Optimal Statistical Estimation and Dynamic Adaptive Control of
Airline Seat Protection Levels for Several Nested Fare Classes under
Parametric Uncertainty of Customer Demand Models
NICHOLAS NECHVAL1, GUNDARS BERZINS1, KONSTANTIN NECHVAL2
1BVEF Research Institute, University of Latvia, Riga LV-1586, LATVIA
2Transport and Telecommunication Institute, Riga LV-1019, LATVIA
Abstract: - Assigning seats in the same compartment to different fare classes of passengers is a major problem
of airline seat allocation. Airlines sell the same seat at different prices according to the time at which the
reservation is made and other conditions. Thus the same seat can be sold at different prices. The problem is to
find an optimal policy that maximizes total expected revenue. To solve the above problem, this paper presents
the novel computational approach to optimization and dynamic adaptive prediction of airline seat protection
levels for multiple nested fare classes of single-leg flights under parametric uncertainty. It is assumed that time
T (before the flight is scheduled to depart) is divided into h periods, namely a full fare period and h-1
discounted fare periods. The fare structure is given. An airplane has a seat capacity of N. For the sake of
simplicity, but without loss of generality, we consider (for illustration) the case of nonstop flight with two fare
classes (business and economy). The proposed airline's inventory management policy is based on the use of the
proposed computational models. These models emphasize pivotal quantities and ancillary statistics relevant
for obtaining statistical predictive limits for anticipated quantities under parametric uncertainty and are
applicable whenever the statistical problem is invariant under a group of transformations that acts
transitively on the parameter space. The proposed technique is based on a probability transformation and
pivotal quantity averaging. It is conceptually simple and easy to use. Finally, we give illustrative examples,
where the proposed analytical methodology is illustrated in terms of the two-parameter exponential
distribution. Applications to other log-location-scale distributions could follow directly.
Key-Words: - Airline seat protection levels, statistical optimization, dynamic adaptive control, pivotal
quantities, ancillary statistics, unknown (nuisance) parameters, elimination
Received: October 14, 2022. Revised: April 21, 2023. Accepted: May 14, 2023. Published: May 25 2023.
1 Introduction
Basically, there have been two static models of
airline seat reservation: nested and non-nested. In
non-nested model, distinct numbers of seats called
buckets are exclusively assigned to each fare class.
The sum of these buckets adds up to the total
airplane seat capacity. In nested model, each fare
class is assigned a booking limit, which is the total
number of seats assigned to that fare class
(protection level) plus the sum of all seat allocations
to its lower fare classes.
Earlier revenue management models considered
non-nested seat allocations. However, a major
difficulty with non-nested seat allocation is that if
the limit for a fare class is reached, a booking
request to that class is denied, while a lower fare
bucket remains open. In a nested seat allocation, this
booking denial does not happen as the inventories
are shared among each fare class and its lower
classes. The problem of constructing optimal airline
seat protection levels for multiple nested fare classes
of single-leg flights has been considered in
numerous papers.
In [1], the author was the first to propose a
solution method of the airline seat allocation
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 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
21 11
Pr( ),cc Zn (1)
where c1 and c2 are the high and low fare levels
respectively, Z1 denotes the demand for the high fare
(or business) class, n1 is the number of seats to
protect for the high fare class and Pr(Z1>n1) is the
probability of selling more than n1 protected seats to
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high fare class customers. It should be noted that
an analytical proof of (1) is not given.
Now we describe how it can be determined
protection levels for multiple nested fare classes of
single-leg flight when we deal with l=2 nested fare
classes. The performance index which can be used
to determine the optimal allocation of seats between
l=2 dependent (i.e., nested) fare classes, subject to
the total airplane seat capacity constraint, is given as
follows.
Maximize the total expected revenue for a
single-leg flight with l=2 nested fare classes,
12 22 2 11 2 2
(, ) ( ) ( ) Qn n n E Q n n ZQ
2
22 22 2
0
()
n
cn Fzdz
2122
1122 111222
00
() ()
nnnz
cnnz Fzdzfzdz
1
2
11 111222
0
() () ,
n
n
c n F z dz f z dz
(2)
subject to
2
1
, 0 for {1, 2},
jj
j
nNn j
(3)
where
() min(, )
jj j j j j
Qn E c nZ
0
() ()
j
j
n
j
jjjj jjjj
n
czfzdz nfzdz
0
0
() () 1 ()
j
j
n
n
jjjj jj j j jj
c zFz Fzdz n Fn
0
()
j
n
j
jjjj
cn Fzdz
(4)
represents the expected revenue from the jth fare
class, cj is the fare level for the jth fare class, nj
denotes the protection level for the jth fare class, Zj
denotes the customer demand for the jth fare class,
()
jj
f
zis the probability density function of Zj.
Theorem 1. If the performance index is given by
(2), (3), then the optimal protection levels have to
satisfy the following system of equations:
121112 1
arg ( ) , max 0,nccFnn Nn
(5)
Proof. A simple application of the Lagrange
multipliers technique leads to the optimal solution
satisfying
12 12
21
(, ) (, )
,
Qn n Qn n
nn
(6)
where
2
12
2222 1122 1222
20
(, ) () () ()
n
Qn n ccFn cnfn cfzdz
n
12
1 11122 11122222
00
() ( ) ( ) ( )
nn
cFzdzfncFnnzfzdz
1
1122 1 11 1 22
0
() () (),
n
cn f n c F z dz f n
(7)
22
12
122211122222
100
(, ) () ( )()
nn
Qn n cfzdzcFnnzfzdz
n
22
1222111222
() ()() .
nn
cfzdzcFnfzdz
(8)
It follows from (6) that
222122 11
1() ()1()cFncFn Fn
(9)
or
2111
().ccFn (10)
This ends the proof.
