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Economic Control Chart

CpM + Ci (-t + nE + ft(ARL2) + S\ T\ + frTf) sK/ARLl + W ^C ~ ECL ⁺ ECL

\(a + bri)lh\\\IX- t+ nE + /t(ARL2) + S\T\ + frTi]

where

ECL = IM + (1 - £i)s7b/ARLl -r+nE + /»(ARL2) + T\ + Ti [2]

The term ECL represents the expected length of a "ąuality cycle", namely the time between the start of successive in control periods. The end of this chapter contains a complete description of all model parameters, while the cost function derivation and the assumptions madę in developing the model can be found in Lorenzen and Vance (1986). The three ratios comprising eąuation [1] represent, respectively, costs due to the production of nonconforming items, the cost of false alarms and for locating and repairing a true assignable cause, and finally the cost of sampling.

The generał applicability of this model is due in part to the use of indicator variables S\ and dl to denote whether production continues during a search for an assignable cause or during process repair, respectively. Also, average run length values ARL1 and ARL2 can be calculated for a variety of applications, allowing this expected cost function to apply to various types of control charts such as X -charts, p- or wp-charts, c-charts and CUSUM charts.

Comparing and Relating the Models

Due to its generality, the Lorenzen-Vance model includes as specific cases most single assignable cause economic control chart models which appear in the ąuality control literaturę. However, due to differences in assumptions and notation, the relationship between a less generał model and the Lorenzen-Vance model may not be obvious. To address this problem, I consider eleven economic control chart models which appeared in nine articles published in the Journal of the American Statistical Association, Technometrics, Management Science, and the Journal of Quality Technology. Chiu (1975, 1976), Chiu and Wetherill (1974), Duncan (1956, 1971, and 1978), Gibra (1978, 1981), and Montgomery (1982). All are single assignable cause models except for Chiu (1976), Duncan (1971), and Gibra (1981). While this article does not deal directly with multiple assignable cause models, I include these articles because in each case the author presents a "matched" single assignable cause model and concludes that the design generated by the matched model agrees closely with