The Defense Communications Agency is responsible for operating and maintaining a world-wide communications system. It thinks of costs as being proportional to the ‘‘message units’’ transmitted in one direction over a particular link in the system. Hence, under normal operating conditions it faces the following minimum-cost flow problem:
cij = cost per message unit over link (i − j).
bi = message units generated (or received) at station i,
uij = upper bound on number of message units that can be transmitted over link (i − j).
Suppose that the agency has been given a budget of B dollars to spend on increasing the capacity of any link in the system. The price for increasing capacity on link (i − j) is pij .
a) Formulate a linear program (not necessarily a network) with only (n + 1) constraints, that will minimize operating costs subject to this budget constraint. (Hint. You must allow for investing in additional capacity for each link.)
b) How can the ‘‘near’’ network model formulated in (a) be analyzed by network-programming techniques? (Hint. How could parametric programming be used?)