# Optimal design of degradation tests in presence of cost constraint

## Optimal design of degradation tests in presence of cost constraint

Reliability Engineering and System Safety 76 (2002) 109±115 www.elsevier.com/locate/ress Optimal design of degradation tests in presence of cost con...
Reliability Engineering and System Safety 76 (2002) 109±115

www.elsevier.com/locate/ress

Optimal design of degradation tests in presence of cost constraint Shuo-Jye Wu*, Chun-Tao Chang Department of Statistics, Tamkang University, Tamsui, Taipei 251, Taiwan, ROC Received 24 November 2000; accepted 16 August 2001

1. Introduction With today's high technology, manufacturers face increasingly intense global competition. To remain pro®table, they are challenged to design, develop test and produce high reliability products. Thus, some life tests result in few or no failures in a short life testing time. In such cases, it is dif®cult to assess reliability with traditional reliability studies that record only failure time. This circumstance applies to a variety of materials and products such as metals, insulations, semiconductors and electrical devices. To study this type of reliability data, accelerated life tests that record failure and censoring times subjected to elevated stress are used widely. However, this approach may not provide help for highly reliable products, which are not likely to fail during an experiment with severe time constraint. A recent approach is to observe degradation measurements of product performance during the experiment. Product performance is usually measured in terms of physical properties. We call these kinds of physical properties, degradation mechanisms'. From an engineering point of view, the common degradation mechanisms include fatigue, cracks, corrosion and oxidation. Examples are loss of tread on rubber tires and degradation of the active ingredient of a drug because of chemical reactions with oxygen and water, * Corresponding author. Tel.: 1886-2-2621-5656; fax: 1886-2-26209732. E-mail address: [email protected] (S.-J. Wu).

