WO2005050410A2 - Revenue management of flexible products - Google Patents

Abstract

Methods and systems are provided that use flexible products to enhance the profitability of perishable/constrained capacity and/or inventory providers such as airlines and hotels. Algorithms or formulae are also provided with which the conditions of maximizing revenue of flexible products can be determined in relation to specific products, and in relation to situations involving repeated transactions where demand varies. The revenue management methods provide lower priced flexible products that attract additional customers who would otherwise not choose to make a purchase, and enable companies to wait for uncertainty on specific product demand to be at least somewhat resolved before assigning alternative products (thereby enabling maximized use of capacity/inventory and generation of revenue).

Description

REVENUE MANAGEMENT OF FLEXIBLE PRODUCTS

CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit under 35 U.S.C. § 119(e) of United States Provisional Patent Application No. 60/520,974, filed November 18, 2004, which is hereby incorporated by reference herein in its entirety.

FIELD OF THE INVENTION

[0002] The present invention relates to products whereby the particular good, property, or service being purchased is not know until after the purchase of the product. More particularly, the present invention relates to methods and systems that use such products to enhance the profitability of perishable/constrained capacity and/or inventory providers such as airlines and hotels.

BACKGROUND OF THE INVENTION

[0003] In a great number of industries, sellers have a perishable/constrained capacity and/or constrained inventory of a good or service for sale. Examples of businesses having perishable/constrained capacity include airlines selling seats on a flight, hotels renting rooms, theaters selling seats to an event, and manufacturers selling capacity slots in a pure order-to- delivery environment. An example of a business having constrained inventory relating to goods for sale is a fashion retailer, where once inventory is ordered, it cannot be replenished because lead times are longer than the sale horizon.

[0004] Airlines, hotels, and other good or service providers have, of course, long been aware of the risks presented by the combination of uncertain demand and immediately perishable capacity, and a number of mechanisms have been proposed to help manage this risk. One such mechanism applying particularly to airlines is the practice of overbooking (e.g., accepting more bookings on a flight than available capacity).

[0005] Typically, overbooking has been considered a way for airlines to hedge against the risks of cancellations and no-shows. For this reason, overbooking models usually assume that the denied boarding cost of refusing a booked passenger is greater than the highest fare. If, on the other hand, denied boarding cost is less than the highest fare, an optimal policy allows overbooking even in the absence of cancellations or no-shows. In this case, the practice of overbooking is used with the purpose of improving revenues by bumping lower fare passengers in favor of higher fare passengers. However, this bumping strategy is inconvenient for passengers, inflexible for airlines, and may result in high involuntary boarding denial and reaccommodation costs if bumped passengers need to be rebooked on competing flights.

[0006] Another mechanism to manage risk, which also applies particularly to airlines, is the sale of deeply discounted "stand-by" tickets. Stand-by passengers are only accommodated if the number of shows from guaranteed bookings is less than the available capacity. If a stand-by passenger is not accommodated on the flight he booked, he will be accommodated on a future departure for the same (or a nearby) destination that does have available capacity. However, stand-bys are merely a hedge against no-shows and overbooking, rather than a strategy to improve capacity utilization.

[0007] Thomas Cook, formally of American Airlines, suggested a "replane" strategy. In this strategy, once airlines observe a high demand for high-fare tickets in connection with a particular flight, they can call many customers to see whether any of them are willing to give up their seats for an alternative flight plus some compensation, even if the flight is not overbooked. However, this strategy for maximizing revenue suffers from (i) the operational costs of identifying the suitable customers willing to be shifted to alternative flights, (ii) the operational costs of shifting customer to alternative flights in an ad hoc manner (such as customer service costs, scheduling coordination costs, etc.), and (iii) the intangible costs of ill-will among low-fare customers who do not have the flexibility to give up their seats.

[0008] Yet another mechanism used in various industries involves the offering of flexible products for sale. Generally speaking, a flexible product is a product that has a set of two or more "alternative products" that serve the same commercial market, where the purchaser of the flexible product does not know at the time of purchase which of the alternative products will ultimately be received, and where the seller of flexible products is potentially subject to perishable/constrained capacity and/or constrained inventory. By contrast, a specific product is a single product or single product unit. Accordingly, a purchaser of a specific product (e.g., a specific flight, automobile, etc.) knows the particular product, or at least the details of the particular product, being purchased at the time of the purchase. It will be understood that, as used herein, the term "product" refers not only to goods, but also to property and services.

[0009] One example of an industry currently using flexible products is the internet advertising industry. Companies such as YAHOO, MSN, and LYCOS sell capacity on different properties to advertisers, where the properties consist of pages devoted to topics such sports, finance, travel, weather, maps, etc. Currently, an advertiser can purchase space on a specific property (a specific product). This type of purchase is generally made by advertisers that have a strong preference for a certain property. For example, American Airlines might want its ads to appear on the travel page of a specific Web site. Alternatively, an advertiser can buy capacity on a generally cheaper, "run-of -network" basis (a flexible product). If an advertiser purchases space on a run-of-network basis, then the seller (e.g., YAHOO) is the one that chooses the property (which may be, for example, any Web page and on any Web site within the advertising network) to host the advertisement.

[0010] Another example of an industry currently using flexible products is the air cargo industry. The majority of air cargo is sold on a reservation basis, similar to passenger sales. Some shippers (primarily forwarders and consolidators) book capacity for their shipments on specific flights. This is known as a "flight-specific" booking. However, in addition to flight-specific bookings, some carriers offer flexible products (referred to as "time-definite" products) in which the carrier specifies only the pick-up time and the delivery time. In this case, the carrier has the option to choose what flights carry the shipment, subject only to the pick-up and delivery requirements.

[0011] The tour operator industry has also been known to use flexible products. In this industry, tour operators (e.g., Airtours and Thompson, both of Europe) often sell tour packages including both air transportation and lodging. In some (or most) cases, a customer can specify a particular hotel within the resort at which he desires to stay. For a popular destination such as Ibiza or the Costa del Sol, however, a tour operator will generally have space agreements with many different hotels. In this case, a customer can, for a discount, specify a desired quality level (e.g., three stars), and the tour operator will choose the property for him based on availability.

