# An optimal arithmetico-geometric process replacement problem for a two unit cold standby repairable system.

Introduction

In the fields of maintenance problems many replacement models were developed based on the assumption that the system after repair is 'as good as new'. This model is referred as perfect repair model. Barlow and Hunter introduced [1] a minimal repair model in which a minimal repair does not change the age of the system. Thereafter an imperfect repair model was developed by Barlow and Proschan [2] under which a repair with probability p as perfect repair and with probability 1-p as minimal repair. Many others worked in this direction and developed corresponding optimal replacement polices e.g. Black et al [3], Park [4], Kijima et al [5], Stadje and Zuckerman [6] etc.

In general, for a deteriorating system, it is reasonable to assume that the successive working times are stochastically decreasing while the consecutive repair times after failures are stochastically increasing, due to the ageing and accumulated wearing many systems. Thus a monotone process model should be a natural model for a deteriorating system. Ultimately, such systems can't work any longer. Neither can it be repaired any more.

To model such simple repairable deteriorating system Lam [7,8,] first introduced a geometric process repair model under the assumptions that the system after repair is not 'as good as new' and the successive working times {X , n=1, 2,....} of a system form a decreasing geometric process while the consecutive repair times {Y , n=1, 2,. ...} form a increasing geometric process. Under these assumptions, he considered two kinds of replacement polices-one based on the working age T of the system and other based on the number of failures N of the system. An explicit expression is derived for the long-run-average cost per unit time and also determined corresponding optimal replacement policy N* such that the long-run-expected average cost per unit time is minimized.

Other replacement policies under geometric process repair model were reported by Lam [9, 10], Stanley [11], Wang and Zhang [12, 13] Zhang [14-16], Zhang [18, 19].

All the research works discussed above are related to one component repairable system. However on practical application, the standby techniques are usually used for improving the reliability or raising the availability of the system. Thus Zhang et al [17] applied the geometric process repair model to a two-identical component cold standby repairable system with one repairman. It is assumed that the system after repair is not 'as good as new' and the successive working times form a decreasing geometric process while the consecutive repair times form an increasing geometric process. Under these assumptions they determined an optimal replacement policy N* such that the long-run-average cost per unit time is minimized.

The purpose of this paper is to apply arithmetico-geometric process model for two unit cold standby system with one repairman to generalize Zhang's et al work [17]. An arithmetico-geometric process approach is considered to be more relevant, realistic and direct to the modeling of the deteriorating system of maintenance problems that are encountered in most situations other than perfect or minimal repair models.

Therefore for a deteriorating system, it is assumed that the successive operating times of each component form a decreasing arithmetico-geometric process while the consecutive repair times form an increasing arithmetico-geometric process and each component after repair is not 'as good as new'. Under these assumptions, we study a

repair replacement policy N based on the number of failures of the component 1. An explicit expression for the long -run average cost per unit time is derived and corresponding optimal replacement policy N* is determined such that the long -run average cost per unit time is minimum. Finally, numerical results are provided to highlight the theoretical results. To study the model we consider the following two definitions.

Definition 1: Given a sequence of random variables [H.sub.1], [H.sub.2] ... if for some real number d and some real positive number r,{[H.sub.n] +(n -1) d}[r.sup.n-1],n=1,2,. ... form a renewal process (RP), then {[H.sub.n], n=1, 2, ...} is an arithmetico-geometric process (AGP). The two parameters d and r are called the common difference and the common ratio of the arithmetico-geometric process respectively.

Definition 2: If r > 1 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where n=2, 3,. .... And [mu][H.sub.1] is the mean of the first random variable H1, then the process is called a decreasing AGP. If d<0 and 0<r<1, then the AGP is called an increasing AGP, if d=0, r=1, then AGP becomes RP. Thus the general term of an AGP is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. See Leung [20].

Model

In this section we develop a model for two component cold standby repairable system with one repairman using arithmetico-geometric process and exposing to Weibull failure law in such a way that the long-run average cost per unit time is minimized with the following assumptions.

Assumptions

At the beginning two components are both new and assume that the component 1 is in working state while the component 2 is in cold standby state.

When the two components in the system are both good, one is in working state and the other is under cold standby. When working one fails it is immediately repaired by the repairman. At the same time standby one begins to work i.e., components appear alternatively in the system. When the repair of failed one is completed, either it begins to work or under cold standby. If one fails while other is still under repair then the system breaks down.

