A variety of concepts and analytical tools fall
under the label yield management. The term is used
in many service industries to describe techniques to allocate
limited resources, such as airplane seats or hotel rooms,
among a variety of customers, such as business or leisure
travelers. By adjusting this allocation a firm can optimize
the total revenue or "yield" on the investment in capacity.
Since these techniques are used by firms with extremely perishable
goods, or by firms with services that cannot be stored at
all, these concepts and tools are often called perishable
asset revenue management.
The techniques of yield management are relatively
new - the first research to deal directly with these issues
appeared less than 20 years ago. These days, yield management
has been an enormously important innovation in the service
industries. American Airlines credits yield management techniques
for a revenue increase of $500 million/year and Delta Airlines
uses similar systems to generate additional revenues of $300
million per year2
While the airlines are the oldest and most sophisticated users
of yield management, these practices also appear in other
service industries. For example, Marriott Hotels credits its
yield management system for additional revenues of $100 million
per year, with relatively small increases in capacity and
costs. Broadcasting companies use yield management to determine
how much inventory (advertising slots) to sell now to the
"upfront market" and how much to reserve and perhaps sell
later at a higher price to the "scatter market." Even manufacturers
are using yield management techniques to increase profits.
After all, manufacturing capacity is as perishable as an airline
seat or an advertising slot - if it is not used when it is
available, that opportunity to use the capacity is gone forever.
The purpose of this note is to introduce the
fundamental concepts and trade-offs of yield management and
to describe the parallels between yield management and the
newsvendor framework that is an important model for inventory
management. In particular, this note focuses on how a manager
might allocate perishable inventory among a variety of customer
segments. To acquire some intuition about the problem we will
spend most of our time with an application that involves two
types of customers. Consider an establishment we will call
the Eastman Towers Hotel of Rochester, NY. The hotel has 210
King/Queen rooms used by both business and leisure travelers.
The hotel must decide whether to sell rooms well in advance
at a relatively low price (i.e., to leisure travelers), or
to 'hold out' and wait for a sale at a higher price to late-booking
Our solution to this problem will be simple
- we will find a single 'cap' or 'booking limit' for the number
of rooms to sell in advance to leisure travelers. There are
also variations of the problem and the solution that are more
complicated, and these variations can improve the performance
of the yield management system. For example, we might change
the booking limit up or down as time passes, or we might assume
that some rooms may be used only for some customers while
other rooms are more flexible (think of 'economy' and 'deluxe'
rooms). We will discuss these extensions in the last section
of this note.
In general, the yield management/booking limit
decision is just one aspect of a more general business question:
How should a firm market and distribute goods to multiple
customer segments? To answer this question, a firm must use
tools for pricing and forecasting as well as for inventory
management. All of these tools are often grouped under the
term revenue management, although the boundaries between
yield management and revenue management are often ambiguous.
In this note we will specifically focus on just a single aspect
of revenue management - yield management.
We next identify the environments in which yield
management techniques are most successful. Sections 3-5 examine
a specific solution to the basic Eastman Towers problem, while
Section 6 will discuss a similar important problem, 'overbooking.'
In Section 7 we describe complications and extensions of the
basic problem. Appendix A is a
list of suggested additional readings on yield management,
Appendix B contains exercises that
test our understanding of the yield management problem3
2. Where and Why Firms Practice Yield Management
Business environments with the following five
characteristics are appropriate for the practice of yield
management (in parentheses we apply each characteristic to
the Eastman Towers Hotel):
- It is expensive or impossible to store excess resource
(we cannot store tonight's room for use by tomorrow night's
- Commitments need to be made when future demand is uncertain
(we must set aside rooms for business customers - "protect"
them from low-priced leisure travelers - before we know
how many business customers will arrive).
- The firm can differentiate among customer segments, and
each segment has a different demand curve (purchase restrictions
and refundability requirements help to segment the market
between leisure and business customers. The latter are more
indifferent to the price.).
