Show steps $(2)$ Find an optimal solution for the following transportation problem. To Valley Town Junction Burg FROM $(1)($J$)($B$)$ Supply Madison $$5$ $$7$ $$6$ $$3($M$)30050350$ Yonkers $$2$ $$9$ $$1$ $$8($Y$)150250200600$ Pittsburgh $$11$ $$3$ $$8$ $$10($P$)250250$ Demand $3002002504501,200$ The above feasible plan is obtained by applying the Northwest Corner Rule $($e$.$g$.,300$ from M to V$,$ etc.$),$ and its total cost $=300($$$5)+50($$$7)+150($$$9)+250($$$1)+200($$$8)+250($$$10)=$ $$7,550.$ Using the Stepping$-$Stone method to find an optimal plan. The improvement indices below $\backslash$Tau $\backslash$omicron Valley Town Junction Burg are determined as follows: FROM $(6)(1)($B$)$ Supply $($row M first, then rows Y and P$)$ Madison $$5$ $$7$ $$6$ $$3$ MJ: $6-1+9-7=7($M$)30050350$ MB: $3-8+9-7=-3$ Yonkers $$2$ $$9$ $$1$ $$8$ YV: $2-9+7-5=-5($V$)150250200600$ PV: $11-10+8-9+7-5=2$ Pittsburgh $$11$ $$3$ S$8$ $$10$ PT: $3-10+8-9=-8*($P$)250250$ PB: $8-10+8-1=5$ Demand $3002002504501,200*$ the "most negative index" We focus on cell PT index$=-8).$ 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: Junction Town $(1)$ $$750$ $$9$ Burg $($B$)$ $$3$ Part$($b$)$ Find the indices $($show steps$)$ Supply Row M first, then rows Y and P $$6350$ si To Valley FROM Madison $$5($M$)300$ Yonkers $$2($Y$)$ Pittsburgh $$11($P$)$ Demand $300$ Part $($a$)$ Find the new TC $=250600$ $$3$ S$8$ $$8350$ $$101004501502002501,200250$ Asterisk the "most negative index" $($show steps$).$ Burg $($B$)$ $$3$ Part$($e$)$ Find the indices $($show steps$)$ Supply Row M first, then rows Y and P $350$ $$8600$ Part $($c$)$ Determine the next improved plan. $\backslash$Tau $\backslash$omicron Valley Town Junction FROM $($V$)($T$)$ Madison $$5$ $$7$ $$6($M$)$ Yonkers $$2$ $$9$ si V Pittsburgh $$11$ $$3$ $$8($P$)$ Demand $300200250$ Part $($d$)$ Find the new TC $=$ Part $($f$)$ Is the current solution optimal? Explain $$102501,200450$ Asterisk the "most negative index" $($show steps$)($hint: see p$\mathrm{.}352)$