Optimizing Behavior on the part of a decision maker involves trying to maximize or minimize an *objective function*. For a manager of a firm, the **objective function** is usually profit, which is to be maximized. For a consumer, the objective function is the satisfaction derived from consumption of goods, which is to be maximized. For a city manager seeking to provide adequate law enforcement services, the objective function might be cost, which is to be minimized. For the manager of the marketing division of a large corporation, the objective function is usually sales, which are to be maximized. The objective function measures whatever it is that the particular decision maker wishes to either maximize or minimize.

If the decision maker seeks to maximize an objective function, the optimization problem is called a **maximization problem**. Alternatively, if the objective function is to be minimized, the optimization problem is called a **minimization problem**. As a general rule, when the objective function measures a benefit, the decision maker seeks to maximize this benefit and is solving a maximization problem. When the objective function measures a cost, the decision maker seeks to minimize this cost and is solving a minimization problem.

The value of the objective function is determined by the level of one or **more activities **or** choice variables**. For example, the value of profit depends on the number of units of output produced and sold. The production of units of the good is the activity that determines the value of the objective function, which in this case is profit. The decision maker controls the value of the objective function by choosing the levels of the activities or choice variables.

The choice variables in the optimization problems discussed in this text will at times vary *discretely* and at other times vary *continuously*. A** discrete choice variable** can take on only specified integer values, such as 1, 2, 3, . . . , or 10, 20, 30, . . . Examples of discrete choice variables arise when benefit and cost data are presented in a table, where each row represents one value of the choice variable. In this text, all examples of discrete choice variables will be presented in tables. A **continuous choice variable** can take on any value between two end points. For example, a continuous variable that can vary between 0 and 10 can take on the value 2, 2.345, 7.9, 8.999, or any one of the infinite number of values between the two limits. Examples of continuous choice variables are usually presented graphically but are sometimes shown by equations. As it turns out, the optimization rules differ only slightly in the discrete and continuous cases.

In addition to being categorized as either maximization or minimization problems, optimization problems are also categorized according to whether the decision maker can choose the values of the choice variables in the objective function from an unconstrained or constrained set of values. **Unconstrained optimization** problems occur when a decision maker can choose *any* level of activity he or she wishes in order to maximize the objective function. We show how to solve only unconstrained *maximization* problems since all the *unconstrained* decision problems we address in this text are maximization problems. **Constrained optimization** problems involve choosing the levels of two or more activities that maximize or minimize the objective function subject to an additional requirement or constraint that restricts the values of A and B that can be chosen. An example of such a constraint arises when the total cost of the chosen activity levels must equal a specified constraint on cost. In this text, we examine both constrained maximization and constrained minimization problems.

As we show later, the constrained maximization and the constrained minimization problems have one simple rule for the solution. Therefore, you will only have one rule to learn for all constrained optimization problems.

Even though there are a huge number of possible maximizing or minimizing decisions, you will see that all optimization problems can be solved using the single analytical technique: *marginal analysis*. **Marginal analysis** involves changing the value(s) of the choice variable(s) by a small amount to see if the objective function can be further increased (in the case of maximization problems) or further decreased (in the case of minimization problems). If so, the manager continues to make incremental adjustments in the choice variables until no further improvements are possible. Marginal analysis leads to two simple rules for solving optimization problems, one for unconstrained decisions and one for constrained decisions. We turn first to the unconstrained decision.