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IEOR 460
1
Assignment 9
1. In class we considered a network problem with two resources and three itineraries, with
two fares per itinerary and illustrated a variety of heuristics for that problem. Here
we will consider an expanded model with the same three itineraries but with five fares
per itinerary. Itinerary one, consumes one unit of resource one and has five fares:
185, 160, 150, 1
3
0, 100. Itinerary two, consumes one unit of resource two and has five
fares: 130, 110, 95, 80, 75. Itinerary three, consumes one unit of each resource and has
five fares:
2
60, 240, 220, 190, 170. Using the single index model the incidence matrix A
is given by
A =
(
1 1 1 1 1 0 0 0 0 0 1 1 1 1 1
0 0 0 0 0 1 1 1 1 1 1 1 1 1 1
)
and the fare vector by
p = ( 185 160 150 130 100 130 110 95 80 75 260 240 220 190 170 ) .
Assume a discrete time model with T = 1, 000 periods and time varying arrival rates
given by
λt = ( .00 .00 .015 .036 .054 .00 .00 .015 .04 .06 .00 .00 .01 .03 .03 )
for 501 ≤ t ≤ 1000 and
λt = ( .02 .04 .015 .00 .00 .03 0.04 .015 .00 .00 .04 .02 .01 .00 .00 )
for 1 ≤ t ≤ 500. The initial vector of capacities is c = ( 100, 120 ) .
All vectors should be interpreted as column vectors (although for convenience we write
them as row vectors).
a) Find the aggregate arrival rate ΛTj =
∫ T
0 λtjdt,j = 1, . . . , 15 for the 15 ODFs.
b) Write the corresponding Deterministic Linear Program (dual of the ADP formula-
tion) to determine how much capacity y should be allocated to each ODF.
1
c) Use Excel solver or any other linear programming solver to determine an optimal
capacity allocation ȳj,j = 1, . . . , 15 and use the sensitivity report to obtain the
dual or shadow prices z̄ = z(T,u) for the two resources. Describe how you would
implement the unmodified bid-price heuristic?
d) Partition the 15 ODFs into sets F = {j : ȳj = ΛTj}, P = {j : 0 < ȳj < ΛTj} and R = {j : ȳj = 0}. Describe how you would handle the ODFs in set P under the PAC heuristic.
e) Use the vector of bid-prices obtained in part c) to obtain the net contribution of
the fares to each resource. For example, for ODF 12, p1,12(z̄) = 240 − z̄2 and
p2,12(z̄) = 240− z̄1. Then map all ODFs for each resource into one of three buckets:
bucket 1 for net fares that are at least 120, bucket two for net fares between 70 and
120, and bucket 3 for net fares below 70. Once the buckets are assigned, compute
the weighted average fare and the aggregate demand for each bucket for each leg.
Finally, for each leg, apply the EMSR-b heuristic to obtain protection levels for the
higher fare buckets.
2
e) Estimate the performance of the heuristics of parts c, d and e. Report the approx-
imate expected revenues and the standard error of your simulations.
3
Submitted to Operations Research
manuscript XX
A General Attraction Model and Sales-based Linear
Program for Network Revenue Management under
Customer Choice
Guillermo Gallego
Department of Industrial Engienering and Operations Research, Columbia University, New York, NY
1
0027,
gmg2@columbia.edu
Richard Ratliff and Sergey Shebalov
Research Group, Sabre Holdings, Southlake, TX 76092, richard.ratliff@sabre.com
This paper addresses two concerns with the state of the art in network revenue management with dependent
demands. The first concern is that the basic attraction model (BAM), of which the multinomial logit (MNL)
model is a special case, tends to overestimate demand recapture in practice. The second concern is that the
choice based deterministic linear program, currently in use to derive heuristics for the stochastic network
revenue management problem, has an exponential number of variables. We introduce a generalized attraction
model (GAM) that allows for partial demand dependencies ranging from the BAM to the independent
demand model (IDM). We also provide an axiomatic justification for the GAM and a method to estimate its
parameters. As a choice model, the GAM is of practical interest because of its flexibility to adjust product-
specific recapture. Our second contribution is a new formulation called the Sales Based Linear Program
(SBLP) that works for the GAM. This formulation avoids the exponential number of variables in the earlier
choice-based network RM approaches, and is essentially the same size as the well known LP formulation
for the IDM. The SBLP should be of interest to revenue managers because it makes choice-based network
RM problems tractable to solve. In addition, the SBLP formulation yields new insights into the assortment
problem that arises when capacities are infinite. Together these two contributions move forward the state of
the art for network revenue management under customer choice and competition
.
Key words : pricing, choice models, network revenue management, dependent demands, O&D, upsell,
recapture
1. Introduction
One of the leading areas of research in revenue management (RM) has been incorporating dem
and
dependencies into forecasting and optimization models. Developing effective models for suppliers
to estimate how consumer demand is redirected as the set of available products changes is critical
1
Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization
2 Article submitted to Operations Research; manuscript no. XX
in determining the revenue maximizing set of products and prices to offer for sale in industries
where RM is used. These industries include airlines, hotels, and car rental companies, but the issue
of how customers select among different offerings is also important in transportation, retailing and
healthcare. Several terms are used in industry to describe different types of demand dependen-
cies. If all products are available for sale, we observe the first-choice, or natural, demand, for each
of the products. However, when a product is unavailable, its first-choice demand is redirected to
other available alternatives (including the ‘no-purchase’ alternative). From a supplier’s perspective,
this turned away demand for a product can result in two possible outcomes: ‘spill’ or ‘recapture’.
Spill refers to redirected demand that is lost to competition or to the no-purchase alternative, and
recapture refers to redirected demand that results in the sale of a different available product. Depen-
dent demand RM models lead to improved revenue performance because they consider recaptured
demand that would otherwise be ignored using traditional RM methods that assume independent
demands. The work presented in this paper provides a practical mechanism to incorporate both
of these effects into the network RM optimization process through use of customer-choice models
(including methods to estimate the model parameters in the presence of competition).
An early motivation for studying demand dependencies was concerned with incorporating a
special type of recapture known as ‘upsell demand’ in single resource RM problems. Upsell
is
recapture from closed discount fare classes into open, higher valued ones that consume the same
capacity (called ‘same-flight upsell’ in the airline industry). More specifically, there was a perceived
need to model the probability that a customer would buy up to a higher fare class if his preferred fare
was not available. The inability to account for upsell demand makes the widely used independent
demand model (IDM) too pessimistic, resulting in too much inventory offered at lower fares and
in a phenomenon known as ‘revenue spiral down’; see Cooper et al. (2006). To our knowledge, the
first authors to account for upsell potential in the context of single resource RM were Brumelle
et al. (1990) who worked out implicit formulas for the two fare class problem. A static heuristic
was proposed by Belobaba and Weatherford (1996), who made fare adjustments while keeping
the assumption of independent demands. Talluri and van Ryzin (2004) develop the first stochastic
Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization
Article submitted to Operations Research; manuscript no. XX 3
dynamic programming formulation for choice-based, dependent demand for the single resource case.
Gallego, Ratliff and Li (2009) developed heuristics that combined fare adjustments with dependent
demand structure for multiple fares that worked nearly as well as solving the dynamic program of
Talluri and van Ryzin (2004). Their work suggest that considering same-flight upsell has significant
revenue impact, leading to improvements ranging from 3-6%.
For network models, in addition to upsell, it is also important to consider ‘cross-flight recapture’
that occurs when a customer’s preferred choice is unavailable, and he selects to purchase a fare
on a different flight instead of buying up to a higher fare on the same flight. Authors that have
tried to estimate upsell and recapture effects include Andersson (1998), Ja et al. (2001), and Ratliff
et al. (2008). In markets in which there are multiple flight departures, recapture rates typically
range between 15-55%; see Ja et al. (2001). Other authors have dealt with the topic of demand
estimation from historical sales and availability
data.
The most recent methods use maximum like-
lihood estimation techniques to estimate primary demands and market flight alternative selection
probabilities (including both same-flight upsell and cross-flight recapture); for details, see Vulcano
et al. (2012), Meterelliyoz (2009) and Newman, Garrow, Ferguson and Jacobs (2010).
Most of the optimization work in network revenue management with dependent demands is based
on formulating and solving large scale linear programs that are then used as the basis to develop
heuristics for the stochastic network revenue management (SNRM) problem; e.g. see Bront et al.
(2009), Fiig et al. (2010), Gallego et al. (2004), Kunnumkal and Topaloglu (2010), Kunnumkal
and Topaloglu (2008), Liu and van Ryzin (2008), and Zhang and Adelman (2009). Note that
preliminary versions of Fiig et al. (2010) were discussed as early as 2005, and the method is one
of the earliest practical heuristics for incorporating dependent demands into SNRM optimization.
However, as described in section 5.4, the applicability of the approach requires severe assumptions
and is restricted to certain special cases and fundamentally differs from the discrete choice models
described in this paper.
Our paper makes two main contributions. The first contribution is to discrete choice modeling
and should be of interest both to people working on RM, as well as to others with an interest
Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization
4 Article submitted to Operations Research; manuscript no. XX
in discrete choice modeling. The second contribution is the development of a tractable network
optimization formulation for dependent demands under the aforementioned discrete choice model.
Our first contribution is a response to known limitations of the multinomial choice model (MNL)
which is often used to handle demand dependencies. The MNL developed by McFadden (1974) is a
random utility model based on independent Gumbel errors, and a special case of the basic attraction
model (BAM) developed axiomatically by Luce (1959). The BAM states that the demand for a
product is the ratio of the attractiveness of the product divided by the sum of the attractiveness
of the available products (including a no purchase alternative). This means that closing a product
will result in spill, and a portion of the spill is recaptured by other available products in proportion
to their attraction values. In practice, we have seen situations where the BAM overestimates or
misallocates recapture probabilities. In a sense, one could say that the BAM tends to be too
optimistic about recapture. In contrast, the independent demand model (IDM) assumes that all
demand for a product is lost when the product is not available for sale. It is fair to say, therefore,
that the IDM is too pessimistic about recapture as all demand spills to the no-purchase alternative.
To address these limitations, we develop a generalized attraction model (GAM) that is better at
handling recapture than either the BAM or the IDM. In essence, the GAM adds flexibility in
modeling spill and recapture by making the attractiveness of the no-purchase alternative a function
of the products not offered. As we shall see, the GAM includes both the BAM and the IDM as
special cases. We justify the GAM axiomatically by generalizing one of the Luce Axioms to better
control for the portion of the demand that is recaptured by other products. We show that the GAM
also arises as a limiting case of the nested logit (NL) model where the offerings within each nest
are perfectly correlated. In its full generality, the GAM has about twice as many parameters than
the BAM. However, a parsimonious version is also presented that has only one more parameter
than the BAM.
We adapt and enhance the Expectation-Maximization algorithm (henceforth EM algorithm)
developed by Vulcano et al. (2012) for the BAM to estimate the parameters for the GAM , with
the estimates becoming better if the firm knows its market share when it offers all products. The
Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization
Article submitted to Operations Research; manuscript no. XX 5
estimation requires only data from the products offered by the firm and the corresponding selections
made by customers.
