produce up to $6,500$ cars per month, and the Dallas plant can produce up to $6,000$ cars per

month. Producing a car costs $$2,000$ in Detroit and $$1,800$ in Dallas. Cars must be shipped to

three cities. Each month, Los Angeles must receive $5,000$ cars, Atlanta must receive $4,000$

cars, and St$.$ Louis must receive $3,000$ cars. Cars may also be shipped through Memphis and

Denver. The cost of shipping a car between cities is given in the table below. At most $2,200$

cars may be sent directly from any one city to any other city. The goal is to find the

production and distribution plan that minimizes total cost and feasibly delivers cars from the

production locations to the customer locations.

i$.$ Formulate this problem as a minimum cost network flow. Draw the network, label all

nodes, and specify the upper bounds and costs of all arcs, as well as the net supplies

of all nodes.

ii$.$ Now suppose that at most $4,000$ cars may be shipped through Memphis, and at most

$3,000$ cars may be shipped through Denver. How would you modify your network in

part $($i$)$ to include this new restriction?

iii. Suppose the company wanted to add a restriction that the number of cars shipped

from Memphis to Atlanta plus the number of cars shipped from Denver to Los

Angeles could be at most $3000.$ Can this constraint be enforced in a network flow

model?