Question $3.(25$ points$)$: $($MCNFP$)$ Each year, Data Corporal produces $400$ computers in Boston and $300$ computers in Raleigh. Los Angeles customers must receive $400$ computers, and Austin customers must receive $300$ computers. Data Corporal can ship computers from each production city to each demand city directly, or through Chicago. The costs of sending a computer between pairs of cities are shown below.

$($a$)$ Suppose that no more than $200$ computers can be shipped between each pair of cities.

$($i$)$ Formulate an MCNFP to minimize the total shipping cost, and provide a graph to include all information about the MCNFP$.$

$($ii$)$ Solve this problem by Excel, Matlab, GAMS, or Python. You can modify the code provided by the instructor to solve it$.$

$($b$)$ Suppose now that the total amount of computers shipped through Chicago cannot exceed $100($That is$,$ the total amount of computers entering Chicago cannot exceed $100).$ How do you formulate the problem as an MCNFP$?$ Note that an MCNFP is only allowed to have two types of constraints: bound constraints on arcs, and balance equations for nodes. If you add a constraint that does not belong to one of these two types, then the new model is not a MCNFP$.$

For part $($b$),$ do not hand in your GAMS code. Just explain how to modify the model $($or the graph$),$ and interpret the optimal solution and optimal value you find. You need to provide your code, a screen shot of the results when you run the code, and explain the meaning of solutions you obtain. Convert everything into PDF and submit it together with your HW solution in a single file.

Hint: Two ways to solve part $2.$ You add an additional constraint for the flow limit at Chicago, but the problem you obtain is no longer an MCNFP$.$ Another way to solve it is to add a dummy node called

Chicago $2,$ and create an arc between Chicago and Chicago $2$ with the maximum flow is $100.$