Non-Linearity

We tend to have good intuition about things when there is a linear relationship between the controlling variables. For example, all other things being held equal, we expect that if we have three times as much work, it will take us three times as long to get the job done. However, our intuition starts to work less well when the variables involved have a non-linear relationship.

With the problem at hand, we note that once we have computed the multiplier, M, there is a linear relationship between the number of useful additional hours devoted to the product and the growth rate, G. That is, we gain useful hours in direct proportion to growth rate, with M as the constant of proportionality. Unfortunately, this is the only linear relationship in the whole model.

To compute the key ingredient M, the multiplier, we used a relationship that is non-linear in the variables P and D:

M = P - (1 - P) D.

This leads to Figure 22.1, where different bands correspond to different ranges of M. Note that we have only weak intuition as to how those bands come out as a function of P and D; we know that low P and high D are bad in combination, but it is hard to judge just how bad. That is why the graph is so helpful.

Similarly, both the overall productivity and increased cost to produce the product are non-linear in M and G. Recall that the relationship for the new overall productivity is

N = (1 + GM) / (1 + G).

This is clearly once again a non-linear relationship. We don't have a good feel for how overall productivity goes, other than the notion that large G and small M must be bad. The increase in cost is just the inverse of N, so that relationship is non-linear as well. The two graphs for these quantities, Figures 22.3 and 22.4, show a banded structure similar to that when computing M. And, as M is non-linear in P and D, we have a non-linearity on top of a non-linearity! So trying to infer what N might be as a function of P, D, and G is a difficult leap indeed; the relationship we are trying to intuit is

N = (1 + G [P - (1 - P) D]) / (1 + G).

Most of us would freely admit that we have little or no intuition for N as a function of these three variables. Yet that is implicitly the task we start out with.

The moral of the story is:

Even simple natural phenomena are sometimes non-linear.

When this occurs, our intuition can be weak. In order to understand what is really happening and make intelligent predictions, having a simple mathematical model and some graphical visualization tools can be invaluable.