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I am being asked to give an explanation what happens if, when pivoting, we chose the right entering variables and the wrong leaving variables if we chose positive pivot element, and why?

and what happens if we chose the right entering variables and the wrong leaving variables but with negative pivot element, and why?

My LP- problem is a regular maximisation of profit problem where where i have three variables (dictionaries) and 3 constraints. I know how to perform simplex method. My only problem is understanding it both graphically. I have tried to build the intuition around it through reading many books and a lot of wiki pages but i havent fully grasped it just yet. Please do help me

Thank you in advance.

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When you pivot, you go from one extreme point to another. When you select an entering variable, you select a variable with positive reduced cost (for maximization), that will typically take a positive value (unless you face degeneracy), and when you select a a leaving variable, you select a variable that will take value $0$.

Geometrically, what happens is you increase the value of the entering variable along an edge of the polygon (the direction is given by the reduced cost), until it reaches another extreme point. This extreme point is caracterized by the leaving variable. At this point, if you continue in the same direction, you exit the polygon. So if you select the wrong leaving variable, you will typically end up on an extreme point outside of the polygon, i.e., you will have an infeasible basis.

If you select the wrong entering variable but pick the right leaving variable, in the worst case you degrade your current solution.

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