The Simplex Method
Pivoting, an Animation
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 1
Assume the Following Initial Basic Feasible
Tableau for a Maximization Problem.
x1
x2
s1
s2
s3
Basis
Cj
10
20
0
0
0
Bi
s1
0
0.7
2
1
0
0
200
s2
0
0.4
0.6
0
1
0
72
s3
0
1.5
0.6
0
0
1
180
Zj
0
0
0
0
0
0
Cj - Zj
10
20
0
0
0
Ratio Test
• Use the down pointer key to move forward in the
animation and the up pointer key to backtrack.
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 2
Step 1. Identify the Entering Variable.
x1
x2
s1
s2
s3
Basis
Cj
10
20
0
0
0
Bi
s1
0
0.7
2
1
0
0
200
s2
0
0.4
0.6
0
1
0
72
s3
0
1.5
0.6
0
0
1
180
Zj
0
0
0
0
0
0
Cj - Zj
10
20
0
0
0
Ratio Test
• Step
So,
the
Entering
as 20
is Variable
the mostis
Rulex2
1,1,is
InWhat
a Maximization,
the Entering
is theVariable
Entering
Variable?
+ve
The +ve
x2 Column
the “Pivot
the (Cj-Zj)
one withvalue!
the most
(Cj-Zj) is
value!
Column” for this Pivot!
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 3
Step 2. Identify the Leaving Variable.
x1
x2
s1
s2
s3
Basis
Cj
10
20
0
0
0
Bi
Ratio Test
s1
0
0.7
2
1
0
0
200
200/2
100
s2
0
0.4
0.6
0
1
0
72
72/0.6
120
s3
0
1.5
0.6
0
0
1
180
180/0.6
300
Zj
0
0
0
0
0
0
Cj - Zj
10
20
0
0
0
• Step
Top
Rule
Take
So,
the
row
2,
the
the
minimum
B1
BiLeaving
=B2
value
200.
Ratio
inValue
Variable
each
Test
in
row
the
value
is
and
the
pivot
is
divide
one
100
column
with
and
it
by
the
is
S1
the
2,is
Bottom
row
B3
180.
Value
inthe
the
pivot
column
is
Middle
==is
72.
Value
in
pivot
column
is
2,row
What
the
Leaving
Variable?
so
least
value
thethe
Leaving
inRatio
the
Ratio
Variable.
row
Test
Test
inTest
value
the
result!
Pivot
is 200/2
Column.
or 100.
180/0.6
or120.
300.
0.6,
so+ve
the
Ratio
value
is
72/0.6
or
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 4
Naming of Parts
x1
x2
s1
s2
s3
Basis
Cj
10
20
0
0
0
Bi
Ratio Test
s1
0
0.7
2
1
0
0
200
100
s2
0
0.4
0.6
0
1
0
72
120
s3
0
1.5
0.6
0
0
1
180
300
Zj
0
0
0
0
0
0
Cj - Zj
10
20
0
0
0
• The x2 Column is the “Pivot Column”, the S1
Row is the “Pivot Row” and the orange box
is the “Pivot Element”.
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 5
The Pivot Row
x1
x2
s1
s2
s3
Basis
Cj
10
20
0
0
0
Bi
Ratio Test
s1
0
0.7
0.35
0.7/ 2
21
2/2
10.5
1/ 2
0 /02
0 0/ 2
200
200/ 2
100
100
s2
0
0.4
0.6
0
1
0
72
120
s3
0
1.5
0.6
0
0
1
180
300
Zj
0
0
0
0
0
0
Cj - Zj
10
20
0
0
0
To achieve
this Pivot
we willRow
divide
the values
the
• We
need the
toallhave
a 1 in in
the
Pivot Row
(except the Cj on the left)! by the value
Pivot
Column.
in the Pivot Element.
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 6
The Second Row
x1
x2
s1
s2
s3
Basis
Cj
10
20
0
0
0
Bi
Ratio Test
s1
0
0.35
1
0.5
0
0
100
100
s2
0
0.4
0.6
0
1
0
72
120
s3
0
1.5
0.6
0
0
1
180
300
Zj
0
0
0
0
0
0
Cj - Zj
10
20
0
0
0
• We need the Second Row to have a 0 in the
Pivot Column.
