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

TO

FROM

Valley

$($V$)$

Town

$($T$)$

Junction

$($J$)$

Burg

$($B$)$ Supply

Macon

$($M$)$

$$6$ $$8$ $$7$ $$4$

$30050350$

York

$($Y$)$

$$3$ $$10$ $$2$ $$9$

$150250200600$

Perry

$($P$)$

$$14$ $$4$ $$9$ $$12$

$250250$

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($$$6)+50($$$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

TO

FROM

Valley

$($V$)$

Town

$($T$)$

Junction

$($J$)$

Burg

$($B$)$ Supply

are determined as follows:

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

Macon

$($M$)$

$$6$ $$8$ $$7$ $$4$

$350$

MJ: $72+108=7$

$30050$ MB: $49+108=3$

York

$($Y$)$

$$3$ $$10$ $$2$ $$9$

$600$

YV: $310+86=5$

$150250200$ PV: $1412+910+86=3$

Perry

$($P$)$

$$14$ $$4$ $$9$ $$12$

$250$

PT: $412+910=9*$

$250$ PJ: $912+92=4$

Demand $3002002504501,200*$ 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:

TO

FROM

Valley

$($V$)$

Town

$($T$)$

Junction

$($J$)$

Burg

$($B$)$ Supply

Part $($b$)$ Find the indices$($show steps$)$

Row M first, then rows Y and P

Macon

$($M$)$

$$6$ $$8$ $$7$ $$4$

$350$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$30050\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

York

$($Y$)$

$$3$ $$10$ $$2$ $$9$

$600$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$250350\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

Perry

$($P$)$

$$14$ $$4$ $$9$ $$12$

$250$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$150100\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

Demand $3002002504501,200$ Asterisk the $$most negative index$$

Part $($a$)$ Find the new TC $=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_($show steps$).$

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

TO

FROM

Valley

$($V$)$

Town

$($T$)$

Junction

$($J$)$

Burg

$($B$)$ Supply

Part $($e$)$ Find the indices$($show steps$)$

Row M first, then rows Y and P

Macon

$($M$)$

$$6$ $$8$ $$7$ $$4$

$350$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

York

$($Y$)$

$$3$ $$10$ $$2$ $$9$

$600$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

Perry

$($P$)$

$$14$ $$4$ $$9$ $$12$

$250$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

Demand $3002002504501,200$ Asterisk the $$most negative index$$

Part $($d$)$ Find the new TC $=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_($show steps$).$

Part $($f$)$ Is the current solution optimal? $\_\_\_\_\_\_.$ Explain $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_($hint: see Online Module $8,$ p$.$M$8-4).$