) = (/(+ γ ))(γ + γ ln(s /ξ k ) + ln(+ γ )); i =, 2. i i 2 Define the indicator function I I(u) in terms of the censoring time τ at stress ξ i by, if u τ', failure observed by time τ' I =., if u >τ', censored at time τ' The log-likelihood function L from an observation y is u τ' L( γ, γ, α ) = L = I(u) log u 2 log(+e ) σ I(u) log(+e ). Then, the log-likelihood, L, for n independent observations is, L = L L n. 3. Asymptotic Variance of the Median Log-Lifetime The asymptotic variance of median log-lifetime is obtained using the inverse of Fisher information matrix, F. Thus, for the plan with a sample of n independent items at two stress levels wherein nφ units are tested under stress rate factor ξ = k /k 2, and the nφ remaining units under stress rate factor ξ 2 =, F is given as: F= nφf ( γ, γ, α ) + nφf ( γ, γ, α ), where ξ ξ E[ L / ] E[ L / γ γ] E[ L / γ α] F ξ ( γ i, γ, α ) = E[ L / γ γ] E[ L / γ ] E[ L / γ α], i =, 2 ((using (5)), E[ L / γ α] E[ L / γ α] E[ L / α ] and the values of these elements are given in Appendix. Based on the asymptotic distribution theory of MLE, ˆµ which is the function of ( ˆ, ˆ, ˆ) γ γ α, say γ w( ˆ, γˆ, αˆ) is asymptotically normally distributed with mean μ, and asymptotic variance Asvar( µ ˆ ) =σ / n, i.e., n ( µ µ ˆ ) N(, σ ), where μ = log(η), and Asvar( µ ˆ ) Asvar(w( γˆ ˆ ˆ, γ, α )) = WF, where W ~ ~ ~ w w w W = ~ ˆ ˆ ˆ. γ γ α (5)

5 Bilevel-Programming Based Time-Censored Ramp-Stress ALTSP with Warranty Cos Uniformly Most Powerful (UMP) Operating Characteristic (OC) Function A statistical hypothesis testing is required to check whether the life time based on median life time of the product in study has achieved a required level or not. The producer may be interested in testing whether the null hypothesis H : µ µ (or η η ), is admissible, where μ = log(η), and μ is the minimum acceptable log of median lifetime, whereas the consumer might also specify some particular value μ 2 less than μ, which can be regarded as a clearly rejectable log of median lifetime, and consider the alternative hypothesis H : µ µ 2 (or η η ). Since normal distribution belongs to exponential family of distributions, and μ = log(η) is a monotone increasing function of η, therefore the test based on ˆµ is a uniformly most powerful size - α test of H : µ µ versus H : µ µ 2 (see [7] (Corollary 2, pg.8), and [8]). The uniformly most powerful (UMP) size - α test is: Reject H if ˆµ < c, where c is given as a solution to P [ µ ˆ < c] = α, and ˆµ is asymptotically normally distributed (see Section 3). Now H α = P ˆ ˆ H [ µ < c] = Φ( n (c µ ) / Asvar( µ )) ; so n (c µ ) / Asvar( µ ˆ) = z α is the critical value of standard normal distribution such that the size of critical region is α. The behavior of UMP life test can be described by its operating characteristic (OC) curve which plots the probabilities of lot acceptance versus log median lifetime, µ. The OC function G n (μ) G ( µ ;n, φξ, ) is given by G n ( µ ) = P[ µ ˆ c µ ] = P[Z n (c µ ) / σ µ ], where Z ~ N(, ) as n, = Φ n (c ) / µ σ, µ >. The lot acceptability constant c is calculated by taking average of the lot acceptability constant w.r.t. producer s risk c and consumer s risk c 2, where c = ( σ / n)z + µ and c = ( σ / n)z + µ. Thus, c = (c + c ) /2. 