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Economic Control Chart Models with Cycle Duration Constraints

The Lorenzen-Vance Model

We base our analysis on the "unified" Lorenzen-Vance economic control chart model, in which a process is initially in control and is subject to the occurrence of a single assignable cause. The random length of the in control period is assumed to have an exponential distribution with mean 1/X. The control charting process involves taking a sample of n observations from the process output every h hours. A search for an assignable cause is undertaken if the calculated process measurement, generally the sample mean or the number of nonconforming items obtained, exceeds control limits. These limits are specified in terms of L, the number of standard deviations above or below the process center linÄ™.

Control chart design parameters n*, h*, and L* are selected as the values of n, h, and L which minimize the expected cost per hour function:

C =

Cp/X + Ci(-x + nE + h(ARL2) + SjTj + 8₂T₂) sY/ARLl + W ECL    ⁺ ECL

(1)

(2)

[(a + bn)/h][l/X - x + nE + h(ARL2) + 61T1 + 8₂T2] ECL

where

ECL = l/X + (1 - 5i)sTq/ARL1 - x + nE + h(ARL2) + Ti + T₂

The term ECL represents the expected length of a "Ä…uality cycle", namely the time between the start of successive in control periods. Appendix A contains a complete description of all model parameters, plus several new parameters introduced in this paper, while the expected cost function derivation can be found in Lorenzen and Vance (1986). The three ratios comprising expression (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 generaÅ‚ applicability of this basie model is due in part to the use of indicator variables 8] and 8₂ 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 from a variety of sampling distributions, allowing this expected cost function to apply to various types of control charts such as X -charts, p-charts, and u-charts.