[0012] Flexible products have also been used by a number of Internet travel sellers, such as PRICELINE and HOTWIRE. Among other things, these companies offer "opaque" fares for hotels and airlines (from which they have received a certain amount of capacity at a discount). Using an airline opaque fare, the purchaser buys a ticket (often at a discount) for a particular origin-destination and flight date without knowing the itinerary, airline, or exact flight-departure and arrival times. She is informed of these details only after the purchase is consummated (and generally within seconds or minutes). Opaque fares were created specifically as an inferior product that could fill capacity without excessively cannibalizing full-fare demand. In general the airlines and hotels offering opaque fares through companies such as PRICELINE and HOTWIRE do not seek to control their total return from their capacity by jointly managing flexible and specific sales.

[0013] While the current use of flexible products (e.g., in the industries described above) provides certain benefits, there remains a need for methods and systems for improving the ways in which flexible products are offered to (and purchased by) customers. There also remains a general need for methods and systems for managing revenue from the selling of products, such as the selling of flexible products and specific products. For example, there remains a need for methods and systems that help a supplier (e.g., airline) determine what kind of booking Umits or nesting structures should be used when the supplier offers a combination of flexible and specific products.

SUMMARY OF THE INVENTION

[0014] The present invention relates to methods and systems for improving the ways in which flexible products are offered to (and purchased by) customers. The present invention also provides methods and systems for, among other things, managing revenue from selling products, including the selling of flexible products and specific products. Particularly for industries with perishable/constrained capacity and/or constrained inventory, the expected revenue with flexible products that are managed using the methods and systems of the present invention can be significantly higher than in the traditional cases where sales at low fares are final or in cases where flexible products are sold without proper revenue management techniques as provided herein.

[0015] According to at least a first embodiment, the invention provides a computer-implemented method for managing revenue from selling a product, the method including selling a flexible product comprising at least two alternative products to a purchaser, wherein one of the alternative products is assigned to the purchaser after purchase of the flexible product and within a specified period of time that is established as a component of the flexible product prior to or upon purchase of the flexible product, the method also including determining by the seller of the flexible product which alternative product is to be assigned to the purchaser, and informing the purchaser which alternative product has been assigned within the specified period of time after the sale of the flexible product.

[0016] According to at least a second embodiment, the invention provides a computer- implemented method for managing revenue from selling a product comprising, the method including selling a flexible product comprising at least two alternative products to a purchaser, wherein one of the alternative products is assigned to the purchaser after purchase of the flexible product, the method also including determining by the purchaser which alternative product is to be assigned to the purchaser, and informing the seller of the flexible product which alternative product has been assigned.

BRIEF DESCRIPTION OF THE FIGURES

[0017] Additional embodiments of the invention, its nature and various advantages, will be more apparent upon consideration of the following detailed description, taken in conjunction with the accompanying figures, in which like reference characters refer to like parts throughout, and in which:

[0018] FIG. 1 is a flow chart illustrating the steps performed according to certain embodiments of the present invention in managing revenue with a flexible product transaction;

[0019] FIG. 2 is a flow chart illustrating the steps performed according to certain embodiments of the present invention where a flexible product's fixed time period for assignment of an alternative product to the purchaser is unilaterally modified following the purchase of the flexible product;

[0020] FIG. 3 is a flow chart illustrating the steps performed according to certain embodiments of the present invention where a customer converts a purchased specific product into a flexible product;

[0021] FIG. 4 is a simplified illustration of how the systems and methods of the present invention can be implemented according to various embodiments of the present invention; [0022] FIG. 5 is a flow chart illustrating the steps performed according to certain embodiments of the present invention in computing the solution to the second period allocation problem;

[0023] FIG. 6 is a graph illustrating the relationship between the gain Gf(T) and the time horizon T according to certain embodiments of the present invention;

[0024] FIG. 7 is a graph illustrating the relationship between revenue and the time horizon T according to certain embodiments of the present invention;

[0025] FIG. 8 is a graph illustrating the relationship between the gain Gf(T) and utility w according to certain embodiments of the present invention;

[0026] FIG. 9 is a graph illustrating the relationship between the gain G/(T) and φ, for the utility w = 1, according to certain embodiments of the present invention;

[0027] FIG. 10 is a graph illustrating the relationship between the gain Gf(T) and φ, for the utility w = 2, according to certain embodiments of the present invention;

[0028] FIG. 11 is a graph illustrating the effects of discount on the gain Gf(T) according to certain embodiments of the present invention;

[0029] FIG. 12 is a graph illustrating the effects of discount on the gain Gf(T) according to certain embodiments of the present invention;

[0030] FIG. 13 is a graph illustrating the relationship between the gains Gf(T) and G_d(T) as a function of the time horizon T according to certain embodiments of the present invention;

[0031] FIG. 14 is a graph illustrating the results of an experiment relating to demand induction versus capacity utilization according to certain embodiments of the present invention; and

[0032] FIG. 15 is a graph illustrating revenue results directed to a case involving generic random networks according to certain embodiments of the invention.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

[0033] The present invention provides various methods and systems for sellers of perishable/constrained capacity and/or inventory, such as airlines and hotels, to use flexible products to enhance revenue (and thus profitability). [0034] As explained above, various forms of flexible products are used in industries relating to internet advertising, air cargo, tour operators, and Internet travel sellers offering opaque fares. Generally speaking, flexible products can be used according to the principles of the present invention in any situation in which a seller offers several products that some customers will consider as close substitutes. For example, in made-to-order manufacturing, sellers could offer an option under which a customer is able to select from either a specific time slot, or a cheaper flexible option under which delivery is guaranteed by some future date but the seller has the choice to choose the actual time slot (and perhaps even day).

[0035] Moreover, there remain industries where flexible products are not believed to be currently offered, but which could benefit from their use. One example of an industry in which flexible products are not believed to be currently offered, but which could benefit from their use, relates to multiple property management. Major hotel chains such as MARRIOTT, SHERATON, and HILTON often operate several properties within a single metropolitan location (for example, New York City or San Francisco). Using the principles of the present invention, these chains can sell a general location-based product (a flexible product) to travelers who are largely indifferent among the specific properties within the general location. The DISNEY corporation faces a similar opportunity at DISNEY WORLD, for example, where it operates a number of different hotels, some having specific themes. There are travelers who strongly prefer a particular hotel, while others may be indifferent as long as the hotel is within DISNEY WORLD. It is this latter class of travelers that would be likely to purchase flexible products if given the option.

