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

#1: What affects our goal?

How much each process

costs to run.

Min Z (cost) = 4 x1 + 1 x2

#1: What constrains our goal?

S. T.

3 x1 + 1 x2 >= 10

1 x1 + 1 x2 >= 5

1 x1 >= 3

#1: Other constraints?

x1, x2 >= 0

#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

Desks Chairs

Wood 4 3 20

$ $40 $25

#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

LP: Graphical Solutions

• Farming Problem

S. T. x1 + x2 = 45

3 x1 + 2 x2 = 100

2 x1 + 4 x2 = 120

LP: Graphical Solutions

x1 + x2 = 45

3 x1 + 2 x2 = 100

2 x1 + 4 x2 = 120

LP: Graphical Solutions

x1 + x2 = 45

3 x1 + 2 x2 = 100

2 x1 + 4 x2 = 120

LP: Graphical Solutions

x1 + x2 = 45

3 x1 + 2 x2 = 100

2 x1 + 4 x2 = 120

LP: Graphical SolutionsMax Z = 200 x1 + 300 x2

35, 0 = $700

0, 30 = $900

20, 20 = $1000

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

Using MS Excel: 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

SEE MS Excel file here