Optimal number and allocation of controls among serial production stages

A production system comprised of multiple stages in tandem is considered. Each stage may be in either of two states: desirable or undesirable. Each stage may be placed under control so that it remains in the desirable state and produces the maximum fraction of conforming production units. Stages remaining uncontrolled may change randomly from the desired state to the undesired state. The controls may be mechanical or manual but involves costs which may or may not be dependent on the stage being controlled. It is desirable to find the optimal number of controls and their allocation among the stages which will maximise the net profit of production. The problem is formulated as a nonlinear mathematical program with binary variables. For the identical control cost case (independent of the stage to which control is applied), an 0(n) linear runtime algorithm is provided, where n is the number of stages. It is shown that a "k or nothing" (where k is the total number of controls applied) control policy is optimal and depends on a critical cost computed from the given parameters of the problem. In addition, conditions are provided under which "all or nothing" control policies are optimal. This is based on the computation of a "critical cost" and is independent of the number of stages in the system. It is shown that when these conditions are met the profit function is pointwise convex in k. Optimal solution techniques are provided and analysed for special cases in terms of the relationships between control cost and production parameters. Sensitivity analysis is provided for each of the parameters of the problem. The solutions are in general robust, with respect to these parameter variations. Numerical examples are provided throughout the paper to illustrate the relevant theorems. The paper ends with a discussion on the general control cost case and some feasible bounds on the optimal solution are offered.