Our second main contribution is the development of a new formulation for the network revenue
management problem under the GAM. This formulation avoids the exponential number of variables
in the Choice Based Linear Program (CBLP) that was designed originally for the BAM; see Gallego
et al. (2004). In the CBLP, the exponential number of variables arises because a variable is needed to
capture the amount of time each subset of products is offered during the sales horizon. The number
of products itself is usually very large; for example, in airlines, it is the number of origin-destination
fares (ODFs), and the number of subsets is two raised to the number of ODFs for each market
segment. This forces the use of column generation in the CBLP and makes resolving frequently
or solving for randomly generated demands impractical. We propose an alternative formulation
called the sales based linear program (SBLP), which we show is equivalent to the CBLP not only
under the BAM but also under the GAM. Due to a unique combination of demand balancing and
scaling constraints, the new formulation, has a linear rather than exponential number of variables
and constraints. This provides important practical advantages because the SBLP allows for direct
solution without the need for large-scale column generation techniques. We also show how to
recover the solution to the CBLP from the SBLP. We show, moreover, that the solution to the
SBLP is a nested collection of offered sets together with the optimal proportion of time to offer
each set. The smallest of these subsets is the core, consisting of products that are always offered.
Larger subsets typically contain lower-revenue products that are only offered part of the time.
Offering these subsets from the largest to the smallest in the suggested portions of the time over
the sales horizon provides a heuristic to the stochastic version of the problem. Moreover, the ease
with which we can compute the optimal offered sets from the SBLP formulation allow us to resolve
the problem frequently and update the offer sets. In addition, other heuristics proposed by the
research community, e.g., Bront et al. (2009), Kunnumkal and Topaloglu (2010), Kunnumkal and
Topaloglu (2008), Liu and van Ryzin (2008), Talluri (2012), and Zhang and Adelman (2009), can
benefit from the ability to quickly solve large scale versions of the CBLP and SBLP formulations.
Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization
6 Article submitted to Operations Research; manuscript no. XX
The solution to the SBLP also provides managerial insights into the assortment problem (a special
case that arises when capacities are infinite).
The remainder of this paper is organized as follows. In §2 we present the GAM as well as
two justifications for the model. We then present an expectation-maximization (E-M) algorithm
to estimate its parameters. In §3 we present the stochastic, choice based, network revenue
management problem. The choice based linear program is reviewed in section §4. The new,
sales based linear program is presented in §5 as well as examples that illustrate the use of the
formulation. The infinite capacity case gives rise to the assortment problem and is studied in §6.
Our conclusions and directions for future research are in §7.
2. Generalized Attraction Model
For several decades, most revenue management systems operated under the simplest type of cus-
tomer choice process known as the independent demand model (IDM). Under the IDM, demand for
any given product is independent of the availability of other products being offered. So if a vendor
is out-of-stock for a particular product (or it is otherwise unavailable for sale), then that demand
is simply lost (to the no-purchase alternative); there is no possibility of the vendor recapturing
any of those sales to alternate products because the demands are treated as independent. Many
practitioners recognized that ignoring such demand interactions lacked realism, but efforts to incor-
porate upsell heuristics into RM optimization processes were hampered by practical difficulties in
estimating upsell rates. It was widely feared that, without a rigorous foundation for estimating
upsell, the use of such heuristics could lead to an overprotection bias in the inventory controls (and
consequent revenue losses). By the late 1990’s, progress on dependent demand estimation methods
began to be made. van Ryzin, Talluri and Mahajan (1999) presented excellent arguments for the
integration of discrete choice models into RM optimization processes, and initial efforts to apply
this approach focused on the multinomial logit model (MNL) which is a special case of the basic
attraction model (BAM) discussed next.
Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization
Article submitted to Operations Research; manuscript no. XX 7
Under the BAM, the probability that a customer selects product j ∈ S when set S ⊂ N =
{1,.. .,n} is offered, is given by
πj(S) =
vj
v0 +
V (S)
, (1)
where V (S) =
∑
j∈S vj. The vj values are measures of the attractiveness of the different choices,
embodied in the MNL model as exponentiated utilities (vj = e
ρuj for utilities uj where ρ > 0 is a
parameter that is inversely related to the variance of the Gumbel distribution). The no-purchase
alternative is denoted by j = 0, and π0(S) is the probability that a customer will prefer not to
purchase when the offer set is S. One could more formally write πj(S+) instead of πj(S) since
the customer is really selecting from S+. However, we will refrain from this formalism and follow
the convention of writing πj(S) to mean πj(S+). Also, unless otherwise stated, S ⊂ N and the
probabilities πj(S) refer to a particular vendor. In discrete choice models, a single vendor is often
implicitly implied. However, in some of our discussions below, we will allow for multiple
vendors.
In the case of multiple vendors, the probabilities πj(S) j ∈S, will depend on the offerings of other
vendors.
There is considerable empirical evidence that the BAM may be optimistic in estimating recapture
probabilities. 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, leave without purchasing,
or go to another store and look for the 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.
We will propose a generalization of the BAM , called the General Attraction Model (GAM) to
better deal with the consequences of not offering a product. Before presenting a formal definition
of the GAM, we will present a simple example and will revisit the red-bus, blue-bus paradox (Ben-
Akiva and Lerman (1994)). Suppose there are two products with attraction values v1 = v2 = v0 = 1.
Under the BAM, πk({1,2}) = 1/3 for k = 0,1,2. Eliminating choice 2 results in πk({1}) = 50% for
Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization
8 Article submitted to Operations Research; manuscript no. XX
k = 0,1, so one half of the demand for product 2 is recaptured by product 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. 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 can now select between the no-purchase alternative (v0 = 1), product 1 (v1 = 1), and
product 2’ (w2 = 0.5), so the probabilities of purchase from the store offering only product 1 are
now
π0({1}) =
1.5
2.5
= 60%, π1({1}) =
1
2.5
= 40%.
By including 2’ as a latent alternative, the probability of choosing product 1 has decreased from
50% under the BAM to 40%. However, this probability is still higher than that of the independent
demand model (IDM) which is 33.3%. So the GAM allows us to obtain estimates that are in-between
the BAM and the IDM by considering product specific shadow attractions. This formulation may
also help with inter-temporal choices, where a choice j /∈S+ may become available at a later time.
In this case wj is the shadow attraction of choice j discounted by time and the risk that it may
not be available in the future. Notice that this formulation is different from what results if we use
the traditional approach of viewing a product and location as a bundle. Indeed, including product
2’ with attraction w2 = 0.5, results in choice set {1,2,2′} for the customer when the vendor offers
assortment {1,2}. This results in π1({1,2,2′}) = 1/3.5 6= 1/3. The problem with this approach, in
terms of the MNL, is that an independent Gumbel random variable is drawn to determine the
random utility of product 2’.
Under the red-bus, blue-bus paradox, a person has a choice between driving a car and taking
either a red or blue bus (it is implicitly assumed that both buses have ample capacity and depart
at the same time). Let v1 represent the direct attractiveness of driving a car and let v2,w2 repre-
sent, respectively, the attraction and the shadow attraction of the two buses. If w2 = v2 then the
probabilities are unchanged if one of the buses is removed (as one intuitively expects because there
are essentially zero search costs in finding a comparable bus). The model with w2 = 0 represents
Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization
Article submitted to Operations Research; manuscript no. XX 9
the case where there is no second bus. The common way to deal with the paradox is to use a nested
logit (NL) model. Under the NL model used to resolve the paradox, a person first selects a mode
of transportation and then one of the offerings within each mode. We are able to avoid the use of
the NL model, but as we shall see later, our model can be viewed as a limit of a NL model.
To formally define the GAM let S̄ = {j ∈N : j /∈S} be the complement of S in N = {1,.. .,n}.
In addition to the attraction values vj, there are shadow attraction values wj ∈ [0,vj],j ∈N such
that for any subset S ⊂N
πj(S) =
vj
v0 + W(S̄) + V (S)
j ∈S, (2)
where for any R ⊂ N, W(R) =
∑
j∈R wj. Consequently, the no-purchase probability is given by
π0(S) = (v0 + W(S̄))/(v0 + W(S̄) + V (S)). In essence, the attraction value of the no-purchase
alternative has been modified to be v0 + W(S̄).
An alternative, perhaps simpler, way of presenting the GAM by using the following transforma-
tion: ṽ0 = v0 + W(N) and ṽj = vj −wj,j ∈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−πS(S). (3)
where πS(S) =
∑
j∈S πj(S). Notice that all choice probabilities are non-negative as long as ṽj =
vj −wj ≥ 0
and ṽ0 =
v0 +
∑
j∈N wj > 0, which holds when v0 > 0 and wj ∈ [0,vj] for all
j ∈N.
The case wk = 0,k ∈N recovers the BAM, because ṽ0 = v0 and ṽj = vj,j ∈N. In contrast, the
case wk = vk,k ∈N, results in ṽ0 + Ṽ (S) = v0 + V (N) for all offered sets S ⊂N. The probability
πj(S), of selecting product j is independent of the set S containing j, resulting in the independent
demand model (IDM).
The parsimonious formulation P-GAM is given by wj = θvj ∀j ∈N for θ ∈ [0,1] can serve to
test H0 : θ = 0 vs H1 : θ > 0 to see whether the BAM applies or to test H0 : θ = 1 against H1 : θ < 1,
to see whether the IDM applies. If either of these tests fail, then a GAM maybe a better fit to the
data.
Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization
10 Article submitted to Operations Research; manuscript no. XX
We provide two justifications for the GAM. The first justification is Axiomatic. The second
is a limiting case of the nested logit (NL) model. Readers who are not interested in the formal
derivation of the GAM can skip to section 2.4 for a method to estimate the GAM parameters.
2.1. Axiomatic Justification
Luce (1959) proposed two choice axioms. To describe the axioms under the assumption that the
no-purchase alternative is always available we will use the notation πS+ (T) =
∑
j∈S+
πj(T) for all
S ⊂T and will use set difference notation T −S = T ∩ S̄ to represent the elements of T that are
not in S. The Luce axioms are given by :
• 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}).
The most celebrated consequence of the Luce Axioms, due to Luce (1959), is that πi(S) satisfies
the Luce Axioms if and only if there exist attraction values vj,j ∈N+ such that equation (1) holds.
Since its original publication, the paper Luce (1959) has been cited thousands of times. Among
the most famous citations, we find the already mentioned paper by McFadden (1974) that shows
that the only random utility model that is consistent with the BAM is the one with independent
Gumbel errors. An early paper by Debreu (1960), criticizes the Luce model as it suffers from the so
called independence of irrelevant alternatives (IIA). The IIA states the relative preference between
two alternatives does not change as additional alternatives are added to the choice set. Debreu,
however, suggests that the addition of an alternative to an offered set hurts alternatives that are
similar to the added alternative more than those that are dissimilar to it. To illustrate this, Debreu
provides an example that is a precursor to the red-bus, blue-bus paradox based on the choice
between three records: a suite by Debussy, and two different recordings of the same Beethoven
Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization
Article submitted to Operations Research; manuscript no. XX 11
symphony. Another famous paper by Tversky (1972), criticizes both the Luce model as well as all
random utility models that assume independent utilities for their inability to deal with the IIA.
Tversky offers the elimination by aspects theory as a way to cope with the IIA. Others, such as n
Domenich and McFadden (1975) and Williams (1977), have turned to the nested logit (NL) model
as an alternative way of dealing with the IIA. Under a nested logit model, for example, a person
will first chose between modes of transportation (car or bus) before selecting a particular car or
bus color.
