1. Chapter 12 Section 12.1 The Geometry of Linear Programming

2. 123456 1 2 3 4 5 6 7 8 9 10 11 12 Linear Programming A linear programming problem solves what the maximum or minimum value for an linear objective function is inside or on the border of a feasible solution (sometimes called constraints). One of the results of theoretical mathematics tells us that the maximum and minimum value of a linear objective function must occur at one of the corner points. Maximize T = 6x - 2y Subject to: x =0 y =0 x =4 Corner Point T = 6 x – 2 y (0,0)6(0)-2(0)=0-0=0 (0,9)6(0)-2(9)=0-18=-18 (4,0)6(4)-2(0)=24-0=24 (4,3)6(4)-2(3)=24-6=18 MAX MIN

3. Using Resources Efficiently In order to make the best use of resources that a company or individual has a linear programming approach is sometimes used. Consider the following example. A person who enjoys woodworking makes patio tables and chairs in their spare time. The can only do woodworking for 10 hours a week. They can only afford $100 in wood supplies a week. They spend 2 hours making a table and 5 hours making a chair. A table uses $40 in supplies while chairs only use $10. A table sells for a profit of $30 and a chair for a profit $20. Write this as a linear programming problem. 1. Let x = number of tables made and y = number of chairs made each week. 2. Then x > 0 and y > 0 3. The number of hours work on tables and chairs is 2 x +5 y that means 2 x +5 y <10 4. The amount of money he spends is 40 x +10 y that means 40 x +10 y <100 5. The amount of money he gets is 30 x +20 y that means T = 30 x +20 y Maximize T = 30 x +20 y subject to:

4. In order to solve the linear programming problem to the right we do the following 3 steps: 1. Graph the region given by the constraints and locate the corner points. 2. Plug all corner points into the objective function. 3. Select which corner point gives the maximum or minimum value depending on what is wanted. Maximize T = 30 x +20 y subject to: 123456 1 2 3 4 5 6 7 8 9 10 11 12 Find the point of intersection of the two lines. Locate all other corner points and fill in values for objective function: Corner Point T = 30 x + 20 y Maximum

5. Example Roger runs his own side business of detailing cars and vans during the evenings. He only has enough time to detail 10 vehicles each week. Each car he details uses 11 dollars worth of cleaning supplies and each van uses 16 dollars of cleaning supplies. He only can spend at most 125 dollars each week on cleaning supplies. Each car he details earns a profit of 28 dollars and each van a profit of 32 dollars. Formulate the above situation as a Linear Programming problem. Let: x = Number of cars detailed in 1 week y = Number of vans detailed in 1 week He only has enough time to detail 10 vehicles each week. He only can spend at most 125 dollars each week on cleaning supplies. Negative numbers do not make sense here. A profit of 15 dollars and each van a profit of 18 dollars. This is written in the following way as a linear programming problem: Maximize: T =28 x +32 y subject to:

6. To solve the linear the linear programming problem to the right (from previous slide). 1. Graph the region described by the inequalities (i.e. this is the feasible region). 2. Find all the corner points (i.e. where the boundary lines intersect). 3. Plug each corner point into the objective function to see what is maximum. Maximize the value T for T =28 x +32 y subject to the constraints: Finding Corner Points Corner Points T =28 x +32 y Maximum