SECTION 11.8 Linear Programming 841 Solution The problem is typical of a linear programming problem . The problem requires that a certain linear function, the income, be maximized. If I represents income, x the amount invested in Treasury bills at 2%, and y the amount invested in corporate bonds at 3%, then = + I x y 0.02 0.03 Assume, as before, that I, x, and y are in hundreds of dollars. The linear function = + I x y 0.02 0.03 is called the objective function . Further, the problem requires that the maximum income be achieved under certain conditions, or constraints , each of which is a linear inequality involving the variables. (See Example 12 in Section 11.7.) The linear programming problem is modeled as = + I x y Maximize 0.02 0.03 subject to the constraints ≥ ≥ + ≤ ≥ ≤ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ x y x y x y 0 0 25 15 5 In general, every linear programming problem has two components: • A linear objective function that is to be maximized or minimized • A collection of linear inequalities that must be satisfied simultaneously DEFINITION Linear Programming Problem A linear programming problem in two variables x and y consists of maximizing (or minimizing) a linear objective function = + z Ax By A B and are real numbers, not both 0 subject to certain constraints, or conditions, expressible as linear inequalities in x and y. 2 Solve a Linear Programming Problem To maximize (or minimize) the quantity = + z Ax By, we need to identify points ( ) x y , that make the expression for z the largest (or smallest) possible. But not all points ( ) x y , are eligible; only those that also satisfy each linear inequality (constraint) can be used. Each point ( ) x y , that satisfies the system of linear inequalities (the constraints) is a feasible point . Linear programming problems seek the feasible point(s) that maximizes (or minimizes) the objective function. Look again at the linear programming problem in Example 1. Analyzing a Linear Programming Problem Consider the linear programming problem = + I x y Maximize 0.02 0.03 subject to the constraints ≥ ≥ + ≤ ≥ ≤ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ x y x y x y 0 0 25 15 5 Graph the constraints. Then graph the objective function for = I 0, 0.3, 0.45, 0.55, and 0.6. EXAMPLE 2 (continued)
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