A discrete-time Markov production–inventory model with machine breakdowns

A machine produces an item at a constant rate, which is assumed to be greater than the demand rate, and the demand is assumed to be known and constant. While operating, the machine can fail, and upon failure it requires service. The machine times-to-failure and repair times are random, and during repairs, demand is backordered as long as the backordering level does not exceed a prescribed amount, after which demand is lost. By considering time to be of discrete units and the times-to-failure and repair times to be geometrically distributed, we model the production–inventory system as a Markov chain and develop an efficient algorithm to compute the potentials that are used to formulate the cost function. The model results are then compared to simulation results where time is treated as a continuous parameter.