Let and [X.sup.(i).sub.n] and [Y.sup.(i).sub.n], for i=1,2 and n=1,2, ... are all S-independent random variables, respectively denoted by working time and repair time of component i , for i=1,2.

Let the sequence {[X.sup.(i).sub.n]n = 1,2,. ....} form a decreasing arithmetico-geometric process exposing to decreasing Weibull failure law, with parameters a>1 and d1>1.

Let the sequence {[Y.sup.(i).sub.n],n = 1,2,.....} form an increasing arithmetico-geometric process exposing to an increasing Weibull failure law with parameters [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let the distribution functions of the random variable [X.sub.n] and [Y.sub.n] be [F.sub.n]([x.sub.n]) and [G.sub.n] ([y.sub.n]) respectively where [X.sub.n] and [Y.sub.n] are in AGP.

Each component after repair is not as good as new and the time interval between the completion of the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] repair on component i and completion of the nth repair on component i is called the nth cycle of component i, where i=1,2 and n=1,2,3. ....

Let E [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The replacement policy N is used.

The components in the system can't produce working reward during cold standby and no cost is incurred during waiting for repair.

The repair cost rate of each component is [C.sub.r] and working reward rate of each component is [C.sub.w], and replacement cost of the system is C.

Optimal Solution

In this section, we determine an optimal solution for a replacement policy N based on the above assumptions. We study a replacement policy N under which the failure number of component 1 reaches N. Since units or components alternatively appears in the system, when failure number of component 1 reaches N, component 2 is either in cold standby state of the Nth cycle or repair state of the [(N-1).sup.th] cycle. Obviously component 1 is not repaired any more when it reaches failure number N, then component 2 works until failure in the Nth cycle.

Let C (N) be the long-run average cost of the system under policy N. Thus, according to renewal reward theorem of Ross [21,22], we have:

C(n)=The expected cost incurred in a renewal cycle/ The expected length of a renewal cycle (3.1)

Let L be the length of the renewal cycle under policy N, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

where the first, second, third, fourth and fifth terms are respectively, the working time of component 1, working time of component 2 in the Nth cycle, repair time, the time length of waiting for repair, the time length of cold standby state of component 1.

Now the expected length of renewal cycle can be determined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

According to the assumptions of the model, definition of probability density function, convolution and using Jacobian transformation the probability density function of [Y.sup.(2).sub.n-1]-[X.sup.(2).sub.n] and [X.sup.(2).sub.n]-[Y.sup.(1).sub.n] respectively are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

Where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

Where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [Y.sup.(i).sub.n]~W(x: [[eta].sub.1][[beta].sub.1]) then the distribution function of [X.sub.n] for, i=1,2. is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

Let [Y.sup.(i).sub.n]~W(x : [[eta].sub.2][[beta].sub.2]) then the distribution function of [Y.sub.n] is :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

From equations (3.4) and (3.5) the distribution functions of [Y.sup.(2).sub.n-1]-[X.sup.(2).sub.n] and [X.sup.(2).sub.n]- [Y.sup.(2).sub.n] respectively is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The expected length of working time is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

The expected length of repair time is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.9)

The expected length of waiting time for repair is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.10)

The expected length of cold standby time is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.11)

According to equation (3.8) to (3.11), we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.12)

Thus, according to equations (3.1) and (3.12), we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

According to equations (3.8) and (3.9); we have where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using C(N), we determine an optimal replacement policy [N.sup.*] such that the long-run average cost per unit time is minimized.

Numerical Results and Conclusions

For given hypothetical values of a, b, Cw, C, Cr,[eta].sub.1],[[beta].sub.1],[eta].sub.1],[[beta].sub.1],[d.sub.1] and [d.sub.2] the optimal replacement policy [N.sup.*] is calculated as follows:

a=1.005, b=0.99, Cw=40, C=6500, Cr=25, d1=0.001, d2=-0.5 [eta].sub.1]=10, [eta].sub.2]= 25[beta].sub.1],= 0.5,[beta].sub.1]= 2

[GRAPHIC OMITTED]

Conclusions

i. From table and the graph we notice that the C (17) = 1.42234 is the minimum i.e, the optimal policy is [N.sup.*] = 17 and we should replace the system at the time of 11th failure.

We notice that for a small increase in 'b' results a corresponding increase in N and for a small increase in 'a' there is a corresponding decrease in N which coincides with practical analogy.

In practice, if a repair man experienced with repair then the successive repair times will form a decreasing AGP while the consecutive working times will form an increasing AGP. Therefore an AGP model can be developed for improving cold standby repairable system.

