9/1 More Linear Programming
• Collect homework
• Roll call
• Review homework
• Lecture - More LP
• Small Groups
• Lecture - Start using MS Excel
• Assign Homework
#1: Leary Chemical• 3 chemicals: A, B, & C.
• 2 production processes: 1 and 2.
• Process 1 per hour: $4, makes 3 A, 1 B, & 1 C.
• Process 2 per hour: $1, yields 1 A & 1 B.
• Must produce 10 A, 5 B, 3 C daily.
• LP to minimize the cost of production.
#1: Leary Chemical• Reorganize the information:
Process 1 Process 2 Req.
A 3 units 1 10
B 1 1 5
C 1 0 3
$ $4 $1
#2: Furnco
...desks and chairs. Each desk: 4 units wood. Each chair: 3 units wood. Each desk contributes $40 to profit, and a chair contributes $25. Must have # chairs at least twice # desks produced. There are 20 units of wood available.
#2: Furnco
• Max Z (profit) = 40 x1 + 25 x2
• S. T. 4 x1 + 3 x2 <= 20
2x1 - x2 <= 0
x1, x2 >= 0
x1, x2 must be integers
#3: Farming • 45 acres. Each acre planted with wheat
yields $200 profit; corn - $300.
• Maximize the profit.
Wheat Corn Available
Labor 3 2 100Fertilizer 2 4 120
#3: Farming Max Z (profit) = 200 x1 + 300 x2
S. T. x1 + x2 <= 45
3 x1 + 2 x2 <= 100
2 x1 + 4 x2 <= 120
x1, x2 >= 0
LP: Graphical Solutions
• Farming Problem
S. T. x1 + x2 <= 45
3 x1 + 2 x2 <= 100
2 x1 + 4 x2 <= 120
x1, x2 >= 0
Small Group Exercise - Graphical Solution
Chemical Co.: must make 1,000 lbs. of a mixture of phosphate & potassium for a customer. Costs: $5/lb. for phosphate, $6/lb. potassium.
Can use no more than 300 lbs. Phosphate. Must use at least 150 lbs. Potassium. Minimize the cost.
Small Group Exercise - Graphical Solution
What are we optimizing? THE COST (MIN)
What affects that? THE PRICE OF Potassium and Phosphate
What constraints are there? HOW MUCH WE CAN USE/ HOW MUCH WE HAVE TO USE
Small Group Exercise - Graphical Solution
Min Z (cost) = 5 x1 + 6 x2
S.T. x1+ x2 = 1000
x1 <= 300
x2 >= 150
x1, x2 >= 0
Min Z (cost) = 5 x1 + 6 x2
S.T. x1+ x2 = 1000x1 <= 300
x2 >= 150
x1, x2 >= 0
0, 1000 = $6000
300, 700 = $5700