A strong Lagrangian relaxation for general discrete-choice
network revenue management
Sumit Kunnumkal∗
Kalyan Talluri†
May 28, 2015
Abstract
Discrete-choice network revenue management (DC-NRM) captures both customer behavior
and the resource-usage interaction of products and is appropriate for airline and hotel revenue
management, dynamic sales of bundles in advertising, and dynamic assortment optimization
in retail. The state-space of the DC-NRM stochastic dynamic program explodes and approximation methods such as the choice deterministic linear program (CDLP ), the affine, and the
piecewise-linear approximation have been proposed to approximate it in practice. The affine
relaxation (and thereby, its generalization, the piecewise-linear approximation) is intractable
even for the simplest choice models such as the multinomial logit (MNL) choice model with a
single segment. In this paper we propose a new Lagrangian relaxation method for DC-NRM
based on an extended set of multipliers. We show that our extended Lagrangian relaxation
obtains a bound that is as tight as the piecewise-linear bound, but with a complexity that is
an order of magnitude smaller than the piecewise-linear approximation. If we assume that the
consideration sets of the different customer segments are small in size—a reasonable modeling
tradeoff in many practical applications—our method is an indirect way to obtain the piecewiselinear approximation on large problems effectively. We show by numerical experiments that our
Lagrangian relaxation method can provide substantial improvements over existing benchmark
methods, both in terms of tighter upper bounds, as well as revenues from policies based on the
relaxation.
1
Introduction and literature review
In industries such as hotels, advertising, and airlines, the products consume bundles of different
resources (multi-night stays, bundles of advertising time-slots, multi-leg itineraries) and each product
has a specific price, set based on the customer segment it is aimed at, the market conditions, and
the product characteristics. The resources are perishable (for instance, for an airline, empty seats
at the moment of departure get no revenue, so the inventory “perishes” at departure) and the firm’s
revenue management function is to decide, at every point in time during the sale period, what
products to open for sale; the tradeoff being selling too much at too low a price early and running
out of capacity, or, rejecting too many low-valuation customers and ending up with excess unsold
∗ Indian
School of Business, Hyderabad, 500032, India, email: sumit [email protected]
and Universitat Pompeu Fabra, Ramon Trias Fargas 25-27, 08005 Barcelona, Spain, email:
[email protected]
† ICREA
1
inventory. Network revenue management (NRM) is control based on the demands for the entire
network. Chapter 3 of Talluri and van Ryzin [21] contains the necessary background on NRM.
NRM incorporating more realistic models of customer behavior as customers choosing from a
set of offered products have recently become popular, initiated in Talluri and van Ryzin [20] for the
single-resource problem. Discrete-choice models are appropriate as they are parsimonious, modeling
the probability of purchase as a function of product characteristics such as price and restrictions.
As customers arrive over time during the sale period (modeled usually as a Poisson process), the
optimization problem can be formulated as a stochastic dynamic program that we call discrete-choice
NRM (DC-NRM).
The DC-NRM dynamic program is computationally intractable and hence many approximation
methods have been proposed, starting with Gallego, Iyengar, Phillips, and Dubey [4] and Liu and
van Ryzin [12] who formulate the choice deterministic linear program (CDLP ) which replaces all the
random quantities by their expected values. Zhang and Adelman [24] propose an affine approximation to the value function, while Meissner and Strauss [13] propose a piecewise-linear approximation.
Kunnumkal and Topaloglu [11] use Lagrangian relaxation ideas to come up with a separable approximation. All of the above mentioned approximation methods obtain upper bounds on the value
function, with the piecewise-linear approximation obtaining the tightest upper bound.
Unfortunately, most of these approximation methods themselves are intractable, even for simple
choice models. Liu and van Ryzin [12] show that CDLP is tractable for the multinomial logit (MNL)
model provided the subset of products of interest to the different customer segments (consideration
sets) are disjoint. However, CDLP is NP-complete when the segment consideration sets overlap
even under the MNL model with two customer segments (Bront, Méndez-Dı́az, and Vulcano [3],
Rusmevichientong, Shmoys, Tong, and Topaloglu [15]). Kunnumkal and Talluri [8] show that the
affine approximation of Zhang and Adelman [24] is NP-hard for the MNL model with even a single
segment, which implies a similar hardness result for the piecewise-linear approximation as well.
These negative results show us the limits of tractability just for the MNL model, let alone general
discrete-choice models.
In this paper, we propose a new Lagrangian relaxation method for DC-NRM. We show that
our extended Lagrangian relaxation obtains a bound that is as tight as the piecewise-linear bound,
but with a complexity that is an order of magnitude smaller than the piecewise-linear approximation. The biggest practical impact of our work is in showing that the complexity of the Lagrangian
relaxation method scales linearly with the resource capacities, while that of the piecewise-linear approximation is exponential—yet they arrive at the same value function approximation. If we assume
that the consideration sets of the different customer segments are small in size—a reasonable modeling tradeoff in many practical applications as we argue next—our method is able to solve relatively
large problems, which would have been impossible attacking the piece-wise linear approximation
directly.
Since we work with general discrete-choice models, all the negative complexity results for the
MNL model carry over for the Lagrangian relaxation as well (as we show that our Lagrangian
relaxation is equivalent to the piece-wise linear relaxation). So naturally we have to make some
assumptions for the sake of tractability. We assume that the consideration sets of the different
segments are small in size. Small consideration sets can be justified in the airline setting where
a segment’s consideration set consists of choices (on one airline) for travel between an origin and
destination, and typically there are only a few alternatives on a given date (Talluri [17]). For hotels,
as the product consists of a multi-night stay and most customers arrive with a fixed duration of stay
in mind, the consideration set consists of the types of rooms and products, usually not a very large
2
number.
Research in the marketing area also gives evidence supporting that customers have relatively
small consideration sets: Hauser and Wernerfelt [6] report average consideration set sizes of 3 brands
for deodorants, 4 brands for shampoos, 2.2 brands for air fresheners, 4 brands for laundry detergents
and 4 brands for coffees. (Note that the study is for brands rather than choices of sizes or colors.)
Another line of marketing research finds great value in deliberately limiting customer choices to a
small number (Iyengar and Lepper [7]). Assuming small consideration set sizes, Talluri [19] and
Meissner, Strauss, and Talluri [14] study tractable approximations to CDLP for DC-NRM. While
our segment-based relaxation has the same underlying motivation, the development is quite different
as we work with the Lagrangian relaxation of the DC-NRM dynamic program.
Our final contribution in this paper is showing by numerical experiments that our Lagrangian
relaxation method can provide substantial improvements over existing methods.
Our work builds on previous research on Lagrangian relaxations for NRM. Topaloglu [22] was
the first paper to propose a time-based Lagrangian relaxation approach for the perfect segmentation
case (also called the independent-class assumption). The Lagrangian relaxation of Topaloglu [22]
associates Lagrange multipliers with each product in each time period, and Kunnumkal and Talluri
[10] show that it obtains an upper bound that coincides with the piecewise-linear bound. In contrast,
for DC-NRM, we give an example which illustrates that the approach proposed by Topaloglu [22]
can be weaker than the piecewise-linear approximation. This motivates our approach that associates
Lagrange multipliers with every offer set, and obtains a bound that is as tight as the piecewise-linear
bound.
To summarize, we make the following research contributions in this paper:
1. We propose a new Lagrangian relaxation method for DC-NRM based on an extended set of
Lagrange multipliers.
2. We show that our Lagrangian relaxation obtains an upper bound on the value function as
tight as the piecewise-linear bound. The result holds for a general discrete-choice model and
is significant because the complexity of the Lagrangian relaxation scales linearly with the
resource capacities, while the piecewise-linear approximation is exponential. The reduction in
complexity can therefore be substantial. This result also implies that the Lagrangian relaxation
bound is stronger than the CDLP and the affine approximation bounds as well. Another
appealing feature of our approach is that it decomposes the network problem into a number
of single resource problems, which can be potentially solved in a distributed fashion.
3. We show how our ideas lead to a tractable method for DC-NRM (for any discrete-choice model)
when the consideration sets of the different segments are small in size.
4. We extract policies from our formulations and show by numerical experiments that they can
lead to substantial improvements over existing methods.
The remainder of the paper is organized as follows: In §2 we describe the DC-NRM model, the
notation, and the stochastic dynamic programming formulation of the problem. §3 reviews the value
function approximation methods in the literature including CDLP , the affine and the piecewiselinear approximation methods. In §4 we describe the Lagrangian relaxation approach and show
that it obtains a bound that is as tight as the piecewise-linear bound; we defer the formal proof to
§5. §6 describes the segment-based Lagrangian relaxation that is tractable under the assumption of
small consideration sets. In §7 we perform numerical experiments to compare the performance of
the Lagrangian relaxation approach with benchmark solution methods.
3
2
Problem formulation
A product is a specification of a price and the set of resources that it consumes. For example, for an
airline a product is an itinerary-fare class combination, where an itinerary is a combination of flight
legs making up a passenger’s journey; for a hotel, a product is a multi-night stay for a particular
room type at a certain price point.
Time is discrete and assumed to consist of τ intervals, indexed by t. The booking horizon begins
at time t = 1 and ends at t = τ ; all the resources perish instantaneously at time τ + 1. We make the
standard assumption that the time intervals are fine enough so that the probability of more than
one customer arriving in any single time period is negligible. We index resources by i, products by
j, and time periods by t. We refer to the set of all resources as I and the set of all products as J .
We let fj denote the revenue associated with product j and Ij ⊂ I the subset of resources it uses.
Similarly, we let Ji ⊂ J denote the subset of products that use resource i.
We use superscripts on vectors to index the vectors and subscripts to indicate their components.
t
t
For example, we write the resource capacity vector associated with time period
t as r and use ri
to represent the capacity on resource i in time period t. Therefore, r 1 = ri1 represents the initial
capacities on the resources and r t = [rit ] denotes the remaining capacities on the resources
at
the
t
1
and
takes
values
in
the
set
R
=
0,
.
.
.
,
r
beginning
of
time
period
t.
The
remaining
capacity
r
i
i
i
R = i Ri represents the state space.
2.1
Demand model
In each time period the firm offers a subset S of its products for sale, called the offer set. A customer
arrives with probability α and given an offer set S, an arriving customer purchases a product j in
the set S or decides not to purchase. The no-purchase option is indexed by 0 and is always present
for the customer. We let Pj (S) denote the probability that the firm sells product j given that a
customer arrives and the offer set is S. Clearly, Pj (S) = 0 if j ∈
/ S. The probability of no sale
given a customer arrival is P0 (S) = 1 − j∈S Pj (S). We assume that the choice probabilities are
given by an oracle, as the model represents a general discrete-choice model; they could conceivably
be calculated by a simple formula as in the case of the multinomial logit (MNL) model (Ben-Akiva
and Lerman [1]).
2.2
DC-NRM dynamic program
The dynamic program (DP) to determine optimal controls is as follows: Let Vt (r t ) denote the
maximum expected revenue to go, given remaining capacity r t at the beginning of period t. Then
Vt (r t ) must satisfy the Bellman equation
⎧
⎫
⎨
⎬
Vt (r t ) = maxt
αPj (S) fj + Vt+1 r t −
ei + αP0 (S) + 1 − α Vt+1 r t
, (1)
⎭
S⊂Q(r ) ⎩
j∈S
i∈Ij
where ei is a vector with a 1 in the ith position and 0 elsewhere and
Q(r) = j | ½[j∈Ji ] ≤ ri , ∀i
represents the set of products that can be offered given the capacity vector r. We use ½[·] as the
indicator function, 1 if true and 0 if false. The boundary conditions are Vτ +1 (r) = Vt (0) = 0 for all
4
r and for all t, where 0 is a vector of all zeroes. V DP = V1 (r 1 ) denotes the optimal expected total
revenue over the booking horizon, given the initial capacity vector r 1 .
