CDAE Class 13 Oct. 10 Last class: Result of Quiz 3 3. Linear programming and applications Class exercise 5 Today: 3. Linear programming and applications Next class: 3. Linear programming and applications Quiz 4 (Sections 3.2 and 3.3) Reading: Linear Programming

CDAE Class 13 Oct. 10 Important dates: Problem set 2 due today Midterm exam:

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 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

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 Step 1. Try to get ride of one variable (e.g., X) and get the solution for the other variable (e.g., Y) Step 2. Substitute the solution for the variable (e.g., Y) back into any equation and get the solution for the other variable (X)

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

Xz The Plastic constraint Feasible Xs Plastic constraint Infeasible Labor constraint Total production constraint Production mix constraint

Recall the feasible region Xz Xs We now demonstrate the search for an optimal solution Start at some arbitrary profit, say profit = $2, Profit = $ 000 2, Then increase the profit, if possible... 3, 4, … and continue until it becomes infeasible Profit =$5040

Xz Xs Let’s take a closer look at the optimal point Feasible region Feasible region

Questions: 1. Why do we need to plot the objective function? The optimal point IS NOT always the intersecting point For example: Maximize P = 9X T + 3X C subject to: 30X T + 20X C < 300 wood 5X T + 10X C < 110 labor X T > 0, X C > 0 2. How do I pick up a starting value for the obj. function?

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 = ?