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The LEGO Furniture Company

The LEGO Furniture Company. Scenario. You manufacture tables and chairs. Tables and chairs are manufactured from small and large bricks. Small brick. Large brick. Furniture Specific Info. Table 2 large bricks 2 small bricks $16 profit. Chair 1 large bricks 2 small bricks $10 profit.

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The LEGO Furniture Company

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  1. The LEGO Furniture Company

  2. Scenario • You manufacture tables and chairs. • Tables and chairs are manufactured from small and large bricks. Small brick Large brick

  3. Furniture Specific Info • Table • 2 large bricks • 2 small bricks • $16 profit • Chair • 1 large bricks • 2 small bricks • $10 profit

  4. Limited Resources: • You have 8 small bricks and 6 large bricks

  5. The Goal • How many tables and how many chairs should be produced to maximize profit?

  6. One possible solution Is this solution optimal? Profit = 3*16 = $48 That is for you to find out!

  7. You BUILD the furniture • GOAL: • Find the combination of tables and chairs that yields the greatest profit GOOD LUCK! Table $16 Chair $10

  8. Linear Programming Vocabulary • Linear Programming: • Method or process used to solve problems that involve using resources most efficiently

  9. Linear Programming Vocabulary • Constraints: • Limitations created by scarce resources (time, $, supplies, equipment)

  10. Furniture Building Activity as a Linear Programming Problem Define variables x – number of tables to produce y – number of chairs to produce

  11. Furniture Building Activity as a Linear Programming Problem (continued) Write the objective function (Remember the OBJECTIVE is to maximize profit.) P(x,y): 16 x + 10 y

  12. Furniture Building Activity as a Linear Programming Problem (continued) Write the constraints x – number of tables to produce y – number of chairs to produce 2x + 1y ≤ 6 Large bricks 2x + 2y ≤ 8 Small bricks x ≥ 0, y ≥ 0 Non-negativity

  13. Graphical Insight 2 x + y ≤ 6 2 x + 2 y ≤ 8

  14. More Linear Programming Vocabulary • Feasible Region: • Set of points that satisfy all the constraints • The shaded region! • Vertices/Corner Points: • Intersection points of the inequality equations (constraints)

  15. More Linear Programming Vocabulary • Optimal Solution: • The outcome of using resources most efficiently to obtain the maximum or minimum. • Occurs at a corner point.

  16. Furniture building optimal solution

  17. Steps to Linear Programming Step 1: Define the variables. Step 2: Write a system of inequalities. Step 3: Graph the system of inequalities. Step 4: Find the coordinates of the vertices of the feasible region. Step 5: Write a function to be maximized or minimized. Step 6: Substitute the coordinates of the vertices into the function. Step 7: Select the greatest or least result. Answer the problem.

  18. Additional Examples The area of a parking lot is 600 square meters. A bus requires 30 square meters. A car requires 6 square meters. The attendant can handle only 60 vehicles. If a bus is charged $7.50 and a car $2.50, how many of each should be accepted to maximize income?

  19. Additional Examples Jerry works no more than 20 hours a week during a school year. He is paid $10 an hour for tutoring geometry students and $7.00 an hour for delivering pizzas for Pizza King. He wants to spend at least 3 hours, but no more than 8 hours a week tutoring. How many hours at each job should he work to make the most money?

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