[0036] Although not known to currently take place, airlines themselves are also able to benefit greatly from offering flexible products to customers. As an example, consider the case of an airline that offers flights from New York's Kennedy airport (JFK) to San Francisco International (SFO). One flight departs at 8:00 AM and arrives at 11:00 AM, the second departs at 9:00 AM and arrives at 12:30 and the third departs at 11 :00 AM and arrives at 2:00 PM. Customers can book seats on any one of the three flights as usual. However, in addition to these three specific products, using the methods and systems of the present invention, the airline might also offer a flexible product (called, e.g., "JFK - SFO morning") at a discount in addition to specific flights serving the same market. According to the invention, the airline would use one or more criteria to assign customers of the "JFK - SFO morning" flight flexible product to a specific flight shortly prior to departure with the goal of maximizing profitability.

[0037] As will become apparent from the explanation provided below, flexible products such as those described above offer at least two advantages to sellers: risk pooling and demand induction.

[0038] With regard to risk pooling, flexible products can improve capacity utilization because customers can be assigned to products after demand uncertainty about specific products has been largely resolved, allowing the supplier to hedge against demand and capacity unbalances. For example, in the "JFK - SFO morning" flight example provided above, the airline would have the luxury of observing specific demand for each of the individual flights before assigning the flexible passengers to the morning departures.

[0039] Moreover, most consumers are likely to view flexible products as less convenient than specific products. This potentially allows flexible products to be sold at a lower price than the specific products without excessively cannibalizing specific product demand. At sufficiently low fares, flexible products may induce demand from a segment of the population that is not likely to purchase a specific product. This result is generally referred to herein as demand induction.

[0040] In light of these and other benefits, various methods and systems for offering flexible products for sale in accordance with the principles of the present invention are now discussed

[0041] FIG. 1 is a flow chart illustrating the steps performed according to certain embodiments of the present invention in managing revenue with a flexible product transaction, and as such, presents concepts generally applicable to the invention. At step 102, a host system, such as a server, presents a selection of products available for sale. For example, the selection can be passively displayed on a seller's Web site. Alternatively, for example, the selection can be displayed in response to a particular inquiry by a potential customer, where this interactive response can be displayed on the seller's Web site or on a third-party Web site (for example, if the products are airplane ticket reservations, then the seller's Web site can be an airline's Web site and the third-party Web site can be a travel agency Web site, such as EXPEDIA.COM, etc.).

[0042] The host system then receives a request from a customer for purchase of a product at step 104. Next, at step 106, the host system determines whether the product that is requested is a specific product. If the product that is requested is determined to be a specific product at step 106, then, at step 108, the host system determines (by programs or algorithms, which can use one or more of the equations described below, or by human decision making) whether selling the specific product as a flexible product can enhance profitability. If it is determined at step 108 that selling the specific product as a flexible product has the potential to enhance profitability, then the host system presents the customer with the choice to purchase a flexible product that has the requested specific product as an alternative product (step 110). At step 112, it is determined whether the customer chooses to purchase the specific product or the flexible product. If it is determined that the specific product has been chosen, then at step 114, the host system provides the customer with confirmation of purchase of the specific product. Similarly, if it is determined at step 108 that selling the specific product as a flexible product is unlikely to enhance profitability, then at step 114, the host system provides the customer with confirmation of purchase of the specific product. In this case, the host system does not present the customer with the choice to purchase a flexible product that has the requested specific product as an alternative product.

[0043] Returning to step 106, if it is determined that the product is not a specific product, then, at step 116, it is determined whether the product is a flexible product. If it is determined at step 116 that the product is not a flexible product, then, assuming specific and flexible products are the only type of products being offered, an error procedure is initiated (step 118). At step 118, the error procedure may take any suitable action, such as causing the process to return to step 102, or bringing the process described by the flow chart of FIG. 1 to an end. However, if it is determined at step 116 that the product is a flexible product, then the host system provides the customer with confirmation of purchase of the flexible product (step 120). The customer will also be provided with a confirmation of purchase of the flexible product at step 120 when it is determined at step 112 that the customer has chosen the flexible product for purchase. Finally, at step 122, the customer is provided with information regarding which alternative product has been assigned. According to various embodiments, this information is provided with a certain fixed time period after the purchase of the flexible product (e.g., in order to provide sufficient notice to the purchaser). It should be noted that, generally speaking, both the seller and the customer are aware of this fixed time period at (or prior to) the time the flexible product is purchased. In other words, the fixed time period may be established as a component or trait of the flexible product. However, according to several embodiments of the invention, the fixed time period associated with the flexible product may be know by the customer and/or the seller only after the actual purchase of the flexible product.

[0044] Moreover, while the flexible product described above in connection with the flow chart of FIG. 1 provides the seller with the discretion in assigning an alternative product, it will be understood that the invention is not limited in this manner. Rather, according to various embodiments (as discussed below), the purchaser can be the one with the power or right to assign one of the alternative products.

[0045] In several embodiments, the fixed time period described above may be modified after sale of the flexible product upon negotiation between the buyer and seller. For example, assume that the fixed time period associated with an airline flexible product is set to one week prior to the earliest departure date of the alternative flights. In this case, the seller (airline) and the passenger may agree to terms (after the purchase of the flexible product) whereby the seller provides some form of compensation to the purchaser in exchange for the assignment deadline being changed from one week to four days prior to the earliest departure date of the alternative flights. Alternatively, for example, the seller (airline) and the purchaser may agree to terms (after the purchase of the flexible product) whereby the purchaser provides some form of compensation to the seller in exchange for the assignment deadline being changed from one week to two weeks prior to the earliest departure date of the alternative flights.

[0046] According to many embodiments of the invention, the fixed time period set forth at the time of purchase of the flexible product is modified unilaterally (pursuant to the terms of the flexible product). Generally speaking, a flexible product that offers the seller the right to unilaterally modify the fixed time period will cost less to the purchaser than one that does not. Conversely, a flexible product that offers the purchaser the right to unilaterally modify the fixed time period will generally cost more to the purchaser than one that does not.