For brevity of notation, we assume that α = 1 in the remaining part of the paper. We note that
this is without loss of generality since this is equivalent to letting P̃j (S)
= αPj (S) and P̃0 (S) =
αP0 (S) + 1 − α, and working with the choice probabilities P̃j (S) | ∀j, S .
2.3
Linear programming formulation of the DC-NRM dynamic program
The value functions can, alternatively, be obtained by solving a linear program, following Schweitzer
and Seidmann [16]. The linear programming formulation of the DC-NRM dynamic program has a
decision variable Vt (r) for each state vector r in each period t and is as follows:
V DP =
min V1 (r 1 )
V
s.t.
(DP )
Vt (r) ≥
Pj (S) fj + Vt+1 r −
ei − Vt+1 (r) + Vt+1 (r)
j
(2)
i∈Ij
∀r ∈ R, S ⊂ Q(r), t
Vt (r) ≥ 0
with the boundary conditions that Vτ +1 (r) = Vt (0) = 0 for all r and for all t. Both the dynamic
program (1) and linear program DP are computationally intractable, but the linear program DP
turns out to be useful in developing value function approximation methods.
3
Value function approximation methods
In this section, we describe three methods in the literature to approximate the DC-NRM dynamic
program value function. We begin with the choice-based deterministic linear program and then
outline the affine and the piecewise-linear approximations.
3.1
Choice-based deterministic linear program (CDLP )
The choice-based deterministic linear program CDLP ), proposed in Gallego et al. [4] and Liu and
van Ryzin [12] is given by
max
[
Pj (S)fj ]ρS,t
V CDLP =
ρ
s.t.
(CDLP )
t
S
j∈S
t
S
Pj (S) ρS,t ≤ ri1
∀i
(3)
j∈Ji
ρS,t = 1
∀t
S
ρS,t ≥ 0 ∀t, S.
In the above linear program, we interpret the decision variable ρS,t as the frequency with which set
S is offered at time period t. The objective function captures the total expected revenue, while the
5
first set of constraints ensure that the total expected capacity consumed on each resource does not
exceed its available capacity. The second set of constraints ensures that the total frequencies add
up to 1 at each time period.
Liu and van Ryzin [12] show that the optimal objective function value of the CDLP gives an
upper bound on the optimal expected revenue. That is, V1 (r 1 ) ≤ V CDLP . Since CDLP has an
exponential number of decision variables it has to be solved using column generation. Liu and
van Ryzin [12] show that column generation is efficient for the MNL model with a single segment.
However, the column generation procedure is intractable in general, NP-complete for MNL with just
two segments (Bront et al. [3] and Rusmevichientong et al. [15]).
3.2
Affine approximation
The affine approximation, proposed
by Zhang and Adelman [24], replaces the value function Vt (r)
with the affine function θt + i Vi,t ri in the linear program DP to obtain the following linear
program:
V AF =
min θ1 +
Vi,1 ri1
θ,V
s.t.
θt +
(AF )
i
Vi,t ri ≥
i
Pj (S) fj −
Vi,t+1 + θt+1 +
Vi,t+1 ri
j
i∈Ij
i
∀r ∈ R, S ⊂ Q(r), t
θt , Vi,t ≥ 0.
Zhang and Adelman [24] show that the optimal objective function value of AF gives an upper
bound on the optimal expected revenue and this bound is tighter than the CDLP bound. While the
number of decision variables in AF is manageable, the number of constraints is exponential both in
the number of resources as well as the products. Vossen and Zhang [23] show that AF has a reduced
formulation where the number of constraints is exponential only in the number of products. Still,
the problem has to be solved by constraint generation and the separation problem is intractable
even for the MNL choice model with a single segment (Kunnumkal and Talluri [8]).
3.3
Piecewise-linear approximation
Meissner and Strauss[13] propose the piecewise-linear approximation, which approximates the value
function Vt (r) with i vi,t (ri ) in DP . The resulting linear program is
V PL =
min
vi,1 (ri1 )
v
s.t.
(P L)
i
vi,t (ri ) ≥
i
j
Pj (S) fj +
(vi,t+1 (ri − 1) − vi,t+1 (ri )) +
vi,t+1 (ri )
(4)
i
i∈Ij
∀r ∈ R, S ⊂ Q(r), t
vi,τ +1 (·) = 0.
Since the piecewise-linear approximation uses a more refined approximation architecture than the
affine approximation, it is natural to expect that it obtains a tighter bound on the value function.
6
Indeed, Meissner and Strauss [13] show V DP ≤ V P L ≤ V AF ≤ V CDLP . However, as with CDLP
and AF , tractability remains
as much, if not more, of an issue for P L as well. Note that linear
program P L has O(2|J | τ i ri1 ) constraints, which is exponential both in the number of products
and the number of resources. Moreover, the separation problem of this linear program is NPcomplete, even for MNL with a single segment; see Kunnumkal and Talluri [8].
4
Lagrangian relaxation using offer-set specific multipliers
In this section, we present our approximation method to obtain an upper bound on the value function
of the DC-NRM dynamic program. We first describe our approach and then show that it obtains
an upper bound that is as tight as that obtained by P L.
We begin by introducing some notation. For an offer set S, we let
IS = {i ∈ I | ∃j ∈ S with j ∈ Ji } ,
the set of resources used by the products in S. We follow the convention that the empty set does
not use any resource. We also let Ci = {S | i ∈ IS } and note that i ∈ IS if and only if S ∈ Ci .
Optimality equation (1) can be equivalently written, in an expanded form, as
i
r
−
−
V
r
+ Vt+1 r
h
P
(S)
f
+
V
e
Vt (r) = max
φ,S,t
j
j
t+1
t+1
S
j∈S
i∈Ij
h
s.t.
hi,S,t = hφ,S,t ∀S, i ∈ IS
(5)
hi,S,t ≤ ri ∀i, S ∈ Ci
hφ,S,t ≤ 1
S
S∈Ci hi,S,t ≤ 1 ∀i
(6)
(7)
(8)
hφ,S,t , hi,S,t ≥ 0 ∀S, i ∈ IS
(9)
where the decision variable hφ,S,t can be interpreted as the frequency with which set S is offered
at time period t. In the expanded formulation, we also have resource level decision variables hi,S,t ,
which can be interpreted as the frequency with which set S is offered on resource i at time period t.
Constraints (5) ensure that for each set S, the frequencies are identical across the resources used by
it, while constraints (6) ensure that a set is offered only if there is sufficient capacity on each resource
that it uses. Constraints (7)-(9) ensure that the frequencies are nonnegative and add up to at most
1. We note that although the expanded formulation has a number of redundant decision variables
and constraints, they turn out to be useful in the development of our approximation method below.
The expanded formulation still has a high dimensional state space and remains intractable. Our
approximation method reduces the complexity by solving a number of single dimensional dynamic
programs, one for each resource. We relax constraints (5) and associate Lagrange multipliers λi,S,t
with them and solve the optimality equation
λ
λ
λ
ϑi,t (ri ) = max
S∈Ci hi,S,t λi,S,t +
j∈Ji Pj (S) ϑi,t+1 (ri − 1) − ϑi,t+1 (ri )
h
+ϑλi,t+1 (ri )
s.t.
(6), (8), hi,S,t ≥ 0 ∀S ∈ Ci
7
(10)
for resource i with the boundary condition that ϑλi,τ +1 (·) = 0. Collecting the terms involving hφ,S,t ,
we solve the optimality equation
λ
(11)
ϑφ,t = max
+ ϑλφ,t+1
S hφ,S,t
j∈S Pj (S)fj −
i∈IS λi,S,t
h
s.t.
(7), hφ,S,t ≥ 0 ∀S
with the boundary condition that ϑλφ,τ +1 = 0. We use
Vtλ (r) =
ϑλi,t (ri ) + ϑλφ,t
(12)
i
as our value function approximation, where the superscript emphasizes the dependence of the value
function approximation on the Lagrange multipliers.
Lemma 1 below shows that the Lagrangian relaxation using offer-set specific multipliers (LRo)
obtains an upper bound on the optimal expected revenue and that this bound is potentially weaker
than the piecewise-linear bound.
Lemma 1.
V DP ≤ V P L ≤ V1λ (r 1 ).
Proof. Appendix.
We obtain the tightest upper bound on the optimal expected revenue by solving
V LRo = min V1λ r 1 .
λ
The following proposition gives the subgradients of ϑλi,t (ri ) and ϑλφ,t (thereby also showing they
are convex functions of λ). It follows that Vtλ (r) is also a convex function of λ and hence the above
optimization problem can be efficiently solved as a convex program.
We introduce some notation for this purpose. Let hλi,S,t (ri ) denote an optimal solution to problem (10) where the arguments emphasize the dependence of the optimal solution on the Lagrange
multipliers and the remaining capacity on the resource. We define hλφ,S,t in a similar manner for
λ
(11). Also, let Xi,t
denote the random variable which represents the remaining capacity on resource
i at time period t when we offer sets according to the optimal solution to problem (10). We have
the following result.
Proposition 1. Let λ and λ̂ be twosets of Lagrange multipliers. Then, ri
τ λ
λ
λ
1. ϑλi,t (ri ) ≥ ϑλ̂i,t (ri ) + k=t S∈Ci
λ̂i,S,k − λi,S,k .
x=0 Pr{Xi,k = x | Xi,t = ri }hi,S,t (x)
τ 2. ϑλφ,t ≥ ϑλ̂φ,t + k=t i S∈Ci hλφ,S,k [λ̂φ,S,k − λφ,S,k ].
Proof. Appendix.
We note that besides showing that V1λ (r) is a convex function of λ, Proposition 1 also gives an
explicit expression for its subgradient. This allows us to use subgradient search to find the optimal
set of Lagrange multipliers (Bertsekas [2]). Proposition 2 below shows that by doing this, we in fact
obtain the piecewise-linear bound.
8
Proposition 2.
V P L = V LRo .
We defer the proof of Proposition 2 to the next section. Here we note some of its implications.
Proposition 2 together with the results in Meissner and Strauss [13] implies that V LRo ≤ V CDLP
and V LRo ≤ V AF . Therefore, LRo obtains an upper bound that is tighter than both CDLP and
AF . It is also quite surprising that the
complexity of LRo
LRo bound is as tight as P L since the
is linear in the resource capacities ( i ri1 ), while that of P L is exponential ( i ri1 ). For typical
DC-NRM instances, the complexity of LRo can be orders of magnitude smaller than that of P L.
Moreover, since LRo decomposes the network problem into a number of single resource problems
that can be solved in parallel, it may also be more suitable for distributed computing techniques.
In closing this section, we note that while Lagrangian relaxation ideas have been applied to
NRM previously, there are some important differences between our approach and previous proposals.
Under the independent-class assumption, Topaloglu [22] proposes relaxing the constraints that the
same acceptance decisions be made for each product across the resources that it uses. The resulting
Lagrangian relaxation associates a multiplier λi,j,t for each product j and each resource i ∈ Ji .
Kunnumkal and Talluri [10] show that the Lagrangian relaxation with product-specific multipliers
(LRp) turns out be equivalent to P L in the independent-class setting. We give an example in
the Appendix which illustrates that the same result fails to hold for DC-NRM. That is, for DCNRM, LRp can be weaker than P L (and by Proposition 2, weaker
than LRo as well). Moreover,
Talluri [18] shows that the optimal multipliers in LRp satisfy i∈Ij λi,j,t = fj . This precludes the
need to solve an optimality equation of the form (11) for LRp. On the other hand, we give an
example
in the
Appendix which illustrates that the optimal multipliers in LRo can be such that
λ
<
i∈IS i,S,t
j∈S Pj (S)fj . Therefore, unlike in the independent-class case, optimality equation
(11) is not redundant for DC-NRM.
