The official setup time for a machine is $40$ minutes. Suppose a workday is $420$ minutes, the

machine must process $200$ parts a day, and the processing time is $0\mathrm{.}5mi\frac{n}{u}$nit.

a$.$ What is the minimum batch size?

b$.$ Suppose the operation is run for one day using the batch size from $($a$),$ and at the end of

the workday only $133$ units have been produced. Assuming the processing time per unit

is $0\mathrm{.}5$ minutes, and no stoppages except for setup, how much time was used for setup?

What was the average time per setup? Explain how the actual setup time could be so

much longer than the official setup time.