Consider the following network representation $of$ a transportation problem.

$40$

$30$

$20$

$10$

$40$

$15$

$9$

$8$

$8$

$10$

$26$

A network diagram between five locations $is$ shown. The left$-$hand side $of$ the graph $is$ labeled "Supplies" and lists Jefferson City and Omaha. The right$-$hand side $of$ the graph $is$ labeled "Demands" and lists Des Moines, Kansas City, and $St.$ Louis. Lines are shown between various locations. The following list contains the numbers placed $on$ the graph.

Jefferson City: $40$

Omaha: $30$

Des Moines: $20$

Kansas City: $10$

$St.$ Louis: $40$

Jefferson City$-$Des Moines: $15$

Jefferson City$-$Kansas City: $9$

Jefferson City$-St.$ Louis: $8$

Omaha$-$Des Moines: $8$

Omaha$-$Kansas City: $10$

Omaha$-St.$ Louis: $26$

The supplies, demands, and transportation costs per unit are shown $on$ the network.

$(a)$

Develop a linear programming model for this problem; $be$ sure $to$ define the variables $in$ your model.

Let

$x11=$ amount shipped from Jefferson City $to$ Des Moines

$x12=$ amount shipped from Jefferson City $to$ Kansas City

$x13=$ amount shipped from Jefferson City $toSt.$ Louis

$x21=$ amount shipped from Omaha $to$ Des Moines

$x22=$ amount shipped from Omaha $to$ Kansas City

$x23=$ amount shipped from Omaha $toSt.$ Louis

Min

$x11+x12+x13+x21+x22+x23$

$s.t.$From Jefferson City

$x11+x12+x1340$

From Omaha

$x21+x22+x2330$

$To$ Des Moines

$x11+x2120$

$To$ Kansas City

$x12+x2110$

$ToSt.$ Louis

$x13+x2340$

$x11,x12,x13,x21,x22,x230$

$(b)$

Solve the linear program $to$ determine the optimal solution.

AmountCostJefferson City$-$Des MoinesJefferson City$-$Kansas CityJefferson City$-St.$ LouisOmaha$-$Des MoinesOmaha$-$Kansas CityOmaha$-St.$ LouisTotal

$(a)$ Develop a linear programming model for this problem; $be$ sure $to$ define the variables $in$ your model.

Let

$x_{11}=$ amount shipped from Jefferson City $to$ Des Moines

$x_{12}=$ amount shipped from Jefferson City $to$ Kansas City

$x_{13}=$ amount shipped from Jefferson City $toSt.$ Louis

$x_{21}=$ amount shipped from Omaha $to$ Des Moines

$x_{22}=$ amount shipped from Omaha $to$ Kansas City

$x_{23}=$ amount shipped from Omaha $toSt.$ Louis

Min

$s.t.$

From Jefferson City

From Omaha

$To$ Des Moines

$To$ Kansas City

$ToSt.$ Louis

$x_{{11}^{\text{'}}}{x}_{{12}^{\text{'}}}{x}_{{13}^{\text{'}}}{x}_{{21}^{\text{'}}},x_{{22}^{\text{'}}}{x}_{23}0$

$(b)$ Solve the linear program $to$ determine the optimal solution.