5
Tightness of the Lagrangian bound
In this section we formally show that the upper bound obtained by LRo is as strong as the P L bound.
We begin with some preliminary results in §5.1 before proceeding to the proof of Proposition 2 in
§5.2.
5.1
Preliminaries
We note that optimality equation (10) can be equivalently written as
½[S∈Ci ] λi,S,t +
Pj (S) ϑλi,t+1 (ri − 1) − ϑλi,t+1 (ri ) + ϑλi,t+1 (ri )
ϑλi,t (ri ) = max
S⊂Qi (ri )
(13)
j∈Ji
where Q
i (ri ) = j | ½[j∈Ji ] ≤ ri . On the other hand, letting R(S) =
j∈S Pj (S)fj and λφ,S,t =
R(S) − i∈IS λi,S,t , we can write optimality equation (11) equivalently as
ϑλφ,t = max λφ,S,t + ϑλφ,t+1 .
S⊂J
9
(14)
If we let λ = {λφ,S,t , λi,S,t |∀t, S, i ∈ IS } and
λi,S,t = R(S), ∀S ⊂ J , t ,
Λ = λ | λφ,S,t +
(15)
i∈IS
then we can write V LRo = minλ∈Λ V1λ (r 1 ). The above minimization problem can be represented as
the linear program
V LRo = min
vi,1 (ri1 ) + vφ,1
λ,v
s.t.
(LRo)
i
vi,t (ri ) ≥ ½[S∈Ci ] λi,S,t +
Pj (S) [vi,t+1 (ri − 1) − vi,t+1 (ri )]
j∈Ji
+vi,t+1 (ri ) ∀t, i, ri ∈ Ri , S ⊂ Qi (ri )
vφ,t ≥ λφ,S,t + vφ,t+1 ∀t, S
λi,S,t + λφ,S,t = R(S) ∀t, S
i∈IS
vi,τ +1 (·) = 0, vφ,τ +1 = 0.
Linear program LRo turns out to be useful when we make the connection between the LRo and P L
bounds.
5.2
Proof of Proposition 2
We give a formal proof of Proposition 2; we include a heuristic, but simpler, calculus-based explanation in the Appendix. Lemma 1 implies that V P L ≤ minλ∈Λ V1λ (r 1 ) = V LRo . So it only remains
to show the more difficult part: V P L ≥ V LRo . Let ψi,t+1 (ri ) = vi,t+1 (ri ) − vi,t+1 (ri − 1) and
Δi,t (ri ) = vi,t+1 (ri ) − vi,t (ri ). The proof proceeds by considering the P L separation problem (27),
which can be written as
⎡
⎤
max
Pj (S) ⎣fj −
ψi,t+1 (ri )⎦ +
Δi,t (ri ).
Φt (v) =
r∈R,S⊂Q(r)
j
i∈Ij
i
/ S, and ½[S∈Ci ] = 1 for j ∈ S
Using R(S) = j∈S Pj (S)fj and the facts that Pj (S) = 0 for j ∈
and i ∈ Ij , we can write Φt (v) as
max
R(S) −
Pj (S)
½[S∈Ci ] ψi,t+1 (ri ) +
Δi,t (ri ).
(16)
Φt (v) =
r∈R,S⊂Q(r)
j∈S
i∈Ij
i
Now consider the following Lagrangian relaxation of Φt (v), where for each time period t, we use
offer set-specific multipliers to allocate the revenue associated with each offer set across the different
resources and solve the optimization problem
⎧
⎧
⎫
⎫
⎨
⎨
⎬
⎬
max
½[S∈Ci ] λi,S,t −
Pj (S)ψi,t+1 (ri ) + Δi,t (ri ) + max{λφ,S,t }
Πt (v) = min
λ∈Λ ⎩
⎭ S⊂J
⎭
ri ∈Ri ,S⊂Qi (ri ) ⎩
i
j∈Ji
We show below that Φt (v) = Πt (v) and use this result to show that V P L ≥ V LRo . We begin with
some preliminary results.
10
Lemma 2. Φt (v) ≤ Πt (v).
Proof. The proof is similar to that of Lemma 1. Let r ∗ , S ∗ be an optimal solution to Φt (v). Since
S ∗ ⊂ Q(r ∗ ) ⊂ Qi (ri∗ ), we have
⎧
⎫
⎨
⎬
max
½[S∈Ci ] λi,S,t −
Pj (S)ψi,t+1 (ri ) + Δi,t (ri ) ≥ ½[S ∗ ∈Ci ] λi,S ∗ ,t
⎭
ri ∈Ri ,S⊂Qi (ri ) ⎩
j∈Ji
−
Pj (S ∗ )ψi,t+1 (ri∗ ) + Δi,t (ri∗ ).
j∈Ji
Therefore, for any λ ∈ Λ, we have
⎧
⎫
⎨
⎬
max
½[S∈Ci ] λi,S,t −
Pj (S)ψi,t+1 (ri ) + Δi,t (ri ) + max {λφ,S,t }
⎭ S⊂J
ri ∈Ri ,S⊂Qi (ri ) ⎩
i
j∈Ji
⎡
⎤
⎣½[S ∗ ∈Ci ] λi,S ∗ ,t −
≥
Pj (S ∗ )ψi,t+1 (ri∗ ) + Δi,t (ri∗ )⎦ + λφ,S ∗ ,t
i
=
j∈Ji
λi,S ∗ ,t + λφ,S ∗ ,t −
i∈IS ∗
= R(S ∗ ) −
Pj (S ∗ )
j
Pj (S ∗ )
j
ψi,t+1 (ri∗ ) +
ψi,t+1 (ri∗ ) +
Δi,t (ri∗ )
i
i∈Ij
Δi,t (ri∗ ) = Φt (v).
i
i∈Ij
It follows that Φt (v) ≤ Πt (v).
It remains to show that Φt (v) ≥ Πt (v). The following lemma shows that we can restrict ourselves
to sets in Ci = {S | i ∈ IS } when solving the optimization problem for resource i.
Lemma 3.
max
⎧
⎨
ri ∈Ri ,S⊂Qi (ri ) ⎩
½[S∈Ci ] λi,S,t −
⎧
⎨
= max
⎩
j∈Ji
Δi,t (0),
⎫
⎬
Pj (S)ψi,t+1 (ri ) + Δi,t (ri )
⎭
max
ri ∈{1,...,ri1 },S∈Ci
{λi,S,t −
j∈Ji
Pj (S)ψi,t+1 (ri ) + Δi,t (ri )}
⎫
⎬
⎭
.
/ Ji for all j ∈ S and we have S ∈
/ Ci . Therefore,
Proof. Note that if S ⊂ Q
i (0), then j ∈
maxS⊂Qi (0) {½[S∈Ci ] λi,S,t − j∈Ji Pj (S)ψi,t+1 (0) + Δi,t (0)} = Δi,t (0).
11
For ri ∈ {1, . . . , ri1 }, Qi (ri ) = J = Ci ∪ Cic . We have
⎫
⎧
⎬
⎨
max ½[S∈Ci ] λi,S,t −
Pj (S)ψi,t+1 (ri ) + Δi,t (ri )
⎭
S⊂J ⎩
j∈Ji
⎧
⎧
⎫
⎫
⎨
⎨
⎬
⎬
= max max λi,S,t −
Pj (S)ψi,t+1 (ri ) + Δi,t (ri ) , maxc {Δi,t (ri )}
⎩S∈Ci ⎩
⎭ S∈Ci
⎭
j∈Ji
⎧
⎫
⎫
⎧
⎨
⎬
⎬
⎨
Pj (S)ψi,t+1 (ri ) + Δi,t (ri ) , Δi,t (ri ) .
= max max λi,S,t −
⎭
⎭
⎩S∈Ci ⎩
j∈Ji
Putting everything together
max
ri ∈Ri ,S⊂Qi (ri )
{½[S∈Ci ] λi,S,t −
Pj (S)ψi,t+1 (ri ) + Δi,t (ri )}
j∈Ji
⎧
⎨
⎫
⎫⎫
⎬
⎬⎬
= max Δi,t (0), max 1
Pj (S)ψi,t+1 (ri ) + Δi,t (ri ) , Δi,t (ri )
max λi,S,t −
⎩
⎭
⎭⎭
ri ∈{1,...,ri } ⎩ S∈Ci ⎩
j∈Ji
⎧
⎫⎫
⎧
⎨
⎬⎬
⎨
λ
max
−
P
(S)ψ
(r
)
+
Δ
(r
)
= max Δi,t (0),
i,S,t
j
i,t+1
i
i,t
i
⎭⎭
⎩
ri ∈{1,...,ri1 },S∈Ci ⎩
⎧
⎨
⎧
⎨
j∈Ji
where the last equality follows from Δi,t (·) being a decreasing function. We note that Δi,t (ri ) ≤
Δi,t (ri − 1) is equivalent to vi,t (ri ) − vi,t (ri − 1) ≥ vi,t+1 (ri ) − vi,t+1 (ri − 1), and the latter inequality
follows Kunnumkal and Talluri [9]. Therefore Δi,t (0) dominates Δi,t (ri ). Note that we do not make
an assumption that vi,t (0) = 0 for any t, and consequently cannot immediately assume Δi,t (0) =
0 ≥ Δi,t (ri ).
Lemma 3 implies that we can write Πt (v) as the linear program
wφ,t +
wi,t
Πt (v) = min
λ,w
i
s.t.
(LPΠt (v) )
wi,t ≥ Δi,t (0) ∀i
wi,t ≥ λi,S,t −
Pj (S)ψi,t+1 (r) + Δi,t (r)
(17)
∀i, r ∈
{1, . . . , ri1 }, S
∈ Ci (18)
j∈Ji
wφ,t ≥ λφ,S,t ∀S
wφ,t ≥ 0
λφ,S,t +
λi,S,t = R(S) ∀S.
(19)
(20)
(21)
i∈IS
In the following, we analyze the structure of a particular optimal solution to LPΠt (v) that allows
us to construct a feasible solution to Φt (v), which in turn shows that Φt (v) ≥ Πt (v). We introduce
some notation that will be useful for this purpose. Given a solution (λ, w) to LPΠt (v) , let
⎡
⎤
Pj (S)ψi,t+1 (r) + Δi,t (r)⎦
ξi,S,t (r) = wi,t − ⎣λi,S,t −
j∈Ji
12
denote the slack in constraint (18) for resource i, offer set S ∈ Ci and r ∈ {1, . . . , ri1 }. Let
Bi,S (λ, w) = r ∈ {1, . . . , ri1 } | ξi,S,t (r) = 0
denote the set of capacity levels for which constraint (18) is binding for resource i and offer set S.
We use the arguments (λ, w) emphasize the dependence of the set of binding constraints on the given
solution. Observe that if Bi,S (λ, w) is empty, then wi,t > λi,S,t − j∈Ji Pj (S)ψi,t+1 (r) + Δi,t (r) for
all r ∈ {1, . . . , ri,1 }.
From now on, we use a concept of optimal solutions with minimal set of binding constraints.
The linear program (LPΠt (v) ) has a finite optimal solution and possibly multiple ones. Naturally,
any optimal solution has a set of binding constraints out of (17)–(21). Given any optimal solution,
we can look for another optimal solution whose set of binding constraints is a strict subset of those
of the previous optimal solution. If there is no such optimal solution, we consider that as having a
minimal set of binding constraints. We have the following lemma.
