Revenue management (RM) is a fast growing branch in operations research (OR) and has been credited for 3–7% revenue improvement in the airline, hotel, and car rental industries [
In many service industries, capacity of supply is often fixed while demand is volatile. Hence, it is challenging for service companies to achieve a balance between supply and demand. To optimize revenues, RM models need to project demand first based on historical data. It then manages supply and demand through pricing, inventory control, and overbooking. Obviously, reliable forecasting is essential to the success of the revenue management system (RMS). Erroneous demand forecast may seriously impede the performance of RMS. Lee [
Forecasting, however, has not advanced as much as other RM components which ultimately depend on accurate forecasting [
With
To overcome these problems, it is necessary to extrapolate the true demand distribution parameters from censored booking data before putting them into the forecasting models. In the airline and hotel industries, this process is called
Wickham [
In view of the revenue benefit, one can see that demand unconstraining deserves attention from both researchers and practitioners. Forecasting based on unconstrained data can overcome the limitation of truncated demand due to the booking limits, better reflect true demand, and improve the accuracy of forecasting. In addition, demand unconstraining is also helpful to the allocation of fleet capacity.
General reviews of the RM literature can be found in [
Over the last decade, there has been extensive research on data unconstraining. Despite significant development in the area, more research seems needed compared to the advancement in other components of RM. The objective of this paper, thus, is to review most recent development of data unconstraining that has appeared in the literature and offer a broad overview of the technique in various industries. The paper cites numerous published journal articles, technical reports, working papers, and conference proceedings. It also examines important areas for future research. The comprehensive survey includes a bibliography of 132 articles.
The rest of this paper is organized as follows. Section
When customers’ booking requests for a certain class are accepted, the recorded booking data show true demand (see Table
Review point to days prior to mapping and booking matrix without demand censoring of a fare class.
Review point  1  2  3  4  5  6  7  8  9  10 

Days prior to departure  50  45  40  35  25  17  10  5  2  0 
 
True demand  0  14  38  50  71  91  103  108  104  102 
Booking limit  115  115  115  115  115  115  115  115  115  115 
Observed bookings  0  14  38  50  71  91  103  108  104  102 
 
Fare class status  Available  Available  


Review point to days prior to mapping and booking matrix with demand censoring of a fare class.
Review point  1  2  3  4  5  6  7  8  9  10 

Days prior to departure  50  45  40  35  25  17  10  5  2  0 
 
True demand  0  14  38  50  71  91  103  108  104  102 
Booking limit  85  85  85  85  85  85  85  85  85  85 
Observed bookings  0  14  38  50  71  85  85  85  81  79 
 
Fare class status  Available  Not available  Available  



Booking pace curve of a fare class.
An observation is considered censored (or constrained) if the booking limit in a given fare class at a specified review point in the history of the service product is less than or equal to the number of bookings present at that time.
A fare class is considered constrained if the observed demand in a given fare class at any review point in the life of the service product is censored (or constrained).
In light of the fact that firms really have a record of the actualnumber of bookings, a challenge faced by them is to estimate how many true demands would have been accepted without any constraint for their products. This process has commonly been called
In practice, forecasting consumes major resources of development, maintenance, and implementation time of an RMS [
Factors impacting on forecasting of RM.
No.  Factors 

1  Seasonality 
2  Dayofweek and timeofday variations 
3  Special events 
4  Sensitivity to pricing actions 
5  Demand dependencies between fare classes 
6  Group bookings 
7  Cancellations 
8 

9  Defections from delayed flights 
10  No shows 
11  Recapture 
Forecasting issues in RMS.
No.  Issues 

1  What to forecast 
2  Level of aggregation 
3  Unconstraining methods 
4  Number of periods to include in forecast 
5  Which data to use 
6  Outliers 
7  Reporting forecast accuracy 
8  Measurement and impacts on revenue 
As presented in [
A revenue management process.
Revenue management forecasting steps.
There are two reasons for obtaining unconstrained data using unconstraining methods. First, the number of forecasting models producing unbiased estimates from censored data is limited. Second, different units within a firm may use various forecasting techniques with no coordination. Although these forecasting models may deal explicitly with censored data, it would be preferable to unconstrain the data collectively and then have all the forecasting models that use the same unconstrained data [
Generally speaking, a firm facing censored sales data has five options: (1) directly observe and record latent demand, (2) leave data constrained, ignoring the fact of censorship, (3) use unconstrained data only and discard censored ones, (4) replace censored data using imputation methods, or (5) statistically unconstrain the data. These alternatives and related methods are illustrated in Table
Research on unconstraining methods used in RMS.
Approach  Model  Reference  Year  

