# Operations Management Homework Assignment 2014 Solution

# Problem 12: Batching

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.

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:

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 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:

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:

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:

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?

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:

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)?

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:

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):

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:

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:

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:

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:

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 ($) __________________

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 ($) __________________

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