The worst case complexity of the simplex algorithm is exponential. Specifically, in the worst case the simplex algorithm visits all 2n vertices, where n is the number of variables, and this takes exponential time. However, in practice simplex algorithms usually perform much better than the worst case. The specific pivot rule used can also affect the performance and complexity - some rules tend to work faster than others on average. Linear programming and the simplex algorithm are used in areas like resource allocation, production scheduling, transportation/logistics, and finance to optimize objectives subject to constraints.

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?