Consider an LP with five variables x$\_1,...,$x$\_5$ and two constraints. At an iteration of the Simplex method, we have basic variables x$\_$B $=[$x$\_1$ x$\_2]^$ and non$-$basic variables x$\_$N $=[$x$\_3$ x$\_4$ x$\_5]^,$

B$^(1)$b $=[1525]^$ and the objective function value c$^\_$B$*$B$^(1)$b $=60.$ Addionally, we have the

following information at the current basic feasible solution $($BFS$)$:

c$\_3$c$^\_$B$*$B$^(1)$A$\_3=3,$ B$^(1)$A$\_3=[21]^$

c$\_4$cc$^\_$B$*$B$^(1)$A$\_4=4,$ B$^(1)$A$\_4=[20]^$

c$5$ c$^\_$B$*$B$^(1)$A$\_5=0,$ B$^(1)$A$\_5=[12]^$

Suppose that the original problem is a minimization problem. Is the current solution optimal?

$$ If no$,$ which variable you will enter the basis and which variable you will exit to improve

the objective value? Then update the current BFS by performing one iteration of pivot.

$$ If yes, explain why it is optimal and how many optimal solutions there are. Provide an

optimal solution