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Analysis of production decisions under budget limitations

The issue of when to intervene in the evolution of a production system is the focus of this study. The interventions take the form of changes to production depending on the current value of the products. Each change incurs a charge representing costs such as physical expansion, overtime or new hiring when production increases and costs such as severance or shut down when production decreases. The goal is to maximize the expected return subject to these intervention costs over at most a finite number of intervention cycles. This paper determines for a large class of problems an explicit formula for the value function and a set of optimal times at which to increase and to decrease production. The optimization is over a very general class of stopping times and proves that an optimal set of times in this general class is given as the hitting times of various levels, depending on the number of remaining interventions. These optimal hitting levels are characterized as a maximizing point for a high-dimensional nonlinear function and can be efficiently and iteratively determined as the solutions of successive 1D nonlinear maximization problems. The solution method is illustrated on some examples, including mean-reverting processes.