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CHAPTER

4. CHAPTER. Linear Programming with Two Variables. Section 4.1 p 1. 4.1 Systems of Linear Inequalities. Section 4.1 p 2. 4.1 Systems of Linear Inequalities.

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  1. 4 CHAPTER Linear Programming with Two Variables Section 4.1 p1

  2. 4.1 Systems of Linear Inequalities Section 4.1 p2

  3. 4.1 Systems of Linear Inequalities The graph of a linear inequality in two variables x and y is the set of all points (x, y) for which the inequality is satisfied. Section 4.1 p3

  4. 4.1 Systems of Linear Inequalities An inequality is nonstrictwhen the corresponding line is part of the graph of the inequality. If the inequality is strict, the corresponding line is not part of the graph of the inequality. We will indicate a strict inequality by using dashes to graph the line. For a nonstrict inequality we will use a solid line to graph the line. Section 4.1 p4

  5. 4.1 Systems of Linear Inequalities Steps for Graphing a Linear Inequality STEP 1 Graph the corresponding linear equation, a line L. If the inequality is nonstrict, graph L using a solid line; if the inequality is strict, graph L using dashes. STEP 2 Select a test point P not on the line L. STEP 3 Substitute the coordinates of the test point P into the given inequality. If the coordinates of this point P satisfy the linear inequality, then all points on the same side of L as the point P satisfy the inequality. If the coordinates of the point P do not satisfy the linear inequality, then all points on the opposite side of L from P satisfy the inequality. STEP 4 Shade (or strike) the region to be included. Section 4.1 p5

  6. 4.1 Systems of Linear Inequalities The set of points belonging to the graph of a linear inequality (the shaded region) is called a half-plane. Section 4.1 p6

  7. 4.1 Systems of Linear Inequalities A system of linear inequalities is a collection of two or more linear inequalities. To graph a system of inequalities containing two variables x and y we locate all the points (x, y) whose coordinates satisfy each of the linear inequalities of the system. Since the graph of each linear inequality of the system is a half-plane, the graph of the system of linear inequalities is the intersection of these half-planes. Section 4.1 p7

  8. 4.1 Systems of Linear Inequalities The graph in Figure 11b is said to be unbounded in the sense that it extends infinitely far in some direction. The graph in Figure 11a is bounded in the sense that it can be enclosed by some rectangle of sufficiently large dimensions. FIGURE 11(a) Bounded graph (b) Unbounded graph The point of intersection of two line segments that form the boundary is called a corner point* of the graph. *sometimes called vertex Section 4.1 p8

  9. 4.2 A Geometric Approach to Linear Programming Problems with Two Variables Section 4.2 p9

  10. 4.2 A Geometric Approach to Linear Programming Problems with Two Variables Linear Programming Problem The problem requires that a certain linear expression be maximized or minimized. This linear expression is called the objective function. Furthermore, the problem requires that the linear expression be maximized or minimized under certain restrictions or constraints, each of which are linear inequalities involving the variables. Section 4.2 p10

  11. 4.2 A Geometric Approach to Linear Programming Problems with Two Variables Definition Linear Programming Problem A linear programming problem in two variables, x and y, consists of maximizing or minimizing an objective function z = Ax + By where A and B are given real numbers, not both zero, subject to certain conditions or constraints expressible as a system of linear inequalities in x and y. The points that satisfy all the constraints are called feasible points. Section 4.2 p11

  12. 4.2 A Geometric Approach to Linear Programming Problems with Two Variables By a solution to a linear programming problem we mean a feasible point (x, y), together with the value of the objective function at that point, which maximizes (or minimizes) the objective function. If none of the feasible points maximizes (or minimizes) the objective function, or if there are no feasible points, then the linear programming problem has no solution. Section 4.2 p12

  13. 4.2 A Geometric Approach to Linear Programming Problems with Two Variables Theorem Some Conditions Relating to the Solution of a Linear Programming Problem Consider a linear programming problem with the set R of feasible points and objective function z = Ax + By. 1. If R is the empty set, then the linear programming problem has no solution and z has neither a maximum nor a minimum value. 2. If R is bounded, then z has both a maximum and a minimum value on R. 3. If R is unbounded, A > 0, B > 0, and the constraints include x ≥ 0 and y ≥ 0, then z has a minimum value on R but not a maximum value. Section 4.2 p13

  14. 4.2 A Geometric Approach to Linear Programming Problems with Two Variables Theorem Fundamental Theorem of Linear Programming with Two Variables Consider a linear programming problem with the set R of feasible points and objective function z = Ax + By where x ≥ 0 and y ≥ 0. If a linear programming problem has a solution, it is located at a corner point of the set R of feasible points; if a linear programming problem has multiple solutions, at least one of them is located at a corner point of the set R of feasible points. In either case the corresponding value of the objective function is unique. Section 4.2 p14

