Optimal burn-in with random minimal repair cost

Journal of the Korean Statistical Society 39 (2010) 245–249

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Optimal burn-in with random minimal repair cost Yong Man Kwon a , Richard Wilson b , Myung Hwan Na c,∗ a

Department of Computer Science and Statistics, Chosun University, Gwangju, 501-759, South Korea

b

School of Mathematics and Physics, University of Queensland, Brisbane, Australia

c

Department of Statistics, Chonnam National University, Gwangju, 500-757, South Korea

article

info

Article history: Received 5 August 2008 Accepted 13 August 2009 Available online 4 September 2009 AMS 2000 subject classifications: primary 62G10 secondary 62G20

abstract Burn-in is a method used to eliminate early failures of components before they are put into field operation. Maintenance policy, such as block replacement policy with minimal repair at failure, is often used in field operation. In this paper burn-in and maintenance policy are taken into consideration at the same time. It is assumed that the cost of a minimal repair to the component which fails at age t is a continuous non-decreasing function of t. We consider the problems of determining optimal burn-in times and optimal maintenance policy. © 2009 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.

Keywords: Burn-in Maintenance policy Bathtub-shaped failure rate Random minimal repair cost

1. Introduction Burn-in is a method widely used to eliminate early failures. The burn-in procedure is stopped when a preassigned reliability goal is achieved, e.g. when the mean residual life is long enough. Since burn-in is usually costly, one of the major problems is to decide how long the procedure should continue for. The best time to stop the burn-in process for a given criterion to be optimized is called the optimal burn-in time. An introduction to this important area of reliability can be found in Jensen and Petersen (1982). In the literature, certain cost structures have been proposed and the corresponding problem of finding the optimal burn-in time has been considered. See, for example, Block and Savits (1997), Cha (2000), Clarotti and Spizzichino (1991), and Mi (1994) for a review of the burn-in procedure. Let F (t ) be the distribution function of a lifetime X . If X has density f (t ) on [0, ∞), then its failure rate function h(t ) is defined as h(t ) = f (t )/F¯ (t ), where F¯ (t ) = 1 − F (t ) is the survival function of X . The following is one definition of a bathtub-shaped failure rate function which we shall use in this article. Definition. A real-valued failure rate function h(t ) is said to be a bathtub-shaped failure rate (BTR) with change points t1 and t2 if there exist change points 0 ≤ t1 ≤ t2 < ∞ such that h(t ) is strictly decreasing on [0, t1 ), constant on [t1 , t2 ) and then strictly increasing on [t2 , ∞). The time interval [0, t1 ] is called the infant mortality period; the interval [t1 , t2 ], where h(t ) is constant, is called the normal operating life or the useful life; the interval [t2 , ∞) is called the wear-out period. Mi (1994) considers a fixed burn-in time b. If the component fails before burn-in time b, then it is repaired completely with shop repair cost, then burn-in the repaired component again and so on. If the component survives the burn-in time b,

∗

Corresponding author. E-mail address: [email protected] (M.H. Na).

1226-3192/$ – see front matter © 2009 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jkss.2009.08.002

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Y.M. Kwon et al. / Journal of the Korean Statistical Society 39 (2010) 245–249

then it is put into field operation. Cha (2000) considers the case when the failed component is only minimally repaired rather than being completely repaired during a burn-in period. For a burned-in component he adopts block replacement policy with minimal repair at failure, assuming that the cost of minimal repair at failure is constant. In this paper, it is assumed that the cost of a minimal repair to the component which fails at age t is a continuous nondecreasing function of t. Hence, as the component ages it becomes more expensive to perform minimal repair. It is shown that the optimal burn-in time b∗ must occur before the change point t1 of h(t ) under the assumption of a bathtub-shaped failure rate function. Explicit solutions for the optimal burn-in time and preventive maintenance policy are given for the Weibull-exponential distribution.

