6. Planes

In this section we introduce a plane, the natural extension of a straight line.

We will examine a cost function that depends on 2 costs drivers, labor costs and transportation costs, plus a fixed cost.

Use z to denote Total Costs, x the number of labor hours, and y the number of cubic foot miles of transportation purchased.

$z = 1,000,000 + 20\, x + 0.5\, y.$

Interpretation of coefficients.

Based on this we can see that fixed costs are $1,000,000, labor rate is $20/hour and transportation costs $0.50 per cubic foot mile. Each multiplicative factor, sometimes called a partial slope, or weight, (depending on the context) measures the unique contribution of each cost driver to the overall cost function.

Consider the cost function evaluated at 100,000 labor hours, then the relationship between costs and transportation is

$z = 1,000,000 + 20\times 100,000 x + 0.5\, y.$

$z = 3,000,000 + 0.5\, y,$

so we see that by taking one of the input variables as fixed at a particular value, we obtain the equation of a straight line again. This means that a one unit change in purchased cubic foot miles results in a $0.50 change in costs, and that this is true, both regardless of the number of cubic foot miles purchased and and the number of labor hours.

Generically, we express the relationship as $z = f(x,y) = c + a\, x + b\, y$, which is the equation of a plane.

Again, we would describe the cost function as linear, because each of the cost drivers (x,y) is multiplied by a constant weight factor, that does not depend on the level of the cost driver. Total costs are derived from a fixed cost, plus the contribution from each cost driver.

This linear cost function assumes that the labor rate is constant, regardless of the number of labor hours purchased, and similarly for the transportation rates. It assumes that no discounts can be negotiated for bulk purchase of either labor or transportation. Finally, it assumes that the total cost is simply the sum of each individual cost component, that is that the costs are additive.

Problem solving. The approach here is an example of a common tactic. Break the problem apart into simpler ones, so that the whole can be obtained as the sum of the simpler parts.

Notice that this strategy, directly leads to making models that are linear.