# Consider the following variant of the replacement model of Sec. 6.10.4 which includes the…

Consider the following variant of the replacement model of Sec. 6.10.4 which includes the possibility of equipment failure. Let s denote the age of the equipment. For any s, there is a probability ps that the equipment fails and a probability 1 – ps that it ages by one period. In addition to the costs in Sec. 6.10.4, there is a fixed cost of a failure F. If the equipment fails, the system moves to state 0 during the period in which the failure occurs, it must be replaced instantaneously at cost K, and then starts the subsequent period in state 0. As before, the actions in any period are to replace or operate as is. Assume that the operating cost h(s) = hs’.

a. Formulate this as a discounted Markov decision problem by defining r(s, a)

b. Give the optimality equation and provide a bounding function w for which

c. Provc that a finite-state approximation converges to u:.

d Find a priori bounds on the precision of the N-state approximation.

e. Show that a control limit policy is optimal.

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