[0047] In a slight variation of the above example, according to the principles of the present invention, while the fixed time period set forth at the time of purchase of the flexible product can be modified unilaterally, such a modification carries with it a predetermined cost to the modifier. For example, a "fee schedule" may be agreed to at the time of purchase of the flexible product by both the airline and passenger, whereby the change in the fixed time period determines the fee charged to either the airline or the passenger (depending on which is the one modifying the fixed time period). For example, a fee schedule may be in place that requires the airline to pay the passenger $25 for each day that is removed from the fixed time period. It should be noted that, even where the fixed time period can be modified unilaterally, there may still be restrictions in place according to some embodiments that limit the modification (e.g., to no less than two days prior to the earliest departure date of the alternate flights).

[0048] FIG. 2 is a flow chart illustrating the steps performed according to certain embodiments of the present invention where a flexible product' s fixed time period for assignment of an alternative product to the purchaser is unilaterally modified following the purchase of the flexible product. At step 202, a flexible product having a set fixed time period in which assignment of an alternative product is to take place is purchased by a customer (e.g., an airline passenger). At some point after the flexible product purchase (but within the time period permitted by the terms of the flexible product), at step 204, either the seller or the purchaser unilaterally modifies the fixed time period associated with the flexible product. At this time (or shortly thereafter), some form of notification regarding the modification is made to the non- modifying party. Next, at step 206, compensation according to the fee schedule that was part of the original flexible product transaction is provided from the modifying party to the non- modifying party. As will be appreciated by persons versed in the art, the end result of the above steps is a flexible product having a modified fixed period of time within which the seller must assign one of the alternative products to the purchaser.

[0049] As flexible products often include alternative products that are perishable, the alternative products themselves often provide the fixed time period. For example, assume that a flexible product is purchased March 1st, and this flexible product comprises three alternative products X, Y, and Z. Moreover, assume that alternative product X is an airplane ticket reservation for a flight departing on March 29th, that alternative product Y is an airplane ticket reservation for a flight departing on April 4th, and that alternative product Z is an airplane ticket reservation for a flight departing on April 5th. By the nature of the alternative products, the seller would generally assign one of X, Y, or Z at a reasonable time prior to March 29th, the earliest of the departure dates for products X, Y, and Z. However, the invention is not limited in this manner. As an example, the terms of the purchased flexible product may allow the assignment of one of products X, Y, and Z to take place after March 31st (the date of departure for product X), but not after April 3^rd (the day before the departure for product Y, and two days before the departure for product Z).

[0050] According to several embodiments of the invention, a seller or a purchaser of a specific product can permit the other party to convert the specific product into a flexible product. Another scenario, which is now explained in greater detail, involves the seller permitting (e.g., through instructions to a host system) a customer who has purchased a specific product to convert the specific product into a flexible product. In this case, for example, the seller can offer the conversion for free, or at a discount with relation to the purchase price of the specific product (such that the customer thereby receives a monetary disbursement or account credit).

[0051] Alternatively, the seller of a specific product may permit the purchaser to convert the flexible product, where it is the purchaser (and not the seller) that has the power or right to assign one of the alternative products. For instance, the buyer may be interested in paying additional money for the conversion due to buyer's uncertainty of whether he/she would be able to use the specific product in the future. As an example, assume that the buyer has purchased a specific product A, where A is an airplane ticket reservation for a flight departing on June 15th. The buyer, after his purchase of A, finds out that he may not be able to fly on June 15^th. To avoid flight cancellation or flight change costs, the customer may desire to convert A into a flexible product that has various alternative flights, one or more of which the customer knows he/she will be able to make (which can include the original specific product flight). The seller will generally offer the conversion based on a determination of whether the conversion, together with the charged premium, will enhance profitability. The buyer will generally choose the conversion based in part of the probability that he will in fact not be able to fly on June 15th, and in part on a determination that the premium is less than the penalties associated with flight cancellation or flight changes.

[0052] FIG. 3 is a flow chart illustrating the steps performed according to certain embodiments of the present invention where a customer converts a purchased specific product into a flexible product. At step 302, the seller receives a request from a customer to convert a previously purchased specific product into a flexible product. If at step 304 it is determined that the requested conversion should be permitted, then, at step 306, the specific product is converted into a flexible product. As will be appreciated by persons versed in the art, the determination regarding whether to permit the conversion, the choice of alternative products to include in the flexible product should the conversion be permitted, the fee to charge for such a conversion, and the fixed time period of the alternative products, for example, can be determined based on a profitability determination (using, e.g., the equations provided below). Assuming the specific product has been converted into a flexible product, at step 308, a confirmation is provided to the customer to confirm that the conversion has taken place. If it is determined at step 304 that the specific product will not be converted into a flexible product, at step 310, the conversion is denied. In this case, at step 312, confirmation of the denial is sent to the customer, and the process ends.

[0053] While the example provided above in connection with the flow chart of FIG. 3 assumes that it is the customer requesting the conversion, the invention is not limited in this manner. Rather, according to various embodiments of the invention, it is the seller of a specific product that seeks to convert the specific product into a flexible product. In this case, any suitable compensation structure may be used for the conversation to take place. Additionally, for example, rather than requesting the conversion of a single specific product into a flexible product, the seller of the specific product (and other specific products) may extend open-ended offers (e.g., by displaying the offer on a Web site) to convert previously sold specific products to corresponding flexible products (for respective fees).

[0054] The systems and methods described herein in accordance with the principles of the present invention may be implemented using any suitable communication network. For example, the methods can be implemented as a Web site that is hosted on an Internet server, which can be any suitable type of server. A user's computer and servers or databases of sellers/service providers can be connected to a host system's Internet server, or any other suitable server, through any suitable Internet connections.

[0055] For example, FIG. 4 is a simplified illustration of how the systems and methods of the present invention can be implemented. As shown, a group of one or more servers and/or databases 402, maintained, operated and/or owned by a seller or the seller's service provider, is connected to an Internet Web page server 304, which can be any suitable type of server. Servers and/or databases 402 may contain, for example, information on products for sale and the products that have been purchased, and the software routines that enable the determination of how to enhance profitability through the offering of flexible products. The Internet Web page server 404 is a nexus at which information from the seller can be displayed or offered to customers (through the customer's computer 406), and at which customers can make purchase requests, selections, or inquiries to the seller. Moreover, connections 408 and 410 shown in FIG. 4, which are used to connect Internet Web page server 404 to servers and/or databases 402 and the customer's computer 406 to the Internet Web page server 404, respectively, can be any suitable type of Internet connection (e.g., wireless or wire-based). Although not shown in FIG. 4, a user may also access servers and/or databases 402 using a traditional (landline) or wireless telephone, for example. In this case, the user can simply listen to offerings and select products for purchase using the touchtone keypad of his telephone, or by speaking bis choices when voice recognition software is being used, for example. It will be understood by persons versed in the art that the invention is not limited to the particular manners of purchasing described above.

