cdae 266 - class 12 oct. 5 last class: quiz 3 3. linear programming and applications today: result...
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CDAE 266 - Class 12Oct. 5
Last class:
Quiz 3 3. Linear programming and applications
Today:
Result of Quiz 3 3. Linear programming and applications
Next class: 3. Linear programming and applications
Reading: Linear Programming
CDAE 266 - Class 12Oct. 5
Important date: Problem set 2 due Tuesday, Oct. 10
Result of Quiz 3N = 49 Range = 4 – 10 Average = 7.96
1. Derivatives
2. Relations among Q, P, TR, TC, Profit and marginal profit
3. MC < MR increase production MC > MR decrease production
4. Profit function Q* that maximizes total profit
5. TC and TR Break-even
9
10
16
13
3. Linear programming & applications
3.1. What is linear programming (LP)?
3.2. How to develop a LP model?
3.3. How to solve a LP model graphically?
3.4. How to solve a LP model in Excel?
3.5. How to do sensitivity analysis?
3.6. What are some special cases of LP?
3.2. How to develop a LP model?
3.2.1. Major components of a LP model: (1) A set of decision variables. (2) An objective function.
(3) A set of constraints.
3.2.2. Major assumptions of LP: (1) Variable continuity (2) Parameter certainty (3) Constant return to scale (4) No interactions between decision variables
3.2. How to develop a LP model?
3.2.3. Major steps in developing a LP model: (1) Define decision variables (2) Express the objective function
(3) Express the constraints (4) Complete the LP model
3.2.4. Three examples: (1) Furniture manufacturer (2) Galaxy industrials (3) A farmer in Iowa
Table A (example 1):---------------------------------------------------------------
Unit requirementsResources ---------------------- Amount
Table Chair available---------------------------------------------------------------Wood (board feet) 30 20 300Labor (hours) 5 10 110=====================================Unit profit ($) 6 8---------------------------------------------------------------
Develop the LP model
Step 1. Define the decision variables
Two variables: T = number of tables made
C = number of chairs made
Step 2. Express the objective function
Step 3. Express the constraints
Step 4. Complete the LP model
Example 2. Galaxy Industries (a toy manufacturer)
2 products: Space ray and zapper 2 resources: Plastic & time
Resource requirements & unit profits (Table B)
Additional requirements (constraints):
(1) Total production of the two toys should be no more than 800.
(2) The number of space ray cannot exceed the number of zappers plus 450.
Table B (example 2):---------------------------------------------------------------
Unit requirementsResources ---------------------- Amount
Space ray Zapper available---------------------------------------------------------------Plastic (lb.) 2 1 1,200Labor (min.) 3 4 2,400=====================================Unit profit ($) 8 5---------------------------------------------------------------
Example 3. A farmer in Iowa has 500 acres of land which can be used to grow corn and/or soybeans. The per acre net profit is $20 for soybeans and $18 for corn. In addition to the land constraint, the farmer has limited labor resources: 200 hours for planting and 160 hours for cultivation and harvesting. Labor required for planting is 0.6 hour per acre for corn and 0.5 hour per acre for soybean. Labor required for cultivation and harvesting is 0.8 hour per acre for corn and 0.3 hour per acre for soybeans.
If the farmer’s objective is to maximize the total profit, develop a LP model that can be used to determine how many acres of soy and how many acres of corn to be planted.
Class Exercise 5 (Thursday, Oct. 5)
Best Brooms is a small company that produces two difference brooms: one with a short handle and one with a long handle. Suppose each short broom requires 1 hour of labor and 2 lbs. of straw and each long broom requires 0.8 hour of labor and 3 lbs. of straws. We also know that each short broom brings a profit of $10 and each long broom brings a profit of $8 and the company has a total of 500 hours of labor and 1500 lbs of straw. Develop a LP model for the company to maximize its total profit.
3.3. How to solve a LP model graphically? 3.3.1. Review of some basic math
techniques: (1) How to plot a linear equation?
e.g., Y = 2 - 0.5X 2X + 3Y = 6 X = 3 Y = 4 X = 0 Y = 0
3.3. How to solve a LP model graphically? 3.3.1. Review of some basic math
techniques: (2) How to plot an inequality
e.g., 2X + 3Y < 12 3X < 15
4Y > 8 4Y > 8 X > 0 Y > 0
3.3. How to solve a LP model graphically? 3.3.1. Review of some basic math
techniques: (3) How to solve a system of two
equations? e.g., 30X + 20Y = 300
5X + 10 Y = 110
3.3. How to solve a LP model graphically? 3.3.2. Major steps of solving a LP
model graphically: (1) Plot each constraint (2) Identify the feasible region
(3) Plot the objective function (4) Move the objective function to
identify the “optimal point” (most attractive
corner) (5) Identify the two constraints that
determine the “optimal point” (6) Solve the system of 2 equations (7) Calculate the optimal value of the
objective function.
3.3. How to solve a LP model graphically?
3.3.3. Example 1 -- Furniture Co.
XT = Number of tables XC = Number of chairs
Maximize P = 6XT + 8XC
subject to: 30XT + 20XC < 300 (wood) 5XT + 10XC < 110 (labor) XT > 0 XC > 0
3.3. How to solve a LP model graphically? 3.3.3. Example 1
(1) Plot each constraint (a) XT > 0 (b) XC > 0 (c) 30XT + 20XC < 300 (wood) (d) 5XT + 10XC < 110 (labor)
(2) Find the feasible region (3) Plot the objective function (4) Move the objective function to
identify the optimal point (most attractive corner)
3.3. How to solve a LP model graphically? 3.3.3. Example 1
(5) Identify the two constraints that determine the “optimal point”
(6) Solve the system of 2 equations
30XT + 20XC = 300 (wood)
5XT + 10XC = 110 (labor)
Solution: XT = , XC =
(7) Calculate the optimal value of the
objective function.
P = 6XT + 8XC =
3.3. How to solve a LP model graphically? 3.3.4. Example 2 -- Galaxy Industries
XS = Number of space ray XZ = Number of zappers
Maximize P = 8XS + 5XZ
subject to 2XS + 1XZ < 1200 (plastic)3XS + 4XZ < 2400 (labor)XS + XZ < 800 (total)XS < XZ + 450 (mix)XS > 0XZ > 0
Take-home exercise
Solve the following LP model graphically:
XT = Number of tables XC = Number of chairs
Maximize P = 6XT + 8XC
subject to: 40XT + 20XC < 280 (wood) 5XT + 10XC < 95 (labor) XT > 0 XC > 0
XT = ? XC = ? P = ?