Broadway Deli is determining its inventory policy for bags of potato chips. The demand for bags of potato chips at Broadway Deli is $100$ per week. Each time that Broadway Deli places an order for bags of potato chips, they must pay processing fees of $$100$ regardless of the size of the order. A bag of potato chips costs $$0\mathrm{.}75$ and holding costs are $$0\mathrm{.}10$ per bag per week. The lead time for the delivery of an order is $2$ weeks. Suppose that each time that Broadway Deli$$s inventory of bags of potato chips reaches $300,$ they place an order for $350$ bags of potato chips. Assume that Broadway Deli starts with $450$ bags of potato chips on day $0.$

a$)$ What will be their long run average purchasing costs per unit time? What will be their long run average fixed ordering costs per unit time? What will be Broadway Deli$$s long run average inventory holding costs per unit time?

b$)$ What is the optimal order size assuming that Broadway Deli would like to minimize the sum of its average holding, ordering, and purchasing costs per unit time? What should be the optimal reorder point?

c$)$ Suppose now that Broadway Deli is able to obtain a discount on its processing fee if it places a large size order. Specifically, if Broadway Deli places an order for $475$ bags of potato chips or more, the processing fee drops to $$75.$ Holding costs per bag of potato chips per unit time and purchasing costs per bag remain the same regardless of the order size. What is now the optimal order size assuming that Broadway Deli would like to minimize the sum of its average holding, ordering, and purchasing costs per unit time?