Single Resource Revenue Management Problems withDependent De.docx

Single Resource Revenue Management Problems with
Dependent Demands c©
Guillermo Gallego
Spring 2013
Abstract
Providers of perishable capacity, such as airline seats and hotel
rooms, use market
segmentation tools to sell the same capacity to different
customers at difference prices.
They impose time-of-purchase and usage restrictions such as
Saturday Night stays on
low fares to increase demand while limiting bookings to prevent
demand cannibalization
from customers who book late and travel for business. Capacity
providers may also
bundle or unbundle ancillary services such as advance seat
selection, meals, mileage
accrual, and luggage handling to justify price differences for the
same basic service. We
will assume that the fare class structure (a menu of prices,
restrictions and ancillary
services) is given and that the capacity provider’s goal is to
allocate capacity among
fare classes to maximize expected revenues net of the cost of
the ancillary services.
We assume that customer demand is governed by a discrete
choice model and depends
on the products available for sale. Thus, if a customer does not

find his preferred
product, he may decide not to purchase or to purchase an
alternative product in the
offer set. The problem is to decide, at each state, which set of
products to make available
for sale. Since there are an exponential number of sets that can
be offered, we study
the structure of the optimization problem to define and
characterize efficient sets with
the purpose of reducing the dimensionality of the optimization
problem and to obtain
insights into the type of products, e.g., nested-by-fares, that
may be optimal to offer. We
provide a dynamic programming formulation for the revenue
management problem and
use fare and demand transformations to relate it to the
formulation of the independent
demand model. We present an upper bound on the value
function throughout the use
of approximate dynamic programming with affine functions. A
formulation that does
not allow fares to be opened once they are closed is also
presented. We then turn
our attention to static models, where the time element is
suppressed. We present an
optimal solution to the two fare problem and an efficient
heuristic for the multi-fare
case. Numerical examples show that the proposed heuristic
works almost as well as the
optimal solution for the restricted dynamic programming model
where fares cannot be
reopened once they are closed.
1

1 Introduction
Suppliers of perishable capacity often offer a menu of products
that vary in terms of price and
quality. If the products differ only on price we would expect
most, if not all, customers to
buy the lowest priced product that gives him a positive surplus
and to walk away otherwise.
When differences in quality are also present, customers do not
always buy the lowest priced
product. As an example, a customer may be willing to pay an
extra $50 for a room with an
ocean view or $29 for advance seat selection and priority
boarding.
When the difference between products in menu is small, there is
a real likelihood that a
customer will buy a different product if his preferred product is
not available. This is known
as demand recapture. Demand that is lost because a product is
not offered is called spilled
demand. Under the independent demand model, all demand for
an unavailable product is
spilled. In contrast, under a discrete choice model, part of this
demand may be recaptured
by other available products. In the context of revenue
management, the independent demand
model results in low protection levels for higher fare classes as
it ignores demand recapture.
This causes higher spill rates among the higher fare classes and
leads to a downward spiral in
revenues as estimates of demands for high fare classes become
lower over time as explained in
Cooper, Homem de Mello and Kleywegt [4].

Revenue mangers have been struggling for decades with the
problem of finding optimal
control mechanisms for fare class structures with dependent
demands. Many attempts have
been made by research groups in industry to cope with this
problem. At the same time aca-
demic research has moved vigorously to tackle the problem of
capacity allocation under choice
models. A key difference between is how the notion of time is
handled. Practitioners tend to
prefer to model time implicitly, e.g., through static models that
are extensions of Littlewood’s
rule and EMSR type heuristics. However, finding the right way
to extend Littlewood’s rule
proved to be more difficult than anticipated. The key
complication is that the marginal value
of capacity is more difficult to compute due to the fact that
protecting an additional unit of
capacity for later sale also alters the number of potential
customers (as some of their demand
may be recaptured). Nevertheless, it is possible to overcome
this difficulty and find static
heuristic policies that are easy to implement and perform almost
as well as dynamic policies.
Academic researchers tend to prefer models where time is
handled explicitly. In contrast to
the independent demand formulations, where static models are
relatively easier to understand
and implement, the opposite seems to be true for dependent
demand models. Indeed, dynamic
formulations of the dependent demand model are only
marginally more complicated to set
up. The problem is a bit more difficult because it requires
specifying a choice model and
identifying the collection of “efficient” sets of products that

may be offered at different states
of the system. Once the efficient sets are identified, either
exactly or approximately, the
problem can be reformulated to be essentially equivalent to that
of the independent demand
model.
In this chapter we will explore both formulations. We start by
introducing discrete choice
models as a tool to model demand dependencies. After
introducing choice models and provid-
ing a variety of examples, we discuss the sales and revenue
rates associated with each set and
give a formal definition of efficient sets. Essentially efficient
sets maximize the revenue rate
2
among all convex combinations of sets whose sales rate is
bounded by the sales rate of the
efficient set. The notion of efficient sets helps simplify the the
dynamic programming formu-
lations so only efficient sets need to be considered. Structural
results are presented that are
useful both in terms of computations and in understanding
dynamic policies. In particular,
we relate the formulation to that of the independent demand
model and to dynamic pricing
with finite price menus. We then present an upper bound on
revenues, and a variant of the
dynamic model where fares cannot be opened once they are
closed. We end the chapter with
a discussion of static models that handle time implicitly. As the
reader will see, these models

are complicated by the fact that changing the protection level
also changes the number of
potential customers for higher fare classes.
2 Discrete Choice Models
We will assume that the set of potential products to be offered
is N = {1, . . . ,n}. For any
subset S ⊂ N, denote by πj(S) the probability that a customer
will select j ∈ S with πj(S) = 0
if j ∈ S′ = {j ∈ N : j /∈ S}. Let π(S) =
∑
j∈ S πj(S) denote the probability of sale when subset
S is offered. The complement π0(S) = 1−π(S) denotes the
probability that a customer selects
an outside alternative. An outside alternative may mean either
that the customer does not
purchase or that he purchases from another vendor. Let S+ =
S∪ {0} denote the set of offered
products together with the outside alternative. Then πS+ (S) =
π(S) + π0(S) = 1. Notice
that this notation implies that the no-purchase alternative j = 0
is always available. For this
reason, it would be more appropriate to write πj(S+) for j ∈ S+.
However, we follow here the
convention of writing πj(S) instead of the more cumbersome
πj(S+) with the understanding
that the no-purchase alternative j = 0 is always available. This
choice of notation makes it
easier to refer to the set S ⊂ N of offered products, but the
reader needs to remember that
implicitly the no-purchase alternative is always available to the
customer. We will describe a
few commonly used choice models, define efficient sets, and

then move on to the problem of
dynamic capacity allocation.
One reason to look at discrete choice models is to better
understand how customers make
choices. Another important reason for our purposes is to find a
subset of products that
maximizes the expected profit or expected revenues among all S
⊂ N.
2.1 The Independent Demand Model (IDM)
Under this model πj(S) is independent of the offer set S
(containing j). Under the IDM all
the demand for fare j is lost (to the no-purchase alternative) if j
is removed from S. This
implies that there are non-negative constants, say v0,vj,j ∈ N,
such that
πj(S) =
vj
v0 + V (N)
j ∈ S
where V (N) =
∑
j∈ N vj ( the constants can be normalized so that v0 + V (N) =
1). In most
practical situations, it is reasonable to expect that some of the
demand for fare j may be
3

recaptured by other products in S. The IDM is pessimistic in the
sense that it ignores recap-
ture. This may lead to incorrect decisions in the context it is
used. In Revenue Management
applications it usually leads to offering too much capacity at
discounted fares. In assortment
planning, it may lead to offering excessive product variety.
2.2 The Basic Attraction Model and the Multinomial Logit
Model
The Basic Attraction Model (BAM) is a discrete choice model
where each fare j ∈ N has an
attraction value vj > 0 and v0 > 0 represents the attractiveness
of the no-purchase alternative.
The choice model is given by
πj(S) =
vj
v0 + V (S)
∀ j ∈ S, (1)
where V (S) =
∑
j∈ S vj. Consequently, products with higher attraction values are
more likely
to be selected.
The BAM was first proposed by Luce [12] who postulated two
choice axioms and demon-
strated that a discrete choice model satisfies the axioms if and
only if it is of the BAM form.
To adequately describe the Luce Axioms we need additional

notation. For any S ⊂ T we let
πS(T) =
∑
j∈ S πj(T) denote the probability that a customer selects a
product in S when the
set T is offered. Also, πS+ (T) = πS(T) + π0(T) = 1 −πT−S(T),
where set differences T −S
mean T ∩S′. The Luce Axioms can be written as:
• Axiom 1: If πi({i}) ∈ (0, 1) for all i ∈ T, then for any R ⊂ S+,
S ⊂ T
πR(T) = πR(S)πS+ (T).
• Axiom 2: If πi({i}) = 0 for some i ∈ T, then for any S ⊂ T
such that i ∈ S
πS(T) = πS−{i}(T −{i}).
Axiom 1 implies that the probability of selecting any set R ⊂
S+, when set T is offered, is
equal to the probability of selecting R when S is offered times
the probability of selecting S+
when T is offered assuming that S ⊂ T. Axiom 2 implies that if
alternative i has no probability
of being chosen, then it can be deleted from S without affecting
the choice probabilities (Luce
[13]).
The celebrated multinomial logit (MNL) model, see McFadden
[16], is a special case of
the BAM that arises from a random utility model. Under a
random utility model, each
product has random utility Ui, i ∈ N+ and the probability that j

∈ S is selected is given
by πj(S) = P(Uj ≥ Ui, i ∈ S+). In words, product j ∈ S is
selected if it gives as much
utility as any other product in S+. More specifically, McFadden
models the random utility of
product i, as Ui = µi + �i, where µi is the mean utility of
product i and depends on the price
and quality of product i. The term �i is modeled as an extreme
value distribution known as
a Gumbel random variable with parameter φ. The �is are
assumed to be independent and
4
identically distributed. We can think of �i as an idiosyncratic
variation on the mean utility
or as errors in measuring the utility. Under these assumptions,
πj(S) is of the BAM form
with vi = e
φµi, i ∈ N. The parameter φ is inversely related to the variance
of the �js. As
φ becomes large, the variance of the �js, becomes small, and
customers will gravitate to the
product in S with the largest mean utility µi (what is known as
the maximum utility model).
On the other hand, when φ becomes small the probability of
selecting any i ∈ S converges
to a uniform distribution where each product is equally likely to
be selected. This is because
when the variance is much larger than the mean, the customer
loses the ability to reliability
select products with higher mean utility.

