Abstract: This paper studies a bivariate repairable system with one repairman is studied. Assume that the system after repair is not ‘as good as new’ and also the successive working times form a decreasing -series process, the successive repair time’s form an increasing geometric process and both the processes are exposing to Weibull failure law. Under these assumptions, we study an optimal replacement policy N in which we replace the system when the number of failures of the system reaches N. We derive an explicit expression for the long-run average cost per unit of time for the bivariate replacement policy (T,N) under which we replace the system when the number of failures reaches N or the age of the system reaches T or whichever occurs first. Under some mild conditions, we determine an optimal repair replacement policy N* such that the long run average cost per unit time is minimized. Numerical results are provided to support the theoretical results.

Most maintenance models developed under the common assumption is that repair is perfect, i.e., repair restores the system to a “good as new” condition, and it is called a perfect repair model. However, perfect repair is not always satisfied. In practical application, most repairable systems are deteriorating due to ageing and accumulative wear. Barlow and Hunter [1] introduced a minimal repair model in which the repair activity doesn’t change the failure rate of the system. Brown and Proschan [3] proposed an imperfect repair model in which the repair is equivalent to a perfect repair with probability ‘p’ and to a minimal repair with probability 1-p (0£p£1). Much research work has been carried out by Lam [7], Stadje and Zukerman [9], Stanley [10], Wang and Zhang [11, 12], Zhang et.al [15, 17] , Zhang [13,16], Zhang and Wang [18] and others along this direction. In application, because of the ageing effect and the accumulated wearing, most systems are degenerative in the sense that the consecutive working time between failures will be shorter and shorter, while the consecutive repair time after failures are getting longer and longer. In other words, the successive operating time are stochastically non- increasing, while the consecutive repair time are stochastically non- decreasing. To model such a deteriorating system, Lam [5,6] proposed a geometric process repair model. Using this model, Lam discussed two kinds of replacement polices known as policy T which based on the working age of the system while the other is called policy N which is based on the cumulative number of failures of the system. Under these two kinds of replacement polices explicit expressions are developed by minimizing the long run average cost per unit of time. Further he also showed that policy N is better than policy T. Zhang [14] also generalized Lam’s [5,6] work using a bivariate replacement policy, called policy(T,N), under which the system is replaced at the working age T or at the time of the N^{th} failure, whichever occurs first, and proved that the bivariate policy is better than the uni-variate replacement policies, policy N and policy T. Later, Zhang [18] proposed optimal replacement policies for the maintenance problems by generalizing Zhang’s [14] work.

Braun et.al [2] explained the increasing geometric process grows at most logarithmically in time, while the decreasing geometric process is almost certain to have a time of explosion. The -series process grows either as a polynomial or exponential in time. It also noted that the geometric process doesn’t satisfy a central limit theorem, while the -series process does. Further, Braun et.al [4] presented that both the increasing geometric process and the -series process have a finite first moment under certain general conditions. Thus the decreasing -series process may be more appropriate for modeling system working times while the increasing geometric process is more suitable for modeling repair times of the system.

Based on this understanding, in this paper, we proposed two different monotone processes, which is a generalization to geometric processes proposed by Lam [5,6].Further, we discussed the model and optimal solution for a bivariate policy (T,N) under which the system is replaced at working age T or at the time of N^{th} failure, whichever occurs first exposing to two monotone process. The objective is to determine the optimal replacement policy (T,N)* such that the long run average cost per unit time is minimized. The explicit expression of the long run average cost per unit time is derived and the corresponding optimal replacement policy can be determined analytically or numerically. We prove that the optimal policy (T,N)* is better than the optimal policy N* for a simple repairable system.

In modeling of these deteriorating systems we utilize definitions given in Lam [5] .

Definition 1: Given two random variables X and Y, if P(X>t) > P(Y>t) for all real t, then X is called stochastically larger than Y or Y is stochastically less than X. This is denoted by X >_{st} Y or Y <_{st} X respectively.

Definition 2: Assume that {Y_{n}, n=1,2,….}, is a sequence of independent non-negative random variables. If the distribution function of X_{n} is F_{n} (t) = F(a^{n-1}t) for some a > 0 and all n=1,2,3,…., then {Y_{n}, n=1,2,…,} is called a geometric process, ‘a’ is the ratio of the geometric process.

