Randomization Approaches for Network Revenue Management with
Customer Choice Behavior
Sumit Kunnumkal
Indian School of Business, Gachibowli, Hyderabad, 500032, India
sumit [email protected]
March 9, 2011
Abstract
In this paper, we present new approximation methods for the network revenue management
problem with customer choice behavior. Our methods are sampling-based and so we require only
minimal assumptions regarding the underlying customer choice model. The starting point for our
methods is a dynamic program that allows randomization. An attractive feature of this dynamic
program is that the size of its action space is linear in the number of itineraries, as opposed to
exponential. It turns out that this dynamic program has a structure that is similar to the dynamic
program for the network revenue management problem under the so called independent demand
setting. Our approximation methods exploit this similarity and build on ideas developed for the
independent demand setting. We present two approximation methods. The first one is based on
relaxing the flight leg capacity constraints using Lagrange multipliers, whereas the second method
involves solving a perfect hindsight relaxation problem. We show that both methods yield upper
bounds on the optimal expected total revenue. Computational experiments indicate that our methods
can generate tighter upper bounds and higher expected revenues when compared with the standard
deterministic linear program that appears in the literature.
Network revenue management with customer choice behavior is well-studied and has many applications in the airline, hotel and car rental industries. In the context of airlines, a representative example,
it involves controlling the sale of itineraries over a flight network. Customers arrive over the booking
period to purchase itineraries. The airline has to decide which itineraries to make available for sale
at each point in time taking into account the remaining capacities on the flight legs. This is a crucial
decision to make since the customer’s purchasing decision is influenced by the set of itineraries that are
offered. Depending on the offer set, the customer may purchase one of the offered itineraries, or may
not purchase anything and simply leave. The airline’s goal is to determine the set of itineraries to offer
at each point in time that maximizes the expected total revenues over the booking period. The airline’s
decision problem can be formulated as a dynamic program. However, computing the value functions
and the optimal policy quickly become intractable and one has to resort to approximation methods.
Many of the approximation methods for the network revenue management problem with customer
choice build on methods developed for network revenue management under the assumption that the
customer’s purchasing decision is not influenced by the set of offered itineraries. This is the so called
independent demand setting, where we assume that customers arrive with the intention of purchasing
a fixed itinerary. If the itinerary is available, they make the purchase. Otherwise, they leave without
making any purchase. Even with the independent demand assumption, the network revenue management problem becomes intractable as the size of the state space increases exponentially with the number
of flight legs. Consequently, the approximation methods for the network revenue management problem
with independent demand have mainly been concerned with reducing the dimensionality of the state
space. Incorporating customer choice behavior adds another layer of complexity since the size of the
action space also increases exponentially with the number of itineraries. This is because of the combinatorial nature of the problem of deciding which subset of itineraries to offer for sale from the set of all
possible itineraries. So, while many of the approximation methods for the network revenue management
problem with customer choice are able to handle the dimensionality of the state space quite well, they
are less effective in dealing with the complexity of the action space. As a result, the tractability of many
of the existing methods depends on the underlying model of customer choice. It is usually assumed that
the customer choices are governed by the multinomial logit model and that the consideration sets, the
sets of itineraries of interest to the different customer segments, are disjoint.
In this paper, we propose new approximation methods that remain tractable for a large class of
choice models. We assume that a customer’s choice decision is governed by a simple utility maximization
principle. That is, a customer has a utility for purchasing each of the itineraries and to not purchasing
anything. Of the available alternatives, the customer chooses the one with the highest utility. The
starting point for our methods is a dynamic program that allows randomization. We generate a sample
path of customer arrivals along with their utilities for the different itineraries and formulate a dynamic
program in order to compute the optimal offer sets. We show that it is possible to reformulate this problem as a dynamic program where the number of decision variables is linear in the number of itineraries.
As a result, the size of the action space becomes manageable. In fact, the resulting formulation is
similar to the dynamic programming formulation of the network revenue management problem with
independent demand. Consequently, we use ideas from the independent demand setting to reduce the
2
size of the state space. We particularly focus on two approximation methods. One is based on the
Lagrangian relaxation idea developed in Kunnumkal and Topaloglu (2010a) and the second is based on
the randomized linear programming approach developed in Talluri and van Ryzin (1999).
The methods that we propose have a number of appealing features. Since they are sampling-based,
they can handle a much broader class of choice models. In particular, we do not require the assumption
that customer choices come from a multinomial logit choice model with disjoint consideration sets. Our
methods yield upper bounds on the optimal expected revenue and estimates of the expected marginal
values of capacity on the flight legs. The marginal value of capacity on a flight leg, referred to as
its bid price, is useful in constructing control policies. On the other hand, upper bounds are useful
when assessing the suboptimality of heuristic control policies. Another useful feature of our approach
is that the randomized dynamic program we propose has a similar structure to the dynamic program
for the network revenue management problem with independent demand. This allows us to draw
upon the rich literature around the network revenue management problem with independent demand.
The two approximation methods that we propose require solving only linear programs, which most
commercial optimization packages are capable of. Moreover, since the linear programs we solve have
only a polynomial number of variables and constraints, it minimizes the need for customized coding
in the way of column generation techniques. This may further enhance the practical appeal of our
methods.
Our work builds on previous research. Liu and van Ryzin (2008) propose a deterministic linear
program for the network revenue management problem with customer choice. Zhang and Adelman
(2009), Meissner and Strauss (2008) and Zhang (2011) use the linear programming approach to approximate dynamic programming to come up with different value function approximations, where as
Kunnumkal and Topaloglu (2008) and Kunnumkal and Topaloglu (2010b) use Lagrangian relaxation
ideas. Kunnumkal and Topaloglu (2010c) propose a linear integer program that allows randomization
and show how it can be used to compute bid prices. The tractability of the above mentioned methods
depends on the assumptions that the customers’ choices are governed by the multinomial logit model
and that the consideration sets of the different customer segments are disjoint. Bront, Mendez-Diaz
and Vulcano (2009) analyze the case where the consideration sets overlap and show that the column
generation subproblem in the deterministic linear program of Liu and van Ryzin (2008) is NP-hard.
Bront et al. (2009) and Meissner and Strauss (2010) propose heuristic methods for column generation.
Talluri (2010) proposes a concave program for general choice models and describes a way to randomize
it. Meissner, Strauss and Talluri (2011) build on this concave program and show how it can be strengthened by adding additional constraints. van Ryzin and Vulcano (2008) and Chaneton and Vulcano (2009)
use stochastic approximation to respectively compute protection levels and bid prices for general choice
models.
The utility maximization criterion to model customer choice behavior has appeared in the literature. For example, van Ryzin and Vulcano (2008) and Chaneton and Vulcano (2009) use it to model
customer choice in network revenue management while Mahajan and van Ryzin (2001) use it in the context of optimizing retail assortments. Our work also builds on approximation methods for the network
3
revenue management problem with independent demand. The papers closest to ours are Kunnumkal
and Topaloglu (2010a) and Talluri and van Ryzin (1999). Kunnumkal and Topaloglu (2010a) propose
a linear program that yields time dependent bid prices. Talluri and van Ryzin (1999) propose a randomized linear program that works with samples of the demand random variables. We refer the reader
to Talluri and van Ryzin (2004) for a comprehensive review of the revenue management literature.
We make the following research contributions in this paper. 1) We present a new dynamic programming approximation for the network revenue management problem with customer choice behavior.
This dynamic programming formulation is attractive because it allows randomization and the size of
its action space is linear in the number of itineraries. 2) We further build on this randomized dynamic
program to obtain tractable approximation methods. As our methods are sampling-based, we are not
constrained by the underlying customer choice model. We are able to handle a variety of choice models;
all we require is the ability to generates samples of the customers’ utilities for the different alternatives. 3) We show that our approximation methods generate upper bounds on the optimal expected
revenues. Upper bounds are useful when assessing the suboptimality of heuristic control policies. We
also show how our methods can be used to obtain bid prices. 4) Computational experiments indicate
that our methods can yield significantly tighter upper bounds and higher revenues than the standard
deterministic linear program.
The rest of the paper is organized as follows. Section 1 describes the network revenue management
problem with customer choice behavior and formulates it as a dynamic program. In Section 2, we
describe the linear program proposed by Liu and van Ryzin (2008). In Section 3, we present the
randomized dynamic program and in Section 4 we describe two tractable approximation methods based
on it. The first method is based on relaxing the flight leg capacity constraints whereas the second
method solves a perfect hindsight relaxation. Section 5 presents our computational experiments. The
proofs of all the propositions and lemmas are deferred to the Appendix.
1
Problem Formulation
We have an airline network consisting of a set of flight legs that we can use to serve the customers
that arrive over time with the intention of purchasing itineraries. We use ℒ to denote the set of flight
legs in the airline network. The initial capacity on flight leg 𝑖 is 𝑐𝑖 . We use 𝒥 to denote the set of
all itineraries. An itinerary 𝑗 has a revenue associated with it, which we denote by 𝑟𝑗 . If we accept
a request for itinerary 𝑗, then we consume capacity on one or more flight legs. We use 𝑎𝑖𝑗 to denote
the number of units of capacity consumed by itinerary 𝑗 on flight leg 𝑖. Naturally, we have 𝑎𝑖𝑗 = 0
if itinerary 𝑗 does not include flight leg 𝑖. We discretize the planning horizon into a finite number of
time periods 𝒯 = {1, . . . , 𝜏 } and assume that the discretization is fine enough so that there is at most
one customer arrival at each time period. The probability of a customer arrival at time period 𝑡 is
𝜆. The fact that the arrival probability is constant over time is only for ease of exposition and it is
straightforward to allow the arrival probability to depend on the time period 𝑡.
