You order two products from the same supplier. The annual demand for Product 1 is 10,000 units and
the annual demand for Product 2 is 20,000 units. Note that demand for both products is constant
throughout the year. The holding cost is the same for both products, $1 per unit per year. However, you
incur a fixed cost of $200 each time you order Product 1, and a fixed cost of $100 each time you order
Product 2. These fixed order costs are independent of the size of the order.
a.

How many units of Product 1 and Product 2 should be ordered at a time in order to minimize
total holding + order cost?
Number of Units of Product 1 to Order _____________
Number of Units of Product 2 to Order _____________
Supporting Work:
b. Suppose that the supplier insists that orders for Product 1 and Product 2 be coordinated so that
they can be shipped at the same time (still incurring the fixed cost of $200 and $100,
respectively). Given this requirement, how many units of Product 1 and Product 2 (to the nearest
integer) should be ordered at a time in order to minimize total holding + order cost? (Hint: the
products must be ordered the same number of times per year)
Number of Units of Product 1 to Order _____________
Number of Units of Product 2 to Order _____________
Supporting Work:

Problem 3: Queuing
A professor holds online office hours all day Saturday and Sunday the weekend prior to final exams. He is
available from 8AM to 8PM both days to answer student questions. Students who arrive while the
professor is busy with another student simply wait until their turn comes up. Students are processed in the
order they arrive, and all students tolerate whatever wait is necessary to get their questions answered. The
professor notes from past experience that students arrive randomly with questions – the average time
between arrivals is 36 minutes and the coefficient of variation of interarrival times is 1. Similarly, the time
required to answer student questions is randomly distributed with an average of 24 minutes and a
coefficient of variation of 1.

a. On average how long does a student have to wait to get in to see the professor?
Average waiting time (minutes) ___________
Supporting Work:
b. Suppose the professor would prefer the average waiting time to be no more than 40 minutes. By
how much would the average interarrival time have to grow in order to meet this standard?
Increase in Average Interarrival Time (minutes) __________
Supporting Work:

c. Assume again an average interarrival time of 36 minutes and suppose the professor is considering
reducing student waiting time by answering questions faster. How much faster would the professor
have to answer questions in order to reduce the average waiting time to 40 minutes?
Decrease in the Average Processing Time (minutes) __________
Supporting Work:

d. Again assume an average interarrival time of 36 minutes and an average processing time of 24
minutes. What would be the average waiting time if the professor could clone himself (thereby
creating a system with two servers and a single queue)?
Average Waiting Time (minutes) __________
Supporting Work:

Problem 14: Newsboy
Susan sells snow cones from a pushcart. Snow cones come in two flavors, Redeye Raspberry (RR) and
Boozy Banana (BB). Susan’s cost for each cone is the same, $0.50/unit, and she charges $2.00/unit for
cones of either flavor. From experience, Susan knows that the daily demand for RR cones is normally
distributed with mean 100 and standard deviation 30, while demand for BB cones is normally distributed
with mean 120 and standard deviation 60.
Assume that the demand for RR cones is independent of the demand for BB cones (and vice versa), and that
demand in excess of supply is lost (no substitutions). Leftover snow cones are discarded at the end of the
day.
a. How many RR and BB cones should be stocked at the beginning of the day to maximize Susan’s
expected profit? What is the expected profit of this policy?

Number of RR Cones ____________
Number of BB Cones ____________
Expected Profit ____________
Supporting Work:

b. If Susan can stock no more than 250 snow cones, how many RR and BB cones should be stocked at
the beginning of the day to maximize Susan’s expected profit? What is the expected profit of this
policy (Hint: the maximum order size makes the cost of overstocking cones of either flavor
increase)?

Number of RR Cones ____________
Number of BB Cones ____________
Expected Profit ____________
Supporting Work:

Problem 5: Mismatch Costs
Purchasing road salt for towns in the Northeast is a challenging task. The town of Homer, New York has
calculated a forecast of their annual salt needs using historical data. The forecast is summarized in the table
below (Q is the quantity needed):

For example, there is a 60.6% chance they will need 50,000 tons or fewer, there is a 3.3% chance they will
need exactly 100,000 tons and there is a very small chance they will need more than 200,000 tons. Suppose
Homer wants to minimize the amount of inventory it purchases while at the same time having no more than
a 6% change of running out of salt (which would force it to purchase salt on the spot market for a premium).
a. At the start of the season, how much salt (in tons) should Homer have available in its storage sheds?
Assume salt must be purchased in increments of 10,000.
Order Quantity (tons) __________
Supporting Work:

b. Now suppose Homer has been offered the following deal from American Salt Mine (ASM). ASM
will sell Homer salt options for $30 per option with an exercise price of $40 for each option Homer
purchases in advance of the season, Homer can “exercise” an option during the season to receive one
ton of salt during the season. For example, if Homer purchases 100,000 options before the season
starts, then it pays ASM $30 x 100,000 = $3,000,000 for those options. As Homer needs salt during
the season, ASM will deliver up to 100,000 tons for a price of $40 per ton. Options are good only for
this season – any unexercised options at the end of the season have no value. Finally, if Homer
exercises all of its options and still needs more salt, then it will have to purchase salt in the spot
market, for an estimated $80 per ton. Given this deal, how many options should Homer purchase
from ASM? Assume options must be purchased in increments of 10,000 tons.
Number of Options to Purchase __________
Supporting Work:

Problem 26: Risk Pooling
The FAMU Bookstore stocks two types of cashmere sweaters. The two sweaters are identical in every
way except on the first sweater is stitched FAMU FOOTBALL while on the second is stitched
RATTLERS FOOTBALL (we’ll refer to these two types as FAMU sweaters and RATTLERS sweaters).
Both sweaters retail for $100 apiece and cost the Bookstore $40 to procure. Because the procurement
lead time is long relative to the length of the football season, the Bookstore places a single order to cover
anticipated sales for the entire season. Any sweaters left over at the end of the season are shipped to a
reseller for $20 apiece. The demand for FAMU sweaters is normally distributed with a mean of 1000
and a standard deviation of 400. The demand for RATTLERS sweaters is normally distributed with a
mean of 800 and a standard deviation of 300. It’s been noted that in previous years when the demand
for one type of sweater is high, the demand for the other type of sweater is low, leading the Bookstore to
estimate the correlation between the two sweaters at -0.40.
a. How many FAMU sweaters should the Bookstore order for the season to maximize expected
profit? What is the expected profit?
Number of FAMU Sweaters to Order (units) __________________
Expected Profit ($) __________________
Supporting work:

b. How many RATTLERS sweaters should the Bookstore order for the season to maximize
expected profit? What is the expected profit?
Number of RATTLERS Sweaters to Order (units)__________________
(4 points)
Expected Profit ($) __________________
Supporting work:

c. The Bookstore’s manager is offered the opportunity to replace the FAMU and RATTLERS
sweaters with a new sweater on which FAMU RATTLERS FOOTBALL is stitched (which we’ll
refer to as a FAMU RATTLERS sweater). The manager believes that all customers who would
otherwise have demanded a FAMU or a RATTLERS sweater will instead buy the FAMU
RATTLERS sweater, i.e., no sales will be lost by stocking FAMU RATTLERS sweaters and
discontinuing the FAMU and RATTLERS lines. In addition, all of the cost and demand
information previously given remains the same. How many FAMU RATTLERS sweaters
should the Bookstore order for the season to maximize expected profit? What is the expected
profit?
Number of FAMU RATTLERS sweaters to Order (units) __________________
Expected Profit ($) __________________