AIRLINE SEAT ALLOCATION WITH MULTIPLE NESTED FARE CLASSES

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AIRLINE SEAT ALLOCATION WITH MULTIPLE NESTED FARE
CLASSES

• Presentation by Nilesh Bagani
  
• Advisor Dr. Eylem
  Tekin
• Reference BRUMELLE, S., McGILL, J. 1993. Opns.
  Res. 41, 127-137.

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MOTIVATION

• ?With an increase in price competition and
  proliferation of discount fare booking, airlines
  are presented with a tactical problem of revenue
  management.
• ?If too many seats are sold at discount fares
  then, we may loose high fare passengers because
  of lack of seats
• ?On the other hand, if too many seats are
  reserved for high fare passenger then flight may
  remain empty!

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Airline Seat Allocation Problem

• Involves determining booking policies for
  optimal assignment of seats in various fare
  classes.
• Complications
• Low fare customers arriving before high fare
  ones.
• Multiple-flight passenger itineraries.
• Cancellation/ Overbooking.
• Dynamic nature of booking process.

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The Problem

• ?Concerns with seat allocation problem when
  multiple fare classes are booked into a common
  seating pool.
• Following Assumptions
• Single flight leg
• Independent demands
• Low before high demands
• No Cancellations
• Limited information
• Nested Classes

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Nesting of fare classes

• Common pool of seats multiple fare classes.
• A fixed upper limit for the lowest fare class,
  next higher for lowest two classes.
• Any higher class passenger can be booked into a
  seat for lower fare class.
• British airways nested fare classes

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Notation and Booking Policy

• f1 Fare of the highest fare class
• fi Fare of the ith highest class
• p1 Protection level for the highest fare
  Class
• p2 Protection level for the first two
  classes together
• pi of seats protected )for the first
  i classes together
• p vector of protection levels
• X1, X2 Xn are the demand in each of the fare
  Classes 1,2 n
• X demand vector
• If at any stage there are s seats remaining and
  there is fare class k
• demand the seats are booked iff s gtpk-1

p3
p2
f5
Class 3
p1
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Littlewoods Rule

• Just two fare Classes Full fare and discount
  fare
• Accept immediate return from selling an
  additional discount seat as long as the discount
  revenue equals or exceeds the Expected full fare
  revenue.
• The EMSRa method (Belobaba, 1987) gives a
  heuristic
• generalization of the above for any number of
  fare classes.
• This paper gives the correct optimal conditions
  for the same
• Problem.

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The Revenue Function
Rk s p x Revenue generated by k highest
fare classes when s seats
are available and the demand vector is x.
p0,p1.pk-1
x1,x2.xk
for 0 s lt x1 for x1 s
for 0 s lt pk for pk s lt xk1 pk for
pkxk1 s
for k 1,2, .
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Methodology

• Objective To find a vector p that maximizes the
  Expected
• Revenue ERks p x for all
  k
• ERks p x is continuous and piecewise linear
  on s gt0 and not
• differentiable at points s pk.
• Finding optimal solution to a concave function
  is easier

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Methodology

• Objective To find a vector p that maximizes the
  Expected
• Revenue ERks p x for all k
• ERks p x is continuous and piecewise linear
  on s gt0 and not
• differentiable at points s pk.
• Treat s and pk as continuous and use non
  smooth optimization
• techniques.
• Finding optimal solution to a concave function
  is easier
• Find the conditions under which the function is
  concave
• and thereafter try to find optimal solution!

