Problem $3$: Star Flows

The Alliance needs to transport supplies through a network of planets and space stations

while minimizing transportation costs. The network consists of seven nodes as follows:

$$ Node $1$: Yavin $4($Supply: $50$ units$)$

$$ Node $2$: Hoth

$$ Node $3$: Endor

$$ Node $4$: Tatooine

$$ Node $5$: Naboo

$2$

$$ Node $6$: Coruscant

$$ Node $7$: Alderaan $($Demand: As much as possible$)$

The first problem is to model this network to determine the maximum flow that can be

supplied to Alderaan from Yavin $4,$ and the second problem is to formulate a minimum cost

flow for sending these supplies from Yavin $4$ to Alderaan.

The following table shows the upper bounds on the flow $($in units$)$ and the costs $($in credits$)$

for each arc:

From Node To Node Flow Upper Bound Cost $($Credits$)$

$1220100$

$1330150$

$241080$

$2520120$

$3625140$

$4715110$

$5720100$

$6730130$

Please do the following:

$$ Draw the network representation for this problem.

$$ Formulate the models needed to solve these two problem in algebraic form.

$$ Solve the models and provide the solution.