Transcript Betting on Uncertain Demand: The Newsvendor Model Chapter 9

What do the following
goods/services have in common?
• Flu vaccines
• A seat on SWA 136
• Fresh flowers
Newsvendor model implementation steps
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Gather economic inputs
Generate a demand model
Choose an objective
Choose a quantity to order
“Too much” and “Too little” Costs
• Co = c – v (= MC = overage cost)
• CU = p – c (= MB = underage cost)
Weekday lunch demand for the spicy black bean burritos at The
Kiosk, a local snack bar, is normally distributed with a mean of 35
and a standard deviation of 9. The Kiosk charges $5.00 for each
burrito, which are all made by the owner/operator in his home
kitchen before departing for work. Virtually all burrito customers
also buy a soda that is sold for 60¢. The burritos cost The Kiosk
$2.00, while the sodas cost 5¢. The owner is very sensitive about
the quality of the food he serves. Thus, he maintains a strict “No
Old Burrito” policy, so any burrito left at the end of the day is fed
to his chickens.
Suppose burrito customers buy their snack elsewhere if The
Kiosk is out of stock. How many burritos should he make for his
lunch crowd?
Newsvendor Model Performance Measures
– Expected lost sales
– Mismatch cost
– Expected sales
– Expected fill rate
– Expected left over inventory
– In-stock probability
– Expected profit
– Stockout probability
a)
b)
c)
d)
11.6 Teddy Bower is an outdoor clothing and accessories chain that
purchases a line of parkas at $10 each from its Asian supplier,
TeddySports. Unfortunately, at the time of order placement, demand is still
uncertain. Teddy Bower forecasts that its demand is normally distributed
with a mean of 2,100 and standard deviation of 1,200. Teddy Bower sells
these parkas at $22 each. Unsold parkas have little salvage value; Teddy
Bower simply gives them away to charity.
What is the probability that this parka turns out to be a dog, defined as a
product that sells less than half of the forecast?
How many parkas should Teddy Bower buy from TeddySports to maximize
expected profit?
If Teddy Bower wishes to ensure a 98.5% fill rate, how many parkas
should it order?
If Teddy Bower wishes to ensure a 98.5% in-stock probability, how many
parkas should it order?
Now assume Teddy Bower orders 3,000 parkas
a) Evaluate Teddy’s expected profit
b) Evaluate Teddy’s fill rate
c) Evaluate Teddy’s stockout probability
a)
b)
c)
d)
e)
f)
11.5 Fashionables is a franchisee of The Limited, the well-known retailer of fashionable clothing.
Prior to the winter season, The Limited offers Fashionables the choice of 5 different colors of a
particular sweater design. The sweaters are knit overseas by hand, and because of the lead
times involved, Fashionables will need to order its assortment in advance of the selling season.
As per the contracting terms offered by The Limited, Fashionables also will not be able to cancel,
modify, or reorder sweaters during the selling season. Demand for each color during the season
is normally distributed with a mean of 500 and a standard deviation of 200. Further, you may
assume that the demands for each sweater are independent of those for a different color.
The Limited offers the sweater to Fashionables at the wholesale price of $40 per sweater, and
Fashionables plans to sell each sweater at the retail price of $70 per unit. The Limited delivers
orders places by Fashionables in truckloads at a cost of $2,000 per truckload. The transportation
cost of $2,000 is borne by Fashionables. Assume unless otherwise specified that all the
sweaters ordered by Fashionables will fit into one truckload. Also assume that all other
associated costs, such as unpacking and handling are negligible.
The Limited does not accept any returns of unsold inventory. However, Fashionables can sell all
of the unsold sweaters at the end of the season at the fire-sale price of $20 each.
How many units of each type of sweater should Fashionables order?
If Fashionable wishes to ensure a 97.5% in-stock probability, what should its order quantity be?
Now assume that Fashionables orders 725 of each kind of sweater
What is Fashionables’ expected profit?
What is Fashionables’ expected fill rate for each type of sweater?
What is the stockout probability for each type of sweater?
Now suppose that The Limited announces that the unit of truckload capacity is 2,500 units. What
now is Fashionables’ optimal order quantity for each sweater?
What if supply is fixed?
• Examples of fixed supply:
• Revenue management is a solution:
– If adjusting supply is impossible – adjust the demand!
– Segment customers into high willingness to pay and
low willingness to pay.
– Limit the number of tickets sold at a low price, i.e.,
control the average price by changing the mix of
customers.
Revenue Management: Booking limits and protection levels
• The Park Hyatt Philadelphia at the Bellevue.
• 118 King/Queen rooms.
• Hyatt offers a rL= $159 (low fare) discount fare for a mid-week stay
targeting leisure travelers.
• Regular fare is rH= $225 (high fare) targeting business travelers.