0951-8320/02/$- see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0951-832 0(01)00123-5 110 S.-J. Wu, C.-T. Chang / Reliability Engineering and System Safety 76 (2002) 109±115 considered a random coef®cient degradation model with random sample size and used maximum likelihood for parameter estimation. A data set from a semiconductor application was used to illustrate their methods. Wu and Shao [11] established the asymptotic properties of the (weighted) least squares estimators under the nonlinear mixed-effect model. They used these properties to obtain point estimates and approximate con®dence intervals for percentiles of the failure time distribution. They applied the proposed methods to metal ®lm resistor and metal fatigue crack length data sets. Wu and Tsai [12] used the optimal fuzzy clustering method to modify the two-stage method. Their procedure can get more accurate estimation results when the patterns of a few degradation paths are different from those of most degradation paths in a test. These papers all focus on estimating the parameters in the degradation model and the percentile of failure time distribution. In this study, we use the general degradation path models to model degradation data, and we will focus on the nonlinear mixed-effect model proposed by Lu and Meeker [4]. There are some important questions about how to design a degradation test to provide a more precise estimate of the percentile of failure time distribution. Important questions include how to determine the number of units that should be tested, the number of inspections on each unit, and a termination time for the degradation experiment. Boulanger and Escobar [1] presented a method for determining the selection of stress levels, sample size at each level and the times at which to measure the devices. However, their method is discussed under a predetermined termination time. Yu and Tseng [13] explained why it is not appropriate to ®x the termination time in advance and proposed an intuitively appealing procedure to determine an appropriate termination time for an accelerated degradation test. One practical problem arising from designing a degradation test is the budget of experiment. The size of budget always affects the decisions of number units to test, number of inspections and termination time and hence, affects the precision of estimating the failure time distribution. In this study, we are going to integrate these factors and the restricted cost of experiment to construct a mathematical model. Then we will use the method of nonlinear mixed integer programming to ®nd the optimal solution of the total number of units to test, the times at which to measure the units, and the termination time of a degradation experiment. Thus, we can set up an optimal degradation test and then obtain an estimate of the percentile of failure time distribution with minimum variance. We will apply the proposed method to a numerical example and discuss its sensitivity analysis. The rest of the paper is organized as follows: Section 2 describes a nonlinear mixed-effect model for degradation data and gives the estimation procedure. Section 3 proposes a procedure to determine the number of units to test, the number of inspections on each unit and the termination time for a degradation test. Section 4 applies the proposed procedure to a numerical example, and Section 5 studies the sensitivity analysis of the proposed procedure. Some conclusions are given in Section 6. 2. Degradation model and parameter estimation In a degradation test, product performance is obtained as it degrades over time and different product units may have different performance. Thus, a statistical model for a degradation test consists of (1) a relationship between degradation measurement and time, and (2) a distribution that describes an individual product unit's characteristics. The general approach is to model the degradation of the individual units using the same functional form and the differences between individual units using random effects. The nonlinear mixedeffect model proposed by Lu and Meeker [4] is yij  h tij ; a; bi  1 1ij ; i  1; ¼; n; j  1; ¼; mi # s; 1 where h is the actual level of degradation of the unit under study and is a given function nonlinear in (a,bi); tij is the time of the jth measurement for the ith unit; yij represents the level of degradation actually observed at time tij ; a denotes the vector of ®xed-effect parameters; bi represents the vector of the ith unit random effects; 1ij 's are i.i.d. measurement errors with mean 0 and variance s 12 ; s is the prespeci®ed largest number of measurements for all units. We assume that Eh tij ; a; bi h ti 0 j 0 ; a; bi 0  , 1 for all i; i 0  1; ¼; n; j; j 0  1; ¼; mi or mi 0 : In addition, since whether or not a unit fails should not depend on measurement error, we assume that {1 ij} and {bi} are independent. Suppose that h in Eq. (1) is a continuous and differentiable function for any ®xed a and b. We assume that the degradation is not reversible. Hence, without lost of generality, we assume that h t; a; b is a strictly increasing function of t, i.e.   2 P 0, h t; a; b , 1  1: 2 2t We further de®ne that failure occur when actual degradation h reaches a prespeci®ed critical value h c. Therefore, the failure time of a product, denoted by T  T hc ; is equal to the solution of h t; a; b  hc : Fig. 1 illustrates this fact. Furthermore, we assume that there exist constants h 1 and h 2 satisfying h2 , hc , h1 such that   P lim h t; a; b , h2  1; 3 t!0 and   P lim h t; a; b . h1  1: t!1 4 Eq. (3) means that no failure occurs before the test starts, and Eq. (4) indicates that lifetime of a unit is ®nite. Based on S.-J. Wu, C.