5. Cost Structure α 2 β 2 2 In this section cost structure that takes into consideration rejection, acceptance, testing, and warranty costs for the adoption of an optimal test planning has been obtained. The commonly used warranty policies are free replacement warranty (FRW) which charges consumers no fee during the term of warranty, and pro-rata warranty (PRW) which charges a pre-set proportion of the cost for each repair during the term of warranty. Occasionally, FRW, and PRW can be combined as one policy, called a general rebate warranty (GRW), to provide consumers with more choices (see [6]). The company takes responsibility for the product warranty once the batch has been accepted by the company; and the periods of FRW and PRW are predetermined as c f, and c p, respectively. However, the pro-rata warranty of the producer between c f, and c p is assumed to vary in time instead of being constant. As a result, the warranty costs of the company could be expressed by a piecewise function of life time X which is given as: C w, if x < cf Cost w(x) = C w(cp x) / (cp c f), if c f x < c P., if x cp

6 286 Preeti Wanti Srivastava and Deepmala Sharma Since X follows log-logistic distribution with parameters (α, γ ), therefore the expected warranty cost per product is given by cf cp C( α, γ ) = C f (x)dx + C ((c x) / (c c ))f(x)dx. w w p p f cf Thus, the expected warranty cost resulting from selling the acceptance lot is given by W(α, γ ) = (N n) C( α, γ ˆ )P[ µ c µ ], and the expected rejection cost resulting from the rejecting the lot is given by R(α, γ ) = (N n) C ˆ rep[ µ< c µ ]. Therefore, the expected total cost per lot for the given sampling plan is given by TCost(n α,γ ) = nc s + τc τ + W(α, γ ) + R(α, γ ), (6) where the sampling cost nc s and the testing cost τc τ are obtained by multiplying the unit cost of sampling with the sample size, and multiplying unit cost of testing with the censoring time. 6. Design of Optimal Sampling Plan using Bilevel-Programming Bilevel-programming problem is a hierarchical mathematical optimization problem that contains an optimization problem in the constraints. Various approaches have been developed in different branches of numerical optimization to solve bilevel-programs. In this paper, in the first stage, optimum stress rates and sample proportion allocated to each stress level are obtained by minimizing n/ F ; followed by the second stage in which optimal sample size has been obtained by minimizing the expected total cost per lot of the sampling plan using non-linear mixed integer programming. To obtain an optimal total sample size, it is necessary to make sufficiently low size of risk that the producer and consumer are willing to accept, say at most α and β, respectively ( α, β <.5 where < α <, < β < ) evaluated at the corresponding acceptable and rejectable μ and μ 2. Thus, the minimum sample size is determined by optimally designing ramp-stress ALT so that the constraints G n( µ ) α and G n( µ 2) β are satisfied. Mathematica 8. has been used to formulate the optimal plan. Formulation of an Optimization Problem The optimal design problem can be formulated as: Min n F φξ, s.t. φ, φ+ φ=, <ξ <, where ( φξ, ) solves Min TCost(n αγ, ) n s.t. G n( µ 2) β, + N n,n Ζ,

7 Bilevel-Programming Based Time-Censored Ramp-Stress ALTSP with Warranty Cos 287 Let + Ψ = {( φ, ξ ) : φ, φ+ φ =, <ξ < }, Ψ = {(n) : G ( µ ) β, N n, n Ζ }, and 2 n,c 2 ( φ, ξ ) = Argmin{n / F : ( φ, ξ ) Ψ }. Then the optimization problem reduces to: * * φξ, n * * { 2 } Min TCost(n α, γ ) : n Ω, ( φ, ξ ) = ( φ, ξ ) (using (6)). Since, the optimum ramp-test depends on τ, s, α, γ, γ and k 2; one must obtain their values from experience, similar data, or a preliminary test. 7. An Illustrative Example ABC Corporation is an IC manufacturing company. Since IC products are relatively expensive, it is worthwhile to conduct a more cost saving life testing program in which an appropriate sample size can be find out to determine the acceptance or rejection of a batch of IC products with the objective of cost minimization for the company. However, given that various relevant costs would influence the profits of ABC Corporation in the