Simplex Algorithm method - Definitions, minimization and maximization example and pseudo code

1. 07 December, 2017 Simplex Algorithm 1
Linear Programming
Simplex Algorithm
Yunus Hatipoğlu
201671209
Çankaya University
CENG511 – Advanced Algorithm

2. 07 December, 2017 The Simplex Algorithm 2
Content
• Linear Programming– what is that?
• The Simplex Algorithm – background
• Usage Areas in real life
• How it works (Pseudo)
• Example Question – Maximize

3. 07 December, 2017 The Simplex Algorithm 3
Linear Programming– what is
that? A technique to
achieve optimal
distribution of
resources,
minimize costs and
maximize profits.

4. 07 December, 2017 The Simplex Algorithm 4
The Simplex Algorithm
• based on algebraic
iteration.
• the initial simplex
table is edited
• then the operations
are continued until
it reaches the
optimal solution

5. 07 December, 2017 The Simplex Algorithm 5
The Simplex Algorithm
• Simplex algorithm visits all 2n vertices in
worst case and this turns out to be true for
and deterministic pivot rule.
• Pivot rule takes exponential time in worst
case.
• The performance of simplex algorithm
depend on the specific pivot rule used.

6. 07 December, 2017 The Simplex Algorithm 6
Usage Areas in real life
• limited use of resources (capacity, raw
material, labor force, etc.) and maximizing
profits or minimizing costs.
Determination of optimum
production schedule:

7. 07 December, 2017 The Simplex Algorithm 7
Usage Areas in real life
• tries to ensure optimum application in
determining the sales, storage or purchases
of the warehouse capacity determined
within a certain time to maximize profits (or
minimize costs).
Storage problems:

8. 07 December, 2017 The Simplex Algorithm 8
How it works?

9. 07 December, 2017 The Simplex Algorithm 9
Maximize & Minimize
• Subjected to constraints :
• 0<= ax + by + cz + ... <= P1
• 0<= dx + ey + fz + ... <= P2
• ...
To maximize:
To minimize:
f = c1x+c2y+c3z ...
We maximize
g = -f = -(c1x+c2y+c3z ...)

10. 07 December, 2017 The Simplex Algorithm 10
Example Question
• Constraints:
2x1 + x2 <= 14
5x1 + 5x2 <=40
x1 + 3x2 <= 18
Maximize the following
n= 50x1 + 30x2
x1, x2 >= 0

11. 07 December, 2017 The Simplex Algorithm 11
Example Question(Cont.)
• 1. Create simplex table
– transform them into equations by adding
increasing variables to the inequalities
2x1 + x2 + s1 = 14
5x1 + 5x2+ s2 = 40
x1 + 3x2 + s3 = 18
Maximize the following
n= 50x1 + 30x2

12. 07 December, 2017 The Simplex Algorithm 12
Example Question(Cont.)
– Write constraint equations in matrix form
2 1 1 0 0
5 5 0 1 0 =
1 3 0 0 1
Maximize the following
n= 50x1 + 30x2
x1
x2
s1
s2
s3
14
40
18

13. 07 December, 2017 The Simplex Algorithm 13
Example Question(Cont.)
x1 x2 s1 s2 s3
2 1 1 0 0 14
5 5 0 1 0 40
1 3 0 0 1 18
-50 -30 0 0 0 0
having the greatest absolute value of negative value
indicates pivot column --> -50
Fixed values are divided
to pivot line and minimum
value determines pivot
line
14/2 = 7 which is minumum
40/5 = 8
18/1 = 18
So 2 is pivot line

14. 07 December, 2017 The Simplex Algorithm 14
Example Question(Cont.)
x1 x2 s1 s2 s3
2 1 1 0 0 14
5 5 0 1 0 40
1 3 0 0 1 18
-50 -30 0 0 0 0
multiply the first line by 1/2 and make the pivot line 1
x1 x2 s1 s2 s3
1 1/2 1/2 0 0 7
5 5 0 1 0 40
1 3 0 0 1 18
-50 -30 0 0 0 0

15. 07 December, 2017 The Simplex Algorithm 15
Example Question(Cont.)
• Multiply the first line by 5 and subtract from the
second line
• Substract first line from 3. line
• First line multiplied by 50 and added to forth line
• Clean pivot column.
x1 x2 s1 s2 s3
1 1/2 1/2 0 0 7
0 5/2 -5/2 1 0 5
0 5/2 -1/2 0 1 11
0 -5 25 0 0 350
x1 x2 s1 s2 s3
1 1/2 1/2 0 0 7
5 5 0 1 0 40
1 3 0 0 1 18
-50 -30 0 0 0 0

16. 07 December, 2017 The Simplex Algorithm 16
Example Question(Cont.)
• Change basis and repeat pivotting.
• Second column pivot
• Second line pivot line
x1 x2 s1 s2 s3
1 1/2 1/2 0 0 7
0 5/2 -5/2 1 0 5
0 5/2 -1/2 0 1 11
0 -5 25 0 0 350
7 / 1/2 = 14
5 / 5/2 = 2
11 / 5/2 = 4.4
So pivot element is 5/2 which is
in second line

17. 07 December, 2017 The Simplex Algorithm 17
Example Question(Cont.)
multiply the second line by 2/5 and make the pivot
line 1
x1 x2 s1 s2 s3
1 1/2 1/2 0 0 7
0 1 -1 2/5 0 2
0 5/2 -1.2 0 1 11
0 -5 25 0 0 350
x1 x2 s1 s2 s3
1 1/2 1/2 0 0 7
0 5/2 -5/2 1 0 5
0 5/2 -1/2 0 1 11
0 -5 25 0 0 350

18. 07 December, 2017 The Simplex Algorithm 18
Example Question(Cont.)
• Multiply the second line by 1/2 and subtract from
the first line
• Multiply the second line by 5/2 and subtract from
third line
• Second line is multiplied by 5 and added to forth
line
x1 x2 s1 s2 s3
1 0 1 -1/5 0 6
0 1 -1 2/5 0 2
0 0 2 -1 1 6
0 0 20 2 0 360
x1 x2 s1 s2 s3
1 1/2 1/2 0 0 7
0 1 -1 2/5 0 2
0 5/2 -1/2 0 1 11
0 -5 25 0 0 350

19. 07 December, 2017 The Simplex Algorithm 19
Example Question(Cont.)
• Since there are no negative indicators left, this is
the last table
x1 x2 s1 s2 s3
1 0 1 -1/5 0 6
0 1 -1 2/5 0 2
0 0 2 -1 1 6
0 0 20 2 0 360
x1 = 6
x2 = 2
s3 = 6
n = 360

20. 07 December, 2017 The Simplex Algorithm 20
Question
• What is worst case complexity of Simplex Algorithm
and what kind of areas complexity algorithm can be
used?