An optimal running policy for buffered continuous production processes subject to random breakdown

We present a general stochastic production model that is useful for examining the behaviour of complex systems. The model is made up of a linked system of production processes (units) together with buffer storages between the units. The productive capacity of the units are subject to random breakdown and periodic maintenance. Branching and recycling of intermediate products and materials, as well as inter-dependence of operation between units, are allowed. It is not always clear how best to operate such a system or to check that a given operating policy is well-defined. The model is rigorously developed and analysed using the theory of maximal solutions of differential inequalities with discontinuous right-hand sides. It is shown that when the capacities of the units possess a certain quasi-monotone increasing property, then there exists a well-defined, greedy 'full-on' operating policy. Moreover the policy maximizes throughput globally so that it is thus never optimal to try to balance the output of units to try to smooth production or to 'save for a rainy day'. Several examples are given showing the flexibility of the model.