- The same unit of capacity can be used to deliver many
different products or services (rooms are essentially the
same, whether used by business or leisure travelers).
- Producers are profit-oriented and have broad freedom
of action (in the hotel industry, withholding rooms from
current customers for future profit is not illegal or morally
irresponsible. On the other hand, such practices are controversial
in emergency wards or when allocating organs for transplantation).
Given these characteristics, how does yield
management work? Suppose that our hotel has established two
fare classes or buckets: full price and discount
price. The hotel has 210 rooms available for March 29
(let us assume that this March 29 is a Monday night). It is
now the end of February, and the hotel is beginning to take
reservations for that night. The hotel could sell out all
210 rooms to leisure travelers at the discount price, but
it also knows that an increasing number of business customers
will request rooms as March 29 approaches and that these business
customers are willing to pay full price. To simplify our problem,
let us assume that leisure demand occurs first and then business
demand occurs. Hence we must decide how many rooms we are
willing to sell at the leisure fare or in other words, how
many rooms shall we protect (i.e., reserve) for the full price
payers. If too many rooms are protected, then there may be
empty rooms when March 29 arrives. If too few are protected,
then the hotel forgoes the extra revenue it may have received
from business customers.
Notice we assume that the hotel can charge
two different prices for the same product (a room). To separate
the two customer segments with two different demand curves,
the hotel must practice market segmentation. In the hotel
example, we assume that the firm can charge different prices
to business and leisure customers4
. In order to differentiate between these two groups, a firm
often introduces booking rules that create barriers or "fences"
between market segments. For example, a Saturday-night stay
may be required to receive a discounted room on Monday, because
most business travelers prefer to go home on the weekend while
leisure customer are more likely to accept, or may even prefer,
the weekend stay. Again, leisure customers are more price-sensitive
so the hotel wishes to sell as many rooms to business customers
at a higher price as possible while keeping room utilization
3. Booking Limits and
Before we look at the mathematics that will
help us to make this decision, some vocabulary would be helpful.
We define a booking limit to be the maximum number
of rooms that may be sold at the discount price. As we noted
previously, we assume that leisure customers arrive before
business customers so the booking limit constraints the number
of rooms that these customers get: once the booking limit
is reached, all future customers will be offered the full
price. The protection level is the number of rooms
we will not sell to leisure customers because of the
possibility that business customers might book later in time.
Since there are 210 rooms available in the hotel, and just
two fare classes, in our example,
booking limit = 210 - the protection level
Therefore, the hotel's task is to determine
either a booking limit or a protection level, since knowing
one allows you to calculate the other. We shall evaluate the
protection level. Suppose the hotel considers protection level
'Q' instead of current protection level Q+1
(Q might be anything from 0 to 209). Further, suppose that
210-Q-1 rooms have already been sold (see Figure 1).
Now a prospective customer calls and wishes to reserve the
first 'protected' room at the discount price.
Figure 1: Protection Level and Booking
Limit in the Hotel
Should the hotel lower the protection level
from Q+1 to Q and therefore allow the booking
of the (Q+1)th room at the discount price? Or should
it refuse the booking to gamble that it will be able to sell
the very same room to a full price customer in the future?
The answer, of course, depends on (i) the relative sizes of
the full and discount prices and (ii) the anticipated demand
for full price rooms. The decision is illustrated as a decision
tree in Figure 2.
Figure 2: Deciding the protection level
In the next section we associate numbers with
this decision and find the optimal protection level, Q*.
4. Solving the Problem
To determine the value of each branch of the
decision tree in Figure 2, we need to know the probability
for each 'chance' branch and the values at the end of the
branches. Suppose that the discount price is $105 per night
while the full price is $159 per night. To find the probability
on each branch, define random variable D to represent
the anticipated demand for rooms at the full price.