References

[1] Barlow, R.E., and Hunter, L.C., 1959, "Optimum Preventive Maintenance Policies," Operations Research, 8, pp. 90-100.

[2] Barlow, R.E., and Proschan, F., 1975, "Statistical theory of reliability and life testing," Holt, Rinehart, Winston.

[3] Block, H.W., Borges, W.S., and Savits, T.H., 1985, "Age dependent minimal repair," Journal of Applied Probability, 22, pp. 370-385.

[4] Park, K.S., 1979, "Optimal number of minimal repairs before replacement," IEEE Trans. on Reliability, 28, pp.137-140.

[5] Masaki Kijima, Hidenori Morimura and Yasusuke Suzuki.,1988, "Periodical replacement problems without assuming minimal repair," European journal of Operational Research, 37, pp.194-2003.

[6] 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.

[7] Lam, Y., 1988a, "A note on the optimal replacement problem, Adv. Appl. Prob.,"20,pp. 479-482.

[8] Lam, Y., 1988 b, "Geometric Process and Replacement Problem," Acta Mathematicae Applicatae Sinica, 4, pp. 366-377.

[9] Lam, Y., 1990, "A repair replacement model," Adv. Appl. Prob., 22, 494-497.

[10] Lam, Y., 2003, "A geometric process maintenance model," South East Asian Bulletin of Mathematics, 27, pp.295-305.

[11] Stanley, A.D.J.,1993, "On geometric process and repair replacement problems," Microelectronics and Reliability, 33,pp. 489-491.

[12] Wang, G.J., and Zhang, Y.L., 2006, "Optimal periodic preventive repair and replacement policy assuming geometric process repair," IEEE Trans. on Reliability, 55, pp.118-121.

[13] Wang, G.J., and Zhang, Y.L., 2007, "An optimal replacement policy for a two component series system assuming geometric process repair," Comp. and Math. With Appl.,54, pp.192-202.

[14] Zhang,Y.L.,1994, "A bivariate optimal replacement policy for a repairable system," Journal of Appl. Proba., 31, pp.1123-1127.

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

[16] Zhang,Y.L., 2002, "A geometric process repair model with good-as-new preventive repair," IEEE Trans. on Reliability, 51,pp. 223-228.

[17] Zhang, Y.L.,Wang, G.J., and. Ji, Z.C., 2006, "Replacement problems for a cold standby repairable system," International Journal of Sys. Sci., 37,pp. 17-25.

[18] Zhang, Y.L., and Wang,G.J.,2006, "A bivariate optimal repair-replacement model using geometric process for a cold standby repairable system," Eng. Opt., 38, pp.609-619.

[19] Zhang, Y.L., and Wang, G.J., 2007, "A geometric process repair model for a series repairable system with k-dissimilar components," Appl. Math. Model., 31, pp.1997-2007.

[20] Leung, K.N.F., 2001, "Optimal replacement policies determined using arithmetico-geometric processes," Engineering Optimization, 33, pp. 473-484.

[21] Ross., S.M., 1970, "Applied Probability Models with Optimization Applications," San-Franciso, Holden-Day.

[22] Ross, S.M., 1996, Stochastic Process, 2nd Edition, Wiley, New York.

(1) B. Venkata Ramudu and (2) Y. Krishna Reddy

(1) Lecturer in Statistics, S.S.B.N. Degree & P.G. College, Anantapur, India E-mail: venkataramudussbn@gmail.com

(2) Director, A.P. IIIT, RGUKT, R.K. Valley, Idupulapaya, Kadapa-516329, A.P., India E-mail: dryaddula@gmail.com
```Table 6.1

N C(N) N C(N)

2 35.581154 27 2.294587
3 18.978964 28 2.425085
4 12.078785 29 2.559237
5 8.382009 30 2.696467
6 6.133282 31 2.836276
7 4.660766 32 2.978235
8 3.65191 33 3.121973
9 2.94161 34 3.267159
10 2.43444 35 3.413515
11 2.071416 36 3.560787
12 1.814116 37 3.708754
13 1.636455 38 3.857227
14 1.52009 39 4.00603
15 1.451767 40 4.155015
16 1.421661 41 4.304048
17 1.422344 42 4.453007
18 1.448096 43 4.601789
19 1.494443 44 4.7503
20 1.557837 45 4.898453
21 1.635431 46 5.046178
22 1.724914 47 5.193405
23 1.824394 48 5.340077
24 1.932302 49 5.486141
25 2.047339 50 5.63155
26 2.168408
```