-T. Chang / Reliability Engineering and System Safety 76 (2002) 109±115 111 Fig. 1. Use the degradation path h t; a; b and critical value h c to de®ne the time to failure t(h c). Eqs. (2)±(4), h t; a; b  hc has a unique and ®nite solution for any given a and b. There is an important relation between the failure time distribution and the degradation distribution, which is useful for estimating percentiles of the failure time distribution. Lemma 1. Let Fh xut be the degradation distribution of h given time t and FT tux be the failure time distribution of T  T x given degradation value x. Under Eqs. (2)±(4), FT tux  1 2 Fh xut: Proof. Note that h t; a; b is an increasing function of t. Then, the inverse transformation, say h 21, exists. For such a transformation, we have Fh xut  Ph t; a; b # x  Pt # h21 x; a; b  1 2 Ph21 x; a; b # t  1 2 FT tux: A Remark 1. There are some products with decreasing degradation functions. For example, light emitting diodes (LEDs) have become widely used in a variety of industrial products. A key quality characteristic of LED is its light intensity. Fig. 1 in Yu and Tseng [13] shows a typical degradation path of an LED product. From the plot, it is seen that the standardized light intensity is a decreasing function of time. Therefore, if a product with decreasing degradation function is studied, then the assumptions (2)±(4) become that P21 , 2h t; a; b=2t , 0  1; and that there exist constants h 1 and h 2 satisfying h2 , hc , h1 such that Plimt!0 h t; a; b . h1   1; and Plimt!1 h t; a; b , h2   1; respectively. Thus, the relation between the degradation distribution of h given time t and the failure time distribution of T given degradation value x is FT tux  Fh xut: The percentiles of the failure time distribution are very important characteristics and elemental in reliability analysis. Let tp be the 100pth percentile of the failure time distribution. One can obtain percentiles of the failure time distribution using Lemma 1, i.e. tp can be obtained by solving 1 2 Fh hc ut  p in t. Suppose that the solution is tp  g hc ; u; where g is a known function and u is a vector of unknown parameters. If u is estimated by u^ n based on yijs, then tp is estimated by t^p  g hc ; u^ n : Thus, it is essential to estimate u. Suppose that the distribution of b is p buf; where p is a known function and f is an unknown q £ 1 parameter vector. Let u  a; f 0 and h tij ; u  Eb h tij ; a; b: De®ne yi  yi1 ; ¼; yimi  0 ; hi u  h ti1 ; u; ¼; h timi ; u 0 and ei  ei1 ; ¼; eimi  0 : Then, model (Eq. (l)) can be written as the following heteroscedastic nonlinear model: yi  hi u 1 ei ; i  1; ¼; n: Denote the parameter space by Q and the unknown true parameter by u0. The least squares estimator of u0 based on 112 S.-J. Wu, C.-T. Chang / Reliability Engineering and System Safety 76 (2002) 109±115 data {y i }ni1 is a vector u^ n [ Q minimizing Q n u  1 n n X i1 yi 2 hi u 0 yi 2 hi u: Under some regularity conditions, Wu and Shao [11] showed the following theorem. Theorem 1. (i) u^ n ! u0 almost surely as n ! 1: (ii) Dn21=2 u0  u^ n - u0  !PN 0; Ip1q  in distribution, n where Dn u  A21 Hi u Si uH 0i uA21 n u i1P n u; 0 Hi u  2h i u=2u; An u  ni1 Hi uH 0i u; and Si u is the covariance matrix of e i : The (j,k)th entry of Si u is 8 < s h tij ; u 1 s 12 if j  k ; s ijk u  : s t ; t ; u if j ± k h ij ik where s h tij ; u  Varbi h tij ; a; bi  and s h tij ; tik ; u  Covbi h tij ; a; bi ; h tik ; a; bi : (iii) n D^ n 2 DnP u0  ! 0 almost surely as n ! 1; where n 21 ^ 0 ^ ^ ^ P^ 0 ^ D^ n  A21 n un  i1 Hi un  i H i un An un ; Si  r i r i ^ and ri  yi 2 hi un  is the ith residual vector. We now return to the original statistical inference in degradation analysis, i.e. using the degradation data to make inference for percentiles of the failure time distribution. From Lemma 1, tp is a solution of 1 2 Fh hc ut  p and is a function of h c and u  a; f 0 ; say tp  g hc ; u: Substituting the least squares estimate u^ n for u, we obtain an estimate t^p  g hc ; u^ n : Since tp is a function of u, the asymptotic normality of t^p can be derived by the Taylor expansion and the asymptotic normality of u^ n : Hence, Wu and Shao [11] shows that the asymptotic variance of t^p is  0   2g hc ; u 2g hc ; u Dn u Var t^p   : 5 2u 2u The estimate of the asymptotic variance of t^p can be obtained by substituting u^ n into Eq. (5). 3. Optimal design of degradation tests To obtain a precise estimate of the percentile of the failure time distribution, frequently asked questions include how many units do I need to test?', how long do I need to run the degradation test?' or how many times do I need to inspect the units in an experiment?' Simply put, more test units, more test time, and more number of measurements will generate more information, which improves the precision of estimates. However, the restricted cost of experiment does not allow us to do so. Thus, the problem of obtaining a precise estimate of the percentile of failure time distribution under a restricted cost of experiment is an important issue to the reliability analyst. There are a lot of decision variables that affect the cost of experiment and the precision of the estimation of the percentile of failure time distribution. The most important three decision variables are: (1) the number of test units, (2) the number of inspections on each unit, and (3) the termination time of the degradation experiment. Let n denote the number of units on test. For each unit, let t be the interval between inspections (or inspection frequency) and let k be the number of inspections. Then the termination time is k 2 1t: The cost of experiment consists of the following three parts. 1. Sample cost: This is the cost of test units. Let Cs be the cost of a test unit. Then the total sample cost is nCs. 2. Inspection cost: The inspection cost includes the cost of using inspection equipment and material. It also depends on the number of test units and the number of inspections. Let Cm denote the cost of one inspection on one test unit. Since the total number of inspections is nk; the total inspection cost is nkCm : 3. Operation cost: The cost consists of the salary of operators, utility, and depreciation of test equipment, etc. It is proportional to the degradation-testing time. Let Ce be the operation cost in the time interval between two inspections. Then the total operation cost is t k 2 1Ce : Therefore, the total cost of experiment is: CT  nCs 1 nkCm 1 t k 2 1Ce : The objective is to obtain a precise estimate of the percentile of failure time distribution; that is, to minimize the asymptotic variance of t^p : Note that the asymptotic variance of t^p in Eq. (5) is a function of t, k and n. For simplicity, we can write G t; k; n  Var t^p : Then, the optimal degradation design problem consists of ®nding t, k and n that minimize the asymptotic variance of t^p : However, the determination of t, k and n is restricted to the budget of experiment, say, Cr. Hence, the optimal degradation design problem can be expressed as follows. minimize G t; k; n subject to nCs 1 nkCm 1 t k 2 1Ce # Cr ; k; n [ N; k$ 2; and t . 0;

