1. Improved Forecast Accuracy in Airline Revenue Management by Unconstraining Demand Estimates from Censored Data Richard H. Zeni

2. DISSERTATION.COM Improved Forecast Accuracy in Airline Revenue Management by Unconstraining Demand Estimates from Censored DataCopyright 2001 Richard H. Zeni All rights reservedDissertation.com USA *2001 ISBN: 1-58112-141-5 www.Dissertation.com/library/1121415a.htmChapter 1 Introduction Accurate forecasts are crucial to a revenue management system Poor estimates of demand lead to incorrect decision making and less-than-optimal revenue performance Revenue management was invented by the U S. airlines in the 1980s in response to a newly deregulated industry and to the increased competition that was created. Since then, it has been adopted by a variety of industries, and the list is constantly growing. But the basic concepts have been around for quite a long time. Consider the grocer who has a supply of perishable fruit. He must price the fruit so that he maximizes his revenues and avoids having the fruit spoil. If he prices it too low, he will sell out too soon and will need to turn away customers when his stock runs out. He would have been better off setting a higher price. If he prices the fruit too high, he will not sell his entire stock and will be forced to dispose of some fruit and take a loss on that inventory. The grocer makes his pricing decision based on his forecast of demand for the fruit. He must predict the quantity of fruit demanded at different price levels. As time goes by and the fruit approaches its final sale date, the grocer will reforecast demand and adjust his price accordingly For example, he might lower his price if he sees that the fruit is not selling as fast as he thought and is about to spoil. Forecasting the demand for his product is more difficult when he runs out of fruit It is difficult for him to estimate how much fruit he would have sold if he had more in stock. In this sense, his data is censored. 3. 2- -Tlie example of the grocer illustrates a low-tech version of revenue management and of the challenge of forecasting demand with censored data, A more sophisticated application of revenue management is practiced by the airlines. They amass large numbers of reservations booking data, and they use it to produce forecasts of demand for airline travel at different price levels. Then they use complicated optimization models to set seat inventory levels for different products. They repeat the forecasting and optimization until the flight departs and the leftover seats spoil, just as the perishable fruit spoils. Hotels use revenue management to overbook rooms and discount rates for nonpeak periods.1 Rental car companies, theaters, and radio stations all use revenue management to manage their perishable inventory.2 Revenue management was developed by the airlines to improve revenue performance in the face of increasing competition. It was obvious to the airlines that passengers could be divided into two broad categories, based on their travel behavior and their sensitivity' to prices. There were business travelers and leisure travelers. Business passengers tended to make their travel arrangements close to their departure date and stay at their destination for only a short time There was usually little flexibility in their plans, and they were willing to pay higher prices for tickets. Leisure tras'elers, on the other hand, booked their flights well in advance of their travel date They stayed longer at their destinations and had much more flexibility in their plans. They would often decide not to travel rather than pay high fares, and flights often departed with empty seats The challenge to the innovators of revenue management was to devise a plan that would make the empty seats available at the lower fair while preventing passengers who were willing to pay the higher fare from buying low-fare seats. Since low-fare 4. -3-passengers typically book in before higher fare customers, the revenue management system must forecast how many business passengers will want to book on a flight. Then it must set aside or protect these seats so that they will be available when the business passengers request them. Accurate forecasts of passenger demand are crucial. If the forecast for business passengers is too high, then too many seats will be protected for these passengers. The flight will depart with empty seats that could have been sold to leisure passengers. If the forecast for business passengers is too low, then too few seats will be protected. Seats will be sold to leisure passengers that could have been sold to higher-fare business passengers. Forecasting demand accurately is inherently difficult since the historical data upon which forecasts are based often do not reflect the true demand. Once an airline stops selling tickets at a particular fare, due to the limits set by the revenue management system, it also stops collecting data. The airline may receive many more requests for a particular fare, but these requests are not recorded the data is censored and does not represent true demand. When censored data is used to represent historical demand, it often results in forecasts with a negative bias. If a revenue management system uses these biased forecasts, then the resulting inventory controls will tend to save too few seats for high- fare passengers. Seats that could be sold to high-fare passengers may be sold to low-fare passengers, and revenue will be lost. It has been estimated that up to 3.0% of potential revenue may be lost if the forecasts used by a revenue management system have a negative bias.1 Tlierefore, some attempt should be made to transform the censored data into more accurate estimates of actual historical demand. Various methods exist that take the observations that have been constrained and unconstrain them so they represent the actual demand. These methods range from 5. 