$[7\mathrm{.}39/16$ Points$]$ Develop a linear programming model to minimize cost. $($Let $x_{ij}$ be the number of square yards of carpet which flows from node $i$ to node $j.)$

Min $0x16+2x26+5x37+3x48+3x59+0\mathrm{.}25x67+0\mathrm{.}25x78+0\mathrm{.}25x89+0\mathrm{.}25x910$

s$.$t$.$

Beginning Inventory Flow $x_{16}=50$

Quarter $1$ Production Flow $x_{26}600$

Quarter $2$ Production Flow $x_{37}300$

Quarter $3$ Production Flow $x_{48}500$

Quarter $4$ Production Flow $x_{59}<mn\; id="EditorElement\_3162773">4</mn><mn\; id="EditorElement\_3162774">0</mn><mn\; id="EditorElement\_3162775">0</mn>$

Quarter $1$ Demand Flow $x_{16}+x_{26}-x_{67}=400$

Quarter $2$ Demand Flow $x_{37}+x_{67}-x_{78}=500$

Quarter $3$ Demand Flow $x_{48}+x_{78}-x_{89}=400$

Quarter $4$ Demand Flow $x_{59}+x_{89}-x_{910}=400$

Ending Inventory Flow $x_{910}=100$

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

Solve the linear program to find the optimal solution $($in dollars$).$

Contois Carpets is a small manufacturer of carpeting for home and office installations. Production capacity, demand, production cost per square yard $($in dollars$),$ and inventory holding cost per square yard $($in dollars$)$ for the next four quarters are shown in the network diagram below.

Production

Production cost

Demand