[0056] While flexible products offer many advantages, there is the risk that poorly managed flexible products can lead to revenue deterioration, for example, through cannibalization of higher fare demands. The present invention therefore also provides revenue management methods and systems for a variety of situations, including the case of a supplier with fixed perishable or constrained capacity/inventory that is offering a combination of flexible and specific products.

[0057] The present invention provides conditions and algorithms for proper revenue management of flexible products. In "Analysis 1" provided below, the invention provides such conditions and algorithms in the simple case of a single flexible product consisting of two specific products (i.e., two alternative products), which can be applied to more complex scenarios (as described in "Analysis 2" provided further below). Finally, in "Analysis 3," strategic pricing in constrained markets with repeated transactions is explained.

[0058] The present invention often poses analyses in terms of an airline. However, one should bear in mind that the present invention s much more broadly applicable.

[0059] It is to be understood and expected that variations in the principles of the invention herein disclosed in an exemplary embodiment can be made by one skilled in the art and it is intended that such modifications, changes, and substitutions are included within the scope of the present invention. The information and examples presented in the following analyses are for illustrative purposes only, and are not meant to be limiting

ANALYSIS 1: REVENUE MANAGEMENT OF FLEXIBLE PRODUCTS AND THE TWO-PRODUCT PROBLEM

[0060] Analysis is now provided of the two-period, two-flight case for an airline offering a flexible product in addition to specific products. Among other things, this analysis compares different control structures for flexible and specific bookings, and provides algorithms for determining booking limits. Additionally, for example, the analysis shows that a carrier can achieve financial gain under certain conditions by allowing overbooking when managing flexible bookings, even in the absence of no-shows or cancellations. The following analysis also presents a consumer choice approach that models both the demand induction and cannibaUzation that may result from offering a flexible product. A simulation is also used to compare results under the various control structures, and for different pricing scenarios, to provide insight into both the demand induction and risk pooling benefits that an airline could achieve from offering flexible products. In Analysis 2, these concepts are extended to cover full airline networks with arbitrary specifications of flexible products. It should be noted that, although this analysis develops a model in the context of passenger airlines, it will be appreciated by persons versed in the art that the results extend directly to any industry that accepts bookings for multiple products or services using perishable/constrained capacity and/or constrained inventory.

[0061] Assume that an airline has two flights A and B serving the same market. For example, both of flights A and B may be from the same origin and scheduled for the same destination. Moreover, assume that passengers purchase their tickets in two periods. In the first period, the airline sells flexible product "(A,B)" in addition to specific products at discounted fares. In the second period, the airline sells specific products "(A)" and "(B)," but not flexible product (A,B).

[0062] At some time after the end of the first period, the airline allocates the passengers who purchased (A,B) to either of flights A or B as it wishes. However, passengers who purchased (A,B) must be accommodated on either flight A or flight B, or the airline pays a denied boarding penalty to each passenger denied space. [0063] In this example, it is assumed that flexible products are not sold during the second period. This corresponds to the assumption that flexible customers (purchasers of a flexible product) will be informed of their flight assignment some time (e.g., 24-72 hours) prior to departure of the earlier of the alternate flights (in this example, A and B). In this case, the airline would not sell flexible products during the last 24-72 hours. This assumption also reflects the fact that the preference of customers for specific products over flexible products relies in part on the time lag between the booking of a flexible product and resolution of allocation, as well as the fact that offering flexible products to customers otherwise willing to pay a premium (in the form of a higher fare) to purchase a ticket in the second period is likely to cannibalize full-fare demand.

[0064] According to the principles of the present invention, analysis is now provided relating to the above-described two-product problem. As will be appreciated by persons versed in the art in light of the following, this analysis may be used by an airline to decide how many units to make available for flexible and specific bookings during the first period, and how to manage the remaining capacity during the second period (both when the airline allows overbooking, and when it does not).

[0065] The following variables are used below: c^A, c^B = capacity of flights; g = fare paid by flexible passengers during first period; g^Λ, g^B = fare paid by specific passengers during first period; f^A, f^B = fare paid by specific passengers during second period; Y= demand for flexible passengers during first period; Y^A, Y^B = demands for specific passengers during first period; D^A, D^B = demands for specific passengers during second period; b = maximum number of flexible bookings accepted during first period; b_\ ^A, bι^B = maximum number of flight specific bookings accepted during first period; b^A, b^B = maximum number of flight specific bookings accepted during second period; and d = gross penalty for each passenger denied boarding. [0066] Because of the general preference for specific products over flexible products, it can be assumed that the fare paid for flexible products is less than that paid for specific products (i.e., 0 < g < min(g^A, g^B)). Moreover, it should be noted that the overbooking penalty (d) has been specified as a gross penalty, and includes the sum of the ill-will cost, reaccommodation cost, and direct cost paid per denied boarding. It is assumed for simplicity that this cost is independent of the number and the mix of denied boardings. Additionally, it is assumed that the gross denied boarding cost exceeds all fares, i.e., d >f' > g^J fo j = A, B. Otherwise, an airline would be likely to set no booking limit on a fare that exceeds the overbooking penalty.

[0067] It is also assumed that booking limits are set at the beginning of a period and cannot change during that period. Moreover, because airlines typically reserve at least some capacity to satisfy the demand for higher fare products during the second-period, it is assumed that the airline does not overbook in the first period, i.e., b^jι ≤c^j, j - A,B, b >0 and b + b ^A + b ^B ≤ c^A + c^B.

[0068] The expected revenue during the first period is given by g^AE min(Y^A, b ) + g^BE min(Y^B, b ^B) + gE min(Y, b), where E min is the expected minimum. Let s^j = min(Y^j, bO, j = A, B denote the number of seats booked by flight specific passengers during the first period. Additionally, let s = min(Y, b) denote the number of flexible seats booked during the first period. Thus, (s^A, s^B, s) is known at the beginning of the second period.

