1. 5.5 Solution Possibilities
Multiple Optimal Solutions:
A zero reduced cost of a nonbasic variable in the final simplex tableau indicates the existence of multiple optimal solutions.
Observation:
If there are two optimal solutions, then there must be infinitely many optimal solutions [why?]

2. Example
There is a nonbasic variable (x5) whose reduced cost is equal to zero. If we put it in the basis the resulting solution will also be optimal (why?)
What is the “geometric” explanation ?

3. Unbounded Solutions:
This case refers to situation where the new basic solution can be increased indefinitely without causing any one of the old basic variables to become negative.
This will be the case if the Ratio Test fails to identify a variable to be taken out of the basis.
The symptom is that the column of the new basic variable consists of non-positive elements (thus the Ratio Test fails).

4. Example According to the Greedy Rule, x2 is the new basic variable.
If we conduct the Ratio Test on the x2 column we fail to find a variable to be taken out of the basis.
This means that x2 can be increased indefinitely.

5. Recipe
If every coefficient in the column of the new basic variable is non-positive, the solution is unbounded.

6. Non Feasible Solution:
Some linear programming problems do not have feasible solutions (the feasible region is empty).
How does this show in the simplex tableau? Namely, how do we determine whether or not the problem under consideration is feasible?

7. Observations:
This is a very important issue
Problems in standard form always have feasible solutions.

8. Cycling
Is it possible that the simplex procedure we described will never stop??????
The answer is Yes!
Reason: If there is a change in the basis but not in the value of the objective function, i.e. a basic variable whose value is zero leaves the basis, we can cycle indefinitely between the two solutions.
A basis with one or more of the basic variables equal zero is called degenerate.

9. 5.5.2 Example
Table 1
Table 2

10. Table 3
Table 4

11. Table 5
Table 6

12. Table 7 =
Table 1 1 1 1
Remark:
In practice this is not an issue because there are simple remedies to prevent cycles, eg. Bland’s Rule.
Corrections: