Optimal replacement of a tool subject to random failure

The paper analyzes the problem of obtaining the optimal initial level and the optimal resetting time of a ‘tool-wear’ process with a positive shift in the mean value subject to random failure. The cost includes the resetting cost, penalty for failure and a cost due to deviation of the quality characteristic from its target value, which is a quadratic loss function. The failure mechanism is described by the proportional hazards model. A formula for the expected average cost per time is derived and it is shown that the optimal solution can be obtained by solving a system of two nonlinear equations. A numerical example is presented.