In [2], the authors showed that in the presence of
l tariff classes under certain conditions of continuity,
the conditions for optimal nested protection levels
are reduced to the following set of probabilistic
statements:
21 11
Pr( ),cc Zn
31 11 1 2 12
Pr ,cc ZnZZ nn
11 1 21
11
1
2
11
Pr .
ll
l
ii
ii
XnZZn
cc nZn
(11)
These statements have an intuitive interpretation,
much like Littlewood’s rule. To illustrate the
method of [2], consider a single-leg flight with l=3
nested fare classes. In [2], the authors show that (for
the case of l=3) the conditions for the optimal
nested protection levels reduce to the following set
of probability statements:
21 11
Pr( ),cc Zn
(12)
31 11 1 2 12
Pr ,cc ZnZZ nn (13)
where (12) has to be transformed (in terms of
probability distributions) to
21 1
(),ccFn (14)
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(13) has to be transformed (in terms of probabilities)
to
112
31 1112
212 1
Pr
,
Pr
Znn
cc nZnn
ZnnZ
(15)
(15) has to be transformed (in terms of probability
distributions) to
12
1
11 2
2
31
21111
1
()
.
()
nn
i
i
n
Fn n
cc
F
nzfzdz
(16)
In other words, the method of [2], needs the
system of equations (in terms of probabilities),
21 11
Pr( ),cc Zn
112
31 111
22121
Pr
,
Pr
Znn
cc nZn
nZnnZ
(17)
which has to be transformed to the system of
equations (in terms of probability distributions),
21 1
(),ccFn
12
1
11 2
2
31
21111
1
()
.
()
nn
i
i
n
Fn n
cc
F
nzfzdz
(18)
The complex empirical transformations of the
system of equations (12), (13) (set of probability
statements) to the system of equations (18) (in terms
of probability distributions), in order to determine
optimal protection levels for l=3 nested fare classes,
show that the method of [2], is not suitable for
practical applications if the number of nested fare
classes l 4.
Unfortunately, we did not find a numerical
example in the literature for the case when the
number of nested fare classes l 4.
2 Optimization of Airline Seat
Protection Levels for Nested Fare
Classes of Single-Leg Flights
The performance index which can be used to
determine the optimal allocation of airline seats
between l dependent (i.e., nested) fare classes,
subject to N (the total airplane seat capacity), is
given as follows.
Maximize the total expected revenue for a
single-leg flight with l nested fare classes (say, l=4)
1234 44
(, , , ) ( )Qn n n n Q n
4
43 4
3
j
j
EQ n Z
44
432
23
jj
jj
EEQ n Z
44
4321
12
,
jj
jj
EEEQ n Z
(19)
subject to
4
1
, 0 for 1(1)4,
jj
j
nNn j
(20)
where
0
() min(, ) ()
l
u
ll l l l l l l ll l
Qn E c nZ c u Fzdz
(21)
represents the expected revenue from the lth fare
class, cl is the fare level for the lth fare class,
(cl < cl-1 < … <c1), nl denotes the booking limit
for the lth fare class, Zl denotes the customer
demand for the lth fare class, ()
ll
f
z is the
probability density function of Zl, ()
ll
F
z is the
cumulative distribution function of Zl,
1
1
l
ll j l
jl
EQ n Z
1
1111
1
00
() ()
l
jl
jl
l
nz
nl
ljl llllll
jl
cnz Fzdzfzdz
1
11 111
0
() ()
l
l
n
ll llllll
n
c n F z dz f z dz
(22)
represents the expected revenue from the (l1)th
fare class, 1l
n
denotes the protection level for the
(l1)th fare class, and so on.
Theorem 2. The optimal solution for the above
performance index (19) is given as follows:
2111
()ccFn)
12 12
21
(, ) (, ,)
from ,
Qn n Qn n
nn
(23)
22
3222 11 2222
1
0
() + ()
n
j
j
ccFn cF nz
f
zdz
or
12
1
2
31112 2 1111
1
()
nn
j
j
n
ccFnn F nzfzdz
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123 123
32
(,,) (,,)
from ,
Qn n n Qn n n
nn
(24)
12
1
123
12
3
1
1
3
4 1 2 111 1
1
3
3 111 1
1
j
j
nn
j
j
n
nn n
j
j
nn
Fn
cc F nzfzdz
F
nzfzdz
1234 1234
43
(, , , ) (, , , )
from .
Qn n n n Qn n n n
nn
(25)
Proof. The proof follows using the technique of
Lagrange multipliers. Here it is omitted and will
appear elsewhere.
For example, consider again a single-leg flight
with l=3 nested fare classes. It follows immediately
from (23) and (24) that the optimal protection levels
have to satisfy the following system of two
equations:
2111
(),ccFn
22
3222 11 2222
1
0
() + () .
n
j
j
ccFn cF nz
f
zdz
(26)
Theorem 3. It can be shown that the system of
two equations (26) can be transformed to (18).
Proof.
