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Simplex Method

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Simplex Method. Starting Simplex Tableau. The Reddy Mikks Model. Maximize z=5x1 4x2 0s1 0s2 0s3 0s4 ... x1,x2,s1,s2,s3,s40. Slack variables: s1, s2, s3, s4 ... – PowerPoint PPT presentation

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Title: Simplex Method


1
Simplex Method
2
Starting Simplex Tableau
  • The Reddy Mikks Model
  • Maximize z5x14x20s10s20s30s4
  • subject to
  • 6x1 4x2 s1 24
  • x1 2x2 s2 6
  • -x1 x2 s3
    1
  • x2
    s4 2
  • x1,x2,s1,s2,s3,s40
  • Slack variables s1, s2, s3, s4
  • Express the objective function z-5x1-4x20
  • Basic z x1 x2
    s1 s2 s3 s4
    Solution
  • z 1 -5 -4
    0 0 0 0 0
    z-row
  • s1 0 6 4
    1 0 0 0 24
    s1-row
  • s2 0 1 2
    0 1 0 0 6
    s2-row
  • s3 0 -1 1
    0 0 1 0 1
    s3-row
  • s4 0 0 1
    0 0 0 1 2
    s4-row

3
Entering leaving variable
  • Entering variable x1
  • Leaving variable s1
  • Entering
    Ratio
  • Basic x1 Solution
    (or intercept)
  • s1 6 24
    x1 24/6 4 minimum
  • s2 1 6
    x1 6/1 6
  • s3 -1 1
    x1 1/-1 -1 (Ignore)
  • s4 0 2
    x1 2/0 8 (Ignore)

4
The graph
x2
Maximize z 5x1 4x2 subject to 6x1 4x2
s1 24 x1 2x2 s2 6 -x1 x2 s3
1 x2 s4 2 x1, x2 0
1
6
2
5
1
3
s1 0
4
3
4
s3 0
3
2
s2 0
4
2
C
s2 0
1
B
A
x1
1
4
5
6
2
-2
-1
0
24/6 4
1/-1 -1
6/1 6
5
Next tableau
  • Increase in the value of the objective z is (5 x
    4 tons) 20
  • The result of swapping the entering and the
    leaving variables the nonbasic and basic
    variables become
  • Nonbasic (zero)
    variables (s1, x2)
  • Basic variables (x1,
    s2, s3, s4)

Basic z x1 x2 s1
s2 s3 s4 Solution
z 1 -5 -4 0
0 0 0 0
s1 0 6
4 1 0 0 0
24 Pivot row s2
0 1 2 0
1 0 0 6
s3 0 -1 1
0 0 1 0
1 s4 0
0 1 0 0 0
1 2
Pivot column
6
The new tableau
  • Apply Gauss-Jordan to produce the new basic
    solution
  • The new tableau corresponding to the new basic
    solution (x1, s1, s2, s3, s4)
  • Basic z x1 x2
    s1 s2 s3 s4 Solution
  • z 1 0 -2/3
    5/6 0 0 0 20
  • x1 0 1 2/3
    1/6 0 0 0 4
  • s2 0 0 4/3
    -1/6 1 0 0 2
  • s3 0 0 5/3
    1/6 0 1 0 5
  • s4 0 0 1
    0 0 0 1 2
  • Set the nonbasic variables x2 and s1 to zero
  • x1 4, s2 2, s3 5, s42
  • New objective value is z 20

7
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