CDAE Class 12 Oct. 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 Class 12 Oct. 5 Important date: Problem set 2 due Tuesday, Oct. 10

Result of Quiz 3 N = 49 Range = 4 – 10Average = 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

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? Major components of a LP model: (1) A set of decision variables. (2) An objective function. (3) A set of constraints 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? 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 Three examples: (1) Furniture manufacturer (2) Galaxy industrials (3) A farmer in Iowa

Table A (example 1): Unit requirements Resources Amount TableChairavailable Wood ( board feet ) Labor ( hours ) ===================================== Unit profit ($)

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 requirements Resources Amount Space ray Zapper available Plastic (lb.) 2 1 1,200 Labor (min.) 3 4 2,400 ===================================== Unit profit ($)

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? Review of some basic math techniques: (1) How to plot a linear equation? e.g., Y = X 2X + 3Y = 6 X = 3 Y = 4 X = 0 Y = 0

3.3. How to solve a LP model graphically? Review of some basic math techniques: (2) How to plot an inequality e.g., 2X + 3Y < 12 3X < 15 4Y > 8 X > 0 Y > 0

3.3. How to solve a LP model graphically? 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? 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? Example 1 -- Furniture Co. X T = Number of tables X C = Number of chairs Maximize P = 6X T + 8X C subject to: 30X T + 20X C < 300 (wood) 5X T + 10X C < 110 (labor) X T > 0 X C > 0

3.3. How to solve a LP model graphically? Example 1 (1) Plot each constraint (a) X T > 0 (b) X C > 0 (c) 30X T + 20X C < 300 (wood) (d) 5X T + 10X C < 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? Example 1 (5) Identify the two constraints that determine the “optimal point” (6) Solve the system of 2 equations 30X T + 20X C = 300 (wood) 5X T + 10X C = 110 (labor) Solution: X T =, X C = (7) Calculate the optimal value of the objective function. P = 6X T + 8X C =

3.3. How to solve a LP model graphically? Example 2 -- Galaxy Industries X S = Number of space ray X Z = Number of zappers Maximize P = 8X S + 5X Z subject to 2X S + 1X Z < 1200 (plastic) 3X S + 4X Z < 2400 (labor) X S + X Z < 800 (total) X S < X Z (mix) X S > 0 X Z > 0

Take-home exercise Solve the following LP model graphically: X T = Number of tables X C = Number of chairs Maximize P = 6X T + 8X C subject to: 40X T + 20X C < 280 (wood) 5X T + 10X C < 95 (labor) X T > 0 X C > 0 X T = ? X C = ? P = ?