Railroad Car Rumba $-$ The Rock Island Railroad $($RIRA$)$ has $100$ empty railcars at its Sioux City rail yards and $150$ rail cars at its North Platte location available for retro$-$ fitting for a new project. Ultimately, RIRA must get $75$ retro$-$fitted cars to Chicago, $85$ retro$-$fitted cars to Peoria, and $55$ retro$-$fitted cars to St Louis.

To retro$-$fit the cars, two different stages of maintenance must be performed on the rail cars. Stage $1$ retro$-$fitting occurs either at Lincoln, Omaha or Des Moines. Stage $2$ retro$-$ fitting occurs at Davenport, IA or Kansas City. Cars must pass first through Stage $1,$ then Stage $2,$ before arriving at the final destination.

We are not interested in retro$-$fitting costs at the $5$ different maintenance locations. Our goal is to find the least distance way to transform the railcars and get them $($in their retro$-$ fitted format$)$ to Chicago, Peoria and St$.$ Louis.

Stage $1$ and Stage $2$ locations have capacities for retro$-$fitting. They are listed below.

Location Lincoln Omaha Des Moines Davenport, IA Kansas City

Capacity Stage $11519015019521352$

Relevant distances are shown below. Using an appropriate linear programming model, find the optimal way of satisfying the rail car transformation problem for RIRA. Explicitly explain why your solution says in terms of cars transportation.

Des Lincoln Omaha Moines

$188162209214248372320281156163167177$

Chicago Peoria

StLouis

Sioux City N Platte Davenport Kcity

$15276203529285234$