1. 1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 8 Systems of Equations and Inequalities

2. OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Linear Programming State linear programming problems. Solve linear programming problems. Use linear programming in applications. SECTION 8.5 1 2 3

3. 3 © 2010 Pearson Education, Inc. All rights reserved Definitions If a function f with domain [a, b] has a largest and a smallest value, the largest value is called the maximum value and the smallest value is called the minimum value. The process of finding a maximum or minimum value of a quantity is called optimization.

4. 4 © 2010 Pearson Education, Inc. All rights reserved A linear programming problem satisfies the following two conditions: 1.The quantity f to be maximized or minimized can be written as a linear expression in x and y. That is, f = ax + by, where a ≠ 0, b ≠ 0 are constants. 2.The domain of the variables x and y is restricted to a region S that is determined by (is a solution set of) a system of linear inequalities. Definitions

5. 5 © 2010 Pearson Education, Inc. All rights reserved The inequalities that determine the region S are called constraints, the region S is called the set of feasible solutions, and f = ax + by is called the objective function. A point in S at which f attains its maximum (or minimum) value, together with the value of f at that point, is called an optimal solution. Definitions

6. 6 © 2010 Pearson Education, Inc. All rights reserved 1.If a linear programming problem has a solution, it must occur at one of the vertices of the feasible solution set. 2. A linear programming problem may have many solutions, but at least one of them occurs at a vertex of the feasible solution set. 3. In any case, the value of the objective function is unique. SOLUTION OF A LINEAR PROGRAMMING PROBLEM

7. 7 © 2010 Pearson Education, Inc. All rights reserved OBJECTIVE Solve a linear programming problem. Step 1 Write an expression for the quantity to be maximized or minimized. This expression is the objective function. Step 2 Write all constraints as linear inequalities. 7 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving a Linear Programming Problem EXAMPLE Maximize f = 2x + 3y subject to the constraints 1 and 2. as noted above

8. 8 © 2010 Pearson Education, Inc. All rights reserved OBJECTIVE Solve a linear programming problem. Step 3 Graph the solution set of the constraint inequalities. This set is the set of feasible solutions. 8 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving a Linear Programming Problem EXAMPLE Maximize f = 2x + 3y subject to the constraints 3.

9. 9 © 2010 Pearson Education, Inc. All rights reserved OBJECTIVE Solve a linear programming problem. Step 4 Find all vertices of the solution set in Step 3 by solving all pairs of equations corresponding to the constraint inequalities. 9 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving a Linear Programming Problem EXAMPLE Maximize f = 2x + 3y subject to the constraints 4. Solve to find A(0, 0), to find B(5, 0), to find C(0, 3).

10. 10 © 2010 Pearson Education, Inc. All rights reserved OBJECTIVE Solve a linear programming problem. Step 5 Find the values of the objective function at each of the vertices of Step 4. 10 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving a Linear Programming Problem EXAMPLE Maximize f = 2x + 3y subject to the constraints 5. Solve At A(0, 0), f = 2(0) + 3(0) = 0 At B(5, 0), f = 2(5) + 3(0) = 10 At C(0, 3), f = 2(0) + 3(3) = 9

11. 11 © 2010 Pearson Education, Inc. All rights reserved OBJECTIVE Solve a linear programming problem. Step 6 The largest of the values (if any) in Step 5 is the maximum value of the objective function, and the smallest of the values (if any) in Step 5 is the minimum value of the objective function. 11 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Solving a Linear Programming Problem EXAMPLE Maximize f = 2x + 3y subject to the constraints 6. At B(5, 0) the objective function f has a maximum value of 10. Although you are not asked for the minimum value, it is 0 at A(0, 0)

12. 12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Solving a Linear Programming Problem Having a Nonunique Solution Minimize f = 60x + 40y subject to the constraints x + y ≤ 8, x + 2y ≥ 4, 3x + 2y ≥ 6, x ≥ 0, y ≥ 0. Solution 1. objective function: f = 60x + 40y 2. constraints:

13. 13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Solving a Linear Programming Problem Having a Nonunique Solution Solution continued 3. 4. Solving the pairs of equations corresponding to the constraint inequalities, we get the coordinates of the vertices.

14. 14 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Solving a Linear Programming Problem Having a Nonunique Solution Solution continued 5.

15. 15 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Solving a Linear Programming Problem Having a Nonunique Solution Solution continued 6. We observe from the table that the minimum value of f is 120 and that it occurs at two vertices, D and E. In fact, the unique minimum value occurs at every point on the line segment DE. So the linear programming problem has infinitely many solutions, each of which satisfies 3x + 2y = 6.

16. 16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Nutrition; Minimizing Calories Fat Albert wants to go on a crash diet and needs your help in designing a lunch menu. The menu is to include two items: soup and salad. The vitamin units (milligrams) and calorie counts in each ounce of soup and salad are given in the table. ItemVitamin AVitamin CCalories Soup Salad 1111 3232 50 40

17. 17 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Nutrition; Minimizing Calories Find the number of ounces of each item in the menu needed to provide the required vitamins with the fewest number of calories. The menu must provide at least: 10 units of Vitamin A 24 units of Vitamin C Solution a. State the problem mathematically. Step 1Write the objective function. x = ounces of soup y = ounces of salad

18. 18 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Nutrition; Minimizing Calories Minimize the total number of calories. Solution continued Step 2Write the constraints. 1 ounce soup provides 1 unit vitamin A 1 ounce salad provides 1 unit vitamin A So the total vitamin A is x + y and this must be at least 10 units so x + y ≥ 10. 50 calories per ounce of soup: 50x 40 calories per ounce of salad: 40y Total calories, f = 50x + 40y

19. 19 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Nutrition; Minimizing Calories Solution continued Step 2 continued 1 ounce soup provides 3 units vitamin C 1 ounce salad provides 2 units vitamin C So the total vitamin C is 3x + 2y and this must be at least 24 units so 3x + 2y ≥ 24. x and y cannot be negative so x ≥ 0 and y ≥ 0.

20. 20 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Nutrition; Minimizing Calories Solution continued Find x and y such that the value of f = 50x + 40y is a minimum, with the restrictions b. Solve the linear programming problem. Summarize the information.

21. 21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Nutrition; Minimizing Calories Solution continued Step 3Graph the set of feasible solutions. The set is bounded by

22. 22 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Nutrition; Minimizing Calories Solution continued Step 4Find the vertices. The vertices of the feasible solutions are A(10, 0), B(4, 6) and C(0, 12). A (10, 0) is obtained by solvingB (4, 6) is obtained by solvingC (0, 12) is obtained by solving

23. 23 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Nutrition; Minimizing Calories Solution continued Step 5Find the value of f at the vertices. Vertex (x, y)Value of f = 50x + 40y (10, 0) (4, 6) (0, 12) 50(10) + 40(0) = 500 50(4) + 40(6) = 440 50(0) + 40(12) = 480 Step 6Find the maximum or minimum value of f. The smallest value of f is 440, which occurs when x = 4 and y = 6.

24. 24 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Nutrition; Minimizing Calories Solution continued c. State the conclusion The lunch menu for Fat Albert should contain 4 ounces of soup and 6 ounces of salad. His intake of 440 calories will be as small as possible under the given constraints.