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:

#1: What affects our goal? How much each process costs to run. Min Z (cost) = 4 x x 2

#1: What constrains our goal? S. T. 3 x x 2 >= 10 1 x x 2 >= 5 1 x 1 >= 3

#1: Other constraints? x 1, x 2 >= 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

Max Z (profit) = 40 x x 2 S. T. 4 x x 2 <= 20 2x 1 - x 2 <= 0 x 1, x 2 >= 0 x 1, x 2 must be integers

#3: Farming 45 acres. Each acre planted with wheat yields $200 profit; corn - $300. Maximize the profit.

#3: Farming Max Z (profit) = 200 x x 2 S. T. x 1 + x 2 <= 45 3 x x 2 <= x x 2 <= 120 x 1, x 2 >= 0

LP: Graphical Solutions Farming Problem S. T. x 1 + x 2 <= 45 3 x x 2 <= x x 2 <= 120 x 1, x 2 >= 0

LP: Graphical Solutions Farming Problem S. T. x 1 + x 2 = 45 3 x x 2 = x x 2 = 120

LP: Graphical Solutions x 1 + x 2 = 45 3 x x 2 = x x 2 = 120

LP: Graphical Solutions x 1 + x 2 = 45 3 x x 2 = x x 2 = 120

LP: Graphical Solutions x 1 + x 2 = 45 3 x x 2 = x x 2 = 120

LP: Graphical Solutions Max Z = 200 x 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 x x 2 S.T. x 1 + x 2 = 1000 x 1 <= 300 x 2 >= 150 x 1, x 2 >= 0

Min Z (cost) = 5 x x 2 S.T. x 1 + x 2 = 1000 x 1 <= 300 x 2 >= 150 x 1, x 2 >= 0 0, 1000 = $ , 700 = $5700

Using MS Excel: Farming Max Z (profit) = 200 x x 2 S. T. x 1 + x 2 <= 45 3 x x 2 <= x x 2 <= 120 x 1, x 2 >= 0 SEE MS Excel file here