$(2)$ Find an optimal solution for the following transportation problem.

The above feasible plan is obtained by applying the Northwest Comer Rule $($e$.$g$.,300$ from $M$ to $V,$ etc.$),$ and

its total cost $=300($$$6)+50($$$8)+150($$$10)+250($$$2)+200($$$9)+250($$$12)=$$$9,000.$

Using the Stepping$-$Stone method to find an optimal plan.

The improvement indices below

are determined as follows:

$($row $M$ first, then rows $Y$ and $P$

$MJ$:$7-2+10-8=1$

MB: $4-9+10-8=-3_{?}$

YV: $3-10+8-6\frac{?}{b}=ar(-5)$

PV: $14-12+9-10+8-6$

PJ: $9-12+9-2=4$

the "most negative index"

$"$ the "most negative index"

We focus on cell $PT($index $=-9).$ Its closed path is: $PT$ to $PB$ to $YB$ to $YT,$ and back to $PT.$ To increase the quantity in

$PT,$ we need to adjust $PB$ down, $YB$ up$,$ and $YT$ down. Therefore, the next improved plan is:

Part $($a$)$ Find the new TC $=$

$($show steps$).$

Part $($c$)$ Determine the next improved plan.

Part $($d$)$ Find the new TC $=$

$($show steps$).$

Partin Is the current solution optimal?

Explain