To our knowledge no one has altered the Luce Axioms with the explicit goal of modifying the
recapture probabilities. Yellott (1977), however, gives a characterization of the Luce model by an
axiom which he called invariance under uniform expansion of the choice set, which is weaker than
Luce’s Axiom but implies Luce’s Axiom when the choice model is assumed to be an independent
random utility model. The reader is also refereed to Dagsvik (1983), who expands Yellott’s ideas
to choices made over time.
Notice that Axiom 1 holds trivially for R = ∅ since π∅(S) = 0 for all S ⊂N. For ∅ 6= R⊂S the
formula can be written as the ratio of the demand captured by R when we add T −S to S. The
ratio is given by
πR(T)
πR(S)
= πS+ (T) = 1−πT−S(T),
and is independent of R⊂S. Our goal is to generalize Axiom 1 so that the ratio remains indepen-
dent of R⊂S but in such a way that we have more control over the proportion of the demand that
goes to T −S. This suggests the following generalization of Axiom 1.
• Axiom 1’: If πi({i}) ∈ (0,1) for all i∈T, then for any ∅ 6= 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.
We will call the set of Axioms 1’ and 2 the Generalized Luce Axioms (GLA). We can think of
θj ∈ [0,1] as a relative measure of the competitiveness of product j in the market place. The next
Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization
12 Article submitted to Operations Research; manuscript no. XX
result states that a choice model satisfies the GLA if and only if it is a GAM and links θj to the
ratio of wj to vj.
Theorem 1. A choice model πi(S) satisfies the GLA if and only if it is of the GAM form. More-
over, wj = θjvj for all j ∈N.
A proof of this theorem may be found in the Appendix. Notice that it is tempting to relax the
constraint wj ≥ 0 as long as ṽ0 > 0. However, this relaxation would lead to a violation of Axiom 2.
2.2. GAM as Limit of Nested Logit Model under Competition
We now show that the GAM also arises as the limit of the nested logit (NL) model. The NL model
was originally proposed in Domenich and McFadden (1975), and later refined in McFadden (1978),
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 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 correspond to individual products offered in the
market. Customers first select a product, and then one of the vendors offering the selected product.
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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 v1/γiij∑
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 vij’s 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 γ’s
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 γ’s
go to zero.
It is easy to see that πi(Sj) is a BAM for each vendor 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 γi ↓ 0 for all i is a GAM. More precisely, there are attraction
values akj, ãkj ∈ [0,akj], and ã0j ≥ 0, such that the discrete choice model for vendor j is given by
πi(Sj) =
aij
ã0j +
∑
k∈Sj
ãkj
∀i∈Sj.
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14 Article submitted to Operations Research; manuscript no. XX
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 select 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.
2.3. Limitation and Heuristic Uses of the GAM
One limitation of the GAM as we propose it is that, in practice, the attraction and shadow attrac-
tion values may depend on a set of covariates that may be customer or product specific. This can be
dealt with either by assuming a latent class model on the population of customers or by assuming
a customer specific random set of coefficients to capture the impact of the covariates. This results
in a mixture of GAMs. The issues also arise when using the BAM. In practice, often a single BAM
is used to try to capture the choice probabilities of customers that may come from different market
segments. This stretches both models, as this should really be handled by adding market seg
ments.
However the GAM can cope much better with this situation as the following example illustrates.
Example 1: Suppose there are two stores (denoted l), and each store attracts half of the
population. We will assume that there are two products (denoted k) and that the GAM parameters
vlk and wlk for the two stores are as given in Table 1:
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Article submitted to Operations Research; manuscript no. XX 15
v10 v11 v12 v20 v21 v22
w11 w12 w21 w22
1 1 1 1 0.9 0.9
0.9 0.8 0.8 0.7
Table 1 Attraction Values (v,w)
With this type of information we can use the GAM to compute choice probabilities πj(S) that
a customer will purchase product j ∈S from Store 1 when Store 1 offers set S, and Store 2 offers
set R = {1,2}. Suppose Store 1 has observed selection probabilities for different offer sets S as
provided in Table 2:
S π0(S) π1(S) π2(S)
∅ 100% 0% 0%
{1} 82.1% 17.9% 0%
{2} 82.8% 0.0% 17.2%
{1,2} 66.7% 16.7% 16.7%
Table 2 Observed πj (S,R) for R = {1,2}
To estimate these choice probabilities using a single market GAM we found the values of
vk,wk,k = 1,2 with v0 = 1 that minimizes the sum of squared errors of the predicted probabilities.
This results in the GAM model: v0 = 1, v1 = v2 = 0.25, w1 = 0.20 and w2 = 0.15. This model per-
fectly recovers the probabilities in Table 2. However, when limited to the BAM, the largest error
is 1.97% in estimating π0({1,2}).
This improved ability to fit the data comes at a cost because the GAM requires estimating w in
addition to v. As we will see in the next section, it is possible to estimate the v and w from the
store’s offer sets and sales data. We will assume, as in any estimation of discrete choice modeling,
that the same choice model prevails over the collected data set. If this is not the case, then the
estimation should be done over a subset of the data where this assumption holds. As such, we do
not require information about what the competitors are offering, only that their offer sets do not
vary over the data set. This is an implicit assumption made when estimating any discrete choice
model. The output of the estimation provides the store manager an indication of which products
face more competition by looking at the resulting estimates of the ratios θi = wi/vi. Over time,
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16 Article submitted to Operations Research; manuscript no. XX
a firm can estimate parametric or full GAM models that take into account what competitors are
offering and then update their choice models dynamically depending on what the competition is
doing.
2.4. GAM Estimation and Examples
It is possible to estimate the parameters of the GAM by refining and extending the Expectation
Maximization method developed by VvRR (Vulcano et al. (2012)) for the BAM. Readers not
interested in estimation can skip to Section 3. We assume that demands in periods t = 1,.. .,T arise
from a GAM with parameters v and w and arrival rates λt = λ for t = 1,.. .,T . The homogeneity
of the arrival rates is not crucial. In fact the method described here can be extended to the case
where the arrival rates are governed by co-variates such as the day of the week, seasonality index,
and indicator variables of demand drivers such as conferences, concerts or other events that can
influence demand. Our main purpose here is show that it is possible to estimate demands fairly
accurately. The data are given by the offer sets St ⊂N and the realized sales zt ⊂Zn+, excluding the
unobserved value of the customers who chose the no-purchase alternative, for t = 1,.. .,T . VvRR
assume that the market share s = πN(N), when all products are offered, is known. While we do
not make this assumption here, we do present results with and without knowledge of s.
VvRR update estimates of v by estimating demands for each of the products in N when some of
them are not offered. To see how this is done, consider a generic period with offer set S ⊂N,S 6= N
and realized sales vector Z = z. Let Xj be the demand for product j if set N instead of S was
offered. We will estimate Xj conditioned on Z = z. Consider first the case j /∈ S. Since Xj is
Poisson with parameter λπj(N), substituting the MLE estimate λ̂ = e
′z/πS(S) of λ we obtain the
estimate X̂j = λ̂πj(N) =
πj (N)
πS (S)
e′z. We could use a similar estimate for products j ∈S but this would
disregard the information zj. For example, if zj = 0 then we know that the realization of Xj was
zero as customers had product j available for sale. We can think of Xj, conditioned on (S,z) as a
binomial random variable with parameters zj and probability of selection πj(N)/πj(S). In other
words, each of the zj customers that selected j when S was available, would have selected j anyway
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Article submitted to Operations Research; manuscript no. XX 17
with probability πj(N)/πj(S). Then X̂j =
πj (N)
πj (S)
zj is just the expectation of the binomial. Notice
that X̂[1,n] =
∑m
i=1
X̂j =
e′z
πS (S)
πN(N) = λ̂s. We can find X̂0 by treating it as j /∈ S resulting in
X̂0 =
π0(N)
πS (S)
e′z = λ̂(1−s). We summarize the results in the following proposition:
Proposition 1. (VvRR extended to the GAM): If j ∈S then
X̂j =
πj(N)
πj(S)
zj.
If j /∈S or j = 0 we have
X̂j =
πj(N)
πS(S)
e′z.
Moreover, X̂[1,n] =
∑
j∈N X̂j =
πN (N)
πS (S)
e′z.
VvRR use the formulas for the BAM to estimate the parameters λ,v1,.. .,vn under the assump-
tion that the share s = v(N)/(1 + v(N)) is known. From this they use an E-M method to iterate
between estimating the X̂js given a vector v and then optimizing the likelihood function over v
given the observations. The MLE has a closed form solution so the updates are given by v̂j = X̂j/X̂0
for all j ∈N, where X̂0 = rX̂[1,n] and r = 1/s−1. VvRR use results in Wu (1983) to show that
the method converges. We now propose a refinement of the VvRR method that can significantly
reduce the estimation errors.
Using data (St,zt),t = 1… ,T, VvRR obtain estimates X̂tj that are then averaged over the T
observations for each j. In effect, this amounts to pooling estimators from the different sets by
giving each period the same weight. When pooling estimators it is convenient to take into account
the variance of the estimators and give more weight to sets with smaller variance. Indeed, if we
have T different unbiased estimators of the same unknown parameter, each with variance σ2t , then
giving weights proportional to the inverse of the variance minimizes the variance of the pooled
estimator. Notice that the variance of X̂tj is proportional to 1/πj(St) if j ∈St and proportional to
1/πSt (St) otherwise. Let atj = πj(St) if j ∈St and atj = πSt (St) if j /∈St. We can select the set the
weights for product j as
ωtj =
atj∑T
s=1
asj
.
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18 Article submitted to Operations Research; manuscript no. XX
Our numerical experiments suggests that using this weighting scheme reduces the variance of the
estimators by about 4%.
To extend the procedure to the GAM we combine ideas of the EM method based on MLEs and
least squares (LS). It is also possible to refine the estimates by using iterative re-weighted least
squares (Green (1984) and Rubin (1983)) but our computational experience shows that there is
little benefit from doing this. To initialize the estimation we use formulas for the expected sales
under St,t = 1,. ..,T for arbitrary parameters λ,v and w. More precisely, we estimate E[Ztj] by
λπj(St) = λ
vj
v0 + V (S) + W(S̄)
j ∈St, t∈{1,.. .,T}.
We then minimize the sum of squared errors between estimated and observed sales (zts):
T∑
t=1
∑
i∈St
[
λ
vj
v0 + V (S) + W(S̄)
−zti
]2
,
subject to the constraints 0 ≤ w ≤ v and λ ≥ 0. This is equivalent to maximizing the likelihood
function ignoring the unobserved values of the no-purchase alternative zt0.
This gives us an initial estimate λ(0),v(0),w(0) and an initial estimate of the market share s(0) =
v(0)(N)/(1 + v(0)(N)). Let r(0) = 1/s(0) − 1. If s is known then we add the constraint v(N) = r =
1/s−1. Given estimates of v(k) and w(k) the procedure works as follows:
E-step: Estimate the first choice demands X̂
(k)
tj and then aggregate over time using the current
estimate of the weights α
(k)
tj to produce X̂
(k)
j =
∑T
t=1
X̂
(k)
tj α
(k)
tj and X̂
(k)
0 = r
(k)X̂(k)[1,n] and update
the estimates of v by
v
(k+1)
j =
X̂
(k)
j
X
(k)
0
j ∈N.
M-step: Feed v(k+1) together with arbitrary λ and w to estimate sales under St,t = 1,.. .,T and
minimize the sum of squared errors between observed and expected sales over λ and w
subject to
w≤v(k+1) and non-negativity constraint to obtain updates λ(k+1), w(k+1) and s(k+1).
