Operations Management

Homework $6$

$1.$ An electronics retailer needs to manage its inventory of keyboards. The weekly demand for keyboards is $50$ units. The retailer incurs a fixed ordering cost of $$200$ each time it places an order for a new shipment of keyboards regardless of the size of the order. Each keyboard costs the retailer $$20$ to purchase. The holding cost for storing a keyboard in inventory for one week is $$1.$ The lead time for delivery of an order is $1$ week.

a$.$ If the retailer orders keyboards in quantities of $150$ units at a time and places a new order each time the retailer$$s inventory reaches $100.$ Draw a graph of the retailer$$s inventory level for the first $12$ weeks. Assume that the retailer starts with $200$ units of keyboards on day $0.$ Label the points in time at which the retailer places an order and when they receive their orders.

b$.$ Assume that the retailer uses the policy described in part a above. 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 the retailer$$s long run average inventory holding costs per unit time?

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

d$.$ Suppose now that it takes $2$ weeks for each order of keyboards to be delivered. Assuming that the retailer orders according to its optimal ordering quantity from part c above, what should be the retailer$$s reorder point for a shipment of keyboards?

e$.$ Suppose now that the retailer is able to obtain a discount on its ordering cost if it places a large size order. Specifically, if the retailer places an order for $200$ units of keyboards or more, the ordering cost drops to $$100.$ Holding costs per unit of keyboard per unit time and purchasing costs per unit remain the same regardless of the order size. As an example, an order for $200$ units of keyboard would cost $$100$ in ordering cost and $200\backslash$times $$20=$ $$4,000$ in purchasing costs. What is now the optimal order size assuming that the retailer would like to minimize the sum of its average

holding, ordering, and purchasing costs per unit time? Show all your work.

$2.$ A bakery needs to decide how many loaves of bread to bake each day. Each loaf costs the bakery

$$2$ to make and sells for $$5.$ Unsold loaves at the end of the day can be sold to a third party at a

value of $$1$ per loaf. The daily demand for the bread has the following distribution:

Demand $($Number of loaves$)51015203040$

Probability $0\mathrm{.}050\mathrm{.}150\mathrm{.}250\mathrm{.}250\mathrm{.}250\mathrm{.}05$

a$.$ Suppose that the bakery has made $30$ loaves. What is the marginal value from a $31$st loaf?

b$.$ What is the optimal number of loaves for the bakery to make in order to maximize its average

profit?

$3.$ A beverage shop is considering how much lemonade to prepare each day. It costs $$0\mathrm{.}30$ per oz to make lemonade, which sells for $$1\mathrm{.}00$ per oz$.$ Customers can buy lemonade in any number of ounces that they wish. Unsold lemonade at the end of the day is discarded. Daily demand for lemonade is normally distributed with a mean of $1,500$ oz and a standard deviation of $100$ oz$.$ Calculate the optimal amount of lemonade the shop should prepare daily. You may use a z$-$table for this question.