Lemma 4. Let (λ̂, ŵ) be an optimal solution to LPΠt (v) with a minimal number of binding constraints. Fix a set S. Either
(i) Bi,S (λ̂, ŵ) is nonempty for all i ∈ IS and ŵφ,t = λ̂φ,S,t , or
(ii) Bi,S (λ̂, ŵ) is empty for all i ∈ IS and ŵφ,t > λ̂φ,S,t .
Proof. Suppose that the statement of the lemma is false. First consider the case where Bi,S (λ̂, ŵ)
w)
is nonempty but Bl,S (λ̂, ŵ) is empty for i, l ∈ IS . Let = minr∈{1,...,rl1 } {ξl,S,t (r)} > 0. Let (λ,
i,S,t
l,S,t
i,j,k
be given by λ = λ̂ − δe
+ δe
and w
= ŵ for some δ ∈ (0, ), where e
is a vector with a 1
w)
in component (i, j, k) and zeroes everywhere else. Note that (λ,
is identical to (λ̂, ŵ) except that
i,S,t = λ̂i,S,t − δ and λ
l,S,t = λ̂l,S,t + δ. We show that (λ,
w)
λ
is an optimal solution with a strictly
fewer number of binding constraints which gives us a contradiction.
Notice that we only need to check constraint (18) for resources i and l and offer set S as all
other resources and offer sets continue to have the same λ’s and w’s as before.
For resource i, since
Bi,S (λ̂, ŵ) is nonempty, there exists r ∈ {1, . . . , ri1 } such that ŵi,t = λ̂i,S,t − j∈Ji Pj (S)ψi,t+1 (r) +
Δi,t (r). We have
i,S,t −
Pj (S)ψi,t+1 (r) + Δi,t (r) > λ
Pj (S)ψi,t+1 (r) + Δi,t (r),
w
i,t = ŵi,t = λ̂i,S,t −
j∈Ji
j∈Ji
which means that the number of binding constraints (18) for resource i and offer set S decreases
l,S,t − by at least one. For resource l and offer set S, w
l,t − [λ
j∈Jl Pj (S)ψl,t+1 (r) + Δl,t (r)] =
ŵl,t − [λ̂l,S,t − j∈Jl Pj (S)ψl,t+1 (r) + Δl,t (r)] − δ > 0, for all r ∈ {1, . . . , rl1 }, where the inequality
follows from the definition of δ. Therefore, all constraints (18) continue to be nonbinding for resource
w)
l and offer set S. Overall, (λ,
has strictly fewer binding constraints than (λ̂, ŵ), which gives a
contradiction.
The above arguments imply that either Bi,S (λ̂, ŵ) is nonempty for all i ∈ IS or Bi,S (λ̂, ŵ) is
empty for all i ∈ IS . Suppose the Bi,S (λ̂, ŵ) is nonempty for all i ∈ IS but ŵφ,t > λ̂φ,St . In this
= λ̂ − δei,S,t + δeφ,S,t and w
case, pick a resource i ∈ IS and let λ
= ŵ, where δ ∈ (0, ) and
= ŵφ,t − λ̂φ,S,t . It can be verified that the number of binding constraints (18) for resource i and
w),
offer set S strictly decreases from (λ̂, ŵ) to (λ,
while the number of binding constraints (19)
remains unchanged, leading to a contradiction. This proves part (i) of the lemma.
13
On the other hand, if Bi,S (λ̂, ŵ) is empty for all i ∈ IS but ŵφ,t = λ̂φ,S,t . Pick a resource i ∈ IS .
= λ̂ + δei,S,t − δeφ,S,t ,
Since Bi,S (λ̂, ŵ) is empty, we have = minr∈{1,...,ri1 } {ξi,S,t (r)} > 0. Let λ
where δ ∈ (0, ), and w
= ŵ. It can be verified that the number of binding constraints (18) for
w),
resource i and offer set S is exactly the same for (λ̂, ŵ) and (λ,
while the number of binding
w)
constraints (19) decreases by one (w
φ,t = ŵφ,t = λ̂φ,S,t > λφ,S,t ). As a result, (λ,
has strictly fewer
binding constraints than (λ̂, ŵ), which gives a contradiction. This proves part (ii) of the lemma.
In the remainder, we focus on an optimal solution to LPΠt (v) with a minimal number of binding
constraints. Letting (λ̂, ŵ) denote such a solution, we define
Cˆi = S ∈ Ci | Bi,S (λ̂, ŵ) is nonempty ,
Cˆφ = S ⊂ J | ŵφ,t = λ̂φ,S,t ,
and
Î + = {i | ŵi,t > Δi,t (0)} ,
where theˆon the sets is to remind the reader of the dependence on (λ̂, ŵ).
Lemma 5. Let (λ̂, ŵ) be an optimal solution to LPΠt (v) with a minimal number of binding constraints.
(i) Cˆi is nonempty for all i ∈ Î + .
(ii) If S ∈ Cˆi , then S ∈ Cˆφ (note that by definition the empty set does not consume any resources
and so ∅ ∈
/ Cˆi ).
Proof. For part (i), for i ∈ Î + , ŵi,t > Δi,t (0). Since (λ̂, ŵ) is optimal there exists S ∈ Ci and
r ∈ 1, . . . , ri1 such that ŵi,t = λ̂i,S,t − j∈S Pj (S)ψi,t+1 (r) + Δi,t (r) > Δi,t (0) (otherwise, we can
reduce ŵi,t contradicting optimality). Therefore Bi,S (λ̂, ŵ) is nonempty and so S ∈ Cˆi and Cˆi is
nonempty.
For part (ii), S ∈ Cˆi implies that i ∈ IS . So we have a set S with Bi,S (λ̂, ŵ) nonempty. By
Lemma 4, ŵφ,t = λ̂φ,S,t and so S ∈ Cˆφ .
Lemma 6. Let (λ̂, ŵ) be an optimal solution to LPΠt (v) with a minimal number of binding constraints. If Î + is nonempty, then ∩i∈Î + Cˆi is nonempty.
Proof. If |Î + | = 1, then the statement holds trivially by part (i) of Lemma 5. Consider the case
|Î + | > 1. If ∩i∈Î + Cˆi is empty, then this implies the following: Fix a resource i ∈ Î + . Part (i) of
/ Cˆl .
Lemma 5 implies that Cˆi is nonempty. Then for every S ∈ Cˆi there exists l ∈ Î + such that S ∈
+
Note that since l ∈ Î , ŵl,t > Δl,t (0).
So let i ∈ Î + , Ŝ ∈ Cˆi and l ∈ Î + with Ŝ ∈
/ Cˆl . If Ŝ ∈
/ Cˆl , there are two possibilities. First, Ŝ ∈ Cl
but Bl,Ŝ (λ̂, ŵ) is empty. But since Ŝ ∈ Cˆi , Bi,Ŝ (λ̂, ŵ) is nonempty, which this contradicts part (i) of
Lemma 4.
The other possibility is that Ŝ ∈
/ Cl . Let
= min ŵl,t − Δl,t (0),
min
S∈Cl \Ĉl ,r∈{1,...,rl1 }
14
{ξl,S,t (r)}
>0
where we define the minimum over an empty set to be infinity (the second term could be empty).
l,S,t
w)
= λ̂ − δ Let δ ∈ (0, ) and (λ,
be given by λ
+ δ S∈Ĉl eφ,S,t and w
= ŵ − δel,t + δeφ,t .
S∈Ĉl e
φ,S,t = λ̂φ,S,t + δ for all S ∈ Cˆl . Similarly, w
l,S,t = λ̂l,S,t − δ, λ
l,t = ŵl,t − δ and
Therefore, we have λ
w
φ,t = ŵφ,t + δ. All other terms remain the same.
w)
We check that (λ,
is feasible and look at the set of binding constraints associated with this
solution. We look at the constraints in LPΠt (v) one by one and compare the the number of binding
w).
constraints in (λ̂, ŵ) with the number in (λ,
w)
and the
Constraints (20) and (21): Since w
φ,t > ŵφ,t , constraint (20) continues to hold for (λ,
number of binding constraints do not increase. By construction (λ, w)
satisfies constraint (21).
Constraints (17): Note that we need to check constraints (17) and (18) only for resource l. For
resource l, we have w
l,t = ŵl,t − δ > Δl,t (0) and so constraint (17) continues to be nonbinding.
l,t = ŵl,t − δ > ŵl,t − ξl,S,t (r) =
Constraints (18): For S ∈ Cl \Ĉl and r ∈ {1, . . . , rl1 }, we have w
l,S,t − λ̂l,S,t − j∈Ji Pj (S)ψl,t+1 (r) + Δl,t (r) = λ
P
(S)ψ
l,t+1 (r) + Δl,t (r). Note that the last
j∈Ji j
So constraint (18) remains nonbinding.
equality holds by definition of λ.
l,t = ŵl,t − δ > λ̂l,S,t − j∈Ji Pj (S)ψl,t+1 (r) +
For S ∈ Cˆl and r ∈ {1, . . . , rl1 }\Bl,S (λ̂, ŵ), w
l,S,t − Δl,t (r) − δ = λ
j∈Ji Pj (S)ψl,t+1 (r) + Δl,t (r). Therefore, constraint (18) continues to be
ˆ
l,t = ŵl,t −δ = λ̂l,t −δ− j∈Ji Pj (S)ψl,t+1 (r)+Δl,t (r) =
nonbinding. For S ∈ Cl and r ∈ Bl,S (λ̂, ŵ), w
l,t − Pj (S)ψl,t+1 (r) + Δl,t (r). So constraints (18) are binding for all such S and r. Note
λ
j∈Ji
w)
that these constraints, by definition, were also binding in (λ̂, ŵ). So, (λ,
satisfies constraints (17)
and (18) for resource l and the number of binding constraints is exactly the same as in (λ̂, ŵ).
Constraint (19): For S ∈ Cˆl , by definition Bl,S (λ̂, ŵ) is nonempty. Part (i) of Lemma 4 implies that
ŵφ,t = λ̂φ,S,t , which means that constraint (19) is binding. We have w
φ,t = ŵφ,t + δ = λ̂φ,St + δ =
φ,S,t and so constraint (19) holds and continues to be binding. For S ∈
l,S,t = λ̂l,S,t . Therefore,
λ
/ Cˆl , λ
w
φ,t = ŵφ,t + δ ≥ λl,S,t . So constraint (19) holds and the number of binding constraints do not
increase.
w).
Now we argue that the number of binding constraints (19) strictly decreases from (λ̂, ŵ) to (λ,
ˆ
For the set Ŝ, since Ŝ ∈ Ci , Bi,Ŝ (λ̂, ŵ) is nonempty. By, part (i) of Lemma 4, ŵφ,t = λ̂φ,Ŝ,t and
=λ
and the constraint is
so the constraint is binding in (λ̂, ŵ). But w
φ,t = ŵφ,t + δ > λ̂
φ,Ŝ,t
φ,Ŝ,t
w).
w)
nonbinding in (λ,
Overall, (λ,
has strictly fewer number of binding constraints (19) and they
are a subset of the set of binding constraints of (λ̂, ŵ) contradicting minimality.
w)
φ,t + i w
i,t , (λ,
is optimal and this gives a contradiction.
Since ŵφ,t + i ŵi,t = w
We are now ready to show that Φt (v) ≥ Πt (v).
Proposition 3. Φt (v) ≥ Πt (v).
Proof. Let (λ̂, ŵ) be an optimal solution to LPΠt (v) with a minimal number of binding constraints.
We consider two cases.
Case 1: Suppose that Cˆi is empty for all i. This means that for all S ∈ Ci , Bi,S (λ̂, ŵ) is empty.
It follows that ŵi,t = Δi,t (0) for all i (otherwise we can reduce ŵi,t contradicting optimality).