2. Expected minimal repair cost Suppose that the burn-in time is fixed as b and we begin to burn in a new component. If the component fails before burn-in time b, it is repaired minimally, and the burn-in procedure is continued with the repaired component. After the fixed burn-in time b, the component is put into field operation. For a burned-in component, block replacement policy with minimal repair at failure is adopted. Under this policy the component is replaced by a burned-in component at planned time T , where T is a fixed number, and is minimally repaired at failure between planned replacements. Let Nx (y) be the random variable denoting the number of minimal repairs for a single component performed R t on the interval [x, x + y]. Then Nx (y) has a Poisson distribution with parameter Hx (y) = H (x + y) − H (x) where H (t ) = 0 h(s)ds. Let C (t ) be the cost of minimal repair to the component which fails at time t, where C (t ) is a continuous non-decreasing function of t. Now suppose that Nx (y) = k, and T1 , . . . , Tk are the times of the minimal repairs; then the total minimal repair cost on Pk the interval [x, x + y] is i=1 C (Ti ). Given Nx (y) = k, it is well known that τ1 = H (T1 ), . . . , τk = H (Tk ) are distributed as the order statistics of a random sample of size k from the uniform distribution on [H (x), H (x + y)]. Hence the expected minimal repair cost in the interval [x, x + y] is Nx (y)

E

X

Nx (y)

! C (Ti )

=E E

X

i =1

i=1

!! . C (Ti ) Nx (y)

Here, Nx (y)

E

X i=1

! C (Ti ) Nx (y) = k i=1 ! k X −1 =E C (H (τi )) Nx (y) = k i=1 ! k X −1 =E C (H (U(i) ))

! C (Ti ) Nx (y) = k = E

k X

i =1

=E

k X

! C (H

−1

(Ui ))

i =1

= kE C (H −1 (U1 )) =k


Z

1 H (x + y) − H (x)

H (x+y)

H (x)

C (H −1 (t ))dt ,

where U(1) , U(2) , . . . , U(k) are the order statistics of i.i.d. random variables U1 , U2 , . . . , Uk which are uniformly distributed on the interval [H (x), H (x + y)]. Therefore, E

NX x (y)

C (Ti )



 =

1 H (x + y) − H (x)

i=1

H (x+y)

Z =

H (x)

H (x+y)

Z

H (x)

C (H

−1



(t ))dt E (Nx (y))

C (H −1 (t ))dt

x+y

Z

C (t )h(t )dt .

= x

(1)

Y.M. Kwon et al. / Journal of the Korean Statistical Society 39 (2010) 245–249

247

3. Optimal burn-in Let C1 (t ) and C2 (t ) denote the costs of a minimal repair which fails at time t during burn-in period and in field operation, respectively. We assume that C1 (t ) ≤ C2 (t ) for all t ≥ 0, so that the cost of a minimal repair during a burn-in procedure is lower than that of a minimal repair in field operation for all time t. Then from (1) the expected minimal repair costs during burn-in period [0, b] and field operating time interval [b, b + T ] are given by b

Z

C1 (t )h(t )dt

b +T

Z

C2 (t )h(t )dt ,

and

0

b

respectively. Let cr be the replacement cost and c0 (b) be the cost for burn-in, where c0 (b) is a continuous non-decreasing function of b. Hence the long-run average cost per unit time C (b, T ) is given by C ( b, T ) =

1 T



c0 (b) +

b

Z

C1 (t )h(t )dt + cr +

b +T

Z



C2 (t )h(t )dt .

(2)

b

0

Theorem 1. Suppose the failure rate function h(t ) and C2 (t ) are differentiable and h(t ) is BTR with change points t1 and t2 . Then the optimal burn-in time b∗ and the corresponding optimal age T ∗ = Tb∗∗ satisfy 0 ≤ b∗ ≤ t1

T ∗ = Tb∗∗ > 0,

and

where Tb∗ is either one of solutions of Eq. (3) or equal to ∞ according to whether (3) has solutions or not: TC2 (b + T )h(b + T ) −

b +T

Z

C2 (t )h(t )dt = cr + c0 (b) +

b

Z

b

C1 (t )h(t )dt .