It follows from (26) that
2
222
12222122
11
00
() ()
n
n
ii
ii
FnzfzdzFnzFz
22
221 2 2 11 22
1
0
() ()()
n
i
i
F
zF n z dz FnFn
22
22 1 2 2 11 22
1
0
(1 ( )) ( ) 1 ( )
n
i
i
F
zF nzdz Fn Fn
22
22
12222122
11
00
()
nn
ii
ii
F
nzdz FzF nzdz
2
2
11 11 2 2 1 2
10
() () ()
n
i
i
Fn FnF n F n z
1
12
2
21111
1
n
i
i
nn
F
nzFzdz
11 11 2 2 11 11 2
() () ( ) () ( )
F
nFnFnFnFnn
12
1
2
21111
1
nn
i
i
n
F
nzfzdz
11 2 2 11 2
() () ( )
F
nFn Fn n
12
1
2
2 111 1
1
,
nn
i
i
n
F
nzfzdz
(27)
where
2
21
1
,
i
i
nz z
(28)
2
21
1
.
i
i
znz
(29)
Substitution (27) into (26), we have
22
3222 11 2222
1
0
() + ()
n
j
j
ccFn cF nzfzdz
11 1 2 2 11 1 2 2 11 1 2
() () () ( ) ( )cF n F n cF n F n cF n n
12
1
2
12 1111
1
nn
i
i
n
cF nzfzdz
12
1
2
111 2 2 111 1
1
() ,
nn
i
i
n
cFn n F n zfzdz
(30)
where
21 1
().ccFn (31)
Thus, using two different analytical approaches, the
same result (18) was obtained. This indicates the
correctness of the used analytical approaches and
completes the proof.
For example, it follows immediately from (23)
and (24) that the optimal protection levels n1 and n2
can be determined as
1
2
2
111
1
arg min ( ) ,
n
c
nFn
c
1
2
2
11
1
arg min ( ) 1 ,
n
c
Fn c
(32)
12
2
1
2
11 2
2
3
221
11
11 1
()
arg min .
nn
j
nj
n
Fn n
c
nFnz
c
fzdz
(33)
For example, in the case of a single-leg flight
with l=3 nested fare classes, the performance index
is given as follows.
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Maximize the total expected revenue for a
single-leg flight with l=3 nested fare classes,
3
123 3 3 33 3
0
(, , ) ()
n
Qn n n c n F z dz
3233
2233 222333
00
() ()
nnnz
c nnz Fzdzfzdz
2
3
22 222333
0
() ()
n
n
c n F z dz f z dz
3233
112332
00
nnnz
cnnnzz
12332
11 1 2 2 2 3 3 3
0
() () ()
nn n z z
F
zdz f z dz f zdz
31
233
11111222333
00
() () ()
nn
nnz
c n F z dz f z dz f z dz
2122
3
1 122 111222
00
() ()
nnnz
n
cnnz Fzdzfzdz
33 3
()
f
zdz
1
32
11111222333
0
() ( ) () ,
n
nn
c n F z dz f z dz f z dz
(34)
subject to
3
1
, 0 for {1, 2, 3}.
jj
j
nNn j
(35)
A simple application of the Lagrange multipliers
technique leads to the optimal solution satisfying
(35) and
2111
123 123
21
( )
,
(, , ) (, , )
from
ccFn
Qn n n Qn n n
nn
2
3222 11122222
0
123 123
32
() + ( )()
.
(,,) (,,)
from
n
ccFn cFnnz
f
zdz
Qn n n Qn n n
nn
(36)
3 Exponential Distribution
Let U= (U1 ... Un) be the n ordered observations
(order statistics) in a sample of size n from the two-
parameter exponential distribution with the
probability density function (pdf)
1
() exp , 0, ,
u
fu u
(37)
and the cumulative distribution function (cdf)
() 1 exp , () 1 (),
u
F
uFuFu
(38)
where (, ),
is the shift parameter and
is
the scale parameter. It is assumed that these
parameters are unknown. In Type II censoring,
which is of primary interest here, the number of
survivors is fixed and Ur is a random variable. In
this case, the likelihood function is given by
1
(, ) ( )
n
j
j
L
fu
1
1exp
n
j
n
j
u
11
1
1exp
n
i
n
j
uuu
1
1
1
1exp ( )
n
i
n
j
uu
1
()
1exp nu
1
1
()
11
exp exp ,
n
n
sns
(39)
where
11 1
1
, ( )
n
nj
j
SUS UU
S (40)
is the complete sufficient statistic for ω. The
probability density function of S=(S1, Sn) is given by
1
(, )
n
f
ss
1
1
2
1
1
21
00
()
11
exp exp
()
11
exp exp
n
n
n
nn
n
nn
n
sns
ss ns
n
ds ds
sn
1
1
2
()
11
exp exp
(1)1
n
n
n
n
sns
n
sn
21
1
()
1exp exp
(1)
nn
n
n
sns
n
s
n
1,
n
f
sfs
(41)
where
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1
11
()
exp , ,
ns
n
fs s
(42)
2
1
1exp , 0.
(1)
nn
nnn
n
s
fs s s
n
(43)
1
1
S
V
(44)
is the pivotal quantity, the probability density
function of which is given by
11 1 1
() exp , 0,fv n nv v
(45)
n
n
S
V
(46)
is the pivotal quantity, the probability density
function of which is given by
2
1exp , 0.
(1)
n
nn n n n
fv v v v
n
(47)
3.1 Pivot-Based Elimination of Unknown
(Nuisance) Parameters from the Two-
Parameter Exponential Distribution
Let us suppose that U is a future observation from
the same distribution (38), independent of U = (U1
... Un). Then a statistical estimate of (38) can be
determined as follows.
