$4\mathrm{.}5-1.$ Consider the following statements about linear program$-$

ming and the simplex method. Label each statement as true or false,

and then justify your answer.

$($a$)$ In a particular iteration of the simplex method, if there is a tie for

which variable should be the leaving basic variable, then the next

BF solution must have at least one basic variable equal to zero.

$($b$)$ If there is no leaving basic variable at some iteration, then the

problem has no feasible solutions.

$($c$)$ If at least one of the basic variables has a coefficient of zero

in row $0$ of the final tableau, then the problem has multiple op$-$

timal solutions.

$($d$)$ If the problem has multiple optimal solutions, then the prob$-$

lem must have a bounded feasible region.