(3)

0

Proof. For any fixed b ≥ 0,

   Z b ∂ 1 C (b, T ) = 2 ηb (T ) − cr + c0 (b) + C1 (t )h(t )dt , ∂T T 0 where

ηb (T ) = TC2 (b + T )h(b + T ) −

b +T

Z

C2 (t )h(t )dt . b

Hence, ∂ C (b, T )/∂ T = 0 if and only if Eq. (3) holds. Note that ηb (0) = 0 and

∂ηb (T ) ∂ = T [C2 (b + T )h(b + T )]. ∂T ∂T Define

 B1 =

Z

T →∞

 B2 =

b ≥ 0 : lim ηb (T ) > cr + c0 (b) +

b

C1 (t )h(t )dt



0

b ≥ 0 : lim ηb (T ) ≤ cr + c0 (b) + T →∞

b

Z



C1 (t )h(t )dt . 0

For any b ∈ B1 such that b ≥ t1 , Eq. (3) has a unique solution Tb∗ , since ηb (T ) is increasing for all T ≥ 0. On the other hand, for any b ∈ B1 such that 0 ≤ b ≤ t1 , Eq. (3) has at least one solution and C (b, T ) has at least one local minimum since the behavior of ηb (T ) depends on that of the function C2 (t )h(t ) for t ≤ t1 − b. We take Tb∗ to be the largest point having minimum value of C (b, T ) among these local minima. This shows that the solution Tb∗ must satisfy Tb∗ > 0 for any given b ∈ B1 . The fact that Tb∗ satisfies Eq. (3) gives Tb∗ C2 (b + Tb∗ )h(b + Tb∗ ) −

b+Tb∗

Z

C2 (t )h(t )dt = cr + c0 (b) +

b

b

Z

C1 (t )h(t )dt ,

(4)

0

for any b ∈ B1 . Combining (2) and (4), we obtain C (b, Tb∗ ) = C2 (b + Tb∗ )h(b + Tb∗ ).

(5)

Thus, minimizing C (b, Tb ) is equivalent to minimizing C2 (b + Tb )h(b + Tb ) for b ≥ 0. By taking the derivative with respect to b on both sides of (4), we obtain ∗

Tb∗

∗

∗

∂ [C2 (b + Tb∗ )h(b + Tb∗ )] = C2 (b + Tb∗ )h(b + Tb∗ ) − C2 (b)h(b) + c00 (b) + C1 (b)h(b) ∂b

(6)

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Y.M. Kwon et al. / Journal of the Korean Statistical Society 39 (2010) 245–249

where c00 (b) is the derivative of c0 (b) with respect to b. From (5) and (6), we have

  ∂ C (b, Tb∗ ) 1 ∗ ∗ 0 = ∗ C2 (b + Tb )h(b + Tb ) − C2 (b)h(b) + c0 (b) + C1 (b)h(b) . ∂b Tb

(7)

If b ≥ t1 , then C2 (b + Tb∗ )h(b + Tb∗ ) − C2 (b)h(b) > 0. Since c00 (b) + C1 (b)h(b) > 0 for all b ≥ 0, consequently, from (7)

∂ C (b, Tb∗ ) > 0. ∂b Therefore the minimum value of C (b, Tb∗ ) can be only be achieved on the interval [0, t1 ], i.e., for any b ∈ B1 , 0 ≤ b∗ ≤ t 1 .

(8)

For any b ∈ B2 , note that ∂ C (b, T )/∂ T < 0 for all T ≥ t1 − b and C (b, T ) can have local minimum points for T ≤ t1 − b. Thus C (b, T ) has its minimum value at either Tb∗ , a solution of Eq. (3), or Tb∗ = ∞. This shows that the solution Tb∗ must satisfy Tb∗ > 0 for any given b ∈ B2 . We now consider two separate cases. Case 1: C (b, T ) has its minimum value at finite Tb∗ . In this case, we can show that 0 ≤ b∗ ≤ t1 using the same argument as for b ∈ B1 . Case 2: C (b, T ) has its minimum value at Tb∗ = ∞. In this case, C2 (t )h(t ) converges to a finite (not too large) limit, say C2 (∞)h(∞), at a faster rate than T → ∞ (for example, this would be true if C2 (t )h(t ) = k for all t > T ∗ for some constant k with k < C (b, T ∗ − b)). Note that C2 (∞)h(∞) = C (b, ∞) < C (b, T ),

(9)

for all T ≥ 0 and b > t1 , since C (b, T ) is non-decreasing in b and non-increasing in T , provided b > t1 . Hence, if there exists some b ∈ [0, t1 ] which is also in B2 with Tb∗ = ∞, then the minimum value of C (b, Tb∗ ) can be achieved on the interval [0, t1 ]. Define g (b) = cr + c0 (b) +

b

Z

C1 (t )h(t )dt − ηb (∞). 0

Note that the derivative of g (b) with respect to b is g 0 (b) = c00 (b) + C1 (b)h(b) + C2 (∞)h(∞) − C2 (b)h(b).