(6)

where N is the set of positive integers. Since the objective function and constraint are both nonlinear functions of decision variables k, n, and t, it is dif®cult to obtain a closed form of the solution. Therefore, in order to ®nd the optimal solution for the problem of nonlinear mixed integer programming, we set up the following algorithm. 3.1. Algorithm for determining the optimal design of degradation test Step 0: At the beginning, set the values of cost parameters

S.-J. Wu, C.-T. Chang / Reliability Engineering and System Safety 76 (2002) 109±115

Cs, Cm, Ce, and Cr, and give the values of model parameters u and s 12 : Step 1: Calculate the upper bound of the number of test units. Under the constraint of total experimental cost, the upper bound is n~  Cr =Cs 1 2Cm ; where x is the greatest integer that is less than or equal to x. Step 2: Set n  2: Step 3: Compute the upper bound of the number of inspections for a given n. Using the constraint of total experimental cost and a given value of n, compute the upper bound k~n  Cr 2 nCs =nCm : Step 4: Compute the inspection frequency. Using the constraint of total experimental cost, for all k [ N; 2 # k # k~n ; compute the inspection frequency tkn  Cr 2 nCs 2 nkCm =k 2 1Ce  and calculate the corresponding value of objection function Gtkn ; k; n: Step 5: Let function Fn  Gt~kn ; kn ; n  min2#k#k~n Gtkn ; k; n: Step 6: Set n  n 1 1: If n # n~ go to Step 3, else go to Step 7. Step 7: Compute the optimal value of objective function. That is, Fnp   min2#n#n~ Fn  Gtp ; kp ; np   min2#n#n~ Gt~kn ; kn ; n: Step 8: The optimal degradation design (t p,k p,n p) is obtained. Remark 2.

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ratio of resistance at time t to initial resistance that is beyond 1.02. Wu and Shao [11] used the following model to describe the degradation over time tj of a resistor i: yij  bi tja 1 1ij ; i  1; ¼; n; j  1; ¼; m; where yij  (observed resistance at time tj / observed resistance at time t1) 2 1, tj is the measurement time (1000 h),a is a ®xed effect parameter, and 1ij is random error with mean 0 and variance s 12 : The random effects b is are independently distributed as an exponential distribution with mean l . Following Section 2, tp has an explicit form:  1=a 2hc tp  ; l logp where hc  0:02: The estimates of l , a and s 12 obtained by Wu and Shao [11] are l^  0:0021; a^  0:4626 and s^ 1  0:00001; respectively. Thus, the point estimate of tp is  1=0:4626 20:02 t^p  ; 0:021 logp and the asymptotic variance of t^p can be obtained by using Eq. (5). Now, let us discuss the problem of designing an experiment which minimizes the asymptotic variance of t^p : From Wu and Shao [11], we have the estimates of l and a as l^  0:0021 and a^  0:4626: We use their results in the design of our new experiment. Suppose further that the values of cost parameters are as follows: Cs  $80=unit; Cm $0:07=unit; Ce  $10=1000 h; and Cr $4000: Consider the objective function to be the asymptotic variance of the 10th percentile of failure time distribution. That is, Gt; k; n  Vart^0:1 : Thus, the optimal degradation design problem is