2.3 The Generalized Attraction Model
There is considerable empirical evidence that the basic
attraction model (BAM) may be too
optimistic in estimating demand recapture probabilities when
the customer’s first choice is
not part of the offer set S. The BAM assumes that even if a
customer prefers j ∈ S′, he must
select among k ∈ S+. This ignores the possibility that the
customer may look for products
j ∈ S′ elsewhere or at a later time. As an example, suppose that
a customer prefers a certain
wine, and the store does not have it. The customer may then
either buy one of the wines
in the store, go home without purchasing, or drive to another
store and look for the specific
wine he wants. The BAM precludes the last possibility; it
implicitly assumes that the search
cost for an alternative source of product j ∈ S′ is infinity, or
equivalently that there is no
competition.
As an illustration, suppose that the consideration set is N = {1,
2} and that v0 = v1 =
v2 = 1, so πk({1, 2}) = 33.3̄% for k = 0, 1, 2. Under the BAM,
eliminating choice 2 results in
πk({1}) = 50% for k = 0, 1. Suppose, however, that product 2 is
available across town and
that the customer’s attraction for product 2 from the alternative
source is w2 = 0.5 ∈ [0,v2].
Then his choice set, when product 2 is not offered, is in reality
S = {1, 2′} with 2′ representing
product 2 in the alternative location with shadow attraction w2.
A customer who arrives to
the store has attractiveness v0 = v1 = 1 and w2 = 0.5, resulting
in

π0({1}) =
1.5
2.5
= 60%, π1({1}) =
1
2.5
= 40%.
This formulation may also help mitigate the optimism of the
BAM in inter-temporal models
where a choice j ∈ S′ may become available at a later time. In
this case w2 is the shadow
attraction of choice 2 discounted by time and the risk that it
may not be available in the
future.
To formally define the general attraction model (GAM),
introduced by Gallego, Ratliff
and Shebalov [9], we assume that in addition to the attraction
values vk,k ∈ N = {1, . . . ,n},
there are shadow attraction values wk ∈ [0,vk],k ∈ N such that
for any subset S ⊂ N
πj(S) =
vj
v0 + W(S′) + V (S)
j ∈ S, (2)
5

where W(R) =
∑
j∈ R wj for all R ⊂ N. For the GAM,
π0(S) =
v0 + W(S
′)
v0 + W(S′) + V (S)
is the probability of the no-purchase alternative. The case wk =
0,k ∈ N recovers the BAM,
while the case wk = vk,k ∈ N recovers the independent demand
model (IDM). As with the
BAM it is possible to normalize the parameters so that v0 = 1
when v0 > 0. The parsimonious
GAM (p-GAM) is given by wj = θvj ∀ j ∈ N for some θ ∈ [0,
1]. The p-GAM can serve to test
the competitiveness of the market, by testing the hypothesis H0
: θ = 0 or H0 : θ = 1 against
obvious alternatives to determine whether one is better off
deviating, respectively, from either
the BAM or the IDM.
There is an alternative, perhaps simpler, way of presenting the
GAM by using the following
transformation: ṽ0 = v0 + w(N) and ṽk = vk −wk,k ∈ N. For S
∈ N, let Ṽ (S) =
∑
j∈S ṽj.
With this notation the GAM becomes:

πj(S) =
vj
ṽ0 + Ṽ (S)
∀ j ∈ S and π0(S) = 1 −π(S). (3)
For S ⊂ T we will use the notation πS+ (T) =
∑
j∈ S+ πj(T).
Gallego, Ratliff and Shebalov [9] proposed the following
generalization to the Luce Axioms:
• Axiom 1’: If πi({i}) ∈ (0, 1) for all i ∈ T, then for any non-
negative R ⊂ S ⊂ T
πR(T)
πR(S)
= 1 −
∑
j∈ T−S
(1 −θj)πj(T)
for some set of values θj ∈ [0, 1],j ∈ N.
• Axiom 2: If πi({i}) = 0 for some i ∈ T, then for any S ⊂ T
such that i ∈ S
πS(T) = πS−{i}(T −{i}).
They call the set of Axioms 1’ and 2 the Generalized Luce
Axioms (GLA). The special
case θj = 0 for all j recovers the original Luce Axiom 1,

resulting in the BAM, while the
case θj = 1 for all j reduces to the IDM. Gallego et al [9]
establish the following connection
between the GLA and the GAM. The proof of Theorem 1 can be
found in the Appendix.
Theorem 1 A Discrete Choice Model satisfies the GLA if and
only if is of the GAM form.
We will now show that the GAM also arises as the limit of the
nested logit (NL) model.
The NL model was originally proposed in Domencich and
McFadden (1975) [5], and later
refined in McFadden(1978) [15], where it is shown that it
belongs to the class of Generalized
Extreme Value (GEV) family of models. Under the nested
choice model, customers first select
a nest and then an offering within the nest. The nests may
correspond to product categories
6
and the offerings within a nest may correspond to different
variants of the product category.
As an example, the product categories may be the different
modes of transportation (car vs.
bus) and the variants may be the different alternatives for each
mode of transportation (e.g.,
the blue and the red buses). As an alternative, a product
category may consist of a single
product, and the variants may be different offerings of the same
product by different vendors.
We are interested in the case where the random utility of the
different variants of a product

category are highly correlated. The NL model, allows for
correlations for variants in nest i
through the dissimilarity parameter γi. Indeed, if ρi is the
correlation between the random
utilities of nest i offerings, then γi =
√
1 −ρi is the dissimilarity parameter for the nest. The
case γi = 1 corresponds to uncorrelated products, and to a BAM.
The MNL is the special
case where the idiosyncratic part of the utilities are independent
and identically distributed
Gumbel random variables.
Consider now a NL model where the nests corresponds to
individual products offered in
the market. Customers first select a product, and then one of the
vendors offering the selected
product. Let Ok be the set of vendors offering product k, Sl be
the set of products offered
by vendor l, and vkl be the attraction value of product k offered
by vendor l. Under the NL
model, the probability that a customer selects product i ∈ Sj is
given by
πi(Sj) =
(∑
l∈ Oi v
1/γi
il
)γi
v0 +

∑
k
(∑
l∈ Ok v
1/γk
kl
)γk v
1/γi
ij∑
l∈ Oi v
1/γi
il
, (4)
where the first term is the probability that product i is selected
and the second term the
probability that vendor j is selected. Many authors, e.g.,
[Greene(1984)], use what is know
as the non-normalized nested models where vijs are not raised
to the power 1/γi. This is
sometimes done for convenience by simply redefining vkl ← v
1/γk
kl . There is no danger in using
this transformation as long as the γs are fixed. However, the
normalized model presented here
is consistent with random utility models, see [Train(2002)], and
we use the explicit formulation
because we will be taking limits as the γs go to zero.

It is easy to see that πi(Sj) is a BAM for each j, when γi = 1 for
all i. This case can be
viewed as a random utility model, where an independent
Gumbel random variable is associated
with each product and each vendor. Consider now the case
where γi ↓ 0 for all i. At γi = 0, the
random utilities of the different offerings of product i are
perfectly and positively correlated.
This makes sense when the products are identical and price and
location are the only things
differentiating vendors. When these differences are captured by
the deterministic part of the
utility, then a single Gumbel random variable is associated with
each product. In the limit,
customers select among available products and then buy from
the most attractive vendor. If
several vendors offer the same attraction value, then customers
select randomly among such
vendors. The next result shows that a GAM arises for each
vendor, as γi ↓ 0 for all i.
Theorem 2 The limit of (4) as γl ↓ 0 for all l is a GAM. More
precisely, there are attraction
values akj, ãkj ∈ [0,akj], and ã0j ≥ 0, such that
πi(Sj) =
aij
ã0j +
∑
k∈Sj ãkj
∀ i ∈ Sj ∀ j.
7

This shows that if customers first select the product and then
the vendor, then the NL
choice model becomes a GAM for each vendor as the
dissimilarity parameters are driven down
to zero. The proof of this result is in the Appendix. Theorem 2
justifies using a GAM when
some or all of the products offered by a vendor are also offered
by other vendors, and customers
have a good idea of the price and convenience of buying from
different vendors. The model
may also be a reasonable approximation when products in a nest
are close substitutes, e.g.,
when the correlations are high but not equal to one. Although
we have cast the justification
as a model that arises from external competition, the GAM also
arises as the limit of a NL
model where a firm has multiple versions of products in a nest.
At the limit, customers go
for the product with the highest attraction within each nest.
Removing the product with the
highest attraction from a nest shifts part of the demand to the
product with the second highest
attraction , and part to the no-purchase alternative.
Consequently a GAM of this form can
be used in Revenue Management to model buy up behavior
when there are no fences, and
customers either buy the lowest available fare for each product
or do not purchase.
2.4 Mixtures of BAMs
It can be shown that any discrete choice model that arrises from
a random utility model can

be approximated to any degree of accuracy by a mixture of
BAMs, see McFadden and Train
(2000) [17]. Unfortunately, as we will later see, the mixture of
logits model leads to difficulties
in optimization when used in RM or assortment planning.
2.5 Markov Driven Discrete Choice Models
Blanchet, Gallego and Goyal (2013), see [2], consider a Markov
Driven Discrete Choice Model
(MDDCM) that is characterized by the probability distribution
qi = πi(N), i ∈ N+ and by a
product substitution matrix Q = (qij), i,j ∈ N+. The vector q of
probabilities (known as first
choice probabilities) corresponding to the choice distribution
that arises when all the products
are available. The substitution probability qij is the probability
that a customer whose first
choice demand is i will substitute to product j when i is
unavailable, with qii = 0 for i 6= 0.
For i 6= 0, qi0 representing the probability that the customer
substitutes i for the no-purchase
alternative. We define q0i = 0 for all i 6= 0 and q00 = 1 as there
is no substitution from the
no-purchase alternative. Mathematically, we can define the
substitution matrix for i,j ∈ N,
i 6= j, via the equation qij = (πj(N − i) − qj)/qi, where N − i = {j
∈ N : j 6= i}. Notice that
qij is the rate at which customers substitute to j when the find i
unavailable. A discrete choice
model arises under the Markovian assumption that if j is also
unavailable, then the customer
(who originally preferred i) will now substitute according to qjk
for k 6= j. As we will explain
later, it is relatively easy to compute the sales rate π(S) and the
revenue rate r(S) associated

with any subset S ⊂ N. Moreover, it is also easy to find the
assortment that maximizes r(S).
One advantage of the MDDCM is that it is very easy to do
assortment planning and RM
optimization with it. Also, the MDDCM can be used to
accurately approximate the behavior
of a mixture of BAMs. This approximation, together with the
ease of optimization, allows for
efficient heuristics for RM and assortment planning for the
mixture of BAMs.
8
2.6 Revenues and Efficient Assortments
We assume without loss of generality that the products are
labeled so that their associated
fares are decreasing1 in j. More precisely, the fares are given by
p1 ≥ p2 ≥ . . . ≥ pn. For any
subset S ⊂ N, let r(S) =
∑
k∈ S pkπk(S) denote the expected revenue when set S is offered.
We will now present a number of examples, and for each
example, we will look into the
collection (π(S),r(S)),S ⊂ N, representing the sales probability
and the expected revenue
from offering set S ⊂ N. Certain of these sets will be special
and play an important role in
revenue management.
Example 1. (BAM) Assume that customers have linear
sensitivities to price and quality:

βp = −1 and βq = 1000. Then a product with price p and quality
q has mean utility
µ = βpp + βqq = −p + 1000q. If φ = .01 then the attractiveness
of the product is v =
exp(.01(−p + 1000q)). Table 1 shows the price and quality of
three different products as well
as µ and v. Table 2 shows the sale probability π(S) and the
expected revenue r(S) of all
the eight subsets, in increasing order of π(S) assuming that v0 =
0. Some subsets in Table 2
are in bold and correspond to the efficient sets: E0 = ∅ , E1 =
{1} and E2 = {1, 2}. We
will later give a precise definition of efficiency but roughly
speaking the efficient sets can be
graphically represented as those that lie in the least, increasing
concave majorant of the graph
(π(S),r(S)),S ⊂ N as can be seen in Figure 1, where a graph of
the (π(S),r(S)) in increasing
order of π(S) as well as the upper concave majorant that goes
through the efficient sets.
(p1,q1) (p2,q2) (p3,q3)
(1000, 1) (850, 0.9) (650, 0.5)
(µ1,v1) (µ2,v2) (µ3,v3)
(0, 1.0) (50, 1.65) (−150, 0.22)
Table 1: Parameters of MNL model with βp = −1, βq = 1000 and
v0 = 1
S π(S) r(S)
∅ 0% $0.00
{3} 18% $118.58
{1} 50% $500.00
{1,3} 55% $515.06
{2} 62% $529.09
{2,3} 65% $538.48

{1,2} 73% $658.15
{1,2,3} 74% $657.68
Table 2: Sets, Sale Rates and Revenue Rates (Efficient Sets are
in Bold)
Example 2. (GAM) We reconsider Example 1 with w1 = 0, w2 =
0.5 and w3 = 0.2. This
means that there are negative externalities associated with not
offering products 2 and 3. The
sales rates and revenue rates are given by Table 3 and also in
Figure 2. The efficient sets are
now E0 = ∅ and Ei = Si = {1, . . . , i} for i = 1, 2, 3. It is
interesting to contrast Tables 2
1We will use increasing and decreasing in the weak sense unless
noted otherwise.
9
$0.00
$100.00
$200.00
$300.00
$400.00

$500.00
$600.00
$700.00
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Re
ve
nu
e
Ra
te
Demand