Step 1. Invariant embedding of S1 in (38) to
isolate the unknown parameter
from the problem
through V1 (44),
() 1 exp u
Fu
11
1exp zs s
1
11
1exp exp , ,
us vus
(48)
Step 2. Averaging (48) over the probability
distribution of the pivotal quantity V1 to eliminate
unknown parameter
from the problem. It follows
from (48) and (46) that the pivot-based estimate of
the cumulative distribution function (38) (obtained
through the pivot-based method) is given by
1,111
0
() () ( )
s
F
uFufvdv
1
111
0
1exp exp exp
us vn nvdv
1
11
0
1exp exp [ 1)
us nvndv
1
1
1exp , , .
1
us nus
n
(49)
Since
1, 1
() 1exp
1
s
dF u us
n
du n
(50)
and
1
1
1
1
1exp
11exp ,
1exp
1
s
us
n
nus
us
ndu
n
(51)
it follows from (51) that the probability density
function (pdf) of U is given by
1
1
,1
1
() exp , ,
s
us
f
uus
(52)
with the cumulative distribution function
1
1
,() 1 exp .
s
us
Fu
(53)
Step 3. Invariant embedding of Sn in (53) to
isolate the unknown parameter
from the problem
through Vn (46),
1
1
,() 1 exp
s
us
Fu
1
1exp n
n
s
us
s
1
1
1exp , .
n
n
us
vus
s
(54)
Step 4. Averaging (54) over the probability
distribution of the pivotal quantity Vn to eliminate
unknown parameter
from the problem. It follows
from (54) and (47) that the pivot-based estimate of
the cumulative distribution function (38) (obtained
through the pivot-based method) is given by
1
1
,
00
() 1 exp
snnn n
n
us
Fufvdv v
s
2
1
exp
(1)
n
nnn
vvdv
n
(1)
1
11 ().
n
n
us Fu
s
s (55)
where
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(1)
1
() 1 () 1 .
n
n
us
Fu Fu s
ss (56)
The pivot-based estimate of the probability density
function (37) is given by
1
1
() 1
() 1 , .
n
nn
dF u us
n
f
uus
du s s
s
s (57)
It follows from (55) that the cumulative distribution
function of the ancillary statistic
1
n
US
XS
(58)
is given by
1
1
() 1 .
1n
Fx
x
(59)
The probability density function of the ancillary
statistic (58) is given by
() 1
() , x 0.
(1 ) n
dF x n
fx dx x
(60)
3.2 Constructing Shortest Length or Equal
Tails Confidence Intervals for Future
Observations from the Two-Parameter
Exponential Distribution under
Parametric Uncertainty
Using (58) and (59), it can be obtained a 100(1
)%
confidence interval for U from
1
12 1 2
Pr Pr
n
US
x
Xx x x
S
11 2 1
Pr 1 .
rn
xS S U xS S
(61)
by suitably choosing the decision variables 1
x
and
2
x
. Hence, the statistical confidence interval for U
is given by
1121
,.
nn
x
ssxss
(62)
The length of the statistical confidence interval for
U is given by
12 2 1 2 1
(, | ) .
nnn n
L
xx s xs xs x x s (63)
In order to find the shortest length confidence
interval 12
(, | )
n
Lx x s , we should find a pair of
decision variables 1
x
and 2
x
such that 12
(, | )
n
Lx x s is
minimum.
It follows from (60) and (63) that
221
100
() () ()
xxx
x
f
xdx f xdx f xdx
21
() () 1 1 ,Fx Fx p p
(64)
where p (0 )p
is a decision variable,
2
2
0
() ( ) 1
x
f
xdx F x p
(65)
and
1
1
0
() ( ) .
x
f
xdx F x p
(66)
Then 2
x
represents the
1p
- quantile, which
is given by
1/( 1)
21
11,
n
p
xq p
(67)
1
x
represents the
p
- quantile, which is given by
1/( 1)
1
11.
1
n
p
xq p
(68)
The shortest length confidence interval for U can
be found as follows:
Minimize
2
2
2
12 2 1 1
(, | )
nnppn
Lxx s x xs q q s
2
1/( 1) 1/( 1)
2
11
.
1
nn
n
s
pp
(69)
subjectto
0,p
(70)
The optimal numerical solution minimizing L(x1,
x2 | sn) can be obtained using the standard computer
software “Solver” of Excel 2016. If, for example, n
= 4,
= 0.05, then the optimal numerical solution is
given by
0p (71)
with the 100(1
)% shortest-length confidence
interval
12
(, | )1.1.
nn
Lx x s s (72)
The 100(1
)% equal tails confidence interval is
given by
12
(, | ; /2)1.5
nn
Lx x s p s
(73)
with
5.0.02p (74)
Relative efficiency. The relative efficiency of
12 ;,/2(| )
n
spLx x
as compared with L(x1,x2| sn)
is given by
2112
rel.eff. ( | ; (,/2,| ), )
Ln n
Lxs Lx
s
xxp
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12
21
()1.