1. In Step 0, the values of u and s 12 are usually unknown. Thus, we can use prior information or data from a pilot test to get their estimates by using the estimation method in Section 2. 2. In Step 1, the upper bound of the number of test units n~ is obtained as follows. From Eq. (6) we have nCs 1 nkCm 1 tk 2 1Ce # Cr ; k \$ 2; which leads to nCs 1 2nCm 1 2 2 1tCe # Cr : Since t . 0; it is easily seen nCs 1 2Cm  # Cr : This implies n # Cr =Cs 1 2Cm : Thus, the upper bound of n is n~  Cr =Cs 1 2Cm : We also can use the similar argument to obtain the upper bound of k for a given value of n and the inspection frequency tkn for given values of k and n.

By using the algorithm in Section 3, we can obtain the optimal degradation design as follows.

4. Numerical example

tp  0:614;

The example consists of designing a degradation test for a metal ®lm resistor. No failures are expected to occur during an experiment. However, there is some concern about changes over time in resistance, an important characteristic of the metal ®lm resistor. The resistance is an increasing function of time. If resistance increases too far from its original value, the failure of metal ®lm resistor could occur. Wu and Shao [11] reported and analyzed the results of a previous degradation test. A total of 200 metal ®lm resistors were tested, and the resistance of each resistor was measured at ®ve different times during the experiment. There were no failures, where a failure was de®ned as the

That is, the optimal number of test units is 49, the optimal number of inspections is 9, and the optimal termination time is 0:614 £ 1000 £ 9 2 1  4912 h: It should be noted that we can use the same procedure to minimize the asymptotic variances of other percentiles of interest, say the 50th and 90th. The optimal solutions might be different. However, since the degradation measurements provide much more information about the lower tail of the failure time distribution, we are only interested in the precision of estimating the 10th percentile of failure time distribution. Hence, we do not discuss other percentiles in this study.

minimize Vart^0:1  subject to 80n 1 0:07nk 1 10tk 2 1 # 4000:

kp  9;

np  49:

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S.-J. Wu, C.-T. Chang / Reliability Engineering and System Safety 76 (2002) 109±115

5. Sensitivity analysis We now study sensitivity of the optimal solution to change in the values of the different parameters associated with the degradation design. These parameters can be divided into two parts: (1) the parameters in degradation model, i.e. a and f; and (2) the parameters in the cost of experiment, i.e. Cs, Cm, Ce, and Cr. We will study how the optimal solution is in¯uenced by the estimates of model parameters and by the cost parameters of experiment in Sections 5.1 and 5.2, respectively.

Table 1 The optimal values of t, k and n under various combinations of l and a

l

a

n

k

t

0.002032

0.452361 0.4626 0.472838

49 49 49

9 9 9

0.614 0.614 0.614

0.0021

0.452361 0.4626 0.472838

49 49 49

9 9 8

0.614 0.614 0.750

0.002168

0.452361 0.4626 0.472838

49 49 49

9 9 8

0.614 0.614 0.750

5.1. The effect of estimated model parameters As we mentioned in Section 3, in practice, the values of model parameters, a and f, are usually unknown. We have to use prior information or data from a pilot test to get their estimates. The estimation method was discussed in Section 2. However, no one can guarantee that the estimates are exactly equal to the unknown model parameters. Thus, in this subsection, we will discuss the in¯uence of changing values of estimated model parameters on the optimal solutions. From Section 4, we obtain the optimal degradation design tp  0:614; kp  9; and np  49 when the estimates of model parameters l^ ; a^ ; s^ 1   0:002 1; 0:4626; 0:00001 and the cost parameters of experiment Cs ; Cm ; Ce ; Cr   80; 0:07; 10; 4000: By Theorem 1 in Section 2, we can also compute the con®dence intervals for l and a . The 95% con®dence intervals for l and a are l^ L ; l^ U   0:002032; 0:002168 and a^ L ; a^ U   0:45236 1; 0:472838; respectively. Thus, we choose various combinations of l and a in their 95% con®dence intervals, respectively, for sensitivity analysis. Set Cs ; Cm ; Ce ; Cr   80; 0:07; 10; 4000; the cost parameters used in Section 4. Now, the optimal solutions t p, k p, and n p are given in Table 1. It shows the optimal solution (t p,k p,n p) is slightly sensitive to the changes in these values of model parameters.

cost parameters Cs, Cm, and Cr. We also can ®nd that k is a decreasing function of Cr and Cm. The inspection frequency t is sensitive to all cost parameters. It is an increasing function of Cr and Cm, and is a decreasing function of Ce. Table 2 also indicates that the termination time k 2 1t is sensitive to all parameters, and it is a decreasing function of Cr and Ce.