We assume that customer choice is governed by a simple utility maximization principle. That is,
4
the customer’s utilities for the different alternatives are random variables and the customer chooses
the alternative with the highest utility. We note that the utility maximization principle is essentially
equivalent to a choice model where customers have an ordered list of preferences and pick the most
preferred alternative from the ones available. We let 𝑈𝑗𝑡 be the random variable which denotes the
utility for purchasing itinerary 𝑗 at time period 𝑡 and let 𝑈𝜙𝑡 be the random variable which denotes
the utility for not purchasing any itinerary at time period 𝑡. We let 𝑈𝒥 = {𝑈𝑗𝑡 : 𝑗 ∈ 𝒥 , 𝑡 ∈ 𝒯 } and
𝑈𝜙 = {𝑈𝜙𝑡 : 𝑡 ∈ 𝒯 }. We allow the random variables {𝑈𝑗𝑡 : 𝑗 ∈ 𝒥 ∪ {𝜙}} to be dependent within each
time period, but assume that they are independent across time periods. In other words, the purchasing
decisions of the different customers are assumed to be independent of each other. Given an offer set
𝒮, the customer chooses the alternative 𝑗𝑡 = argmax𝑗∈𝒮∪{𝜙} {𝑈𝑗𝑡 } with the highest utility. We assume
that there are no ties with probability 1. The probability that the customer chooses itinerary 𝑗 at time
period 𝑡 given the offer set 𝒮 is
Pr{𝑗𝑡 = 𝑗 ∣ 𝒮} = Pr{𝑈𝑗𝑡 = max {𝑈𝑘𝑡 }} for 𝑗 ∈ 𝒮.
𝑘∈𝒮∪{𝜙}
We have Pr{𝑗𝑡 = 𝑗 ∣ 𝒮} = 0 for 𝑗 ∈
/ 𝒮 and the probability of purchasing nothing is
∑
Pr{𝑗𝑡 = 𝜙 ∣ 𝒮} = Pr{𝑈𝜙𝑡 = max {𝑈𝑘𝑡 }} = 1 −
Pr{𝑗𝑡 = 𝑗 ∣ 𝒮}.
𝑘∈𝒮∪{𝜙}
(1)
𝑗∈𝒥
Here we emphasize that the customer’s utilities for the different alternatives do not depend on the set of
itineraries made available for sale. While there are choice models that do not satisfy this assumption, it
covers many of the commonly used choice models in the literature; see Mahajan and van Ryzin (2001)
and Zhang and Cooper (2005).
At each time period, we have to decide which itineraries to make available for sale taking into
account the state of the remaining leg capacities. Using 𝑥𝑖𝑡 to denote the remaining capacity on flight
leg 𝑖 at time period 𝑡, 𝑥𝑡 = {𝑥𝑖𝑡 : 𝑖 ∈ ℒ} captures the state of the remaining leg capacities. We let
𝒬(𝑥𝑡 ) = {𝑗 ∈ 𝒥 : 𝑎𝑖𝑗 ≤ 𝑥𝑖𝑡
∀ 𝑖 ∈ ℒ},
(2)
denote the itineraries that can be potentially offered given the remaining leg capacities. The decision
problem is to determine the set of itineraries to offer to the customers at each time period so as
to maximize the expected total revenue over the planning horizon. Under the assumption that the
customer arrivals in the different time periods and the purchasing decisions of the different customers
are independent of each other, we can obtain the value functions {𝑉𝑡 (⋅) : 𝑡 ∈ 𝒯 } through the optimality
equation
{
}
[
] [
]
∑
∑
𝜆 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮} 𝑟𝑗 + 𝑉𝑡+1 (𝑥𝑡 − 𝑖∈ℒ 𝑎𝑖𝑗 𝑒𝑖 ) + 1 − 𝜆 + 𝜆 Pr{𝑗𝑡 = 𝜙 ∣ 𝒮} 𝑉𝑡+1 (𝑥𝑡 )
𝑉𝑡 (𝑥𝑡 ) = max
𝒮⊂𝒬(𝑥𝑡 )
{
= max
𝒮⊂𝒬(𝑥𝑡 )
𝑗∈𝒥
∑
}
[
]
∑
𝜆 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮} 𝑟𝑗 + 𝑉𝑡+1 (𝑥𝑡 − 𝑖∈ℒ 𝑎𝑖𝑗 𝑒𝑖 ) − 𝑉𝑡+1 (𝑥𝑡 ) + 𝑉𝑡+1 (𝑥𝑡 ),
(3)
𝑗∈𝒥
where 𝑒𝑖 is the ∣ℒ∣-dimensional unit vector with a one in the element corresponding to 𝑖 ∈ ℒ and the
second equality follows from (1). The boundary condition for the optimality equation above is 𝑉𝜏 +1 (⋅) =
5
0. Throughout the rest of the paper, we assume that 𝜆 = 1 for notational brevity. We note that this is
equivalent to letting P̃r{𝑗𝑡 = 𝑗 ∣ 𝒮} = 𝜆 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮} and P̃r{𝑗𝑡 = 𝜙 ∣ 𝒮} = 1 − 𝜆 + 𝜆 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮} and
working with the probabilities {P̃r{𝑗𝑡 = 𝑗 ∣ 𝒮} : 𝑗 ∈ 𝒥 ∪ {𝜙}}.
Solving the above dynamic program for practical problem instances becomes difficult for two reasons. One is that the size of the state space increases exponentially with the number of flight legs in
the airline network. For, if we let 𝒞𝑖 = {0, . . . , 𝑐𝑖 }, then the state space of the above dynamic program
∏
is 𝑖∈ℒ 𝒞𝑖 , which is exponential in the number of flight legs. Secondly, the size of the action space also
increases exponentially with the number of itineraries in the flight network since the number of potential
offer sets is of the order of 2∣𝒥 ∣ . In the following sections, we look at relaxations of problem (3) that are
computationally tractable.
2
Choice Based Deterministic Linear Program
The choice based deterministic linear program, proposed by Liu and van Ryzin (2008), is an approximation that replaces all random quantities by their expected values. If set 𝒮 is offered at time period
∑
𝑡, then the expected revenue obtained is 𝑗∈𝒥 𝑟𝑗 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮}, while the expected capacity consumed
∑
on flight leg 𝑖 is 𝑗∈𝒥 𝑎𝑖𝑗 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮}. The choice based deterministic linear program assumes that
the revenue generated and the capacities consumed by offering set 𝒮 take on their expected values. It
determines the optimal choice of offer sets at each time period by solving
𝑧𝐶𝐷𝐿𝑃
= max
∑∑
𝑟𝑗 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮}ℎ𝑡 (𝒮)
(4)
𝑎𝑖𝑗 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮}ℎ𝑡 (𝒮) ≤ 𝑐𝑖 ∀𝑖 ∈ ℒ
(5)
𝑡∈𝒯 𝑗∈𝒥
subject to
∑∑
𝑡∈𝒯 𝑗∈𝒥
∑
ℎ𝑡 (𝒮) ≤ 1 ∀𝑡 ∈ 𝒯
(6)
𝒮⊂𝒥
ℎ𝑡 (𝒮) ≥ 0 ∀𝑡 ∈ 𝒯 .
(7)
In the above linear program, the decision variable ℎ𝑡 (𝒮) denotes the frequency with which set 𝒮 is offered
at time period 𝑡. The first set of constraints ensure that the expected capacity consumed on each flight
leg does not exceed the available capacity. The second set of constraints ensure that the total frequency
with which we offer the sets at each time period is at most one. Note that the number of decision
variables in the above linear program is exponential in the number of itineraries. So in general, one has
to resort to column generation to solve the problem (4)-(7). Liu and van Ryzin (2008) show that column
generation can be efficiently carried out provided the choice probabilities come from the multinomial
logit model. Gallego, Ratliff and Shebalov (2010) show that problem (4)-(7) can be reformulated as a
linear program with only a polynomial number of variables provided the choice probabilities come from
a general attraction model, of which the multinomial logit is a special case.
There are two main uses of the choice based deterministic linear program. First, Liu and van
Ryzin (2008) show that its optimal objective value gives an upper bound on the optimal expected total
6
revenue. That is, we have 𝑉1 (𝑐) ≤ 𝑧𝐶𝐷𝐿𝑃 . Second, we can use the dual solution of the choice based
deterministic linear program to construct heuristic control policies. Let 𝜋
ˆ = {ˆ
𝜋𝑖 : 𝑖 ∈ ℒ} denote the
optimal values of the dual variables associated with constraints (5). Noting that 𝜋
ˆ𝑖 approximates the
marginal value of capacity on flight leg 𝑖, we use 𝜋
ˆ𝑖 as its bid price. We can use these bid prices to come
up with different control policies. Zhang and Adelman (2009) propose approximating the value function
∑
𝑉𝑡 (𝑥𝑡 ) by 𝑖∈ℒ 𝜋
ˆ𝑖 𝑥𝑖𝑡 and solving problem (3) using this value function approximation to decide on the
offer set. That is, we solve the problem
[
]
∑
∑
max
Pr{𝑗𝑡 = 𝑗 ∣ 𝒮} 𝑟𝑗 −
𝑎𝑖𝑗 𝜋
ˆ𝑖
(8)
𝒮⊂𝒬(𝑥𝑡 )
𝑗∈𝒥
𝑖∈ℒ
to decide on the set of itineraries to offer at time period 𝑡. We note that the above maximization
problem is combinatorial in nature and can be potentially difficult to solve for a general choice model.
Bront et al. (2009) and Meissner and Strauss (2010) propose heuristic methods for solving problem (8).
Chaneton and Vulcano (2009) propose a simpler alternative, where we make an itinerary available for
sale provided its fare exceeds the sum of the bid prices on the flight legs it uses and there is sufficient
capacity.
