So far, we have analyzed data by building models of the data and then interpreted those models. We have worked with models as equations that take one or more variables as input and have even worked with nonlinear functions. But analyzing the data and building the model is only part of the process. It is important that our model be useful for answering questions about the underlying situation and that we be able to use our model to make decisions. One of the most common uses of a model is in optimization, where we seek to make some quantity (such as profit or cost) either as large as possible (for profit) or as small as possible (for cost). In an earlier chapter, we did this with functions of a single variable, making use of a concept from calculus: the derivative. We found that when the derivative of a function is zero, the function is at a critical point, and that critical points are the only candidates for being optimum values of the function.
But this process ignores two things. The first is that most functions or models have several independent variables. Consider, for example, the commuter rail system examples we have used before. In that case, we built a model with a total of four variables. Our one-variable optimization process won’t work here. The second thing we have ignored is that we are seldom free to choose just any values of the independent variables in order to achieve our optimum results. We are often constrained by resources. These resource constraints could involve time, money, personnel or just about anything that could limit our ability to reach certain values or combinations of values for the independent variables.
To correctly deal with the first problem, multiple-variable functions, we need to use partial derivatives (one for each independent variable) and solve several equations simultaneously. The idea is similar to the one variable case, but we now need all of the partial derivatives to be zero at the same exact point (set of values of the independent variables). We will not be looking into this here, because most of the common multivariable functions, linear and multiplicative, do not have critical points, and so we find no optimum solutions. Instead, we’ll focus on the second aspect of optimization, applying constraints.
To begin, we must learn how to formulate the constraints. Typically, these will take the form of inequalities, rather than equations. After all, if the most you have to spend on production is $100,000 and you can achieve a slightly higher profit by using on $95,000, why not do it? So, rather than forcing our constraints to be equations, where quantities are equal to each other, we will use inequalities, where some quantity is either less than, greater than, less than or equal to, or greater than or equal to, some other quantity. We’ll also see that most optimization problems involve multiple constraint conditions. For example, one constraint may involve time, one might involve cost of raw materials, one might involve equipment, and one might involve distribution.