The hotel may estimate the distribution of D from historical
demand, as well as from forecasts based on the day of the
week, whether there is a holiday, and other predictable events.
Here we will assume that the distribution is derived directly
from 123 days of historical demand, as shown in Table 1 below.
In the table, the 'Cumulative Probability'
is the fraction of days with demand at or below the number
of rooms in the first column (Q).
Table 1. Historical demand for rooms
at the full fare.
Now consider the decision displayed in Figure
2. If we decide to protect the (Q+1)th room
from sale, then that room may, or may not, be sold later.
It will be sold only if demand D at full fare is greater than
or equal to Q+1, and this event has probability
1-F(Q). Likewise, the protected room will not be sold
if demand is less than or equal to Q, with probability F(Q).
Figure 3 shows our decision with these values included.
Figure 3: Booking Limit Decision with Data
Given Figure 3, we can calculate the value
of lowering the protection level from Q+1 to Q.
Lowering the protection level results in selling the (Q+1)th
room at a discount which guarantees revenue of $105. Protecting
Q+1 rooms has an expected value equal to:
(1 - F(Q))($159) +F(Q)($0) = (1 - F(Q))
Therefore, we should lower the protection level
to Q as long as:
(1 - F(Q))($159)
- $105) / $159 = 0.339.
Now, F(Q) is the third column in Table 1. We
simply scan from the top of the table towards the bottom until
we find the smallest Q with a cumulative value greater than
or equal to 0.339. The answer here is that the optimal protection
level is Q*=79 with a cumulative value of 0.341. We can now
evaluate our booking limit: 210 - 79 = 131. If we choose a
larger Q*, then we would be protecting too many rooms thereby
leaving too many rooms unsold on average. If we set Q* at
a smaller value, we are likely to sell too many rooms at a
discount thereby turning away too many business customers
5. A General Formula
The solution described above is an example
of a standard technique that was developed for the airline
industry. The technique was named "Expected Marginal Seat
Revenue" (EMSR) analysis by Peter Belobaba at MIT5. In our
example we had two fare classes with prices rL
and rH (rH is the higher
price, $159, while rL is the lower price,
$105). We also had a random variable, D, to represent the
distribution of demand at the high fare. Since there are just
two fare classes, the optimal booking limit for low fare class
is equal to the total capacity minus Q*.
The EMSR analysis can actually be described
as a newsvendor problem. We make a fixed decision, Q,
the protection level, then a random event occurs, the number
of people requesting a room at the full fare. If we protect
too many rooms, i.e., D<Q, then we end up earning
nothing on Q-D rooms. So the overage penalty C
is rL per unsold room. If we protect too few rooms, i.e.,
D<Q, then we forgo rH-rL in potential revenue
on each of the D-Q rooms that could have been sold
at the higher fare so the underage penalty B is rH-rL.
The critical ratio is
From the newsvendor analysis, the optimal protection
level is the smallest value Q* such that
Another important component in the yield management
toolbox is the use of overbooking when there is a chance
that a customer may not appear. For example, it is possible
for a customer to book a ticket on an airline flight and not
show up for the departure. If that is the case, the airline
may end up flying an empty seat resulting in lost revenue
for the company. In order to account for such no-shows, airlines
routinely overbook their flights: based on the historical
rate of no-shows the firm books more customers than available
seats. If, by chance, an unusually large proportion of the
customers show up, then the firm will be forced to 'bump'
some customers to another flight. Hotels, rental car agencies,
some restaurants, and even certain non-emergency health-care
providers also overbook. When determining the optimal level
of overbooking, the calculation is similar to the calculation
used for yield management. The optimal overbooking level balances
(i) lost revenue due to empty seats and (ii) penalties (financial
compensation to bumped customers) and loss of customer goodwill
when the firm is faced with more demand than available capacity.