3
Randomized Dynamic Program
In this section, we present a randomized dynamic program for the network revenue management problem
with customer choice behavior. Letting 𝑈𝒥 = {𝑈𝑗𝑡 : 𝑗 ∈ 𝒥 , 𝑡 ∈ 𝒯 } be a sample of the customers’ utilities
for the different itineraries at the different time periods, we solve the optimization problem
}
{
]
[
∑
∑
𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ) =
max
Pr{𝑗𝑡 = 𝑗 ∣ 𝒮, 𝑈𝒥 } 𝑟𝑗 + 𝑉𝑡+1 (𝑥𝑡 − 𝑖∈ℒ 𝑎𝑖𝑗 𝑒𝑖 ∣ 𝑈𝒥 ) − 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 )
𝒮⊂𝒬(𝑥𝑡 )
𝑗∈𝒥
+𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 ), (9)
with the boundary condition that 𝑉𝜏 +1 (⋅ ∣ 𝑈𝒥 ) = 0. We use the argument 𝑈𝒥 to emphasize that the
solution to the above optimality equation depends on the sampled utilities 𝑈𝒥 and therefore is a random
variable. We also note that 𝑈𝒥 only specifies the utilities for purchasing the itineraries; the utilities for
not purchasing anything 𝑈𝜙 = {𝑈𝜙𝑡 : 𝑡 ∈ 𝒯 } are still random. The following proposition shows that
𝔼{𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 )} is an upper bound on 𝑉𝑡 (𝑥𝑡 ), where the expectation is with respect to 𝑈𝒥 .
Proposition 1 We have 𝑉𝑡 (𝑥𝑡 ) ≤ 𝔼{𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 )} for all 𝑡 ∈ 𝒯 .
Note that Proposition 1 implies that 𝑉1 (𝑐) ≤ 𝔼{𝑉1 (𝑐 ∣ 𝑈𝒥 )} and so we get an upper bound on the
optimal expected revenue by solving problem (9). Besides giving an upper bound on the value function,
the randomized dynamic program also simplifies the optimization problem by reducing the size of the
action space. We show below that instead of optimizing over subsets of itineraries, it is sufficient to
optimize over the individual itineraries. We introduce some notation first. We let
𝑝𝑗𝑡 (𝑈𝒥 ) = Pr{𝑗𝑡 = 𝑗 ∣ {𝑗}, 𝑈𝒥 } = Pr{𝑈𝑗𝑡 > 𝑈𝜙𝑡 ∣ 𝑈𝒥 }
7
be the probability that the customer purchases itinerary 𝑗 when it is the only itinerary that is offered
at time period 𝑡. Note that the last equality follows from the fact that the customer will purchase the
itinerary only if its utility exceeds the utility of not purchasing anything. We use the argument 𝑈𝒥 to
emphasize that this probability is conditional on the sampled utilities. The following lemma shows that
at each time period, we can solve an optimization problem involving ∣𝒥 ∣ decision variables as opposed
to 2∣𝒥 ∣ decision variables.
Lemma 2 Consider the optimization problem
{
[
∑
∑
𝑝𝑗𝑡 (𝑈𝒥 )𝑦𝑗𝑡 𝑟𝑗 + 𝑉˜𝑡+1 (𝑥𝑡 −
𝑉˜𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ) = max
}
]
˜
+ 𝑉˜𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 ),
𝑖∈ℒ 𝑎𝑖𝑗 𝑒𝑖 ∣ 𝑈𝒥 ) − 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 )
𝑗∈𝒥
(10)
subject to
∑
𝑎𝑖𝑗 𝑦𝑗𝑡 ≤ 𝑥𝑖𝑡 ∀𝑖 ∈ ℒ
(11)
𝑦𝑗𝑡 ≤ 1
(12)
𝑗∈𝒥
∑
𝑗∈𝒥
𝑦𝑗𝑡 ∈ {0, 1} ∀𝑗 ∈ 𝒥 ,
(13)
with the boundary condition 𝑉˜𝜏 +1 (⋅ ∣ 𝑈𝒥 ) = 0. We have 𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ) = 𝑉˜𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ) for all 𝑥𝑡 , 𝑡 ∈ 𝒯 .
Although the number of decision variables in problem (10)-(13) is manageable, the size of the state
space is still exponential in the capacities of the flight legs. On the other hand, noting that the decision
variables in problem (10)-(13) are only over the itineraries, this problem has a similar structure to the
network revenue management problem with independent demand. This allows us to use approximation
ideas developed for the independent demand setting to reduce the complexity of the state space. We
present two approximation methods in the following section.
4
Relaxations of the Randomized Dynamic Program
In this section, we describe two tractable relaxations of problem (10)-(13). The first method is based
on relaxing the flight leg capacity constraints using Lagrange multipliers. This yields an upper bound
on the value function of the randomized dynamic program. We find the set of Lagrange multipliers
which yields the tightest upper bound by solving a linear program. This idea is similar to that pursued
in Kunnumkal and Topaloglu (2010a). The second method we propose is based on solving a perfect
hindsight relaxation, where we have access to the customers’ utilities for not purchasing anything also.
This method is similar to the randomized linear programming method of Talluri and van Ryzin (1999).
We note that other approximation methods developed for the network revenue management problem
with independent demand can also be applied to problem (10)-(13). In this paper, we particularly focus
on the above mentioned two methods because they involve solving linear programs, which can be done
quickly and efficiently. Speed is an important factor since we have to resolve the problems for many
different samples.
8
4.1
Capacity Relaxation
∑
Letting 𝒴 = {𝑦 ∈ {0, 1}∣𝒥 ∣ : 𝑗∈𝒥 𝑦𝑗 ≤ 1} and 𝑦𝑡 = {𝑦𝑗𝑡 : 𝑗 ∈ 𝒥 }, we consider relaxing constraints (11)
by introducing Lagrange multipliers 𝜆𝑖𝑡 and solve the optimization problem
{
[
]
∑
∑
𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 , 𝜆) = max
𝑝𝑗𝑡 (𝑈𝒥 )𝑦𝑗𝑡 𝑟𝑗 + 𝑉𝑡+1 (𝑥𝑡 − 𝑖∈ℒ 𝑎𝑖𝑗 𝑒𝑖 ∣ 𝑈𝒥 , 𝜆) − 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 , 𝜆)
𝑦𝑡 ∈𝒴
+
𝑗∈𝒥
∑
𝜆𝑖𝑡 (𝑥𝑖𝑡 −
𝑖∈ℒ
∑
}
𝑎𝑖𝑗 𝑦𝑗𝑡 )
+ 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 , 𝜆)
(14)
𝑗∈𝒥
with the boundary condition that 𝑉𝜏 +1 (⋅ ∣ 𝑈𝒥 , 𝜆) = 0. The following proposition shows that as long as
the Lagrange multipliers are nonnegative, 𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 , 𝜆) is an upper bound on 𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ).
Proposition 3 If 𝜆 = {𝜆𝑖𝑡 : 𝑖 ∈ ℒ, 𝑡 ∈ 𝒯 } ≥ 0, then we have 𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ) ≤ 𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 , 𝜆).
Note that Propositions 1 and 3 together imply that as long as the Lagrange multipliers are nonnegative, we have 𝑉1 (𝑐) ≤ 𝔼{𝑉1 (𝑐 ∣ 𝑈𝒥 , 𝜆)}. So we are naturally interested in finding the set of Lagrange
multipliers that gives the tightest upper bound. That is, for each sample 𝑈𝒥 , we are interested in
solving the problem
min 𝑉1 (𝑐 ∣ 𝑈𝒥 , 𝜆).
𝜆≥0
We next show that the above minimization problem reduces to solving a linear program and therefore
is tractable. We begin with the following result, which gives a closed form expression for 𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 , 𝜆).
Lemma 4 We have
𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 , 𝜆) =
𝜏
∑
Λ𝑠 +
𝑠=𝑡
where Λ𝑡 = max𝑗∈𝒥
𝜏
∑∑
(
𝜆𝑖𝑠 )𝑥𝑖𝑡 ,
𝑖∈ℒ 𝑠=𝑡
{
}+
[
] ∑
∑
∑
and we use {⋅}+ = max{0, ⋅}.
