Dead band adjustment charts with asymmetric off-target costs, deterministic process drift and delayed dynamics

Feed-back control systems are a type of process control method in which a quantitative quality characteristic is measured at regular intervals of time, successive measures are used to estimate the current mean deviation of the quality characteristic from the target and, when necessary, adjustments of an input compensatory variable are made to bring the estimated process mean back to the target. Important problems to be solved are how often to sample the quality characteristic and when and by how much to apply an adjustment, i.e. to determine the optimal sampling interval, action limits and amount of adjustment. This determination may be done to minimize the long run cost resulting from the combined effects of the costs of taking each observation, of making each adjustment and of being off target. When the adjustment cost is zero, adjustments are made at each sampling point (this is a repeated adjustment scheme). However, when the cost of making an adjustment is important, two different action limits must be chosen and an adjustment must be made only when the current estimate of the mean deviation of the quality characteristic from target jumps outside the action limits (this is a dead band scheme). We consider three situations requiring asymmetric action limits; these are deterministic process drift, asymmetric off-target costs and asymmetric underlying distributions. Deterministic process drift is often caused by tool wear. Asymmetric costs may be due to asymmetric tolerances and/or to different costs of exceeding positive and negative tolerances. Asymmetric distributions sometimes appear in essentially positive quality characteristics. In addition, we consider the possibility that there is a delay in taking appropriate actions; this delay is often caused by the tests and procedures that are required to sample the process.