• Demand for low fare rooms is abundant.
• Let D be uncertain demand for high fare rooms.
– Suppose D has Poisson distribution with mean 27.3.
• Assume most of the high fare (business) demand occurs only within a
few days of the actual stay.
• Objective:
– Maximize expected revenues by controlling the number of low fare
rooms you sell.
Yield management decisions
• The booking limit is the number of rooms you are willing to sell in a
fare class or lower.
• The protection level is the number of rooms you reserve for a fare
class or higher.
• Let Q be the protection level for the high fare class.
• Q is in effect while you sell low fare tickets.
• Since there are only two fare classes, the booking limit on the low fare
class is 118 – Q:
– You will sell no more than 118-Q low fare tickets because you are
protecting (or reserving) Q seats for high fare passengers.
0
118
Sell no more than the low
fare booking limit, 118 - Q
Q seats protected for
high fare passengers
Related calculations
• How many high-fare travelers will be refused a
reservation?
• How many high-fare travelers will be accommodated?
• How many rooms will remain empty?
• What is the expected revenue?
Revenue management challenges …
• Demand forecasting
• Dynamic decisions
• Variable capacity
• Group reservations
• How to construct good “fences” to differentiate customers?
• Multi-leg passengers/multi-day reservations
• Cancellations & no-shows
Revenue Management: Overbooking at the Hyatt
• The forecast for the number of customers that do not show up (X) is
Poisson with mean 8.5.
• The cost of denying a room to the customer with a confirmed
reservation is $350 in ill-will and penalties.
• How many rooms (Y) should be overbooked (sold in excess of
capacity)?
• Newsvendor setup:
– Single decision when the number of no-shows is uncertain.
– Underage cost if X > Y (insufficient number of seats overbooked).
– Overage cost if X < Y (too many seats overbooked).
On a given Philadelphia-LA flight, there are 200 seats.
Suppose the ticket price is $475 on average and the
number of passengers who reserve a seat but do not
show up for departure is ~N(30,15). You decide to
overbook the flight and estimate that the average loss
from a passenger who will have to be bumped is $800.
a) What is the maximum number of reservations that
should be accepted?
b) Suppose you allow 220 reservations. How much money
do you expect to pay out in compensation to bumped
passengers?
c) Suppose you allow 220 reservations. What is the
probability that you will have to deal with bumped
passengers?
Newsvendor/Yield Mgt. Summary
• The model can be applied to settings in which …
– There is a single order/production/replenishment opportunity.
– Demand is uncertain.
– There is a “too much-too little” challenge:
• Firm must have a demand model that includes an expected demand
and uncertainty in that demand.
• Order at the “balancing point”
• Yield management and overbooking give demand flexibility where
supply flexibility is not possible.
• These are powerful tools to improve revenue
Recall that the full fare of the Park Hyatt Philly is $225, the
expected full-fare demand is ~P(27.3), the discount fare is
$159, and there are 118 king/queen rooms. Now suppose the
cost of an occupied room is $45 per night. That cost includes
the labor associated with prepping and cleaning a room, the
additional utilities used and the wear and tear on the furniture
and fixtures. Suppose the Park Hyatt wishes to maximize
expected profit rather than expected revenue. What is the
optimal protection level for the full fare?
Kiosk Redux
Suppose that any customer unable to purchase a burrito
settles for a lunch of a strawberry Pop-Tart and a soda. PopTarts sell for 75¢ and cost the Kiosk 25¢. (As Pop-Tarts and
soda are easily stored, the Kiosk never runs out of these
essentials.) How many burritos should Kiosk management
prepare?
a)
b)
c)
d)
Flextrola is developing a new product and Solectrics can produce one of the
key components for $72 per unit as long as Flextrola submits an order well
in advance of the selling season. Flextrola’s demand forecast is normally
distrbuted with a mean of 1000 and a standard deviation of 600. Flextrola
sells the unit after integrating some software for $121. Leftover units are
sold for $50 after the season. Xandova Electronics (XE) wants a piece of
this action and is able to offer 100% fill rate and one day delivery no matter
when the orders are submitted. Flextrola promises a one-week lead time, so
the one day lead from XE would allow Flextrola to operate with make-toorder production. XE’s price is $83.50 per unit.
What is Flextrola’s expected profit if they use XE as their sole supplier?
How many units should Flextrola buy from each if they use both suppliers?
Solectrics comes back to the table and offers an option contract of Q
options for $25 each. During the selling season Flextrola can exercise up to
the Q options with a one day lead time and the exercise price is $50 per
unit. If Flextrola wants more units beyond the options already purchased,
Solectrics will provide them at XE’s price of $83.50. How many options
should Flextrola purchase?
Given your answer to c) what is Flextrola’s profit?