CHAPTER 4 Linear Programming with Two Variables

Transcription

1 CHAPTER 4 Linear Programming with Two Variables In this chapter, we will study systems of linear inequalities. They are similar to linear systems of equations, but have inequalitites instead of equalities. We will optimize (maximize or minimize) a linear function under certain conditions, given in the form of linear inequalities. Such problems are called linear programming problems and the corresponding mathematical formulation is called a linear program. In chapter 4, we solve linear programming problems in two variables by graphing. In chapter 5, we will solve linear programming problems with two or more variables using a matrix method. First, we will model linear programming problems, but not solve them. 175

2 176 HELENE PAYNE, FINITE MATHEMATICS Exercise 162. An appliance dealer, Andrea Kurtz, wants to purchase a combined total of no more than 100 refrigerators, and dishwashers for inventory. Refrigerators weigh 200 pound each, and dishwashers weigh 100 pounds each. Suppose that the dealer is limited to a total of 12, 000 pounds for these two items. If a profit of $35 for each refrigerator and $20 on each dishwasher is projected, how many of each should be purchased and sold to make the largest profit? (a) Organize the data given into a table. (b) From the table, formulate the linear programming problem by writing the objective function and constraints.

3 CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 177 The inequalities in the above problem, x + y x + 100y are called structural constraints. There will always be some structural constraints in linear programming problems. The other two inequalities, x 0, y 0 are called non-negativity constraints and they are always added to the structural constraints in linear programming problems. The profit function, P = 35x + 20y, is called the objective function. That is the function we want to optimize, in this case maximize.

4 178 HELENE PAYNE, FINITE MATHEMATICS Exercise 163. Write the mathematical formulation of the linear programming problem, and identify the objective function and all the constraints. A particular salad contains 4 units of vitamin A, 5 units of vitamin B complex, and 2 mg of fat per serving. A nutritious soup contains 6 units of vitamin A, 2 units of vitamin B complex, and 3 mg of fat per serving. If a lunch consisting of these two foods is to have at least 10 units of vitamin A and at least 10 units of vitamin complex, how many servings of each should be used to minimize the total number of milligrams of fat?

5 CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 179 Exercise 164. Write the mathematical formulation of the linear programming problem, and identify the objective function and all the constraints. An investment club has at most $30, 000 to invest in either junk bonds or premium-quality bonds. Each type of bond is bought in $1, 000 denominations. The junk bonds have an average yield of 12%, and the premium-quality bonds yield 7%. The policy if the club is to invest at least twice the amount of money in premium-quality bonds as in junk bonds. How should the club invest its money in these bonds to receive the maximum return on its investment?

6 180 HELENE PAYNE, FINITE MATHEMATICS 4.1. Systems of Linear Inequalities In linear programming problems in two variables, the structural constraints consist of a system of inequalities. Solutions of a system of inequalities, when graphing, corresponds to a shaded region in the plane. In this section, we will learn how to graph a system of linear inequalities, finding which region to shade. In the following section, we will solve linear programming problems graphically. Any linear inequality on two variables has one of the forms ax + by < c, ax + by c, ax + by > c, ax + by c, where a, b, and c are real numbers, where a and b are not both zero. Examples of linear inequalities in two variables are, 3x + 2y 5, x 3y > 7, and y 0. The laws of inequalities are listed below. Laws of Inequalities 1. If a < b and c is any number, then a + c < b + c. 2. If a < b and c is any positive number, then ac < bc. 3. If a < b and c is any negative number, then ac > bc.

7 CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 181 Exercise 165. Solve the inequality 2x y 4 for y. Solutions of a Linear Inequality The solutions of a linear inequality in two variables consists of all ordered pairs (x, y) that, when substituted into the inequality, result in a true statement. Exercise 166. For the inequality in the above problem, y 2x 4. Check which of the following points are solutions to the inequality. (a) (3, 3) (b) (2, 1) (c) (4, 5)

8 182 HELENE PAYNE, FINITE MATHEMATICS Exercise 167. Graph the line y = 2x 4 and the points from the previous exercise, (3, 3), (2, 1), (4, 5). Free Multi-Width Graph Paper from (a) y > 2x 4 represents points above the line. (b) y = 2x 4 represents points on the line. (c) y < 2x 4 represents points below the line. Use this information to shade the area which corresponds to the solutions of the inequality y 2x 4

9 CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 183 Graphing the Solution of a Linear Inequality 1. Replace the inequality symbol with = to obtain the equation of the boundary line. 2. Plot the line represented by the boundary line. If the inequality is or, plot a solid line. If the inequality is > or <, plot the line dashed. 3. Select a test point, not on the boundary line and substitute it into the original inequality. If the resulting inequality is true, shade the side of the line where the test point is located. If the resulting inequality is false, shade side of the line opposite to where the test point is located.

10 184 HELENE PAYNE, FINITE MATHEMATICS Exercise 168. Graph the following linear inequalities carefully following the steps listed above. (a) 3x + y 9 Free Multi-Width Graph Paper from (b) 3x 2y < 12 Free Multi-Width Graph Paper from

11 CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 185 Exercise 169. Graph the solution of the following system of linear inequalities. List all the corner points of the shaded area. x + 2y 4 x y 1 x 0 y 0 Free Multi-Width Graph Paper from

12 186 HELENE PAYNE, FINITE MATHEMATICS The Feasible Region For any system of linear inequalities, the set of points that makes all inequalities true is called the feasible region. When graphing the solution of a system of linear inequalities, we shade the feasible region. Corner Point of a Feasible Region For a feasible region in the plane, defined by a system of linear inequalities, a point of intersection of two boundary lines that is also part of the feasible region is called a corner point or vertex of the feasible region. For any system of linear inequalities, the set of points that makes all inequalities true is called the feasible region. When graphing the solution of a system of linear inequalities, we shade the feasible region.

