1. The Theory of the Simplex Method Asst. Prof. Dr. Nergiz Kasımbeyli
ISE 203 OR I
Chapter 5
The Theory of the Simplex Method
Asst. Prof. Dr. Nergiz Kasımbeyli

9. x1=0 and x1=4
x2=0 and 2x2=12

22. Fig. 5.3
Not a convex set!

27. Whenever a constraint boundary equation is one of the defining equations for a CP solution, its indicating variable has a value of zero in the augmented form of the problem.
Each such indicating variable is called a nonbasic variable for the corresponding basic solution.

29. Degenerate solution
A BF solution is a basic solution where all m basic variables are nonnegative (≥ 0).
A BF solution is said to be degenerate if any of these m variables equals zero.
Thus, it is possible for a variable to be zero and still be a basic variable for the current BF solution (Another constraint boundary equation is satisfied in addition to its n defining equations).

38. The Matrix Form of Simplex

39. The Matrix Form of Simplex

47. Fundamental Insight or Z rhs
Decision variables
Slack variables
rhs
All you need to know is B-1 and cbB-1. You can calculate these from the initial tableau. or
You can read them directly off the final tableau.

48. Fundamental Insight We replace cBB-1 with y*; and B-1 with S* Z rhs
Decision variables
Slack variables
rhs
We replace cBB-1 with y*; and B-1 with S*

49. Fundamental Insight Z
Decision variables
Slack variables
rhs
y* plays a very special role. These are shadow prices. We will often write the final tableau like this.
We can use the fundamental insight for sensitivity analysis.

50. Apply Fundamental Insight
Here is part of the final Tableau for Wyndor
Use the fundamental insight to find the values of the decision variables and the profit.

51. cBB-1b
cBB-1
B-1b
B-1

52. b

53. Apply Fundamental Insight
Here is part of the final Tableau for Wyndor
Use the fundamental insight to find the values of the decision variables and the profit.

54. Revised Simplex MethodZ
Decision variables
Slack variables
rhs
Find entering basic variable
Calc top row
Find leaving basic variable
Calc rhs and column under entering basic variable
New Solution
Calc new B-1

55. A more efficient Simplex Method
Revised Simplex
A more efficient Simplex Method
Instead of computing B-1 in every iteration,
it derives B-1new from B-1old .

56. Revised Simplex