Rate
Revenue
Efficient
Fron<er
Figure 1: Example 1: The inefficient sets are the diamonds
below the curve
and 3. For the BAM of Table 2 offering set S2 = {1, 2} results
in revenue r(S2) = $658.15.
However offering S2 = {1, 2} in Example 2 results in a
significant reduction of revenue since
more of the demand from Product 3 is lost under the GAM than
under the BAM. Notice that
offering set S3 = {1, 2, 3} hurts revenues, relative to offering
set S2, under the BAM but helps
under the GAM. This is because under the BAM the incremental
revenue from adding fare 3
to set S2 is smaller than the loss from demand cannibalization.
In contrast, under the GAM,
the incremental revenue from adding fare 3 to set S2 is larger
than the loss from demand
cannibalization.
S π(S) r(S)
∅ 0% $0.00
{3} 13% $84.17
{1} 37% $370.37
{1,3} 45% $420.48
{2} 58% $491.94
{2,3} 65% $538.48
{1,2} 69% $623.95
{1,2,3} 74% $657.68

Table 3: Sets, Sale Rates and Revenue Rates (Efficient Sets are
in Bold)
Figure 1 shows a graph of the (π(S),r(S)) in increasing order of
π(S) as well as the efficient
frontier (upper concave majorant that goes through the efficient
sets).
Example 3. (Mixtures of BAMs) Consider the following mixture
of BAMs with three products
and two customer classes taken from Rusmevichientong,
Shmoys and Topaloglu [18]. The fares
are p1 = 80,p2 = 40 and p3 = 30. An arriving customer may
follow one of two BAMs, each
with probability 0.5. BAM-1 has attractiveness (1, 5, 20, 1),
respectively, for the no-purchase
alternative and for the three products. BAM-2 has attractiveness
(5, 1, 50, 50), respectively,
for the no-purchase alternative and for the three products. Let
πi(S) and ri(S) denote,
respectively, the sales probability and the expected revenue
associated with BAM-i, for i =
1, 2. Given any offer set S, let π(S) = 0.5π1(S) + 0.5π2(S) be
the probability of sale for the
10
$-‐

$100.00
$200.00
$300.00
$400.00
$500.00
$600.00
$700.00

0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Re
ve
nu
e
Ra
te
Demand
Rate
Revenue
Efficient
Fron=er
below the solid curve

mixture of the two BAMs and r(S) = 0.5r1(S) + 0.5r2(S) be the
expected revenue for the
mixture of the two BAMs. Table 4 shows (S,π(S),r(S)) for all S
⊂{1, 2, 3}. Notice that the
efficient sets are E0 = ∅ , E1 = {1} and E2 = {1, 3}. Figure 3
shows a graph of the (π(S),r(S))
in increasing order of π(S) as well as the efficient frontier.
S π(S) r(S)
∅ 0% $0.00
{1} 50% $40.00
{3} 70% $21.14
{1,3} 88% $44.82
{2} 93% $37.23
{1,2} 94% $41.65
{2,3} 95% $35.53
{1,2,3} 96% $39.66
Table 4: Sets, Sale Rates and Revenue Rates for Example 3
Here is another example of a mixture of BAMs where the
efficient sets are not nested.
Example 4. (Mixtures of BAMs) Consider the following mixture
with four products and
three customer classes. The fares are p1 = 11.50, p2 = 11.00, p3
= 10.80 and p4 = 10.75. The
attractiveness of the BAMs are, respectively, (1, 5, 2, 300, 1),
(1, 6, 4, 300, 1) and (1, 0, 1, 300, 7),
with the first component representing the attractiveness of the
no-purchase alternative. An
arriving customer has 1/6 probability of belonging to market
segment 1, 1/3 probability to

market segment 2 and 1/2 probability of belonging to market
segment 3. Table 5 lists the
efficient sets and the corresponding sales and revenue rates. The
efficient sets are E0 = ∅ ,
E1 = {1}, E2 = {1, 2}, E3 = {1, 4}, E4 = {1, 2, 4} and E5 = {1,
2, 3}. Notice that E3 does
not contain E2 and E5 does not contain E4. Moreover, this is an
instance where the number
of efficient sets m = 5 > n = 4.
11
$-‐
$5.00
$10.00
$15.00
$20.00

$25.00
$30.00
$35.00
$40.00
$45.00
$50.00
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%

Re
ve
nu
e
Ra
te
Demand
Rate
Revenue
Efficient
Fron=er
below the solid curve
S π(S) r(S)
∅ 0% $0.00
{1} 42.5% $4.88
{1,2} 69.9% $7.83
{1,4} 87.2% $9.65
{1,2,4} 89.8% $9.90
{1,2,3} 99.7% $10.77

Table 5: Sets, Sale Rates and Revenue Rates for Example 4
3 Efficient Sets and Assortment Optimization
In this section we formally define efficient sets and show that
for certain choice models the
efficient sets have desirable properties such as being nested or
nested by fares. We also look
at assortment optimization, which is essentially the problem of
finding the assortment S ⊂ N
with highest revenue r(S). The reader not interested in the
technical details may skip this
section on first reading.
For any ρ ∈ [0, 1] consider the linear program
R(ρ) = max
∑
S⊂N
r(S)t(S) (5)
subject to
∑
S⊂N
π(S)t(S) ≤ ρ
∑
S⊂N
t(S) = 1
t(S) ≥ 0 ∀ S ⊂ N.
The linear program selects a convex combination of all possible
actions (subsets of N) to

maximize the expected revenue that can be obtained subject to
the bound ρ on the probability
of sale. Later we will see that ΛR(c/Λ) is an upper bound on the
expected revenue when we
12
expect Λ customers to arrive and the capacity is c. Thus the
ratio of capacity to potential
demand c/Λ will play the role of ρ.
The decision variables in the linear program (5) are the
proportion of time t(S) ≥ 0 that
each subset S ⊂ N is offered for sale. The following results
follow from the standard theory
of parametric linear programming.
Proposition 1 R(ρ) is increasing, concave, and piece-wise linear.
We can trace the efficient frontier (ρ,R(ρ)), 0 ≤ ρ ≤ 1, by
solving the linear program (5)
parametrically (perhaps using the dual formulation and column
generation), for all 0 ≤ ρ ≤ 1.
In some cases the column generation step may be NP-hard, but
for now we will assume that
it is possible to solve the problem for all ρ ∈ [0, 1]. Sets S ⊂ N
such that r(S) < R(π(S)) are
said to be inefficient as the lie below the efficient frontier
(ρ,R(ρ)) at ρ = π(S). Sets S such
that R(π(S)) = r(S) are said to be efficient as the pair (π(S),r(S))
lies in the efficient frontier
at ρ = π(S). Equivalently, a set S is efficient if t(S) = 1 is an
optimal solution to the linear
program for ρ = π(S), as then R(ρ) = r(S). For any choice model,

let C = {E0,E1, . . . ,Em}
be the collection of efficient sets. Let πj = π(Ej) and rj = r(Ej)
for all j ∈ M = {0, 1, . . . ,m}.
We will assume from now on that the efficient sets are sorted in
increasing order of πj,j ∈ M.
Let uj = (rj − rj−1)/(πj −πj−1) be the slope joining (πj−1,rj−1)
and (πj,rj). Because R(ρ)
is increasing concave and linear between the points (πj,rj), it
follows that u1 > u2 > ... >
um ≥ 0, and that
R(ρ) = rj−1 + uj(ρ−πj−1) ∀ ρ ∈ [πj−1,πj] ∀ j = 1, 2, . . . ,m.
Notice also that R(ρ) = R(πm) = rm for all ρ > πm, so the
definition of R(ρ) can be extended
to all ρ ≥ 0.The upper concave envelopes of Figures 1, 2 and 3
are all of these form.
When Ej−1 ⊂ Ej, then uj is the marginal contribution to revenue
of the fare or fares added
to Ej−1 to form Ej. It is possible to show, through dual
feasibility, that Ej+1 is a maximizer
of the ratio (r(S) − rj)/(π(S) −πj) over all sets S such that π(S) >
πj if the resulting ratio
is non-negative (otherwise Ej = Em is the last efficient set).
Let us reconsider the choice models of the previous section to
look more closely into the
efficient sets. For Example 1, the efficient sets are E0 = ∅ , E1 =
{1} and E2 = {1, 2}. For
Example 2, the efficient sets are E0 = ∅ , E1 = {1}, E2 = {1, 2}
and E3 = {1, 2, 3}, while for
Example 3, the efficient sets are E0 = ∅ , E1 = {1} and E2 = {1,
3}. Notice that in these
examples the efficient sets are nested, and in Examples 1 and 2
the efficient sets are also

nested-by-fare. In Example 3, the efficient sets are not nested-
by fare since the efficient set
E2 = {1, 3} skips product 2 with p2 > p3. Notice that the
number of efficient sets is at most
n in Examples 1,2 and 3. In contrast, the efficient sets of
Example 4 are E0 = ∅ , E1 = {1},
E2 = {1, 2}, E3 = {1, 4}, E4 = {1, 2, 4} and E5 = {1, 2, 3}, so
the efficient sets are not nested
and the number of efficient sets is greater than n.
Talluri and van Ryzin [19], who first identified efficient sets for
discrete choice models, used
a slightly different definition. Under their definition, a set S is
inefficient, if it is possible to
form a convex combination of other sets that result in a strictly
higher revenue with the same
sale probability or if it is possible to form a convex
combination of other sets that results in
13
a strictly lower sale probability but the same revenue. They
define efficient sets as those that
are not inefficient. The two definitions are essentially
equivalent, although it is possible for
our definition to include sets that would be deemed inefficient
by the definition in [19]. As an
example, suppose that rm = rm−1 with πm > πm−1. Then
(πm,rm) is in the efficient frontier,
but it is deemed inefficient by [19], because the same revenue
rm−1 = rm can be obtained
with a lower sale probability. This is an innocuous subtlety,
because in this case the efficient
set Em will not be selected by an optimization algorithm unless

we specify that we want to
provide better customer service without sacrificing revenue.
The following result is key in simplifying the optimization
problem resulting from the
dynamic programs in the next section. In essence it reduces the
optimization from 2n subsets
S ⊂ N to just m, the number of efficient sets.
Theorem 3 For any z ≥ 0,
max
S⊂N
[r(S) −zπ(S)] = max
S∈ C
[r(S) −zπ(S)] = max
j∈ M
[rj −zπj].
Proof: It is enough to show that any maximizer of [r(S)−zπ(S)]
must be efficient. Suppose
for a contradiction that r(T) − zπ(T) > r(S) − zπ(S) for all S 6=
T ⊂ N and that T is
not an efficient set. Then there exist a set of non-negative
weights t(S) : S ⊂ N such
that R(π(T)) =
∑
S⊂N t(S)r(S) > r(T) with
∑
S⊂N t(S) = 1 and

