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EMSR vs. EMSU: Revenue or Utility?

2003 Agifors Yield Management Study Group

Honolulu, Hawaii

Larry Weatherford,PhD

University of Wyoming

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Outline of Presentation

• Classic EMSR Model for Seat Protection– Example Calculations

• New Utility Model (EMSU)– Example Calculations

• Comparison of Decision Rules

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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)

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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.

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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

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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

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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

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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!

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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

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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

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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!

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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

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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.

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• 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?

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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.

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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)

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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

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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

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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

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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!

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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

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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

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• 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.

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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!

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• 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