15
Part (ii) of Lemma 4 implies that ŵφ,t > λ̂φ,S,t for all S. It follows that ŵφ,t = 0. Therefore,
Πt (v) = i Δi,t (0). Note that r
= 0 and S = ∅ is feasible for Φ
t (v) and the objective function value
associated with this solution is i Δi,t (0). Therefore Φt (v) ≥ i Δi,t (0) = Πt (v).
Case 2: Suppose that Cˆi is nonempty for some i. We consider two subcases.
Case 2.a. Î + is empty. We choose a resource l such that Cˆl is nonempty and choose Ŝ ∈ Cˆl such
that Bl,Ŝ (λ̂, ŵ) is nonempty. By part (i) of Lemma 4, Bi,Ŝ (λ̂, ŵ) is nonempty for all i ∈ IŜ and
ŵφ,t = λ̂φ,Ŝ,t . So, for all i ∈ IŜ , we have ŵi,t = λ̂i,Ŝ,t − j∈Ji Pj (Ŝ)ψi,t+1 (r̂i ) + Δi,t (r̂i ), where
r̂i ∈ Bi,Ŝ (λ̂, ŵ). Note that r̂i ≥ 1 for all i ∈ IŜ . On the other hand, since Î + is empty. we have
/ IS . Putting everything together,
ŵi,t = Δi,t (0) for all i. In particular, ŵi,t = Δi,t (0) for all i ∈
ŵi,t +
ŵi,t
Πt (v) = ŵφ,t +
i∈IŜ
=
λ̂φ,Ŝ,t +
i∈I
/ Ŝ
⎡
⎣λ̂
i,Ŝ,t −
i∈IŜ
=
R(Ŝ) −
⎤
Pj (Ŝ)ψi,t+1 (r̂i ) + Δi,t (r̂i )⎦ +
j∈Ji
j∈S i∈Ij
≤
Δi,t (0)
i∈I
/ Ŝ
½[Ŝ∈Ci ] Pj (Ŝ)ψi,t+1 (r̂i ) +
Δi,t (r̂i ) +
Δi,t (0)
i∈I
/ Ŝ
i∈IŜ
Φt (v)
where the last equality follows since (λ̂, ŵ) satisfies constraint (21) and
Pj (S)ψi,t+1 (ri ) =
½[i∈IS ] Pj (S)ψi,t+1 (ri ) =
½[i∈IS ] Pj (S)ψi,t+1 (ri )
i
i∈IS j∈Ji
j
j∈Ji
=
i∈Ij
½[S∈Ci ] Pj (S)ψi,t+1 (ri ).
j∈S i∈Ij
The inequality follows from (16) by noting that Ŝ and r, where ri = r̂i for i ∈ IŜ and ri = 0
otherwise, is feasible to Φt (v). To see this, observe that for all j ∈ Ŝ, if ½[i∈Ij ] = 1, then i ∈ IŜ and
so ri = r̂i ≥ 1. Therefore, for all j ∈ Ŝ, ½[i∈Ij ] ≤ ri for all i and so Ŝ ⊂ Q(r).
Case 2.b. Î + is nonempty. Let Ŝ = ∩i∈Î + Cˆi , which by Lemma 6 is nonempty. Note that Ŝ ∈ Cˆi ⊂ Ci
for all i ∈ Î + . Now every i ∈ Î + satisfies ŵi,t > Δi,t (0). Therefore if i ∈
/ IŜ then ŵi,t = Δi,t (0).
Since Ŝ ∈ Cˆl for some l, BlŜ (λ̂, ŵ) is nonempty. Part (i) of Lemma 4 implies that Bi,Ŝ (λ̂, ŵ) is
nonempty for all i ∈ IŜ and that ŵφ,t = λ̂φ,Ŝ,t . Since Bi,Ŝ (λ̂, ŵ) is nonempty for i ∈ IŜ , there exists
r̂i ∈ 1, . . . , ri1 such that ŵi,t = λ̂i,Ŝ,t − j∈Ji Pj (Ŝ)ψi,t+1 (r̂i ) + Δi,t (r̂i ) (r̂i need not be unique,
we can pick any r ∈ Bi,Ŝ (λ̂, ŵ)). We have
⎡
⎤
⎣λ̂
Πt (v) = λ̂φ,Ŝ,t +
Pj (Ŝ)ψi,t+1 (r̂i ) + Δi,t (r̂i )⎦ +
Δi,t (0)
i,Ŝ,t −
i∈IŜ
=
R(Ŝ) −
j∈Ŝ
≤
Pj (Ŝ)
j∈Ji
½[Ŝ∈Ci ] ψi,t+1 (r̂i ) +
i∈Ij
i∈IŜ
i∈I
/ Ŝ
Δi,t (r̂i ) +
Δi,t (0)
i∈I
/ Ŝ
Φt (v)
where the inequality follows from (16) by noting that Ŝ and r is feasible to Φt (v) where ri = r̂i ≥ 1
for i ∈ IŜ and ri = 0 otherwise.
16
Lemma 2 and Proposition 3 together imply that Φt (v) =
Πt (v). 1We are now ready to show that
V P L ≥ V LRo . Note that we can write P L as V P L = minv i vi,1 (r
i ) subject to0 ≥ Φt (v) for all t
with the boundary condition that vi,τ +1 (·) = 0. Letting V = minv i vi,1 (ri1 ) + t Φt (v) subject to
0 ≥ Φt (v) for all t with the same boundary condition, it follows that V P L ≥ V . Using the fact that
Φt (v) = Πt (v) together with the linear programming formulation LPΠt (v) , of Πt (v), we can write V
as
!
= min
vi,1 (ri1 ) +
wi,t
wφ,t +
V
v
t
i
s.t.
wi,t + vi,t (ri ) ≥
⎧
⎨
max
S⊂Qi (ri ) ⎩
i
½[S∈Ci ] λi,S,t +
j∈Ji
⎫
⎬
Pj (S) [vi,t+1 (ri − 1) − vi,t+1 (ri )] + vi,t+1 (ri )
⎭
∀t, ri ∈ Ri
wφ,t ≥ max {λφ,S,t }
S⊂J
wφ,t ≥ 0
λφ,S,t +
∀t
λi,S,t = R(S) ∀t, S ⊂ J
i∈IS
vi,τ +1 (·) = 0.
With the change of variables ϑi,t (ri ) = vi,t (ri ) +
above linear program as
= min
ϑi,1 (ri1 ) + ϑφ1
V
ϑ
τ
s=t
wi,s and ϑφ,t =
τ
s=t
wφ,s , we can write the
i
s.t.
ϑi,t (ri ) ≥
⎧
⎨
max
S⊂Qi (ri ) ⎩
½[S∈Ci ] λi,S,t +
j∈Ji
⎫
⎬
Pj (S)[ϑi,t+1 (ri − 1) − ϑi,t+1 (ri )] + ϑi,t+1 (ri )
⎭
∀t, ri ∈ Ri
ϑφ,t ≥ max {λφ,S,t } + ϑφ,t+1 ∀t
S⊂J
λφ,S,t +
λi,S,t = R(S) ∀t, S ⊂ J
i∈IS
ϑi,τ +1 (·) = 0.
which is exactly LRo. Therefore, V P L ≥ V LRo
6
Segment-based Lagrangian relaxation
Although the complexity of LRo scales linearly with the resource capacities, it is still exponential
in the number of products. In this section, we present a tractable approximation that applies to
settings where the total demand is comprised of multiple customer segments with small consideration
sets. We first describe the demand model and then present the tractable variant of LRo, which we
refer to as segment-based Lagrangian relaxation (sLRo).
17
6.1
Demand model with multiple customer segments
We consider the case where the total demand is comprised of demand from multiple customer segments. Modeling demand in this manner can be a way to capture heterogenous customer preferences.
Moreover, each customer segment is interested only in a small subset of the products (its consideration set). Small consideration sets are a reasonable modeling tradeoff in many situations that
also finds support in the empirical and behavioral literature. We let G denote the set of customer
segments. Customer segment g ∈ G has a consideration set J g ⊂ J of products that it considers for
purchase. A segment g customer is indifferent to a product outside its consideration set, in the sense
that the customer’s choice probabilities are not affected by products offered outside its consideration
set. We assume that the consideration sets of the different customer segments are known to the firm
by a previous process of estimation and analysis. We also assume that the consideration sets of the
different segments are small enough for its power set to be enumerable.
In each period, we have exactly one customer arrival and an arriving
customer belongs to segment
g with probability αg . Since the total arrival rate is 1, we have g αg = 1. We let Pjg (S) denote the
probability that a segment g arrival purchases product j when S is the offer set. Since a segment
g customer is indifferent to products outside its consideration set, we have Pjg (S) = Pjg (S ∩ J g ) =
Pjg (S g ), where S g = S ∩ J g .
6.2
Tractable Lagrangian relaxation
We modify the Lagrangian relaxation approach for the case with multiple customer segments in the
following manner: We introduce segment-based decision variables hgi,S g ,t , which denote the frequency
with which set S g ⊂ J g is offered to segment g on resource i at time period t. As in LRo, we relax the
condition that the frequencies be identical across the resources by associating Lagrange multipliers
λgi,S g ,t with them and solve the optimality equation
λ
g
g
g
g
g
λ
λ
ϑλi,t (ri ) = max
α
h
+
P
(S
)
ϑ
(r
−
1)
−
ϑ
(r
)
+ ϑλi,t+1 (ri )
i,t+1 i
i,t+1 i
i,S g ,t
g∈Gi
S g ∈Ci i,S g ,t
j∈Ji j
h
s.t.
hgi,S g ,t ≤ ri ∀g ∈ Gi , S g ⊂ J g , S g ∈ Ci
g
S g ∈Ci hi,S g ,t ≤ 1 ∀g ∈ Gi
hgi,S g ,t ≥ 0 ∀g ∈ Gi , S g ⊂ J g , S g ∈ Ci
for resource i, with the boundary condition that ϑλi,τ +1 (·) = 0, where Gi = {g|∃j ∈ J g with j ∈ Ji }
can be interpreted as the set of customer segments that use resource i. We modify optimality
equation (11) in a similar manner for the case with multiple segments:
g g
g
g
g
α
h
f
P
(S
)
−
λ
ϑλφ,t = max
+ ϑλφ,t+1
g
g
g
g
j j
g
S
j∈S
i∈IS g i,S ,t
φ,S ,t
h
s.t.
Sg
hgφ,S g ,t ≤ 1 ∀g
hgφ,S g ,t ≥ 0 ∀g, S g ⊂ J g
with the boundary condition that ϑλφ,τ +1 = 0.
We refer to the above described modification as the segment-based Lagrangian
relaxation (sLRo).
Many of the results from §4 carry over to sLRo. In particular, Vtλ (r) = i ϑλi,t (ri ) + ϑλφ,t gives us
18
an upper bound on the value function. We find the tightest upper bound by solving
V sLRo = min V1λ (r1 ).
(22)
λ
Vtλ (r) is a convex function of λ and an expression for its subgradient can be derived in a manner
analogous to Proposition 1; we omit the details. Therefore, we can solve minimization problem
(22) using subgradient search. However, sLRo can be weaker than LRo and we can have V LRo <
V sLRo . Still, an advantage of sLRo is its computational tractability. Note that solving LRo requires
|J |
O(2
Lagrange multipliers which quickly becomes intractable. On the other hand, sLRo requires
) |J
g
O( g 2 | ) Lagrange multipliers, a much more manageable number provided the consideration set
for each segment is small in size. Moreover, when the consideration sets of the segments overlap, it
is possible to further tighten sLRo by adding the product-cut equalities (PC-equalities) described
in Meissner et al. [14] and Talluri [19]; we omit the details.
7
Computational experiments
In this section, we compare the upper bound and revenue performance of sLRo with benchmark
solution methods. In all of our test problems, choice is governed by the MNL model. We begin by
describing the MNL choice model and then describe the different benchmark solution methods and
the experimental setup.