/
=,|
,
1
(| )1.5;2
n
n
n
n
x
s
x
ss
Lx
p
s
Lx
0.7. (75)
Subsect
4 Generalized Pivotal Quantities to
Construct Statistical Predictive
Limits for Order Statistics in the
New Sample
Theorem 4. Suppose we are interested in a new
random sample of m ordered observations
U1…Um from a known distribution with a
probability density function (pdf) (),
f
u
cumulative
distribution function (cdf) (),Fu
where
is the
parameter (in general, vector). Then for constructing
one-sided predictive limits (for the rth order statistic
Ur, r{1, 2, …, m}) with confidence level 1
can be used the following generalized pivotal
quantities.
Generalized Pivotal Quantity GPQ1:
1()
r
GPQ F u
,1
1(1)1
(1 )
~() ,
,1
rm r
rmr
zz
zf z rm r
01,z (76)
where ,1
()
rm r
f
z
is the probability density function
(pdf) of the beta distribution ((,-1))Beta r m r
with the shape parameters r and m-r+1,
Proof. It follows from (76) that
1
,1
0
() ( | )
GPQ
rm r r r
rr
dd
f
zdz P U u m
du du
(77)
with
1
,1
0
() ( | ),
GPQ
rm r r r
f
zdz P U u m
(78)
where
(|)
rr
P
Uum
[()] [1 ()].
m
j
mj
rl
jr
mFu Fu
j
(79)
This ends the proof.
Generalized Pivotal Quantity GPQ2:
21 ()
r
GPQ F u
1,
(1)1 1
(1 )
~() ,
1,
mr r
mr r
zz
zf z mr r
01,z (80)
where 1, ()
mr r
f
z
is the probability density function
(pdf) of the beta distribution (( 1,))
B
eta m r r
with shape parameters mr+1 and r.
Proof. It follows from (80) that
1
1,
2
() ( | )
mr r r r
rr
GPQ
dd
f
zdz P U u m
du du
(81)
with
1
1,
2
() ( | ).
mr r r r
GPQ
f
zdz P U u m
(82)
This ends the proof.
Generalized Pivotal Quantity GPQ3:
()
1
31()
r
r
Fu
mr
GPQ rFu
1
,1 1
1
1
~() ,
,1
11
r
rm r m
r
rz
mr
mr
zz
rm r rz
mr
(0, ),z
(83)
where ,1
()
rm r z
is the probability density function
(pdf) of the F distribution ((, 1))Frm r with
parameters r and m−r+1, which are positive integers
known as the degrees of freedom for the numerator
and the degrees of freedom for the denominator.
Proof. It follows from (83) that
3
,1
0
() ( | )
GPQ
rm r r r
rr
dd
zdz P U u m
du du
(84)
with
3
,1
0
() ( | ).
GPQ
rm r r r
zdz P U u m
(85)
This ends the proof.
Generalized Pivotal Quantity GPQ4:
1()
41()
r
r
F
u
r
GPQ mr Fu
1, 1
1
1
~() ,
1, 1
1
mr
mr r m
mr
mr z
r
r
zz
mr r mr z
r
(0, ),z
(86)
where 1, ()
mr r z
is the probability density
function (pdf) of the F distribution (( 1,)
F
mr r
with parameters m
r +1 and r, which are positive
integers known as the degrees of freedom for the
numerator and the degrees of freedom for the
denominator.
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Proof. It follows from (86) that
1,
4
() ( | )
mr r r r
rr
GPQ
dd
zdz P U u m
du du
(87)
with
1,
4
() ( | ).
mr r r r
GPQ
zdz P U u m
(88)
This ends the proof.
5 Generalized Pivotal Quantities to
Construct Statistical Predictive
Limits for Order Statistics in the
Same Sample
Theorem 5. Suppose we observe some random
sample of m ordered observations U1…Um from a
known distribution with a probability density
function (pdf) (),
f
u
cumulative distribution
function (cdf) (),Fu
where
is the parameter (in
general, vector). The order statistic Ur is known.
Then, for constructing one-sided predictive limits
(for the kth order statistic Uk, k{r+1, …, m}) with
confidence level 1,
the following generalized
pivotal quantities can be used.
Generalized Pivotal Quantity GPQ5:
()
51 ()
r
k
F
u
GPQ
F
u
1(1)1
,1
(1 )
~() ,
,1
rk mr
rkmr
zz
zz
rkmr
01,z (89)
where ,1
()
rkmr z
is the probability density function
of the beta distribution ((, 1))Beta r k m r with
shape parameters r−k and m−r+1.
Proof. It follows from (89) that
5
,1
0
()
GPQ
rkmr
r
dzdz
du
(|;)
rrkk
r
d
P
UuUum
du
(90)
with
5
,1
0
() ( | ; ),
GPQ
rkmr r r k k
zdz P U u U u m
(91)
where
(|;)
rrkk
P
UuUum
() ()
1.
() ()
j
mk j
mk
rr
jrk kk
mk Fu Fz
jFu Fz
(92)
This ends the proof.
Generalized Pivotal Quantity GPQ6:
()
6()
r
k
Fu
GPQ Fu
11 1
1,
(1 )
~() ,
1,
mr rk
mr rk
zz
zz
mr rk
01.z
(93)
where 1, ()
mr rk z
is the probability density function
(pdf) of the beta distribution (( 1,))
B
eta m r r k
with shape parameters m−r+1 and r−k.
Proof. It follows from (93) that
1
1,
6
()
mr rk
rGPQ
dzdz
du
(|;)
rrkk
r
d
P
UuUum
du
(94)
with
1
1,
6
() , ( | ; )
mr rk r r k k
GPQ
zdz P U u U u m
(95)
This ends the proof.