  15. 4.2 A Geometric Approach to Linear Programming Problems with Two Variables Steps for Solving a Linear Programming Problem If a linear programming problem has a solution, follow these steps to find it: STEP 1 Write an expression for the quantity that is to be maximized or minimized (the objective function). STEP 2 Determine all the constraints and graph the set of feasible points. STEP 3 List the corner points of the set of feasible points. STEP 4 Determine the value of the objective function at each corner point. STEP 5 Select the maximum or minimum value of the objective function. Section 4.2 p15

  16. 4.3 Models Utilizing Linear Programming with Two Variables Section 4.3 p16

  17. 4.3 Models Utilizing Linear Programming with Two Variables Maximizing Profit Nutt’s Nuts has 75 pounds of cashews and 120 pounds of peanuts available. These are to be mixed in 1-pound packages as follows: a low-grade mixture that contains 4 ounces of cashews and 12 ounces of peanuts and a high-grade mixture that contains 8 ounces of cashews and 8 ounces of peanuts. The profit is $0.25 on each package of the low-grade mixture and $0.45 on each package of the high-grade mixture. How many packages of each type of mixture should be prepared to maximize the profit? Section 4.3 p17

  18. 4.3 Models Utilizing Linear Programming with Two Variables Solution Let x denote the number of packages of the low-grade mixture and y the number of packages of the high-grade mixture. If P denotes the profit, then P = 0.25x + 0.45y STEP 1 We want to maximize P so P = 0.25x + 0.45y is the objective function. STEP 2 The constraints consist of the system of linear inequalities The cashew constraint The peanut constraint The nonnegativity constraint The nonnegativity constraint Section 4.3 p18

  19. 4.3 Models Utilizing Linear Programming with Two Variables Solution (continued) See Figure 20 for the graph of the set of feasible points. Notice that this set is bounded, so the linear programming problem has a solution. STEP 3 The corner points are (0, 0), (0, 150), (160, 0), and (90, 105). Section 4.3 p19

  20. 4.3 Models Utilizing Linear Programming with Two Variables Solution (continued 2) STEP 4 In Table 3 we evaluate the objective function P at each corner point. TABLE 3 A maximum profit is obtained if 90 packages of low-grade mixture and 105 packages of high-grade mixture are made. The maximum profit obtainable under the conditions described is $69.75. Section 4.3 p20

  21. 4.Extra Multiple Choice Questions Select the best answerfor each of the followingmultiple choice questions. Section 4.MC p21

  22. 4.Extra Multiple Choice Questions 1. List the corner points of the graph of the solution set of the following system of inequalities A. (0, 0), (0, 3), (0, 4) and (5, 0) B. (3, 0), (0, 3), (4, 0) and (0, 5) C. (3, 0), (0, 0), (4, 0) and (0, 5) D. (3, 0), (0, 3), (0, 4) and (5, 0) Section 4.MC p22

  23. 4.Extra Multiple Choice Questions 2. Let P = (1, 5), Q = (10, 0), R = (2, 5) and S = (0, 5). Which of these points are in the solution set of the following system of inequalities? A. P, R and S B. Q, R and S C. P and S D. P and Q Section 4.MC p23

  24. 4.Extra Multiple Choice Questions • 3. Apply the graphical solution method to solve the following linear programming problem: Minimize z = 4x + 5y • A. The minimum value of z is 20 at x = 0 and y = 4. • B. The minimum value of z is 26 at x = 4 and y = 2. • C. The minimum value of z is 24 at x = 6 and y = 0. • D. The minimum value of z is 30 at x = 0 and y = 5. Section 4.MC p24

  25. 4.Extra Multiple Choice Questions • 4. Apply the graphical solution method to solve the following linear programming problem: Maximize P = 5x + 4y • A. The maximum value of P is 36 at x = 4 and y = 4. • B. The maximum value of P is 40 at x = 8 and y = 0. • C. The maximum value of P is 32 at x = 0 and y = 8. • D. The maximum value of P is 48 at x = 0 and y = 12. Section 4.MC p25

  26. 4.Extra Multiple Choice Questions Answers for the Multiple Choice Questions 1. D. (3, 0), (0, 3), (0, 4) and (5, 0) 2. C. P and S 3. B. The minimum value of z is 26 at x = 4 and y = 2. 4. A. The maximum value of P is 36 at x = 4 and y = 4. Section 4.MC p26

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