4- -the simplistic, such as discarding all censored observations, to the complex, such as calculating the expected value of the true demand via the Expectation-Maximization algorithm. However, little research has been done to determine which methods work best. Airlines tend to use the simple heuristic methods. While they recognize that these heuristics are not adequate, they are hesitant to invest in other techniques due to the lack of evidence that alternative unconstraining will produce more accurate forecasts. Airlines use complex revenue management systems to determine the number of seats to make available at different fares. First, a forecast of demand is produced from censored data. Based on that forecast, booking limits at the various fare levels are set. Reservation requests are then either accepted or denied based on the booking limit. The bookings that are accepted become the historical data for the next forecast. The bookings that are denied are not recorded, hence the censored data The censored data is then unconstrained so that it represents the true demand, and the process begins again. This feedback loop is illustrated in Figure I I 6. Figure 1.1: Revenue Management Process1.1Goal of Dissertation This research project examines the challenge of forecasting in the presence of censored data. The 7. -6-goal of this dissertation is to improve forecast accuracy by unconstraining the constrained observations. Each of the unconstraining methods are examined and compared to determine which methods produce the best estimate of true demand The degree of improvement in forecast accuracy that can be gained by unconstraining data were investigated The costs of using this data in a revenue management system are investigated. While this dissertation will focus on forecasting for airline revenue management, the methods discussed and developed here easily generalize to other industries.1.2Structure of Dissertation The remainder of this paper is divided into five chapters. The following sections in this chapter are areview of what revenue management is and how airlines and other transportation companies use it in an attempt to maximize revenue The different levels at which revenue management can be applied are introduced and illustrated. The various optimization methods are surveyed Chapter 2 contains a literature review of forecasting for revenue management, and it surveys the current practices. Also, the discussion of censored data begins here. The challenges that this data present to forecasting are examined, and the cost of ignoring the problem is investigated. The literature is reviewed to present an overview of the available methods for dealing with the censored data problem. Simple ad-hoc and more complex statistical methods are discussed. The strengths and weaknesses of each are examined, and applications outside of the transportation industry are discussed Prior to analyzing methods for unconstraining censored data, a reliable data set needed be built to simulate censored and uncensored data. In Chapter 3, a new technique developed as part of this research project is presented This method transforms uncensored data into data with censored observations Chapter 4 contains a comprehensive analysis and comparison of the available unconstraining methods. Each method is applied to a sample data set. The improvement in forecast accuracy from each of these techniques is evaluated An extension to the EM algorithm developed as part of this research project is presented. 8. Chapter 5 concludes the dissertation. The research findings are summarized, and future research directions are outlined. Application of the analyzed methods is discussed with respect to practical considerations.1.3Revenue Management Defined There are several definitions of revenue management (also referred to as yield management) inthe literature American Airlines (1987) defined the goal of yield management as to maximize passenger revenue by selling the right seats to the right customers at the right time.4 Pfiefer (1989) described airline yield management as the process by which discount fares are allocated to scheduled flights for the purposes of balancing demand and increasing revenues From the hotel industrys perspective it has been defined as charging a different rate for the same sendee to a different individual 6 and controlling the tradeoff between average rate and occupancy. Weatherford and Bodily (1992) have concluded from the above definitions that the term yield management is too limited in describing the broad class of revenue management approaches * After analyzing situations in which yield management was used, they concluded that these situations had the following characteristics in common:1 There is one date on which the product or service becomes available and another after which it is either not available or it spoils. The product cannot be stored for significant periods of time-lt eventually perishes. In the grocery store example, the fruit would spoil. 2There is a fixed number of unils. Capacity cannot be changed in (lie short term In the hotel example, there are only so many rooms that may be sold at a given property location3.There is the possibility of segmenting price-sensitive customers. In the airline 9. 8- -example, vacation travelers are much more sensitive to price than business travelers.Weatherford and Bodily proposed the term perishable-asset revenue management (PARM) to define this class of problems and described it as the optimal revenue management of perishable assets through price segmentation.1.4Origins of Revenue Management The roots of modem revenue management can be traced back to the early days of the U.S. airlineindustry Prior to the Airline Deregulation Act of 1979, fares for airline travel in the United States were regulated by the Civil Aeronautics Board (CAB), The CAB ensured that the airlines operated in a highly controlled environment designed to serve the public convenience and necessity.9 The CAB required economic justification for any fares proposed by the airlines. Thus, there were few fares for customers to choose from. In the 1930s all airlines offered all seats on a flight for the same price. But it was obvious to the airlines that passengers could be divided into two broad categories, based on their travel behavior and their sensitivity to prices. There were business travelers and leisure travelers. Business passengers tended to make their travel arrangements close to their departure date and stay at their destination for only a short time. There was little flexibility in their plans and they were willing to pay higher prices for tickets. Leisure travelers, on the other hand, booked their flights well in advance of their travel date They stayed longer at their destinations and had much more flexibility in their plans. They would often decide not to travel rather than pay high fares. Since there was only one fare offered to both types of passengers, many of the leisure passengers chose not to fly, and many flights departed with empty seats. Airline managers saw an opportunity to increase revenue by lowering fares in certain markets. The first experiment to offer low-fare sendee occurred in California on die San Francisco-Los Angeles route in 10. -9-1940.'" United Airlines began its Sky Coach Sendee using 10-passenger Boeing247s and charging a oneway fare of S13.90. The CAB approved the low fares based on the lower operating cost of the B-247s and fewer amenities offered on board. The experiment was a success but ended shortly thereafter when the airlines fleet was turned over to the armed forces during World W'ar II. Throughout the next few decades, other discount fares were offered with varying degrees of success. First-class and coach-class became standard on all airlines But the airlines were not permitted to offer different fares within the coach cabin and prices were set through a cost-plus pricing formula administered by the CAB. Carriers gradually became less efficient at operating their airlines, and coach fares rose over time as average costs increased. During the 1960s, the CAB began approving new types of fares such as night coach fares and 7-21 day excursion fares based on length of stay. However, the airlines placed no limits on the number of seats that could be sold at these fares, and all were available on a first-come, first-served basis. 