process of the life test, it would be essential for the managers to select the appropriate decision parameters regarding the accelerated life test sampling plan. (See [6], []) Consider a hypothetical data set for a time-censored ramp-stress ALTSP for an electrical insulation of high reliability as α= 3.5, γ = 5, γ =.2, s = 25, k2 = 2, τ = 24. The optimal proportion of the sample allocated to low stress, and optimal low stress rate * * * are ξ =.53, φ =.364 φ =.635. Let the product lot size N be 2 and the manager assesses various cost (in thousands) with regard to the lot of IC products are as follows: C s = $25, C re = $2, C w = $25, C τ = $5, c f = $ and c p = $45. An agreement between the producer and the consumer is considered to obtain an optimal failure censored acceptance sampling plan such that the respective probabilities of rejecting a good item and accepting a bad item are at most α =. and β =., respectively. In addition, a lot of items are acceptable if the log of median lifetime of items is longer than or equal to μ = days (say, i.e., 24 hours). When the log of median lifetime of items reduces to μ 2 = 5 days (say, i.e., 2 hours) or less, the lot considered is rejected. In this case, the ratio of mean is μ 2 /μ is considered as.5. In * Table, the value of optimal sample size n is 25 which is precisely the minimal number of failures and the optimum lot acceptability constant c * is The optimal cost is $ 2,49. Thus, the manager must accept a lot if µ ˆ The optimal number of samples at low stress rate * nφ = 9 and at high stress rate * nφ = 4. Figure depicts the plot of G n (μ ) against different values of μ for given pairs ( α, β ) = (.,.5), (.,.), (.5,.5) and (.5,.).

8 288 Preeti Wanti Srivastava and Deepmala Sharma Figure : UMP OC Curve Table presents optimum total cost and optimum sample size and lot acceptability constant for selected values of α, β and μ 2 /μ. In view of Table, it is observed that as μ 2 /μ increases, the optimal acceptability constant and sample size increase. This is quite evident because, if the probabilities of classifying good lots as bad ones and bad lot as good ones reduce, more empirical information is needed to judge the lots. Table : Optimal sample size and the corresponding acceptability constant for selected values 8. Conclusion α β of α, β and μ 2 /μ Optimal Values Acceptability Constant μ 2 /μ Sample Size Total Cost c c 2 c , , , , , , , , , , , , , , , , , , , , In this paper, we have obtained an optimum time-censored ramp-stress accelerated life test sampling plan using log-logistic distribution assuming inverse power law and a cumulative exposure model. The life test sampling plan for variables based on hypothesis testing is used, and the operating characteristic curve of UMP test is obtained. The

9 Bilevel-Programming Based Time-Censored Ramp-Stress ALTSP with Warranty Cos 289 optimum plan yields optimum total sample size and sample proportions allocated to each stress by minimizing expected total cost of the sampling plan that helps in formulating an effective decision-making process.. The procedure developed has been explained using a numerical example. Acknowledgement :This research is supported by R & D grant received from University of Delhi. The authors are grateful to the Editor-in-chief and referees for their valuable comments. References [] Bhattacharyya, G. K., and Z. Soejoeti. A Tampered Failure Rate Model for Step-Stress Accelerated Life Test. Commun. Statist. Theor. Meth., 989; 8: [2] Chiodo, E., and G. Mazzanti. The Log-Logistic Model for Reliability Characterization of Power System Components subjected to Random Stress. In Proceedings of the International symposium SPEEDAM 4, Capri, Italy, 24; (also IEEE