DYNAMIC BOOKING POLICY FOR AIRLINE SEAT
INVENTORY CONTROL
Kristine Rozite, Nicholas A. Nechval, Konstantin N. Nechval, Edgars K. Vasermanis
Department of Mathematical Statistics, University of Latvia, Raina Blvd 19, Riga LV-1019, Latvia
Keywords: Passenger demand, Fare classes, Airline seat inventory control, Dynamic booking policy
Abstract: A dynamic booking policy for multiple fare classes that share the same seating pool on one leg of an airline
flight, when seats are booked in a nested fashion and when lower fare classes book before higher ones, is
determined. The dynamic policy of airline booking makes repetitive use of an optimal static policy over the
booking period, based on the most recent demand and capacity information. It allows one to allocate seats
dynamically and anticipatory over time.
1 INTRODUCTION
It is common practice for airlines to sell a pool of
identical seats at different prices according to
different booking classes to improve revenues in a
very competitive market. In other words, airlines sell
the same seat at different prices according to
different types of travellers (first class, business and
economy). The question then arises whether to offer
seats at a relatively low price at a given time with a
given number of seats remaining or to wait for the
possible arrival of a higher paying customer.
Assigning seats in the same compartment to
different fare classes of passengers in order to
improve revenues is a major problem of airline seat
inventory control. This problem is usually
considered in three stages according to increasing
difficulty. First is the one-leg problem, which deals
with one airplane for one takeoff and landing and
ignores the potential revenue impact of other links of
the passengers' itineraries. Second is the multihop
problem, which deals with one airplane having
multiple takeoffs and landings (still ignoring the
impact of other links). The third is the origin-
destination network (OD network) problem, which
considers many airplanes having many takeoffs and
landings on a routing network.
This paper deals with the above problem under
the following assumptions: (i) Single flight leg:
Bookings are made on the basis of a single departure
and landing. No allowance is made for the
possibility that bookings may be part of larger trip
itineraries; (ii) Independent demands: The demands
for the different fare classes are stochastically
independent; (iii) Low before high demands: The
lowest fare reservations requests arrive first,
followed by the next lowest, etc.; (iv) No
cancellations: Cancellations, no-shows and
overbooking are not considered; (v) Limited
information: The decision to close a fare class is
based only on the number of current bookings; (vi)
Nested fare classes: Any fare class can be booked
into seats not taken by bookings in lower fare
classes. We seek the dynamic policy of airline
booking that makes repetitive use of an optimal
static policy over the booking period, based on the
most recent demand and capacity information. It
allows one to allocate seats dynamically and
anticipatory over time.
2 STATIC BOOKING POLICY
Littlewood (1972) was the first to propose a static
solution method for the seat inventory control
problem for a single leg flight with two fare classes.
The idea of his scheme is to equate the marginal
revenues in each of the two fare classes. He suggests
closing down the low fare class when the certain
revenue from selling another low fare seat is
exceeded by the expected revenue of selling the
same seat at the higher fare. That is, low fare
booking requests should be accepted as long as
{
}
,Pr
1112
uXcc >≥ (1)
where c
1
and c
2
are the high and low fare levels
314
Rozite K., A. Nechval N., N. Nechval K. and K. Vasermanis E. (2004).
DYNAMIC BOOKING POLICY FOR AIRLINE SEAT INVENTORY CONTROL.
In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 314-317
DOI: 10.5220/0001133003140317
Copyright
c
 SciTePress

respectively, X
1
denotes the demand for the high fare
class, u
1
is the number of seats to protect for the high
fare class and Pr{X
1
>u
1
} is the probability of selling
all protected seats to high fare passengers. The
smallest value of u
1
that satisfies the above condition
is the number of seats to protect for the high fare
class, and is known as the protection level of the
high fare class. The concept of determining a
protection level for the high fare class can also be
seen as setting a booking limit, a maximum number
of bookings, for the lower fare class. Both concepts
restrict the number of bookings for the low fare class
in order to accept bookings for the high fare class.
Richter (1982) gave a marginal analysis, which
proved that (1) gives an optimal allocation
(assuming certain continuity conditions).
Optimal policies for more than two classes have
been presented independently by Curry (1990),
Wollmer (1992), Brumelle and McGill (1993), and
Nechval et al. (2002a, 2002b). Curry uses
continuous demand distributions and Wollmer uses
discrete demand distributions. The approach
Brumelle and McGill propose, is based on
subdifferential optimization and admits either
discrete or continuous demand distributions. They
show that an optimal set of nested protection levels,
u(1), u(2), . . . , u(m-1), where the fare classes are
indexed from high to low, must satisfy the
conditions:
))},(({))}(({
1
juREcjuRE
jjj −++
≤≤
δ
δ
(2)
for each j=1, 2, …, m-1, where is the
expected revenue from the j highest fare classes
when u(j) seats are protected for those classes and
δ
))}(({ juRE
j
+
and
δ
−
are the right and left derivatives with respect
to u(j) respectively. These conditions express that a
change in u(j) away from the optimal level in either
direction will produce a smaller increase in the
expected revenue than an immediate increase of c
j+1
.