11. -II-In the early 1970s, the CAB responded to demand for more discount fares by easing regulations for charter airlines. With their lower operating costs, the charter carriers w'ere able to offer low fares and still earn a profit. For example, in the winter of 1976, passengers could travel from New York to Florida on a charter for as little as $99." This fare was less than the average cost for a major airline to fly that market. So if the airline matched the charter fare, then it would lose money on the Bight, even if it filled every seat. This situation caused concern among the managers at the major airlines. Their initial thought was to figure out a way to reduce costs so they could remain competitive. But that was impractical. The costs of operating a major airline u'ith its staffing and airport needs were simply much higher than the cost of running a charter operation But then the executives at American Airlines realized something. On average, their planes were departing with half their seats empty'. While the average cost of these seats was higher than the charter fares, the marginal cost was close to zero. So if they could find a w>ay to sell just the empty' seats at the charter fares, profits would increase dramatically. The challenge was to devise a plan that would make the empty seats available at the lower fair, while preventing passengers who w'ere willing to pay the higher fare from buying low-fare scats. American Airlines response to this challenge was the introduction of Super Saver Fares in 1977 With these fares came the beginning of modern day revenue management.12 12. 12--1.5Seat Inventory Control The Super Saver Fares were the first capacity-controlled, restricted discount fares That is, theywere offered in limited numbers and certain conditions had to be met for the fare to be valid. For example, the tickets needed to be purchased at least 21 days in advance of travel, and the itinerary had to include a Saturday night stay These restrictions were meant to prevent the high-fare passengers from purchasing the low-fare tickets. It reflected the airlines belief that business travelers did not have enough flexibility' in their plans to meet the restrictions and, therefore, would continue to pay the higher fares. American began its Super Saver Fares by offering approximately 30% of the seats on each flight to these fares." But they soon found the number of seats needed to be controlled carefully to increase total revenue. If too many discount seats were sold, then the airline would turn away late-booking, high-fare business passengers. If too few seats were sold to discount passengers, then the planes would depart with empty' seats. The correct number of seats to be allocated to the discount passengers could only be calculated from an accurate forecast of demand for high-fare tickets Research thus began to develop the appropriate models to forecast demand and calculate discount seat allocations. Since the first Super Saver Fare appeared on the market, the airline pricing structure has changed dramatically Airlines publish a variety of fares in an attempt to segment the market. Their goal is to design what is referred to as a fare product for each segment of the market. These fare products are differentiated by advance purchase restrictions, minimum stay requirements and penalties for refunds. The fare products correspond to the price elasticity airlines have identified among their customers. For example, discount passengers who desire a low price must be willing to purchase their tickets weeks in advance of travel and stay at their destination for at least one Saturday night. If they cancel or change their plans, then they will be charged a penalty. On the other hand, business travelers place a high value on flexibility. They may purchase their tickets at any time and change their reservations without penalty. There are no restrictions on the amount of lime they must stay at their destination. For this flexibility, 13. -13the business traveler is willing to pay a higher fare than the leisure passenger. 14 So while these two types of passengers may be seated next to each other on a flight, they are paying different prices and receiving different products. Airlines use single-letter class codes to distinguish between different fare products For example Y might be used for full-fare coach, M and Q for discounts, V for deeper discounts, and other classes which vary by airline.15 There are often six or more different fare classes offered by a singe airline in a given origin and destination (O&D) market in the coach cabin of the aircraft. An example of a typical airline fare class structure is given in Table II.Fare ClassFare Product TypeYFull coach fare w'ith no restrictionsB MUnrestricted discounted one-way fares Seven-day advance purchase with minimum stay requirements0 VFourteen date advance purchase with minimum stay requirements Deeply discounted twenty-one-day advance purchase with minimum stay requirementsTabic 1.1: A Typical Airline Fare Class StructureModern revenue management systems forecast demand for each one of these fare classes by using historical booking data from the same fare class of similar flight departures. This data is usually aggregated by departure time, day of week and time of day.16 These forecasts are then used as inputs to optimization models that calculate booking limits and control the number of seats available at various fare levels. Obviously, an airline would like to carry' as many of the high-fare business passengers as possible. Only those seats that cannot be sold to business passengers should be made available to the leisure passengers. The problem is that leisure passengers tend to book their reservations first. And even if they did not, the advance- purchase restrictions most airlines place on leisure-fare tickets often force 14. -14-this behavior. So, before any seats are sold, the revenue management system must forecast how many business passengers will still want to book on a flight after the leisure passengers have made their