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metode kuantitantif simplex minimization
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Budi Harsanto blogs.unpad.ac.id/budiharsanto
2012
LP Simplex Minimization
Course : Quantitative Method / Operations Research
Add slack variable (+S) to constraint equation.
Slack : unused resources.
Slack coefficient in objective function is 0.
Example:
6X1 + 7X2 < 150
Convert,
6X1 + 7X2 + S1 = 150
Reduce slack/ surplus (-S) and Add artifical (+A) variable to constraint equation.
Surplus Negative Slack.
Surplus coefficient in objective function is 0.
Example
5X1 + 10X2 + 8X3 ≥ 210
Convert,
5X1 + 10X2 + 8X3 – S1 + A1 = 210
Convert =
Add artificial variable (+A).
Artificial coefficient on objective function
Maximization: -M
Minimization: +M
When optimal solution found, all artificial variable must be leaving from solution mix.
Example (Minimization)
25X1 + 30X2 = 900
Convert,
25X1 + 30X2 + A2 = 900
Minimize cost Z = $5X1 + $9X2 + $7X3
Subject to:
5X1 + 10X2 + 8X3 ≥ 210
25X1 + 30X2 = 900
Minimize cost Z =
$5X1 + $9X2 + $7X3 + $0S1 + $MA1 + $MA2
subject to:
5X1 + 10X2 + 8X3 - 1S1 + 1A1 + 0A2 = 210
25X1 + 30X2 + 0X3 + 0S1 + 0A1 + 1A2 = 900
Example
Exercise 1
Convert into the proper form for use in the simplex method.
Minimize cost Z= 4X1 + 1X2
Subject to: 3X1 + X2 = 3
4X1 + 3X2 > 6
X1 + 2X2 < 3
Exercise 2
Convert into the proper form for use in the simplex method.
Maximize earnings Z= 0,80X1 + 0,40X2 + 1,20X3 - 0,10X4
Subject to: X1 + 2X2 + X3 + 5X4 < 150
X2 - 4X3 + 8X4 = 70
6X1 + 7X2 + 2X3 - X4 > 120
Minimization Procedure
Maximization Minimization i. Entering variable: C-Z biggest
positive.
ii. Leaving variable: smallest non negative ratio.
iii. Calculate new pivot row.
iv. Calculate the other rows.
v. If still positive number in C-Z , repeat step i-v.
i. Entering variable: C-Z biggest negative (most negative).
ii. Leaving variable: smallest non negative ratio.
iii. Calculate new pivot row.
iv. Calculate the other rows.
v. If still negative number in C-Z, repeat step i-v.
EXAMPLE
Solve this Problem!
Minimize Cost Z = 5X1 + 6X2
Subject to:
X1 + X2 = 1000
X1 < 300
X2 > 150
Example: Simplex Equation
Minimize Cost Z = 5X1 + 6X2 + 0S1 + 0S2 + MA1 + MA2
Subject to:
X1 + X2 + A1 = 1000
X1 + S1 = 300
X2 - S2 + A2 = 150
Example: Iteration 1
C 5 6 0 0 M M
Solution Mix X1 X2 S1 S2 A1 A2 Quantity
M A1 1 1 0 0 1 0 1000
0 S1 1 0 1 0 0 0 300
M A2 0 1 0 -1 0 1 150
Z M 2M 0 -M M M 1150M
C-Z -M+5 -2M+6 0 M 0 0 -
Example: Iteration 2
C 5 6 0 0 M M
Solution Mix X1 X2 S1 S2 A1 A2 Quantity
M A1 1 0 0 1 1 -1 850
0 S1 1 0 1 0 0 0 300
6 X2 0 1 0 -1 0 1 150
Z M 6 0 M-6 M -M+6 850M+900
C-Z -M+5 0 0 -M+6 0 2M-6 -
Example: Iteration 3
C 5 6 0 0 M M
Solution Mix X1 X2 S1 S2 A1 A2 Quantity
M A1 0 0 -1 1 1 -1 550
5 X1 1 0 1 0 0 0 300
6 X2 0 1 0 -1 0 1 150
Z 5 6 -M+5 M-6 M -M+6 550M+2400
C-Z 0 0 M-5 -M+6 0 2M-6 -
Example: Iteration 4
C 5 6 0 0 M M
Solution Mix X1 X2 S1 S2 A1 A2 Quantity
0 S2 0 0 -1 1 1 -1 550
5 X1 1 0 1 0 0 0 300
6 X2 0 1 -1 0 1 0 700
Z 5 6 -1 0 6 0 5700
C-Z 0 0 1 0 M-6 M -
Referensi Utama
1. Taylor III, Bernard W, Introduction to Management Science, Latest Edition.
2. Render, Barry; Stair, Jr Ralph M & Hanna, Michael E, Quantitative Analysis for Management, Latest Edition.
3. Taha, Hamdy A., Operation Research An Introduction, Latest Edition.
4. Internet