𝑝𝑗𝑡 (𝑈𝒥 ) 𝑟𝑗 − 𝑖∈ℒ 𝑎𝑖𝑗 ( 𝜏𝑠=𝑡+1 𝜆𝑖𝑠 ) − 𝑖∈ℒ 𝑎𝑖𝑗 𝜆𝑖𝑡
Using the result in Lemma 4, we have that the problem min𝜆≥0 𝑉1 (𝑐 ∣ 𝑈𝒥 , 𝜆) can be solved as the
linear program
𝑧𝐶𝑅 (𝑈𝒥 ) = min
∑
Λ𝑡 +
𝑡∈𝒯
subject to
Λ𝑡 +
∑∑
(
𝜆𝑖𝑡 )𝑐𝑖
𝑖∈ℒ 𝑡∈𝒯
∑
[∑
]
𝑎𝑖𝑗 𝜆𝑖𝑡 + 𝑝𝑗𝑡 (𝑈𝒥 )
𝑎𝑖𝑗 (𝜆𝑖,𝑡+1 + . . . + 𝜆𝑖𝜏 ) ≥ 𝑟𝑗 𝑝𝑗𝑡 (𝑈𝒥 ) ∀𝑗 ∈ 𝒥 , 𝑡 ∈ 𝒯
𝑖∈ℒ
𝑖∈ℒ
Λ𝑡 ≥ 0 ∀𝑡 ∈ 𝒯
𝜆𝑖𝑡 ≥ 0 ∀𝑖 ∈ ℒ, 𝑡 ∈ 𝒯 ,
9
with the understanding that 𝜆𝑖,𝜏 +1 = 0. Taking the dual of this linear program, we get
𝑧𝐶𝑅 (𝑈𝒥 ) = max
∑∑
𝑟𝑗 𝑝𝑗𝑡 (𝑈𝒥 )𝑦𝑗𝑡
(15)
𝑡∈𝒯 𝑗∈𝒥
subject to
∑
𝑎𝑖𝑗 𝑦𝑗𝑡 +
𝑗∈𝒥
∑
∑
[
]
𝑎𝑖𝑗 𝑝𝑗1 (𝑈𝒥 )𝑦𝑗1 + . . . + 𝑝𝑗,𝑡−1 (𝑈𝒥 )𝑦𝑗,𝑡−1 ≤ 𝑐𝑖 ∀𝑖 ∈ ℒ, 𝑡 ∈ 𝒯(16)
𝑗∈𝒥
𝑦𝑗𝑡 ≤ 1∀𝑡 ∈ 𝒯
(17)
𝑦𝑗𝑡 ≥ 0 ∀𝑗 ∈ 𝒥 , 𝑡 ∈ 𝒯 ,
(18)
𝑗∈𝒥
with the understanding that 𝑦𝑗0 = 0. In the above linear program, we can interpret the decision variable
𝑦𝑗𝑡 as the frequency with which we offer itinerary 𝑗 for sale at time period 𝑡. Since 𝑝𝑗𝑡 (𝑈𝒥 )𝑦𝑗𝑡 represents
the expected sales of itinerary 𝑗 at time period 𝑡, we can interpret the first set of constraints as saying
that the capacity consumed by the itineraries offered at time period 𝑡 should not exceed the expected
[
]
∑
capacity consumed up to time period 𝑡, which is 𝑐𝑖 − 𝑗∈𝒥 𝑎𝑖𝑗 𝑝𝑗1 (𝑈𝒥 )𝑦𝑗1 + . . . + 𝑝𝑗,𝑡−1 (𝑈𝒥 )𝑦𝑗,𝑡−1 .
We emphasize that the expectations are conditional on the sampled utilities 𝑈𝒥 . The second set of
constraints ensure that the total frequency with which we offer the individual itineraries at each time
period is at most one.
ˆ 𝒥 ) = argmin 𝑉1 (𝑐 ∣ 𝑈𝒥 , 𝜆), we have that 𝔼{𝑉1 (𝑐 ∣ 𝑈𝒥 , 𝜆(𝑈
ˆ 𝒥 ))} = 𝔼{𝑧𝐶𝑅 (𝑈𝒥 )} is an
Letting 𝜆(𝑈
𝜆≥0
ˆ
ˆ 𝑖𝑡 = 𝔼{𝜆
ˆ 𝑖𝑡 (𝑈𝒥 )}, we use ∑𝜏 𝜆
upper bound on the optimal expected revenue. Letting 𝜆
𝑠=𝑡 𝑖𝑠 as the bid
∑
∑𝜏 ˆ
price of flight leg 𝑖 at time period 𝑡. We approximate 𝑉𝑡 (𝑥𝑡 ) by 𝑖∈ℒ 𝑠=𝑡 𝜆𝑖𝑠 𝑥𝑖𝑡 and solve the problem
max
∑
𝒮⊂𝒬(𝑥𝑡 )
𝜏
]
[
∑
∑
ˆ 𝑖𝑠
𝜆
Pr{𝑗𝑡 = 𝑗 ∣ 𝒮} 𝑟𝑗 −
𝑎𝑖𝑗
𝑗∈𝒥
𝑖∈ℒ
(19)
𝑠=𝑡
to decide on the set of itineraries to offer at time period 𝑡. As it becomes difficult to analytically compute
ˆ 𝑖𝑡 (𝑈𝒥 )}, we resort to Monte Carlo simulation to estimate these
the expectations 𝔼{𝑧𝐶𝑅 (𝑈𝒥 )} and 𝔼{𝜆
quantities. In particular, we generate 𝐾 samples of the customers’ utilities for the different itineraries
𝑘 : 𝑗 ∈ 𝒥 , 𝑡 ∈ 𝒯 } are the utilities generated in the 𝑘th sample. We solve
𝑈𝒥1 , . . . , 𝑈𝒥𝐾 where 𝑈𝒥𝑘 = {𝑈𝑗𝑡
linear program (15)-(18) for each sample. Letting 𝑧𝐶𝑅 (𝑈𝒥𝑘 ) denote the optimal objective value and
ˆ 𝑘 : 𝑖 ∈ ℒ, 𝑡 ∈ 𝒯 } denote optimal values of the dual variables corresponding to constraints (16), we
{𝜆
𝑖𝑡
∑𝐾 ˆ 𝑘
∑
𝑘
ˆ
ˆ
use 𝐾
𝑘=1 𝜆𝑖𝑡 /𝐾 as the sample estimates of 𝔼{𝑧𝐶𝑅 (𝑈𝒥 )} and 𝜆𝑖𝑡 = 𝔼{𝜆𝑖𝑡 (𝑈𝒥 )},
𝑘=1 𝑧𝐶𝑅 (𝑈𝒥 )/𝐾 and
respectively.
4.2
Perfect Hindsight Relaxation
We consider another relaxation of problem (9) where we allow access to the customers’ utilities for
not purchasing anything as well. In particular, letting 𝑈𝒥 = {𝑈𝑗𝑡 : 𝑗 ∈ 𝒥 , 𝑡 ∈ 𝒯 } be a sample of the
customers’ utilities for the different itineraries at the different time periods and 𝑈𝜙 = {𝑈𝜙𝑡 : 𝑡 ∈ 𝒯 }
be a sample of the customers’ utilities for not purchasing anything and 𝑈 = 𝑈𝒥 ∪ 𝑈𝜙 , we solve the
10
optimization problem
{
𝑉𝑡 (𝑥𝑡 ∣ 𝑈 ) =
max
𝒮⊂𝒬(𝑥𝑡 )
∑
}
[
]
∑
Pr{𝑗𝑡 = 𝑗 ∣ 𝒮, 𝑈 } 𝑟𝑗 + 𝑉𝑡+1 (𝑥𝑡 − 𝑖∈ℒ 𝑎𝑖𝑗 𝑒𝑖 ∣ 𝑈 ) − 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈 )
𝑗∈𝒥
+𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈 ).
(20)
Note that we can interpret problem (20) as determining the set of itineraries to offer at each time period
after knowing the entire sample path: the customers’ utilities for the different itineraries as well as for
not purchasing anything. Not surprisingly, it also gives an upper bound on problem (9).
Proposition 5 We have 𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ) ≤ 𝔼{𝑉𝑡 (𝑥𝑡 ∣ 𝑈 ) ∣ 𝑈𝒥 } for all 𝑡 ∈ 𝒯 .
Proposition 5 together with Proposition 1 imply that 𝑉1 (𝑐) ≤ 𝔼{𝑉1 (𝑐 ∣ 𝑈 )}. Therefore, we obtain
another upper bound on the optimal expected revenue by solving problem (20). Noting that Pr{𝑗𝑡 =
𝑗 ∣ 𝒮, 𝑈 } = 1(𝑈𝑗𝑡 > 𝑈𝜙𝑡 ) if 𝑗 = argmax𝑘∈𝒮 {𝑈𝑘𝑡 } and is zero otherwise, we have
}
{
[
]
∑
𝑉𝑡 (𝑥𝑡 ∣ 𝑈 ) = max 1(𝑈𝑗𝑡 > 𝑈𝜙𝑡 ) 𝑟𝑗 + 𝑉𝑡+1 (𝑥𝑡 − 𝑖∈ℒ 𝑎𝑖𝑗 𝑒𝑖 ∣ 𝑈 ) − 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈 ) + 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈 ).
𝑗∈𝒬(𝑥𝑡 )
It follows that we can solve problem (20) as the following linear binary integer program:
𝑧𝑃 𝐻 (𝑈 ) = max
∑∑
𝑟𝑗 1(𝑈𝑗𝑡 > 𝑈𝜙𝑡 )𝑦𝑗𝑡
(21)
𝑎𝑖𝑗 1(𝑈𝑗𝑡 > 𝑈𝜙𝑡 )𝑦𝑗𝑡 ≤ 𝑐𝑖 ∀𝑖 ∈ ℒ
(22)
𝑡∈𝒯 𝑗∈𝒥
subject to
∑∑
𝑡∈𝒯 𝑗∈𝒥
∑
𝑦𝑗𝑡 ≤ 1∀𝑡 ∈ 𝒯
(23)
𝑗∈𝒥
𝑦𝑗𝑡 ∈ {0, 1} ∀𝑗 ∈ 𝒥 , 𝑡 ∈ 𝒯 .
(24)
In the above problem, the decision variable 𝑦𝑗𝑡 indicates whether we offer itinerary 𝑗 at time period 𝑡.
The first set of constraints ensure that the total capacity consumed by the itinerary requests on each
flight leg does not exceed its available capacity. The second set of constraints ensure that we offer at
most one itinerary at each time period.