The proof of convergence is a combination of the arguments in Wu, as in VvRR, and the fact
that the least squared errors problem is a convex minimization problem. The case when s is known
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Article submitted to Operations Research; manuscript no. XX 19
works exactly as above, except that we do not need to update this parameter. The M-step can be
done by solving the score equations:
∑
t∈Ai
πi(St)[e
′zt + λπ0(St)]−
∑
t∈Ai
zti = 0 i = 1,.. .,n,
∑
t/∈Ai
π̂i(St)[e
′zt + λπ0(St)]−λ
∑
t/∈Ai
π̂i(St) = 0 i = 1,.. .,n.
and
T∑
t=1
πSt (St)−
T∑
t=1
e′zt/λ = 0,
where Ai is the set of periods where product i is offered and
π̂i(St) =
vi
1 + V (St) + W(S̄t)
t /∈Ai.
This is a system of up to 2n + 1 equations on 2n + 1 unknowns. Attempting to solve the score
equations directly or through re-weighted least squares does not materially improve the estimates
relative to the least square procedure described above.
We now present three examples to illustrate how the procedure works in three different cases:
1) w = 0 corresponding to the BAM, 2) w = θv corresponding to the (parsimonious) p-GAM, and
3) 0 6= w≤v corresponding to the GAM. These examples are based on Example 1 in VvRR. The
examples involve five products and 15 periods where the set N = {1,2,3,4,5} was offered in four
periods, set {2,3,4,5} was offered in two periods, while sets {3,4,5},{4,5} and {5} were each
offered in three periods. Notice that product 5 is always offered making it impossible to estimate
w5. On the other hand, estimates of w5 would not be needed unless the firm planned to not offer
product 5.
In their paper, VvRR present the estimation results for a specific realization of sales zt,t =
1,.. .,T = 15 corresponding to the 15 offer sets St,t = 1,. ..,15. We instead generate 500 simulated
instances of sales, each over 15 periods, and then compute our estimates for each of the 500
simulations. We report the mean and standard deviation of λ, v and either θ or w over the 500
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simulations. We feel these summary statistics are more indicative of the ability of the method
to estimate the parameters. We report results when s (historical market share) is assumed to be
known, as in Vulcano et al. (2012), and some results for the case when s is unknown (see 1. in the
endnotes).
Estimates of the model parameters tend to be quite accurate when s is known and the estimation
of λ and v become more accurate when w 6= 0. The results for s unknown tend to be accurate when
w = 0 but severely biased when w 6= 0. The bias is positive for v and w and negative for λ. Since it is
possible to have biased estimates of the parameters but accurate estimates of demands, we measure
how close the estimated demands λ̂π̂i(S) = λ̂
v̂i
1+v̂(S)+ŵ(S̄)
i∈S, S ⊂N are from the true expected
demands λπi(S), i∈S, S ⊂N and also how close the estimated demands λ̂π̂i(St) i∈St,t = 1,.. .,T
are from sales zti,i∈St,t = 1,.. .,T . These measures are reported, respectively, as the Model Mean
Squared Error (M-MSE ) and the traditional Mean Squared Error (MSE) between the estimates
and the data. Surprisingly the results with unknown s, although strongly biased in terms of the
model parameters, often produce M-MSEs and MSEs that are similar to the corresponding values
when s is known.
Example 2 (BAM) In the original VvRR example, the true value of w is 0. The estimates with
s known and s unknown as well as the true values of the parameters are given in Table 3.
s λ v1 v2 v3 v4 v5
known 50.27 0.988 0.700 0400 0.200 0.050
unknown 51.26 1.212 0.818 0.449 0.219 0.053
true 50.00 1.000 0.700 0.400 0.200 0.050
Table 3 Estimated values of λ and v with s known
The MSE and the M-MSE were respectively 247.50 and 58.44, for both the case of s known and
unknown.
Example 3 (P-GAM) For this example v is the same as above but w = θv with θ = 0.20. The
results are given in Table 4. Notice that the estimate of θ is more accurate for the case of s unknown.
Clearly the estimated values with s known are much closer to the actual values. However, the fit
to the model and to data for the case of s unknown is similar to the case of s known. Indeed, the
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Article submitted to Operations Research; manuscript no. XX 21
s λ v1 v2 v3 v4 v5 θ
known 50.12 0.988 0.702 0.397 0.201 0.051 0.238
unknown 4.390 0.111 0.079 0.062 0.035 0.014 0.219
true 50.00 1.000 0.700 0.400 0.200 0.050 0.200
Table 4 Estimated values of λ,v and θ with s known
MSE and the M-MSE were respectively 226.2 and 50.37 for the case of s known and 228.54 and
53.84, respectively, for the case of s unknown. This suggests that the parameter values that can be
far from the true parameters and yet have a sufficiently close fit to the model and to the data.
Example 4 (GAM) For this example λ and v are as above, and w = (0.25,0.35,0.15,0.05,0.05).
The results are given in Table 5.
s λ v1 v2 v3 v4 v5 w1 w2 w3 w4 w5
known 50.19 0.994 0.700 0.404 0.202 0.051 0.321 0.288 0.186 0.106 0.010
true 50.00 1.000 0.700 0.400 0.200 0.050 0.250 0.350 0.150 0.050 0.050
Table 5 Estimated values of λ,v and w with s known and unknown
Notice that the estimated values of λ and v for the case of v known are better under the GAM
than for the case w = 0 or w = θv. The MSE and M-MSE are respectively 204.71 and 48.56 for the
case of s known. The estimates of the parameters are severely biased when s is unknown so are not
reported. However, the MSE and M-MSE were respectively, 203.70 and 54.33, a surprisingly good
fit to data. This again suggests that there is a ridge of parameter values that fit the data and the
model well.
3. Stochastic Revenue Management over a Network
In this section we present the formulation for the stochastic revenue management over a network.
We will assume that there are L customer market segments and that customers in a particular
market segment are interested in a specific set of itineraries and fares (see 2. in the endnotes) from
a set of origins to one or more destinations. For example, a market segment could be morning
flights from New York to San Francisco that are less than $500, or all flights from New York
City to the Mexican Caribbean. Notice that there may be multiple fares with different restrictions
associated with each itinerary. At any time the set of origin-destination-fares (ODFs) offered for
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22 Article submitted to Operations Research; manuscript no. XX
sale to market segment l is a subset Sl ⊂Nl,l∈L = {1,.. .,L} where Nl is the consideration set for
market segment l∈L with associated fares plk, for each product k in nest Nl. If the consideration
sets are non-overlapping over the market segments then there is no need to add further restrictions
to the offered sets. This is also true when consideration sets overlap if the capacity provider is free
to independently decide what fares to offer in each market. As an example, round-trips from the
US to Asia are sold both in Asia and the US. If the carrier can charge different fares depending
on whether the customer buys the fare in the US or in Asia, then no further restrictions are
needed. Otherwise, he needs to impose restrictions so that the same set of products is offered in
both markets. One way to handle this is to post a single offer set S at a time, so then use the
projection subsets Sl = S ∩Nl for all markets l that are constrained to offers the same price for
offered fares. Customers in market segment l select from Sl and the no-purchase alternative. We
will abuse notation slightly and write πlk(Sl) to denote the probability of selecting product k∈Sl+
from market segment l. For convenience we define πlk(Sl) = 0 if k /∈Sl+.
The state of the system will be denoted by (t,x) where t is the time-to-go and x is an m-
dimensional vector representing remaining inventories. The initial state is (T,c) where T is the
length of the horizon and c is the initial inventory capacity. For ease of exposition we will assume
that customers arrive to market segment l as a Poisson process with time homogeneous rate λl.
We will denote by J(t,x) the maximum expected revenues that can be obtained from state (t,x).
The Hamilton-Jacobi-Bellman (HJB) equation is given by
∂J(t,x)
∂t
=
∑
l∈L
λl max
Sl⊂Nl
∑
k∈Sl
πlk(Sl) (plk −∆lkJ(t,x)) (5)
where ∆lkJ(t,x) = J(t,x)−J(t,x−Alk), where Alk is the vector of resources consumed by k∈Nl
(e.g. flight legs on a connecting itinerary class). The boundary conditions are J(0,x) = 0 and
J(t,0) = 0. Talluri and van Ryzin (2004) developed the first stochastic, choice-based, dependent
demand model for the single resource case in discrete time. The first choice-based, network
formulation is due to Gallego et al. (2004).
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Article submitted to Operations Research; manuscript no. XX 23
4. The Choice Based Linear Program
An upper bound J̄(T,c) on the value function J(T,c) can be obtained by solving a deterministic
linear program. The justification for the bound can be obtained by using approximate dynamic
programming with affine functions.
We now introduce additional notation to write the linear program with value function J̄(T,c) ≥
J(T,c). Let πl(Sl) be the nl = |Nl|-dimensional vector with components πlk(Sl),k ∈ Nl. Clearly
πl(Sl) has zeros for all k /∈Sl. The vector πl(Sl) gives us the probability of sale for each fare in Nl
per customer when we offer set Sl. Let rl(Sl) = p
′
lπl(Sl) =
∑
k∈Nl
plkπlk(Sl) be the revenue rate per
customer associated with offering set Sl ⊂Nl where pl = (plk)k∈Nl is the fare vector for segment
l∈L. Notice that πl(∅) = 0 ∈ with offering set Sl ⊂Nl where Al is a matrix with columns Alj associated with ODF j ∈Nl.
The choice based deterministic linear program (CBLP) for this model can be written as
J̄(T,c) = max l∈LλlT Sl⊂Nl subject to l∈LλlT Sl⊂Nl Sl⊂Nl αl(Sl) ≥ 0 Sl ⊂Nl l∈L.
(6)
The decision variables are αl(Sl),Sl ⊂ Nl,l ∈L which represent the proportion of time that set
Sl ⊂Nl is used. The case of non-homogeneous arrival rates can be handled by replacing λlT by∫ T ∑ nl decision variables. Gallego et al. (2004) showed that J(T,c) ≤ J̄(T,c)
and proposed an efficient column generation algorithm for a class of attraction models that
includes the BAM. This formulation has been used and analyzed by other researchers including
Bront et al. (2009), Kunnumkal and Topaloglu (2010), Kunnumkal and Topaloglu (2008), Liu and
van Ryzin (2008) and Zhang and Adelman (2009).
5. The Sales Based Linear Program for the GAM
In this section we will present a linear program that is equivalent to the CBLP for the GAM, but
is much smaller in size (essentially the same size as the well known IDM) and easier to solve (as Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization it avoids the need for column generation). The new formulation is called the sales based linear
program (SBLP) because the decision variables are sales quantities. The number of variables in
the formulation is l∈L(nl + 1), where xlk,k = 0,1.. .,nl,l∈L represents the sales of product k in
market segment l. Let the aggregate market demand for segment l be denoted by Λl = λlT,l∈L
(or 0 this notation the formulation of the SBLP is given by:
R(T,c)= max l∈L k∈Nl subject to capacity : l∈L k∈Nl balance : vl0
xl0 +
∑ ṽlk xlk = Λl l∈L
scale : xlk − xl0
vl0 non-negativity: xlk ≥ 0 k∈Nl l∈L
(7)
Theorem 3. If each market segment πlj(Sl) is of the GAM form (2) then J̄(T,c) = R(T,c).
We demonstrate the equivalence of the CBLP and SBLP formulations by showing that the SBLP
is a relaxation of the CBLP and then proving that the dual of the SBLP is a relaxation of the dual
of the CBLP. Full details of this proof may be found in the Appendix.
