Opportunistic replacement policies when the changes of states are markovian

Abstract

Consider a series system of two units, named 1 and 2, respectively. Two units are observed at the beginning of discrete time periods t = 0,1,2,.... and classified as being in one of a countable number of states. Let (i, r) be a state of the system at time t, when the state of unit 1 is i and state of unit 2 is r at time t. Under some conditions, the opportunistic replacement policy that minimizes the expected total discounted cost or the average cost of maintenance is shown to be characterized by the control limits i* (r) (a function of r) and r* (i) (a function of i): (a) in observed state (i,r), the optimal policy for unit 1 is to replace if i$\geqslant$ i*(r) and no action otherwise; (b) in observed state (i, r), the optimal policy for unit 2 is to replace if r$\geqslant$r* (i) and no action otherwise. This study also develops optimal policies in the finite time horizon case, where time horizon is fixed or a finite integer valued random variable with known probability mass function. Finally, some special cases and extensions of the basic model are discussed.