∑
S⊂N t(S)π(S) ≤ π(T).
Consequently,
r(T) −zπ(T) < R(π(T)) −zπ(T) ≤
∑
S⊂N
[r(S) −zπ(S)]t(S) < r(T) −zπ(T),
where the first inequality follows from the inefficiency of T, the
second from
∑
S⊂N t(S)π(S) ≤
π(T) and the third from the claimed optimality of T and the fact
that t(T) < 1 as T is not
efficient. Since the displayed equation is a contradiction, it
must be that T is efficient, and
consequently we can restrict the maximization to C resulting in
maxS∈ C[r(S) − zπ(S)] =
maxj∈ M [rj −zπj].
Theorem 3 tell us that to find the best set to offer when the
marginal cost is z, we should
search for an efficient set with the largest value of rj − zπj. Now
rj − zπj > rj−1 − zπj−1 if
and only if z < uj, where uj is the slope joining (πj−1,rj−1) and
(πj,rj). From Proposition 1
the uj’s are non-negative and decreasing in j = 1, . . . ,m. To
visualize this graphically, draw
the efficient points (πj,rj),j ∈ M and a line with slope z ≥ 0
from the origin. An optimal
solution is to select the efficient set with index

j(z) = max{j ∈ M : uj > z}.
From this it is clear that j(z) < m for all z ≥ 0 whenever rm =
rm−1, because um = 0 ≤ z.
This shows that it is indeed innocuous to use our more liberal
definition of efficient sets,
which is more consistent with the definition of efficiency used
in portfolio theory in Finance.
However, if z = 0, we may want to offer Em instead of Em−1
when the slope um = 0. This is
because more customers are served by offering Em resulting in
the same revenue as Em−1. To
do this, we just modify the definition of j(z) to j(z) = max{j ∈
M : uj ≥ z}.
14
Consider the linear program
RC(ρ) = max
∑
j∈ M
rjtj
subject to
∑
j∈ M
πjtj ≤ ρ
∑
j∈ M
tj = 1

tj ≥ 0 ∀ j ∈ M.
Clearly RC(ρ) ≤ R(ρ) for all ρ ∈ [0, 1] as the linear program
RC(ρ) is more restricted. The
next result shows that in fact RC(ρ) = R(ρ) so the optimization
problem can be reduces to
the collection C of efficient sets.
Theorem 4 RC(ρ) = R(ρ) for all ρ ∈ [0, 1].
Proof: Consider the dual of the linear program defining R(ρ) in
terms of variables z and β.
The problem is to minimize ρz +β subject to π(S)z +β ≥ r(S),
∀ S ⊂ N, z ≥ 0. The solution
in terms of β is to set β = maxS⊂N [r(S) − zπ(S)]. By Theorem
3, β = maxj∈ M [rj − zπj], so
the dual problem is reduced to minimizing ρz + β subject to πjz
+ β ≥ rj,j ∈ M. The result
follows by recognizing that this is the dual of the liner program
defining RC(ρ).
3.1 Nested-by-Fare Efficient Sets
A choice model is said to have the nested-by-fare property if p1
> p2 > .. . > pn and the
collection of efficient sets C is contained in the collection
{S0,S1, . . . ,Sn} where S0 = ∅ and
Sj = {1, . . . ,j} for j = 1, . . . ,n. The next theorem, due to
Talluri and van Ryzin [19],
provides necessary and sufficient conditions for the nested-by-
fare property to hold. This
result is important because it justifies in some important cases
the optimality of the nested-
by-fare structure and because it reduces the number of efficient
fares to at most n. The

theorem requires the following simple idea that compares the
revenue performance of two
n-dimensional vectors x and y with the same sums x[1,n] =
y[1,n] = ρ, where we use the
notation x[1, i] =
∑i
j=1 xj. It turns out that if the partial sums of x dominate the
partial
sums of y, then p′x ≥ p′y for all vectors p with decreasing
components. More precisely, if
x[1, i] ≥ y[1, i] for all i = 1, . . . ,n − 1, and p1 ≥ p2 ≥ . . . ≥ pn,
then
∑n
i=1 pixi ≥
∑n
i=1 piyi.
This idea is related to the powerful concept of majorization
popularized by Marshal and Olkin
[14]. Here we will use it to compare the revenues associated
with the sale probability vector
yi = πi(T), i = 1, . . . ,n induced by a set T ⊂ N to the sales
induced by a convex combination of
the nested sets S0,S1, . . . ,Sn. More precisely, consider a
convex combination α0,α1, . . . ,αn of
non-negative values adding to one such that
∑n
k=0 αkπ(Sk) = π(T) and let xi =
∑n
k=0 αkπi(Sk)

be the sale of product i induced by this combination. Then x[1,
i] ≥ y[1, i] i = 1, . . . ,n − 1
and x[1,n] = y[1,n] imply that the revenue from the convex
combination will dominate the
revenue under set T.
15
Theorem 5 (Taluri and van Ryzin [19])A choice model has the
nested-by-fare order property
if and only if π(S) is increasing in S, and for every subset T
there is a convex combination
of π(S0),π(S1), . . . ,π(Sn) with partial sums of sales that
dominate the partial sums of sales
under T with total sales equal to total sales under T .
When such a convex combination exists, it favors sales at
higher fares and thus results in
higher expected revenues. Talluri and van Ryzin [19] show that
the conditions of Theorem 5
are satisfied by the MNL and the independent demand model.
Here we show that the nested-
by-fare property holds more generally by showing that it holds
for the parsimonious GAM
model (P-GAM) with wi = θvi for all i ∈ N for some θ ∈ [0, 1].
This model reduces to the
BAM when θ = 0 and to the IDM when θ = 1. The proof of this
result is in the Appendix.
Proposition 2 The efficient sets are nested-by-fares for the P-
GAM.
While Theorem 5 assures us that C ⊂ {S0,S1, . . . ,Sn}, the

converse does not necessarily
hold as there may be some sets in the collection C that are not
efficient.
Proposition 3 Consider the P-GAM for a system with strictly
decreasing fares p1 > p2 >
... > pn. Then a necessary and sufficient condition for all nested
sets to be efficient is
pn ≥ (1 −θ)r(Sn−1).
Proof: Consider the plot (π(Sj),r(Sj)),j = 0, . . . ,n. For the P-
GAM model we can write
r(Sj) as a convex combination of r(Sj−1) and pj/(1−θ) of the
form r(Sj) = αjr(Sj−1) + (1−
αj)pj/(1 −θ), where αj = (ṽ0 + Ṽ (Sj−1))/(ṽ0 + Ṽ (Sj)).
Consequently r(Sj) ≥ r(Sj−1) if and
only if pj ≥ (1 − θ)r(Sj−1). Suppose now that pn ≥ (1 −
θ)r(Sn−1). We will first show that
the plot is increasing, or equivalently that pj ≥ (1 − θ)r(Sj−1)
for all j. Suppose that there
is a j such that pj < (1 − θ)r(Sj−1). Then pj/(1 − θ) < r(Sj) <
r(Sj−1) and consequently
pj+1 < pj < (1 −θ)r(Sj). We can then repeat the argument to
show that pk < (1 −θ)r(Sk−1)
for all k ≥ j, but for k = n, this contradicts pn ≥ (1 −θ)r(Sn−1).
Now, π(Sj) = αjπ(Sj−1) + (1−αj)/(1−θ). This allow us to write
the marginal revenue as
uj =
r(Sj) − r(Sj−1)
π(Sj) −π(Sj−1)
=
pj − (1 −θ)r(Sj−1)
1 − (1 −θ)π(Sj−1)

(6)
To establish concavity, it is enough to show that uj is
decreasing in j. By using the known
convex combinations of r(Sj) and of π(Sj) we see that
uj+1 = uj −
pj −pj+1
αj(1 −πj−1)
< uj
where the inequality follows from pj > pj+1.
16
For the BAM, Proposition 3 implies that all nested sets are
efficient as long as pn ≥ r(Sn−1).
At the other extreme, for the IDM all nested sets are efficient as
long as pn ≥ 0. Another
consequence is that for the P-GAM, the efficient sets are S0,S1,
. . . ,Sj∗ where
j∗ = max{j ∈ N : pj ≥ (1 −θ)r(Sj−1)}.
All fares pj,j > j
∗ are inefficient and need not be considered.
3.2 Assortment Optimization under the MDDCM
Suppose we want to an offer set S that maximizes r(S) − zπ(S)
over all S ⊂ N, where the
choice model is the MDDCM characterized by (q,Q). This

Markovian assumption greatly
simplifies the optimization problem and allows the use of
successive approximations to find
the optimal revenues and the optimal offer set under the
Markovian assumption. Indeed, let
g0i = pi − z for i = 1, . . . ,n, where for convenience we define
g00 = p0 = 0. Now consider
the successive approximation scheme gk+1 = max(Qgk,p),
where now p is the vector of fares
with p0 = 0. Blanchet, Gallego and Goyal [2] show that g
k is monotonically increasing and
converges to a vector g. The vector g represents the maximum
revenue that can be obtained
from each customer request under the Markov Chain discrete
choice model. Blanchet, Gallego
and Goyal [2] show that under the Markovian assumption an
optimal offer set is is given by
S = {i : gi = g0i}, and that q′g = r(S) −zπ(S). This method can
be used to identify efficient
sets by solving the problem for all z ≥ 0, resulting in Ej,j ∈ M.
The method can be used as
a heuristic to approximately compute the collection of efficient
sets for discrete choice models,
such as the mixture of BAMs, for which the optimization is NP-
hard.
4 Dynamic Capacity Allocation Models
In this section we consider models where time is considered
explicitly. Modeling time explicitly
allows for time-varying arrival rates and time-varying discrete
choice models. We assume that
customers arrive to the system according to a time heterogenous
Poisson process with intensity
λt, 0 ≤ t ≤ T where T is the length of the horizon, and t

represents the time-to-go. Then the
number of customers that arrive during the last t units of time,
say Nt, is a Poisson random
variable with mean Λt =
∫ t
0 λsds.
We assume that customers arriving at time t select product k
from the offered set, say
S with probability πkt(S) given by the discrete choice model
prevalent at time-to-go t. Let
V (t,x) denote the maximum expected revenue that can be
attained over the last t units of the
sale horizon with x units of capacity. Assume that at time t we
offer set S ⊂ N = {1, . . . ,n}
and keep this set of fares open for δt units of time. If λtδt << 1,
the probability that a
customer arrives and requests product k ∈ S is approximately
λtπkt(S)δt so
V (t,x) = max
S⊂N
∑
k∈ S
{λtδtπkt(S)[pk + V (t− δt,x− 1)] + (1 −λtδt
∑
k∈ S
πkt(S))V (t− δt,x)} + o(δt)
17

= V (t− δt,x) + λtδt max
S⊂N
∑
k∈ S
πkt(S)[pk − ∆V (t− δt,x)] + o(δt)
= V (t− δt,x) + λtδt max
S⊂N
[rt(S) − ∆V (t− δt,x)πt(S)] + o(δt) (7)
for x ≥ 1 with boundary conditions V (t, 0) = V (0,x) = 0, t ≥ 0,
x ∈ N where rt(S) =∑
k∈ S pkπkt(S), πt(S) =
∑
k∈ S πkt(S) and ∆V (t− δt,x) = V (t− δt,x) −V (t− δt,x− 1).
The value function V (t,x) is often computed approximately by
solving a discrete time
dynamic program based on (7) that involves rescaling time and
the arrival rates, using δt = 1,
and dropping the o(δt) term. Time can be rescaled by a positive
real number, say a, such
that T ← aT is an integer by setting λt ← 1aλt/a, πjt(S) ←
πj,t/a(S). The resulting dynamic
program is given by
V (t,x) = V (t− 1,x) + λt max
S⊂N
[rt(S) − ∆V (t− 1,x)πt(S)], (8)

with the same boundary conditions. Formulation (8) is due to
Talluri and van Ryzin [19].
It reduces to the Lee and Hersh [11] model when demands are
independent. Alternatively,
instead of rescaling (7) we can subtract V (t−δt,x) from both
sides of equation (7), divide by
δt and take limits as δt ↓ 0 to obtain the Hamilton-Jacobi-
Bellman equation:
∂V (t,x)
∂t
= λt max
S⊂N
[rt(S) − ∆V (t,x)πt(S)] (9)
with the same boundary conditions.
The generic optimization problem in formulations (8) and (9) is
of the form
max
S⊂N
[rt(S) −zπt(S)]
where z ≥ 0 is the marginal value of capacity. Since there are 2n
subsets S ⊂ N = {1, . . . ,n}
the optimization requires the evaluation of a large number of
subsets, and the problem has
to be solved for different values of z as the marginal value of
capacity changes with the state
(t,x) of the system. One may wonder if there is a way to
simplify the problem by reducing the
number of subsets that we need to look at. From Theorem 3, we

know that the optimization
problem can be reduces to maxj∈ Mt [rjt − zπjt], where Mt is the
index of the collection of
efficient sets at time t, rjt = rt(Ejt) and πjt = πt(Ejt) and Ejt is
the jth efficient for the
discrete choice model prevalent at time-to-go t. This greatly
simplifies the optimization step
in the dynamic programs (8) and (9). For convenience we will
refer to action j when offering
efficient set Ejt,j ∈ Mt. If the problem of finding the efficient
sets is NP-hard, then the
Markov Chain approximation suggested in [2] can be used for
positive values of z ≥ 0, to
heuristically identify a collection of sets that are nearly
efficient and then use this collection
in the calculation.
Using the notion of efficient sets, formulation (8) reduces to
V (t,x) = V (t− 1,x) + λt max
j∈ Mt
[rjt − ∆V (t− 1,x)πjt], (10)
18
while formulation (9) reduces to
∂V (t,x)
∂t
= λt max
j∈ Mt

[rjt − ∆V (t,x)πjt]. (11)
Just as in the case of the independent demand model, the
marginal value ∆V (t,x) is
increasing in t and decreasing in x. For formulation (10), let
a(t,x) = arg max
j∈ Mt
[rjt − ∆V (t− 1,x)πjt]. (12)
For formulation (11) the definition of a(t,x) is the same except
that we use ∆V (t,x) instead
of ∆V (t− 1,x) in the right hand side.
Notice that rjt − ∆V (t − 1,x) > rj−1,t − ∆V (t − 1,x), so offering
action j (set Ejt) is
better than offering action j−1 if and only if ujt =
(rjt−rj−1,t)/(πj,t−πj−1,t) > ∆V (t−1,x).
Moreover, since the ujts are non-negative and decreasing in j, it
follows that it is optimal to
offer action
a(t,x) = max{j : ujt > ∆V (t− 1,x)},
for formulation (8). The formula for a(t,x) for formulation (9)
requires ∆V (t,x) instead of
∆V (t− 1,x). The following result is valid for both formulations
(10) and (11).
Theorem 6 (Taluri and van Ryzin ([19])) It is optimal to offer
action j = a(t,x) (efficient
set Ea(t,x),t) at state (t,x). Moreover, a(t,x) is decreasing in t
and increasing in x over every
time interval where the choice model is time invariant.

