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1
EMSR vs. EMSU: Revenue or Utility?
2003 Agifors Yield Management Study Group
Honolulu, Hawaii
Larry Weatherford,PhD
University of Wyoming
2
Outline of Presentation
• Classic EMSR Model for Seat Protection– Example Calculations
• New Utility Model (EMSU)– Example Calculations
• Comparison of Decision Rules
3
EMSR Model for Seat Protection:Assumptions
• Basic modeling assumptions for serially nested classes:a) demand for each class is separate and independent of
demand in other classes.
b) demand for each class is stochastic and can be represented by a probability distribution
c) lowest class books first, in its entirety, followed by the next lowest class, etc.
d) all demands arrive in a single booking period (i.e., static optimization model)
4
EMSR Model for Seat Protection:Assumptions
• Another key assumption:e) your company is risk-neutral (that is, you’re indifferent
between a sure $100 and a 50% chance of $200 (50% chance of 0).
EMSR has been used for over a decade as the industry standard for leg seat control.
5
EMSR Model Calculations
• Because higher classes have access to unused lower class seats, the problem is to find seat protection levels for higher classes, and booking limits on lower classes
• To calculate the optimal protection levels:Define Pi(Si ) = probability that Xi > Si,
where Si is the number of seats made available to class i, Xi is the random demand for class i
6
EMSR Calculations (cont’d)
• The expected marginal revenue of making the Sth seat available to class i is:EMSRi(Si ) = Ri * Pi(Si ) where Ri is the average
revenue (or fare) from class i
• The optimal protection level, 12, for class 1 from class 2 satisfies:EMSR1(12 ) = R1 * P1(12 ) = R2
• Once 12 is found, set BL2 = Capacity - 12 . Of course, BL1 = Capacity
7
Example Calculation
• Consider the following flight leg example :
Fare Class Avg. Demand Std. Dev. Fare
Y 40 10 500
B 50 15 300
M 60 20 100
• To find the protection for the Y fare class, we want to find the largest value of Y for which
EMSRY(Y ) = RY * PY(Y ) > RB
8
Example (cont’d)
EMSRY(Y ) = 500 * PY(Y ) > 300 PY(Y ) > 0.60
where PY (Y ) = probability that XY > Y.
• If we assume demand in Y class is normally distributed with mean, std. dev. given earlier, then we can calculate that Y = 37 is the largest integer value of Y that gives a probability > 0.6 and therefore we will protect 37 seats for Y class!
9
Joint Protection for Classes 1 and 2
• How many seats to protect jointly for classes 1 and 2 from class 3?
• The following calculations are necessary:
)Pr()(
**
ˆˆˆ
212,1
2,1
22112,1
22
212,1
212,1
SXXSP
X
XRXRR
XXX
10
Protection for Y+B Classes
• To find the protection for the Y and B fare classes from M, we want to find the largest value of YB that makes
EMSRYB(YB ) =RYB * PYB(YB ) > RM
• Intermediate Calculations:RYB = (40*500 + 50 *300)/ (40+50) = 388.89
03.183251510ˆˆˆ
905040
2222,
,
BYBY
BYBY XXX
11
Example: Joint Protection
• The protection level for Y+B classes satisfies: 388.89 * PYB(YB ) > 100
PYB(YB ) > .2571
• Again, we can calculate that YB = 101 is the largest integer value of YB that gives a probability > 0.2571 and therefore we will jointly protect 101 seats for Y and B class from class M!
12
Joint Protection for Y+B
• Suppose we had an aircraft with capacity 150 seats, our Booking Limits would be:
BLY = 150
BLB = 150 - 37 = 113
BLM = 150 - 101 = 49
13
New Utility Model (EMSU)
• What if you’re a smaller company and not willing to take as many risks?
• That is, instead of being risk-neutral, you are actually risk-averse.
• First step is to quantify how risk averse you are.
14
• There are several ways to do this, but one pretty simple way is to look at the following gamble:– Situation 1: You have a 50-50 chance of winning
either $100 or $0.– Situation 2: A certain cash payoff of $x.
– How big would x have to be to make you indifferent between the 2 situations?
15
Risk neutral vs. Risk averse
• If you said x would have to be $50, then you are risk-neutral.
• If you picked a value for x that is less than $50 (e.g., $40), then you a risk-averse. Obviously, the lower the value for x, the more risk-averse you are.
• If you picked a value for x that is more than $50, you are risk-seeking.
16
Utility Calculation
• One of the easiest ways to convert from a $ amount to a utility is to use an exponential curve
• U(x) = 1 - exp (-x/riskconstant)
17
Sample curves
•
0
0.2
0.4
0.6
0.8
1
0 100 200 300 400 500
Dollar amt
Uti
lity
Riskconstant = 50
Riskconstant =100
Riskconstant =150
18
EMSU Calculations
• The expected marginal utility of making the Sth seat available to class i is:EMSUi(Si ) = U(Ri) * Pi(Si ) where U(Ri) is the utility of the
average revenue (or fare) from class i
• The optimal protection level, 12, for class 1 from class 2 satisfies:EMSU1(12 ) = U(R1) * P1(12 ) = U(R2)
Once 12 is found, set BL2 = Capacity - 12 . Of course, BL1 = Capacity
19
Example Calculation
• Consider the same flight leg example from before:
Fare Class Avg. Demand Std. Dev. Fare
Y 40 10 500
B 50 15 300
M 60 20 100
• To find the protection for the Y fare class, we want to find the largest value of Y for which
EMSUY(Y ) = U(RY)* PY(Y ) > U(RB)
• Assume our risk constant is $50
20
Example (cont’d)
EMSUY(Y ) =U(500)* PY(Y ) > U(300)
= 0.999955 * PY(Y ) > 0.997521
PY(Y ) > 0.99757
where PY (Y ) = probability that XY > Y.
• If we assume demand in Y class is normally distributed with mean, std. dev. given earlier, then we can calculate that Y = 11 is the largest integer value of Y that gives a probability > 0.99757 and therefore we will protect 11 seats for Y class!
21
Probability Calculations
• Using similar joint protection logic as before yields the following:
The protection level for Y+B classes satisfies: U(388.89) * PYB(YB ) > U(100)
0.999581 * PYB(YB ) > 0.864665
PYB(YB ) > .865027
22
Joint Protection for Y+B
• We can calculate that YB = 70 is the largest integer value of YB that gives a probability > 0.865 and therefore we will jointly protect 70 seats for Y and B class from class M!
• Suppose we had an aircraft with capacity 150 seats, our Booking Limits would be: BLY = 150
BLB = 150 - 11 = 139
BLM = 150 - 70 = 80
23
• As you can see, these seat allocation decisions are much more conservative (more risk-averse) in that they protect many fewer seats for the upper classes and allow more to be sold to the more “sure” lower fare class.
24
Comparison of Decision Rules
• Now, what revenue and utility impact does this decision have?
• Using the 3 fare class example (data already shown), assume a plane with capacity = 150
• In 10,000 iterations (random draws of demand), EMSR generated an average utility of 127.26, while EMSU generated an average utility of 132.79, for a 4.17% increase!
25
• The average # booked in each class were:– EMSR EMSU
– Y 38.9 32.5– B 49.2 49.5– M 45.5 58.8
– LF 89.0% 93.9%– Yld $290.01 $264.21