Let X be the number of no-shows and
suppose we forecast the distribution of no-shows with distribution
function F(x). Let Y be the number of seats
we will overbook, i.e., if the airplane has S seats
then we will sell up to S+Y tickets6
. As in the newsvendor model, define the underage penalty
by B and the overage penalty by C. In this case
C represents ill-will and the net penalties that are
associated with refusing a seat to a passenger holding a confirmed
reservation (the 'net' refers to the fact that the airline
may collect and hold onto some revenue from this passenger
as well). Here, B represents the opportunity cost of
flying an empty seat. To explain further, if X>Y then
we could have sold X-Y more seats and those passengers
would have seats on the plane. So B equals the price
of a ticket. If X<Y then we need to bump Y-X
customers and each has a net cost of C. Thus, the formula
for an optimal number overbooked seats takes following familiar
form: is smallest value Y* such that
As an example, suppose the Eastman Towers Hotel
described above estimates that the number of customers who
book a room but fail to show up on the night in question is
Normally distributed with mean 20 and standard deviation 10.
Moreover, Eastman Towers estimates that it costs $300 to "bump"
a customer (the hotel receives no revenue from this customer,
and the $300 is the cost of alternative accommodation plus
a possible gift certificate for a one-night stay at Eastman
in the future). On the other hand, if a room is not sold then
the hotel loses revenue equal to one night sold at a discount
since at the very least this room could have been sold to
. How many bookings should Eastman allow? The critical ratio
From the Normal distribution table we get (-0.65)
= 0.2578 and (-0.64)
= 0.2611, so the optimal z* is approximately -0.645
and the optimal number of rooms to overbook is Y*=20-0.645*10=13.5
(one can also use Excel's Norminv function for
this calculation: "=Norminv(0.2592,20,10)" gives us
an answer of 13.5). If we round up to 14, this means
that up to 210+14=224 bookings should be allowed.
7. Complications and
There are a wide variety of complications we
face when implementing a yield management system. Here we
discuss a few of the more significant challenges.
In the examples above, we used historical demand
to predict future demand. In an actual application, we may
use more elaborate models to generate demand forecasts that
take into account a variety of predictable events, such as
the day of the week, seasonality, and special events such
as holidays. In some industries greater weight is given to
the most recent demand patterns since customer preferences
change rapidly. Another natural problem that arises during
demand forecasting is censored data, i.e., company often does
not record demand from customers who were denied a reservation.
In our example in Sections 3-5 we used a discrete,
empirical distribution to determine the protection level.
A statistical forecasting model would generate a continuous
distribution, such as a Normal or t distribution. Given a
theoretical distribution and its parameters, such as the mean
and variance, we would again place the protection level where
the distribution has a cumulative probability equal to the
Dynamic Booking Limits
By observing the pattern of customer arrivals,
firms can update their demand forecasts, and this may lead
to changes in the optimal booking limit. In fact, many airline
yield management systems change booking limits over time in
response to the latest demand information. For example, it
is possible that an airline may raise a booking limit as it
becomes clear that demand for business-class seats will be
lower than was originally expected. Therefore, during one
week a leisure customer may be told that economy-class seats
are sold out, but that same customer may call back the next
week to find that economy seats are available.
Variation and Mobility of Capacity
Up to now, we have assumed that all units of
capacity are the same; in our Eastman example we assumed that
all 210 hotel rooms were identical. However, rooms often vary
in size and amenities. Airlines usually offer coach and first-classes.
Car rental firms offer subcompact, compact, and luxury cars.
In addition, car rental firms have the opportunity to move
capacity among locations to accommodate surges in demand,
particularly when a central office manages the regional allocation
of cars. The EMSR framework described above can sometimes
be adapted for these cases, but the calculations are much
more complex. Solving such problems is an area of active research
in the operations community.
Mobility of Customer Segments
In the example of Sections 3-5 we assumed that
a leisure customer who is not able to book a room at the discount
price (because of the booking limit) does not book any room
from that hotel. In fact, some proportion of leisure customers
who are shut out from discount rooms may then attempt book
a room at the full price. The possibility that a customer
may 'buy-up' complicates the model but, most modern yield
management systems take such customer movements into account.
