1. Associate Professor of Computers & Informatics - Benha University
Operations Research 1
Dr. El-Sayed Badr
Associate Professor of Computers & Informatics - Benha University
Dr. El-Sayed Badr

2. 2. All the variables are non-negative.
The Simplex Method and Sensitivity Analysis
The development of the simplex method computations is facilitated by imposing two requirements on the constraints of the problem:
All the constraints (with the exception of the non-negativity of the variables) are equations with nonnegative right-hand side.
2. All the variables are non-negative.
Dr. El-Sayed Badr

3. How to convert max problem to min problem ?
Dr. El-Sayed Badr

4. The Slack Variables
Dr. El-Sayed Badr

5. The Surplus Variables
Dr. El-Sayed Badr

6. Graph all constraints, including non-negativity restriction
TRANSITION FROM GRAPHICAL TO ALGEBRAIC SOLUTION
Algebraic Method
Graphical Method
Represent the solution space by m equations in n variables and restrict all variables to non-negativity m < n.
Graph all constraints, including non-negativity restriction
The system has infinity of feasible solutions.
Solution space consists of infinity of feasible points.
Determine the feasible basic solution of the equations.
Identity feasible corner points of the solution space.
Candidates for the optimum solution are given by a finite number of basic feasible solutions.
Candidates for the optimum solution are given by a finite number of corner points.
Use the objective function to determine the optimum basic feasible solution from among all the candidates
Use the objective function to determine the optimum corner point from among all the candidates. 6

7. Example:
n = 4
m = 2 7

8. Basic Solutions and Basic Feasible Solutions
Objective value, z
Feasible?
Corner point
Basic solution
Basic variables
Non-Basic variables
Yes A
(4, 5)
(x3, x4)
(x1, x2)=(0,0) - No F
(4, -3)
(x2, x4)
(x1, x3) =(0,0)
7.5 B
(2.5, 1.5)
(x2, x3)
(x1, x4) =(0,0) 4 D
(2, 3)
(x1, x4)
(x2, x3) =(0,0) E
(5, -6)
(x1, x3)
(x2, x4) =(0,0) 8 C
(1, 2)
(x1, x2)
(x3, x4) =(0,0) 8

9. Iterative Nature of the Simplex Method9

10. The Simplex Method (Reddy Mikks Problem):10

11. 1- Entering Variable 2- Leaving Variable solution x6 x5 x4 x3 x2 x1 z
(المقام صفرا،يهمل)
(المقام سالب،يهمل)
(المقام صفرا،يهمل)
(المقام سالب،يهمل)
solution x6 x5 x4 x3 x2 x1 z
Basic
z-row -4 -5 1
x3-row 24 4 6
x4-row 2
x5-row -1
x6-row
1- Entering Variable
2- Leaving Variable 11

12. solution x6 x5 x4 x3 x2 x1 z Basic -4 -5 1 الصف المحورى 24 4 6 2 -1-4 -5 1
الصف المحورى 24 4 6 2 -1 12

13. solution x6 x5 x4 x3 x2 x1 z Basic 20 1 4 2 5
5/6
-2/3 1 4
1/6
2/3 2
-1/6
4/3 5
5/3 13

14. solution x6 x5 x4 x3 x2 x1 z Basic 21 ½ ¾ 1 3 -1/2 ¼ 3/2 -1/8 5/2 -5/41 3
-1/2
3/2
-1/8
5/2
-5/4
3/8
1/2
-3/4
1/8 14

15. Another Example : Minimization Problem
solution x5 x4 x3 x2 x1 z
Basic -5 -3 3 1
Pivot row -2 2 -1
Dr. El-Sayed Badr

16. solution x5 x4 x3 x2 x1 z Basic 1.5- -1.5 -9.5 1 0.5 1.5 -1 -0.5 -3.5
-1.5
-9.5 1
0.5
1.5 -1
-0.5
-3.5 16

17. Simplex Algorithm 17

18. Question1: Solve the following Linear Problem
Klee-Minty
Question1: Solve the following Linear Problem 18

19. Question2: What is the situation if our problem
Simplex Algorithm
Question2: What is the situation if our problem
Contains constraints of the kind
( <= , >= and = = ) ?
Answer1:
1- Dual Simplex Algorithm Two-Phase Method.
3- Big M-Method ……. 19