reservations. Then it must set aside or protect these seats so that they will be available when the business passengers request them The seat inventory' control problem has been approached from a variety' of perspectives. Seat inventories can be controlled over individual flight legs (the takeoff and landing of one flight) or over the airlines entire route network Most airlines manage seat inventories by fare class at the leg level. That is, they attempt to maximize revenue on each individual flight leg. Reservation requests are evaluated by the airline based on the availability of a particular fare class on each flight leg. A passengers entire origin and destination itinerary' is not taken into account when the decision is made. 17 In a leg-based system, the inventory controls limit the number of seats that may be booked in a particular fare class on a given flight. Of course, the inventory' controls will only be as good as the forecasts of demand for these fare classes. In general there are two consequences to a bad forecast. If the forecast for business passengers is too high, then too many seats will be protected for these passengers. The flight will depart with empty seats that could have been sold to leisure passengers. If the forecast for business passengers is too low, then too few seats will be protected. Seats will be sold to leisure passengers that could have been sold to higher-fare business passengers.1.6Booking Limits and Nesting The objective of a revenue management system is to set booking limits at different levels ofcontrol in an attempt to maximize revenue As noted above, first a forecast of demand is made, and then optimization is performed to calculate protection levels. Thus, a certain number of seats is protected from being sold to low-fare passengers. The logic is that if a certain number of high-fare passengers are 15. -15-expected to book, then seats should be set aside so that they will be available when the requests are made. For example, suppose there is a forecast for the highest value fare class (Y) and the subsequent optimization produces a protection level of 40. At least 40 seats should be protected for these Y-fare passengers and not sold to anyone else. However, what happens if more than 40 Y-fare passengers request seats? The airline would not want to deny these requests. To eliminate the possibility of turning away high-fare passengers when there are seats available, airline reservation systems usually nest the booking limits.18 Nesting allows high-fare passengers to book seats that are available to lower-fare passengers. Any seat available at a particular fare should also be available at a higher fare. In the example illustrated in Table 1.2 there are 100 seats available to be sold on the flight leg. Forty seats are being protected for Y passengers, but the entire inventory of seats (100) is available to be booked by these passengers. So the booking limit for Y is 100. To arrive at the booking limit for M, the protection level for Y is subtracted from the total capacity on the aircraft (100-40=60). Now suppose the protection level for the Y/M nest is 65. So 65 seats are protected for sale to Y or M passengers. To arrive at the booking limit for V, the Y/M protection level is subtracted from the remaining capacity (100-65=35). Therefore, the booking limit is given byBooking Limit, =(C-0H) whereC is the remaining capacity in the aircraft cabin and O, is the protection level of the /"' fare class nest. 16. -16Fare class(I)Total Protection Level (0,)Nested Booking Limit (C-0..,)1 (Y)401002 (M)65603(V)-35Table 1.2: Nesting1.7on a Flight Leg with O 0 anil C'-lOOLevels of Seat Inventor}' Control The above example illustrates the concepts of booking limits and nesting at the flight leg level.That is, all the forecasting and optimization was done for an individual flight making one departure and arrival While many airlines operate their revenue management systems at the leg level, other systems have been developed to provide more detailed levels of control. Some of the different levels of control are: First-Come, First-Served (no control)Leg LevelVirtual NestingOrigin and Destination Itinerary LevelIn the following sections, the various levels of inventory control will be discussed by reviewing the available literature in this area. They will also be illustrated using a simple hub and spoke network example. Consider the sample network in Figure 1.2. There are four airports; Los Angeles (LAX), Pittsburgh (PIT), Boston (BOS) and Miami (MIA). Pittsburgh is the hub airport and the others are the spokes. Most major airlines use this type of network structure to maximize connecting opportunities. There are three possible itineraries, each of which may be booked in one of three fare classes, for a total of nine origin/destination/fare class (ODF) itineraries. For example, the LAX-PIT-MIA itinerary booked in the Y class has a fare of S400 and requires the passenger to connect in Pittsburgh. The PIT-MIA itinerary' is non-stop, which is also referred to as a local itinerary. 17. -17-Figure 1.2: A Simple Airline NetworkNow suppose there is only one seat left to sell on the PIT-MIA leg, and assume that low-fare passengers make their reservations strictly before high-fare passengers. Also assume that there is deterministic demand greater than one for all itineraries and fare classes. Since there is demand for nine origin/destination/fare class (ODF) itineraries and only one seat left to satisfy the demand, the question is, who will get the seat under the various levels of control?1.7.1First-Come, First-Served Airlines began charging different fares for the same seats in the coach cabin in the I960s whenthe Civil Aeronautics Board began approving new types of discount fares that could be sold to passengers flying at off-peak hours or who purchased their tickets 7 to 21 days before departure. At this time the airlines placed no limit on the number of seats that could be sold at any particular fare. So all fares were available on a first-come, first-served basis.1 While this provided a revenue opportunity for the airlines to sell seats that would otherwise go empty, it also created a problem. The demand for low-fare tickets was often quite large. And that low-fare demand materialized before the high-fare demand Therefore, airplanes 18. -18-were being filled with low-fare leisure travelers and the high-fare business travelers were being turned away because there were no seats available when they made their requests. This situation is illustrated in the example network in Figure 1.2. First-come, first-served means there are no inventory controls at all. Whoever requests the seat first will get the seat. Under the low before high assumption, the seat will be sold to a V- fare passenger. There is no information on which Vfare passenger will book first, so revenue will be$150, $175 or $200.1.7.2Leg-level First come, first served obviously did not provide the means for airlines to meet their goal ofmaximizing revenue. They needed to limit the amount of seats that were available at discount fares. American Airlines was the first to place leg-level limits on the number of seats that could be sold in any given fare class. In 1977, the airline limited the number of its new Super Saver Fares to 30% of the seats on each flight leg.20 But they soon found that a more sophisticated method of determining booking limits was needed On some flights, too many discount seats were being sold, and the airline would turn away the late-booking high-fare business passenger On other flights, too few seats were sold to discount passengers, and the planes would depart with empty seats. The 19. 