Nonlinear Costs for Overbooking
In some cases the unit cost of overbooking
increases as the number of 'extra' customers increases. This
is particularly true if capacity is slightly flexible. For
example, a rental car agency faced with an unexpected surge
of customers may be able to obtain, for a nominal price, a
few extra cars from a nearby sister facility. However, at
some point the agency's extra supply will run out - and then
customers will be lost, resulting in a much greater cost.
Customers in a Fare Class Are Nor All Alike
While two leisure travelers may be willing
to pay the same price for a particular night's stay, one may
be staying for just one day while the other may occupy the
room for a week. A business traveler on an airplane flight
may book a ticket on just one leg or may be continuing on
multiple legs. Not selling a ticket to the latter passenger
means that revenue from all flight legs will be lost.
In each of these cases, the total revenue
generated by the customer should be incorporated into the
yield management calculation, not just the revenue generated
by a single night's stay or a single flight leg. There are
additional complications when code-sharing partners (distinct
airlines that offer connecting flights among one another)
operate these flight legs. If code-sharing occurs, then each
of the partners must have an incentive to take into consideration
the other partners' revenue streams.
A similar complication occurs when there are
group bookings. How should the hotel consider a group booking
request for 210 rooms at a discount rate? Clearly, this is
different from booking 210 individual rooms since denying
a reservation to one out of 210 group customers may mean that
all 210 will be lost to the hotel. Frequently, companies restrict
group reservations during the peak seasons but up till now
there are no general rules for handling group reservations.
Table 2 summarizes the impact of these issues
on three industries: airlines, hotels and car rental firms8
. Sophisticated yield management tools have been developed
in all three industries, and these tools take industry-specific
factors into account. However, all of these tools are based
on the basic EMSR model described above.
Table 2: Comparison of
yield management applications
Appendix A. Suggested further readings and
Handbook of Airline Economics. 1995.
Aviation Week Group, a division of McGraw-Hill Companies,
Chapters 47 - 50.
Handbook of Airline Marketing. 1998. Aviation Week
Group, a division of McGraw-Hill Companies, Chapters 23 -
R. G. Cross. 1996. Revenue Management: Hard-Core Tactics
for Market Domination. Broadway Books.
Dhebar, A. and A. Brandenburger. 1993. American Airlines,
Inc.: Revenue Management. Harvard Business School Case.
M. K. Geraghty and E. Johnson. 1997. Revenue Management Saves
National Car Rental. Interfaces. Vol.27, pp. 107-127.
J. I. McGill and G. J. Van Ryzin. 1999. Revenue Management:
Research Overview and Prospects. Transportation Science.
Vol.33, pp. 233-256.
R. D. Metters. Yield Management at Pinko Air. Southern Methodist
B. C. Smith, J. F. Leimkuhler, and R. M. Darrow. 1992. Yield
Management at American Airlines. Interfaces. Vol.22,
K. T. Talluri and G. J. van Ryzin. 2002. The Theory and Practice
of Revenue Management. Kluwer Academic Publishers, Dordrecht,
The Netherlands. To be published.
G. J. van Ryzin. 1998. Transportation National Group. Columbia
Appendix B. Suggested exercises.
Recall the yield management problem faced by
the Eastman Towers Hotel, as described in the note "Introduction
to the Theory and Practice of Yield Management." A competing
hotel has 150 rooms with standard Queen-size beds and two
rates: a full price of $200 and a discount price of $120.
To receive the discount price, a customer must purchase the
room at least two weeks in advance (this helps to distinguish
between leisure travelers, who tend to book early, and business
travelers who value the flexibility of booking late). You
may assume that if a leisure traveler is not able to get the
discount rate, she will choose to book at another hotel.