Generalized Pivotal Quantity GPQ7:
1()()
71
() ()
rr
kk
m r Fu Fu
GPQ Fu Fu
rk
,1
~()
rkmr
zz
1
1
1
1,
(, 1)
11
rk
mk
rk
rk z
mr
mr
rkmr rkz
mr
z(0,),
(96)
where ,1
()
rkmr z
is the probability density function
(pdf) of the F distribution (( , 1))Fr km rwith
parameters r−k and m−r+1, which are positive
integers known as the degrees of freedom for the
numerator and the degrees of freedom for the
denominator.
Proof. It follows from (96) that
7
,1
0
()
GPQ
rkmr
r
dzdz
du
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(|;)
rrkk
r
dPU u U u m
du
(97)
with
7
,1
0
() ( | ; ).
GPQ
rkmr r r k k
zdz P U u U u m
(98)
This ends the proof.
Generalized Pivotal Quantity GPQ8:
() ()
81
() ()
1
rr
kk
r k Fu Fu
GPQ Fu Fu
mr
1,
~()
mr rk
zz
1
1
1
1
,
(1,) 1
1
mr
mk
mr
mr z
rk
rk
mr rk mr z
rk
(0, ),z (99)
where 1, ()
mr rk z
is the probability density
function (pdf) of the F distribution (( 1, ))Fm r r k
with parameters mr+1 and rk, which are
positive integers known as the degrees of freedom
for the numerator and the degrees of freedom for the
denominator,
Proof. It follows from (99) that
1,
8
()
mr rk
rGPQ
dzdz
du
(|;)
rrkk
r
dPU u U u m
du
(100)
with
1,
8
() ( | ; ).
mr rk r r k k
GPQ
zdz P U u U u m
(101)
This ends the proof.
6 Illustrative Example
For the sake of simplicity but without loss of
generality, consider the problem of optimal
allocation of seats between two dependent (i.e.,
nested) fare classes. The performance index which
can be used to determine the optimal allocation of
seats between two dependent (i.e., nested) fare
classes, subject to the total airplane seat capacity
constraint, is given as follows.
Maximize the total expected revenue for a
single-leg flight with two nested fare classes
(business and economy),
12 22
(, ) ( )Qn n Q n
211 2 22
{( min(, ))},EQn n nZ
(102)
where
22 22 2 2
() {min(, )}Qn Ec nZ
2
22 22 2
0
() ,
n
cn Fzdz
(103)
211 2 22
{ ( min( , ))}EQn n nZ
2122
1 122 111222
00
() ()
nnnz
c n n z F z dz f z dz
1
2
11 111222
0
() ( ) .
n
n
c n F z dz f z dz
(104)
subject to
12 , 0 for 1, 2,
j
nn Nn j
(105)
where c1 and c2 are the high and low fare levels
respectively (c1>c2), nj denotes the booking limit for
the jth fare class, Zj denotes the customer demand
for the jth fare class, ()
j
j
f
zis the probability density
function of Zj , N is the total capacity of the cabin to
be shared among the two fare classes. A simple
application of the Lagrange multipliers technique
leads to the optimal solution satisfying
2
12111 11
1
arg Pr arg c
nccZn Fn
c
1
2
12
11
1
arg min ( ) ,
n
cc
Fn c
21
min(0, ),nNn
(106)
where n1 denotes the optimal protection level for the
high fare class, and n2 denotes the optimal booking
limit for the low fare class. Thus, (106) suggests
closing down the low fare class when the certain
revenue from selling low fare seats is exceeded by
the expected revenue of selling the same seat at the
higher fare. It should be remarked that there is no
protection level for the low fare (or economy) class;
n2 is the booking limit, or the number of seats
available, for the low fare class; the low fare class is
open as long as the number of bookings in this class
remains less than this limit. Thus, (n1+n2) is the
booking limit or number of seats available for the
high fare class at the time. The high fare class is
open as long as the number of bookings in this and
low classes remains less than this limit.
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6.1 Model of Optimal Statistical Estimation
of Airline Seat Protection Levels for
Nested Fare Classes under Parametric
Uncertainty
The model is given as follows:
Step 1. It follows from (56) that
(1)
11
11 1 1
() ()1 () 1 .
n
n
ns
Fn Fn Fn s
ss (107)
Step 2. It follows from (106) and (107) that
(1)
2211
111
11
arg arg 1
n
n
ccns
nFn
ccs
1/
1
1
2
1.
n
n
c
ss c
(108)
Step 3.
1/
2112
min 0, 1 .
n
n
nNsscc
(109)
The proposed policies of the dynamic adaptive
airline seat inventory control are based on the use of
order statistics of cumulative customer demand,
which have such properties as bivariate dependence
and conditional predictability. Dynamic adaptation
of the airline seat reservation system to airline
customer demand is carried out via the bivariate
dependence of order statistics of cumulative
customer demand. Dynamic anticipatory adaptive
optimization of the airline seat allocation includes
dynamic anticipatory adaptive nested optimization
of protection levels over time T. It is carried out via
the conditional predictability of order statistics. The
airline seat reservation system makes online
decisions as to whether to accept or reject any
customer request using established decision rules
based on order statistics of the current cumulative
customer demand. The computer simulation results
are promising.
