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Simpson & Keats

that minimizes expected hourly cost. Because they are inputs to (1) and (2), the ARL's for the design configuration are calculated prior to optimization. These calculations for the X chart were relatively simple, but those for the CUSUM control chart reąuired the computationally intensive Markov chain approach. Because this algorithm requires inversions of large matrices, we first developed ARL tables using the Markov approach for various combinations of n, k, H, and A, and then used a table lookup techniąue for the optimization runs.

In the first section of this chapter, through a sensitivity analysis, we suggested simplifying the LV model with the CUSUM by using only four input variables in model. Sensitivity studies have been performed on economic models developed prior to Lorenzen and Vance for both the X (Chiu and Wetherill, 1974; Goel et al., 1968) and CUSUM procedures (Chiu, 1974; Goel and Wu, 1973. The findings show that a smaller subset of model inputs significantly impact the expected cost per time unit and the chart design parameters. For example, X sensitivity studies using the Duncan model were summarized by Montgomery (1980) and revealed that A largely determines the optimum sample size. The cost of producing out-of-control (Ci) drives the optimum sampling interval, and the cost to search and repair the assignable cause (W) has the most impact on the control interval width.

Goel (1968) performed an economic comparison of the X and CUSUM charts using Duncan’s model and found little difference between them relative to optimum cost. Lorenzen and Vance (1986b) compared the X, CUSUM and EWMA control charts using their model. They performed a limited sensitivity analysis. For instance, their first sensitivity fixed all twelve inputs, then varied sample size and control limits to optimize cost. Their second analysis fixed the sample size at the optimal cost value from the previous test and varied the size of the process shift. Finally, some cost and time inputs are varied one factor at a time in order to determine the response impacts.

We chose to design several experiments in which all the inputs and design parameters were varied simultaneously, so that a comprehensive study of the model dynamics could be madę. Several experiments were designed to accommodate the many levels of the process shift, different control chart types and different example input scenarios.

In making economic comparisons of the X and CUSUM procedures we first performed sensitivity analyses to determine the elements that most influence variability in cost. We then studied various optimal-cost responses to determine differences between the two techniąues. The sensitivity and response analyses were performed under the same two scenarios, representative of