As t increases ∆V (t− 1,x) increases and the optimal solution
shifts to efficient sets with
a smaller sale probability. In contrast, as x increases, ∆V
(t−1,x) decreases, and the optimal
solutions shifts to efficient sets with larger sale probability. In
general, we cannot say that
we close lower fares when t is large (or open lower fares when x
is large) because the efficient
sets need not be nested-by-fare. There is a large class of choice
models, however, for which
the efficient sets are nested-by-fare. We have shown in Section
3 that for the parsimonious
GAM (p-GAM) with wj = θvj,j ∈ N, θ ∈ [0, 1], the efficient
sets are of the form Ej = Sj =
{1, . . . ,j} for j = 0, . . . ,m ≤ n, so the efficient sets are nested-
by-fares. Since the IDM and
the BAM correspond to the cases θ = 0 and θ = 1, this implies
that the collection of efficient
sets are nested-by-fares for both the IDM and the BAM. If the
efficient sets are nested-by-
fare, then we can talk of opening and closing fares as the state
dynamics change with the
understanding that if a fare is open, then all higher fares will be
open at the same time.
4.1 Dynamic Pricing Formulation with a Finite Price Menu
While πjt and rjt are respectively, the sales rate and the revenue
rate under action j at time t,
the average fare per unit sold under action j is qjt = rjt/qjt,
where for convenience we define
q0t = 0 for E0 = ∅ . From the increasing concavity of R(ρ), it
follows that qjt is decreasing in
j ∈ Mt. This suggest that if t is large relative to x we should
offer action 1 as this maximizes

19
the revenue per unit of capacity. However, offering action 1
may result in very slow sales and
may lead to capacity spoilage. At the other extreme, offering
action m, maximizes the revenue
rate and this is optimal when capacity is abundant as this
maximizes the revenue over the
sales horizon. For other cases, a tradeoff is needed between the
average sales price qjt and
the demand rate λtπjt associated with action j that takes into
account the marginal value of
capacity at state (t,x). This leads to the following equivalent
pricing formulation:
Then
V (t,x) = V (t− 1,x) + max
j∈ Mt
λtπjt[qjt − ∆V (t− 1,x)]. (13)
The equivalent formulation for the continuous time model (11)
is
∂V (t,x)
∂t
= max
j∈ Mt
λtπjt[qjt − ∆V (t,x)] (14)
These formulation suggest that we are selecting among actions j
∈ Mt to maximize the

sales rate λtπjt times the average fare qjt net of the marginal
value of capacity. We can think
of λtπjt as the demand rate associated with average fare qjt,
which reduces dynamic capacity
allocation models to dynamic pricing models with finite price
menus.
5 Formulation as an Independent Demand Model
Recall from Chapter 1, that the discrete-time, independent
demand, formulation with n fares
pt1 > pt2 > .. . > ptn and demand rates λ1t, . . . ,λtn, is of the
form:
V (t,x) = V (t− 1,x) +
n∑
j=1
λjt[ptj − ∆V (t− 1,x)]+, (15)
with boundary conditions V (t, 0) = V (0,x) = 0. One may
wonder, whether it is possible to
transform formulation (10) for dependent demands into
formulation (15) for demand formula-
tion, by transforming data λt and (πjt,rjt), j = 0, 1, . . . ,m for
the dependent demand model
into data (p̂ jt, λ̂tj), j = 1, . . . ,m for the independent demand
model.
One reason to wish for this is that such a transformation would
allow computer codes
available to solve (15) to solve (10). Another may be to try to
use the transformed data as
input to static models and then use static solutions or heuristics
such as the EMSR-b.

While it is indeed possible to do the transformation from (10)
into (15), there are really
no computational benefits as the running time of both (10) and
(15) are O(nct). Moreover,
as we shall see later, using the transformed data as input for the
static model is fraught with
problems.
The data transformation is to set λ̂jt = λt[πjt −πj−1,t] and p̂ jt =
ujt = (rjt −rj−1,t)/(πjt −
πj−1,t) for j ∈ Mt. The reader can verify that
∑j
k=1 λ̂jt = λtπjt and that
∑j
k=1 λ̂jtp̂ jt = λtrjt.
This transformation is in effect creating artificial products with
independent demands λ̂jt and
20
prices p̂ jt for j ∈ Mt.
Theorem 7 Formulation (10) is equivalent to
V (t,x) = V (t− 1,x) +
∑
j∈ Mt
λ̂jt[p̂ jt − ∆V (t− 1,x)]+ (16)

with boundary conditions V (t, 0) = V (0,x) = 0.
Proof of Theorem 7: Let k(t,x) = max{j : p̂ jt > ∆V (t − 1,x)}. It
is therefore optimal
to accept all of the artificial products such that j ≤ k(t,x). Notice
that a(t,x) = k(t,x) on
account of p̂ jt = ujt. Aggregating artificial products 1, . . .
,a(t,x), results in action k(t,x) =
a(t,x), corresponding to efficient set Ea(t,x),t. More precisely,
∑
j∈ Mt
λ̂jt[p̂ jt − ∆V (t− 1,x)]+ =
∑
j≤k(t,x)
λ̂jt[p̂ jt − ∆V (t− 1,x)]
=
∑
j≤a(t,x)
λ̂jt[p̂ jt − ∆V (t− 1,x)]
= ra(t,x),t −πa(t,x),t)∆V (t− 1,x)
= max
j∈ Mt
[rjt −πjt∆V (t− 1,x)].
Consequently,
V (t,x) = V (t− 1,x) +
∑
j∈ Mt

λ̂jt[p̂ jt − ∆V (t− 1,x)]+ = V (t− 1,x) + λt max
j∈ Mt
[rjt −πjt∆V (t,x)].
The fare and demand transformations that map λt and (πjt,rjt),j
∈ Mt into (p̂ jt, λ̂jt),j ∈
M have appeared in many papers but the original ideas go back
to Kincaid and Darlin [10] as
documented in Walczack et al. [20]. The fact that the
transformation works for the dynamic
program has lead some practitioners to conclude that the
transformed data can be used as an
input to static models where demands are treated as
independent. In particular, Fiig et al. [6],
and Walczack et al. [20], proposed feeding the transformed data
and applying the EMSR-b
heuristic, treating demands as independent under the low-to-
high arrival pattern assumption.
While this is a tempting heuristic, it would be wrong to expect
that controls obtained this way
will be close to optimal if demand dependencies are ignored. In
particular, the low-to-high
demand arrival pattern, is tantamount to assuming that Poisson
demands with parameters
λ̂jt will arrive low-to-high, but this are artificial demands from
customers willing to buy under
action j but not under action j−1. When capacity is allocated to
this marginal customers, we
cannot prevent some degree of demand cannibalization from
customers willing to buy under
action j − 1 into some of the fares in action j. We will return to
this issue in Section 8.
21

6 Upper Bound on the Value Function
We will present an an upper bound on the value functions (7) an
(8) for the case where the
choice models are time invariant. The idea is based on
approximate dynamic programming
(ADP) using affine functions and it can be easily extended to
the case where the choice models
vary over time. It is well known that a dynamic program can be
solved as a mathematical pro-
gram by making the value function at each state a decision
variable. This leads to the formula-
tion V (T,c) = min F(T,c) subject to the constraints ∂F (t,x)/∂t ≥
λt[rj−∆F(t,x)πj] ∀ (t,x)
for all j ∈ M, where the decision variables are the class of non-
negative functions F(t,x) that
are differential in x with boundary conditions F(t, 0) = F(0,x) =
0 for all t ∈ [0,T] and all
x ∈ {0, 1, . . . ,c}.
While this formulation is daunting, it becomes easier once we
restrict the functions to
be of the affine form F̃(t,x) =
∫ t
0 βs(x)ds + xzt, zt ≥ 0. We will restrict ourselves to the
case βs(x) = βs for x > 0, βs(0) = 0, zt = z for t > 0 and z0 = 0.
With this restriction,
the partial derivative and marginal value of capacity have
simple forms and the boundary
conditions are satisfied. More precisely, ∂F̃(t,x)/∂t = βt for x >
0, ∆F̃(t,x) = z, for t > 0,

F̃(t, 0) = F̃(0, t) = 0. This reduces the program to Ṽ (T,c) = min
F̃(T,c) = min
∫T
0 βtdt + cz,
subject to βt ≥ λt[rj − zπj] ∀ j ∈ M. Since we have restricted the
set of functions F(t,x)
to be affine, it follows that V (T,c) ≤ Ṽ (T,c). Since this is a
minimization problem, the
optimal choice for βt is βt = λt maxj∈ M [rj − zπij] = λtg(z)
where g(z) = maxj∈ M [rj − zπj]
is a decreasing convex, non-negative, piecewise linear function
of z. Consequently the overall
problem reduces to
Ṽ (T,c) = min
z≥0
[
∫ T
0
λtg(z)dt + cz] = min
z≥0
[ΛTg(z) + cz], (17)
where ΛT =
∫T
0 λtdt is the aggregate arrival rate over the sales horizon [0,T].
Notice that
ΛTg(z) + cz is convex in z. If the discrete choice model is time
varying, then gt(z) =
maxj∈ Mt [rjt −zπjt], result in

