Comment on Makis and Jardine (1992): Optimal replacement policy for a general model with imperfect repair
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Viewpoint Comment on Makis and Jardine (1992): Optimal replacement policy for a general model with imperfect repair In this viewpoint we obtain a new way of finding the optimal policy of a maintenance model with minimal repair and increasing repair cost which depends on the number of the repair in a cycle. Recall that a cycle is the time between two neighbouring replacements. Boland and Proschan1 proposed this model and considered a periodic replacement policy. However, this policy was not optimal. Later, Makis and Jardine2 considered a general model with imperfect repair which included the previous model as a special case. In this article the optimal replacement policy is given. To achieve this goal, Makis and Jardine formulated the replacement problem in the framework of semi-Markov decision processes. Using well-known results from optimal stopping theory, we can find the same optimal policy. It is hoped that the result of the present analysis, which is based on more elementary arguments, will complement Makis and Jardine’s paper. We first fix the notation to be used: cr ¼ replacement cost; cn ¼ repair cost at the nth failure in a cycle; mðxÞ ¼ mean residual life; Tn ¼ nth failure time in a replacement cycle.
Obviously, EðRT Þ 5 V EðTÞ, for every T 2 G, and T is optimal if EðRT ÿ V T Þ ¼ inf EðRT ÿ V T Þ: T 2G
ð3Þ
Using (3) and noting again that only replacement policies which stop at failure have to be taken into account, we can formulate the optimal policy problem for the model examined by Boland and Proschan as follows. To find the optimal stopping rule for the stochastic sequence Zn ¼ c r þ
nP ÿ1
ci ÿ V T n
n51
ð4Þ
i¼1
First, we shall show that we are in the monotone case (see eg Chow et al3), ie Zn 4 EðZnþ1 j Z1 ; . . . ; Zn Þ ) Znþ1 4 EðZnþ2 j Z1 ; . . . ; Znþ1 Þ
ð5Þ
It is clear that, Znþ1 ¼ Zn þ cn ÿ V ðTnþ1 ÿ Tn Þ. Then, because of the above assumptions, Zn 4 EðZnþ1 j Z1 ; . . . ; Zn Þ is equivalent to cn 5 V mðTn Þ. Also, since cnþ1 5 cn and mðTn Þ 5 mðTnþ1 Þ, we have Znþ1 4 EðZnþ2 j Z1 ; . . . ; Znþ1 Þ, as desired. Thus the stopping rule N ¼ inf fn 5 1 j Zn 4 EðZnþ1 j Z1 ; . . . ; Zn Þg
ð6Þ
The following will be assumed throughout:
is optimal. In other words, the optimal replacement policy is
1. cr > 0:cn is a non-decreasing sequence. 2. mð0Þ < þ1. 3. The failure rate is increasing.
N ¼ inf fn 5 1 j cn 5 V mðTn Þg
It should be noted that only policies replacing at failure need be considered. In order to find the optimal policy, we adopt the usual long-run average cost criterion. By classical arguments from renewal theory, the long-run average reward R of the system under a replacement policy T is given by VT ¼ EðRT Þ=EðT Þ
ð1Þ
Let V ¼ inf T 2G RT , where G is the set of all policies, where replacement is made at failure. A replacement policy T is optimal if VT ¼ inf RT T 2G
ð2Þ
ð7Þ
The above policy is equal to the policy found by Makis and Jardine
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