Question $5$

$(40$ points$)$ Provide explanations for your answers below. An answer that may seem correct with an incorrect or missing explanation will receive zero credit.

$($a$)(5$ points$)$ A problem with an unbounded feasible region never has a unique optimal solution. True or false? Explain your answer or provide an example.

$($b$)(5$ points$)$ If we have a degenerate solution at an iteration of the simplex method, we may or may not have a degenerate solution at the next iteration. True or false? Explain your answer or provide an example.

$($c$)(5$ points$)$ Cunvert the following set of constraints for a problem into a set of equality constraints with nonnegative variables.

$x_1-2x_2+3x_{3}5$

$x_1+x_{3}8$

$x_{1}2$

$x_{2}0,$

$($d$)(5$ points$)$ "A variable who enter the basis in an iteration cannot leave the basis in the next iteration." True or false? Explain your answer on a simplex tableau.