7.1
MNL choice model with multiple customer segments
The total demand is comprised of multiple customer segments and within each segment choice is
according to the MNL model. The MNL model associates a preference weight ωjg with product j
that is in the consideration set of segment g. Similarly it associates a preference weight ω0g with
a segment g arrival not purchasing anything. The probability that a segment g arrival purchases
product j when S is the offer set is (Ben-Akiva and Lerman [1])
Pjg (S)
=
ω0g +
ωg
j
k∈S∩J g
ωkg
,
while the probability it purchases none of the offered products is
P0g (S) =
ω0g
+
ωg
0
k∈S∩J g
ωkg
.
In our computational experiments, we have J g ∩ J g = ∅ so that the consideration sets of the
segments are disjoint.
7.2
Benchmark methods
Choice Deterministic Linear Program (CDLP) This is the solution method described in §3.1. Since
customer choice is according to the MNL model and we have disjoint consideration sets, we use
the equivalent sales based formulation of CDLP described in Gallego, Ratliff, and Shebalov [5] and
solve it as a compact linear program.
19
Affine Approximation (AF) This is the solution method described in §3.2. We use the reduced
formulation described in Vossen and Zhang [23] to solve AF . While the number of variables in
the reduced formulation is manageable, it still has a large number of constraints. We solve AF by
generating constraints on the fly and stop when we are within 1% of optimality. For the MNL choice
model, the separation problem of AF can be solved as a linear mixed integer program (Zhang and
Adelman [24]).
Segment-based Lagrangian Relaxation (sLRo) This is the solution method described in §6. In our
computational
experiments, we use subgradient search to solve problem (22). We use a step size
√
of 250/ k at iteration k of the subgradient algorithm and run the algorithm for 2500 iterations.
Although, our step size selection does not guarantee convergence, it provided good solutions and
stable performance in our test problems.
7.3
Hub-and-spoke network
We have a network with a single hub serving N spokes. There is one flight from each spoke to the
hub and one flight from the hub to each spoke, so that there are 2N flights in total. Figure 1 shows
the structure of the network with N = 6. Note that the flight legs correspond to the resources in
our DC-NRM formulation.
The total number of fare-products is 2N (N + 1). There are 4N fare-products connecting hubto-spoke and spoke-to-hub origin-destination pairs. Of these, half are high fare-products whose
revenues are drawn from the Poisson distribution with a mean of 30, while the remaining are low
fare-products whose revenues are drawn from the Poisson distribution with a mean of 10. There are
2N (N − 1) fare products connecting spoke-to-spoke origin destination pairs. Half of them are high
fare-products whose revenues are drawn from the Poisson distribution with a mean of 300, while
the remaining are low fare-products whose revenues are drawn from the Poisson distribution with a
mean of 100.
Each origin-destination pair is associated with a customer segment and a segment is only interested in the fare-products connecting its origin-destination pair. Within each segment, choice is
according to the MNL model. The preference weights of the high fare-products are drawn from the
Poisson distribution with a mean of 200, while that of the low fare-products are drawn from the
Poisson distribution with a mean of 80. The no-purchase preference weights are drawn from the
Poisson distribution with a mean of 10. We remark that we set the problem parameters in the same
manner as Meissner and Strauss [13].
We measure the tightness of the leg capacities using the nominal load factor. Letting
Pj (S g )fj
S g∗ ∈ argmax S g ⊂J g
j∈S g
be the offer set that maximizes expected revenues from segment g when there is ample capacity on
all the flight legs, the nominal load factor is
g g
g∗
t
i
gα [
j∈Ji Pj (S )]
1
ζ=
.
i ri
We have τ = 200 time periods in all of our test problems. We label our test problems by the pair
(N, ζ) where N ∈ {6, 8, 10} and ζ ∈ {0.8, 1.0, 1.2, 1.6}, which gives us a total of 12 test problems.
20
Table 1 gives the upper bounds obtained by the different solution methods. The first column
in the table describes the problem characteristics by using (N, ζ). The second to fourth columns,
respectively, give the upper bounds obtained by CDLP, AF and sLRo. The last two columns,
respectively, give the percentage gap between the upper bounds obtained by CDLP and AF with
respect to sLRo. sLRo obtains significantly tighter upper bounds that CDLP and AF . The average
gap between the upper bounds obtained by CDLP and sLRo is around 6%, while that between AF
and sLRo is around 3%. We observe that the gaps tend to increase with the load factor ζ and the
size of the network as measured by N .
Table 2 gives the CPU seconds required by the three solution methods for different numbers of
spokes in the network and different numbers of time periods in the booking horizon. All of our
computational experiments are carried out on a Core i7 desktop with 3.4-Ghz CPU and 16-GB
RAM. We use CPLEX 12.2 to solve all the linear programs. Since CDLP has a compact linear
programming representation, it solves in a fraction of a second. The running times of AF and sLRo
are in minutes and are generally comparable. We observe that sLRo, which decomposes the network
problem into a number of single resource problems, seems to have an advantage over AF for the
larger networks.
Table 3 compares the revenue performance of the three solution methods. We evaluate revenues
through simulation and use common random numbers in our simulations. All the three solution
methods obtain value function approximations of the form Vt (r) that we use to make the control
decisions. In the case of CDLP , we use Vt (r) = i μ̂i ri , where μ̂ = {μ̂i |∀i} are the optimal values of
the dual variables associated with constraints (3). For AF , we use Vt (r) = θ̂t + i V̂i,t ri , where θ̂ =
{θ̂t |∀t}, V̂ = {V̂i,t |∀i, t} is an optimal solution to AF . For sLRo, we use Vt (r) = i ϑλ̂i,t (ri ) + ϑλ̂φ,t ,
where λ̂ is an optimal solution to problem (22). For each solution method, we use the corresponding
value function approximation Vt (r), and if r t is the vector of remaining resource capacities at time
period t, we use the policy of offering the set that attains the maximum in the optimization problem
⎧
⎫
⎨
t i t ⎬
max
Pj (S) fj + Vt+1 r −
e + P0 (S)Vt+1 r
.
⎭
S⊂Q(r t ) ⎩
j∈S
i∈Ij
We refresh the respective value function approximations periodically by resolving the solution methods at five equally spaced intervals over the booking horizon. Table 3 reports the expected revenues
obtained by the three solution methods. The columns have a similar interpretation as in Table 1,
except that they give revenues. In the last two columns, we use to indicate that sLRo generates
higher revenues than the corresponding benchmark method at the 95% level. sLRo on average
generates revenues that are 2.15% higher than CDLP and the revenue gaps tend to increase with
the load factor. sLRo generates revenues that are on average 1.82% higher than AF , although we
observe instances where the gap is almost 3%.
Finally, we test the sensitivity of the revenue performance of the solution methods to the number
of times they are reoptimized over the booking horizon. For each solution method, we consider the
percentage change in revenues when we carry out the optimization only once at the beginning of the
booking horizon and use the resulting value function approximation for making the control decisions
(as opposed to refreshing the value function approximation by reoptimizing the solution method five
times over the booking horizon). Table 4 reports the percentage change in the revenues when the
number of reoptimizations changes from five to one. It is known that the revenue performance of
CDLP can suffer if not reoptimized periodically over the booking horizon and our results are in line
with this observation. On the other hand, it is interesting to note that the revenue performance of
sLRo is remarkably insensitive to the number of reoptimizations.
21
B
A
H
C
Figure 1: Structure of the airline network with a single hub and six spokes.
Problem
(N, ζ)
(6, 0.8)
(6, 1.)
(6, 1.2)
(6, 1.6)
(8, 0.8)
(8, 1.0)
(8, 1.2)
(8, 1.6)
(10, 0.8)
(10, 1.0)
(10, 1.2)
(10, 1.6)
Upper bound
CDLP
AF
sLRo
15,182
15,033
14,793
14,830
14,681
14,316
14,493
14,187
13,581
12,624
12,238
11,942
15,235
15,020
14,697
14,757
14,542
14,029
14,372
13,944
13,269
12,606
12,030
11,661
15,191
14,922
14,550
14,705
14,436
13,841
14,371
13,879
13,057
11,712
10,968
10,811
avg.
% Gap with sLRo
CDLP
AF
2.63
1.62
3.59
2.55
6.71
4.47
5.71
2.48
3.66
2.20
5.19
3.65
8.32
5.09
8.10
3.16
4.41
2.56
6.24
4.30
10.07
6.30
8.33
1.44
6.08
3.32
Table 1: Comparison of the upper bounds on the optimal expected total revenue.
8
Conclusions
In this paper we develop a new Lagrangian relaxation approach for DC-NRM with strong theoretical
properties and numerical performance. We show that the Lagrangian relaxation equals the piecewiselinear approximation with a complexity scaling linearly with the resource capacities, compared to
the exponential complexity of solving the piecewise-linear approximation directly. We build on these
ideas and proposed a segment-based relaxation that is tractable when the consideration sets of the
different customer segments are small. Our numerical experiments show that the proposed approach
can provide significant benefits, both in terms of tighter upper bounds and higher expected revenues.
Finally, we note that our results apply at the highest level of generality, for any discrete-choice model
of customer demand behavior. An interesting future research direction is to see if improvements in
complexity can be obtained by specializing to specific models such as MNL or nested logit.
22
No. of
spokes
6
8
10
12
CDLP
0.1
0.2
0.2
0.4
CPU secs.
AF
sLRo
166
130
328
183
653
257
1,124
333
No. of time
periods
100
200
300
400
CPU secs.
CDLP
AF
0.1
64
0.1
166
0.1
246
0.2
379
sLRo
34
130
287
498
Table 2: CPU seconds for the benchmark solution methods as a function of the number of spokes
in the airline network and the number of time periods in the booking horizon.
Problem
(N, ζ)
(6, 0.8)
(6, 1.)
(6, 1.2)
(6, 1.6)
(8, 0.8)
(8, 1.0)
(8, 1.2)
(8, 1.6)
(10, 0.8)
(10, 1.0)
(10, 1.2)
(10, 1.6)
Expected revenue
CDLP
AF
sLRo
14,476
14,553
14,738
13,947
14,006
14,200
13,019
13,082
13,285
10,977
11,080
11,413
14,362
14,335
14,551
13,565
13,503
13,755
12,635
12,677
12,905
10,670
10,731
11,060
14,207
14,272
14,416
13,335
13,259
13,590
12,478
12,462
12,637
9,658
9,818
10,012
avg.
% Gap with sLRo
CDLP
AF
1.78 1.26 1.78 1.37 2.00 1.53 3.81 2.92 1.30 1.49 1.38 1.83 2.09 1.77 3.53 2.97 1.45 1.00 1.88 2.44 1.26 1.38 3.54 1.94 2.15
1.82
Table 3: Comparison of the expected revenues.
Problem
(N, ζ)
(6, 0.8)
(6, 1.)
(6, 1.2)
(6, 1.6)
(8, 0.8)
(8, 1.0)
(8, 1.2)
(8, 1.6)
(10, 0.8)
(10, 1.0)
(10, 1.2)
(10, 1.6)
avg.
% Change in expected revenues
5 reopt vs 1 reopt
CDLP
AF
sLRo
-7.44
-2.58
-0.14
-1.82
-2.09
-0.10
-4.58
-4.49
-0.31
-13.95
-6.30
-0.31
-3.54
-4.03
-0.05
-2.61
-1.95
-0.01
-4.55
-7.98
0.04
-6.76
-7.75
0.17
-3.19
-4.16
0.07
-1.69
-0.86
0.04
-3.52
-5.09
-0.02
-10.51
-5.84
-0.09
-5.35
-4.43
-0.06
Table 4: Sensitivity of the solution methods to the number of reoptimizations.
