Download - 2b Gauss Simplek Dasar
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GOOD MORNING CLASS!
In Operation Research Class,WE MEET AGAINWITH A TOPIC OF :
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LINEAR PROGRAMMING
THE SIMPLEX METHOD :
GAUSSIAN ELIMINATION
SETTING UP THE INITIAL SOLUTION
DEVELOPING THE SECOND SOLUTION DEVELOPING THE THIRD SOLUTION
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Gaussian Elimination :
IS CHANGING :
5 4 -3
2 -4 3
4 -3 1
X
Y
Z
230
120
140= TO
1 0 0
0 1 0
0 0 1
X
Y
Z
50
40
60
=
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HOW TO CHANGE ?5 X+ 4 Y+ -3 Z = 230
2 X+ -4 Y+ 3 Z = 120
4 X+ -3 Y+ 1 Z = 140
5 4 -3 = 230 5
2 -4 3 = 120 2
4 -3 1 = 140 4
1 0,8 -0,6 = 46 0,8
0 -5,6 4,2 = 28 -5,6
0 -6,2 3,4 = -44 -6,2
1 0 0 = 50 0
0 1 -0,8 = -5 -0,75
0 0 -1,3 = -75 -1,25
1 0 0 = 50
0 1 0 = 40
0 0 1 = 60
http://gauss.xls/http://gauss.xls/http://gauss.xls/ -
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LP as Three Stage Process :
Problem Formulation
Problem Solution
Solution Interpretation and Implementation
Maximize : Profit = 8 T + 6 C
Subject to the constraint :
4 T + 2 C = 0
2 T + 4 C = 0
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Problem Formulation
Maximize : Profit = 8 T + 6 C + 0 S1 + 0 S2
Subject to :
4 T + 2 C + 1 S1 + 0 S2 = 60
2 T + 4 C + 0 S1 + 1 S2 = 48
All variables >= 0
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Parts of the Simplex Tableau
Cj Product 8 6 0 0 Non-Negative
Mix Quantity T C S1 S2 Ratio
0 S1 60 4 2 1 0 150 S2 48 2 4 0 1 24
Cj column (profits per unit)
Productmix column
Constant Column (quantities
of product in the mix)
Variable columns
Real Product Slack Time
Cj row
Variable
row
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The Initial Simplex Tableau
Cj Product 8 6 0 0 Non-Negative
Mix Quantity T C S1 S2 Ratio
0 S1 60 4 2 1 0 15
0 S2 48 2 4 0 1 24
Zj 0 0 0 0 0
Cj - Zj 8 6 0 0
Pivot Point
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The Replacing Row & Second Simplex
Cj Product 8 6 0 0Mix Quantity T C S1 S2
8 T 15 1 0,5 0,25 0
0 S2 48 2 4 0 1
Zj
Cj - Zj
Cj Product 8 6,0 0,0 0,0
Mix Quantity T C S1 S2
8 T 15 1 0,5 0,3 0,0
0 S2 18 0 3,0 -0,5 1,0
Zj 120 8 4,0 2,0 0,0
Cj - Zj 0 2,0 -2,0 0,0
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Replacing Row of the Third Tableau
Cj Product 8 6,0 0,0 0,0 Non-NegativeMix Quantity T C S1 S2 Ratio
8 T 15 1 0,5 0,3 0,0 30
0 S2 18 0 3,0 -0,5 1,0 6
Zj 120 8 4,0 2,0 0,0
Cj - Zj 0 2,0 -2,0 0,0
Cj Product 8.0 6.0 0.0 0.0
Mix Quantity T C S1 S2
8 T 12 1.0 0.0 0.3 -0.2
6 C 6 0.0 1.0 -0.2 0.3Zj 132 8.0 6.0 1.7 0.7
Cj - Zj 0.0 0.0 -1.7 -0.7
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Summary Of Step InThe Simplex Maximization Procedure
Set up the inequalities describing the problem constraint
Convert the inequalities to equation by adding slack variables
Enter the equation in the simplex table
Calculate the Zj and Cj Zj values for this solution
Determine the entering variable (optimal column) by choosingthe one with highest Cj Zj value
Determine row to be replaced by dividing quantity columnvalues by their corresponding optimal column values andchoosing the smallest non negative ratio
Compute the values for the replacing rows
Compute the values for the remaining rows
Calculate Zj and Cj Zj values for this solution
If there is a positive Cj Zj value return to step 5.
If there is no positive Cj Zj value , the optimal solution hasbeen obtained
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Exercise for you :
The Tekno Fertilizer Company makes two types of fertilizerwhich are manufactured in two departments. Type Acontribution $3 per ton, and type B contributes $4 per ton
Department
Hours per ton Maximum hours
Type A Type B worked per week
1 2 3 40
2 3 3 75
Set up a linear programming problem to determine howmuch of the two fertilizer to make in order to maximizeprofits. Use simplex algorithm to solve your problem.
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THANK YOU !!!
SEE YOU NEXT WEEK
!!
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SOLUTION
Maximize : Profit = $3 A + $4 B
Subject to constraint :
2A + 3B