PROBLEM STATEMENT
Build a mathematical model for airline flight overbooking to find an optimal
overbooking strategy
Consider:
 Ways to handle bumped passengers
 Current situation:
― Fewer flights by airlines from point A to point B
― Heightened security at and around airports
― Passengers’ fear
― Loss of revenue
Write a memo to the airline’s CEO, summarizing your findings
REPORT LAYOUT
Real Data
•Statistical “break
even” point (58%
of capacity) and
rate of no-shows
(12% of
reservations)
Binomial/Poisson
•Models the
probability of
getting 𝑋
customers to
show up in time
for departure
Static Simulation
“Bump functions”
•Use 5 different
models of bump
cost to decide the
best “fixed
booking limit”
Dynamic Simulation
“Firesale Model”
•More realistic
profit simulation
with cancelled
tickets being resold to improve
the “fixed booking
limit”
THE STATIC MODEL
Cancelled tickets cannot be resold
System:
 Binomial random variable for number of passengers who show up
 Define total profit function
 Apply function to consumer behavior patterns
 Compute optimal number of passengers to overbook
STATIC MODEL: BINOMIAL RANDOM VARIABLE
APPROACH
Number of passengers = 𝑋~Binomial (𝐵, 𝑝)
Pr 𝑖 passengers arrive at the gate = Pr 𝑋 = 𝑖
=
𝐵
𝑖
𝑝𝑖 1 − 𝑝
𝐵−𝑖
STATIC MODEL: PROFIT FUNCTION
𝑇𝑝 𝑋 = 𝐵 − 𝑋 𝑅 +
𝐴𝑖𝑟𝑓𝑎𝑟𝑒 ∗ 𝑋 − 𝐶𝑜𝑠𝑡𝑓𝑙𝑖𝑔ℎ𝑡
𝑋 ≤ 𝐶$
𝐴𝑖𝑟𝑓𝑎𝑟𝑒 − 𝐶𝑜𝑠𝑡𝐴𝑑𝑑 ∗ 𝑋 − 𝐶$
𝐶$ < 𝑋 ≤ 𝐶
𝐴𝑖𝑟𝑓𝑎𝑟𝑒 − 𝐶𝑜𝑠𝑡𝐴𝑑𝑑 ∗ 𝑋 − 𝐶$ − 𝐵𝑢𝑚𝑝 𝑋 − 𝐶
𝑋>𝐶
Where,
𝐵 − 𝑋 =# of no shows
𝑋 − 𝐶$ =profitable passengers
𝑋 − 𝐶 =excess passengers
BUMP FUNCTION
No Overbooking Model
 Wastes 16 seats on average
Bump Threshold Model:
Pr 𝑋 > flight capacity < 𝐵𝑇
 Independent of revenue and cost
 Gives number of ticket sales 𝐵, expecting bumping to occur on less than 5% of
flights.
BUMP FUNCTION
Linear Compensation Plan
𝐵 𝑋 − 𝐶 = 𝐵$ ∗ (𝑋 − 𝐶)
 Fixed cost for each bumped passenger
 Where:
 𝑋 − 𝐶 = # of bumped passengers
 𝐵$ = cost of handling each bumped passenger
BUMP FUNCTION
Nonlinear Compensation Plan
BumpNL 𝑋 − 𝐶 = 𝐵$ ∗ 𝑋 − 𝐶 ∗ 𝑒 𝑟∗
 Exponential
 Where:
 𝑟 = exponential rate based on actual data
 𝐵$ = cost of handling each bumped passenger
𝑋−𝐶
BUMP FUNCTION
Time-Dependent Compensation Plan (Auction)
Compensation =
316
105.33𝑒 0.07324𝑡
0 ≤ 𝑡 ≤ 15min
15min < 𝑡 ≤ 30 min
 Airline offers flight vouchers to those willing to be bumped based on time
 They used a Chebychev weighting function to
determine when people will accept a bump
TESTING STATIC MODEL:
BUMP FUNCTIONS
No Overbooking, 𝐵 = 118
Bump Threshold, 𝐵 = 145
Linear Compensation, 𝐵 = 162
Nonlinear Compensation, 𝐵 = 154
Time-Dependent Compensation, 𝐵 = 154
DYNAMIC/FIRESALE MODEL
Accounts for multiple tickets being bought at once, “group tickets”
Uses cancellation times to sell all possible tickets.
Ran simulations to find optimal number of bookings
TESTING DYNAMIC MODEL:
BUMP FUNCTIONS
Linear Compensation, 𝐵 = 155
(𝑡𝑦𝑝𝑜: 162 + 3 = 155)
Nonlinear Compensation, 𝐵 = 154
Time-Dependent Compensation, 𝐵 = 155
STRENGTHS
Used actual data
Used multiple models for bump functions
Graphs are clear and relevant
Similar results at various levels of sophistication
 Which also provides good approximation methods
WEAKNESSES
No stability analysis (e.g. different no-show rate)
Assumed infinite (and static) demand
“Dynamic” model is not a “real-time” strategy
Required “Memo” not included
Unclear Presentation
 Typo in their total profit function
 Unclear hierarchy of headings
 Graphs and charts placed under the wrong headings (leading to typos)
 Details of “Firesale” model are missing
 Precise implementation of binomial and Chebychev distributions are not explained