[0069] Let r^j ≡ c^j - s^j, j = A, B denote the residual capacity of the flights at the beginning of the second period, and let c(s^A, s^B, s) ≡ r^A + r⁸ - s ≥O denote the residual total capacity at the beginning of the second period. For simplicity, "c" will often be written below in place of "c(s^A, , s)X

[0070] Given the vector r - ( , r³, c) of residual capacities, the airline must determine how many seats, b ^j>0, to make available for sale for flight j = A, B during the second period. For this analysis, a static control policy is used, whereby the parameters b^j , j = A, B are decided at the beginning of the second period (observing D^j, j = A, B).

[0071] Assume the following definitions: H(b^A,b^B) = f^AErmn(D^A,b^A) + f^BEram(D^B,b^B) (1)

H(b^A,b^B,c) = H(b^A,b^B) -dE(rmn(D^A,b^A) + rmn(D^s,b^B) -c)⁺ (2)

As used herein, the "+" superscript indicates that the expression to which it is affixed will be considered to be zero when it has a negative value. [0072] When overbooking is not allowed, the optimal expected profit during the second period is given by h(r^a , r^b ,c) = max H(b^A,b^B) , where 0 < b^j < r^j for j=A,B, b^A+b^B<p, and b^A,b^B are b ,b integers.

[0073] Let F^j(x) ≡ Pr(D ^J ≥ x), where Pr(D ≥ ) refers to the probability that D ^j is greater than or equal to x, and consider the expected marginal seat revenues EMSR (x) ≡ f^j Emin (O\ x) -f^j E min(D^j , x-l) =f F^j(x) fo x s {1, ..., r^j },j = A, B. Also, let EMSR(k;r^A,r^B) be the k^th largest value in set {EMSR^j(x),l<x<r^j,j=A,B}.

[0074] Following the convention that sums over empty sets are zero, from what is above, the optimal expected profit during the second period is given by: h(r^A,r^B,c) = T EMSR(k, , r^B). (5) lc=l

[0075] By contradiction, it can be shown that that EMSR^A(£); k = 1, ..., b^A and EMSR⁵(fc); k = 1, ..., b^B represent, collectively, the c largest EMSR values in the set {EMSR^x); 1 < x ≤ r^J ' ; j = A, BJ.

[0076] As an example, assume the following:/"¹ = $350; f^B - $330, D^ follows a Poisson distribution with parameter λ = 50, D^B follows a Poisson distribution with parameter λ = 40, r⁴ = 60, r^B = 38, and c = 83. Using the above equations to solve this problem, (b^A, b^B) = (47, 36) is an optimal allocation, and h(60, 38, 83) = $350 E min(D^A, 47) + $330 E min(Z)^s, 36) = $27,470.09.

[0077] Moreover, EMSR^A(47) = $239.16, EMSR^β(36) = $250.00, EMSR^A(48) = $220.62 and EMSR^S(37) = $232.21. Thus, h(60 + 1, 38 + j, 84) = h(6Q, 38, 83) + EMSR⁵(37) = $27, 702.30 for all non-negative integers i and;. Finally, h(60, 38, 82) = /Ϊ(60, 38, 83) - EMSR^A(47) = $27,230.93.

[0078] If s = 0 (i.e., only specific products are offered), then b^j = r^J,j = A, B is optimal.

Moreover, as will be appreciated by persons versed in the art, equation (5) provides two algorithms to find an optimal allocation depending on the size of s relative to c + s. In particular, if s is small relative to c + s, then the algorithm involves seeking the ,s smallest EMSR values, and subtracting those from the allocation r³. On the other hand, if c is small, then the algorithm calls for finding the c largest EMSR values. [0079] When overbooking is allowed, the optimal expected profit during the second period is given by h(r^A ,r^B,c) ~ max H(b^A ,b^B ,c) , where 0 < b^j< r^j for j=A,B, and b^A,b^B are integers. b ,b

[0080] For b^A<r^A,b^B<r^B such that b^A+b^B>c, we can write the following difference equations. For b^A<r^A: A_AH(b^A,b^B ,c) ≡ H(b^A,b^B ,c) - H(b^A -l,b^B ,c) = F^A(b^A)[f^A -dF^B(c + \-b^A)] (

For ^≤r^6'- A_BH(b^A,b^B,c) ≡ H(b^A,b^B,c) -H(b^A,b^B -l,c) = F^B(b^B)[f^B -dF^A(c + l-b^B)] (7)

where the difference equations represent the marginal contribution of accepting one more booking of the specific products.

[0081] Because ^_A ^H(^b '^{b > C})_1S independent of b^B and is decreasing in b^A<r^A we can find the largest b^A≤r^A such that ^^AHΦ ^{>b > c})>0. Similarly, we can find the largest b^B≤r^B such that A_BH(b^A,b^B,c)_≥Q

[0082] Two ways of obtaining an optimal solution for determining how many seats to allocate to the second period when there is overbooking are now presented. One solution uses an algorithm as will not be explained in with reference to the flow chart of FIG. 5.

[0083] At step 502, an optimum solution (b^A,b^B) is computed to the problem without overbooking as in equation (5). Let π =f^AEmm(b^A,D^A')+f^BEmin(b^B,D^B) be the corresponding expected profit.

[0084] At step 504, if it is determined that b^A=r^A and b^B=r^B then the algorithm is complete. Otherwise, at step 506, < ' ^{, c} for i=A,B is calculated such that b^l ≤ r^l according to equations (6) and (7).

[0085] Next, at step 508, it is determined if both b^A and b^B are less than or equal to zero. If they both are less than or equal to zero, then the algorithm is complete. Otherwise, the process algorithm continues with step 510. [0086] Finally, at step 510, the highest ^Δ< (^{& >b} '^c) is added to π, and the allocation is updated to (b^A,b^B)<^(b^A+l,b^B) if the highest EMSR was from A, or the allocation is updated to (b^A,b^B)<-(b^A,b^B+l) if the highest EMSR was from B. Afterwards, the algorithm returns to step 504 and the process continues until one of the stopping conditions described above are met. [0087] Consider the example provided above, in which = 60, r⁸ = 38, c = 83. If d = 450, then (b ^A,b ^B) = (48,38) . Note that the residual capacity at the beginning of the second period is c = 83 seats, whereas the optimal allocation will allow up to 48 + 38 = 86 bookings, meaning that it is optimal for the airline to accept the possibility of up to three denied boardings. The calculation of (b ^A,b ^B) = (48,38) and the associated expected profit b(60,38,83) can be illustrate in terms of the steps from the optimal solution without overbooking, (b^A,b^B) = (47,36) as shown in the following table. Table 1

[0088] In this example, the additional expected revenue net of expected denied boarding costs from allowing overbooking is equal to $21.63, or about 0.08% of the expected revenue without overbooking. [0089] Another solution for determining how many seats to allocate to the second period where there is overbooking is more direct because it does not require calculation of the optimal solution to the allocation problem without overbooking. However, it also does not directly compute the gain from allowing overbooking. I x^A+x^B< c+l, this solution is b ^J = min(r ^J , c + 1 - x^}), j = A, B , where x^ is the smallest integer such that F^B(x)<f /d, and x³ is the smallest integer such that F^A(x)<f^B/d.

