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Linear Programming and Applications: Theory and Practice

Enhance your understanding of linear programming through practical applications. Learn how to develop, solve, and analyze LP models using graphical methods and Excel. Dive into sensitivity analysis and special cases of LP.

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Linear Programming and Applications: Theory and Practice

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  1. CDAE 266 - Class 16 Oct. 18 Last class: 3. Linear programming and applications Quiz 4 Today: Result of Quiz 4 3. Linear programming and applications Group project 2 Next class: 3. Linear programming Review for the midterm exam

  2. CDAE 266 - Class 16 Oct. 18 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. 30 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. 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 = 9XT + 3XC • subject to: 30XT + 20XC< 300 wood • 5XT + 10XC< 110 labor • XT> 0, XC > 0 • (b) There could be more than two straight-line constraints (see example 2) • How do I pick up a starting value to draw the objective • function?

  5. 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) Class Exercise 7 (Thursday, Oct. 11)

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

  7. 3.4. How to solve a LP model in Excel? (follow the class handout on Oct. 18) 3.4.1. Check the Excel program

  8. 3.4. How to solve a LP model in Excel? 3.4.2. Enter the data & formulas (Example 2 -- Galaxy Industries) 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

  9. 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

  10. 3.5. How to interpret computer reports and conduct sensitivity analysis? 3.5.1. Answer report (1) Optimal solution for X1 and X2 (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”)

  11. 3.5. How to interpret computer reports and conduct sensitivity analysis? 3.5.2. Sensitivity report (1) Optimal solution for X1 and X2 (2) “Reduced cost” (3) “Objective coefficient” (4) “Allowable increase” & “allowable decrease” for each objective coefficient (5) Calculate the “range of optimality” for each objective coefficient (6) Interpretation of “range of optimality”

  12. 3.5. How to interpret computer reports and conduct sensitivity analysis? 3.5.2. Sensitivity report (7) Use the “range of optimality” to answer questions regarding a change in an objective coefficient? (8) What is the “100% rule” and how to use that? (9) “Final value” = “LHS” value (10) Interpretation of the “shadow price”

  13. 3.5. How to interpret computer reports and conduct sensitivity analysis? 3.5.2. Sensitivity report (11) “Constraint R.H side” (12) “Allowable increase” & “allowable decrease” for each constraint (13) Calculate the “range of feasibility” for each constraint (14) Use the “range of feasibility” and “shadow” price to answer questions regarding a change in the RHS value (15) The “100% rule”

  14. 3.5. How to interpret computer reports and conduct sensitivity analysis? 3.5.3. Summary of sensitivity analysis (1) One objective function coefficient changes. (2) Two objective function coefficients change at the same time -- 100% rule (3) One RHS value changes (4) Two RHS values change at the same time -- 100% rule

  15. 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

  16. 3.6. What are some special cased of LP? 3.6.1. Multiple optimal solutions 3.6.2. Infeasible problems 3.6.3. Unbounded problems

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