V
̄ (T,c) = min
z≥0
[∫ T
0
λtgt(z)dt + cz
]
,
where the objective function is also convex in z.
We can linearize (17) by introducing a new variable, say y, such
that y ≥ rj − zπj for all
j ∈ M and z ≥ 0, which results in the linear program:
Ṽ (T,c) = min
z≥0
[ΛTy + cz],
subject to ΛTy + ΛTπjzj ≥ ΛTrj j ∈ M
z ≥ 0,
where for convenience we have multiplied the constraints y +
πjz ≥ rj,j ∈ M by ΛT > 0.
22
The dual of this problem is given by
Ṽ (T,c) = ΛT max
∑

j∈ M
rjtj
subject to ΛT
∑
j∈ M
πjtj ≤ c
∑
j∈ M
tj = 1
tj ≥ 0 ∀ j ∈ M.
This linear program decides the proportion of time , tj ∈ [0, 1],
that each efficient set Ej is
offered to maximize the revenue subject to the capacity
constraint. The reader can verify that
Ṽ (T,c) is closely related to the functions RC(ρ) and R(ρ) used
to define efficient sets. The
following result establishes this relationship.
Proposition 4
Ṽ (T,c) = ΛTRC(c/ΛT ) = ΛTR(c/ΛT ).
Example 5 Suppose that p1 = 1000,p2 = 600 and a BAM with v0
= v1 = v2 = e
1. We will
assume that the aggregate arrival rate over the sales horizon
[0,T] = [0, 1] is ΛT = 40. We
know that for this problem the efficient sets are E0 = ∅ ,E1 =
{1} and E2 = {1, 2}. For c = 24,
the optimal solution is t1 = 6/15 and t2 = 9/15 with 8 units of

sales under action S1 and 16
units of sales under action S2 and revenue $20,800. Table 6
provides the upper bound Ṽ (T,c)
for different values of c. Notice that sales under action S1, first
increase and then decrease as
c increases. Table 6 also reports sales under fare 1 and fare 2,
and the reader can see that
sales under fare 1 first increase and then decrease.
c Ṽ (T,c) t1 t2 sales S1 sales S2 fare 1 sales fare 2 sales
12 12,000 0.6 0.0 12 0 12 0
16 16,000 0.8 0.0 16 0 16 0
20 20,000 1.0 0.0 20 0 20 0
22 20,400 0.7 0.3 14 8 18 4
24 20,800 0.4 0.6 8 16 16 8
26 21,200 0.1 0.9 2 24 14 12
28 21,333 0.0 1.0 0 26.6 13.3 13.3
32 21,333 0.0 1.0 0 26.6 13.3 13.3
Table 6: Upper Bound and Optimal Actions for Fluid Model
7 Uni-directional Dynamic Programming Models
Like the Lee and Hersh model, formulations (8) and (9),
implicitly assume that the capacity
provider can offer any subset of fares at any state (t,x). This
flexibility works well if customers
are not strategic. Otherwise, customers may anticipate the
possibility of lower fares and
23
postpone their purchases in the hope of being offered lower
fares at a later time. If customers

act strategically, the capacity provider may counter by imposing
restrictions that do not allow
lower fares to reopen once they are closed. Actions to limit
strategic customer behavior are
commonly employed by revenue management practitioners.
Let VS(t,x) be the optimal expected revenue when the state is
(t,x), we are allowed
to use any subset U ⊂ S of fares and fares cannot be offered
once they are closed. The
system starts at state (T,c) and S = N. If a strict subset U of S is
used then all fares in
U ′ = {j ∈ N : j /∈ U} are permanently closed and cannot be
offered at a later state regardless
of the evolution of sales. To obtain a discrete time counterpart
to (8), let
WU (t,x) = VU (t− 1,x) + λt[rt(U) −πt(U)∆VU (t− 1,x)].
Then the dynamic program is given by
VS(t,x) = max
U⊂S
WU (t,x) (18)
with boundary conditions VS(t, 0) = VS(0,x) = 0 for all t ≥ 0, x
∈ N and S ⊂ N. The goal
of this formulation is to find VN (T,c) and the corresponding
optimal control policy.
Notice that formally the state of the system has been expanded
to (S,t,x) where S is
the last offered set and (t,x) are, as usual, the time-to-go and the
remaining inventory. For-
mulation (8) requires optimizing over all subsets S ⊂ N, while

formulation (18) requires an
optimization over all subsets U ⊂ S for any given S ⊂ N.
Obviously the complexity of these
formulations is large if the number of fares is more than a
handful. Airlines typically have over
twenty different fares so the number of possible subsets gets
large very quickly. Fortunately,
in many cases we do not need to do the optimization over all
possible subsets. As we have
seen, the optimization can be reduced to the set of efficient
fares Ct at time t. For the p-GAM,
we know that Ct ⊂ {S0,S1, . . . ,Sn} where Sj = {1, . . . ,j} are
the nested-by-fares sets. For
the p-GAM, and any other model for which the nested-by-fare
property holds, the state of
the system reduces to (j,t,x) where Sj is the last offered set at
(t,x). For such models the
formulation (18) reduces to
Vj(t,x) = max
k≤j
Wk(t,x) (19)
where Vj(t,x) = VSj (t,x) and
Wk(t,x) = Vk(t− 1,x) + λt[rkt −πkt∆Vk(t− 1,x)],
rkt =
∑
l∈ Sk plπlt(Sk) and πkt =
∑
l∈ Sk πlt(Sk).
The structural results obtained for the Independent Demand

Model carry on to the p-
GAM model and for other models where the efficient sets have
the nested-by-fare property.
Let
aj(t,x) = max{k ≤ j : Wk(t,x) = Vj(t,x)}. (20)
Theorem 8 At state (j,t,x) it is optimal to offer set Saj (t,x) =
{1, . . . ,aj(t,x)}, where
aj(t,x), given by (20), is decreasing in t and increasing in x over
time intervals where the
choice model is time invariant.
24
The proof of this results follows the sample path arguments of
the corresponding proof in
the independent demand chapter. Clearly V1(t,x) ≤ V2(t,x) ≤
Vn(t,x) ≤ V (t,x) ≤ Ṽ (t,x) as
V (t,x) is not restricted, as Vj(t,x) is, to monotone offerings.
c Load Factor V3(T,c) V (T,c) Ṽ (T,c)
4 4.59 3,769 3,871 4,000
6 3.06 5,356 5,534 6,000
8 2.30 6,897 7,013 7,477
10 1.84 8,259 8,335 8,950
12 1.53 9,304 9,382 10,423
14 1.31 9,976 10,111 10,846
16 1.15 10,418 10,583 11,146
18 1.02 10,803 10,908 11,447
20 0.92 11,099 11,154 11,504
22 0.84 11,296 11,322 11,504
24 0.77 11,409 11,420 11,504

26 0.71 11,466 11,470 11,504
28 0.66 11,490 11,492 11,504
Table 7: Value Functions for Dynamic Allocation Policies
Example 6 Table 7 presents the value functions V3(T,c), V (T,c)
and the upper bound Ṽ (T,c)
that result from solving the dynamic programs (10) and (19) for
the MNL model with fares
p1 = $1, 000,p2 = $800,p3 = $500 with price sensitivity βp =
−0.0035, schedule quality
si = 200 for i = 1, 2, 3 with quality sensitivity βs = 0.005, and
an outside alternative with
p0 = $1, 100 and schedule quality s0 = 500, Gumbel parameter
φ = 1, arrival rate λ = 25
and T = 1. Recall that for the MNL model, the attractiveness vi
= e
φµi where µi is the
mean utility. In this case µi = βppi + βssi. The computations
were done with time rescaled
by a factor a = 25, 000. Not surprisingly V3(T,c) ≤ V (T,c) as
V3(T,c) constraints fares to
remain closed once they are closed for the first time. However,
the difference in revenues is
relatively small except for load factors around 1.15 where the
difference can be as large as
1.5% in revenues.
8 Static Capacity Allocation Models
We will assume that we are working with a choice model with
efficient sets that are nested:
E0 ⊂ E1 . . . ⊂ Em, even if they are not nested by fare. We
continue using the notation πj =∑
k∈ Ej πk(Ej) and rj =

∑
k∈ Ej pkπk(Ej), so the slopes uj = (rj−rj−1)/(πj−πj−1),j = 1, . . .
,m
are positive and decreasing. We will denote by qj = rj/πj the
average fare, conditioned on a
sale, under efficient set Ej (action j) for j > 1 and define q0 = 0.
Suppose that the total number of potential customers over the
selling horizon is a random
variable D. For example, D can be Poisson with parameter Λ.
Let Dj be the total demand if
only set Ej is offered. Then Dj is conditionally binomial with
parameters D and πj, so if D
is Poisson with parameter Λ, then Dj is Poisson with parameter
Λπj.
25
We will present an exact solution for the two fare class capacity
allocation problem and
a heuristic for the multi-fare case. The solution to the two fare
class problem is, in effect,
an extension of Littlewood’s rule for discrete choice models.
The heuristic for the multi-fare
problem applies the two fare result to each pair of consecutive
actions, say j and j + 1 just as
in the case of independent demands.
8.1 Two Fare Classes
While capacity is available, the capacity provider will offer
either action 2 associated with

efficient set E2 = {1, 2}, or action 1, associated with efficient
set E1 = {1}. If the provider
runs out of inventory he offers action 0, corresponding to E0 =
{0}, which is equivalent to
stop selling. We will assume action 2 will be offered first and
that y ∈ {0, . . . ,c} units of
capacity will be protected for sale under action 1.
Under action 2, sales are min(D2,c−y) and expected revenues
are q2E min(D2,c−y) as
q2 is the average fare per unit sold under action 2. Of the (D2 −
c + y)+ customers denied
bookings, a fraction β = π1/π2 will be willing to purchase under
action 1. Thus, the demand
under action 1 will be a conditionally binomial random variable,
say U(y), with a random
number (D2 − c + y)+ of trials and success probability β. The
expected revenue that results
from allowing up to c−y bookings under action 2 is given by
W2(y,c) = q1E min(U(y), max(y,c−D2)) + q2E min(D2,c−y),
where the first term corresponds to the revenue under action 1.
Notice that if D2 ≤ c − y
then U(y) = 0 since U(y) is a binomial random variable with
zero trials. Conditioning the
first term on the event D2 > c−y, allow us to write
W2(y,c) = q1E[min(U(y),y)|D2 ≥ c−y)]P(D2 > c−y) + q2E
min(D2,c−y).
The reader may be tempted to follow the marginal analysis idea
presented in Chapter 1 for the
independent demand case. In the independent demand case, the
marginal value of protecting
one more unit of capacity is realized only if the marginal unit is

sold. The counterpart here
would be P(U(y) ≥ y|D2 > c−y) and a naive application of
marginal analysis would protect
the yth unit whenever q1P(U(y) ≥ y|D2 > c−y) > q2.
However, with dependent demands, protecting one more unit of
capacity also increases
the potential demand under action 1 by one unit. This is because
an additional customer
is denied capacity under action 2 (when D2 > c − y) and this
customer may end up buying
a unit of capacity under action 1 even when not all the y units
are sold. Ignoring this
can lead to very different results in terms of protection levels.
The correct analysis is to
acknowledge that an extra unit of capacity is sold to the
marginal customer with probability
βP (U(y − 1) < y − 1|D2 > c−y). This suggest protecting the yth
unit whenever
q1[P(U(y) ≥ y|D2 > c−y) + βP (U(y − 1) < y − 1|D2 > c−y)] >
q2.
To simplify the left hand side, notice that conditioning on the
decision of the marginal customer
26
results in
P(U(y) ≥ y|D2 > c−y) = βP (U(y−1) ≥ y−1|D2 >
c−y)+(1−β)P(U(y−1) ≥ y|D2 > c−y).
Combining terms leads to protecting the yth unit whenever

q1[β + (1 −β)P(U(y − 1) ≥ y|D2 > c−y)] > q2.
Let
r = u2/q1 =
q2 −βq1
(1 −β)q1
, (21)
denote the critical fare ratio. In industry, the ratio r given by
(21) is know as fare adjusted
ratio, in contrast to the unadjusted ratio q2/q1 that results when
β = 0.
The arguments above suggests that the optimal protection level
can be obtained by se-
lecting the largest y ∈ {1, . . . ,c} such that P(U(y − 1) ≥ y|D2 >
c − y) > r provided that
P(U(0) ≥ 1|D2 ≥ c) > r and to set y = 0 otherwise.
To summarize, an optimal protection can be obtain by setting
y(c) = 0 if P(U(0) ≥ 1|D2 >
c) ≤ r and setting
y(c) = max{y ∈ {1, . . . ,c} : P(U(y − 1) ≥ y|D2 > c−y) > r}
otherwise. (22)
Formula (22) is due to Gallego, Lin and Ratliff [7]. This is
essentially a reinterpretation
of the main result in Brumelle et al. originally developed for
fares instead of actions.
One important observation is that for dependent demands the
optimal protection level

y(c) depends in c in a more complicated way than in the
independent demand model. For the
dependent demand model y(c) is first increasing and then
decreasing in c. The reason is that
for low capacity it is optimal to protect all the inventory for
sale under action 1. However, for
high capacity it is optimal to allocate all the capacity to action
2. The intuition is that action
2 has a higher revenue rate, so with high capacity we give up
trying to sell under action 1.
This is clearly seen in Table 6 of Example 5. Heuristic solutions
that propose protection levels
of the form min(yh,c), which are based on independent demand
logic, are bound to do poorly
when c is close to Λπ2. In particular, the heuristics developed
by Belobaba and Weatherford
[1], and more recently by Fiig et al. [6] and by Walczak et al.
[20] are of this form and are
therefore not recommended.
One can derive Littlewood’s rule for discrete choice models
(22) formally by analyzing
∆W2(y,c) = W2(y,c)−W2(y−1,c), the marginal value of
protecting the yth unit of capacity
for sale under action 1.
Proposition 5
∆W2(y,c) = [q1(β + (1 −β)P(U(y − 1) ≥ y|D2 > c−y) − q2]P(D2
> c−y). (23)
Moreover, the expression in brackets is decreasing in y ∈ {1, . .
. ,c}.
27

