Maintenance and Replacement

The problem of simultaneously determining the lifetime of an asset or an activity along with its management during that life is an important problem in practice. Section 9.1 considers a single machine whose resale value gradually declines over time. Its output is assumed to be proportional to its resale value. By applying preventive maintenance, it is possible to slow down the rate of decline of the resale value. An optimal control problem with free terminal time is formulated to simultaneously determine the optimal rate of preventive maintenance and the sale date of the machine and then explicitly solved. Section 9.2 assumes that the production rate of the machine is independent of its age, while its probability of failure increases with its age. The purpose of preventive maintenance is to influence the failure rate of the machine Also allowed is the sale of the machine at any time, provided it is still in running condition and its disposal as junk if it breaks down before it is sold. The optimal control problem is therefore to find an optimal maintenance policy for the period of ownership and an optimal sale date at which the machine should be sold, provided that it has not yet failed. A deep and illuminating interpretation of the transversality condition is provided. Section 9.3 extends the problem of maintenance and replacement to a chain of machines. The problem is to find the optimal number of machines over a finite horizon and the optimal times of their replacements such that the existing machine will be replaced by a new machine at these times. At the end of the given horizon, the last machine purchased will be salvaged. Moreover, the optimal maintenance policy for each of the machines in the chain must be found. The problem is solved by a mixed optimization technique. The subproblems dealing with the maintenance policy are solved by appealing to the discrete maximum principle. These subproblem solutions are then incorporated into a Wagner and Whitin framework to solve the full problem. The procedure is illustrated by a numerical example. There are many exercises at the end of the chapter.