Network simplex method, leaving and entering variables

Question

Could someone give me a hint on this question, which is a past exam question:

Under what circumstances will an entering variable in the network simplex method be the same as the leaving variable?

Thank you for your help.

Answer 1

Is it possible this is a trick question?

The entering basic variable is a nonbasic variable that you increase from $0$, while keeping the other nonbasics constant. Doing this will turn it into a basic variable for the next iteration. The leaving basic variable is determined by the minimum ratio test. This is determined by which basic variable drops to $0$ first, as the entering basic variable is increased. Doing this will convert it into a nonbasic variable for the next iteration.

So, entering is a nonbasic variable, and leaving is a basic variable, and they are converted after the process is complete. By this definition, I'm not sure that the entering and leaving can be the same. There may be a unique situation I am unaware of. I wanted to leave this as a comment but it was far too long. If someone more experienced with linear programming comes along and answers your question more appropriately I'll be happy to remove this.

Answer 2

`

All basic variables are algebraically eliminated before applying optimality check or finding entering basic variables.

` This is done because we want to know if the objective function can be increased by improving values of variables which are kept aside to have a value of zero(i.e non-basic variables).So a basic variable cannot be an entering variable.