Develop a linear programming model to minimize cost and meet demand exactly. $($Let $x_{ij}$ be the number of square yards of carpet which flows from node $i$ to node $j.)$

Min

s$.$t$.$

Beginning Inventory Flow

Quarter $1$ Production Flow

Quarter $2$ Production Flow

Quarter $3$ Production Flow

Quarter $4$ Production Flow

Quarter $1$ Demand Flow

Quarter $2$ Demand Flow

Quarter $3$ Demand Flow

Quarter $4$ Demand Flow

Ending Inventory Flow

$x_{lj}0$ for all $i,j.$

Solve the linear program to find the optimal solution.

$x_{16}=$

$x_{26}=$

$x_{37}=$

$x_{48}=$

$x_{59}=$

$x_{67}=$

$x_{78}=$

$x_{89}=$

$x_{910}=$

Report the cost $($in dollars$)$ associated with the optimal solution.

$Production Production Cost