Linear Programming Models Tran Van Hoai Faculty of Computer Science & Engineering HCMC University of Technology Tran Van Hoai

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 – … Tran Van Hoai2 LP model =model to optimize a linear objective function subject to linear constraints LP model =model to optimize a linear objective function subject to linear constraints

D, C integers(Discrete) First power (e.g., 5X, -2Y, 0Z) MAXIMIZE50D+ 30C+ 6M SUBJECT TO 7D+ 3C+ 1.5M≤ 2000 D≥ 100 C≤ 500 D,C,M≥ 0 Example (NetOffice) Tran Van Hoai3 (Total profit) (Raw steel) (Contract) (Cushions) (Nonnegativity) ILP model =LP model in which variables are integers

Why LP is important ? Many problems naturally modelled in LP/ILP models – Or possibly approximated by a LP/ILP models Tran Van Hoai4 Efficient solution techniques exists Output is easily understood as “what-if” information

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 Hoai5

Simplex method Tran Van Hoai6

Interior point methods Tran Van Hoai7

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 Hoai8

Case study Galaxy Industries MAX8X 1 +5X 2 (Total weekly profit) S.T.2X 1 +X2X2 ≤1000(Plastic) 3X 1 +4X 2 ≤2400(Production time) X1X1 +X2X2 ≤700(Total production) X1X1 -X2X2 ≤350(Mix) X1X1 X2X2 ≥0(Nonnegativity) Tran Van Hoai9 - Which combinations are possible (feasible) for Galaxy Industries ? -Which maximizes the objective function ? HAVE A LOOK AT GRAPHICAL REPRESENTATION - Which combinations are possible (feasible) for Galaxy Industries ? -Which maximizes the objective function ? HAVE A LOOK AT GRAPHICAL REPRESENTATION

Tran Van Hoai10 We can remove C3 (X 1 +X 2 ≤700) without eliminating any of feasible region C3 is redundant constraint We can remove C3 (X 1 +X 2 ≤700) without eliminating any of feasible region C3 is redundant constraint Feasible region Infeasible point feasible point Extreme points

Tran Van Hoai11 Feasible region 8X+5X=50008X+5X=

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, Tran Van Hoai12

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 – hard deadline: 11 Oct, 2010 – Lose 20% for 1 st week late, 50% for 2 nd week late, 100% for 3 rd week late Tran Van Hoai13

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? Tran Van Hoai14 Model can be re-solved if change made Sensitive analysis can tell us at a glance on change Model can be re-solved if change made Sensitive analysis can tell us at a glance on change

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 Hoai15

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 Hoai16

Duality Tran Van Hoai17 MAX8X 1 +5X 2 (Total weekly profit) S.T.2X 1 +X2X2 ≤1000(Plastic) 3X 1 +4X 2 ≤2400(Production time) X1X1 +X2X2 ≤700(Total production) X1X1 -X2X2 ≤350(Mix) X1X1 X2X2 ≥0(Nonnegativity) MIN1000Y Y Y Y 4 S.T.2Y 1 +3Y 2 +Y3Y3 +Y4Y4 ≥8 Y1Y1 +4Y 2 +Y3Y3 -Y4Y4 ≥5 Y1Y1 Y2Y2 Y3Y3 Y4Y4 ≥0 Each LP has a dual problem Dual problem provides upper bound for primal problem Each LP has a dual problem Dual problem provides upper bound for primal problem MAXC T X S.T.AX ≤ B X ≥ 0 MAXC T X S.T.AX ≤ B X ≥ 0 MINB T Y S.T.A T Y ≥ C Y ≥ 0 MINB T Y S.T.A T Y ≥ C Y ≥ 0