Page 860 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Geometric Solution of Linear Programming Problems (1)

Each of the preceding three examples is a special case of the following problem.
Problem Find values of and that either maximize or minimize

subject to

(2)

and

(3)

In each of the m conditions of 2, any one of the symbols ≤, ≥, and = may be used.

The problem above is called the general linear programming problem in two variables. The linear function z in 1 is called the
objective function. Equations 2 and 3 are called the constraints; in particular, the equations in 3 are called the nonnegativity
constraints on the variables and .

We shall now show how to solve a linear programming problem in two variables graphically. A pair of values that satisfy

all of the constraints is called a feasible solution. The set of all feasible solutions determines a subset of the -plane called the

feasible region. Our desire is to find a feasible solution that maximizes the objective function. Such a solution is called an optimal

solution.

To examine the feasible region of a linear programming problem, let us note that each constraint of the form

defines a line in the -plane, whereas each constraint of the form

defines a half-plane that includes its boundary line

Thus the feasible region is always an intersection of finitely many lines and half-planes. For example, the four constraints

of Example 1 define the half-planes illustrated in parts (a), (b), (c), and (d) of Figure 11.3.1. The feasible region of this problem is
thus the intersection of these four half-planes, which is illustrated in Figure 11.3.1e.