The proof of Proposition 5 is due to Gallego, Lin and Ratliff [7]
and can be found in the
Appendix. As a consequence, ∆W2(y,c) has at most one sign
change. If it does, then it must
be from positive to negative. W2(y,c) is then maximized by the
largest integer y ∈ {1, . . . ,c},
say y(c), such that ∆W2(y,c) is positive, and by y(c) = 0 if
∆W2(1,c) < 0. This confirms
Littlewood’s rule for discrete choice models (22).
8.2 Heuristic Protection Levels
In terms of computations, notice that
P(U(y − 1) ≥ y|D2 > c−y) = lim
L→∞
∑L
k=0 P(Bin(y + k,β) ≥ y)P(D2 = c + k + 1)
P(D2 > c−y)
,
where we have been careful to only include terms that can be
potentially positive in the
numerator. It is often enough to truncate the sum at a value, say
L, such that P(D2 >
L) ≤ �P(D2 > c − y) for some small �, as those terms do not
materially contribute to the
sum. While the computation of y(c) and V2(c) = W2(y(c),c) is
not numerically difficult, the
conditional probabilities involved may be difficult to
understand conceptually. Moreover, the
formulas do not provide intuition and do not generalize easily to

multiple fares. In this section
we develop a simple heuristic to find near-optimal protection
levels that provides some of the
intuition that is lacking in the computation of optimal
protection levels y(c). In addition, the
heuristic can easily be extended to multiple-fares.
The heuristic consists of approximating the conditional
binomial random variable U(y−1)
with parameters (D2−c+y−1)+ and β by its conditional
expectation, namely by (Bin(D2,β)−
β(c+ 1−y))+. Since Bin(D2,β) is just D1, the approximation
yields (D1−β(c+ 1−y))+. We
expect this approximation to be reasonable if ED1 ≥ β(c + 1−y).
In this case P(U(y−1) ≥
y|D2 > c−y) can be approximated by P(D1 ≥ (1 −β)y + β(c + 1)).
Let
yp = max{y ∈ N : P(D1 ≥ y) > r)}.
We think of yp = (1 − β)y + β(c + 1) as a pseudo-protection
level that will be modified
to obtain a heuristic protection level yh(c) when the
approximation is reasonable. Let y =
(yp −β(c + 1))/(1 −β). Then ED1 ≥ β(c + 1 −y)) is equivalent to
the condition
c < yp + d2
where d2 = E[D2] − E[D1] = Λ(π2 − π1) = Λπ2(1 − β), where
for convenience we write
β = β1 = π1/π2. Consequently, if c < y
p + d2 we set
yh(c) = max

{
y ∈ N : y ≤
yp −β(c + 1)
(1 −β)
}
∧ c.
When the condition c < yp +d2 fails, we set y
h(c) = 0. Thus, the heuristic will stop protecting
capacity for action 1 when c ≥ yp + d2. In a deterministic
setting yp = E[D1], so it stops
protecting when c ≥ E[D1] + d2 = E[D2], i.e., when there is
enough capacity to satisfy the
28
expected demand under action 2.
Notice that the heuristic involves three modifications to
Littlewood’s rule for independent
demands. First, instead of using the first choice demand D1,2
for fare 1, when both fares
are open, we use the stochastically larger demand D1 for fare 1,
when it is the only open
fare. Second, instead of using the ratio of the fares p2/p1 we use
the modified fare ratio
r = u2/q1 based on sell-up adjusted fare values. From this we
obtain a pseudo-protection level
yp that is then modified to obtain yh(c). Finally, we keep yh(c)
if c < yp +d2 and set y

h(c) = 0
otherwise. In summary, the heuristic involves a different
distribution, a fare adjustment, and a
modification to the pseudo-protection level. The following
example illustrates the performance
of the heuristic.
Example 7 Suppose that p1 = 1000,p2 = 600 and a BAM with v0
= v1 = v2 = e
1 and that
Λ = 40 as in Example 5. We report the optimal protection level
y(c), the heuristic protection
level yh(c), the upper bound V
̄ (c), the optimal expected
revenue V (c) of the uni-directional
formulation (18), the performance V2(c) of y(c) and the
performance V
h
2 (c) = W2(y
h(c),c)
of yh(c), and the percentage gap between (V2(c) − V h2
(c))/V2(c) in Table 8. Notice that the
performance of the statiheuristic, V h2 (c), is almost as good as
the performance V2(c) of the
optimal policy under the uni-directional formulation (18) where
fares classes cannot be opened
once they are closed.
c y(c) yh(c) V
̄ (c) V (c) V2(c) V
h
2 (c) Gap(%)
12 12 12 12,000 11,961 11,960 11,960 0.00

16 16 16 16,000 15,610 15,593 15,593 0.00
20 20 20 20,000 18,324 18,223 18,223 0.00
24 21 24 20,800 19,848 19,526 19,512 0.07
28 9 12 21,333 20,668 20,414 20,391 0.11
32 4 0 21,333 21,116 21,036 20,982 0.26
36 3 0 21,333 21,283 21,267 21,258 0.05
40 2 0 21,333 21,325 21,333 21,322 0.01
Table 8: Performance of Heuristic for Two Fare Example
8.3 Theft versus Standard Nesting and Arrival Patterns
The types of inventory controls used in the airline’s reservation
system along with the demand
order of arrival are additional factors that must be considered in
revenue management opti-
mization. If y(c) < c, we allow up to c−y(c) bookings under
action 2 with all sales counting
against the booking limit c−y(c). In essence, the booking limit
is imposed on action 2 (rather
than on fare 2). This is known as theft nesting. Implementing
theft nesting controls may be
tricky if a capacity provider needs to exert controls through the
use of standard nesting, i.e.,
when booking limits are only imposed on the lowest open fare.
This modification may be
required either because the system is built on the philosophy of
standard nesting or because
users are accustomed to thinking of imposing booking limits on
the lowest open fare. Here
we explore how one can adapt protection levels and booking
limits for the dependent demand
model to situations where controls must be exerted through
standard nesting.
29

A fraction of sales under action 2 correspond to sales under fare
p2. This fraction is given
by ω = π2(E2)/π2. So if booking controls need to be exerted
directly on the sales at fare p2,
we can set booking limit ω(c−y(c)) on sales at fare p2. This is
equivalent to using the larger
protection level
ŷ(c) = (1 −ω)c + ωy(c) (24)
for sales at fare 1. This modification makes implementation
easier for systems designed for
standard nesting controls and it performs very well under a
variety of demand arrival patterns.
It is possible to combine demand choice models with fare
arrival patterns by sorting cus-
tomers through their first choice demand and then assuming a
low-to-high demand arrival
pattern. For the two fare case, the first choice demands for fare
1 and fare 2 are Poisson
random variables with rates Λπ1(E2) and Λπ2(E2). Assume now
that customers whose first
choice demand is for fare 2 arrive first, perhaps because of
purchasing restrictions associated
with this fare. Customers whose first choice is fare 2 will
purchase this fare if available. They
will consider upgrading to fare 1 if fare 2 is not available. One
may wonder what kind of
control is effective to deal with this arrival pattern. It turns out
that setting protection level
ŷ(c) given by (24) for fare 1, with standard nesting, is optimal
for this arrival pattern and is

very robust to other (mixed) arrival patterns (Gallego, Li and
Ratliff [8]).
8.4 Multiple Fare Classes
For multiple fare classes, finding optimal protection levels can
be very complex. However, if
we limit our search to the best two consecutive efficient sets we
can easily adapt the results
from the two-fare class to deal with the multiple-fare class
problem. For any j ∈ {1, . . . ,n−1}
consider the problem of allocating capacity between actions j
(corresponding to efficient set
Ej) and action j + 1 (corresponding to efficient set Ej+1) where
action j + 1 is offered first.
In particular, suppose we want to protect y ≤ c units of capacity
for action j against action
j + 1. We will then sell min(Dj+1,c−y) units under action j + 1
at an average fare qj+1. We
will then move to action j with max(y,c−Dj+1) units of capacity
and residual demand Uj(y),
where Uj(y) is conditionally binomial with parameters (Dj+1 −
c + y)+ and βj = πj/πj+1.
Assuming we do not restrict sales under action j the expected
revenue under actions j + 1
and j will be given by
Wj+1(y,c) = qjE min(Uj(y), max(y,c−Dj+1)) + qj+1E
min(Dj+1,c−y). (25)
Notice that under action j we will either run out of capacity or
will run out of customers.
Indeed, if Uj(y) ≥ y then we run out of capacity, and if Uj(y) <
y then we run out of
customers. Let Wj+1(c) = maxy≤c Wj+1(y,c) and set W1(c) =
q1E min(D1,c). Clearly,

Vn(c) ≥ max
1≤j≤n
Wj(c), (26)
so a simple heuristic is to compute Wj(c) for each j ∈ {1, . . .
,n} and select j to maxi-
mize Wj(c). To find an optimal protection level for Ej against
Ej+1 we need to compute
∆Wj+1(y,c) = Wj+1(y,c)−Wj+1(y−1,c). For this we can repeat
the analysis of the two fare
case to show that an optimal protection level for action Ej
against action Ej+1 is given by
30
yj(c) = 0 if ∆Wj+1(1,c) < 0 and by
yj(c) = max{y ∈ {1, . . . ,c} : P(Uj(y − 1) ≥ y|Dj+1 > c−y) > rj},
(27)
where
rj =
uj+1
qj
=
qj+1 −βjqj
(1 −βj)qj
.

Alternatively, we can use the heuristic described in the two fare
section to approximate
Uj(y−1) by Dj−βj(c+ 1−y) and use this in turn to approximate
the conditional probability
in (27) by P(Dj ≥ y + β(c−y + 1)). This involves finding the
pseudo-protection level
y
p
j = max{y ∈ N : P(D1 ≥ y) > rj}.
If c < y
p
j + dj+1, then
yhj (c) = max
{
y ∈ N+ : y ≤
y
p
j −βj(c + 1)
1 −βj
}
∧ c, (28)
and set yh(c) = 0 if c ≥ ypj + dj+1.
We will let V hn (c) be the expected revenues resulting from
applying the protection levels.
Example 8 Consider now a three fare example with fares p1 =
1000,p2 = 800,p3 = 500,

schedule quality si = 200, i = 1, 2, 3, βp = −.0035, βs = .005, φ
= 1. Then v1 = .082,v2 =
0.165,v3 = 0.472. Assume that the outside alternative is a
product with price p0 = 1100 and
schedule quality s0 = 500 and that the expected number of
potential customers is Poisson with
parameter Λ = 25. Table 9 reports the protection levels yj(c)
and y
h
j (c) as well as V3(c) and
V h3 (c) for c ∈ {4, 6, . . . , 26, 28}. As shown in the Table, the
heuristic performs very well with
a maximum gap of 0.14% relative to V3(c) which was computed
through exhaustive search.
It is also instructive to see that V h3 (c) is not far from V3(c,T),
as reported in Table 7, for the
dynamic model. In fact, the average gap is less than 0.5% while
the largest gap is 1.0% for
c = 18.
Example 7 suggest that the heuristic for the static model works
almost as well as the
optimal dynamic program Vn(T,c) for the case where efficient
sets are nested-by-fare and
fares cannot be opened once they are closed for the first time.
Thus the multi-fare heuristic
described in this section works well to prevent strategic
customers from gaming the system
provided that the efficient fares are nested-by-fare as they are in
a number of important
applications. While the heuristic for the static model gives up a
bit in terms of performance
relative to the dynamic model, it has several advantages. First,
the static model does not

need the overall demand to be Poisson. Second, the static model
does not need as much detail
in terms of the arrival rates. These advantages are part of the
reason why people in industry
have a preference for static models, even thought dynamic
models are easier to understand,
easier to solve to optimality, and just as easy to implement.
31
c Load y1(c) y2(c) y
h
1 (c) y
h
2 (c) V3(c) V
h
3 (c) Gap(%)
4 4.59 4 4 4 4 3,769 3,769 0.00
6 3.06 3 6 3 6 5,310 5,310 0.00
8 2.30 1 8 1 8 6,845 6,845 0.00
10 1.84 0 10 0 10 8,217 8,217 0.00
12 1.53 0 12 0 12 9,288 9,288 0.00
14 1.31 0 14 0 14 9,971 9,971 0.00
16 1.15 0 13 0 14 10,357 10,354 0.02
18 1.02 0 9 0 10 10,700 10,694 0.05
20 0.92 0 5 0 6 11,019 11,019 0.00
22 0.84 0 4 0 2 11,254 11,238 0.14
24 0.77 0 3 0 0 11,391 11,388 0.03
26 0.71 0 2 0 0 11,458 11,450 0.08
28 0.66 0 2 0 0 11,488 11,485 0.03

