This chapter presents a series of other important mathematical programming models that arise when some of the basic assumptions of LP are made more or less restrictive. For example, one assumption of LP is that decision variables can take on fractional values such as $X_1=0.33,X_2=1.57,\text{ or }{X}_3=109.4$. Yet a large number of business problems can be solved only if variables have integer values. When an airline decides how many Boeing 757s or Boeing 777s to purchase, it can’t place an order for 5.38 aircraft; it must order 4, 5, 6, 7, or some other integer amount. In this chapter, we present the general topic of integer programming, and we specifically consider the use of special variables that must be either 0 or 1.

Another major limitation of LP is that it forces the decision maker to state one objective only. But what if a business has several objectives? Management may indeed want to maximize profit, but it might also want to maximize market share, maintain full employment, and minimize costs. Many of these goals can be conflicting and difficult to quantify. South States Power and Light, for example, wants to build a nuclear power plant in Taft, Louisiana. Its objectives are to maximize power generated, reliability, and safety and to minimize the cost of operating the system and the environmental effects on the community. Goal programming is an extension to LP that can permit multiple objectives such as these.

Lastly, linear programming can, of course, be applied only to cases in which the constraints and objective function are linear. Yet in many situations, this is not the case. The price of various products, for example, may be a function of the number of units produced. As more units are produced, the price per unit decreases. Hence, an objective function may read as follows:

Because of the squared terms, this is a nonlinear programming problem.

Let’s examine each of these extensions of LP—integer, goal, and nonlinear programming—one at a time.