5.2. The effect of cost parameters of experiment Changes in cost parameters of experiment can affect the determination of the optimal degradation design. Let us consider the values of model parameters l; a; s 1   0:0021; 0:4626; 0:00001 and the cost parameters of experiment Cs ; Cm ; Ce ; Cr   80; 0:07; 10; 4000: The solutions of the optimal degradation design are shown in Section 4. Using the same values of model parameters, the sensitivity of each decision variables n, k, and t to changes in the cost parameters of experiment is examined. Table 2 shows that the number of test units n is insensitive to changes in Ce and Cm, and is highly sensitive to changes in Cs and Cr. In addition, n is an increasing function of Cr and a decreasing function of Cs. The number of inspections k is slightly sensitive to changes in Ce, and it is highly sensitive to changes in the

6. Conclusions Determining appropriate number of test units, number of inspections, and termination time under restricted cost of experiment is an important decision problem for experimenTable 2 The optimal values of t, k and n, the termination time k 2 1t; and the asymptotic variance of the percentile of failure time distribution under various combinations of Cs, Cm, Ce, and Cr Cs

Cm

Ce

Cr

n

k

t

k 2 1t

Vart^p 

80

0.07

10

2000 4000 6000 8000 10000

24 49 74 99 124

17 9 6 5 4

0.321 0.614 0.978 1.134 1.509

5.136 4.912 4.890 4.536 4.527

90.19 44.17 29.25 21.86 17.45

80

0.07

10 20 30 40 50 60

4000

49 49 49 49 49 49

9 9 10 10 10 10

0.614 0.307 0.169 0.126 0.100 0.084

4.912 2.456 1.521 1.134 0.900 0.756

44.17 44.17 44.17 44.18 44.18 44.18

80

0.05 0.06 0.07 0.08 0.09 0.10

10

4000

49 49 49 49 49 49

12 10 9 8 7 6

0.460 0.560 0.614 0.694 0.819 1.012

5.060 5.040 4.912 4.858 4.914 5.060

44.17 44.17 44.17 44.17 44.17 44.17

75 80 85 90 95 100

0.07

10

4000

53 49 46 44 42 39

4 9 10 6 7 13

0.338 0.614 0.642 0.430 0.524 0.537

1.014 4.912 5.778 2.150 3.144 6.444

40.86 44.17 47.05 49.20 51.54 55.50

S.-J. Wu, C.-T. Chang / Reliability Engineering and System Safety 76 (2002) 109±115

ters when conducting a degradation test. We use the method of nonlinear mixed integer programming and propose an algorithm to set up an optimal degradation design. Under this optimal degradation design, we obtain an estimate of the percentile of failure time distribution with minimum variance. Moreover, we ®nd the following characteristics for the sensitivity analysis: 1. The proposed optimal degradation design is quite robust to the estimates of model parameters l and a . 2. The budget of experiment has a high in¯uence on all decision variables. Its increment leads to the increment of the number of test units and the inspection frequency, but the decrement of the termination time and the asymptotic variance of the percentile of failure time distribution. 3. Among all decision variables n, k and t the number of test units n has the most important in¯uence on the value of asymptotic variance. Thus, if we wish to reduce the asymptotic variance, we should increase the number of test units. That is, we have to raise the budget of experiment or we can reduce the cost of a test unit. Finally, the proposed method can lead to better designs for conducting degradation tests. It provides the most ef®cient use of one's resources and to achieve the precision that one can expect to have with such a design. This approach is very intuitive, and can be very useful to engineers.\hskip24pt\hbox{(1)} Acknowledgements The comments of referees led to improvement of the paper and are greatly appreciated. This work was partially

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supported by the National Science Council of ROC grant NSC89-2118-M-032-004. We also thank Ching-Chyi Yang and Fang-Yih Chen for their help.