The intuition behind the SBLP formulation is as follows. Our decision variables represent the
amount of inventory to allocate for sales across ODFs for different market segments. The objective
is to maximize revenues. Three types of constraints are required. The capacity constraints limit
the sum of the allocations. The balance constraints ensure that the sale allocations are in certain
hyperplanes, so when some service classes are closed for sale, the demands for the remaining open
service classes (including the null alternative) changes dependently; they must be modified to offset
the amount of the closed demands. The scale constraints operate on the principle that demands
are rescaled depending on both the availability and the attractiveness of other items in the choice
set.
The decision variables xl0 play an important role in the SBLP. Since the null alternative has
positive attractiveness and is always available, the dependent demand for any service class is Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization upper bounded based on its size relative to the null alternative. The scale constraints imply that
xlk/vlk ≤ xl0/vl0. Therefore, the scale constraints favor a large value of xl0. On the other hand,
the balance constraint can be written as k∈N ṽlkxlk/vlk ≤ Λl − ṽl0xl0/vl0, favoring a small value
of xl0. The LP solves for this tradeoff taking into account the objective function and the capacity
constraint. We will have more to say about xl0 in the absence of capacity constraints in Section 6.
The combination of the balance and scale constraints ensures that GAM properties are compactly
represented in the optimizer and that the correct ratios are maintained between the sizes of each
of the open alternatives.
For the BAM the demand balance constraint simplifies to xl0 + k∈Nl pendent demand case, ṽl0 = vl0 + k∈Nl to xl0 = (vl0/ṽl0)Λl, and the scale constraint becomes xlk ≤ (vlk/ṽl0)Λl,k ∈ Nl, so clearly xl0 +∑ xlk ≤ Λl.
We remark that the CBLP and therefore the SBLP can be extended to multi-stage problems
since the inventory dynamics are linear. This allows for different choice models of the GAM family
to be used in different periods and is the typical approach in practice.
5.1. Recovering the Solution for the CBLP
So we have learned that the SBLP problem is much easier to solve and provides the same revenue
performance as the CBLP. However, there are some circumstances (described later in this section)
in which it is more desirable to use the CBLP solution. The reader may wonder whether it is
possible to recover the solution to the CBLP from the solution to the SBLP. In other words, is it
possible to obtain a collection of offer sets {S : α(S) > 0} that solves the CBLP from the solution
xlk,k ∈Nl,l ∈L of the SBLP? The answer is yes, and the method presented here for the GAM
is inspired by Topaloglu et al. (2011) who used our SBLP formulation and presented a method
to recover the solution to the CBLP for the BAM. When used with the GAM, recovering the
CBLP solution is important because solutions in terms of assortments tend to be more robust then
solutions in terms of sales. Indeed, our experience is that when GAM parameters are estimated, Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization the solution in terms of sales can be fairly far from the solution with known parameters, whereas
the solution in terms of assortments tends to be much closer.
In this section we will show how to obtain the solution of the CBLP from the SBLP and will also
show that for each market segment the offered sets are nested in the sense that there exist subsets
Slk increasing in k with positive weights. The products in the smallest set are core products that
are always offered. Larger sets contain additional products that are only offered part of the time.
The sorting of the products is not by revenues, but by the ratio of xlk/vlk for all k∈Nl ∪{0} for
each l∈L. We then form the sets Sl0 = ∅ and Slk = {1,. ..,k} for k = 1,.. .,nl, where nl = |Nl| is
the cardinality of Nl. For k = 0,1,.. .,nl, let
αl(Slk) =
( ) Λl where for convenience we define xl,nl+1/vl,nl+1 = 0. Set αl(Sl) = 0 for all Sl ⊂Nl not in the col-
lection Sl0,Sl1,.. .,Slnl . It is then easy to verify that αl(Sl) ≥ 0, that Sl⊂Nl Sl⊂Nl are satisfied, and the objective function coincides with that of the SBLP. Notice that for each
market segment the offered sets are nested Sl0 ⊂Sl1 ⊂ .. .⊂Slnl , and that set Slk is offered for a
positive fraction of time if and only if xlk/vlk > xl,k+1/vl,k+1. Unlike the BAM and the IDM, the
sorting order of products under the GAM is not necessarily revenue ordered. This is because fares
that face more competition (wlj close to vlj) are more likely to be offered even if their corresponding
fares are low.
5.2. Bounds and Numerical Examples
In this section we present two results that relate the solutions to the extreme cases w = 0 and w = v
to the case 0 ≤w≤v. The proofs of the propositions are in the Appendix.
The reader may wonder whether the solution that assumes wj = 0 for all j ∈N is feasible when
wj ∈ (0,vj) j ∈N . The following proposition confirms and generalizes this idea. Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization Proposition 2. Suppose that αl(Sl),Sl ⊂ Nl,l ∈L is a solution to the CBLP for the GAM for
fixed vlk,k ∈Nl+,l∈L and wlk ∈ [0,vlk],k ∈Nl,l∈L. Then αl(Sl),Sl ⊂Nl,l∈L is a feasible, but
suboptimal, solution for all GAM models with w′lk ∈ (wlk,vlk],k∈Nl,l∈L.
An easy consequence of this proposition is that if αl(Sl),l ∈L is an optimal solution for the
CBLP under the BAM, then it is also a feasible, but suboptimal, solution for all GAMs with
wk ∈ (0,vk]. This implies that the solution to the BAM is a lower bound on the revenue for the
GAM with wk ∈ (0,vk] for all k.
As mentioned in the introduction, the IDM underestimates demand recapture. As such optimal
solutions for the SBLP under the IDM are feasible but suboptimal solutions for the GAM. The
following proposition generalizes this notion.
Proposition 3. Suppose that xlk,k ∈Nl,l∈L is an optimal solution to the SBLP for the GAM
for fixed vlk,k∈Nl+ and wlk ∈ [0,vlk],k∈Nl,l∈L. Then xlk,k∈Nl+,l∈L is a feasible solution for
all GAM models with w′lk ∈ [0,wlk],k∈Nl,l∈L with the same objective function value.
An immediate corollary is that if xlk,k ∈Nl,l∈L is an optimal solution to the SBLP under the
independent demand model then it is also a feasible solution for the GAM with wk ∈ [0,vk] with
the same revenue as the IDM. This implies that the solution to the IDM is another lower bound
on the GAM with wk ∈ [0,vk] for all k.
Example 5: The following example problem was originally reported in Liu and van Ryzin (2008).
There are three flights AB,BC and AC. All customers depart from A with destinations to either
B or C. The relevant itineraries are therefore AB,ABC and AC. For each of these itineraries
there are two fares (high and low) and three market segments. Market segment 1 are customers
who travel to B. Market segment 2 are customers who travel to C but only at a high fare. Market
segment 3 are customers who travel to C but only at the low fare. The six fares as well as vlj,j ∈Nl
and vl0 for l = 1,2,3 are given in Table 6. The arrivals are uniformly distributed over a horizon of
T = 15, and the expected market sizes are Λ1 = 6,Λ2 = 9,Λ3 = 15 and that capacity c = (10,5,5)
for flights AB, BC and AC. Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization 1 2 3 4 5 6 0 Fares $1,200 $800 $600 $800 $500 $300 (AChigh,v2j) 10 5 5 Table 6 Market Segment and Attraction Values v
In this example we will consider the parsimonious p-GAM model wlj = θvlj for all j ∈ Nl,l ∈
{1,2,3} for values of θ ∈ [0,1]. We will assume that the vlj values are known for all j ∈ Nl,l ∈
{1,2,3}, but the BAM and the IDM use the extreme values θ = 0 and θ = 1, while the GAM uses
the correct value of θ. This situation may arise if the model is calibrated based on sales when all
products are offered, e.g., Sl = Nl for all l∈{1,2,3}. The objective here is to study the robustness
of using the BAM and the IDM to the p-GAM for values of θ ∈ (0,1). The situation where the
parameters are calibrated from data that includes recapture will be presented in the next example.
The solution to the CBLP for the BAM is given by α({1,2,3}) = 60%, α({1,2,3,5}) = 23.3% and
α({1,2,3,4,5}) = 16.7% with α(S) = 0 for all other subsets of products. Equivalently, the solution
spends 18, 7 and 5 units of time, respectively, offering sets {1,2,3}, {1,2,3,5} and {1,2,3,4,5}. The
solution results in $11,546.43 in revenues (see 3. in the endnotes). By Proposition 2, the solution
α({1,2,3}) = 60%, α({1,2,3,5}) = 23.3%, α({1,2,3,4,5}) = 16.7% is feasible for all θ∈ (0,1] with a
lower objective function value. The corresponding SBLP solution is shown in Table 7 and provides
the same revenue outcome as the CBLP.
1 2 3 4 5 6 0 (AChigh,x2j) 4.50 2.25 2.25 $7,200.00 Total 4.50 2.25 4.29 0.50 2.75 0.0 15.71 $11,546.43 The case θ = 1 corresponds to the IDM. The solution to the SBLP for the IDM and the cor-
responding revenue is given in Table 8. By Proposition 3, the solution to the IDM is a feasible
solution for all θ∈ [0,1) with the same objective value function. Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization 1 2 3 4 5 6 0 (AChigh,x2j) 4.50 2.25 2.25 $7,200.00 Total 4.50 2.25 2.00 0.50 2.75 3.00 9.05 $11,075.00 R en $10,100
$10,200
$10,300
$10,400
$10,500
$10,600
$10,700
$10,800
$10,900
$11,000
$11,100
$11,200
$11,300
$11,400
$11,500
$11,600
Theta
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Legend
gam bam idm
Figure 1 GAM, BAM and IDM Revenue for Known θ ∈ [0,1] Values
Figure 1 shows how the two extreme solutions (CBLP for θ = 0) and (SBLP for θ = 1) perform
for the attraction values given in Table 6 when the true model is GAM with θ∈ (0,1). Notice that
the BAM solution does well only for small values of θ, say θ≤ 20% and performs poorly for large
values of θ. Indeed, the BAM solution gives up about 10% of the optimal profits at the extreme
value of θ = 1. The case θ = 1 corresponds to the independent demand model, and in this case
the BAM anticipates demand recapture when there is none. A related finding was reported by
Hartmans (2006) of Scandinavian Airlines; his simulation studies showed clear positive revenue
potential of including recapture effects, but assuming strong recapture in cases where it does note
exist can lead to revenue losses. The solution for the IDM gives up significant profits only for small
values of θ. In particular, the IDM solution loses over 4% of the optimal profits for θ = 0 but less Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization than 0.25% of profits for θ≥ 40%.
The optimal value function for the GAM is only marginally better than the maximum of the
value functions for the BAM and the IDM. Notice that these results apply to the special case of the
p-GAM, where all products in the choice set have had their shadow attractions downgraded by the
same relative amount. In a more typical application, only products that face competition will have
positive shadow attractions, so we expect the revenue differences between the BAM and GAM not
to be as large as in Example 5. Nonetheless, the example does highlight the asymmetry of revenue
performance associated with overestimating versus underestimating recapture probabilities.
Example 6: One may wonder how the three models would compare, if the attraction values in
Example 5 were estimated from data. Estimating the parameters would give the BAM and the
IDM more flexibility to compensate for using the wrong value of θ. On the other hand, the burden
of estimating a p-GAM as either a BAM or an IDM, can put stress on the estimated parameter.