(10)

If t1 6∈ B2 , so that g (t1 ) < 0, there exists u > t1 such that [u, ∞) ⊂ B2 . Hence, C (b, Tb ) is increasing on [t1 , u) and ∗

C (t1 , Tt∗1 ) < C2 (∞)h(∞). This contradicts Eq. (9). Therefore, the minimum value of C (b, Tb∗ ), with Tb∗ = ∞, can be achieved on the interval [0, t1 ]. This completes the proof.  Corollary 2. Suppose the failure rate function h(t ) is differentiable and BTR with change points t1 and t2 . If C2 (t )h(t ) is nondecreasing and not eventually constant, then the optimal burn-in time b∗ is 0 and the corresponding optimal age T ∗ exists uniquely. 4. Examples Suppose that the failure rate of a component has a Weibull-exponential distribution, i.e.

  β−1  t   αβ α h(t ) =  β−1  t1   αβ α

0 ≤ t ≤ t1 t ≥ t1 ,

where 0 < β < 1 is the shape parameter, α is the scale parameter and t1 is the change point. The failure rate function is strictly decreasing on [0, t1 ] and stays at a constant for t ≥ t1 . We take c0 (b) = c0 b, Ci (t ) = ci exp(aH (t )) for i = 1, 2 with c1 ≤ c2 . Table 1 presents the optimal burn-in time b∗ and maintenance policy T ∗ when α = 1, β = 2/3, a = 0.5, c0 = 0.5, c1 = 1, c2 = 2 and for various choices of cr and t1 . From Table 1, it can be seen that the optimal replacement time is given by a finite value. Intuitively, it may be guessed that it would be infinite given that the distribution is Weibull-exponential. However, a finite optimal replacement time is obtained as the minimal repair cost is increasing and hence so is C2 (t )h(t ). Also, it should be noted that the optimal burn-in time b∗ decreases and T ∗ increases as cr increases, while b∗ and T ∗ increase with t1 .

Y.M. Kwon et al. / Journal of the Korean Statistical Society 39 (2010) 245–249

249

Table 1 Optimal burn-in time b∗ and maintenance policy T ∗ for Weibull-exponential distribution. t1 1

2

3

cr

b∗

T∗

b∗

T∗

b∗

T∗

3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

0.0098 0.0080 0.0066 0.0057 0.0049 0.0043 0.0038 0.0034

2.9849 3.2052 3.3980 3.5696 3.7249 3.8669 3.9977 4.1193

0.0142 0.0118 0.0100 0.0086 0.0076 0.0066 0.0058 0.0054

3.7578 4.0117 4.2360 4.4376 4.6207 4.7892 4.9451 5.0901

0.0165 0.0141 0.0120 0.0105 0.0090 0.0081 0.0072 0.0066

4.3012 4.5702 4.8101 5.0268 5.2253 5.4079 5.5778 5.7363

Acknowledgements This study was financially supported by Chonnam National University, 2008. The authors are greatly indebted to the referees for helpful comments, which have improved the presentation of this paper considerably. References Block, H., & Savits, T. (1997). Burn-in. Statistical Science, 12(1), 1–19. Cha, J. H. (2000). On a better burn-in procedure. Journal of Applied Probability, 37(4), 1099–1103. Clarotti, C. A, & Spizzichino, F. (1991). Bayes burn-in decision procedures. Probability in the Engineering and Informational Science, 4, 437–445. Jensen, F., & Petersen, N. E. (1982). Burn-in. New York: John Wiley. Mi, J. (1994). Burn-in and maintenance policies. Advances in Applied Probability, 26, 207–221.