1. XYZ office supplies is trying to manage its inventory of paper that it sells. The average demand for paper at XYZ is 30,000 sheets of paper per week. Each time that XYZ places an order for a new shipment of paper it must pay \$40 in processing fees. Paper costs XYZ \$.03 to purchase per sheet. The cost for XYZ to keep 1,000 sheets of paper in its store for one year is \$5.00. Assume that the leadtime for the delivery of paper is 0 weeks and that there are 52 weeks in a year.

a. Suppose that XYZ places orders for paper in quantities of 90,000 sheets at a time and places a new order for paper each time that it runs out. Draw a graph showing the amount of inventory that XYZ has on-hand at each point in time up until the time when it places its fourth order. Label the points in time at which XYZ places a new order, showing the time at which each order is placed. Assume that XYZ places its first order for 90,000 sheets of paper on day 0.

b. Suppose again that XYZ places orders for 90,000 sheets of paper at a time. What will be XYZ’s average holding costs per year? What about their average ordering costs per year?

c. What is the optimal amount of paper for XYZ to order each time it places an order? As-suming that XYZ orders according to its optimal ordering quantity, what will be XYZ’s average ordering costs over the course of a year? What about their average holding costs over the course of a year?

d. Suppose now that the leadtime for the delivery of a shipment of paper increases to 2 weeks. Assuming that XYZ orders according its optimal order quantity from part (c), what should be XYZ’s reorder point for the delivery of paper?

e. The demand for paper at XYZ has become more variable and follows a normal distribution. The average demand for paper at XYZ is still 30,000 sheets per week but the standard deviation of the demand is now 5,000 sheets per week. Leadtime for a shipment of paper is still 2 weeks and all other costs parameters for the problem remain the same. Suppose that XYZ would like to operate at a service level of 99%. What should XYZ set its new reoder point to be? Also, what will be the additional holding costs that XYZ incurs over the course of a year as a result of adjusting its reorder point?

f. Suppose now that the demand for paper is constant again (no variability) with an average demand of 30,000 sheets per week. The leadtime for the delivery of paper is still 2 weeks. However, rather than ordering paper when XYZ reaches its reorder point calculated from part (d), XYZ decides to place a new order for paper whenever it has 110,000 sheets of paper of left. How much 1 additional holding costs (over the EOQ costs) will XYZ incur as a result of this ordering policy?

2. A bakery needs to determine how many loaves of bread to bake each morning. A loaf of

bread costs \$3.00 to make and sells for \$6.00. Any unsold loaves of bread can be sold the next day at a discounted price of \$1.50. The following is the daily demand distribution for loaves of bread.

Number of loaves 40 45 50 55 60 65 70 75

Probability .05 .05 .10 .20 .30 .20 .05 .05

a. Suppose that the bakery is considering baking 55 loaves of bread. What is the marginal value from a 56th loaf of bread?

b. What is the optimal number of loaves of bread for the bakery to bake each morning?

3. A pastry shop is considering how much hot chocolate to make each morning. Hot choco-late costs \$0.15 per oz to make and sells for \$0.375 per oz. Customers can buy hot chocolate in any number of ounces that they wish. Any hot chocolate not sold by the end of the day is discarded.

The daily demand for hot chocolate is normally distributed with a mean of 220 oz and a standard deviation of 20 oz. How much hot chocolate should the pastry shop make each morning?