Transcript QM B Lego Simplex

The LEGO Furniture Company
Scenario
• You manufacture
tables and chairs.
Small brick
• Tables and chairs
are manufactured
from small and
large bricks.
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
2x + 2y ≤ 8
x ≥ 0, y ≥ 0
Large bricks
Small bricks
Non-negativity
Graphical Insight
Lego
7
2x+y≤6
6
Chairs
5
4
Small Bricks
3
Large Bricks
2x+2y≤8
2
1
0
0
1
2
3
Tables
4
5
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.
Furniture building optimal solution
Corner Point
Coordinates (x,y)
P(x,y) = 16x +10y
(0,4)
$40
(3,0)
$48
(2,2)
$32 + $20 = $52
(0,0)
$0
Steps to Linear Programming
Step
Step
Step
Step
1: Define the variables.
2: Write a system of inequalities.
3: Graph the system of inequalities.
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?