1. CDAE 266 - Class 15 Oct. 16 Last class: Result of group project 1 3. Linear programming and applications Class Exercise 7 Today: 3. Linear programming and applications Quiz 4 (sections 3.1, 3.2 and 3.3) Next class: 3. Linear programming Group project 2 (read the project and develop a LP model before the class)

2. CDAE 266 - Class 15 Oct. 16 Important dates: Problem set 3: due Tuesday, Oct. 23 Problems 8-1, 8-2, 8-3, 8-5 and 8-14 (pp. 3-29 and 3-30) Group project 2 (case 1 on page 3-31): due Tuesday, Oct. 23 Midterm exam: Thursday, Oct. 25

3. 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?

4. 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.

5. 3.3. How to solve a LP model graphically? 3.3.3. 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

6. 3.3. How to solve a LP model graphically? 3.3.3. 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)

7. 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 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 =

8. 3.3. How to solve a LP model graphically? 3.3.4. 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 + 450 (mix) X S > 0 X Z > 0

9. 1200 600 Xz The Plastic constraint Feasible Xs Plastic constraint Infeasible Labor constraint 600 800 Total production constraint Production mix constraint

10. Recall the feasible region 600 800 1200 400600800 Xz Xs We now demonstrate the search for an optimal solution Start at some arbitrary profit, say profit = $2,000... Profit = $ 000 2, Then increase the profit, if possible... 3, 4, … and continue until it becomes infeasible Profit =$5040

11. 600 800 1200 400600800 Xz Xs Let’s take a closer look at the optimal point Feasible region Feasible region

12. Questions: 1. Why do we need to plot the objective function? (a) The optimal point IS NOT always the intersecting point when you have two straight-line constraints 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 (b) There could be more than two straight-line constraints (see example 2) 2.How do I pick up a starting value to draw the objective function?

13. Class Exercise 7 (Thursday, Oct. 11) Solve the following LP model graphically: X = Number of product A, Y = number of product B Maximize P = 3X + 9Y subject to 30X + 20Y < 300 5X + 10Y < 110 X > 0, Y > 0 X* = ? Y* = ? P = ? (Try P = 27 to draw the objective function)

14. Take-home Exercise (Thursday, Oct. 11) 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 = ?

15. 3.4. How to solve a LP model in Excel? 3.4.1. Check the Excel program

16. 3.4. How to solve a LP model in Excel? 3.4.2. Enter the data & formulas (Example 2 -- Galaxy Industries) X 1 = Number of space ray X 2 = Number of zappers Maximize P = 8X 1 + 5X 2 subject to 2X 1 + 1X 2 < 1200 (plastic) 3X 1 + 4X 2 < 2400 (labor) X 1 + X 2 < 800 (total) X 1 - X 2 < 450 (mix) X 1 > 0 X 2 > 0

17. 3.4. How to solve a LP model in Excel? 3.4.3. Solve the model and obtain computer reports -- Answer report -- Sensitivity report -- Limits report

18. 3.5. How to interpret computer reports and conduct sensitivity analysis? 3.5.1. Answer report (1) Optimal solution for X 1 and X 2 (2) Optimal value of the objective function (3) “Original value” (4) “Cell value” or LHS value (5) “Status” of each constraint (6) “Slack” of each constraint (Relation between “status” & “slack”)

19. Take home exercise Solve the following LP model in Excel and obtain the “answer report” and “sensitivity report” X1 = Number of space ray X2 = Number of zappers Maximize P = 8X1 + 5X2 subject to 2X1 + 1X2 < 1200 (plastic) 3X1 + 4X2 < 2400 (labor) X1 + X2 < 800 (total) X1 - X2 < 450 (mix) X1 > 0 X2 > 0