3.1.1 Example 1, production time

Consider a production line run by a particular manager. The line takes 50 hours to set up, and once set up takes 15 minutes (0.25 hours) to produce each item.

In this problem we are interested in the relationship (function) that links how many items have been produced , and the time taken to produce those items.

To get a feel for the relationship, it's simple to consider some ``candidate'' values for the number of items produced and figure out how long it took to produce them. We'll look at how long it takes to produce 100, 200, 300 and 400 items.

To produce 100 items there's a 50 hours set up time, then at 0.25 hours per item that's another $0.25 \times 100 = 25$ hours of production time, making a total of 75 hours.

The other times for 200,300 and 400 items follow from a similar calculation. The calculations can be displayed in a table:

 

Table 1: Tabular representation of the relationship between the number of items produced and the time to produce those items.

x = # of Items	y = # of Hours
100	75
200	100
300	125
400	150

Now we'll take a look at what the relationship looks like graphically.

From the formula perspective, denoting time in hours as H and numbers of items produced as I, the relationship can be written as

$$\fbox{$H = 50 + 0.25 \times I,$ }$$

which you should now recognize as the formula for a straight line.

Interpretation:

Notice that the 50 in the formula, generically called the intercept, is measured in Hours and the slope, 0.25 has units of Hours/Item, so that the 2 sides balance in units.

In this example both the intercept and slope of the line have clear interpretations.

Key point: the time to produce an additional unit stays the same, regardless of how many units have already been produced.