00471 $17d33eb6862cf36c8a862921ea9ad8

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An Algorithm and a Graphical Approach for Short Run Processes

The objective fiinction to minimize is the expected cost per cycle, given by

CC(«,fc,/) = [(C₀ + C,n)f + C₅]    + C₂E\A2] + C^aĄAl] + C_AP_r {restor-

ing the process}    [1]

where C₀ is the fixed sampling cost, C, is the variable sampling cost, C₂ is the cost of running out-of-control, C₃ is the cost of a false alarm, C₄ is the cost of restoring the process, and C₅ is the setup cost of the process. The probability of restoring the process is F₀P{R < T] + (1 - F₀)P{O < T}.

Constrained X Chart Designs for Short Runs

In order to make the resulting model morę realistic and to obtain designs with better statistical properties, constraints may be introduced in the model. The purpose of the finał model is to obtain a chart design (n.kj) that minimizes cost Equation [1] subject to the constraints discussed in this section.

An aspect that is often neglected in the economic design of control charts is the relationship between the sample size and the production ratę P. It is not uncommon to find designs where the number of parts sampled is reąuired to be larger than the number of parts produced, a clearly impossible result. Thus we add the constraint

[2]

TP

n <—

/

which implies that the number of parts produced every Tj f time units must be larger than the number of parts taken in the sample.

Statistical constraints can also be added to the model. Following Saniga (1989), we can limit the power of the chart and the probability of false alarms by introducing the constraints

1 - P_a = Power > Power_lb    [3]

14]

The average time to signal (ATS) can be limited in our model with respect to the length of the production run, i.e.,