On the Optimal Operating Policy for the Machine Repair Problem When Failure and Repair Times Have Erlang Distribution

In the machine repair problem we are given m + n identical machines, m operating and n spares. All machines are independent of one another. When an operating machine fails, it is immediately sent to one of K repair facilities and immediately replaced by a spare machine, if one is available. Repair facility i has s_j independent and identical repairmen, 1 ≦ i ≦ k. Repairmen work on machines on a one-to-one basis. The cost of repairing a failed machine is a constant depending upon the facility performing the repair. Moreover, if the number of operating machines falls below m, a penalty is accrued at a rate depending on the shortage.

This paper treats the following decision problem: to assign (at the time of each failure) the failed machine to a repair facility so as to minimize the long-run average cost when K ≧ 2 and failure and repair times have Erlang distribution. We show that determining a stationary policy to minimize long-run average cost and long-run average expected cost is equivalent to solving a certain nonlinear program. The nonlinear program has a simple structure and an optimal solution.