Direct Observation  Directly Observe  Orkin [ 
1998  
and Record Latent Demand  Queenan et al. [ 
2007  
Ignore the censored data  Naïve #1 (N1)  
Discard the censored data  Naïve #2 (N2)  Saleh [ 
1997  
Imputation unconstraining  Naïve #3 (N3)  
 


Spill Model  Swan [ 
1979–1990  
Maximum Likelihood Estimation (MLE)  Brummer et al. [ 
1988 1990  
Booking Profile (BP)  Wickham [ 
1995  
Projection Detruncation (PD)  Hopperstad [ 
1995  
Pickup Detruncation (Pickup)  Skwarek [ 
1996  
Expectation Maximization (EM)  Salch [ 
1997  
Life Table (LT)  van Ryzin and McGill [ 
2000  
Observed Load Factor (OLF) table  Li and Oum [ 
2000  
Nonlinear Programming  Gao and Zhu [ 
2005  
Gao [ 
2006  
MultidistributionBased EM and PD  Guo [ 
2008  
Guo et al. [ 
2011  
Singleclass  Airlines 


BP, PD, and pickup  Skwarek [ 
1996  
N2, N3, BP, and PD  Skwarek [ 
1996  
Hopperstad [ 
1997  
N1, N2, N3, BP, and EM  Pölt [ 
2000  
N1, N2, N3, BP, PD, and EM  Weatherford and Pölt [ 
2002  
Zeni [ 
2001  
Zeni and Lawrence [ 
2004  
EM and PD  Chen and Luo [ 
2005  