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 7
0.35 *
0.5 *
0.21 1 0.6
* 0.6 0.3 0 *00.6 0 *00.6
0.6
0.6
100 *
60
0.6
The Second Row
x1
x2
s1
s2
s3
Basis
Cj
10
20
0
0
0
Bi
Ratio Test
s1
0
0.35
1
0.5
0
0
100
100
s2
0
0.4 0.19
0.4
0.21
0 0- 0
7212
72
- 60
120
s3
0
1.5
0.6
0
0
1
180
300
Zj
0
0
0
0
0
0
Cj - Zj
10
20
0
0
0
0.6 0
0.6
0-0.3
-00.3 1 1- 0
0.6
• Multiply the Pivot Row (somewhere off the
Tableau!!) by the value in the Pivot Column and Row
• 2.
AND then subtract the result from row 2.
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 8
The Third Row
x1
x2
s1
s2
s3
Basis
Cj
10
20
0
0
0
Bi
Ratio Test
s1
0
0.35
1
0.5
0
0
100
100
s2
0
0.19
0
-0.3
1
0
12
120
s3
0
1.5
0.6
0
0
1
180
300
Zj
0
0
0
0
0
0
Cj - Zj
10
20
0
0
0
• We need the Third Row to have a 0 in the
Pivot Column.
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 9
0.35 *
0.5 *
0.21 1 0.6
* 0.6 0.3 0 *00.6 0 *00.6
0.6
0.6
100 *
60
0.6
The Third Row
x1
x1
x2
x2
s1
s1
s2
s2
s3
s3
Basis
Basis
Cj
Cj
10
10
20
20
00
00
00
Bi
Bi
Ratio Test
s1
s1
00
0.35
0.35
11
0.5
0.5
00
00
100
100
100
s2
s2
00
0.4
0.19
0.6
0
0
-0.3
11
00
12
72
12
120
s3
s3
00
1.5 1.29
1.5
0.21
0.6 0
0 -0.3
-00.3 0 00
-0
0.6
0.6
Zj
Zj
00
00
00
00
00
Cj -- Zj
Zj
Cj
10
10
20
20
00
00
00
120
1 11
- 0 180
- 60
180
300
00
• Multiply the Pivot Row (somewhere off the
Tableau!!) by the value in the Pivot Column and Row
• 3.
AND then subtract the result from row 3.
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 10
Resetting and Recalculating
x1
x2
s1
s2
s3
Basis
Cj
10
20
0
0
0
Bi
x2
s1
20
0
0.35
1
0.5
0
0
100
s2
0
0.19
0
-0.3
1
0
12
s3
0
1.29
0
-0.3
0
1
120
Zj
0
7
0
20
0
10
0
0
0
2000
Cj - Zj
10
3
20
0
0
-10
0
0
Ratio Test
• Replace S1 with x2. Update the Cj col.
• Calculate the Zj row. And the Cj-Zj.
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 11
The New Entering and
Leaving Variables
x1
x2
s1
s2
s3
Basis
Cj
10
20
0
0
0
Bi
Ratio Test
x2
20
0.35
1
0.5
0
0
100
285.714
s2
0
0.19
0
-0.3
1
0
12
63.1579
s3
0
1.29
0
-0.3
0
1
120
93.0233
Zj
7
20
10
0
0
2000
Cj - Zj
3
0
-10
0
0
• x1’s (Cj-Zj) value is the largest +ve value so x1 is
the Entering Variable and s2 is the leaving variable
because its Ratio Test value is the min +ve value.
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 12
Running The Next Pivot
Should Get You to Here!!
x1
x2
s1
s2
s3
0
0
0
Bi
Basis
Cj
10
20
x2
20
0
1
1.053 -1.84
0
77.895
x1
10
1
0
-1.58 5.263
0
63.158
s3
0
0
0
1.737 -6.79
1
38.526
Zj
10
20
5.263 15.79
0
2189.5
Cj - Zj
0
0
-5.26
0
-15.8
• The optimal solution is x1 = 63.158, x2 = 77.895,
s3 = 38.526, s2 = 0, s1 = 0 and Z = 2,189.5.
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 13
Pivoting In a Minimization
• The only difference is in the choice
of the Entering Variable, which
involves selecting the variable with
the most negative (Cj-Zj) value!
• In all other ways the procedure is the
same.
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 14
The End
Use the ESC key to exit!
© Copyright Andrew Hall, 2002
FOMGT 353 Introduction to
Management Science
Pivoting Slide 15