Suppose a faucet manufacturer imports fixtures from oversees companies. Ships carrying the

fixtures arrive in either New York, Charleston or Miami. The fixtures are then transported

via rail to distribution centers in either Houston, Memphis, and Chicago. The per unit costs

of the transport is given in the network below.

$($a$)$ Give an LP representing the flow of goods from supply centers to warehouses at minimum costs

$($e$)$ Suppose the cost to ship from Miami to Memphis changes from $17$t o $17+,$ find a range

of values $$ so that the optimal shipping routes remain the same. If $$ goes beyond the

bound, describe what happens to produce a new optimal solution.

$($f$)$ Suppose the amount supplied at Miami and demanded at Chicago both go up by $500.$

Describe what happens to the current solution to re$-$establish optimality. Do this without

resolving the problem from scratch.