Arnoff Enterprises manufactures the central processing unit $($CPU$)$ for a line of personal computers. The CPUs are manufactured in Seattle, Columbus, and New York and shipped to warehouses in Pittsburgh, Mobile, Denver, Los Angeles, and Washington, D$.$C$.,$ for further distribution. The following table shows the number of CPUs available at each plant, the number of CPUs required by each warehouse, and the shipping costs $($dollars per unit$)$:

$\backslash$table$[[,,,,,,],[,,$Warehouse$],[$Plant$,$Pittsburgh,Mobile,Denver,$\backslash$table$[[$Los$],[$Angeles$]],$Washington,Available$],[$Seattle$,10,20,5,9,10,9000],[$Columbus$,2,10,8,30,6,4000],[$New York,$1,20,7,10,4,8000],[$CPUs Required,$3000,5000,4000,6000,3000,21,000]]$

$28348002012/11/07168\mathrm{.}20\mathrm{.}218\mathrm{.}73$

a$.$ Develop a network representation of this problem.

b$.$ Determine the amount that should be shipped from each plant to each warehouse to minimize the total shipping cost.

c$.$ The Pittsburgh warehouse just increased its order by $1000$ units, and Arnoff authorized the Columbus plant to increase its production by $1000$ units. Will this production increase lead to an increase or decrease in total shipping costs? Solve for the new optimal solution.