Tran Van Hoai 1

Linear Programming Models

Tran Van HoaiFaculty of Computer Science & Engineering

HCMC University of Technology

2010-2011

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

– …

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LP model = model to optimize a linear objective function subject to linear constraints

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D, C integers (Discrete)

First power (e.g., 5X, -2Y, 0Z)

MAXIMIZE 50D + 30C + 6MSUBJECT TO 7D + 3C + 1.5M ≤ 2000

D ≥ 100 C ≤ 500

D, C, M ≥ 0

Example (NetOffice)

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(Total profit)(Raw steel)(Contract)(Cushions)(Nonnegativity)

ILP model = LP model in which variables are integers

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Why LP is important ?

• Many problems naturally modelled in LP/ILP models– Or possibly approximated by a LP/ILP models

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• Efficient solution techniques exists

• Output is easily understood as “what-if” information

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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

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Simplex method

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Interior point methods

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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

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Case studyGalaxy Industries

MAX 8X1 + 5X2 (Total weekly profit)S.T. 2X1 + X2 ≤ 1000 (Plastic)

3X1 + 4X2 ≤ 2400 (Production time)X1 + X2 ≤ 700 (Total production)X1 - X2 ≤ 350 (Mix)X1 X2 ≥ 0 (Nonnegativity)

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- Which combinations are possible (feasible) for Galaxy Industries ?-Which maximizes the objective function ?

HAVE A LOOK AT GRAPHICAL REPRESENTATION

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We can remove C3 (X1+X2≤700) without eliminating any of feasible region

C3 is redundant constraint

Feasible region

Infeasible point

feasible pointExtreme points

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Feasible region

8X+5X=5000

8X+5X=4000

4360

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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

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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 late2010-2011

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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?

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Model can be re-solved if change made

Sensitive analysis can tell us at a glance on change

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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

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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

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Duality

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MAX 8X1 + 5X2 (Total weekly profit)S.T. 2X1 + X2 ≤ 1000 (Plastic)

3X1 + 4X2 ≤ 2400 (Production time)X1 + X2 ≤ 700 (Total production)X1 - X2 ≤ 350 (Mix)X1 X2 ≥ 0 (Nonnegativity)

MIN 1000Y1 + 2400Y2 + 700Y3 + 350Y4

S.T. 2Y1 + 3Y2 + Y3 + Y4 ≥ 8Y1 + 4Y2 + Y3 - Y4 ≥ 5Y1 Y2 Y3 Y4 ≥ 0

Each LP has a dual problem

Dual problem provides upper bound for primal problem

MAX CTXS.T. AX ≤ B

X ≥ 0

MIN BTYS.T. ATY ≥ C

Y ≥ 0