19--correct number of seats to allocate to the discount passengers could only be calculated from an accurate forecast of demand for the higher fare tickets. In a leg-level revenue management system, seats are protected for each fare class on a flight leg, based only on the revenue contribution from selling the seat on that leg. In other words, the passengers full origin and destination itinerary is not considered when the booking limits are calculated.*1 In the above example, there is sufficient demand for the Y class on the PIT-MIA leg to protect the seat for a Y-class passenger. There is no information on which Y-class passenger would book first, so revenue would be $325, $375 or $400. The resulting protection levels and booking limits are illustrated in Table 1.3.Fare class(i)Total Protection Level (0,)Nested Booking Limit (C-0,.,)1 (Y)112(M)0 -03 (V)0Tahle 1.3: Protection levels and Booking Limits for a Leg-Level KM System1.7.3Virtual Nesting While leg-level revenue management systems were a vast improvement over first- come, first-served, they did not provide airlines with the ability to discriminate between different passenger types traveling in the same fare class. In the example above, the leg- level booking limits prevented the last seat on the flight from being sold to a low-fare M or V passenger, but they were not able to choose between the various Y-fare passengers. Ideally, the system should allocate the seat to the $400 LAX-PIT-MIA Y-fare passenger and to no one else That is, the inventory controls would be at the itinerary/fare class level. Then revenue 20. 20--would be maximized. However, the enormous number of controls required for an itinerary-level system made it infeasible in the early days of revenue management. In 1983, American Airlines developed a method of clustering the number of fare class into smaller buckets, thus reducing the number of inventory controls to a manageable number. 22 In this virtual nesting system, seats are protected for virtual buckets rather than fare classes. The buckets are defined by a range of values, and the ODF itineraries are mapped to the buckets as in Table 1 4Virtual Bucket 1Value Range S350-S400Mapping LAX-MIA-Y BOS-MIA-Y$300-5349PIT-MIA-Y2 LAX-MIA-M 3$250-52994$200-52495BOS-MIA-M$0-5199PIT-MIA-M LAX-MIA-V PIT-M1A-V BOS-MIA-VTabic 1.4: Virtual Nesting MappingIn our example, the seat would be protected for a bucket 1 passenger Whichever bucket I passenger books first would get the seat. Revenue would be either $375 or $400 Thus, while not yet able to choose the passenger traveling on the highest valued itinerary', You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. 21. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. of holding the Olh seat So if the protection level is say 6, and the probability of selling that 611 seat to a full-fare passenger is only 0.4, then the protection level should be lowered.1.8.1Expected Marginal Seat Revenue (EMSR) Peter Belobaba took Littlewoods rule, generalized it to more than two fare classes, and called it the ExpectedMarginal Seat Revenue (EMSR) method. EMSR calculates protection levels by equating the immediate revenue from accepting a low-fare reservation with the expected revenue from saving that seat for a higher-fare passenger.25 26 The protection level 0, for the highest value fare class is obtained from/,=MSR,(,) = /, Pr [-Y, > 0,)(1.2)where f is the average fare for fare class /, A', is the demand for fare class 1, and EMSR| (0,) is the expected marginal seat revenue of the 0'11 seat in fare class one, which is simply the probability Pr[ ] of selling 0, or more seats multiplied by the fare level in fare class 1 The total protection 0, for the two highest fare classes is obtained from 0,=^+tf,2(1.3)where 0 and 8,2 are the individual protection levels from /,=/, Pr [ X ,>8(1.4)and You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. 22. -22-expected marginal seat revenue from holding each additional seat for fare class 1 As long as the i'.MSR, curve is above the f2 line, the expected value from holding the additional seat is greater than the immediate return from selling it and the protection level should be increased. To see this, start with a protection level of 0. The probability that at least zero seats will be sold in fare class 1 is 1: IXS400 = S400 > $200. So at least zero seats are always protected. Now increase the protection level by I. Now the probability of selling at least 1 seat in fare class I is 0.9: 0.9x $400 = $360 > $200 So the protection level should be increased. When the protection level is 5, the probability of selling at least 5 seats in fare class 1 is 0.5: 0.5x5400 = S200 = /,. The algorithm stops here and the protection level is 5. The booking limit for fare class I is the remaining capacity of the aircraft The booking limit for fare class 2 is the capacity minus the protection level for fare class 1. Notice that theEMSR, curve was never used and therefore the demand forecast for the lowest fare class is irrelevant. 23. -231.8.2Network Formulations While EMSR works well for leg level and virtual nesting revenue management systems, mathematicalprogramming models have received most of the attention for calculating network level controls. The deterministic linear program and the probabilistic nonlinear program are illustrated here. These formulations can be found in Williamson's 1992 thesis. Both the deterministic and probabilistic mathematical programs for network control consist of an objective function to maximize revenue (or expected revenue in the probabilistic case) and a set of constraints. In the linear program there are two types of constraints, the capacity constraint on each flight leg and the demand constraints You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. 