n a $33$ transportation problem, let $$ be the amount shipped from source $$ to

destination $,$ and let $$ be the corresponding transportation cost per unit. The amounts of

supply at sources $1,2,$ and $3$ are $15,30,$ and $85$ units, respectively, and the demands at

destinations $1,2,$ and $3$ are $20,30,$ and $80$ units, respectively. Assume that the starting

northwest$-$corner solution is optimal and that the associated values of the multipliers are given

as $1=2,2=3,3=5,1=2,2=5,$ and $3=10.$

$($a$)$ Find the associated optimal cost.

$($b$)$ Determine the smallest value of $$ for each nonbasic variable that will maintain the

optimality of the northwest$-$corner solution.