# deciding to insert a variable a to the basic set in the next step and exclude 𝒂 basic one 𝒃

Let's say you are in the middle of applying the Simplex Method to an LP problem. You've reached a tableau and by checking the sign of the objective coefficients you decided to insert a variable a to the basic set in the next step and exclude 𝒂 basic one 𝒃.

a. Is it possible that in the next step, the variable 𝒂 will become non-basic again?

b. Is it possible that in the next step, the variable 𝒃 can become basic again?

The answer to (a) is yes. Consider the LP in the image below, whose feasible region is the triangle ABC. The decision variables are $$x$$ and $$y$$, and the slack/surplus variables are $$s_1$$ and $$s_2$$. The simplex method starts at A and (after pivoting an artificial variable out of the basis) has $$x$$ and $$s_2$$ as the basic variables. Since the rate of increase of $$f$$ is faster along edge AB than along edge AC, the next pivot moves from A to B, removing $$s_2$$ from the basis and adding $$y$$. The final pivot moves from B to the optimal vertex C, removing $$y$$ from the basis and adding $$s_1$$.