23
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25
Appendix
Proof of Lemma 1
The first inequality follows from Meissner and Strauss [13]. So, we only show the second inequality.
Let
v̄i,t (ri ) = ϑλi,t (ri ) + ϑλφ,t /|I| ∀t, i, ri ∈ Ri
(23)
where |I| is the total number of resources. Fix t, r and Ŝ ⊂ Q(r). Since Ŝ ⊂ Q(r), ½[j∈Ji ] ≤ ri for
all j ∈ Ŝ, which implies that ½[i∈I ] = maxj∈Ŝ ½[j∈Ji ] ≤ ri . It follows that setting ĥi,Ŝ,t = 1 for all
Ŝ
i ∈ IŜ and ĥi,S,t = 0 for all S = Ŝ and i ∈ IS yields a feasible solution to optimality equation (10)
and we have
λ
λ
λ
ϑi,t (ri ) ≥ ½[i∈I ] λi,Ŝ,t +
Pj (Ŝ) ϑi,t+1 (ri − 1) − ϑi,t+1 (ri )
(24)
+ ϑλi,t+1 (ri ).
Ŝ
j∈Ji
We also have that ĥφ,Ŝ,t = 1 and ĥφ,S,t = 0 for S = Ŝ is feasible to optimality equation (11) and we
have
ϑλφ,t ≥
Pj (Ŝ)fj −
λi,Ŝ,t + ϑλφ,t+1 .
(25)
j∈Ŝ
i∈IŜ
Plugging (24) and (25) into (23)
v̄i,t (ri ) =
ϑλi,t (ri ) + ϑλφ,t
i
≥
j∈Ŝ
=
j∈Ŝ
i
Pj (Ŝ)fj +
λ
Pj (Ŝ) ϑλi,t+1 (ri − 1) − ϑλi,t+1 (ri ) +
ϑi,t+1 (ri ) + ϑλφ,t+1
i
i∈IŜ j∈Ji
Pj (Ŝ) fj +
v̄i,t+1 (ri − 1) − v̄i,t+1 (ri ) +
v̄i,t+1 (ri ).
i
i∈Ij
Therefore,
v̄ = {v̄it (ri )|∀i,
t, ri } satisfies constraints (4) and is feasible for P L. Its objective function
value is i v̄i,1 (ri1 ) = i ϑλi,1 (ri1 ) + ϑλφ,1 = V1λ (r 1 ). Therefore, V P L ≤ V1λ (r 1 ).
Proof of Proposition 1
We show the results by induction over the time periods. We begin with the first part of the proposition. It is easy to show the result for the last time period. So we assume that the result holds for
26
time period t + 1 and show that it holds for time period t. We have
ϑλ̂i,t (ri ) ≥
hλi,S,t (ri )λ̂i,S,t + ϑλ̂i,t+1 (ri − 1)
hλi,S,t (ri )
Pj (S)
S∈Ci
hλi,S,t (ri )
Pj (S)
+ϑλ̂i,t+1 (ri ) 1 −
≥
ϑλi,t (ri )
+
S∈Ci
S∈Ci
j∈Ji
j∈Ji
hλi,S,t (ri )[λ̂i,S,t
− λi,S,t ]
S∈Ci
+
ri
τ
k=t+1 S∈Ci
+
x=0
ri
τ
k=t+1 S∈Ci
= ϑλi,t (ri ) +
τ
λ
λ
λ
λ
Pr{Xi,k
= x | Xi,t+1
= ri − 1}hλi,S,k (x) [λ̂i,S,k − λi,S,k ] Pr{Xi,t+1
= ri − 1 | Xi,t
= ri }
λ
λ
λ
λ
Pr{Xi,k
= x | Xi,t+1
= ri }hλi,S,k (x) [λ̂i,S,k − λi,S,k ] Pr{Xi,t+1
= ri | Xi,t
= ri }
x=0
ri
k=t S∈Ci
λ
λ
Pr{Xi,k
= x | Xi,t
= ri }hλi,S,k (x) [λ̂i,S,k − λi,S,k ],
x=0
where the first inequality holds since {hλi,S,t (ri )|∀S ∈ Ci } is a feasible solution to problem (10)
when the Lagrange multipliers
holds by the induction assumption,
are λ̂. The
second inequality
λ
λ
and uses the facts that S∈Ci hλi,S,t (ri ) j∈Ji Pj (S) = Pr{Xi,t+1
= ri − 1 | Xi,t
= ri } and 1 −
λ
λ
λ
(r
)
P
(S)
=
Pr{X
=
r
|
X
=
r
}.
On
the
other
hand,
the
last equality
h
i
j
i
i
i,t+1
i,t
i,S,t
S∈Ci
j∈Ji
follows by noting that
λ
λ
= x | Xi,t
= ri =
Pr Xi,k
λ
λ
λ
λ
λ
Pr Xi,k
= x | Xi,t
= ri , Xi,t+1
= ri Pr Xi,t+1
= ri | Xi,t
= ri
λ
λ
λ
λ
λ
+ Pr Xi,k
= x | Xi,t
= ri , Xi,t+1
= ri − 1 Pr Xi,t+1
= ri − 1 | Xi,t
= ri
λ
λ
λ
λ
= Pr Xi,k
= x | Xi,t+1
= ri Pr Xi,t+1
= ri | Xi,t
= ri
λ
λ
λ
λ
+ Pr Xi,k
= x | Xi,t+1
= ri − 1 Pr Xi,t+1
= ri − 1 | Xi,t
= ri .
This concludes the proof of the first part of the proposition.
The proof of the second part proceeds in a similar fashion. We assume that the result holds for
time period t + 1 and show that it holds for time period t. We have
hλφ,S,t
Pj (S)fj −
λ̂i,S,t + ϑλ̂φ,t+1
ϑλ̂φ,t ≥
S
≥
ϑλφ,t +
j∈S
τ
k=t
i
i∈IS
hλφ,S,k λ̂φ,S,k − λφ,S,k ,
S∈Ci
where the first inequality holds because {hλφ,S,t |∀S} is feasible for problem (14) when the Lagrange
multipliers are λ̂ and the second inequality follows from the induction assumption.
Lagrangian relaxation using product-specific multipliers
We first describe the Lagrangian relaxation approach using product-specific multipliers (LRp) and
then give an example which illustrates it can be weaker than P L for choice network RM. Under
27
the independent-class assumption, Topaloglu [22] proposes relaxing the constraints that the same
acceptance decisions be made for each product across the resources that it uses by associating
Lagrange multipliers with them. The relaxation associates a multiplier λi,j,t for each product j and
i ∈ Ij , and can be interpreted as the portion of the revenue of product j that is allocated to resource
i. The Lagrangian relaxation approach of Topaloglu [22] can be easily extended to choice network
RM. We solve the optimality equation
λ
λ
λ
λ
(ri ) = max
Pj (S) λi,j,t + νi,t+1
(ri − 1) − νi,t+1
(ri ) + νi,t+1
(ri )
(26)
νi,t
S⊂Qi (ri )
j∈Ji
λ
for resource i, where Qi (ri ) = j | ½[j∈Ji ] ≤ ri . It is possible to show that i νi,t
(ri ) is an upper
bound on Vt (r) (Kunnumkal and Topaloglu [11]). So, we can find the tightest upper bound on the
optimal expected revenue by solving
λ
min
νi,1
(ri1 ),
V LRp = λ|
i∈Ij
λi,j,t =fj , ∀j,t
i
where the constraint i∈Ij λi,j,t = fj ensures that the Lagrange multipliers correspond to a valid
fare allocation. In contrast to the independent demands setting (where V P L = V LRp ), as the
following example illustrates, we can have V P L < V LRp for choice network RM.
Example 1: Consider a network revenue management problem with two products, two resources and
a single time period in the booking horizon. The first product uses only the first resource, while the
second product uses only the second resource, and we have a single unit of capacity on each resource.
Note that in the airline context, this example corresponds to a parallel flights network. The revenues
associated with the products are f1 = 10 and f2 = 1. The choice probabilities are given in Table
5. In the Lagrangian relaxation, since we have Lagrange multipliers
only for j ∈ Ji , we have only
two multipliers λ1,1,1 and λ2,2,1 . Moreover, the constraint i∈Ij λi,j,t = fj implies that λ1,1,1 = f1
and λ2,2,1 = f2 . Noting that there is only a single time period in the booking horizon and that
Qi (1) = J for i = 1, 2,
λ
ν1,1
(1) = max{P1 (S)f1 } = 5
S⊂J
and
λ
ν2,1
(1) = max{P2 (S)f2 } = 10/11
S⊂J
so that V
LRp
= 65/11.
On the other hand, the linear program associated with the piecewise-linear approximation is
V P L = min
v
v1,1 (1) + v2,1 (1)
s.t
v1,1 (1) + v2,1 (1) ≥ max{P1 ({1, 2})f1 + P2 ({1, 2})f2 , P1 ({1})f1 , P2 ({2})f2 , 0}
v1,1 (1) + v2,1 (0) ≥ max{P1 ({1})f1 , 0}
v1,1 (0) + v2,1 (1) ≥ max{P2 ({2})f2 , 0}
v1,1 (0) + v2,1 (0) ≥ 0.
Note that the first, second, third and fourth constraints correspond to the states vectors [1, 1], [1, 0], [0, 1]
and [0, 0], respectively. It is easy to verify that an optimal solution to P L is v1,1 (1) = 5, v1,1 (0) =
10/11, v2,1(1) = 0, v2,1 (0) = 0 and we have V P L = 5 < V LRp .
28
S
{1}
{2}
{1, 2}
P1 (S)
1/2
0
1/12
P2 (S)
0
10/11
10/12
Table 5: Choice probabilities
S
{1}
{2}
{1, 2}
P1 (S)
50/99
0
1/2
P2 (S)
0
51/101
1/2
Table 6: Choice probabilities
Optimal LRo multipliers
Talluri [18] shows that the optimal LRp multipliers satisfy i∈Ij λi,j,t = fj for all j. We give an
example below which illustrates that a similar property fails to hold for LRo.
Example 2: Consider a network revenue management problem with two products, two resources and
a single time period in the booking horizon. The first product uses only the first resource, while the
second product uses only the second resource, and we have a single unit of capacity on each resource.
Note that in the airline context, this example corresponds to a parallel flights network. The revenues
The choice probabilities
associated with the products are f1 = 99 and f2 = 101. are given in Table
=
{1},
S
=
{2}
and
S
=
{1,
2},
we
have
P
(S
)f
=
50,
6. Letting
S
1
2
3
1 j
j∈S1 j
j∈S2 Pj (S2 )fj = 51
and
P
(S
)f
=
100,
I
=
{1},
I
=
{2}
and
I
=
{1,
2}.
If
we
impose
the constraint
3 j
S1
S2
S3
j∈S3 j λ
=
P
(S)f
on
the
Lagrange
multipliers,
then
we
have
λ
=
50,
λ2,S2 ,1 = 51
j
1,S1 ,1
i∈IS i,S,t
j∈S j
λ
λ
and λ1,S3 ,1 + λ2,S3 ,1 = 100. We have ϑ1,1 (1) = max{50, λ1,S3 ,1 } and ϑ2,1 (1) = max{51, λ2,S3,1 }. It
can be verified that
min
λ | λ1,S1 ,1 =50,λ2,S2 ,1 =51,λ1,S3 ,1 +λ2,S3 ,1 =100
ϑλ1,1 (1) + ϑλ2,1 (1) = 101 > 100 = V P L = V LRo ,
where the last equalityfollows from Proposition
2. Therefore, the optimal LRo multipliers in the
above example satisfy i∈IS λi,S,t < j∈S Pj (S)fj .
