Finite Mathematics MAT 141: Chapter 3 Notes

Transcription

1 Finite Mathematics MAT 141: Chapter 3 Notes Linear Programming David J. Gisch Graphing Linear Inequalities Linear Inequalities Graphing with Intercepts Find the -intercept. Substitute 0and solve for. Find the y-intercept. Substitute x 0and solve for. Example: Graph the equation using the intercepts

2 Graphing with Intercepts Example: Graph the equation using the intercepts ,000 Graphing with Intercepts Example: Graph the equation using the intercepts Inequalities Graph the equation using the intercepts Inequalities Graph the equation using the intercepts

3 Which way do you Shade 2 Methods. 1. Test a point. Substitute a point (not on the line) into the equation. If it results in a true inequality shade that side, if not, shade the other side. Solid Line or Dotted Line? If there is an equals sign then it is a solid line. If there is not an equals sign then it is a dotted line. 2. Solve for Y and follow the sign., means you shade up, means you shade down Graphing with Intercepts Example: Graph the inequality Systems of Linear Inequalities You graph all of the inequalities and where all of the shaded regions overlap is the solution. 3

4 Systems of Linear Inequalities 1 Graph Linear Inequalities Example: Graph the system of inequalities Linear Inequalities Example: Graph the system of inequalities Linear Inequalities Example: Graph the system of inequalities

5 Feasible Region Example: The Gillette company produces two popular electric razors, the M3Power and the Fusion Power. Due to demand, the number of M3Power razors is never more than half the number of Fusion Power razors. The factories production cannot exceed more than 800 razors per day. (a) Write a system of inequalities to express the conditions of the Gillette Company. (b) Graph the feasible region. Feasible Region As briefly mentioned in the previous section, the graph of a system of linear inequalities is called referred to as a feasible region. All the points in that region are scenarios that meet the limitations of our constraints. Solving Linear Programming Problems Graphically 5

6 Linear Programming Linear Programming is a method for solving problems in which a particular quantity that must be maximized or minimized is limited by other factors, called constraints. An objective function is an algebraic expression in two or more variables describing a quantity that must be maximized or minimized. Equations with Multiple Variables If an equation has two variables it is an equation that can be graphed in the plane, hence 2D. If the variables have no powers then it is a linear equation. Example, 5 10or 5 10 If an equation has three variables it is an equation that can be graphed in space, hence 3D. If the variables have no powers then it is a plane (flat surface). Example, z 3 12 or, 3 12 Objective Function Example: Bottled water and medical supplies are to be shipped to victims of an earthquake by plane. Each container of bottled water will serve 10 people and each medical kit will aid 6 people. Let x represent the number of bottles of water to be shipped and y the number of medical kits. Write the objective function that describes the number of people that can be helped. Constraints Example: Each plane can carry no more than 80,000 pounds. The bottled water weighs 20 pounds per container and each medical kit weighs 10 pounds. Let x represent the number of bottles of water to be shipped and y the number of medical kits. (a) Write an inequality that describes this constraint. (b) Are there any other constraints? 6

7 Feasible Region Example: Graph the feasible region from example 2. Objective Functions and Feasible Regions The feasible region is the inputs, that fit our constraints, for the objective function. The lowest point (minimum) or highest point (maximum) of the graph of the objective function will be at a corner. Feasible Region Example: What are the corner points of the region in example 2? Check the Points Corner Points Objective Function, 0, , , , , ,000 While it seems odd, we can help the most people if we ship zero containers of water and 8,000 medical kits. Of course if we wanted to include some minimum amount of water that would add another constraint and change our feasible region. 7

8 Steps for Linear Programming Applications of Linear Programming Applications of Linear Programming Did You Know? In the last section we had few constraints and therefore the corner points were easy to find. We now will look at having several constraints and using systems of equations to find the corner points. 8

9 Example: Flowers Unlimited has two spring floral arrangements, the Easter Bouquet and the Spring Bouquet. The Easter Bouquet requires 10 jonquils and 20 daisies and produces a profit of $1.50. The Spring Bouquet requires 5 jonquils and 20 daisies and yields a profit of $1. How many of each type of arrangements should the florist make to maximize the profit if 120 jonquils and 300 daisies are available? (b) Write down the constraints (c) Graph the feasible region. (a) Write an objective function., (d) Find the corner points. (e) Substitute the corner points into the objective function. 9

10 Example: A construction company needs to hire at least 100 employees for a project. They will need at least 30 more unskilled laborers than skilled laborers. At least 20 skilled laborers should be hired. The unskilled laborers earn $8 per hour, and the skilled laborer earns $15 per hour. How many employees should the company hire to minimize its hourly cost while satisfying all of the requirements? (b) Write down the constraints (c) Graph the feasible region. (a) Write an objective functions., 8 15 (d) Find the corner points. (e) Substitute the corner points into the objective function. 10

11 Example: Certain animals at a rescue shelter must have at least 30 g of protein and at least 20 g of fat per feeding period. These nutrients come from food A, which cost 18 cents per unit and supplies 2 g of protein and 4 g of fat; and food B, which cost 12 cents per unit and supplies 6 g of protein and 2 g of fat. Food B is bought under a long term contract requiring at least 2 units of B be used per serving. How much of each type of food must be bought to minimize the cost per serving. (b) Write down the constraints (c) Graph the feasible region. (a) Write an objective functions., (d) Find the corner points. (e) Substitute the corner points into the objective function. 11

12 Example: At the end of every month, after filling orders for its regular customers, a coffee company has some pure Colombian coffee and some special-blend coffee remaining. The practice of the company has been to package a mixture of the two coffees into 1-pound packages as follows: a low-grade mixture containing 4 ounces of Colombian coffee and 12 ounces of special-blend coffee and a high-grade mixture containing 8 ounces of Colombian and 8 ounces of special-blend coffee. A profit of $0.30 per package is made on the low-grade mixture, whereas a profit of $0.40 per package is made on the highgrade mixture. This month, 120 pounds of special-blend coffee and 100 pounds of pure Colombian coffee remain. How many packages of each mixture should be prepared to achieve a maximum profit? Assume that all packages prepared can be sold. (b) Write down the constraints (c) Graph the feasible region. (a) Write an objective functions., (d) Find the corner points. (e) Substitute the corner points into the objective function. Insurance Example: A company is considering two insurance premium plans with the types of coverage and premiums shown in the following figure. *This means $50 buys you one unit of Policy A, giving you $10,00 fire/theft coverage and $180,000 liability coverage. The company wants at least $300,000 fire/theft insurance and at least $3,000,000 liability insurance from these plans. How many of each type should they buy to minimize the cost? (a) Write an objective functions.,

13 (b) Write down the constraints ,000 15, ,000 / 180, ,000 3,000,000 (d) Find the corner points. (c) Graph the feasible region. (e) Substitute the corner points into the objective function. Insurance Example: The Ric Shaw Chair company makes two types of rocking chairs, a plain chair and a fancy chair. Each rocking chair must be assembled and then finished. The plain chair takes 4 hours to assemble and 4 hours to finish. The fancy chair takes 8 hours to assemble and 12 hours to finish. The company can provide at most 160 worker-hours of assembly and 180 worker-hours of assembly per day. If the profit on a plain chair is $25 and the profit on a fancy chair is $40, how many of each type should the company make to maximize profit? (a) Write an objective functions.,

14 (b) Write down the constraints (c) Graph the feasible region. (d) Find the corner points. (e) Substitute the corner points into the objective function. 14