It is not clear, therefore, whether the gaps suggested by Figure 1 are indicative of the difference
in performance when the parameters need to be estimated. On the other hand, the GAM has one
more parameter to estimate. What is important, therefore is to ascertain whether the GAM can
result in significant improvements over the BAM and IDM, and whether the gaps are smaller or
larger than those in the figure.
We tested this for three values of θ: 0.1, 0.5 and 0.9. Based on Figure 1, at θ = 0.1, we expect the
BAM to do very well and the IDM to do poorly. At θ = 0.5, we expect the BAM to struggle and
the IDM to do well. Finally, at θ = 0.9 we expect the BAM to do poorly and the IDM to very well.
To evaluate the performance of the three models, we generated data, estimated the parameters,
solved the SBLP with the estimated parameters for each of the three models, and then evaluated,
via simulation, the performance of the heuristic that uses the offer sets recommended by the SBLP.
We then compute the relative performance of the BAM and the IDM to the GAM. We report the
average of the relative performance over 500 repetitions of the entire procedure.
We simulate 50 instances of demands for each of the three non-trivial subsets for each market
to generate the data to estimate the parameters. For example, market 1 consists of product 3 and Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization 4, with non-trivial subsets {3},{4} and {3,4}. To estimate the parameters we used the procedure
described in Section 2.4. To solve the linear program, we used the SBLP and converted the solution
to the CBLP as indicated in Section 5.1. To evaluate the performance of the heuristic we simulated
1000 repetitions using the offer sets recommended by the heuristic. The implementation of the
heuristic always starts by offering the largest set within a market. All the simulated requests are
accepted on a first-come first-serve basis within capacity limits.
For θ = 0.1, all of the three models do a reasonably good job of estimating the attraction values.
All of the three models, agreed on offering product 3 all of the time. The GAM and the BAM
agreed on offering products 1 and 2 all of the time. The IDM model, however, would some times
offer product 2 by itself for short periods of time. For market 3, the BAM and the GAM, were
remarkably consistent on offering product 5 by itself about 28% of the time and products 4 and
5 about 14% of the time. We were surprised by this, given the often significant differences in
the estimated parameters. The IDM disagreed offered product 5 by itself about 28% of the time,
but rarely offered product 4 at all. The performance of the BAM was essentially identical to the
performance of the GAM, achieving close to 100% of the revenues obtained by the GAM heuristic.
The IDM model, on the other hand, yielded about 98.2% of the revenue of the GAM heuristic.
For θ = 0.5, the estimates of the parameters for both the GAM and the IDM are fairly close to
the actual values. This is not so for the BAM, that makes a huge effort to accommodate for the
partial recapture resulting from θ = 0.5. Table 9 illustrates the true parameters and the estimated
values for the BAM, GAM, and IDM for one instance of the problem. Remarkably, even though
the estimates of the parameters for the BAM are often way off, the recommendations made by the
BAM were not far from those made by the GAM when viewed as solutions of the CBLP. In fact,
for the estimated values in Table 9, the recommendations were nearly identical: Offer products 1,2
and 3 all the time. Offer product 4 and 5 together 16% of the time, and offer product 5 by itself
27% of the time. The IDM solution for the reported instance in Table 9 was consistent with the
BAM and GAM for markets segments 1 and 2, but offered products 4 and 5 together 4% of the
time, and product 5 by itself 34% of the time. Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization Averaging over the 500 instances, the IDM was able to capture 99.45% of the revenues of the
GAM, while the BAM was able to capture 99.58%. This is contrast to Figure 1. These experiments
suggest that estimation helps the BAM relative to having the correct values of the attraction values
and the wrong value of θ, but it does not help the IDM as much.
v13 v16 v10 Λ1 θ v21 v22 v20 Λ2 θ v34 v35 v30 Λ3 θ Table 9 Estimates of Parameters when θ = 0.50
For θ = 0.9, the estimates of the parameters are way off for the BAM, while the estimates for the
GAM and the IDM are quite close to the actual values. Nevertheless, the three models continue
to agree in terms of offering products 1, 2 and 3 all the time for most instances of the problem.
For a particular instance, generated with the same seed as Table 9, the BAM, offers products 4
and 5 together 16% of the time and offers product 5 by itself 10% of the time. The IDM, offers
products 4 and 5 together 14% of the time and product 5 by itself 30% of the time. The GAM
offers products 4 and 5 together 16% of the time and product 5 by itself 27% of the time. Notice
that the GAM and the IDM solutions are very similar. But the BAM suffers because very little
demand is recaptured from product 6 and from product 4 when only product 5 is offered. Averaging
over the 500 instances, the IDM was able to capture 99.9% of the revenue relative to the GAM,
while the BAM was able to capture 98.9% of the revenue relative to the GAM. In summary, with
estimated parameters, the IDM and the BAM do better relative to the GAM than what would
be predicted by Figure 1 when the parameters are known. Our biggest surprise was the relative
modest deterioration of the BAM revenue given how far the estimates of the parameters are, in
particular when θ = 0.9. To illustrate this, the estimates for the attraction values v21,v22,v20 and
Λ2 are 0.81, 0.38, 18.81, and 112.14. In contrast the true values, are 10, 5, 5, and 9.
Example 7: We continue using the running example in Liu and van Ryzin but we now assume
that demand is governed by the specific GAM presented in Table 10. The model reflects positive Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization w values for lower fares. The optimal revenue under this model is $11,225.00, and the optimal
allocations are given in Table 11. We know from Proposition 2 that the CBLP solution for the
BAM is feasible for this GAM. The resulting revenue is $11,185.86. The solution to the IDM for
the SBLP is also feasible for the GAM and results in revenue $11,075.00. It is clear that in this
case the BAM performs better than the IDM solution. The solution for the CBLP is as follows:
For market 1, fare ABH is offered 100% of the time and nothing else. For market 2, fares ACH
and ABCH are both offered 100% of the time. Finally, for market 3, fare ABCL is offered 25%
of the time by itself, while fares ABCL and ACL are offered together 17% of the time. When the
parameters of the problem need to be estimated, the BAM obtained 99.8% of the revenues, and
the IDM obtained 99.0% of the revenues relative to the GAM.
1 2 3 4 5 6 0 Fares $1,200 $800 $600 $800 $500 $300 (AChigh,v2j,w2j) (10, 2) (5, 1) 5 Table 10 Market Segment and Attraction Values (v,w)
1 2 3 4 5 6 0 (AChigh,x2j) 4.50 2.25 2.25 $7,200.00 Total 4.50 2.25 3.75 0.50 2.75 0.00 14.52 $11,225.00 5.3. Overlapping Market Segments
Another point worth noting is that the CBLP and SBLP formulations remain equivalent even if
the market segments have products that overlap. The caveat is that both formulations assume in
this case that each market segment will be managed independently. This works only if the capacity
provider has the ability to offer a product for a market segment without offering it to other market Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization segments. On the other hand, if a fare cannot be offered in one market without offering to other
markets then both the CBLP and the SBLP formulations are upper bounds that can be tightened
by adding additional constraints. This approach was used by Bront et al. (2009) for the CBLP.
They use column generation assuming that each market segment follows a BAM and show that the
column generation step is NP-hard. The SBLP formulation allows for further manipulation as it is
easier to add cuts that can streamline some of the computation for overlapping market segments.
In particular, Talluri (2012) has reported encouraging results using the SBLP in the context of the
BAM with overlapping market segments when the number of different products for each market
segment is relatively small.
5.4. Heuristics for the Stochastic Problem
The compact form of the SBLP formulation and the ease with which we can recover the solution to
the CBLP from the SBLP allows for the practical developments of heuristics that call for frequent
re-solving of the SBLP. For example, a bid-price heuristic can be developed by taking into account
both the dual vector z corresponding to the capacity constraint as well as the dual variable βl
corresponding to the balance constraint for market segment l ∈L. Let plk −z′Alk −βlṽlk/vlk be
the reduced fare for k ∈Nl. Products with positive reduced fares in the SBLP have xlk at their
upper bounds of the scale constraint, which implies that they are offered 100% of the time in the
CBLP. Products with xlk/vlk ∈ (0,xl0/vl0) have reduced fares equal to zero. These corresponds to
fares that are available only part of the time in the solution to the CBLP. Finally, products with
negative reduced fares in the SBLP have xlk = 0 and are not offered at all by the CBLP. As in
any bid-price heuristic, the caveat is that it may accept too many requests for products with zero
reduced fares. The remedy for this, as is the case for independent demand model, is to re-solve the
SBLP frequently as this gives the model an opportunity to close down such fares.
An alternative heuristic is to use the solution to the SBLP, map it into the solution of the CBLP
and use the solution to the CBLP as a guide to admission control heuristics. To see how this Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization can be implemented, suppose that for a certain market the solution to the CBLP, after relabel-
ing the products if necessary, is to offers sets {1,2,3},{1,2,3,4} and {1,2,3,4,5} respectively for
α({1,2,3}) = 0.70,α({1,2,3,4}) = 0.20 and α({1,2,3,4,5}) = .10 fraction of the horizon [0,T ]. It is
typically advisable to offer the largest sets first, as they include the fares with zero reduced costs.
The advantage of this heuristic is that it does not over accept fares with zero reduced costs. The
heuristic can also benefit from re-solving. For example, the largest sets in each market segment with
αl(Slk) > 0,l∈L can be offered for minl∈Lαl(Slk)T units of time and at that point the SBLP can
be re-solved to obtain an updated solution to the SBLP from which a new solution to the CBLP
can be produced. The procedure can be repeated until the end of the horizon. Our experience is
that if the SBLP is solved only once, then this heuristic can perform better than the bid-price
heuristic. However, with frequent resolving, the bid-price heuristic tends to do better.
The compact form of the SBLP allows for solving the problem for random realizations of the
aggregate demand as suggested in Talluri and van Ryzin (1999). Our experience is that this refine-
ment does better than the bid-price heuristic if the SBLP is solved infrequently. However, when
the SBLP is re-solved frequently during the horizon the advantage tends to disappear and even
become negative.
While the above heuristics do not anticipate randomness, they do respond to randomness when
the SBLP is re-solved frequently. A number of authors have tried to incorporate randomness more
directly at either the leg level or the itinerary level. Liu and van Ryzin (2008) project demands into
legs and apply dynamic programming at the leg level to come up with improved heuristics. Variants
of the expected marginal seat revenue (EMSR) heuristic that attempt to deal with dependent
demands are discussed in Weatherford and Ratliff (2010) (e.g. DAVN-MR and choice-based EMSR).
The EMSR-b MR method, proposed in Fiig et al. (2010) , see also Walczack, Mardan, and Kallesen
(2010), uses a combination of fare adjustments and a decomposition of dependent demands into
independent components that allows for the use of techniques developed originally for the problem
with independent demands. While computationally appealing, these methods makes very strong
assumptions about the order of arrivals that are unlikely to hold in practice because the independent Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization demands obtained by the transformation cannot be directly observed and therefore cannot be
adequately controlled by the output of the heuristic. The reason for this is that the independent
demands obtained by the transformation consist of customers willing to buy the marginal fare and
nothing else. In other words, customers willing to pay a higher fare need to wait until their fare is
available and are not allowed to take advantage of the lower fares. In essence, the method works only
if yieldable demand arrives before priceable demand for each fare class. In sharp contrast the choice-
based EMSR heuristic of Gallego, Ratliff and Li (2009), while a bit more onerous computationally,
does almost as well as the dynamic program at the leg level because it takes into account demand
dependency. We refer the reader to Gallego and Li (2012) for an explanation of the potential issues
arising from using fare transformation with the independent demand decomposition suggested by
Fiig et al. (2010) and Walczack, Mardan, and Kallesen (2010).
