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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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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
Limited Resources: • You have 8 small bricks and 6 large bricks
The Goal • How many tables and how many chairs should be produced to maximize profit?
One possible solution Is this solution optimal? Profit = 3*16 = $48 That is for you to find out!
You BUILD the furniture • GOAL: • Find the combination of tables and chairs that yields the greatest profit GOOD LUCK! Table $16 Chair $10
Linear Programming Vocabulary • Linear Programming: • Method or process used to solve problems that involve using resources most efficiently
Linear Programming Vocabulary • Constraints: • Limitations created by scarce resources (time, $, supplies, equipment)
Furniture Building Activity as a Linear Programming Problem Define variables x – number of tables to produce y – number of chairs to produce
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
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
Graphical Insight 2 x + y ≤ 6 2 x + 2 y ≤ 8
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)
More Linear Programming Vocabulary • Optimal Solution: • The outcome of using resources most efficiently to obtain the maximum or minimum. • Occurs at a corner point.
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.
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?
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?