The same conditions apply for discrete and
continuous demand distributions. Notice, that it is
only necessary to set m-1 nested protection levels
when there are m fare classes on the flight leg,
because no seats will have to be protected for the
lowest fare class. Brumelle and McGill (1993) show
that under certain continuity conditions the
conditions for the optimal nested protection levels
reduce to the following set of probability statements:
c
2
=c
1
Pr{X
1
>u(1)},
c
3
=c
1
Pr{X
1
>u(1) I X
1
+X
2
>u(2)},
M
c
m
=c
1
Pr{ X
1
>u(1) I X
1
+X
2
>u(2) I
… I X
1
+X
2
L +X
m−1
>u(m-1)}. (3)
These statements have a simple and intuitive
interpretation, much like Littlewood’s rule. Just like
Littlewood’s rule, this method is based on the idea
of equating the marginal revenues in the various fare
classes. In Nechval et al. (2002a) use is made of a
technique of Lagrange multipliers (Huang et al.,
1970; Nechval, 1982, 1984), which admits
continuous demand distributions and allows one to
obtain results in the form suitable for a practical use.
Robinson (1995) finds the optimality conditions
when the assumption of a sequential arrival order
with monotonically increasing fares is relaxed into a
sequential arrival order with an arbitrary fare order.
Furthermore, Curry (1990) provides an approach to
apply his method to origin–destination itineraries
instead of single flight legs, when the capacities are
not shared among different origin–destinations.
3 DYNAMIC BOOKING POLICY
It will be noted that the solution methods described
above are all static. This class of solution methods is
optimal under the sequential arrival assumption as
long as no change in the probability distributions of
the demand is foreseen. However, information on
the actual demand process can reduce the
uncertainty associated with the estimates of demand.
Hence, repetitive use of a static method over the
booking period, based on the most recent demand
and capacity information, is the general way to
proceed.
In this section, we consider a flight for a single
departure date with T predefined reading dates at
which the dynamic policy is to be updated, i.e., the
booking period before departure is divided into T
readings periods determined by the T reading dates.
These reading dates are indexed in decreasing order,
t=T, …, 1, 0, where t=1 denotes the first interval
immediately preceding departure, and t=0 is at
departure. The T-th reading period begins at the
initial reading date at the beginning of the booking
period, and the t-th reading period begins at t-th
reading date furthest from the departure date. Thus,
the indexing of the reading periods counts
downwards as time moves closer to the departure
date. Typically, the reading periods that are closer to
departure cover much shorter periods of time than
those further from departure. For example, the
reading period immediately preceding departure may
cover 1 day whereas the reading period 1-month
from departure may cover 1 week.
Let us suppose that the total seat demand for fare
class j at the t-th reading date (time t) prior to flight
departure is X
jt
(j∈{1, 2, …, m}), where X
1t
corresponds to the highest fare class; f
jt
(x
jt
; θ
jt
) is the
DYNAMIC BOOKING POLICY FOR AIRLINE SEAT INVENTORY CONTROL
315

probability density function of X
jt
, where θ
jt
is a
parameter (in general, vector). We assume that these
demands are stochastically independent. The vector
of demands is X
t
=(X
1t
, … , X
mt
). Each booking of a
fare class j seat generates average revenue of c
j
,
where c
1
>c
2
> … >c
m
. Let u
jt
, j∈{1, …, m-1} be an
individual protection level for fare class j at time t
prior to flight departure. This many seats are
protected for class j from all lower classes. The
protection for the two highest fare classes is
obtained by summing two individual protection
levels, (u
1t
+u
2t
), and so on. There is no protection
level for the lowest fare class, m; u
mt
is the booking
limit, or number of seats available, for class m at
time t prior to flight departure; class m is open as
long as the number of bookings in class m remains
less than this limit. Thus, (u
jt
+ … +u
mt
) is the
booking limit, or number of seats available, for class
j, j∈{1, …, m}. Class j is open as long as the
number of bookings in class j and lower classes
remain less than this limit. The maximum number of
seats that may be booked by fare classes in the next
at time t prior to flight departure is the number of
unsold seats U
t
. Demands for the lowest fare class
arrive first, and seats are booked for this class until a
fixed time limit is reached, bookings have reached
some limit, or the demand is exhausted. Sales to this
fare class are then closed, and sales to the class with
the next lowest fare are begun, and so on for all fare
classes. It is assumed that any time limits on
bookings for fare classes are prespecified. That is,
the setting of such time limits is not part of the
problem considered here. It is possible, depending
on the airplane capacity, fares, and demand
distributions that some fare classes will not be
opened at all.