Obviously:

if a>1, then {Y_{n}, n=1,2,….} is stochastically decreasing, i.e, Y_{n} >_{st} Y_{n+1} , n=1,2,…;

if 0<a<1, then {Y_{n}, n=1,2,….} is stochastically increasing, i.e, Y_{n} <_{st} Y_{n+1} , n=1,2,;

if a=1, then the geometric process becomes a renewal process.

Definition 3: Assume that {X_{n}, n=1,2,….}, is a sequence of independent non-negative random variables. If the distribution function of X_{n} is for some > 0 and all n=1, 2, 3… then {X_{n}, n=1, 2…} is called an series process, is called exponent of the process. Braun et. al [12].

Obviously:

if >0, then {X_{n}, n=1,2,….} is stochastically decreasing, i.e, X_{n} >_{st} X_{n+1} , n=1,2,…;

if <0, then {X_{n}, n=1,2,….} is stochastically increasing, i.e., X_{n} <_{st} X_{n+1} , n=1,2,;

if =0, then the series process becomes a renewal process.

2. The Model

In this section, we develop a model for the bi-variate optimal replacement policy for the system specializing to two monotone processes by minimizing the long-run expected reward per unit time with the following assumptions:

ASSUMPTIONS

At the beginning, a new system is installed. The system will be replaced some time by a new and identical one, and the replacement time is negligible.

2. Let X_{n} and Y_{n} be the successive working time follows decreasing a -series process and the successive repair time’s form an increasing geometric process respectively and both the processes are exposing to weibull failure law with parameters λ and μ respectively. A sequence {x_{n}, n=1,2,……} and sequence {Y_{n}, n=1, 2, …..} are independent.

Let and be the distribution function of Xn and Y_{n} respectively, and both are distributed according to weibull failure law.

The constant repair cost rate is C_{r} and the constant revenue rate whenever the system working is C_{w} and the replacement cost is C.

The replacement policy (T, N) is used.

3. Optimal Solution

Using the assumptions on the model, it is determined an optimal replacement policy (T, N)* such that the long-run expected reward per unit time is maximum. Let T_{1} be the first replacement time and let T_{n}(n³2)be the time between the (n-1)^{th} replacement and n^{th} replacement. Clearly {T_{1},T_{2},……,} forms a renewal process.

Let C(T,N) be the long-run average cost rate per unit time under the replacement policy (T,N). Since {T_{1},T_{2}….,} forms a renewal process, the inter arrival time between two consecutive replacements is a renewal cycle. Thus according to the renewal reward theorem [see Ross 8] we have:

(3.1)

Let L be the length of a renewal cycle of the system under policy (T.N). Then

(3.2)

(3.3)

(3.4)

I is the indicator function such that

I_{A }= 1, if event A occurs

= 0, if event a doesn’t occurs.

Now the expected length of a renewal cycle L can be evaluated as follows:

. (3.5)

Lemma:Let then the expectation of indicator function I{U_{n}< T < U_{N}} is given by:

(3.6)

(3.7)

(3.8)

Now, using equation (3.6) in the above lemma, we have:

, (3.9)

(3.10)

(3.11)

Substitute the results in equations (3.9) and (3.11), in equation (3.5)

(3.12)

Thus from equation (3.1) and (3.12) the long-run expected average per unit time C(T,N) under policy (T,N) is given by:

(3.13)

When Tà¥, the optimal replacement policy (T,N*) reduces to C(N) and which is given by:

(3.14)

Now the expected length of working time can be obtained as follows:

Let Then the distribution function of , for k=1,2,3,….and i=1,2 is :

By definition the expected length of working time is given by :

(3.15)

(3.16)

The expected length of repair time of component 1 can be obtained as follows:

Let then the distribution function of for i=1,2, and k=1, 2, 3, …., is given as:

By definition, the expected length of repair time is:

(3.17)

(3.18)

Using equations (3.15) and (3.18), we have:

(3.19)

In the next section, using this C(N) equation given in (3.19), we determine the optimal value N* through empirical results.

4. Numerical Results and Conclusions

For the given hypothetical values of the parameters λ, µ,α , b, C_{w}, C_{r}, and C the long-run average cost per unit time is determined and is given in the following table: 4.1.