We use 𝔼{𝑉1 (𝑐 ∣ 𝑈 )} = 𝔼{𝑧𝑃 𝐻 (𝑈 )} as an upper bound on the optimal expected revenue. In order
to obtain a control policy, we solve the linear programming relaxation of problem (21)-(24). Letting
𝜌ˆ(𝑈 ) = {ˆ
𝜌𝑖 (𝑈 ) : 𝑖 ∈ ℒ} denote the optimal values of the dual variables corresponding to constraints
∑
(22), we use 𝜌ˆ𝑖 = 𝔼{ˆ
𝜌𝑖 (𝑈 )} as the bid price of flight leg 𝑖. We approximate 𝑉𝑡 (𝑥𝑡 ) by 𝑖∈ℒ 𝜌ˆ𝑖 𝑥𝑖𝑡 and
solve the problem
[
]
∑
∑
max
Pr{𝑗𝑡 = 𝑗 ∣ 𝒮} 𝑟𝑗 −
𝑎𝑖𝑗 𝜌ˆ𝑖
(25)
𝒮⊂𝒬(𝑥𝑡 )
𝑗∈𝒥
𝑖∈ℒ
11
to decide on the set of itineraries to offer at time period 𝑡. It again becomes difficult to analytically
compute 𝔼{𝑧𝑃 𝐻 (𝑈 )} and 𝔼{ˆ
𝜌𝑖 (𝑈 )} and so we resort to Monte Carlo simulation. In particular, we
generate 𝐾 samples of the customers’ utilities for the different itineraries as well as not purchasing
𝑘 : 𝑗 ∈ 𝒥 ∪ {𝜙}, 𝑡 ∈ 𝒯 } are the utilities generated in the 𝑘th
anything 𝑈 1 , . . . , 𝑈 𝐾 where 𝑈 𝑘 = {𝑈𝑗𝑡
sample. We solve problem (21)-(24) for each sample. Letting 𝑧𝑃 𝐻 (𝑈 𝑘 ) denote the optimal objective
∑
𝑘
value, we use 𝐾
𝜌𝑘𝑖 : 𝑖 ∈ ℒ} denote
𝑘=1 𝑧𝑃 𝐻 (𝑈 )/𝐾 as the sample estimate of 𝔼{𝑧𝑃 𝐻 (𝑈 )}. Letting {ˆ
the optimal values of the dual variables corresponding to constraints (22) in the linear programming
∑
relaxation of problem (21)-(24), we use 𝐾
ˆ𝑘𝑖 /𝐾 as an estimate of 𝜌ˆ𝑖 = 𝔼{ˆ
𝜌𝑖 (𝑈 )}.
𝑘=1 𝜌
We close this section with a comment on the upper bounds obtained by problems (4)-(7), (15)-(18)
and (21)-(24). It turns out that none of the upper bounds uniformly dominates the other. For example,
consider a revenue management problem on a single flight leg for a single time period where we have
two itineraries with 𝑟1 = 11 and 𝑟2 = 5. We have a single unit of capacity on the flight leg and each
itinerary consumes one unit of capacity. Furthermore, we have 𝑈𝜙1 = 0 with probability 1. On the
other hand, we have
⎧
⎨ (1, −1) with probability 1/3
(−1, 1) with probability 1/3
(𝑈11 , 𝑈21 ) =
⎩
(1, 2)
with probability 1/3.
It is easy to verify that we have 𝑉1 (𝑐) = 𝑧𝐶𝐷𝐿𝑃 = 22/3, while we have 𝔼{𝑧𝑃 𝐻 (𝑈 )} = 𝔼{𝑧𝐶𝑅 (𝑈𝒥 )} = 9.
So, in this case we have that 𝑧𝐶𝐷𝐿𝑃 < 𝔼{𝑧𝑃 𝐻 (𝑈 )} = 𝔼{𝑧𝐶𝑅 (𝑈𝒥 )}. It is also possible to come up with
examples where the direction of the inequalities is reversed. On the other hand, in our computational
experiments that we present next, we find that the capacity relaxation method consistently generates
tighter upper bounds than the perfect hindsight relaxation method which in turn is tighter than the
choice based deterministic linear program.
5
Computational Experiments
In this section, we numerically compare the performance of the choice based deterministic linear program, the capacity relaxation method and the perfect hindsight relaxation method. We first describe
the benchmark solution methods. After that we present our experimental setup and the results of the
numerical study.
Choice Based Deterministic Linear Program (CDLP): This is the solution method that we describe in
Section 2. In our practical implementation, we divide the booking horizon into five equal segments. At
the beginning of each segment, we solve problem (4)-(7) after replacing the right hand side of equation
(5) with the remaining capacities on the flight legs and the set of time periods 𝒯 with the current set
of remaining time periods. We get a fresh set of optimal dual values {ˆ
𝜋𝑖 : 𝑖 ∈ ℒ} and we plug them into
decision rule (8) to decide on the set of itineraries to offer. We continue to use this decision rule until
the beginning of the next segment, where we resolve problem (4)-(7).
Capacity Relaxation (CR): This is the solution method that we describe in Section 4.1. In our practical
implementation, we divide the booking horizon into five equal segments. At the beginning of each
12
segment, we solve problem (15)-(18) after replacing the right hand side of equation (16) with the
remaining capacities on the flight legs and the set of time periods 𝒯 with the current set of remaining
ˆ 𝑘 : 𝑖 ∈ ℒ, 𝑡 ∈ 𝒯 , 𝑘 ∈ 𝐾}
time periods. We repeat this for 𝐾 samples to get a fresh set of dual values {𝜆
𝑖𝑡
and use these in decision rule (19) to decide on the set of itineraries to offer. We continue to use this
decision rule until the beginning of the next segment, where we resolve problem (15)-(18). We use
𝐾 = 100 in our computational experiments. Increasing the value of 𝐾 further did not result in any
noticeable changes in performance.
Perfect Hindsight Relaxation (PH): This is the solution method that we describe in Section 4.2. As with
CDLP and CR, in our practical implementation, we divide the booking horizon into five equal segments.
At the start of each segment, we refresh our bid prices by solving the linear programming relaxation of
problem (21)-(24) after replacing the right hand side of equation (22) with the remaining capacities on
the flight legs and the set of time periods 𝒯 with the current set of remaining time periods. We repeat
this for 𝐾 samples and use the fresh set of optimal dual values {ˆ
𝜌𝑘𝑖 : 𝑖 ∈ ℒ, 𝑘 ∈ 𝐾} in decision rule (25)
to decide on the set of itineraries to offer. We continue to use this decision rule until the beginning
of the next segment, where we again resolve problem (21)-(24). As in CR, we use 𝐾 = 100 in our
computational experiments.
We note that all of the above mentioned benchmark methods obtain bid prices that are capacity
independent, in that they do not naturally change with the capacities on the flight legs. It is possible
to obtain capacity dependent bid prices by using the optimal dual values obtained by the benchmark
methods in a dynamic programming decomposition scheme as suggested by Liu and van Ryzin (2008)
or Zhang (2011). We do not pursue that here for a number of reasons. Capacity dependent bid prices
typically come with a higher overhead, both in terms of computation and implementation. Numerical
studies also indicate that the performance gap between capacity independent bid prices remains intact
when we use them in a dynamic programming decomposition scheme; see for example Kunnumkal and
Topaloglu (2010c).
We test the performance of the benchmark solution methods on two groups of test problems. The
first group involves an airline network with a single hub serving multiple spokes, while the second group
of test problems have an airline network with two hubs serving multiple spokes. Our test problems
closely parallel those in Kunnumkal and Topaloglu (2010b).
5.1
Airline Network with a Single Hub
We consider an airline network with a single hub that serves 𝑁 spokes. Half of the spokes have two
flights to the hub, while the remaining half have two flights from the hub. The total number of flights
is 2𝑁 . Figure 1 shows the structure of the airline network with 𝑁 = 8. There are four itineraries
between each spoke-to-hub and hub-to-spoke origin destination pair. On the other hand, we have eight
itineraries between each spoke-to-spoke origin destination pair, so that the total number of itineraries is
2𝑁 (𝑁 + 2). Half of these itineraries are high fare itineraries while the other half are low fare itineraries.
We let 𝛾 denote the ratio between the high fare and the low fare.
13
Each origin destination pair is associated with a customer segment. We let 𝒦 denote the set
of customer segments. At each time period a customer from segment 𝑙 ∈ 𝒦 arrives with probability
𝜆𝑙 . An arriving customer is interested only in the set of itineraries connecting the origin destination
pair that it is associated with. Therefore, the consideration sets of the different customer segments
are disjoint. Customer choice is governed by the multinomial logit model. In the multinomial logit
model, the utility for purchasing itinerary 𝑗 that is in the consideration set of customer segment 𝑙 is
given by 𝑈𝑙𝑗𝑡 = 𝑢𝑙𝑗𝑡 + 𝜉𝑙𝑗𝑡 , where 𝑢𝑙𝑗𝑡 is a constant called the nominal utility and 𝜉𝑙𝑗𝑡 is a Gumbel
random variable with mean zero and scale parameter one. The utility for not purchasing anything for
customer segment 𝑙 is 𝑈𝑙𝜙𝑡 = 𝑢𝑙𝜙𝑡 + 𝜉𝑙𝜙𝑡 , where 𝑢𝑙𝜙𝑡 is the nominal utility for not purchasing anything
and 𝜉𝑙𝜙𝑡 is a Gumbel random variable with mean zero and scale parameter one. The random variables
{𝜉𝑙𝑗𝑡 : 𝑗 ∈ 𝒥 ∪ {𝜙}, 𝑡 ∈ 𝒯 } are independent; see Ben-Akiva and Lerman (1994).
We measure the tightness of the leg capacities in the same manner as Zhang and Adelman (2009).
