Optimal switchover times between two activities utilizing the same resource

The ‘gold-mining’ decision problem is concerned with the efficient utilization of a delicate mining equipment working in a number of different mines. Richard Bellman was the first to consider this type of a problem. The solution found by Bellman for the finite-horizon, continuous-time version of the problem with two mines is not overly realistic since he assumed that fractional parts of the same mining equipment cound be used in different mines and this fraction could change instantaneously. In this paper, we provide some extensions to this model in order to produce more operational and realistic solutions. Our first model is concerned with developing an operational policy where the equipment may be switched from one mine to the other at most once during a finite horizon. In the next extension we incorporate a cost component in the objective function and assume that the horizon length is not fixed but it is the second decision variable. Structural properties of the optimal solutions are obtained using nonlinear programming. Each model and its solution is illustrated with a numerical example. The models developed here may have potential applications in other areas including production of items requiring the same machine or choosing a sequence of activities requiring the same resource.