Trejo Manufacturing makes a variety of household appliances at a single manufacturing facility. The expected demand for one of these appliances during the next $4$ months is shown in the following table along with the expected production costs and the expected capacity for producing these items.

Month

$1234$

Demand $415575315535$

Production Cost $$49\mathrm{.}00$ $$45\mathrm{.}00$ $$46\mathrm{.}00$ $$47\mathrm{.}00$

Production Capacity $495525455545$

Trejo estimates that it costs $$1\mathrm{.}50$ per month for each unit of this appliance carried in inventory at the end of each month. Currently, Trejo has $120$ units in inventory on hand for this product. To maintain a level workforce, the company wants to produce at least $400$ units per month. They also want to maintain a safety stock of at least $50$ units per month. Trejo wants to determine how many of each appliance to manufacture during each of the next $4$ months to meet the expected demand at the lowest possible total cost. $($Let Pi $=$ number of units to produce in month i and Ii $=$ inventory held at the end of month i$.)$

$($a$)$

Draw a network flow model for this problem. $($Note: LB $=$ lower bound$)$