The Lincoln Lock Company manufactures a commercial security lock at plants in Atlanta, Lovisville, Detroit, and Phoenix. The unit cost of production at each plant is $35\mathrm{.}50,37\mathrm{.}50,37\mathrm{.}25,$ and $$36\mathrm{.}25,$ and the annual capacities are $18,000,15,000,25,000,$ and $20,000,$ respectively. The locks are sold through wholesale distributors in seven locations around the country. The unit shipping cost for each plant$-$distributor combination is shown in the following table, along with the forecasted demand from each distributor for the coming year.

$\backslash$table$[[,$Tacoma,$\backslash$table$[[$San$],[$Diego$]],$Dallas,Denver,St Louis,Tampa,Baltimore$],[$Atlanta$,2\mathrm{.}5,2\mathrm{.}75,1\mathrm{.}75,2,2\mathrm{.}1,1\mathrm{.}8,1\mathrm{.}65],[$Louisville$,1\mathrm{.}85,1\mathrm{.}9,1\mathrm{.}5,1\mathrm{.}6,1,1\mathrm{.}9,1\mathrm{.}85],[$Detroit$,2\mathrm{.}3,2\mathrm{.}25,1\mathrm{.}85,1\mathrm{.}25,1\mathrm{.}5,2\mathrm{.}25,2],[$Phoenix$,1\mathrm{.}9,0\mathrm{.}9,1\mathrm{.}6,1\mathrm{.}75,2,2\mathrm{.}5,2\mathrm{.}65],[$Demand$,5500,11500,10500,9600,15400,12500,6600]]$

$1.)$ Determine the least costly way of shipping locks from plants to distributors.

Note that part of the answer is already pre$-$filled $($the zeros$).$ When running this optimization problem, you do not know there are zeros beforehand. The answer is just partially prefilled to facilitate grading.

$\backslash$table$[[,$Tacoma,$\backslash$table$[[$San$],[$Diego$]],$Dallas,Denver,St Louis,Tampa,Baltimore$],[$Atlanta$,0,0,0,0,0,12500,5500],[0,0],[$Louisville$,0,0,0,0,15000,0,0],[0],[$Detroit$,0,0,7500,9600,400,0,$DB$],[0,0,$c$],[$Phoenix$,5500,11500,3000,0,0,0,0],[0,0,0],[,,,,,,,]]$

$2.)$ How much would the least costly way still cost me$?$