$1$

Consider the following network representation of a transportation problem.

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

$x_{11}=$ 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 to St$.$ 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 to St$.$ Louis

Min $x_{11}+x_{12}+x_{13}+x_{21}+x_{22}+x_{23}$

$s.t.x_{11}+x_{12}+x_{13}$

$x_{11}+x_{21}+x_{22}+x_{23}$

$x_{12}+x_{22}=$

$x_{13}+x_{23}=$

$x_{11},x_{12},x_{13},x_{21},x_{22},x_{23}0$

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

$\backslash$table$[[,$Amount,Cost$],[$Jefferson City$-$Des Moines$],[$Jefferson City$-$Kansas City$],[$Jefferson City$-$St$.$ Louis$],[$Omaha$-$Des Moines$],[$Omaha$-$Kansas City$],[$Omaha$-$St$.$ Louis$],[$Total$,,]]$