6. The Assortment Problem
As a corollary to Theorem 3, the infinite capacity formulations for the SBLP and CBLP are
equivalent. With infinite capacity the problem separates by market segment so it is enough to study
a single market segment. To facilitate this we will drop the subscript l both in the formulation
and in the analysis. The resulting problem can be interpreted as that of finding the assortment of
products S ⊂N and the corresponding sales xk,k∈N to maximize profits from a market segment
of size D that makes decisions based on a GAM choice model. Under this interpretation the pk
values are gross unit profits, net of unit costs.
R(T) = max k∈N pkxk
subject to balance : ṽ0 ∑ ṽk scale : v0 non-negativity: xk ≥ 0 k∈N+
(8)
The problem is similar to a bin-packing problem with constraints linking the xk’s to x0. An
analysis of the problem reveals that the allocations should be in descending order of the ratio of Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization pk to ṽk/vk. This ratio can be expressed as pkvk/ṽk. For the BAM the allocation is in decreasing
order of price. Including lower priced products improves sales, but it cannibalizes demands and
dilutes revenue from higher priced products. As such, the allocation of demand to products needs
to balance the potential for more sales against demand cannibalization. The sorting of products
for the GAM depends not only on price but on the ratio vk/ṽk, so a low priced product with a
high competitive factor θk = wk/vk may have higher ranking than a higher priced product with
a low competitive factor. To explain how to obtain an efficient allocation of demand over the
different products we will assume that the products are sorted in decreasing order of pkvk/ṽk.
Let Xj = {1,.. .,j} and consider the allocation xk = vkx0/v0 for all k∈Xj and xk = 0 for k /∈Xj.
Solving the demand constraint for x0 we obtain
x0 = ṽ0 + Ṽ (Xj) and revenue
Rj = ∑ pkπk(Xj)D. (9)
Now consider the problem of finding the revenue Rj+1 for set Xj+1 = Xj ∪{j + 1}. It turns out
that Rj+1 can be written as a convex combination of Rj and pj+1Dvj+1/ṽj+1. This implies that
Rj+1 >Rj if and only if Rj index
j = max{j ≥ 1 : pj ≥ ∑ pkπk(Xj)(1−θj)}. (10)
The following proposition whose proof can be found in the Appendix establishes the optimality
(see 4. in the endnotes) of Xj.
Theorem 4. The assortment Xj with j given by equation (10) is optimal.
Notice that j in equation (10) is independent of D. This means that products 1,.. .,j are offered
for all demand levels D > 0 and products j + 1,.. .,n are closed regardless of the value of D.
Products j + 1,.. .,n are inefficient in the sense they do not increase sales enough to compensate
for the demand they cannibalize. Table 4 illustrates precisely this fact as product 5 priced at $105 Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization is inefficient because p5/(1−θ5) at $100 fare with a competitive factor θ1 = 0.9 than the higher product 5 at $105 with competitive
factor θ5 = 0. The intuition behind this is that closing product 5 is more likely to result in recapture
than closing fare 1 (whose demand will most likely be lost to competition or to the no purchase
alternative).
k pk vk θk xk pkxk Total $107.79 If the problem consists of multiple market segments and each market segment is allowed to post
its own available product set Xlj then the problem reduces to sorting the products for market
segment l in decreasing order of plk/(1−θlk), where θlk = wlk/vlk, and then computing j(l) using
(10) for each l. The resulting sets Xlj(l),l ∈ L are optimal. This has implications to Revenue
Management applications where capacity is finite. When capacity is finite, the fares are reduced
by the marginal value of capacity, and fares that are sub-optimal when capacity is infinite remain
so when capacity is finite. This fact has been used remove inefficient fares from the SBLP by Sabre
resulting in a significant savings in computations.
If a single assortment set S is to be offered to all market segments then the problem is
NP-hard, see Rusmevichientong et al. (2010), even if there are only two market segments and
the choice model is the BAM. However, the problem for the BAM is somewhat easier to analyze
since the preference ordering for all the market segments is p1 > p2 > … > pn whereas the
preference ordering for the GAM may be different for different market segments. Assuming the
order p1 > p2 > … > pn, Rusmevichientong et al. (2010) provide a two market segment example
where it is optimal to offer X1 = {1} to market segment one, to offer X2 = {1,2} to market Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization segment two and to offer X = {1,3} to a composite of 50% of each of the two market segments.
Given the NP-hardness of the assortment problem with overlapping sets, the research activity
has concentrated on bounds, heuristics and performance warranties. We refer the reader to the
working papers by Rusmevichientong et al. (2009) and Rusmevichientong and Topaloglu (2009).
If prices are also decision variables, with choices reflecting price sensitivities, then the problem
involves finding the offer sets and prices for each market segment. We refer the reader to Schön
(2010) who allows for price discrimination, e.g., to offer different prices and offer sets to different
market segments. The problem of offering a single offer set and a single vector of prices for all the
market segments is again NP-hard.
7. Conclusions and Future Research
The general attraction model formulation is a new variation of the basic attraction model. We have
presented an E-M algorithm to estimate the parameters that is an extension and refinement of
Vulcano et al. (2012). Work is currently underway to develop techniques to estimate the parameters
in situations where the arrival rates λlt are driven by a set of covariates such as day-of-the-week,
seasonality indices, and other factors that may affect demand. One known problem in applying the
BAM to typical airline problems is that the IIA property can be violated in cases where there are
large differences in the departure times of alternate flight services; in such cases, the recapture rates
between these widely spread departures would be overestimated. This problem can be mitigated
by segmenting customers by their preferred departure time.
One limitation of our model is that it provides only an upper bound when there are overlapping
market segments, and only a single offer set can be posted at any given time. However, the SBLP
formulation is more amenable to adding cuts and to finding polynomial time approximation algo-
rithms to handle commonality constraints. We are currently working on these extensions for both
the assortment problem and the case of finite capacities. It is also possible to extend the results in
this paper to other objective functions given the equivalence of the solutions space for the CBLP Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization and the SBLP (see 5. in the endnotes). The SBLP provides the basis for many heuristics for the
stochastic network revenue management problem. Since the size of the problem is essentially the
same as that of the independent demand model, applying the randomized linear program (RLP)
of Talluri and van Ryzin (1999) becomes viable for the GAM model. Other heuristics such as those
proposed by Gallego, Ratliff and Li (2009), Jasin and Kumar (2010), Kunnumkal and Topaloglu
(2008), and Liu and van Ryzin (2008) are aided by the ease of solving the SBLP and enriched by
the GAM. We also showed that the SBLP and GAM can be successfully applied to the assortment problem
that arises when the capacities are set at infinity; the assortment problem is important in its own
right both in the context of perishable and non-perishable products. The new insight is that,
under the GAM, it is not necessarily true that the optimal assortment consists of the highest
priced products.
Endnotes
1. The authors are grateful to Anran Li for coding the EM procedure and conducting the experi-
ments. 2. The airfare value used in the RM optimization may be adjusted to account for other associated
revenues (e.g. baggage fees) or costs (e.g. meals or in-flight wireless access charges).
3. This result is a slight improvement on the one reported in Liu and van Ryzin (2008) for which
we calculate a profit of $11,546.20 based on an allocation of 16.35, 2.48, 10.3 and 0.87 units of
time, respectively, on the sets {1,2,3}, {1,2,3,4}, {1,2,3,5}, {1,2,3,4,5}.
4. H. Topaloglu at Cornell independently found the same sorting for the GAM. Personal commu-
nication.
5. H. Topalogulu has used this idea to model the multiclass overbooking problem under discrete
choice models in a presentation at the 2011 Informs Revenue Management Conference at Columbia
University. Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization Appendix A: Proof of Theorem 1
If the choice model πj(S),j ∈S satisfies GAM then there exist constants vj,j ∈N+, ṽj ≥ 0,j ∈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}) = ṽ0+Ṽ (T)−ṽi V (S) ṽ0+Ṽ (T) Conversely, suppose a choice model satisfies the GLA. Then by selecting R = {i}⊂S ⊂T = N
we see from Axiom 1’ that
πi(S) = 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) +∑ πi(S) = 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) = ṽ0 + Ṽ (S) so it satisfies the GAM.
Appendix B: 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 Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization 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(∑ v )γk . (11) j ∈Oi is given by 1/γi l∈Oi 1/γi . (12)
Consequently, the probability that a customer selects product i∈Sj from vendor j is given by(∑ v 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 1/γk γk = maxl∈Oi vil. The limit, say pij, of equation
(12) 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) = v0 + k . 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) = v0 + k∈Sj ∑ V −kj where S̄j = {i∈N : i /∈Sj}. The selection probability can be written as
πi(Sj) = k∈N V ∑ (Vkj −V −kj ) or equivalently, as
πi(Sj) = (Vkj −V −kj )pkj since Vkj −V −kj = (Vkj −V − that πi(Sj) corresponds to a GAM with ã0j = v0 + k∈N V − Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization Appendix C: Proof of Theorem 3
We will first establish that J̄(T,c) ≤R(T,c) by showing that any solution to the CBLP can
be converted into a feasible solution to the SBLP for the GAM. Assume a solution αl(Sl),l ∈
L for the CBLP is given and for k ∈ Nl let xlk = Λl Sl⊂Nl tion and the capacity constraint in terms of xlk are given respectively by l∈L k=1 l∈L k=1 x̃l0 = Λl αl(Sl)πl0(Sl).
Notice that x̃l0 = xl0 + k∈Nl xl0 = Λl αl(Sl) v0 + Wl(S̄l) + Vl(Sl)
and ylk = Λl Sl⊂Nl:k/∈Sl
αl(Sl) wlk
v0 + Wl(S̄l) + Vl(Sl) We now argue that is bounded above by xl0 . This follows because
xlk Sl⊂Nl:k∈Sl
αl(Sl) v0 + Wl(S̄l) + Vl(Sl) while
xl0 and the second equation sums over at least as many subsets.
Notice that if wlk > 0 then
xlk + = or equivalently
ylk = wlk
[ ] since vl0 k∈Nl. Substituting ylk into
xl0 + ylk + xlk = Λl Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization and collecting terms we obtain( ∑ ) ∑ ( ) which is equivalent to the demand balance constraint since ṽl0 = vl0 + k∈Nl for all k∈Nl.
We will now show that R(T,c) ≤ J̄(T,c) by showing that any solution to the dual of the CBLP
can be converted to a feasible solution to the dual of the SBLP. Let z and β̃l,l∈L be an optimal
solution for the dual of the CBLP, where z is the dual vector corresponding to the capacity
constraint, and for each l∈L, β̃l is the dual variable corresponding to the constraint limiting the
proportion of time that sets Sl ⊂Nl are offered.
Let βl = β̃l/Λl. Then the feasibility constraint for the dual of the CBLP takes the form
βl + (z′Alj −plj)πlj(Sl) ≥ 0 Sl ⊂Nl, l∈L. (14)
For each market segment we define the sets:
Fl = {k∈Nl : plk −z′Alk − βl ≥ 0}, l∈L
and the quantities:
γlk = (plk −z′Alk)vlk −βlṽlk ≥ 0 k∈Fl.
For k /∈Fl we set γlk = 0.
