Linear Programming

1. Linear Programming U3.5

2. Linear Programming • The process (technique) of taking linear inequalities relating to some situation, and finding the “best” value obtainable under those conditions. • A typical example would be taking the limitations of materials and labor, then determining the • "best" production levels for maximal profits under those conditions. What is Linear Programming?

3. In "real life", linear programming is part of a very important area of mathematics called "optimization techniques". • This field of study (or at least the applied results of it) are used every day in the organization and allocation of resources. These "real life" systems can have dozens or hundreds of variables, or more. • In algebra, though, you'll only work with the simple (and graph-able) two-variable linear case. What is Linear Programming?

4. Objective Function • A function that describes the goal of the problem. Also called the optimization equation. • It might describe the profits a company earns or the cost associated with a certain manufacturing process. • Constraints • System of linear inequalities that bound the shaded or feasible region. • Constraints may not be stated but must make sense to the situation. • Example: You can’t produce negative products. • They represent bounds on such things as labor and material requirements. Vocabulary

5. Feasible Region • The shaded region on the graph that is in the shape of a polygon. The polygon is bounded by the constraints. • The feasibility region shows all the possible points for which the objective function makes sense. • Boundary Point • The points of intersection of the constraints. • The “vertices” of the shaded region • The optimal value of the objective function, if there is one, will occur at a vertex of the feasible region. Vocabulary

6. Maximum Value • Boundary point of the feasible region yielding the largest solution of the objective function. • Minimum Value • Boundary point of the feasible region yielding the smallest solution of the objective function. Vocabulary

7. Write each inequality (constraint) in graphing form (Remember to define your variables) • Graph each constraint on the same graph and shade. • Identify (shade) the Feasibility Region. • Identify the Boundary points (vertices/corner points) of the feasibility region. • Identify or Write the objective function • Evaluate the boundary points in the objective function (Plug it in plug it in) to find the Maximum/Minimum optimal value. • State your solution. Steps to Linear Programming

8. Step 1: Write in graphing form Given the objective function z = 3x + 4y, find the maximal and minimal value of the problem with the following constraints. Example 1 Step 2: Graph each constraint Step 3: Identify the feasibility region

9. Step 3: Identify the boundary points Given the objective function z = 3x + 4y, find the maximal and minimal value of the problem with the following constraints. (2,6) (6,4) Example 1 • To find the boundary points -- which aren't always clear from the graph, pair the lines (thus forming asystem of equations) and solve using substitution, elimination or graphing: (-1,-3) So the boundary points are (2, 6), (6, 4), and (–1, –3).

10. Step 3: Evaluate the boundary points • To evaluate the boundary points in the objective function z = 3x + 4y. Given the objective function z = 3x + 4y, find the maximal and minimal value of the problem with the following constraints. (2, 6):      z = 3(2)   + 4(6)   =   6 + 24 =   30 (6, 4):      z = 3(6)   + 4(4)   = 18 + 16 =   34 (–1, –3):   z = 3(–1) + 4(–3) = –3 – 12 = –15 To find the Maximum or Minimum compare the results and locate the point that produces the max or min. Example 1 Step 4: State your solution. The maximum of z = 34 occurs at (6, 4), The minimum of z = –15 occurs at (–1, –3).

11. Given the objective function z = -x + 3y, find the maximal and minimal value of the problem with the following constraints. Example 2 • Find the boundary points of the feasible region

12. Find the boundary point that that maximizes and minimizes the feasible region. • What was the maximum value we found? • What was the minimum value we found? • Give your solution to the problem. • The maximum value is 30 and occurs at the point (6,12) • The minimum value is -18 and occurs at the point (6,-4) Example 2

13. Now we will go through the entire process including writing the constraints and objective functions. Here we go…

14. A bakery makes cakes and cookies. Each cake requires preparation time, baking time and icing time. Each process is only available for 8 hours each day. The table below gives the information about process times and expected profits from each cake and each dozen of cookies. How many cakes and dozens of cookies should be produced each day to maximize the profit. Example 3

15. 1. Define Variables: Follow the Process for Linear Programming Example 3 1. Define Constraints:

16. Graph Constraints Shade the Feasibility Region : (0,0), (0,32), (10.91, 26.18), (24,0) Example 3 4. Identify the boundary points:

17. 5. DefineThe Objective Function: 6. Evaluate your Boundary points in the objective function: You can’t have 10.91 cakes or 26.18 dozen cookies so round and make sure your new point is still in the feasible region. If so you have to plug those values back in to get the actual profit. • 10(0) + 7(0) = $0.00 • 10(0) + 7(32) = $224.00 Example 3 • 10(10.91) + 7(26.18) = $292.36 • 10(24) + 7(0) = $240.00 Conclusion: The Maximum profit of $292 will occur when the bakery makes 11 cakes and 26 dozen cookies.

18. Absent Students