Question $2(25$pts$)$

DirectCo sells an item whose demand over the next $4$ months is $100,140,210,$ and $180$ units, respectively. The company can stock just enough supply to meet each month's demand, or it can overstock to meet the demand for $2$ or more successive and consecutive months. In the latter case, a holding cost of $$1\mathrm{.}20$ is charged per overstocked unit per month. DirectCo estimates the unit purchase prices for next $4$ months to be $$15,$$$12,$$$10,$ and $$14,$ respectively. A setup cost of $$200$ is incurred each time a purchase order is placed. The company wants to develop a purchasing plan that will minimize the total costs of ordering, purchasing, and holding items in stock.

a$)$ Provide the network representation that results from interpreting this as a shortest path problem.

b$)$ What are the constraints of the corresponding LP formulation?