Supplementary Material: An intuitive explanation of Proposition 2
In this section we give a simple calculus-based explanation for the equivalence of the Lagrangian
relaxation and the piecewise-linear approximation bounds given in Proposition 2.
Since P L has an exponential number of constraints, we have to solve it by generating constraints
on the fly. The separation problem for P L is, for each period t, given values of {vi,t (ri ) | ∀t, i, ri ∈ Ri },
to check if
⎡
⎤
Φt (v) =
max
Pj (S) ⎣fj +
[vi,t+1 (ri − 1) − vi,t+1 (ri )]⎦ +
[vi,t+1 (ri ) − vi,t (ri )] (27)
r∈R,S⊂Q(r)
j∈S
i
i∈Ij
is less than or equal to zero. If yes, constraint (4) is satisfied at time period t for all r and S ⊂
29
Q(r). Otherwise,
we find a violated constraint and add it to the linear program.Recall that
R(S) =
j Pj (S)fj is the expected revenue from offering set S. Letting i (S) =
j∈Ji Pj (S),
ψi,t+1 (ri ) = vi,t+1 (ri ) − vi,t+1 (ri − 1) and Δi,t (ri ) = vi,t+1 (ri ) − vi,t (ri ), Φt (v) can be equivalently
written as
Φt (v) =
max
R(S) −
ψi,t+1 (ri )i (S) +
Δi,t (ri ).
r∈R,S⊂Q(r)
i
i
Note that i (S) is the expected capacity consumed on resource i when set S is offered, while ψi,t+1 (ri )
is the marginal value of capacity at time period t + 1. The key part of the proof relies on showing
that Φt (v) is equivalent to the optimization problem
Πi,t (v, λ) + Πφ,t (v, λ) .
Πt (v) = min
λ∈Λ
i
where Πi,t (v, λ) = maxri ∈Ri ,S⊂Qi (ri ) {½[S∈Ci ] λi,S,t − ψi,t+1 (ri )i (S) + Δi,t (ri )} and Πφ,t (v, λ) =
maxS⊂J {λφ,S,t }. Once we establish this result, the rest of the proof proceeds by noting the correspondence between Πt (v) and the constraints in LRo.
Here we give a heuristic argument to provide intuition behind the result. Using binary variables
{hS,t | ∀S}, Φt (v) can be written as
#
"
Φt (v) = max max
ψi,t+1 (ri )i (S) hS,t +
Δi,t (ri )
R(S) −
r
s.t
h
S
½[S∈Ci ] hS,t ≤ ri , ∀i
i
i
(28)
hS,t = 1
S
hS,t ∈ {0, 1}.
Now, for a fixed integer r the inner maximization problem (over the hS,t variables) is an integer
linear programming problem. However, note that as the
constraint set has a totally unimodular
structure (if we write the second set of constraints as − S hS,t = −1, each column has exactly one
+1 and one −1). Therefore, for any integer r we can ignore the integrality requirements on hS,t
without affecting the optimal objective function value. For
integer r, let σt∗ denote
any fixed integer
∗
an optimal dual variable associated with the constraint S hS,t = 1 (σt are hence functions of r,
but we suppress the dependence for brevity) and we can write the problem as
Φt (v)
= max
r∈R
S
"
R(S) −
#+
ψi,t+1 (ri )i (S) −
i
σt∗
+
Δi,t (ri ) + σt∗
i
where
[x]+ = max{x, 0} and we use the facts that ψi,t+1 (0) = ∞, and hS,t = 1 if R(S) −
∗
i ψi (ri )i (S) − σt > 0 and hS,t = 0 otherwise.
Moreover, from our argument on total unimodularity, the dual variable σt∗ ’s are such that exactly
one set S has R(S) − i ψi (ri )i (S) − σt∗ > 0 and this is a necessary and sufficient condition for
optimality for the inner maximization problem of (28).
Now note that ψi,t+1 (ri ) can be interpreted as the marginal value of capacity on resource i.
Kunnumkal and Talluri [9] show that the marginal value is a non-increasing function of capacity.
Assuming (this is non-rigorous) it to be strictly decreasing and hence an invertible function of ri ,
30
we can optimize over ψid := ψi,t+1 (ri ) instead of ri , where the superscript indicates that ψid takes
on discrete values since ri is integer. We can write Φt (v) equivalently as
Φt (v)
=
max
{ψ d | ψid ≥ψi,t+1 (ri1 ), ∀i}
"
R(S) −
#+
ψid i (S) − σt∗
+
i
S
Δi,t (ψid ) + σt∗ .
i
Note that σt∗ are now functions of ψid via the inverse map.
We interpolate ψid with a continuous piecewise-linear function ψi and obtain a relaxation of Φt (v)
in the following manner. Letting Δ̄i,t (·) and σ̄t∗ denote linear interpolations of Δi,t (·) and σt∗ , we
write a relaxed problem as
"
#+
∗
R(S) −
Δ̄i,t (ψi ) + σ̄t∗
max 1
ψi i (S) − σ̄t
+
Φ̄t (v) =
{ψ | ψi ≥ψi,t+1 (ri ), ∀i}
=
"
R(S) −
S
#+
ψi∗ i (S) − σ̄t∗
i
S
i
+
i
Δ̄i,t (ψi∗ ) + σ̄t∗ ,
i
where ψ ∗ = {ψi∗ | ∀i} is an optimal solution to the above maximization problem. Φ̄t (v) is a relaxation
since we optimize over a continuous space instead of a discrete grid.
Now assume that ψ ∗ is an interior optimal solution (again non-rigorous), so that it satisfies the
first order set of conditions
−
½[R(S)−i ψi∗ i (S)−σ̄t∗ >0] i (S) + Δ̄i,t (ψi∗ ) = 0 ∀i ∈ I
(29)
S∈Ci
where Δ̄i,t (·) represents the first derivative of Δ̄i,t (·), we use the fact that i (S) = 0 for S ∈
/ Ci , and
we invoke the envelope theorem to treat the optimal multipliers σ̄t∗ as constants,.
Next, we look at Πt (v). Introducing variables {hi,S,t | ∀S} we can write Πi,t (v, λ) equivalently as
Πi,t (v, λ) = maxri ,hi
[½[S∈Ci ] λi,S,t − ψi,t+1 (ri )i (S)]hi,S,t + Δi,t (ri )
S∈Ci
s.t
½[S∈Ci ] hi,S,t ≤ ri
S hi,S,t = 1
ri ∈ Ri , hi,S,t ∈ {0, 1}.
hi,S,t = 1, we get
Π̄i,t (v, λ) = maxri ,hi
[½[S∈Ci ] λi,S,t − ψi,t+1 (ri )i (S)]hi,S,t + Δi,t (ri )
Relaxing the constraint
S
S∈Ci
s.t
½[S∈Ci ] hi,S,t ≤ ri
ri ∈ Ri , hi,S,t ∈ {0, 1},
and Πi,t (v, λ) ≤ Π̄i,t (v, λ). Similarly, introducing variables {hφ,S,t | ∀S} we can write Πφ,t (v, λ) as
Πφ,t (v, λ) = maxhφ S λφ,S,t hφ,S,t
s.t
S hφ,S,t = 1
hφ,S,t ∈ {0, 1}.
31
Relaxing the constraint Shφ,S,t = 1 by associating σ̄t∗ as the corresponding multiplier, we get
Π̄φ,t (v, λ) = maxhφ,S,t ∈{0,1} S [λφ,S,t − σ̄t∗ ]hφ,S,t + σ̄t∗ , and Πφ,t (v, λ) ≤ Π̄φ,t (v, λ). Letting
Π̄t (v) = minλ∈Λ
i Π̄i,t (v, λ) + Π̄φ,t (v, λ) ,
we have Πt (v) ≤ Π̄t (v). Applying the same heuristic arguments as before to Π̄i,t (v, λ), we have
+
Π̄i,t (v, λ) =
max
½
+ Δ̄i,t (ψi ),
(30)
[S∈Ci ] λi,S,t − ψi i (S)
{ψi | ψi ≥ψi,t+1 (ri1 )} S∈Ci
and an interior optimal solution satisfies the first order condition
−
½[λi,S,t −ψi i (S)>0] i (S) + Δ̄i,t (ψi ) = 0.
(31)
S∈Ci
Now, consider the Lagrange multipliers λ̂ = λ̂i,S,t | ∀S, i ∈ IS with
λ̂i,S,t =
ψi∗ i (S)[R(S) − σ̄t∗ ]
,
∗
l∈IS ψl l (S)
Note that λ̂i,S,t − ψi∗ i (S) = ψi∗ i (S)
i ∈ IS and λ̂φ,S,t = σt∗ , ∀S.
R(S)− l∈I ψl∗ l (S)−σ̄t∗
S
.
∗
l∈I ψl l (S)
(32)
Therefore, ∀i
S
½[λ̂i,S,t −ψi∗ i (S)>0] = ½R(S)−
l∈IS
.
ψl∗ l (S)−σ̄t∗ >0
Consequently ψi∗ satisfies the first order optimality condition (31) of the inner maximization problem
of the relaxed Lagrangian problem. Moreover as S is the unique set with ½[λ̂i,S,t −ψ∗ i (S)>0] , we
i
achieve optimality.
So we have
Π̄i,t (v, λ̂) =
+
½[S∈Ci ] λ̂i,S,t − ψi∗ i (S)
+ Δ̄i,t (ψi∗ ).
S
From this we can conclude
Π̄t (v) ≤
Π̄i,t (v, λ̂) + Π̄φ,t (v, λ̂)
i
=
i
[½[S∈Ci ] λ̂i,S,t − ψi∗ i (S)]+ +
Δ̄i,t (ψi∗ ) + σ̄t∗
i
S
[R(S) − l∈IS ψl∗ l (S) − σ̄t∗ ]+ ∗
Δ̄i,t (ψi∗ ) + σ̄t∗
+
=
ψi i (S)
∗ (S)
ψ
l
l∈I
l
S
i
S i∈IS
∗
∗ +
∗
∗
Δ̄i,t (ψi ) + σ̄t
=
[R(S) −
ψl l (S) − σ̄t ] +
S
i
l∈IS
= Φ̄t (v)
where the inequality follows since λ̂ is feasible to Π̄t (v). Putting everything together, Φt (v) ≤
Πt (v) ≤ Π̄t (v) ≤ Φ̄t (v), where the first inequality holds since Πt (v) is a relaxation of Φt (v)
(Lemma 1).
At first glance, it is a bit surprising that we are able to show a strong duality result (Φt (v) =
Φ̄t (v)) for an optimization problem on integers. Next, we outline the reason for this.
32
3itO
0
1
2
r
3
Figure 2: Shape of the relaxed (continuous) Lagrangian value function for a fixed λ for piecewiselinear functions ψ(r) and Δ(r) with integer breakpoints. The function need not be convex or concave,
but is piecewise-linear and convex between integer breakpoints. The maximizer therefore happens
at an integer r and can be found by a simple search over r as it is a one-dimensional function.
The Lagrangian relaxation has a rather peculiar property: it so turns out that all local maxima
of the Lagrangian occur at integers! To see this, consider (30) for a fixed λ and the shape of the
function between two adjacent integer points: the first term in the objective is a convex function
of ri (via ψi which decreases as ri increases as ψ(r) is a decreasing function; hence the slope of
the first term increases with increasing ri ) while the second term is a linear function (Δ̄i,t (·) being
a piece-wise linear function with breakpoints at adjacent integer values of ri ). Consequently, the
objective function is convex between adjacent breakpoints of the piecewise-linear functions and the
maximum value must occur at an integer; see Figure 2.
So the implication of the form seen in Figure 2 is that all the optima happen only at integer
points so it gives us Φt (v) = Φ̄t (v), which implies that Φt (v) = Πt (v).
33