Linear Programming Models

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