∑
Letting 𝒮𝑡∗ = argmax𝒮⊂𝒥 𝑗∈𝒥 𝑟𝑗 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮} be the offer set that maximizes expected revenue at
time period 𝑡 when there is ample capacity on all the flight legs, we use
∑
∑
∑
∑
𝑎𝑖𝑗 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮𝑡∗ }
𝑙∈𝒦 𝜆𝑙
𝑡∈𝒯
𝑖∈ℒ
∑ 𝑗∈𝒥
𝛼=
,
𝑖∈ℒ 𝑐𝑖
to measure the tightness of the leg capacities. We have ∣𝒯 ∣ = 200 time periods in all of our test
problems. We vary 𝑁 , 𝛾 and 𝛼 to obtain different test problems. We label our test problems by the
triplet (𝑁, 𝛾, 𝛼) ∈ {8, 10, 12} × {1.5, 3} × {1.3, 1.6}, where 𝑁 is the number of spokes, 𝛾 is the ratio
between the high and low fare itineraries and 𝛼 measures the tightness of the leg capacities. This gives
us a total of twelve test problems.
Table 1 compares the upper bounds obtained by CR, PH and CDLP. The first column in this table
gives the characteristics of the problem by using (𝑁, 𝛾, 𝛼). The second, third and fourth columns, respectively, give the upper bounds obtained by CR, PH and CDLP. The fifth column gives the percentage
gap between the upper bounds obtained by PH and CR, while the last column gives the percentage gap
between the upper bounds obtained by CDLP and CR. CR performs consistently well in our computational experiments and we use CR as a benchmark. In the last two columns, a “✓ ” indicates that the
gap is significant at the 95% level, while a “⊙” indicates that the gap is not significant at the 95% level.
We observe that CR generates significantly tighter upper bounds than PH and CDLP. On average, the
upper bounds obtained by CR are about 2% tighter than PH and 8% tighter than CDLP.
Table 2 compares the total expected revenues obtained by CR, PH and CDLP. We evaluate the
expected revenues by simulation and use common random numbers in our simulations. The columns
have a similar interpretation as in Table 1 except that they give the expected revenues obtained by
the three methods. The last two columns include a “✓” if CR does better than the respective solution
method at the 95% level, a “×” otherwise and a “⊙” if there does not exist a statistically significant
difference between the two. The average gap between the total expected revenues obtained by CR
and CDLP is around 2%. The performance gaps are statistically significant in ten out of the twelve
test problems. The performance gap between CR and CDLP seems to increase with the fare ratio
and the tightness of the leg capacities. The performance gaps between CR and PH are small in most
14
cases, although we observe one instance where PH performs about 1% better than CR. PH performs
significantly better than CDLP. The average gap between the total expected revenues obtained by PH
and CDLP is around 2%.
5.2
Airline Network with Two Hubs
We consider an airline network with two hubs that serve 𝑁 spokes in total. Half of the spokes have two
flights to the first hub, while the other half have two flights from the second hub. In addition, there
are four flights from the first to the second hub. The total number of flights is 2𝑁 + 4. Figure 2 shows
the structure of the airline network with 𝑁 = 8. We randomly sample from the set of all the possible
itineraries so that the total number of itineraries is around 4𝑁 2 . Half of these itineraries are high fare
itineraries while the other half are low fare itineraries.
Similar to the test problems with a single hub, each origin destination pair is associated with a
customer segment. An arriving customer belongs to one of the segments and is interested only in the set
of itineraries connecting the origin destination pair that it is associated with. We continue to assume
that customer choice is governed by the multinomial logit model with disjoint consideration sets. We
label the test problems by the triplet (𝑁, 𝛾, 𝛼) ∈ {4, 6, 8} × {1.5, 3} × {1.3, 1.6}, which gives us a total
of twelve test problems.
Table 3 compares the upper bounds obtained by CR, PH and CDLP. The columns have the same
interpretation as in Table 1. The results display the same trends that we observed for the airline network
with a single hub. CR consistently generates the tightest upper bounds, followed by PH and CDLP.
On average, the upper bounds obtained by CR are about 2% tighter than PH and 9% tighter than
CDLP. Table 4 compares the total expected revenues obtained by CR, PH and CDLP. CR generates
significantly higher revenues than CDLP. The average gap between the total expected revenues obtained
by CR and CDLP is around 2%, although we observe test problems where the gap is as high as 5%.
We find one test problem where the performance gap between PH and CR is around 2.5%, but the
gaps are quite small and insignificant in the remaining cases. The average performance gap between
PH and CDLP is around 2%. The ratio between the high and low fares and the tightness of the leg
capacities seem to be two factors which contribute to increasing the performance gaps between CDLP
and the other two solution methods . Problems with large differences between the high and low fares
and tight leg capacities tend to be more difficult to solve, because the consequences of offering the
“wrong” set of itineraries tend to be more severe. It is therefore encouraging that CR and PH provide
good performance for such test problems.
All of the computational experiments are carried out on a Pentium Core 2 Duo desktop with 3
GHz CPU and 3 GB RAM running Windows XP. The running time of CDLP is of the order of seconds.
For 𝐾 = 100 samples, the running time of PH is of the order of seconds, while that of CR is in minutes.
CR takes about a minute and a half to solve the largest test problem.
15
6
Conclusions
We presented new methods to obtain upper bounds and bid prices for the network revenue management
problem with customer choice behavior. The starting point for our methods is a dynamic programming
approximation that we solve for a sample of the customers’ utilities for the different itineraries. An
attractive feature of this randomized dynamic program is that the number of decision variables is linear
in the number of itineraries. As a result, we are able to reduce the complexity of the action space. We
build on this randomized dynamic program to obtain two tractable approximation methods. The first
method that we propose involves relaxing the flight leg capacity constraints using Lagrange multipliers.
The second method involves solving a perfect hindsight relaxation. We showed that both methods
give upper bounds on the optimal expected total revenue. Our methods may also be appealing from a
practical standpoint as they involve solving only linear programs. Computational experiments indicate
that our methods can significantly improve upon the upper bounds and expected revenues obtained by
the choice based deterministic linear program.
Appendix
Proof of Proposition 1
We show the result by induction over the time periods. It is easy to show that the result holds at time
period 𝜏 . Assuming the result holds at time period 𝑡 + 1, we show that it holds at time period 𝑡. Letting
𝒮ˆ be an optimal solution for problem (3), we note that 𝒮ˆ is feasible for problem (9). We also note that
for a given offer set 𝒮, Pr{𝑗𝑡 = 𝑗 ∣ 𝒮, 𝑈𝒥 } is a function of {𝑈𝑗𝑡 : 𝑗 ∈ 𝒥 }, while 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 ) is a function
of {𝑈𝑗𝑠 : 𝑗 ∈ 𝒥 , 𝑠 ∈ {𝑡 + 1, . . . , 𝒯 }}. Since the random variables {𝑈𝑗𝑡 : 𝑗 ∈ 𝒥 } are independent across
time, it follows that
{
}
{
} {
}
𝔼 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮, 𝑈𝒥 }𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 ) = 𝔼 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮, 𝑈𝒥 } 𝔼 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 ) ,
(26)
where the expectation is with respect to 𝑈𝒥 . Therefore, we have
∑ {
{
}
}[
}]
{
ˆ 𝑈𝒥 } 𝑟𝑗 + 𝔼 𝑉𝑡+1 (𝑥𝑡 − ∑
𝔼 𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ) ≥
𝔼 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮,
𝑎
𝑒
∣
𝑈
)
𝑖𝑗
𝑖
𝒥
𝑖∈ℒ
𝑗∈𝒥
[
∑ {
}] {
}
ˆ 𝑈𝒥 } 𝔼 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 )
+ 1−
𝔼 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮,
≥
∑
𝑗∈𝒥
[
] [
]
∑
ˆ 𝑟𝑗 + 𝑉𝑡+1 (𝑥𝑡 − ∑
ˆ
Pr{𝑗𝑡 = 𝑗 ∣ 𝒮}
𝑎
𝑒
)
+
1
−
Pr{𝑗
=
𝑗
∣
𝒮}
𝑉𝑡+1 (𝑥𝑡 )
𝑡
𝑖∈ℒ 𝑖𝑗 𝑖
𝑗∈𝒥
𝑗∈𝒥
= 𝑉𝑡 (𝑥𝑡 ),
where the first inequality uses (26) and the fact that 𝒮ˆ is a feasible but not necessarily optimal solution
to problem (9) and the second inequality follows from the induction assumption and the fact that
{
}
{ {
}}
{
}
Pr{𝑗𝑡 = 𝑗 ∣ 𝒮} = 𝔼 1(𝑈𝑗𝑡 = max {𝑈𝑘𝑡 }) = 𝔼 𝔼 1(𝑈𝑗𝑡 = max {𝑈𝑘𝑡 })∣𝑈𝒥
= 𝔼 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮, 𝑈𝒥 } .
𝑘∈𝒮∪{𝜙}
𝑘∈𝒮∪{𝜙}
□
16
Proof of Lemma 2
We show the result by induction over the time periods. It is easy to show that the result holds at time
period 𝜏 . Assuming the result holds at time period 𝑡 + 1, we show that it holds at time period 𝑡. We
first show that 𝑉˜𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ) ≤ 𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ). Let 𝑦˜𝑡 = {˜
𝑦𝑗𝑡 : 𝑗 ∈ 𝒥 } be an optimal solution to problem
˜
(10)-(13) and let 𝒮 = {𝑗 ∈ 𝒥 : 𝑦˜𝑗𝑡 = 1}. Note that since 𝑦˜𝑡 satisfies (11)-(13), the offer set 𝒮˜ is feasible
for problem (9). We have
[
]
∑
∑
𝑉˜𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ) =
𝑝𝑗𝑡 (𝑈𝒥 )˜
𝑦𝑗𝑡 𝑟𝑗 + 𝑉˜𝑡+1 (𝑥𝑡 − 𝑖∈ℒ 𝑎𝑖𝑗 𝑒𝑖 ∣ 𝑈𝒥 ) − 𝑉˜𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 ) + 𝑉˜𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 )
𝑗∈𝒥
=
∑
[
]
˜ 𝑈𝒥 } 𝑟𝑗 + 𝑉𝑡+1 (𝑥𝑡 − ∑
Pr{𝑗𝑡 = 𝑗 ∣ 𝒮,
𝑎
𝑒
∣
𝑈
)
−
𝑉
(𝑥
∣
𝑈
)
+ 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 )
𝑖𝑗
𝑖
𝑡+1
𝑡
𝒥
𝒥
𝑖∈ℒ
𝑗∈𝒥
≤ 𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ),
where the first equality follows from the optimality of 𝑦˜𝑡 , the second equality follows from the fact that
˜ ≤ 1 and Pr{𝑗𝑡 = 𝑗 ∣ 𝒮,
˜ 𝑈𝒥 } = 0 for 𝑗 ∈
∣𝒮∣
/ 𝒮˜ and the induction assumption. The inequality holds since
𝒮˜ is feasible for problem (9). This implies 𝑉˜𝑡 (𝑥𝑡 ) ≤ 𝑉𝑡 (𝑥𝑡 ).