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Methodology

• Problem s seats are unbooked, class k1 is
  being booked
• (ie. is open) and pk is to be
  determined.
• Concavity Right derivative left derivative
• (of fs) dfs
  d-fs
• Subdifferential (dfs) Closed interval df(s),
  d-f(s)
• Optimality Given Concavity, fs will be
  maximized at any
• point for which 0? ds or
  dfs 0 d-fs

for all k
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The Revenue Function
Rk s p x Revenue generated by k highest
fare classes when s seats
available and demand vector is x.
for 0 s lt x1 for x1 s
for 0 s lt pk for pk s lt xk1 pk for
pkxk1 s
for k 1,2, .
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Marginal Value of Extra Seat
for s lt x1 for s x1
0
for s x1 for s gtx1
0
for 0 s lt pk for pk s lt xk1 pk for
pkxk1 s
for 0 lt s pk for pk lt s xk1 pk for
pkxk1 lt s
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Concavity of ERk1s p X

• By Induction ER1spX is concave.
• Let ERkspX be
  concave
• ERk1spX Check concavity only at s pk
  and s pk Xk1
• At s pk
• At s pk Xk1

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Optimal Protection Levels

• Derivatives with respect to pk are found

for 0 s pk for pk lt s xk1 pk for
pkxk1 lt s
for 0 lt s lt pk for pk s lt xk1 pk for
pkxk1 s
?
or
? Optimal Protection levels
(1)
for k 1,2,3
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Non integral solutions???

• What to do if pk comes out to be a fraction??
• If the demand random variables X1, X2.. are
  integer valued then
• there exists an optimal integer policy p
• Reason ERks p x is CLBI (concave and
  linear between integers)
• Covering property If c is a constant such
  that
• for some s1lt s2, then there is an integer n in
  s1,s2 such that c lies in df(n)
• Proof by induction

Hence there is an integer p1 in 0,s such that
For higher order terms authors use expected
marginal value.
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Optimal Stopping and Optimality
Optimal Stopping problems?? We need to check
conditions for monotonicity
Expected gain in revenue by changing protection
Level for the nest of k highest classes from
pk to pk1
If,

• Booking problem for class k is monotone as
• There exist pk such that Gk is nonnegative for
  pk lt pk and
• non positive for pk pk
• Revenue is bounded.

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Alternative Expression

• If demand distributions are continuous then the
  optimal protection is
• The right hand side is simply the probability
  that all the remaining seats are solid
• Proof By Induction!

(2)
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Application of Optimality Conditions

• Previous airline practice Demand estimated with
  continuous distributions (usually normal,
  (Shlifer 1975))
• Continuous joint distributions are can be easily
  found and (2) is guaranteed to have a solution.
• Numerical/ Monte Carlo integration can be easily
  deployed !
• Monitoring Past performance

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Comparison with EMSRa

• Expected Marginal Seat Revenue Method
• A heuristic based on the same assumption,
  (Belobaba, 1987).
• ?The protection level p1 is optimal. Others are
  not !

Table I, EMSRa Vs. Optimal
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Comparison with EMSRa
Capacity Error 82 0.54 100
0.45 120 0.35 140
0.24 160 0.14
Table II Capacity effect
Mean X 40,60,80
s.d X 16,24,32

• The EMSRa consistently underestimates protection
  levels p1 and p2
• The discrepancy between the revenues is of the
  order of 0.5
• EMSRa is much easier computationally
• The EMSRa method can both underestimate as well
  as overestimate
• if the distribution is exponential Open
  question with normal
• distribution

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Conclusions

• ?The paper presents a rigorous formulation of the
  correct optimal conditions for the problem under
  the constraints of the problem using
  subdifferential optimization in a DP framework.
• ?Optimality conditions were shown to reduce to
  simple probability statements.
• ?The connection with the theory of optimal
  stopping is made. The fixed booking policy is
  shown to be optimal over all sets of policies
• ?The results are compared with the existent
  policies in airline industries and it is found
  that improvements of the order of 0.5 can be
  made with in revenue

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Is the problem fully solved?

• Excellent result under the given assumptions!
• Are all assumptions true??
• Single flight leg, No Cancellations, Low before
  high demands
• Limited information, Independent demands Q- Can
  a relationship be found?
• Q- If yes then should this model be extended for
  dynamically changing the
• protection levels??
• Author finds p1?p2?p3?
• But demand comes reverse
• order (x5?x4?x3? x2?) so
• we need p4?p3?p2? p1
• Q- Will it be better to use inverse revenue
  function for dynamic seat allocation?

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• QUESTIONS??