BP and PD  Zickus [ 
1998  
Gorin [ 
2000  
Statistical Model Unconstraining  EM  He and Luo [ 
2006  
DES  Guo et al. [ 
2008  
Parametric Regression (PR)  Liu et al. [ 
2002  
Hotels  Liu [ 
2004  
Double Exponential Smoothing (DES) 
Queenan et al. [ 
2007  
Censored Demand EM Procedure  McGill [ 
1995  
Spill Model  Farkas [ 
1996  
Belobaba and Farkas [ 
1999  
Cumulative Expected Bookings  Mishra and Viswanathan [ 
2003  
Multiclass  Airlines  Q forecasting  Boyd and Kallesen [ 
2001  
Boyd et al. [ 
2004  
Hopperstad et al. [ 
2006  
Hopperstad [ 
2007  
EM  Karmarkar et al. [ 
2011  
RegressionBased Estimation  Ja et al. [ 
2001  
Correlated Demand Forecasting  Stefanescu et al. [ 
2004  
Stefanescu [ 
2009  
EM (Discrete Choice Model)  Talluri and van Ryzin [ 
2004  
Airlines  Vulcano et al. [ 
2010  
Multiflight  Multiflight Recapture Heuristic  Ratliff et al. [ 
2008  
Log Riskratio Estimation Heuristic  Talluri [ 
2009  
EM (Customer Choice Sets)  Haensel and Koole [ 
2011  
EM (Substitution Effects and Indirect Competitor Estimation)  Vulcano et al. [ 
2012  
Hotels  Twostep Decomposition (Marginal Log Likelihood Functions)  Newman et al. [ 
2012 
Direct observations of demand include records of bookings (requests that are met) and rejections (requests that are not met). The method may not be able to uncover true latent demand. Booking data censorship may be caused by availability (denials) or rate (regrets). Bookings declined due to availability are considered latent demand [
Firms invest in systems and train their managers in order to track turndowns directly and depend on these direct observations to unconstrain their sales data. Queenan et al. [
Ignoring censorship and performing estimates as if the censoring never happened, this approach is referred to as method Naïve #1 (N1) in [
As noted in [
The definition given in [
In recent years, statistical methods focusing on solving censored data problem have become a hotspot in research. “These models avoid the ad hoc nature of imputation methods and are built on a foundation of statistics theory. This is done at the cost of additional complexity and model assumptions that must be validated” [
As illustrated in Table
The singleclass methods stem from airline revenue management. Most of the early RM models make a potentially problematic assumption; customer demand for each of the fare classes is independent of the control policy implemented by the seller. That is, demand of any fare class does not depend on the selling status of other fares. Obviously, this may not be the case in reality [
The singleclass algorithms use univariate and disaggregate demand models. Because the RM optimization approaches (e.g., EMSR or EMSRb) require independent demand inputs, the singleclass unconstraining techniques perform best within this framework. The assumption that demands for each flight classes are uncorrelated makes these methods unable to capture demand interactions.
Brummer et al. [
Wickham [
Hopperstad [
In contrast to projection methods, the “Pickup Detruncation (Pickup)” proposed by Skwarek [
Salch [
Van Ryzin and McGill [
Li and Oum [
Gao and Zhu [
Zeni [
Similar to Skwarek [
He and Luo [
Sporadic literature reports have been received for demand unconstraining in the hotel industry. As noted by Orkin [
Liu et al. [
Queenan et al. [
As noted in [
Restrictionfree pricing (RFP) reduces customers’ switching costs between fare types and causes more pronounced downsell. Since then, capacity control becomes the only restrictions conducting fare class sales. In order to avoid multiple counted demands, the multiclass methods are developed. They capture the buyup and buydown interactions (vertical recapture) among different fare classes. Although they do not address crossflight horizontal recapture, compared with the singleclass methods, the multiclass methods are more practical and representative in the real world.
McGill [
Farkas [
Mishra and Viswanathan [
Boyd and Kallesen [
Karmarkar et al. [
Although many researchers have considered a buyup and buydown effect in traditional models, horizontal recapture is a complex function of how attractive different products are viewed in a market, especially the OD environment. Industry practitioners report horizontal recapture rates in the range of 15% to 55% [
Generally speaking, multiflight methods are probably the most difficult to calibrate because of the complexity of underlying demand models (e.g., multinomial logit, MNL, choice model), but they are able to unconstrain demands from bookings through vertical and horizontal recaptures under almost all combinations of open flights and fare classes. To some extent, they could eliminate “double counting” effects.
Ja et al. [
Stefanescu et al. [
Talluri and van Ryzin [
Vulcano et al. [
Ratliff et al. [
Talluri [
Haensel and Koole [
Vulcano et al. [
Newman et al. [
As addressed by Liu et al. [
There has been a great deal of theoretical and applied research on censored data analysis in reliability engineering, biomedical sciences, and econometrics (see [
Research on data unconstraining in RMS has made great advancement in various industries and also unfolds directions for future studies.
Customer choice models allow consumers to make their purchasing decisions on alternatives they are given. In recent years they have gained increasing attention in RM. Literature such as Talluri and van Ryzin [
Although looking promising theoretically, the choice models have not been widely adopted in practice [
In addition, as shown in Table
Ratliff and Research [
Generally speaking, most of the existing unconstraining methods, such as EM, PD and others, assume known booking curve or the distribution. But in reality, firms often do not know a priori the shape of the booking curve. In some situation, the traditional assumption of normal distribution for the nominal demand is usually inappropriate. Some researchers, such as Lee [
As shown in [
If information on customer behavior is available, then choice models with unconstrained demand data provide flexibility, in addition to gaining forecasting or optimization power [
Clearly, robust optimization method is another way to solve the problems resulting from forecasting inaccuracy and data insufficiency. It is worth investigating how limited demand information can be estimated through the use of demand unconstraining methods, and what kinds of robust unconstraining methods make the robust optimization policies more effective.
The forecast method and the procedure used to update forecast parameters are important factors to determine the choice of the unconstraining method in RMS. There exist coordination problems between unconstraining and forecasting methods. Similarly, attention needs to be paid between the unconstrained estimation and seat optimization methods.
Skwarek [
Admittedly the methods reviewed in these studies are still limited. To evaluate latest proposed methods, especially the multiflight unconstraining methods, it is necessary to conduct simulation tests under various market conditions. The PODS tool or other simulation methods [
“Revenue management can be defined as the art of maximizing profit generated from a limited capacity of a product over a finite horizon by selling each product to the right customer at the right time for the right price. It encompasses practices such as pricediscrimination and turning down customers in anticipation of other, more profitable customers.” [
The problem of censored data analysis not only exists in the RMS of airline and hotel industries, but also in many other fields, such as reliability engineering, biomedical sciences, and econometrics. Despite the growing amount of literature, more research of unconstraining methods in RMS is needed for emerging problems. For instance, what are appropriate techniques that fit choice models, especially those applied in an RM network environment; can new robust approaches reduce the number of iterations and counterintuitive results in the process of parameter estimation; will robust optimization policies become more effective if limited demand information can be estimated through the use of unconstraining methods; what are more practical and robust multiflight methods based on multidistribution assumptions; how to evaluate the performance of currently proposed methods by conducting simulation tests under various market conditions.
This work was supported in part by the Major Program of the National Natural Science Foundation of China (71090402), the Program for Changjiang Scholars and Innovative Research Team in University of China (IRT0860), and the Fundamental Research Funds for the Central Universities (SWJTU11ZT32). The authors are grateful to Mark Ferguson for sharing unpublished working papers at various stages of preparation of this manuscript. They also would like to thank the valuable comments and suggestions of two anonymous referees.