6.2 Model of Dynamic Adaptive Control of
Airline Seat Protection Levels for Nested
Fare Classes under Parametric
Uncertainty
For example, consider a single-leg flight with two
fare classes (business and economy) for a single
departure date with predefined reading dates at
which the dynamic policy is to be updated, i.e., the
booking period before departure is divided into h
reading periods: (
0=0,
1], (
1,
2], …, (
h-1,
h]
determined by the h reading dates:
1,
2, …,
h.
These reading dates are indexed in increasing order:
0<
1<
2< <
h, where (
h-1,
h] denotes the reading
period immediately preceding a departure, and
h is
at departure. 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 cumulative customer
demand for the high (business) fare class at the kth
reading date (time
k, 1kh) is Uk representing the
kth order statistic from the underlying distribution
with the probability density function ()
f
u
and
cumulative distribution function (),
F
u
where
is
a parameter (in general, vector). This parameter is
assumed to be unknown, but there is a sample of
order statistics U1 ... Uh (statistical estimates of
cumulative customer demands for the high
(business) fare class of past flights).
Also, suppose that the cumulative customer
demands for the high and low fare classes are
stochastically independent. Each booking of a seat
of the high fare class in the reading period (
k1,
k]
generates revenue of c1. Each booking of a seat of
the low fare class in the reading period (
k1,
k]
generates revenue of c2, where c1 > c2 for all k{1,
…, h). Then the model of optimal statistical
estimation of airline seat protection levels for nested
fare classes under parametric uncertainty includes
the following steps:
Step 1. Let’s assume that Uk , {1, ..., },kh
represents the kth order statistic from the two-
parameter exponential distribution (38), where the
parameter
is unknown. It follows from (91) that
5
,1
0
()
GPQ
rkhr zdz
(|;),
rrkk
PU u U u h
(110)
where
,1;
()
51 ()
r
rkhr
k
Fu
GPQ q
Fu
(α-quantile), (111)
() exp .
u
Fu
(112)
It follows from (111) and (112) that
,1;
1
ln .
1
rk
rkhr
uu q
(113)
Step 2. Assuming r=h, ρ=ρk and α=αk, it follows
from (113) that
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,1;
1
ln .
1k
kkh
hk
uu q
(114)
Step 3. For each k=1, …, h-1 we define ρk and αk
such that
,1;
1
ln 1k
khk
hk
uu q
1/
21
1111
12
arg 1
n
n
cc
nFnss
cc
(115)
It should be noted that equation (115) is used to
determine the exact fragment estimate
,1;
1
ln ,
1k
k
hk
q
1, ..., 1 ,kh (116)
based on accurate statistical information obtained
from the process of selling and reserving air tickets
for the high (business) fare class and past single-leg
flights.
Step 4. For a new flight with new cumulative
customer demand values of new
k
Ufor each k=1, …,
h-1, the exact fragment estimate (116) can be used
for dynamic adaptive control of airline seat
protection level for the high (business) fare class
under parametric uncertainty of customer demand
models during the process of selling and reserving
air tickets for a new future flight as follows:
1
,1;
1
ln ,
1k
new new
kk
hk
NU q
1, ..., 1 ,kh (117)
where 1
new
N
is the dynamic adaptive airline seat
protection level for the high (business) fare class
under parametric uncertainty of customer demand
models during the process of selling and reserving
air tickets for a new future flight.
7 Conclusion
New rigorous formulations of the problems of
statistical optimization and dynamic adaptive
control of airline seat protection levels for several
nested fare classes under parametric uncertainty of
consumer demand models are presented. Several
results useful for practical application have been
obtained. Illustrative examples are given.
The new intelligent analytical technique
proposed in this paper represents the conceptually
simple, efficient, and useful method for constructing
optimal airline seat protection levels for multiple
nested fare classes of single-leg flights with any
practical number l (2) of nested fare classes. The
technique yields the optimal allocation of airline
seats between l dependent (i.e., nested) fare classes,
subject to N (the total airplane seat capacity), that
takes into account not only the past observations but
also the future observation program and the
associated statistics. The optimum procedure is to
consider the situation as a dual optimization of
allocation problem where information and action are
interrelated.
The technique used in this article is based on a
probability transformation and pivotal quantity
averaging, [3], [4], [5], [6], [7], [8], [9], [10], [11],
[12], [13], [14], [15], [16], [17]. It is conceptually
simple and easy to use.
The methodology presented in this article can be
useful for solving problems of the optimal allocation
of resources in physics and engineering.
References:
[1] Littlewood, K., Forecasting and control of
passenger bookings, in Proceedings of the
12th AGIFORS Symposium, American
Airlines, New York, 1972, pp. 95117.
[2] Brumelle, S. L. and McGill, J. I., Airline seat
allocation with multiple nested fare classes,
Operations Research, vol. 41, 1993, pp.
127137.
[3] Nechval, N.A. and Vasermanis,
E.K.,Improved Decisions in Statistics, Riga:
Izglitibas soli, 2004.
[4] Nechval, N.A., Berzins, G., Purgailis, M., and
Nechval, K.N., Improved estimation of state
of stochastic systems via invariant embedding
technique, WSEAS Transactions on
Mathematics, Vol. 7, 2008, pp. 141–159.
[5] Nechval, N.A., Nechval, K.N., Danovich V.,
and Liepins, T., Optimization of new-sample
and within-sample prediction intervals for
order statistics, in Proceedings of the 2011
World Congress in Computer Science,
Computer Engineering, and Applied
Computing, WORLDCOMP'11, Las Vegas
Nevada, USA, CSREA Press, July 18-21,
2011, pp. 9197.
