1. Find Z′(m).

2. Solve the equation Z' (m) = 0.

As a practical matter, it is usually required that m be a whole
number. If m does not come out to be a whole number, then
m⁺ and m^-, the two whole numbers closest to m, must be
chosen. Calculate both Z(m⁺t) and Z(m^-); the smaller of the two provides the optimum value of Z.

3. Suppose a company finds that its demand for trainees is 3 per
month, that a training program requires 12 months, that the
fixed cost of training a batch of trainees is $15,000, that
the marginal cost per trainee per month is $100, and that trainee are paid $900 per month after training but before going to work. Use your result from Exercise 2 and find m.

4. Since m is not a whole number, find m⁺ and m^-.

5. Calculate Z(m⁺) and Z(m^-).

6. What is the optimum time interval between successive batches of trainees? How many trainees should be in a batch?

7. The parameters of this model are likely to change over time; it
is essential that such changes be incorporated into the model
as they change. One way to anticipate this is to create a spread-
sheet that gives the manager the total cost of training a batch
of trainees for various scenarios. Using the data from Exercise 3 as a starting point, create a spreadsheet that varies these
numbers and calculates the total cost of training a group of
employees for each scenario. Graph the total cost of training
with respect to changes in the various costs associated with
training.