Table 9: Performance of Heuristic for Three Fare Example
9 Acknowledgments
I acknowledge the feedback from my students and collaborators.
In particular, I would like
to recognize the contributions and feedback from Anrand Li,
Lin Li, Jing Dong and Richard
Ratliff.
32
10 Appendix
Proof of Theorem 1: If the choice model πj(S),j ∈ S satisfies
GAM then there exist constants
vi, i ∈ N+, ṽi ∈ [0,vi] i ∈ N and ṽ0 = v0 +
∑
j∈N (vj − ṽj) such that πR(S) = v(R)/(ṽ0 + Ṽ (S)
for all R ∈ S. For this choice model, the left hand side of
Axiom 1’ is (ṽ0 +Ṽ (S)/(ṽ0 +Ṽ (T)).
Let θj = 1−ṽj/vj. Then the right hand side of Axiom 1’ is given
by 1−
∑
j∈T−S(1−θj)vj/(ṽ0 +
Ṽ (T)) = (ṽ0 + Ṽ (S)/(ṽ0 + Ṽ (T)). This shows that the GAM
satisfies Axiom 1’. If πi({i}) = 0
then vi = 0 and consequently ṽi = 0. From this it follows that
πS−{i}(T −{i}) =

V (S)−vi
ṽ0+Ṽ (T)−ṽi
=
V (S)
ṽ0+Ṽ (T)
= πS(T), so Axiom 2 holds.
Conversely, suppose a choice model satisfies the GLA. Then by
selecting R = {i} ⊂ S ⊂
T = N we see from Axiom 1’ that
πi(S) =
πi(N)
1 −
∑
j/∈ S(1 −θj)πj(N)
.
Since π0(N) +
∑
j∈ N πj(N) = 1 the denominator can be written as π0(N) +
∑
j∈ N θjπj(N) +∑
j∈ S(1 −θj)πj(N), resulting in
πi(S) =
πi(N)
π0(N) +

∑
j∈ N θjπj(N) +
∑
j∈ S(1 −θj)πj(N)
.
Letting vj = πj(N) for all j ∈ N+, ṽ0 = v0 +
∑
j∈N θjvj and ṽj = (1 −θj)vj ∈ [0, 1] for j ∈ N
the choice model can be written as
πi(S) =
vi
ṽ0 + ṽ(S)
,
so it satisfies the GAM.
Proof of Theorem 2: Let N ′ be the universe of products. The set
of vendors is M =
{1, . . . ,m}. Let vij be the attraction value of product i offered
by vendor j, and let Sj ⊂ N ′
be the products offered by vendor j ∈ M. Let Oi = {j ∈ M : i ∈
Sj} be the set of vendors
offering product i, and N = {i ∈ N ′ : Oi 6= ∅ } be the collection
of products that are offered.
We consider a NL model where customers first select a product
and then a vendor. Let γi
be the dissimilarity parameter of nest i. Under the nested model,
a nest i with Oi 6= ∅ , is
selected with probability (∑

l∈ Oi v
1/γi
il
)γi
v0 +
∑
k
(∑
l∈ Ok v
1/γk
kl
)γk . (29)
The conditional probability that a customer who selects nest i
will buy the product from
vendor j ∈ Oi is given by
v
1/γi
ij∑
l∈ Oi v
1/γi
il
. (30)
Consequently, the probability that a customer selects product i
∈ Sj from vendor j is given
33

by (∑
l∈ Oi v
1/γi
il
)γi
v0 +
∑
k
(∑
l∈ Ok v
1/γk
kl
)γk v
1/γi
ij∑
l∈ Oi v
1/γi
il
.
We are interested in the limit of these probabilities as γk ↓ 0 for
all k. For this purpose
we remark that well know fact: limγk↓0(
∑

l∈ Ok v
1/γk
il )
γk = maxl∈ Oi vil. The limit, say pij, of
equation (30) is equal to 1 if vendor j is the most attractive, i.e.,
if vij > vil for all l ∈ Oi,
l 6= j; pij = 0, if vij < maxl∈ Oi vil; and pij = 1/m if vij =
maxl∈ Oi vil but vendor j is tied with
other m− 1 vendors.
Now,
πi(Sj) =
pij maxl∈ Oi vil
v0 +
∑
k maxl∈ Ok vkl
.
For all k ∈ N, define V −kj = maxl∈ Ok,l 6=j vkl and Vkj =
maxl∈ Ok∪ {j} vkl. Then, for i ∈ Sj,
πi(Sj) =
Vijpij
v0 +
∑
k∈ Sj Vkj +
∑
k∈S
̄ j V

−
kj
,
where S
̄ j = {i ∈ N : i /∈ Sj}. The selection probability can be
written as
πi(Sj) =
Vijpij
v0 +
∑
k∈ N V
−
kj +
∑
k∈ Sj (Vkj −V
−
kj )
,
or equivalently, as
πi(Sj) =
Vijpij
v0 +
∑
k∈ N V
−

kj +
∑
k∈ Sj (Vkj −V
−
kj )pkj
,
since Vkj − V −kj = (Vkj − V
−
kj )pkj for all k. This is because pkj < 1 implies Vkj − V
−
kj = 0.
This shows that πi(Sj) corresponds to a GAM with ã0j = v0 +
∑
k∈ N V
−
kj , akj = pkjVkj, and
ãkj = (Vkj −V −kj )pkj.
Proof of Proposition 2: Notice that π(S) is increasing in S for
the GAM since V (S) ≥
Ṽ (S) ≥ 0 are both increasing in S. Since the P-GAM is a special
case π(S) is increasing in S.
Now, consider a set T that is not nested-by-fare and let ρ =
π(T). Then there exist an index
k such that πk < ρ ≤ πk+1. Let tk = (πk+1 −ρ)/(πk+1 −πk), tk+1
= 1 − tk and tj = 0 for all
other j. Then

∑n
j=1 πjtj = tkπk + tk+1πk+1 = ρ. We will show that the vector x,
defined by
xj = tkπj(Sk) + tk+1πj(Sk+1),j = 1, . . . ,n majorizes the vector
πj(T), j = 1, . . .n. To do this
we will need to establish the following equation:
tkπi(Sk) + tk+1πi(Sk+1) =
vi
ṽ0 + Ṽ (T)
∀ i ≤ k. (31)
The result then follows, because for j ≤ k, equation (31) implies
that
x[1,j] =
v(Sj)
ṽ0 + Ṽ (T)
≥ πSj (T),
34
as not all elements in Sj need to be in T. On the other hand, for
j > k, we have
x[1,j] = π(T) ≥ πSj (T),
since again, not all elements in Sj need to be in T.
To verify equation (31) we will show that tk and tk+1 can be
written as

tk =
ṽ0 + ṽ(Sk)
ṽ0 + ṽ(T)
v(Sk + 1) −v(T)
v(Sk + 1) −v(Sk)
,
and
tk+1 =
ṽ0 + ṽ(Sk+1)
ṽ0 + ṽ(T)
v(T) −v(Sk)
v(Sk + 1) −v(Sk)
,
as then (31) follows from simple algebra. We know from
arguments in the proof of Proposi-
tion 2, that
πk+1 −πk = πk+1(Sk+1)
ṽ0
ṽ0 + ṽ(Sk)
,
so showing the result for tk is equivalent to showing that
πk+1 −ρ =

v(Sk + 1) −v(T)
ṽ0 + ṽ(Sk + 1)
ṽ0
ṽ0 + ṽ(T)
,
which follows from easily from the fact that ṽj = (1−θ)vj for all
j. The proof for tk+1 follows
along similar lines.
Proof of Proposition 5: We need to compute the difference
between E[min(U(y), max(y,c−
D2)) and E[min(U(y − 1), (y − 1,c− D2)]. Since both quantities
are zero when c − D2 ≥ y
and if c − D2 < y the difference reduces to E[min(U(y),y)] −
E[min(U(y − 1),y − 1)]. By
expressing the expectation as the sum of tail probabilities and
conditioning on the demand
from the potential customer we see that
E[min(U(y),y)] =
y∑
z=1
P(U(y) ≥ z)
= P(U(y) ≥ y) + β
y−1∑
z=1
P(U(y − 1) ≥ z − 1) + (1 −β)
y−1∑
z=1

P(U(y − 1) ≥ z).
We now use the fact that E[min(U(y − 1),y − 1)] =
∑y−1
z=1 P(U(y − 1) ≥ z), to express the
difference as
E[min(U(y),y)] −E[min(U(y − 1),y − 1)] = P(U(y) ≥ y) + βP
(U(y − 1) < y − 1).
Since P(U(y) ≥ y) = βP (U(y − 1) ≥ y − 1) + (1 −β)P(U(y − 1) ≥
y), we conclude that
E[min(U(y),y)]−E[min(U(y−1),y−1)] = [β+(1−β)P(U(y−1) ≥
y|D2 > c−y)]P(D2 > c−y).
35
On the other hand, E min(c−y,D2) −E min(c + 1 −y,D2)] =
−P(D2 > c−y). Therefore,
∆W2(c,y) = [q1(β + (1 −β)P(U(y − 1) ≥ y|D2 > c−y) − q2)]P(D2
> c−y)
as claimed.
36
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Page 1Page 2Page 3Page 4
Overview
Introduction
Neoclassical Choice Model
Luce Model
Random Utility Models
Applications to Revenue Management
Discrete Choice Models
Guillermo Gallego
Department of Industrial Engineering and Operations Research
Columbia University
Spring 2013
Guillermo Gallego Discrete Choice Models
Overview
Introduction
Luce Model

Outline
I Introduction
I Neoclassical Theory
I Luce Model
I Random Utility Models
I Applications to Revenue Management
Overview
Introduction
Luce Model
Outline
I Introduction
I Luce Model

Overview
Introduction
Luce Model
Outline
I Introduction
I Luce Model
Overview
Introduction

Luce Model
Outline
I Introduction
I Luce Model
Overview
Introduction
Luce Model
Outline
I Introduction

I Luce Model
Overview
Introduction
Luce Model
Outline
I Introduction
I Luce Model

Overview
Introduction
Luce Model
Introduction
Choice Modeling is a general purpose tool for making
probabilistic
predictions about human decision making behavior. It is
regarded
as the most suitable method for estimating consumers
willingness
to pay in multiple dimensions. The Nobel Prize for economics
was
awarded to Daniel McFadden in 2000 for his work in the area.
We will consider three theories about about how a rational
decision
maker decides among a discrete number of alternatives:
I Neoclassical choice model
I Luce choice model
I Random utility choice models
Overview
Introduction

Luce Model
Introduction
probabilistic
regarded
willingness
was
decision
I Luce choice model
Overview
Introduction

Luce Model
Introduction
probabilistic
regarded
willingness
was
decision
I Luce choice model
Overview
Introduction
Luce Model

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