24. -24Sensitivity to pricing actions: Price increases and decreases result in demand decreases and increases respectively, but different passenger types have different elasticitiesDemand dependencies between fare classes: Passengers who book full-fare seats might have met the restrictions for lower fare seats, but there was no availability. Passengers who were able to book the Iow-fare seats might have been willing to purchase a high-fare ticket had the low'er fare not been available.Group bookings: Groups tend to book and cancel reservations in large numbers, but this data is used to allocate seats to individual passengersCancellations: A revenue management system requires a forecast of how many passengers will book and travel in each fare class. Since some passengers make reservations and subsequently cancel them, this behavior must be considered.Censoring of historical demand data: Aircraft capacity and booking limits constrain the demand seen in the historical dataDefections from delayed flights: When a flight is delayed, some passenger decide not to travel. The data reflects their behavior as cancellations, but their desire was to travel. 25. No-Shows: Some passengers make reservations, decide not to travel, and do not cancel their reservations This occurs often with passengers traveling on tickets with no cancellation penalties.Recapture: Flight cancellations or limited aircraft capacity may cause passengers to travel on flights other than those originally desired.Each of these factors presents a challenge of its own when forecasting for a revenue management system. In 1989 there were 30,000 daily fare changes in the U.S. domestic airline industry. 2 Accounting for price changes alone presents a significant challenge.2.1 Types of Forecasting There are three primary categories for the types of forecasts used in the airline industry: macro-level, passenger choice modeling, and micro-level. A description of each, as well as a comprehensive literature search, can be found in the dissertation by Lee.5 Macro-level forecasts are usually made for aggregate forecasts of total airline passenger demand. For example, a macro-level forecast might be a projection of total annual domestic air travel or future air travel between the United States and Asia Passenger choice models attempt to predict future demand by modeling current passenger behavior based on socioeconomic factors and the characteristics of alternative travel options For example, these models might be used to forecast an individuals choice of air travel as opposed to rail travel or the choice of a particular airline over a competitor. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. 26. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. hand, a seasonal index, a day of week index, and a historical mean of bookings-to-come as the explanatory' variables. Here he obtained better results. However, Sa did not test the forecasting ability' of the models or incorporate the effects of booking limits into the statistical analysis. In 1988, Brummer, et. al. performed a study that explicitly took into account the fact that booking data is constrained by the booking limit.10 The authors were attempting to find the true mean and standard deviation of the uncensored log-normal distribution, given a data set with some constrained observations. Most of the study was spent on the mathematical derivation of the likelihood function of the censored log-normal distribution. It included neither a class by class analysis of bookings nor an attempt to forecast future demand using the proposed model Ben-Akiva et. al. (1987) propose three models for flight-specific, class-specific reservations forecasting: a regression model for advance bookings on a given flight, a time series model for historical booking on previous departures of the same flight, and a combined advance bookings/historical bookings model. 11 The preliminary analysis presented in the paper showed that the combined model outperforms the historical bookings and advance bookings models While their results were promising, the authors did not have sufficient data to validate the results of the estimated models on future flights. Also, they used monthly data rather than the daily data that is required in microlevel forecasting of the booking process. The booking limits placed on fare classes often result in censored data. Various methods have been examined to estimate a forecasting model with censored data. The book by Maddala (1983) contains a chapter on censored and truncated regression You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. 27. -27-Week -4 -3 -2 -1 0 1 2 3DP 0DP 7DP 14DP 21DP 60DP 90DP 360100 979070552565502310 908594 9180 7760 5745 4221 198 7088 -73 7054 5039 3517 156 50--45 --301340251130000Tablc 2.1: Bwikings MatrixThis is an example of the booking history of a flight over an eight week period. Week 0 refers to the most recent departure while weeks with negative numbers correspond to flights that departed in the past. Weeks with positive numbers are for future flights. For example, week -3 represents a departure 3 weeks in the past, and week 3 refers to a departure 3 weeks in the future The DP columns represent the current bookings on hand at the corresponding number of days prior to departure. For example, the DP 7 column indicates the number of bookings on hand 7 days prior to the departure date of the flight. DP 0 is the final count of bookings at departure time. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. ForecastM = ax. Actualt+(-a)x Forecast,. (2.1) 28. Equation 2.1 produces a forecast at time I /by multiplying the smoothing parameter, alpha, by the actual observation at time /, then adding the remaining weight multiplied by the previous forecast. For example, suppose the smoothing parameter has a value of.15. The new forecast will place 15% weight on the new' observation and 85% weight on the old forecast The choice of the smoothing parameter will, therefore, have an effect on the responsiveness of the model For small values of alpha, the forecast will respond slowly to changes in the data, resulting in a relatively stable forecast. For large values of alpha, the forecast will respond quickly. Unfortunately, the changes in the data may reflect a new- trend or it may be a result of random fluctuations The latter could introduce an undesirable level of instability in the forecast. The competing needs for a forecast that responds quickly to new trends and a forecast that is stable must be considered when choosing the smoothing parameter. This is usually done by careful analysis of historical data. It is apparent from Equation (2.1) that exponential smoothing requires only a small amount of data storage. Only the most recent observation, the most recent forecast, and the smoothing parameter need to be stored2.4.2Moving Average This model produces a forecast of future demand by averaging the II most recent historical observations. It istermed a moving average because a new forecast may be You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. 