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Linear Programming Models

Linear Programming Models. Tran Van Hoai Faculty of Computer Science & Engineering HCMC University of Technology. Impact of Linear Programming (LP). Contributing to the success of all operational activities of big names United Airlines (and all airlines around the globe) San Miguel

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Linear Programming Models

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  1. Linear Programming Models Tran Van HoaiFaculty of Computer Science & Engineering HCMC University of Technology Tran Van Hoai

  2. Impact of Linear Programming (LP) • Contributing to the success of all operational activities of big names • United Airlines (and all airlines around the globe) • San Miguel • By 1995, become the first non-Japanese, non-Austrilia firm in 20 Asian food and beverage company • … LP model = model to optimize a linear objective function subject to linearconstraints Tran Van Hoai

  3. Example (NetOffice) MAXIMIZE 50D + 30C + 6M SUBJECT TO 7D + 3C + 1.5M ≤ 2000 D ≥ 100 C ≤ 500 D, C, M ≥ 0 (Total profit) (Raw steel) (Contract) (Cushions) (Nonnegativity) D, C integers (Discrete) First power (e.g., 5X, -2Y, 0Z) ILP model = LP model in which variables are integers Tran Van Hoai

  4. Why LP is important ? • Many problems naturallymodelled in LP/ILP models • Or possibly approximated by a LP/ILP models • Efficient solution techniques exists • Output is easily understood as “what-if” information Tran Van Hoai

  5. Solution techniques • 1940s: Simplex method by Dantzig • A breakthrough in MS/OR, solving LP model numerically • 1970s: Polynomial method by Karmarkar • A breakthrough in MS/OR, solving LP model efficiently • Interior point methods Tran Van Hoai

  6. Simplex method Tran Van Hoai

  7. Interior point methods Tran Van Hoai

  8. Assumptions for LP/ILP • Parameter values are known with certainty • Constant returns to scale (proportionally) • 1 item adds $4 profit, requires 3 hours to product,then 500 items add $4x500, require 3x500 hours • No interactions between decision variables • Additive assumptions: total value of a function = adding linear terms Tran Van Hoai

  9. Case studyGalaxy Industries • - Which combinations are possible (feasible) for Galaxy Industries ? • Which maximizes the objective function ? • HAVE A LOOK AT GRAPHICAL REPRESENTATION Tran Van Hoai

  10. Infeasible point Extreme points feasible point Feasible region We can remove C3 (X1+X2≤700) without eliminating any of feasible region C3 is redundant constraint Tran Van Hoai

  11. 8X+5X=5000 8X+5X=4000 Feasible region 4360 Tran Van Hoai

  12. Assignment 1 (1) • 2 ≤ |group| ≤ 4 • 56 (HTQ2010) + 18 (HTQ2010) = 74 • ~20 groups • 40 problems in Chapter 2 • 2 different groups must solve different problems • List of assigned problems sent to Mr. Hoai before 27 Sep, 2010 Tran Van Hoai

  13. Assignment 1 (2) • Report (in Microsoft Word) the process to solve the assigned problem • Length(Report) ≥ 6 A4-pages, font size ≤ 12 • Model provided in Excel or WinQSB • Report and Model must be sent to Mr. Hoai within 2 weeks by email • hard deadline: 11 Oct, 2010 • Lose 20% for 1st week late, 50% for 2nd week late, 100% for 3rd week late Tran Van Hoai

  14. Sensitive analysis • Input parameters not known with certainty • Approximation • Best estimation • Model formulated in dynamic environment, subject to change • Managers wish to perform “what-if” analysis • What happens if input parameters changes? Model can be re-solved if change made Sensitive analysis can tell us at a glance on change Tran Van Hoai

  15. Objective function coefficients • Range of optimality • All other factors the same • How much objective coefficients change without changing optimal solution • Reduced costs • How much objective coefficient for a variable have to be increased before the variable can be positive • Amount objective function will change per unit increase in this variable Tran Van Hoai

  16. Right-hand side coefficents • Shadow prices • The change of objective function value per unit increase to its right-hand side coefficients • Range of feasibility • The range in which a constraint is still in effect Tran Van Hoai

  17. Duality Each LP has a dual problem Dual problem provides upper bound for primal problem MAX CTX S.T. AX ≤ B X ≥ 0 MIN BTY S.T. ATY ≥ C Y ≥ 0 Tran Van Hoai

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