Chapter 16
Optimization in Several Variables with Constraints¹

In a previous chapter, you explored the idea of slope (rate of change, also known as the derivative) and applied it to locating maxima and minima of a function of one variable (the process was referred to as optimization). However, we know that most functions that model real world data are composed of several variables, so we need slightly different techniques for this. If you recall the one-variable case, we only needed to set that derivative to zero to find the local maxima and minima. When there are n independent variables, there are n different partial derivatives. We can find the location of the maxima and minima by find the points at which all n of these derivatives are zero at the same time (simultaneously). This involves a great deal of algebra, and is not always possible to do without resorting to numerical methods that only find approximate locations.

To make matters worse, we also find that rarely are we optimizing a function by itself. Consider, for example, revenue for selling a certain number of products. The more you sell, the more you earn, so there is no maximum revenue; we can make as many as we want and still earn more revenue. But in the real world, we have to account for the cost of the objects we are selling, which includes raw materials, labor and equipment to produce them, marketing, distribution, and other costs. These extra conditions, known as constraints, make finding an optimum solution much more difficult. In this chapter, we will focus on defining such constraints and phrasing them mathematically. We will then see how to set up a spreadsheet to solve the optimization problem under these constraints.