[6] Nechval, N.A., Nechval, K.N., and Berzins,
G., A new technique for intelligent
constructing exact -content tolerance limits
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Nicholas Nechval, Gundars Berzins, Konstantin Nechval
E-ISSN: 2224-2880
406
Volume 22, 2023
with expected (1 − )-confidence on future
outcomes in the Weibull case using complete
or Type II censored data, Automatic Control
and Computer Sciences (AC&CS), Vol. 52,
2018, pp. 476–488.
[7] Nechval, N.A., Berzins, G., Nechval, K.N.,
and Krasts, J., A new technique of intelligent
constructing unbiased prediction limits on
future order statistics coming from an inverse
Gaussian distribution under parametric
uncertainty, Automatic Control and Computer
Sciences (AC&CS), Vol. 53, 2019, pp. 223–
235.
[8] Nechval, N.A. Berzins, G., and Nechval,
K.N., A novel intelligent technique for
product acceptance process optimization on
the basis of misclassification probability in
the case of log-location-scale distributions, in:
F. Wotawa et al. (Eds.) Advances and Trends
in Artificial Intelligence. From Theory to
Practice. IEA/AIE 2019, Lecture Notes in
Computer Science, vol. 11606, 2019, pp. 801–
818, Springer Nature Switzerland AG.
[9] Nechval, N.A., Berzins, G., and Nechval,
K.N. A novel intelligent technique of
invariant statistical embedding and averaging
via pivotal quantities for optimization or
improvement of statistical decision rules
under parametric uncertainty, WSEAS
Transactions on Mathematics, Vol. 19, pp.
17-38, 2020.
[10] Nechval, N.A. Berzins, G., and Nechval,
K.N., A new technique of invariant statistical
embedding and averaging via pivotal
quantities for intelligent constructing efficient
statistical decisions under parametric
uncertainty, Automatic Control and Computer
Sciences (AC&CS), Vol. 54, 2020, pp. 191-
206.
[11] Nechval, N.A., Berzins, G., and Nechval,
K.N., Cost-effective planning reliability-based
inspections of fatigued structures in the case
of log-location-scale distributions of lifetime
under parametric uncertainty, in Proceedings
of the 30th European Safety and Reliability
Conference and the 15th Probabilistic Safety
Assessment and Management Conference,
Edited by Piero Baraldi, Francesco Di Maio
and Enrico Zio, ESREL2020-PSAM15, 1-6
November, 2020, Venice, Italy, pp. 455-462.
[12] Nechval, N.A., Berzins, G., and Nechval,
K.N., A new technique of invariant statistical
embedding and averaging in terms of pivots
for improvement of statistical decisions under
parametric uncertainty, CSCE'20 - The 2020
World Congress in Computer Science,
Computer Engineering, & Applied
Computing, July 27-30, 2020, Las Vegas,
USA, in: H. R. Arabnia et al. (eds.), Advances
in Parallel & Distributed Processing, and
Applications, Transactions on Computational
Science and Computational Intelligence, pp.
257-274. Springer Nature Switzerland AG
2021.
[13] Nechval, N.A., Nechval, K.N., and Berzins,
G., A new unified computational method for
finding confidence intervals of shortest length
and/or equal tails under parametric
uncertainty, in Proceedings of the 2021
International Conference on Computational
Science and Computational Intelligence
(CSCI), 15-17 December 2021, Las Vegas,
NV, USA, pp. 533 – 539, Publisher: IEEE
2021.
[14] Nechval, N.A., Berzins, G., and Nechval,
K.N., A new simple computational method of
simultaneous constructing and comparing
confidence intervals of shortest length and
equal tails for making ecient decisions
under parametric uncertainty. Proceedings of
Sixth International Congress on Information
and Communication Technology – ICICT
2021, Lecture Notes in Network and Systems
(LNNS, volume 235), Yang X.-S., Sherratt S.,
Dey N., Joshi A. (eds), 25-26 February 2021,
London, United Kingdom, pp. 473-482.
Springer Nature Singapore 2022.
[15] Nechval, N.A., Berzins, G., Nechval, K.N.,
Tsaurkubule, Zh., and Moldovan, M., A new
intelligent method for eliminating unknown
(nuisance) parameters from underlying
models as an alternative to the Bayesian
approach. Journal of Applied Mathematics
and Computation, Vol. 1, 2022, pp. 53-65.
[16] Nechval, N.A., Nechval, K.N., and Berzins,
G., Adequate mathematical models of the
cumulative distribution function of order
statistics to construct accurate tolerance limits
and confidence intervals of the shortest length
or equal tails, WSEAS Transactions on
Mathematics, vol. 20, pp. 154-166, 2023.
[17] Nechval, N.A., Berzins, G., and Nechval,
K.N., A novel computational intelligence
approach to making efficient decisions under
parametric uncertainty of practical models and
its applications to industry 4.0, Advanced
Signal Processing for Industry 4.0, Volume 1,
Chapter 7, pp. 1- 40, IOP Publishing
Ltd.2023. To appear.
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Volume 22, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Nicholas Nechval carried out the proof of theorems,
which are the basis of analytical methodology.
Gundars Berzins, Konstantin Nechval were
responsible for illustrative examples.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflict of interest to declare.
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This article is published under the terms of the
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.47
Nicholas Nechval, Gundars Berzins, Konstantin Nechval
E-ISSN: 2224-2880
408
Volume 22, 2023