indexed paper). [3] Chung, S. W., Y. S. Seo, and W. Y. Yun. Acceptance Sampling Plans based on failure censored step-stress accelerated tests for Weibull distributions. Journal of Quality in Maintenance Engineering, 22; 2(4): [4] David, P. K., and G. C. Montanari. Compensation Effect in Thermal Aging investigated according to Eyring and Arrhenius Models. Euro. Trans. Elect. Power Engg., 992; 2(3): [5] DeGroot, M. H., and P. K. Goel. Bayesian Estimation and Optimal Designs in Partially Accelerated Life Testing. Naval Res. Logist. Quart., 979; 26: [6] Huang, Y. S., C. H. Hseish, and J. W. Ho. Decisions on an Optimal Life Test Sampling Plan with Warranty Considerations. IEEE Transactions on Reliability, 28; 57(4): [7] Lehmann, E. L. Testing Statistical Hypotheses. (second ed.), John Wiley and Sons Inc., 986. [8] Mood, A. M., F. A. Graybill, D. C. Boes. Introduction to the Theory of Statistics. third ed., McGraw-Hill Book Co., 974. [9] Nelson, W. Accelerated Testing-Statistical Models, Test Plans, and Data Analyses. John Wiley, Sons, New York, 99. [] Seo, J. H., M. Jung, and C. M. Kim. Design of Accelerated Life Test Sampling Plans with a Nonconstant Shape Parameter. European Journal of Operational Research, 29; 97: [] Solomon, P., N. Klein, and M. Albert. A Statistical Model for Step and Ramp Voltage Breakdown Tests in Thin Insulators. Thin Solids Films, 965; 35: [2] Srivastava, P. W., and R. Shukla. Optimum Simple Ramp-test for the Log-Logistic Distribution with Censoring. Journal of Risk and Reliability, 29; 223: [3] Starr, W. T., and H. S. Endicott. Progressive Stress A New Accelerated Approach to Voltage Endurance. Trans. AIEE, Power Apparatus and Systems, 96; [4] Wang, G. F., F. Chen, Z. Y. Teng, L. G. Yin, K. J. Zhang, K. J. Jiang, C. F. Jiang, G. Yan, W. X. H. Li, and S. S. K. Chou. Accelerated Lifetime Test for TMR Heads by Ramped Stress. IEEE Transactions on Magnetics, 28; 44(): -3. [5] Wallace, Jr., W.E. Present Practice and Future Plans for MIL-STD-78. Naval Res. Logistics Quart., 985; 32: Appendix The expectations of negative of second order derivatives given L are derived in the following equations. These expectations are calculated with the aid of E[ ln L i / γ i ] =, for i =,, E[ ln L i / α ] = τ 3 E[ L / γ ] = ( α / 3) ( / (+e ) ), τ 2 2u u 4 2τ τ E[ L / γ γ ] = ( α / ( + γ)) ( 2ue /(+e ) du + τ e (( + e ) ) + (T / ( + γ)) (E[ L / γ ]),

10 29 Preeti Wanti Srivastava and Deepmala Sharma E[ L / γ α ] = (T / α) E[ L / γ ] + (( + γ ) / α) E[ L / γ γ ], E[ L / γ ] = / ( + γ ) ( α E[ L / γ α ] + T E[ L / γ γ ]), E[ L / γ α ] = ( α / ( + γ )) E[ L / α ] + (T / α) E[ L / γ γ ] (T / ( α ( + γ ))) E[ L / γ ], τ u u 2 2 2u u 4 2 2τ τ 3 E[ L / α ] = (/ α ) ( (e /(+e ) + 2u e /(+e ) )du + τ e /(+e ) ), E[ L / γ γ ] = E[ L / γ γ ], E[ L / α γ ] = E[ L / γ α], E[ L / α γ ] = E[ L / γ α]. Preeti Wanti Srivastava is Associate Professor in Department of Operational Research, University of Delhi, Delhi-7, India. She received her B.Sc. (Hons.) in Statistics, M.Sc. (Statistics), M.Phil. (Statistics), Ph.D. (Statistics) in 987, 989, 992 and 2; respectively from University of Delhi, India. Her research area is reliability theory. She has published seventeen papers in journals of international repute. Deepmala Sharma is Research Scholar in Department of Operational Research, University of Delhi, Delhi-7, India. She received B.Sc. (Mathematics Honors), M.Sc. (Operational Research) and M.Phil. (Operational Research) in 24, 26 and 2; respectively from University of Delhi, Delhi, India. Currently, she is pursuing her Ph.D. in the Department of Operational Research, University of Delhi, India under the supervision of Dr. Preeti Wanti Srivastava. Her research area is Reliability.