To show the reverse inequality, let 𝒮ˆ be the optimal solution to problem (9), 𝚥ˆ = argmax𝑘∈𝒮ˆ{𝑈𝑘𝑡 }
and 𝑦ˆ𝑡 = {ˆ
𝑦𝑗𝑡 : 𝑗 ∈ 𝒥 } with 𝑦ˆ𝑗𝑡 = 1 for 𝑗 = 𝚥ˆ and 𝑦ˆ𝑗𝑡 = 0 for 𝑗 ∈ 𝒥 ∖{ˆ
𝚥}. Note that since 𝒮ˆ ⊂ 𝒬(𝑥𝑡 ), we
ˆ 𝑈𝒥 } = Pr{𝑈𝑗𝑡 > 𝑈𝜙𝑡 ∣ 𝑈𝒥 } = 𝑝𝑗𝑡 (𝑈𝒥 )
have that 𝑦ˆ𝑡 satisfies constraints (11)-(13). We have Pr{𝑗𝑡 = 𝑗 ∣ 𝒮,
ˆ 𝚥}, we have Pr{𝑗𝑡 = 𝑗 ∣ 𝒮,
ˆ 𝑈𝒥 } = 0 for
for 𝑗 = 𝚥ˆ. On the other hand, since 𝑈𝑗𝑡 < 𝑈𝚥ˆ𝑡 for 𝑗 ∈ 𝒮∖{ˆ
ˆ 𝚥}. Also, note that Pr{𝑗𝑡 = 𝑗 ∣ 𝒮,
ˆ 𝑈𝒥 } = 0 for 𝑗 ∈
ˆ Using the above facts and the optimality
𝑗 ∈ 𝒮∖{ˆ
/ 𝒮.
ˆ we have
of 𝒮,
]
[
∑
ˆ 𝑈𝒥 } 𝑟𝑗 + 𝑉𝑡+1 (𝑥𝑡 − ∑
𝑎
𝑒
∣
𝑈
)
−
𝑉
(𝑥
∣
𝑈
)
+ 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 )
𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ) =
Pr{𝑗𝑡 = 𝑗 ∣ 𝒮,
𝑡+1 𝑡
𝒥
𝒥
𝑖∈ℒ 𝑖𝑗 𝑖
𝑗∈𝒥
[
]
∑
= 𝑝𝚥ˆ𝑡 (𝑈𝒥 ) 𝑟𝚥ˆ + 𝑉𝑡+1 (𝑥𝑡 − 𝑖∈ℒ 𝑎𝑖ˆ𝚥 𝑒𝑖 ∣ 𝑈𝒥 ) − 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 ) + 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 )
]
[
∑
∑
=
𝑝𝑗𝑡 (𝑈𝒥 )ˆ
𝑦𝑗𝑡 𝑟𝑗 + 𝑉˜𝑡+1 (𝑥𝑡 − 𝑖∈ℒ 𝑎𝑖𝑗 𝑒𝑖 ∣ 𝑈𝒥 ) − 𝑉˜𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 ) + 𝑉˜𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 )
𝑗∈𝒥
≤ 𝑉˜𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ),
where the last equality follows from the induction assumption and the fact that 𝑦ˆ𝑗𝑡 = 0 for 𝑗 ∈ 𝒥 ∖{ˆ
𝚥}
and the inequality holds since 𝑦ˆ𝑡 is feasible for problem (10)-(13).
□
Proof of Proposition 3
We show the result by induction over the time periods. It is easy to show that the result holds at time
period 𝜏 . Assuming the result holds at time period 𝑡 + 1, we show that it holds at time period 𝑡. Using
the equivalent representation of 𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ) in Lemma 2 and letting 𝑦˜𝑡 = {˜
𝑦𝑗𝑡 : 𝑗 ∈ 𝒥 } be an optimal
17
solution to problem (10)-(13), we have
[
] [
]
∑
∑
∑
𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ) =
𝑝𝑗𝑡 (𝑈𝒥 )˜
𝑦𝑗𝑡 𝑟𝑗 + 𝑉𝑡+1 (𝑥𝑡 − 𝑖∈ℒ 𝑎𝑖𝑗 𝑒𝑖 ∣ 𝑈𝒥 ) + 1 − 𝑗∈𝒥 𝑝𝑗𝑡 (𝑈𝒥 )˜
𝑦𝑗𝑡 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 )
𝑗∈𝒥
≤
∑
[
] [
]
∑
∑
𝑝𝑗𝑡 (𝑈𝒥 )˜
𝑦𝑗𝑡 𝑟𝑗 + 𝑉𝑡+1 (𝑥𝑡 − 𝑖∈ℒ 𝑎𝑖𝑗 𝑒𝑖 ∣ 𝑈𝒥 , 𝜆) + 1 − 𝑗∈𝒥 𝑝𝑗𝑡 (𝑈𝒥 )˜
𝑦𝑗𝑡 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 , 𝜆)
𝑗∈𝒥
+
∑
𝜆𝑖𝑡 (𝑥𝑖𝑡 −
𝑖∈ℒ
∑
𝑎𝑖𝑗 𝑦˜𝑗𝑡 )
𝑗∈𝒥
≤ 𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 , 𝜆),
where the first inequality uses the induction assumption and the facts that since 𝑦˜𝑡 satisfies constraints
∑
∑
(11)-(13) and 𝜆 ≥ 0, we have [1 − 𝑗∈𝒥 𝑝𝑗𝑡 (𝑈𝒥 )˜
𝑦𝑗𝑡 ] ≥ 0 and 𝜆𝑖𝑡 (𝑥𝑖𝑡 − 𝑗∈𝒥 𝑎𝑖𝑗 𝑦˜𝑗𝑡 ) ≥ 0. The last
inequality holds since 𝑦˜𝑡 is a feasible solution to problem (14).
□
Proof of Lemma 4
We show the result by induction over the time periods. It is easy to show that the result holds at time
period 𝜏 . Assuming the result holds at time period 𝑡 + 1, we show that it holds at time period 𝑡. We
have
𝜏
}
{∑
∑
∑
∑
[
] ∑
𝑎𝑖𝑗 (
𝜆𝑖𝑠 ) +
𝜆𝑖𝑡 (𝑥𝑖𝑡 −
𝑎𝑖𝑗 𝑦𝑗𝑡 )
𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 , 𝜆) = max
𝑝𝑗𝑡 (𝑈𝒥 )𝑦𝑗𝑡 𝑟𝑗 −
𝑦𝑡 ∈𝒴
+
𝑦𝑡 ∈𝒴
=
𝜏
∑
𝑠=𝑡
𝑖∈ℒ
𝑗∈𝒥
𝜏
∑ ∑
Λ𝑠 +
(
𝜆𝑖𝑠 )𝑥𝑖𝑡
𝑠=𝑡+1
= max
𝑠=𝑡+1
𝑖∈ℒ
𝑗∈𝒥
𝜏
∑
𝑖∈ℒ 𝑠=𝑡+1
{∑
𝜏
𝜏
𝜏
(
)}
∑
∑
∑
∑∑
[
] ∑
𝜆𝑖𝑠 )𝑥𝑖𝑡
𝑦𝑗𝑡 𝑝𝑗𝑡 (𝑈𝒥 ) 𝑟𝑗 −
𝑎𝑖𝑗 (
𝜆𝑖𝑠 ) −
𝑎𝑖𝑗 𝜆𝑖𝑡
+
Λ𝑠 +
(
𝑗∈𝒥
Λ𝑠 +
𝑠=𝑡+1
𝑖∈ℒ
𝜏
∑
∑
(
𝑖∈ℒ
𝑠=𝑡+1
𝑖∈ℒ 𝑠=𝑡
𝜆𝑖𝑠 )𝑥𝑖𝑡 ,
𝑖∈ℒ 𝑠=𝑡
where the first
equality (follows from the induction assumption and the)}
last equality {uses the fact
{∑
] ∑
[
[
∑
∑𝜏
= max𝑗∈𝒥 𝑝𝑗𝑡 (𝑈𝒥 ) 𝑟𝑗 −
that max𝑦𝑡 ∈𝒴
𝑖∈ℒ 𝑎𝑖𝑗 𝜆𝑖𝑡
𝑗∈𝒥 𝑦𝑗𝑡 𝑝𝑗𝑡 (𝑈𝒥 ) 𝑟𝑗 −
𝑖∈ℒ 𝑎𝑖𝑗 ( 𝑠=𝑡+1 𝜆𝑖𝑠 ) −
}+
] ∑
∑𝜏
∑
𝑎
(
𝜆
)
−
𝑎
𝜆
= Λ𝑡 .