We will now show that z,βl,l∈L and γlk k ∈Nl,l∈L is a feasible solution to the dual of the
SBLP:
R(T,c) = min l∈LβlΛl + z subject to βl + z′Alk + ≥ plk k∈Nl, l∈L
βl j∈Nl ≥ 0 l∈L
z ≥ 0, γlk ≥ 0 k∈Nl, l∈L
(15)
By construction
βl + z′Alk + Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization for all k∈Nl,l∈L.
Our next task is to show that
βl γlj Since vl0 > 0 and γlj = 0 for j /∈Fl it is enough to show that
βlṽl0 − γlj ≥ 0
Substituting the definition of γlj and rearranging terms, the inequality is equivalent to
βl[ṽl0 + Ṽl(Fl)] + (z′Alj −plj)vlj ≥ 0.
Dividing by ṽl0 + Ṽl(Fl) the inequality is equivalent to
βl + (z′Alj −plj)πlj(Fl) ≥ 0.
However, this inequality holds because of the feasibility of the dual of the CBLP (14) applied to
Fl. Since any optimal solution to the dual of the CBLP has a corresponding solution to the dual
of the SBLP with the same objective function we have R(T,c) ≤ J̄(T,c) completing the proof.
Appendix D: Proof of Proposition 2
To show feasibility it is enough to show that the capacity constraint is satisfied and to do this
it is enough to show that the vectors πl(Sl) are decreasing in wk. To facilitate the notation we will
drop the market segment index l. Then
πj(S) = v0 + W(S̄) + V (S) which is decreasing in wk. As a result, if we compare the demands under model wk,k ∈ N and
model w′k ∈ (wk,vk],k∈N, we see that the demands under the latter model are bounded above by
the demands under the former model. Consequently, strictly less capacity is consumed under the
latter model. Moreover, revenues suffer from lower demands so the original solution is suboptimal. Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization Appendix E: Proof of Proposition 3
Suppose that xlk,k ∈Nl,l ∈L is an optimal solution to the SBLP under the GAM for fixed
vk,k∈N+ and wk ∈ [0,vk],k∈N and consider a GAM model with the same vks and w′k ∈ [0,wk],k∈
N. To see that the xlk remains feasible it is enough to verify the demand balance constraint for
each market segment. To facilitate the exposition we will drop the market segment subscript l. The
demand balance constraint can be written as
x0 + xk + wk( ) = D.
Now consider a GAM with w′k ∈ [0,wk]. Then the demand scaling constraints guarantee that
x0 + w′k( ) ≤D.
This shows that xk,k ∈ N together with some x′0 ≥ x0 form a feasible solution to the demand
balance constraint without violating the demand scaling constraints.
Appendix F: Proof of Theorem 4:
We first verify that x0 and xk,k ∈Xj, with j given by equation (10) is feasible for the SBLP
assortment problem. Suppose that xk = vkx0/v0 for all k∈Xj, Then the demand balance equation
is ṽk Solving for x0 we obtain x0 = v0D/(ṽ0 + Ṽ (Xj)). But now = x0 = 1 D for all k ∈ Xj
together with the non-negativity of x0 and xk,k ∈Xj shows that the solution is indeed feasible.
This feasible solution has objective value Rj given by equation (9).
To prove that Rj is the maximum revenue for the assortment problem we will show that there
is a feasible solution to the dual with the same objective value. Consider the dual with objective
function Dβ and constraint set ṽ0β − k β = Rj/D together with γk = pkvk − ṽkβ for k∈Xj and γk = 0 for k /∈Xj is a feasible solution to Gallego, Ratliff, and Shebalov: A General Choice Model and Network RM Optimization the dual. The non-negative constraint γk ≥ 0 is satisfied by construction for k > j. For k∈Xj the
constraint is equivalent to pkvk ≥ ṽlβ, which in turn is equivalent to pkDvk/ṽk ≥Rj for all k∈Xj.
Since the items are sorted in decreasing order of pkvk/ṽk it is enough to show that pjDvj/ṽj ≥Rj.
However, we know that Rj is a convex combination of Rj−1 and pjDvj/ṽk. Moreover, since
Rj ≥Rj−1 it follows that Rj ≤pjDvj/ṽj. It remains to show that ṽ0β− k by showing that β = Rj/D is a solution to equation ṽ0β = k∈Xj k∈Xj. This is equivalent to [ṽ0 +Ṽ (Xk)]β = k∈Xj ∑ pkπk(Xj) = Rj/D
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∑
∑
rl(Sl)αl(Sl)
∑
∑
Alπl(Sl)αl(Sl) ≤c∑
αl(Sl) = 1 l∈L
0
λlsds. There are
l∈L 2
24 Article submitted to Operations Research; manuscript no. XX
∑
∫ T
λlsds if the arrival rates are time varying) inclusive of the no-purchase alternative. With
∑
∑
plkxlk
∑
∑
Alkxlk ≤ c
ṽl0
k∈Nl
vlk
vlk
≤ 0 k∈Nl l∈L
Article submitted to Operations Research; manuscript no. XX 25
∑
∑
xlk = Λl. For the inde-
∑
vlk and ṽlk = 0 for k∈Nl. The balance constraint reduces
k∈Nl
26 Article submitted to Operations Research; manuscript no. XX
xlk
vlk
−
xl,k+1
vl,k+1
Ṽ (Slk)
,
∑
αl(Sl) = 1, and that∑
αl(Sl)πlk(Sl) = xlk. Applying this to the CBLP we can immediately see that all constraints
Article submitted to Operations Research; manuscript no. XX 27
28 Article submitted to Operations Research; manuscript no. XX
Products ACH ABCH ABH ACL ABCL ABL Null
(AB,v1j) 5 8 2
(AClow,v3j) 5 10 10
Products ACH ABCH ABH ACL ABCL ABL xl0 Profit
(AB,x1j) 4.29 1.71 $2,571.43
(AClow,x3j) 0.50 2.75 11.75 $1,775.00
Table 7 SBLP Allocation Solution for BAM with (θ = 0)
Article submitted to Operations Research; manuscript no. XX 29
Products ACH ABCH ABH ACL ABCL ABL xl0 Profit
(AB,x1j) 2.00 3.00 0.80 $2,100.00
(AClow,x3j) 0.50 2.75 6.00 $1,775.00
Table 8 SBLP Allocation Solution for IDM with (θ = 1)
ev
ue
30 Article submitted to Operations Research; manuscript no. XX
Article submitted to Operations Research; manuscript no. XX 31
32 Article submitted to Operations Research; manuscript no. XX
True 5.00 8.00 2.00 6.00 0.50 10.00 5.00 5.00 9.00 0.50 5.00 10.00 10.00 15.00 0.50
BAM 3.79 8.38 1.87 5.00 0.00 5.19 2.35 12.46 17.54 0.00 3.05 6.77 15.18 23.17 0.00
GAM 4.96 8.08 1.95 5.53 0.39 10.36 4.70 4.94 8.78 0.50 4.74 10.51 10.17 15.17 0.35
IDM 5.34 7.85 1.81 6.40 1.00 10.06 5.04 4.89 9.64 1.00 5.00 9.99 10.00 16.78 1.00
Article submitted to Operations Research; manuscript no. XX 33
Products ACH ABCH ABH ACL ABCL ABL Null
(AB,v1j,w1j) (5, 0) (8, 1) 2
(AClow,v3j,w3j) (5, 1) (10, 0) 10
Products ACH ABCH ABH ACL ABCL ABL xl0 Profit
(AB,x1j) 3.75 0.00 1.50 $2,250.00
(AClow,x3j) 0.50 2.75 10.77 $1,775.00
Table 11 SBLP Allocation Solution with GAM (w > 0)
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36 Article submitted to Operations Research; manuscript no. XX
∑
v0
x0 +
k∈N
vk
xk = D
xk
vk
− x0
≤ 0 k∈N
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v0
D, xk = πk(Xj)D k∈Xj,
k∈Sj
k∈Xj
38 Article submitted to Operations Research; manuscript no. XX
0 1 0.0233
1 $100 15 0.9 0.3488 $34.88
2 $110 6 0.8 0.1395 $15.35
3 $115 9 0.7 0.2093 $24.07
4 $120 12 0.6 0.2791 $33.49
5 $105 9 0 0.0000 $0.00
Table 12 Example Assortment Problem
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∑
∑
V (S)−vi
=
= πS(T), so Axiom 2 holds.
πi(N)
∑
.
∑
∑
j∈S(1−θj)πj(N), resulting in
πi(N)
π0(N) +
∑
∑
.
∑
vi
,
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l∈Oi
1/γi
il
)γi
v0 +
∑
k
(∑
l∈Ok
v
1/γk
kl
The conditional probability that a customer who selects nest i will buy the product from vendor
v
ij∑
v
il
l∈Oi
1/γi
il
)γi
v0 +
∑
k
(∑
l∈Ok
v
1/γk
kl
)γk v1/γiij∑
l∈Oi
v
1/γi
il
.
∑
v
il )
pij maxl∈Oi vil
∑
maxl∈Ok vkl
Vijpij
∑
Vkj +
k∈S̄j
,
Vijpij
v0 +
∑
−
kj +
k∈Sj
,
Vijpij
v0 +
∑
k∈N V
−
kj +
∑
k∈Sj
,
−
kj )pkj for all k. This is because pkj < 1 implies Vkj −V
kj = 0. This shows
∑
−
kj , akj = pkjVkj, and ãkj = (Vkj−V
kj )pkj.
Article submitted to Operations Research; manuscript no. XX 43
∑
αl(Sl)πlk(Sl). The objective func-
∑
∑nk
plkxlk and∑
∑nk
Alkxlk ≤c. Let
∑
Sl⊂Nl
∑
ylk where
∑
Sl⊂Nl
vl0
∑
k∈Nl.
xlk
vlk
vl0
vlk
= Λl
∑
1
vl0
= Λl
∑
Sl⊂Nl
αl(Sl)
1
v0 + Wl(S̄l) + Vl(Sl)
vlk
ylk
wlk
xl0
vl0
,
xl0
vl0
−
xlk
vlk
≥ 0 (13)
xlk
vlk
≤ xl0
. On the other hand, by definition, ylk = 0 if wlk = 0. Thus formula (13) holds for all
∑
k∈Nl
∑
k∈Nl
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vl0 +
k∈Nl
wlk
vl0
xl0 +
k∈Nl
vlk −wlk
vlk
xlk = Λl ∀ l∈L,
∑
wlk and ṽlk = vlk−wlk
∑
j∈Sl
ṽlk
vlk
∑
′c
ṽlk
vlk
γlk
vlk
ṽl0
vl0
−
∑
γlj
vl0
ṽlk
vlk
γlk
vlk
≥plk
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ṽl0
vl0
−
∑
j∈Nl
vl0
≥ 0
∑
j∈Fl
∑
j∈Fl
∑
j∈Fl
vi
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n∑
k=1
n∑
k=1
x0
v0
−
xk
vk
n∑
k=1
xk +
n∑
k=1
x0
v0
−
xk
vk
ṽ0
v0
x0 +
∑
k∈Xj
v0
x0 = D.
xk
vk
v0
ṽ0+Ṽ (Xj )
∑
γk ≥ 0 and ṽkβ + γk ≥ pkvk,k ∈ N. We claim that
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∑
γk ≥ 0. We will do this
∑
γk where γk = pkvk − ṽkβ for
∑
pkvk. We see that β =
k∈Xj
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