Table 4.1: Values of the long-run average cost per unit time

For the given hypothetical values λ=10,µ=25,α=0.25,

C_{w}=100,C_{r}=450,

C=8000 b=0.95

For the given hypothetical values λ=10,µ=25,α=0.25,

C_{w}=100,C_{r}=450,

C=8000 b=0.90

N

C(N)

C(N)

2

401.0486

401.0486

3

368.4716

369.9845

4

359.0672

362.6706

5

356.2385

362.0414

6

356.0372

364.0167

7

357.0761

367.1586

8

358.7558

370.8445

9

360.7815

374.7673

10

362.9947

378.7621

11

365.3047

382.7345

12

367.6569

386.6284

13

370.0176

390.4094

14

372.365

394.0566

15

374.6852

397.5576

16

376.9691

400.905

17

379.2106

404.0958

18

381.4059

407.1289

19

383.5526

410.0056

20

385.6492

412.7282

Conclusions

From the table 4.1 and graph 4.1, it is examined that the long-run average cost per unit time C (6) = 356.0372 is minimum for the given b=0.95. Thus, we should replace the system at the time of 6^{th} failure.

Hor different values of the parameters, we observed that as ‘b’ increases number of failures increases, while ‘’ decreases an increase in the number of failure, which coincides with the practical analogy and helps the decision maker for making an appropriate decision.

Braun, W.J., Li, W., and Zhao, Y.Q., 2005, ‘Properties of geometric and related process’, Naval Research Logistic, 52, pp.607-616.

Brown, M., and Proschan, F., 1983, ‘Imperfect Repair’, Journal of Applied Probability, 28, pp.851-859.

Braun, W.J., Li, W., and Zhao, Y.Q., 2008, ‘Some theoretical Properties of the geometric and -series process’, Communication in Statistics- Theory and Methods, 37, pp.1483-1496.

Lam, Y., 1988a‘A note on the optimal replacement problem’, Advanced Applied Probability, 20, pp.479-482.

Lam, Y., 1988b ‘Geometric Process and Replacement Problem’, Acta, Mathematicae Applicatae Sinica, 4, pp.366-377.

Lam, Y., 2003, ‘A geometric process maintenance model’, South East Asian Bulletin of Mathematics, 27, pp.295-305 .

Ross, S.M., 1970, ‘Applied Probability Models with Optimization Applications’,San-Franciso, Holden-Day.

Stadje, W., and Zuckerman, D., 1992,‘Optimal repair policies with general degree of repair in two maintenance models’, Operations Research Letters, 11,pp.77-80.

Stanley, A.D.J., 1993,‘On geometric process and repair replacement problems’, Microelectronics and Reliability, Vol.33(4), pp.489-491.

Wang, G.J., and Zhang, Y.L., 2006, ‘Optimal periodic preventive repair and replacement policy assuming geometric process repair’, IEEE Transactions on Reliability, 55, pp.118-121.

Wang, G.J., and Zhang, Y.L., 2007, ‘An optimal replacement policy for a two-component series system assuming geometric process repair’, Computers and Mathematics with Applications, .54, pp.192-202.

Zhang, Y.L., 1999, ‘An optimal geometric process model for a cold standby repairable system’, Reliability Engineering and System Safety, 63, pp.107-110.

Zhang, Y.L., 1994, ‘A bivariate optimal replacement policy for a repairable system’, Journal of Applied probability, 31, pp. 1123-1127.

Zhang, Y.L., Yam, R.C.M., and Zuo, M.J., 2001, ‘Optimal Replacement Policy for a deteriorating production system with preventive maintenance’, International Journal of System Science,.32, pp.1193-1198.

Zang, Y.L., 2002, ‘A geometric process repair model with good-as-new preventive repair’,IEEE Transaction on Reliability, 51, pp. 223-228.

Zhang, Y.L., Wang, G.J., and Ji Z.C., 2006, ‘Replacement problems for a cold standby repairable system’, International Journal of System Science, 37, pp.17-25.

Zhang, Y.L., and Wang, G.J., 2006, ‘A bivariate optimal repair-replacement model using geometric process for a cold standby repairable system’, Engineering Optimization,.38, pp.609-619.