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.
Network Flow – Network Simplex Method, Leaving and Entering Variables
network-flowoperations researchsimplex
Best Answer
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.