Simplex Method

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Simplex Method
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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

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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)

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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
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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
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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

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Question?