29. -29-where Bookings un is the final bookings at departure, BookingsDn is total bookings at 7 days prior to departure,BookingsDn4 is total bookings as of 14 days prior to departure and /?, /?, and p. are parameters to be estimated. One variation of this model would be to use bookings in higher-valued fare classes as the independent variables.17 The model would then reflect the fact that demand for all fare classes is interrelated. The drawback of the regression model is the assumption of linearity. Also, notice that there are no economic causal variables in this formulation that one might expect to see in an econometric model of demand The forecast of future bookings is entirely dependent on the prior booking activity' and on the estimated linear trend it will follow Any deviation from that trend will result in a forecast bias2.4.4Additive Pickup Model A "pickup model produces a forecast of total bookings at departure by adding the historical incrementalbookings to the current bookings at a given day prior to departure. This implies that final bookings are a function of current bookings on hand and on the amount picked up between the current day and departure For example, the relationship between bookings at departure and day prior 14 can be expressed asBookings Dn = Bookings Bpu + PUomi(>) You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. departure Since the Classical Pickup Model uses data only from departed flights, the average pickup would be calculated by subtracting the average bookings on DP 14 from the average bookings on DPO for the historical flights. In this example the simple average of 2-week pickup is 33. The forecast of final bookings would therefore be 30. 45+33=78. The average pickup using the Classical Pickup Model is therefore expressed as:PUonxm = BookingsDn - Bookings urxwhere Bookings,lptl is the average number of final bookings at departure and BookingsDFX is the average number of bookings at day prior X. The example illustrated above used the mean of the previous 4 departures to calculate the average as follows:Bookings ^where BookingsaKBookings DK II isois the number of bookings at day prior X for each of the n historical departures Exponentialsmoothing may also be used to calculate the averages. In that case the average would be given by:BookingsDpx = ax BookingsDpx +(1 OC)x Bookings Dpx 31. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. 32. -32-The data to be used in calculating the pickup ratio depends on the methodology chosen. As in the case of the Classical and Advanced Pickup Models, data may be included from only departed flights or from recent booking activity Using the former scenario, return once again to the example illustrated in Table 2.2. Again a forecast of final bookings is to be generated in week 2 at 14 days prior to departure. The average pickup ratio for DP14 is 1.54. The forecast is then 45(1.54)=69. It is important to note that none of the models described above use any information other than historical bookings to predict future demand. There are no causal variables such as population, employment, income or other economic activity. That is why it is critical that the historic bookings data be an accurate representation of demand. Any weakness in the data will be reflected in the accuracy of the forecast. 33. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. 34. -34-revenue resulting from the inventory controls set by the revenue management system, and an evaluation is made But if the demand data is constrained by the very' controls that are being evaluated, then no reliable results can be obtained. These revenue opportunity models are an important part of the evolution of revenue management systems. Revenue management systems are usually developed with the aid of simulation. The simulations are naturally fraught with many assumptions. It is not until the system is actually implemented at an airline that any real results are known. Unfortunately, evaluating the performance of a revenue management system is made difficult by the presence of censored data. In fact, the higher the demand, the greater the potential for a revenue management system to increase revenue. But the higher demand results in more censoring of the data and greater difficulty evaluating the system In the development stages of revenue management systems, censored data plays an important role as well As noted above, simulation is often used to develop and evaluate different models. The simulators use historical data to represent demand If this data is censored, then it does not represent true demand. So the simulation is performed based on an inaccurate assumption of the distribution of demand,2.6.1 Sensitivity Analysis A sensitivity analysis performed by Weatherford (1997) has demonstrated the costs of using a negatively biased forecast in a yield management system.21 When forecasted demand is 12.5% lower than actual demand, revenue can decrease by as much as 7% to 1.2% When the forecast is 25% lower than actual, revenue can be off as much You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. 35. 35--Multiple imputation is a two-stage process In the first stage, m simulated versions of the data are created under a data model. Then, the m versions of the complete data are analyzed by complete-data statistical techniques and the results are combined. Sometimes the complete-data statistical analysis will involve different models from the one used to produce the multiple imputations. For example, when analyzing data from a sample survey, one might impute the missing data on the basis of an elaborate multivariate model. But then one proceeds to analyze the data using classical nonparametric survey methods in which inferences are based entirely on the randomization used to draw the sample (e.g. Cochran, 1977).36 Multiple imputation is a Monte Carlo approach to the analysis of incomplete data Rubin (1987) described it in the context of nonresponse in survey samples The technique is quite general and can be used in many non-survey applications as well.2.7.3Statistical Model Methods Procedures based on models for incomplete data have been used to make inferences on the likelihood under thatmodel, with methods such as maximum likelihood. 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. One of the most common statistical-based models is the Expectation-Maximization (EM) algorithm, which iteratively calculates the maximum likelihood estimates of parameters in incomplete data problems The EM algorithm is discussed at length in section 2 8.6. 36. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. 