[0090] Over a certain range, the optimal overbooking allowance decreases by two seats for each additional flexible seat sold during the first period.

[0091] As an example, let x^A = 36, x^B = 46, and x^A+ x^B = 82 < 84, so (b ^A,b ^B) = (84-36=48, 84- 46=38) is an optimal overbooking allocation that overbooks three seats. If c is reduced to 82, the new optimal allocation is (47,37). On the other hand, if c is reduced to 81, the new optimal allocation changes to (46,36), coinciding with the allocation without overbooking.

[0092] Thus, the second solution simplifies the computation of an optimal policy, but it does not provide the expected optimal profit. However, the expected optimal profit can be computed as follows. Let (b^A,b^B) be an optimal solution when overbooking is not allowed, and let (b ^A,b ^B) ≥ (b^A,b^B) be an optimal solution when overbooking is allowed. Then, the additional expected revenue from allowing overbooking can be calculated simply by adding up the expected marginal seat revenues, that is,

∑[/^A - dF^B(c + l-k)]F^A(k) + ∑[f^B -dF^A(c + l-k)]F^B(k) k=b^Λ+\ k=b^B+l

[0093] In light of the above explanation, it can be seen that it may be optimal for an airline to set allocations that allow overbooking in the second period, even in the absence of no-shows and cancellations. If the airline decides not to allow over-booking, it is constrained to set its booking limits b^A and b^B such that b^A+b^B = c^A+c^B - s. On the other hand, if it allows overbooking, the number of possible booking policies is significantly expanded. Intuitively, allowing overbooking is optimal if the expected incremental gain from expanding the booking limits for the specific products outweighs the expected incremental risk of outcomes that would lead to bookings in the "overbooking region." [0094] For the problem of dynamically allocating seats in the second period, assume that the second period consists of T time intervals, and that at most one request for specific product A or B (but not both) occurs during each time interval.

[0095] Let h(r^A ,r^B,c) = 7(0, r^A,r^B , c) denote the optimal expected revenue of dynamically managing the residual capacity during the second period, where V(t,r) can be calculated recursively according to V(t, r) = V(t + 1, r) + p^J[f^j - A^jV(t + 1, r)]⁺ , where r ≥O is the j=A,B probability of a type; arrival, p +p^B<\, Δ ^jN(t,r)=V(t,r)-Y(t,r-e ^j), e^A=(l,0,l), e^B=(0,l,l), V(T+l,r)=0, V(t,r)=0 if the total residual capacity, i.e., the third component of r, is zero, and V(t,r)= - ∞ if either the first or the second component of r is negative.

[0096] Persons versed in the art will be appreciate that the dynamic formulation can be expanded to time-varying arrival probabilities, to time-varying fares, and to the case where flexible sales are allowed over the entire horizon, or a portion thereof. It should also be noted that the second- period dynamic allocation approach will never result in overbooking.

[0097] The problem of calculating the first period seat allocations is now addressed. At the beginning of the first period, the booking limits (b ^A , b ^B , b) need to be determined. Recall that the demands for discount flight specific products for A and B and flexible products in the first period are denoted by Y^A, Y³ and Y, respectively with corresponding fares, g^A, g^B and g.

[0098] Start with bt* = (c^j - x^j) j = A, B, and search for the optimal number of flexible bookings to allow during the first period. The initial result for the optimal number of flexible bookings to allow during the first period corresponds to the largest b such that E _{A rB} EMSR(r^A + r^B + l-b;r^A,r^δ) ≤ g . From this point, iteration between reducing the values bι^j, j = A, B for a fixed b and increasing b for fixed bι^j, j ~ A, B can take place until total expected profit is no longer increasing. A similar heuristic is used for the case where overbooking is allowed in the second period, and when second-period bookings are managed dynamically.

[0099] Numerical results for various settings of the problem parameters with and without overbooking are now presented in order to provide insight into the benefits that an airline might achieve from flexible products. The numerical results provided below demonstrate the relative magnitude of the risk-pooling and demand induction benefits of flexible products, and the benefits of allowing overbooking.

Risk Pooling Benefits

[00100] Consider the case of two flights A and B, with identical capacities, c^A = c^B = 100, and two booking periods. In the second period, only full-fare specific bookings for A and B are offered to customers. Moreover, assume that, for both flights, the full fare is $200, and full-fare demand follows independent Poisson distributions with λ₂ ^A = 75 and λ₂ ^B = 25, respectively.

[00101] In the base case, it is assumed that only specific discount booking requests are received during the first period. Each discount passenger pays $150, and specific demands follow independent Poisson distributions with λι^A = 80 and λι^B = 40 for flights A and B, respectively. In this case, the optimal booking limits for the two flights are b^A = 31 and b^B = 78, and the associated expected total revenue across both flights is $29,178.

[00102] To test the sensitivity of total expected revenue to the option of offering flexible products in the first period, it can be assumed that the demand for flexible products follows a Poisson distribution with mean λ = λι^A + λι^β = 120. In other words, assume that when a flexible product is offered instead of the two specific products, the expected total demand for the flexible product is equal to the sum of the expectations of the specific products. Given the assumption that specific products are preferred over flexible products, the flexible products would need to be offered at a lower fare to achieve the same level of demand. Assume that to achieve this, the flexible product would need to be sold at fare/^ = α $150, where α, the discount factor, is less than or equal to one. Comparing the computational results of the expected profit for different values of α, and comparing these results to the expected profit of offering only specific products during the first period, a measure of the benefit of risk pooling net the discount needed to keep the aggregate demand constant can be measured.

