Control chart and stochastic control processes

Abstract

In this paper, we shall discuss the relation between the expected cost model for a process and the probabilistic control model, whose mean is controlled by an X chart. T. Odanaka has exploited the problem of minimizing the membership function or the probability that the performance level of the machine will run out of range in a fuzzy environment. This paper considers the same basic problem of the expected cost model, modified to include the set up costs for controlling the action in any period. The expected cost comprises the fixed and variable cost of taking a control action, the cost of sampling, the cost of investigating the process when the control chart indicates that the process parameters exceed the specified bound, and the cost of producing defective units. Dynamic programming can be used to determine optimal control policies for models where control produces economically measurable benefits and/or costs. When the controlling action incurs a set up charge, the optimal policies are found to be characterized by a pair ( s n , S n , S′ n , s′ n ), where the action is made in a period n to state S n or S′ n ( S n < S′ n ), if the state is found to be outside s n or s′ n ( s n < s′ n ). This statement is subject to a set of basic assumptions such that proportional costs of changing the state variable are zero, the two fixed costs are equal, the loss function is symmetric quasi-convex and the problem's probability densities are quasi-concave.