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Consider the communications network shown below.

If there is an arc (i,j) from Node i to Node j, then Node i can transmit messages directly to Node j. Thus, Node 1 can only transmit messages to Node 4 via Nodes 2 and 3. The capacity, in messages per minute, of each arc is given beside the arc in the figure.

(a)
Formulate a Minimum Cost Network Flow Problem (MCNFP) to determine the maximum number of messages per minute that may be sent from Node 1 to Node 5.
(b)
Formulate your MCNFP formulation from part (a) as a linear program (LP).
(c)
Explain why it is not possible to send 10 messages per minute from Node 1 to Node 5.
(d)
Suppose that it possible to increase the capacity of the existing arcs in network by paying $100 per extra unit of capacity per arc. For example, the capacity of arc (1,2) could be increased to 5 for a cost of $200. Explain how you would modify your LP from part (b) to determine a set of minimum-cost arc-capacity increases to the network shown above so that 10 messages per minute may be transmitted from Node 1 to Node 5.


The resulting mathematical program should determine how much capacity to add to each arc as well as how to route the messages through the network. Keep in mind that the capacity increases come in indivisible units.
 
 

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