3.1 Problem Statement
Since the fare requests in each class are independent,
we may find the expected revenue for m classes,
R
mt
(u
1t
, u
2t
, … , u
mt
), in terms of the revenue for class
m, plus the expected revenue of the remaining m-1
classes, accrued from reading period t to departure,
given that U
t
specifies the remaining set capacity at
the beginning of reading period t. Thus, the problem
at time t prior to flight departure is to find an optimal
vector of individual protection levels (for the m-1
highest fare classes) and booking limit (for the
lowest fare class m),
) , ... , ,(
21
∗∗∗
mttt
uuu
), , ... , ,(maxarg
21
) , ... , ,(
21
mtttmt
Duuu
uuuR
tmttt
∈
=
(4)
where
,([) ..., , ,(
1,1
0
t21 ttm
u
mtmmttmt
uRxcuuuR
mt
−
+=
∫
mtmtmtmtmtmttmtm
dxxfxuuu );()] , ...,
1,-2,
θ
−
+
−
∫
∞
−
++
mt
u
mtmtmtmttmttmmtm
dxxfuuRuc ,);()] ..., ,([
1,-1,1
θ
(5)
is the expected revenue, with R
0t
(⋅)=0,
.
1(1) ,0 ,
:) ..., ,(
1
1
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
=∀≥=
=
∑
=
m
j
jttjt
mtt
t
mjuUu
uu
D
(6)
3.2 Optimal Protection Levels
An optimal set of individual protection levels
1,-21
tmtt
must satisfy the conditions given
by the following theorem.
) , ... , ,(
∗∗∗
uuu
Theorem 1.
The optimal protection levels can be
obtained by finding
that satisfy
∗∗∗
tmtt
uuu
1,-21
, ... , ,
∫
∞
∗
=
t
u
tttt
dxxfcc
1
,);(
111112
θ
∫
∞
∗
=
t
u
tttt
dxxfcc
2
222223
);( θ
,);();(
221
2
21111
0
2221
∫∫
∞
−+
∗∗
∗
+
ttt
t
xuu
ttttt
u
ttt
dxdxxfxfc θθ
∶
∫
∞
−−−−−
∗
−
=
tk
u
tktktktkkk
dxxfcc
,1
,1,1,1,11
);( θ
∫
∗
−
−−−−
+
tk
u
tktktkk
xfc
,1
0
,1,1,12
);( θ
∫
∞
−+
−−−−−
−
∗
−
∗
−
×
tktktk
xuu
tktktktktk
dxdxxf
,1,1,2
,1,2,2,2,2
);( θ
∫
∗
−
−−−
++
tk
u
tktktk
xfc
,1
0
,1,1,11
);( ... θ
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316

∫∫
−−−++−+
−−−
−
∗
−
∗
−
∗
−
∗
−
×
ttktkttktktk
xxuu
ttt
xuu
tktktk
xfxf
3,1,12,1,1,2
......
0
222
0
,2,2,2
);();( θθ L
,...);(
,11
......
111
23,1,121
tkt
xxxuuu
ttt
dxdxxf
tttktktt
−
∞
−−−−+++
∫
−
∗
−
∗∗
× θ
(7)
where k∈{2, …, m-1}.
Proof. The proof is a simple application of the
Lagrange multipliers technique. 
One can see that the above equations are solved
recursively for each fare class starting with the first
fare class. This process is continued until we have
the first k such that
∑
=
∗
≤
1-
1
k
j
tjt
Uu
(8)
and
∑
=
∗∗
−∈>>
k
j
jttjt
mkuUu
1
1}. ..., {1, ,0 ,
(9)
Then
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−=
∑
−
=
∗∗
1
1
,0max
k
j
jttkt
uUu
(10)
and for all j>k. If 0=
∗
jt
u
,
1
1
t
m
j
jt
Uu ≤
∑
−
=
∗
(11)
then the optimal booking limit for the lowest fare
class, m, is
(12)
. ,0max
1
1
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−=
∑
−
=
∗∗
m
j
jttmt
uUu
It follows from the above that, in general, an optimal
set of individual protection levels must satisfy the
following conditions:
},Pr{
1112
∗
>=
tt
uXcc
)}],()Pr{(
21211113
∗∗∗
+>+>=
tttttt
uuXXuXcc I
∶
)()Pr{(
2121111
∗∗∗
+>+>=
ttttttk
uuXXuXcc I
)}, ... ... ( ...
,121,121
∗
−
∗∗
−
+++>+++
tktttktt
uuuXXXII
(13)
where k∈{2, …, m-1}.
4 CONCLUSION
This paper considers the airline seat inventory
control problem for a single leg route taking into
account dynamics and uncertainty of booking
process. We show that a booking policy that
maximizes expected revenue can be characterized by
a simple set of conditions that relate the probability
distributions of demand for the various fare classes
to their respective fares.
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