□
𝑖∈ℒ 𝑖𝑗 𝑖𝑡
𝑠=𝑡+1 𝑖𝑠
𝑖∈ℒ 𝑖𝑗
Proof of Proposition 5
We show the result by induction over the time periods. It is easy to show that the result holds at time
period 𝜏 . Assuming the result holds at time period 𝑡 + 1, we show that it holds at time period 𝑡. Letting
𝒮ˆ be an optimal solution for problem (9), we note that 𝒮ˆ is feasible for problem (20). We also note
that for a given offer set 𝒮, Pr{𝑗𝑡 = 𝑗 ∣ 𝒮, 𝑈 } is a function of {𝑈𝑗𝑡 : 𝑗 ∈ 𝒥 ∪ {𝜙}}, while 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈 ) is a
function of {𝑈𝑗𝑠 : 𝑗 ∈ 𝒥 ∪ {𝜙}, 𝑠 ∈ {𝑡 + 1, . . . , 𝒯 }}. Since the random variables {𝑈𝑗𝑡 : 𝑗 ∈ 𝒥 ∪ {𝜙}} are
independent across time, it follows that
{
}
{
} {
}
𝔼 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮, 𝑈 }𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈 ) ∣ 𝑈𝒥 = 𝔼 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮, 𝑈 } ∣ 𝑈𝒥 𝔼 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈 )∣ 𝑈𝒥 ,
(27)
18
where the expectation is with respect to 𝑈𝜙 and we recall that 𝑈 = 𝑈𝒥 ∪ 𝑈𝜙 . Therefore, we have
∑ {
{
}
}[
{
}]
ˆ 𝑈 } ∣ 𝑈𝒥 𝑟𝑗 + 𝔼 𝑉𝑡+1 (𝑥𝑡 − ∑
≥
𝔼 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮,
𝔼 𝑉𝑡 (𝑥𝑡 ∣ 𝑈 )∣𝑈𝒥
𝑖∈ℒ 𝑎𝑖𝑗 𝑒𝑖 ∣ 𝑈 ) ∣ 𝑈𝒥
𝑗∈𝒥
[
∑ {
}] {
}
ˆ 𝑈 }∣𝑈𝒥 𝔼 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈 ) ∣ 𝑈𝒥
+ 1−
𝔼 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮,
≥
∑
𝑗∈𝒥
[
]
ˆ 𝑈𝒥 } 𝑟𝑗 + 𝑉𝑡+1 (𝑥𝑡 − ∑
Pr{𝑗𝑡 = 𝑗 ∣ 𝒮,
𝑎
𝑒
∣
𝑈
)
𝑖𝑗
𝑖
𝒥
𝑖∈ℒ
𝑗∈𝒥
[
]
∑
ˆ 𝑈𝒥 } 𝑉𝑡+1 (𝑥𝑡 ∣ 𝑈𝒥 )
+ 1−
Pr{𝑗𝑡 = 𝑗 ∣ 𝒮,
𝑗∈𝒥
= 𝑉𝑡 (𝑥𝑡 ∣ 𝑈𝒥 ),
where the first inequality follows from (27) and the fact that 𝒮ˆ is a feasible but not necessarily optimal
solution to problem (20) and the second inequality follows from the induction assumption and the fact
that
{
{ {
}
}}
Pr{𝑗𝑡 = 𝑗 ∣ 𝒮, 𝑈𝒥 } = 𝔼 1(𝑈𝑗𝑡 > max {𝑈𝑘𝑡 })∣𝑈𝒥 = 𝔼 𝔼 1(𝑈𝑗𝑡 > max {𝑈𝑘𝑡 })∣𝑈
𝑘∈𝒮∪{𝜙}
𝑘∈𝒮∪{𝜙}
{
}
= 𝔼 Pr{𝑗𝑡 = 𝑗 ∣ 𝒮, 𝑈 } .
□
References
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19
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20
Figure 1: Structure of the airline network with a single hub and eight spokes.
Figure 2: Structure of the airline network with two hubs and eight spokes.
Problem
(𝑁, 𝛾, 𝛼)
(8, 1.5, 1.3)
(8, 1.5, 1.6)
(8, 3, 1.3)
(8, 3, 1.6)
(10, 1.5, 1.3)
(10, 1.5, 1.6)
(10, 3, 1.3)
(10, 3, 1.6)
(12, 1.5, 1.3)
(12, 1.5, 1.6)
(12, 3, 1.3)
(12, 3, 1.6)
Upper Bound
CR
PH
CDLP
2,225
2,448
2,829
2,205
2,350
2,646
5,744
5,915
6,469
5,293
5,343
5,502
4,472
4,723
5,082
4,242
4,357
4,597
10,281 10,414 10,927
9,125
9,172
9,243
5,744
5,828
5,996
5,018
5,052
5,096
11,470 11,585 11,992
10,033 10,087 10,193
% Gap
PH
9.99 ✓
6.57 ✓
2.99 ✓
0.95 ⊙
5.60 ✓
2.73 ✓
1.29 ⊙
0.51 ⊙
1.45 ✓
0.67 ⊙
1.01 ⊙
0.53 ⊙
with CR
CDLP
27.13 ✓
19.99 ✓
12.63 ✓
3.96 ✓
13.62 ✓
8.38 ✓
6.28 ✓
1.29 ✓
4.38 ✓
1.55 ✓
4.56 ✓
1.59 ✓
Table 1: Comparison of the upper bounds on the optimal expected total revenue for test problems on
an airline network with a single hub.
21
Problem
(𝑁, 𝛾, 𝛼)
(8, 1.5, 1.3)
(8, 1.5, 1.6)
(8, 3, 1.3)
(8, 3, 1.6)
(10, 1.5, 1.3)
(10, 1.5, 1.6)
(10, 3, 1.3)
(10, 3, 1.6)
(12, 1.5, 1.3)
(12, 1.5, 1.6)
(12, 3, 1.3)
(12, 3, 1.6)
Expected Revenues
CR
PH
CDLP
1,975
1,977
1,967
1,914
1,933
1,918
5,321
5,336
5,202
4,909
4,930
4,678
3,967
3,982
3,908
3,821
3,830
3,741
9,555
9,596
9,272
8,711
8,719
8,216
5,475
5,457
5,298
4,821
4,793
4,654
10,939 10,921 10,567
9,640
9,592
9,206
% Gap with CR
PH
CDLP
-0.10 ⊙
0.39 ⊙
-1.01 × -0.24 ⊙
-0.28 ⊙ 2.23 ✓
-0.43 ⊙ 4.71 ✓
-0.37 ⊙ 1.49 ✓
-0.24 ⊙ 2.11 ✓
-0.43 × 2.96 ✓
-0.09 ⊙ 5.68 ✓
0.32 ⊙
3.23 ✓
0.57 ✓
3.46 ✓
0.16 ⊙
3.41 ✓
0.50 ✓
4.50 ✓
Table 2: Comparison of the expected total revenues for the test problems on an airline network with a
single hub.
Problem
(𝑁, 𝛾, 𝛼)
(4, 1.5, 1.3)
(4, 1.5, 1.6)
(4, 3, 1.3)
(4, 3, 1.6)
(6, 1.5, 1.3)
(6, 1.5, 1.6)
(6, 3, 1.3)
(6, 3, 1.6)
(8, 1.5, 1.3)
(8, 1.5, 1.6)
(8, 3, 1.3)
(8, 3, 1.6)
Upper Bound
CR
PH
CDLP
2,485
2,667
3,003
2,270
2,367
2,589
5,128
5,446
6,003
4,742
4,836
5,321
4,120
4,261
4,646
3,592
3,664
3,861
8,738
8,860
9,469
7,504
7,566
7,769
5,778
5,983
6,275
4,999
5,103
5,311
12,663 12,741 13,114
10,594 10,648 10,772
% Gap
PH
7.29 ✓
4.28 ✓
6.20 ✓
1.99 ⊙
3.42 ✓
2.01 ✓
1.40 ⊙
0.83 ⊙
3.56 ✓
2.09 ✓
0.61 ⊙
0.51 ⊙
with CR
CDLP
20.84 ✓
14.05 ✓
17.05 ✓
12.21 ✓
12.76 ✓
7.49 ✓
8.36 ✓
3.53 ✓
8.61 ✓
6.25 ✓
3.56 ✓
1.67 ✓
Table 3: Comparison of the upper bounds on the optimal expected total revenue for the test problems
on an airline network with two hubs.
Problem
(𝑁, 𝛾, 𝛼)
(4, 1.5, 1.3)
(4, 1.5, 1.6)
(4, 3, 1.3)
(4, 3, 1.6)
(6, 1.5, 1.3)
(6, 1.5, 1.6)
(6, 3, 1.3)
(6, 3, 1.6)
(8, 1.5, 1.3)
(8, 1.5, 1.6)
(8, 3, 1.3)
(8, 3, 1.6)
Expected Revenues
CR
PH
CDLP
2,213
2,215
2,199
1,972
1,977
1,988
4,458
4,571
4,379
4,209
4,234
4,045
3,668
3,667
3,623
3,232
3,229
3,145
8,091
8,045
7,756
7,043
7,014
6,647
5,268
5,288
5,184
4,614
4,601
4,495
11,939 11,915 11,604
10,121 10,056
9,590
% Gap with CR
PH
CDLP
-0.06 ⊙
0.62 ⊙
-0.24 ⊙ -0.83 ⊙
-2.54 × 1.78 ✓
-0.59 ⊙ 3.91 ✓
0.03 ⊙
1.24 ✓
0.11 ⊙
2.69 ✓
0.57 ✓
4.14 ✓
0.42 ⊙
5.63 ✓
-0.38 ⊙ 1.60 ✓
0.28 ⊙
2.56 ✓
0.21 ⊙
2.81 ✓
0.64 ✓
5.24 ✓
Table 4: Comparison of the expected total revenues for the test problems on an airline network with
two hubs.
22