13 CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 187 Exercise 170. Graph the feasible region given by the system of linear inequalities 2x + y 6 x + 2y 6 x 0 y 0. List all the corner points of the feasible region. Free Multi-Width Graph Paper from

14 188 HELENE PAYNE, FINITE MATHEMATICS 4.2. Solving Linear Programming Problems Graphically In section we will learn how to solve linear programming problems graphically. Consider the linear programming problem, Maximize f = 2x + 3y subject to x + y 9 3x + 2y 24 x + 2y 16 x 0; y 0 Below, we can see what different values of the objective function correspond to graphically on the feasible region arising from the structural constraints.

15 CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 189 f=25 f=24 f=18 f=12 f=6 In the corner point, (0, 8) we have f = 2(0) + 3(8) = 24. There are many points for which f = 24, for example in the point, ( 3 2, 7), which is in the feasible region. In fact, there is a whole line for which f = 24, namely the line 2x+3y = 24. We can also see in the graph that the smaller the values of f, the lower down in the graph the corresponding line is located, and the larger the value of f, the higher up the corresponding line is located. The largest value of the function f on the feasible region is assumed in the corner point (2, 7), where f = 25. This example suggests that if a linear objective function, f has a maximum on the feasible region bounded by the constraints, then such a maximum will occur at a corner point of the feasible region.

16 190 HELENE PAYNE, FINITE MATHEMATICS Feasible regions can be bounded (i.e. completely enclosed) or unbounded. They can also be convex or not convex. Here we will only work with convex feasible regions. unbounded bounded bounded and convex and convex and not convex

17 CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 191 The Fundamental Theorem of Linear Programming If the feasible region for a linear programming problem is nonempty and convex, and if the objective function has a maximum (or minimum) value within that set, then that maximum (or minimum) will always correspond to at least one corner point of the region. For a linear programming problem with a nonempty feasible region, R and an objective function f, 1. If R is bounded, then f has both a minimum and a maximum value at some corner point of R. 2. If R is unbounded and if f has a maximum or a minimum value, on R, then that value will occur at a corner point. However, the nature of f and the shape of R will determine whether a maximum or a minimum exists. The graph below shows a function f = 2x y, which has no maximum value on the unbounded feasible region. 4.3 SOLVING LINEAR PROGRAMMING PROBLEMS GRAPHICALLY 183 v, 8- '7- ),1 14, I IV , 7- v izil 4 5 6'1 8 r I ed x l.(b) bounded and convex (c) bounded and not convex FIGURE 11 Bou n d ed ness o n d Convexity FIGURE 12 An Unbounded Feosible Region The Fundamental Theorem of Linear Programming If the feasible region for a linear programming problem is nonempty and convex, and

18 192 HELENE PAYNE, FINITE MATHEMATICS Exercise 171. Solve the linear programming problem below by following the steps below. Maximize P = 4x + 4y subject to x + 3y 30 2x + y 20 x 0; y 0 (a) Graph the feasible region. (b) Find the coordinates of the corner points. (c) Evaluate P at each corner point, and find the extreme value requested. Free Plain Graph Paper from

19 CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 193 Exercise 172. Solve the linear programming problem below by following the steps below. Minimize C = 5x + 3y subject to x + y 6 6x + y 16 x + 6y 16 x 0; y 0 (a) Graph the feasible region. (b) Find the coordinates of the corner points. (c) Evaluate C at each corner point, and find the extreme value requested. Free Plain Graph Paper from

20 194 HELENE PAYNE, FINITE MATHEMATICS Exercise 173. Solve the linear programming problem below by following the steps below. Minimize C = x 2y subject to x 2 x 4 y 1 y 5 (a) Graph the feasible region. (b) Find the coordinates of the corner points. (c) Evaluate C at each corner point, and find the extreme value requested. Free Plain Graph Paper from

21 CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES Models Utilizing Linear Programming with Two Variables Exercise 174. Two grains, barley and corn, are to be mixed for animal food. Barley contains 1 unit of fat per pound, and corn contains 2 units of fat per pound. The total number of units of fat in the mixture are not to exceed 12 units. No more than 6 pounds of barley and no more than 5 pounds of corn are to be used in the mixture. If barley and corn each contain 1 unit of protein per pound, how many pounds of each grain should be used to maximize the number of units of protein in the mixture? Free Plain Graph Paper from

22 196 HELENE PAYNE, FINITE MATHEMATICS Exercise 175. A sales representative covers territory in Iowa and Kansas. Her daily travel expenses average $120 in Iowa and $100 in Kansas. Her company provides an annual travel allowance of $18, 000. Her company also stipulates that she must spend at least 50 days in Iowa, and 60 days in Kansas per year. If sales average $3, 000 per day in Iowa, and $2, 500 per day in Kansas, how many days should she spend in each state to maximize sales? Free Plain Graph Paper from