For a particular Tuesday night, the hotel estimates
that the average demand by business travelers has a mean of
70 rooms and a standard deviation of 29 rooms. Assume that
demand follows a Normal distribution around the forecast.
- Find the optimal protection level for full price rooms
(the number of rooms to be protected from sale at a discount
- Find the booking limit for discount rooms.
- Suppose that for a short time, the hotel's forecast of
business customer demand is biased upward: the forecast
of 70 rooms is too high and fewer business customers appear,
on average. Qualitatively describe the economic consequences
of using the protection level and booking limit derived
in (a) and (b).
- Suppose that for a short time, the hotel's forecast of
business customer demand is biased downward: the
forecast of 70 rooms is too low and more business customers
appear, on average. Qualitatively describe the economic
consequences of using the protection level and booking limit
derived in (a) and (b).
An airline offers two fare classes for coach
seats on a particular flight: full-fare class at $440/ticket
and economy class at $218/ticket. There are 230 coach seats
on the aircraft. Demand for full-fare seats has a mean of
43, a standard deviation of 8, and the following empirical
Economy-class customers must buy their tickets
three weeks in advance, and these tickets are expected to
- Find the (i) protection level and (ii) booking limit for
- Suppose that unsold seats may sometimes be sold at the
last minute at a very reduced rate (similar to USAirways'
"esavers" for last-minute travel). What effect will this
have on the protection level calculated in (a)?
The protection level (Q*) will be…
Circle one: Higher Lower The same
Sunshine Airlines flies a direct flight from
Baltimore to Orlando. The airline offers two fares: a 14-day
advance purchase economy fare of $96 and a full fare of $146
with no advance purchase requirement. In its yield management
system, Sunshine has two fare classes and, based on historical
demand patterns, has established a booking limit of 90 seats
for the economy fare class.
Because KidsWorld is located near Orlando, Sunshine strikes
a deal with KidsWorld Inc. The deal allows Sunshine to offer
a 5-day KidsWorld pass to anyone who purchases an economy-class
Sunshine ticket (the customer pays $96 for the Sunshine ticket
and an additional $180 for the KidsWorld pass, a discount
from the regular price of $200 for a 5-day pass purchased
directly from KidsWorld). Sunshine passes along the $180 pass
revenue to KidsWorld, but as part of the agreement, Sunshine
receives a $50 incentive or 'kick-back' from KidsWorld for
every pass sold through Sunshine.
Given that Sunshine has accepted the deal with KidsWorld,
should Sunshine change the booking limits and protection levels
for the Orlando flight? Should it make other changes to its
fare class structure? If not, why not? If yes, how? Your answer
should be qualitative, but you should be as specific
WZMU is a television station that has 25 thirty-second
advertising slots during each evening. The station is now
selling advertising for the first few days in November. They
could sell all the slots now for $4,000 each, but because
November 7 will be an election day, the station may be able
to sell slots to political candidates at the last minute for
a price of $10,000 each. For now, assume that a slot not sold
in advance and not sold at the last minute is worthless
To help make this decision, the sales force has created the
following probability distribution for last-minute sales:
- How many slots should WZMU sell in advance?
- Now suppose that if a slot is not sold in advance and
is not sold at the last minute, it may be used for a promotional
message worth $2500. Now how many slots should WZMU sell
An aircraft has 100 seats, and there are two
types of fares: full ($499) and discount ($99).
- While there is unlimited demand for discount fares, demand
for full fares is estimated to be Poisson with mean l=20
(the table below gives the distribution function). How many
seats should be protected for full-fare passengers?
- An airline has found that the number of people who purchased
tickets and did not show up for a flight is normally distributed
with mean of 20 and standard deviation of 10. The airline
estimates that the ill will and penalty costs associated
with not being able to board a passenger holding confirmed
reservation are estimated to be $600. Assume that opportunity
cost of flying an empty seat is $99 (price that discount
passenger would pay). How much should airline overbook the
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