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Main Steps of the Simplex Method1. Put the problem in Row-Zero Form. 2. Construct the Simplex tableau. 3. Obtain an initial basic feasible solution (BFS). 4. If the current BFS is optimal then go to step 9.5. Choose a non-basic variable to enter the basis. 6. Use the ratio test to determine which basic variable must
leave the basis. 7. Perform the pivot operation on the appropriate element of
the tableau. 8. Go to Step 4. 9. Stop.
2
Step 1
LP in Row-0 FormMaximize zs.t. z - 4.5 x1 - 4 x2 = 0
30 x1 + 12 x2 + x3 = 6000 10 x1 + 8 x2 + x4 = 2600
4 x1 + 8 x2 + x5 = 2000 x1, x2, x3, x4, x5 0
Original LPMaximize 4.5 x1 + 4 x2s.t. 30 x1 + 12 x2 6000 10 x1 + 8 x2 2600
4 x1 + 8 x2 2000 x1, x2 0
3
Steps 2 and 3
Initial BFS:BV = {z, x3, x4, x5}, NBV = {x1, x2}z = 0, x3 = 6,000, x4 = 2,600, x5 = 2,000x1 = x2 = 0
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5
4
3
54321
2000100840
26000108100
600000112300
000045.41
basic
x
x
x
z
bxxxxxz
Steps 4 and 5
x1 and x2 are eligible to enter the basis.
Select x1 to become a basic variable
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5
4
3
54321
2000100840
26000108100
600000112300
000045.41
basic
x
x
x
z
bxxxxxz
Step 6
• How much can we increase x1?
• Constraint in Row 1:30 x1 + 12 x2 + x3 = 6000
implies
x3 = 6000 - 30 x1 - 12 x2.
• x2 = 0 (it will stay non-basic)
• x3 0 forces x1 200.
6
Step 6
• How much can we increase x1?
• Constraint in Row 2:10 x1 + 8 x2 + x4 = 2600
implies
x4 = 2600 - 10 x1 - 8 x2
• x2 = 0 (it will stay non-basic)
• x4 0 forces x1 260.
7
Step 6
• How much can we increase x1?
• Constraint in Row 3:4 x1 + 8 x2 + x5= 2000
implies
x5 = 2000 - 4 x1 - 8 x2
• x2 = 0 (it will stay non-basic)
• x5 0 forces x1 500.
8
Step 6• From constraint 1, we see that we can increase x1
up to 200, if simultaneously reduce x3 to zero.• From constraint 2, we see that we can increase x1
up to 260, if we simultaneously reduce x4 to zero.• From constraint 3, we see that we can increase x1
up to 500, if we simultaneously reduce x5 to zero.• Since x3 is the limiting variable, we make it non-
basic as x1 becomes basic.
9
Step 6: Ratio Test for x1
Row 1: 30 x1 + 12 x2 + x3 = 6000 =>
30 x1 + x3 = 6000 => x1 6000/30 = 200.
Row 2: 10 x1 + 8 x2 + x4 = 2600 =>
10 x1 + x4 = 2600 => x1 2600/10 = 260.
Row 3: 4 x1 + 8 x2 + x5 = 2000 =>
4 x1 + x5 = 2000 => x1 2000/4 = 500.
10
Step 6: Ratio Test for x1
The minimum ratio occurs in Row 1.Thus, x3 leaves the basis when x1 enters.
11
500420002000100840
26010260026000108100
200306000600000112300
000045.41
ratiobasic
5
4
3
54321
x
x
x
z
bxxxxxz
Step 7: Pivot x1 in and x3 out
Pivot on the x1 column of Row 1 to makex1 basic and x3 non-basic.
12
5
4
3
54321
2000100840
26000108100
600000112300
000045.41
basic
x
x
x
z
bxxxxxz
First ERO: divide Row 1 by 30
Step 7: Pivot x1 in and x3 out
First ERO: divide Row 1 by 30
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5
4
1
54321
2000100840
26000108100
200000333.04.010
000045.41
basic
x
x
x
z
bxxxxxz
Second ERO: Add –10 times Row 1 to Row 2
Step 7: Pivot x1 in and x3 outSecond ERO: Add –10 times Row 1 to Row 2
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5
4
1
54321
2000100840
600013333.0400
200000333.04.010
000045.41
basic
x
x
x
z
bxxxxxz
Third ERO: Add –4 times Row 1 to Row 3
Step 7: Pivot x1 in and x3 outThird ERO: Add –4 times Row 1 to Row 3
15
5
4
1
54321
1200101333.04.600
600013333.0400
200000333.04.010
000045.41
basic
x
x
x
z
bxxxxxz
Fourth ERO: Add 4.5 times Row 1 to Row 0
Step 7: Pivot x1 in and x3 outFourth ERO: Add 4.5 times Row 1 to Row 0
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5
4
1
54321
1200101333.04.600
600013333.0400
200000333.04.010
9000015.02.201
basic
x
x
x
z
bxxxxxz
Steps 4 and 5
BV = {z, x1, x4, x5}, NBV = {x2, x3}z = 900, x1 = 200, x4 = 600, x5 = 1200Increasing x2 may lead to an increase in z.
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5
4
1
54321
1200101333.04.600
600013333.0400
200000333.04.010
9000015.02.201
basic
x
x
x
z
bxxxxxz
Step 6: Ratio Test for x2
The minimum ratio occurs in Row 2.Thus, x4 leaves the basis when x2 enters.
18
5.1874.612001200101333.04.600
1504600600013333.0400
5004.0200200000333.04.010
9000015.02.201
ratiobasic
5
4
1
54321
x
x
x
z
bxxxxxz
Step 7: Pivot x2 in and x4 Out
BV = {z, x1, x2, x5}, NBV = {x3, x4}z = 1230, x1 = 140, x2 = 150, x5 = 240
19
5
2
1
54321
24016.14.0000
150025.00833.0100
14001.00667.0010
1230055.00333.0001
basic
x
x
x
z
bxxxxxz
Steps 4 and 5
x3 is eligible to enter the basis
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5
2
1
54321
24016.14.0000
150025.00833.0100
14001.00667.0010
1230055.00333.0001
basic
x
x
x
z
bxxxxxz
Step 6: Ratio Test for x3
21
6004.024024016.14.0000
18000833.0150150025.00833.0100
21000667.014014001.00667.0010
1230055.00333.0001
ratiobasic
5
2
1
54321
x
x
x
z
bxxxxxz
15025.00833.0 432 xxx
432 25.00833.0150 xxx
If x3 enters the basis, then x2 will increase as well.
Step 6: Ratio Test for x3
22
6004.024024016.14.0000
150025.00833.0100
21000667.014014001.00667.0010
1230055.00333.0001
ratiobasic
5
2
1
54321
x
x
x
z
bxxxxxz
If x3 enters the basis, then x5 will leave the basis.
Step 7: Pivot x3 in and x5 out
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3
2
1
54321
6005.241000
2002083.00833.00100
1001667.01667.00010
12500833.04167.00001
basic
x
x
x
z
bxxxxxz