Improving Forecast Accuracy by Unconstraining Censored Demand Data Rick Zeni AGIFORS Reservations and Yield Management Study Group May, 2001

Inventory Controls Cause the Censoring Booking Limit

Cost of Using Censored Data Forecasts are too low Too few seats are protected for high-fare passengers Revenue is lost

Methods for Handling Censored Data Ignore the censoring Discard the censored data Unconstrain the Data: Mean Imputation Method Booking Profile Method EM Algorithm Projection-Detruncation Method

Ignore the Censoring +No unconstraining needed +May be appropriate if few observations are censored -Forecasts may have a positive or negative bias

Discard Censored Observations +Simple to implement +Fast processing -Results in negative bias

Mean Imputation Method Compare constrained values with the mean from uncensored observations If the mean is greater than the constrained value, the censored data is replaced (imputed) with the mean

Booking Profile Method Estimate the shape of the booking profile for flights that have no constrained data points Choose a starting point where the booking data represents unconstrained demand Scale the shape of the booking profile to higher levels of demand

Constrained and Unconstrained Booking Profiles

Expectation-Maximization Algorithm Given a distribution assumption and constrained observation C, the best estimate of the unconstrained value is

Expectation-Maximization Algorithm Step 0: Obtain initial estimates of Step 1(E-step): Replace all censored observations with their expected values Step 2: (M-Step): Re-estimate given the new unconstrained data (maximizing the expected likelihood) Repeat steps 1 and 2 until convergence

Projection Detruncation Similar to the EM algorithm Differs mainly in the way the expected values are calculated There is an additional parameter that affects the aggressiveness of the unconstraining

Projection-Detruncation Booking Limit Projection A B A The underlying idea is that the probability of underestimating demand is known and constant Observations that fall to the right of the booking limit represent censored data

Projection-Detruncation Booking Limit Projection A B A An underestimate of the projected value is indicated by area B The probability of an underestimate is given by

Which Method Works Best?

Test Data-Common Approach Choose uncensored data and artificially constrain demand to simulate censored data The choice of constraining techniques will influence which unconstraining method works best

Test Data-My Approach 1 Collect actual demand data that has not been censored 2 Calculate booking limits using a reduced aircraft capacity 3Compare the booking limits with the actual demand and determine where the data has been censored 4Construct a censored data set that is an accurate representation of true demand behavior

Performance Measurement Each method is evaluated based on the reduction of the error from the baseline method (Ignoring the censoring)

Results for the Ignore Method (baseline) Distribution of Errors of the Observations for the Ignore Method Applied to High Demand Flights

Results for the Discard Method Distribution of Errors of the Means for the Discard Method Applied to High Demand Flights

Results for the Mean Imputation Method Distribution of Errors of the Observations for the Mean Imputation Method Applied to High Demand Flights

Results for the Booking Profile Method Distribution of Errors of the Observations for the Booking Profile Method Applied to High Demand Flights

Results for the EM Algorithm Distribution of Errors of the Observations for the EM Algorithm Applied to High Demand Flights

EM Algorithm Convergence Rate

Extended EM Algorithm 1Produce unconstrained estimates of all censored observations using the EM algorithm 2Calculate the mean of demand at the market O&D / fare class / review point level from uncensored observations only 3If all the observations in the sample are censored, compare the unconstrained estimate from step 1 with the mean from step 2. If the mean is greater than the estimate, the mean becomes the estimate. Otherwise, do nothing

Results for the Extended EM Distribution of Errors of the Observations for the Extended EM Algorithm Applied to Low Demand Flights

Results for Projection Detruncation Distribution of Errors of the Observations for the PD Algorithm Applied to High Demand Flights

Overall Comparison

Summary It is better to do nothing than to discard the censored observations EM algorithm produces the best error reduction Simulated data showed similar results