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Capacity allocation
Lecture 4
Capacity allocation
• How many low-fare seats (hotel rooms, rental
cars) to allow to be booked while facing the
possibility of future high-fare demand
• Airlines, rental car companies, hotels, cruise
lines, freight transportation, made-to-order
manufacturing
The two-class problem
• The basic model: all the discount bookings
occur any full-fare passengers seek to book
• Maximizing revenue/taking into account
incremental costs and ancilliary
contribution/taking into account sunk costs
• Determine the discount booking limit
• Tradeoff between setting it too high or too
low (dilution vs. spoilage)
• ***7.1
ff(C-b=y)
1 0
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y
0.8
Ff(C-b=y)
50
0.6
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0
0
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Y
ff(C-b=y)
1 0
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y
0.8
Ff(C-b=y)
50
0.6
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Y
ff(C-b=y)
1 0
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y
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Ff(C-b=y)
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ff(C-b=y)
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y
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Ff(C-b=y)
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Y
• B=60->61
• PlaneC=100
–
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*Dd=50
*Df=45; Df=55
*86%
**Dd=65
**Df=30
**14%..*6.5%
***Dd=65
***Df=45
***14%..*93.5%
=6.5%*190+93.5%*(190-200)=3
=86%*0+14%*3=0.42
=86%*0+14%*(6.5%*190+93.5%*(190-200))
=14%*(6.5%*190+93.5%*(190-200))=>6.5%*190+93.5%*(190-200)
=190—93.5%*200=> Pd—93.5%*Pf>0=>Pd/Pf>93.5%;190/200=95%
50%*10000+50%*20000=
• Discount booking limit for an airplane with
150 seats is 80 seats.
• The airplane is being substituted for one with
100 seats.
• What is the discount booking limit now?
ff(C-b=y)
100
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y
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1-Ff(C-b=y)
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b
100
ff(C-b=y)
1 0
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y
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Ff(C-b=y)
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Y
ff(C-b=y)
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y
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1-Ff(C-b=y)
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b
Kui pd/pf = 0.5, siis paneme me täishinnaga müüki alati mü=66 piletit, olenemata
sellest kui suur on full demand standardhälve
100
ff(C-b=y)
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y
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1-Ff(C-b=y)
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b
Kui diskonteeritud piletid on väga kallid, siis mida riskantsem on täishinnaga piletite
müük, seda vähem me täishinnale pileteid protectime
100
ff(C-b=y)
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y
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1-Ff(C-b=y)
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b
Kui diskonteeritud piletid on väga odavad, siis mida riskantsem on täishinnaga piletite
müük, seda rohkem me täishinnale pileteid protectime
100
Relation to the newsvendor problem,
1882
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Overage and underage costs
O=20
U=5
With respect to the high price class
O=pd
U=pf-pd
Using Littlewood’s (1972) rule
• The algorithm for setting b:
– Set b to 0; set pd=190; set pf=200->pd/pf=95%
– Set b=b+1 (e.g. remainder = 99, if
PlaneCapacity=100)
– Check whether 1-Ff(C-b) > 95%– probability, that
there is too much full demand, is over 95%
– If no, go back to increasing b
– Otherwise stop at previous b
Dynamic programming, Bellman’s
principle
Expected Marginal Seat Revenue
(EMSR) Heuristics, the case with three
price classes, heuristic EMSRa
• Littlewood lookup for the distribution of P1:
– P3/P1
• Littlewood lookup for the distribution of P2:
– P3/P2
• Booking limit b3= PlaneC minus the two
quantities computed above
• The cases with more classes analogous – e.g.
first calculate b4, then b3, then b2
Heuristic EMSRb, this time, for
example, for four classes
• First of all, we want the booking limit b4
• For the classes above it, calculate a following
„weighted average“ class
• AvgNew=Avg3+Avg2+Avg1
• StdevNew=sqrt(Stdev3^2+Stdev2^2+Stdev1^2)
• priceNew=price3*Avg3 /AvgNew +
price2*Avg2/AvgNew + price1*Avg1/AvgNew
• Do a Littlewood lookup on price4/priceNew on a
distribution given by AvgNew and StdevNew to
determing b4
• Proceed analogously for b3, then b2
Comparison of EMSRa, EMSRb and
dynamic programming, see Belobaba
1992
Demand dependence
• Demand in each fare class is independent of
demand in the other fare classes
– E.g. no cannibalization – opening a discount class,
has no effect on full-fare demand
– Also, no buy-up/sell-up – closing a discount fare
class does not lead to increased demand in higher
fare classes
Two-class capacity allocation with
demand dependence
• Instead of =pd/pf, we have the modified
formula =1/(1-alfa)*(pd/pf-alfa)
• to look up using Littlewood’s rule,
• alfa being the fraction of customers, who will
switch to full price, after being denied a
discounted ticket
Modified EMSR Heuristics, Belobaba
and Weatherford 1996
• Allows buy-up to the next highest class only
• Modified EMSRa – use modified formula for
the calculation of the next level only, original
formula for all the classes above that
• Modified EMSRb – use modified formula with
respect to „weighted average“ class created
for EMSRb (create this class in the original
way)
• Though, they are „heuristics grafted on to
another heuristic“
Measuring the effectiveness of
revenue management
•
•
•
•
Total revenue opportunity
No revenue management
Revenue Opportunity Metric
Example:
– ROM = ($45 000 – $35 000)/($50 000 - $35 000)
– =67%
• But, ROM for flights with very low demand will always be close to
100%
• And, see Fig7.5 above, if full-fare demand is high, but uncertain,
ROM will be lower, than when full-fare demand is high, but more
certain (low stdev)
• Thus, as these examples show, changes in ROM, might also be due
to changes in the underlying factors of demand, rather than due to
changes in the effectiveness of revenue management
• LECTURE 4 ENDS HERE
Tactical revenue management
• Calculates and periodically updates the booking limits
• Resources
– Units of capacity (flight departure, hotel room night, rental
car day)
• Products
– Customers are seeking to purchase those (a seat on Flight
130 from St. Louis to Cleveland on Monday, June 30 –
single resource; A two-night stay at the Sheraton
Cleveland, arriving on March 19 and departing on March
21 – two resources)
• Fare classes
– A combination of a price and a set of restrictions on who
can purchase and when (e.g. group and regional pricing)
• The fact that RM operates fare classes, does
not change much from customers view – he
still sees only the lowest available fare
• Since airlines still respond to the offers made
by the competition, RM supplements rather
than replaces pricing
Capacity allocation
• How many seats (hotel rooms, rental cars) to allow
low-fare customers to book – given the possible future
high-fare demand
• Two-class problem
– Discount customers
– Full-fare customers
• BASIC MODEL – all discount bookings happen before
full-fare bookings
• We maximize expected revenue – incremental costs
and ancillary contribution are zero
• In reality companies should maximize expected total
contribution

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