37. I(r, x) is the open/closed indicator at review point r for departure .v where -37 jO (closed) if CB(r,x)> BL(r,x) ,(r x)= ' |l (open) if CB(r,x)< Bl.(r.x) II is the number similar of historical flights in the data sample.The demand curve is inverted and transformed into a demand-to-come curve As illustrated in Figure 2.4, the demand-to-come curve defines what the future additional demand is at any point prior to flight departure For example, the total demand at departure time for the fare class in this example is for 100 seats The demand curve indicates that at 60 days prior to departure, 20 seats had already been sold So at this point, the demand-to-comc is for 80 seats This is the value obtained from the demand- to-come curve as illustrated in Figure 2.4. 38. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. 39. 39--confidence Obtaining reliable demand data is an ongoing process. A more systematic approach must be followed The emerging role of the internet may make capturing demand data directly more practical in the future. As potential passengers visit web sites in search of low fares, they leave behind valuable information whether they make a purchase or not. Airlines can use such information as the markets requested, proportion of requests that resulted in a reservation being made, and interest in the various products offered. In addition, online auction sites provide information regarding the amount of money passengers are willing to pay for tickets.2.8.3Ignore the Censored Data Two common approaches to handling censored data in revenue management systems is to simply ignore ordiscard the cases where the censoring occurs and perform estimates based on the remaining data as if the censorship never existed. These two methods are discussed by Little and Rubin (1987) as available-case analysis and complete-case analysis respectively.18 Complete-data techniques are then used without any consideration for the nature of the data. These simple methods are not really attempts to deal with the problem, but a hope that the problem is not too serious. That is not to say that these methods have no merit. If the censoring is limited to a small portion of the data, then these simple procedures may be appropriate. The additional accuracy resulting from more sophisticated techniques may not be worth the implementation and maintenance cost of the models You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. 40. 40--In others, the censoring is ignored, and the observations are used as is. The method works by comparing the number of reservations at particular points of time in a flights history with the corresponding booking limits. If the number of bookings is at or above the booking limit, then the data is censored and an indicator is set to closed Otherwise the indicator is set to open. The demand estimate at each point is based on the indicator as follows: If the indicator is open, then there was no constraining and this is the true demand. If the indicator is closed, the bookings are compared with the average number of bookings for similar flights that were open at this point If the censored observation is greater than this average, then it is used to represent the true demand. Otherwise it is replaced with the average.More formally, the method can be expressed as follows:fCB(r,x) if l(r,x) = I UD(r,x) = cB(r,x) if I(r,x) = 0and('(/',v) >AB(r) [/(#(/) if I(r,x) = OandCB(r,x) < AB(r).where AB(r)is the average unconstrained bookings at review point r of all previous similar departures and is given by You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. 41. 41--2.8.7 Percentile Imputation Method The third imputation method used in this study is the Percentile Imputation Method. With this method, an estimate of the 75"' percentile is used to replace constrained demand values. Percentile imputation fits into the broad category' of imputation-based procedures that are used to fill in missing values.41 The method works by comparing the number of reservations at particular points of time in a flights history with the corresponding booking limits. If the number of bookings is at or above the booking limit, then the data is censored, and an indicator is set to "closed. Otherwise the indicator is set to "open. The demand estimate at each point is based on the indicator as follows:If the indicator is open, then there was no constraining and this is the true demandIf the indicator is closed, then the bookings are compared with the 75 percentile estimate of bookings for similar flights that were open at this point. If the censored observation is greater than this value, then it is used to represent the true demand Otherwise it is replaced with the percentile estimateMore formally, the method can be expressed as follows:You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. 42. 42--The logic used to predict the high-demand bookings is the same that is used to unconstrain the data Suppose that instead of one low-demand history, we collect a sample of similar flights that were not constrained, and then we average the bookings at each review point The flights in this sample will likely have low demand, hence the lack of constraining. Now suppose one of the flights had constrained data. The unconstrained demand can be estimated from the percentage increases calculated from the uncensored data set. This is analogous to predicting demand in the future. Instead of predicting the outcome of an event that has not yet occurred, we are predicting the outcome that has already occurred, but we can not observe. To complete the example above, suppose the data of one of the flights in our sample is observed as given in Table 2.13 The shaded boxes represent constrained dataDP 0DP 7DP 18DP 30DP 60DP 90DP 360Observed Data141414141250Increase8%30%10%45%50%--Unconstrained452519171250Demand Tabic 2.13: Sample DataThe observations at DP 60, DP 90, and DP 360 are not censored and therefore represent true demand at those review points. Starting at DP 30, the observations are constrained, so the booking level of 14 is an underestimate of demand From a group of similar uncensored flights, the percentage increases were calculated. The unconstrained estimate of demand at DP 30 is then calculated by multiplying 12 by 1.45 which equals 17. The estimate for DP 18 is 17x1.10 = 19, and so on. 43. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. You have either reached a page that is unavailable for viewing or reached your viewing limit for this book. where.?2 is the sample variance, uncorrected for degrees of freedom:",(*, -*)J s~ = -------------------nand.v is the sample mean: 44. nNow the loglikelihood function is differentiated with respect to the parameters, the result is set to zero and solved for jU and d:. 2