[00103] Table 2 lists expected revenue for different flexible fare levels, assuming no demand induction, where the base case is offering only discount flight-specific products in the first period with a corresponding total revenue of $29,178. Table 2

[00104] The "Static-Control Revenue" column in Table 2 shows the expected maximum revenue that can be gained from both flights assuming that only flexible products are offered in the first period, and that static control without overbooking is applied to full-fare bookings in the second period. In other words, booking limits b^A and b^B are set optimally at the beginning of the second period with b^A + b^B = c^A + c^B ~ s, where s is the number of flexible bookings accepted in the first period. Full-fare bookings for each flight in the second period are then given by min[b', D '] for i = A, B, where D ' is unconstrained demand. [00105] The "Dynamic-Control Revenue" column shows the expected maximum revenue that can be achieved from full dynamic control of second period full-fare bookings using the dynamic program described above. Revenue under both control mechanisms is compared against the base case under which specific products are offered at a discount in the first period. As expected, the total expected revenue from dynamic control of full-fare bookings is greater than that achieved from static control. Nevertheless, it should be noted that, in these cases, the vast majority of the benefits from flexible products can be achieved through static control. [00106] Table 2 thus shows that the risk-pooling benefits provided by flexible products can be significant, even in the absence of any induced demand. Under static control, offering flexible products in the first period provides higher revenue than offering specific products, as long as the fare for the flexible products is greater than 70% of the specific fare (assuming that total expected demand remains the same). Demand Induction and CannibaUzation

[00107] The case where flexible products stimulate higher demand, but also cannibalize demand from discount specific products, is now considered. To simulate the effect of offering a flexible product, a simple consumer-choice model that estimates both demand induction and cannibaUzation in a consistent fashion is used. Specifically, it is assumed that the fraction of buyers with a maximum willingness-to-pay (WTP) for specific products has a joint distribution

w^B). It will be understood that, as used herein, "R²" refers to positive real numbers in two dimensions, e.g., the first quadrant of the plane. Moreover, it is further assumed that the total number of buyers is a Poisson random variable with parameter λ₄ and that the WTP distribution is independent of the total number of buyers. Finally, it is assumed that, for each customer, the maximum WTP for the flexible product is a function of his WTP's for the specific products according to: w(w^A, w^B) = pw^A + (1 - p)w^B - p, where p is the customer's probabiUty that he will be assigned to flight A, and p >0 is his reduction in WTP for the flexible product. Conceptually, p is the "value of information" that the buyer of the flexible product would pay to know which flight he would be assigned to at the time of booking. In a fully general model, both p and p would be random variables, possibly correlated with w^A and w^B. However, for simplicity, it is assumed that ? = 1/2 (the maximum-entropy assumption), and that both/? and p are constant across the population.

[00108] For this model, in order to simplify the calculations, it is also assumed that there is no recapture among products (i.e., if a customer does not find his first choice available, he does not purchase any product). This is a relatively conservative assumption, because it tends to reduce the benefits of offering the flexible product (because it is assumed that the customers who seek to buy the flexible product but cannot because of the booking Umit are lost, where in reality some of them would be willing to buy the specific products).

[00109] Consider the case where the flight capacities are set at c^A = c^B = 100. As before, two periods are considered, where in the first period, both flexible products and discount specific products are offered. The total population of buyers in the first period has mean λ = 444. The WTP of buyers for flights A and B are given by independent uniform distributions on (0, W⁴) and (0, W⁸), respectively. For this case, the following is set: W = 186 and W⁸ = 168. In the second period, only full-fare specific products can book. The full fares for each flight are $200 and the discount fares $150. The full-fare demands were assumed Poisson, with parameters λ = 75 and λ = 25.

[00110] Tables 3 and 4 show the results for the above case, where p = 10 and p = 30, respectively. Table 3

Table 4

[00111] In Tables 3 and 4, λ' for i = A, B, fare the mean demands for each product including induction and cannibaUzation, and b b^B, and b are the respective optimal booking first-period booking Umits for A, B, and the flexible product. In each case, offering flexible products at a very low fare leads to a loss in total revenue because the loss from cannibaUzation exceeds the gain from demand induction. [00112] However, at a sufficiently high fare, offering flexible products begins to show positive benefits. These benefits begin to decline when the flexible fare becomes high enough that it is no longer inducing sufficient new demand to outweigh cannibaUzation. When the flexible fare is equal to $150 - p, the benefit from offering flexible products drops to zero, because they no longer induce any additional demand.

[00113] It is noted that, for p = 10, the expected demand for all three products at any flexible fare g is the same as the expected demands for p = 30 at a fare g - $20.00. However, for p = 10, the value of b is higher at g than for p = 30 at g - $20.00, because flexible products have relatively greater value at the higher fare. Consequently, the maximum achievable expected revenue is higher with the lower value of p. Similar reasoning shows that, for this choice model, maximum achievable expected revenue from offering flexible products is a decreasing function of p.

[00114] The present invention encompasses extending the approach used in the above analysis to a full network consisting of many flexible and specific products. The feasibility of serving any portfolio of flexible and specific bookings on a network of constrained capacities can be determined by solving a linear program, as provided below. Also, a column generation approach, such as provided below, can be used to determine the best set of flexible products to offer given a constrained network and a set of customer preferences.

[00115] Furthermore, the analysis provided above in accordance with the principles of the present invention encompass the incorporation of general consumer choice models, such as provided below. A flexible product is not only an economic substitute for each of the specific products it contains, because those specific products are also substitutes for each other. Thus, it can be expected that closing any of the availabilities of the products might increase demand seen for the others.

[00116] It has been assumed that offering low-price flexible products can induce additional demand. The scope of the present invention also extends to the models in accordance with the above analysis where at least some of this induced demand is drawn from competitors who are not offering flexible products. In this case, a competitor that is unable to offer flexible products (e.g., due to a Umitation in its booking system) may compete by lowering its specific fares. The present invention also encompasses the case of two competing carriers, both of which can offer flexible products in a market, but one of which has more flight frequencies.

[00117] It should also be noted that, while the above analysis concerns the benefits of offering flexible product bookings when they are managed by setting a limit on their total availabiUty, the principles of the present invention are not Umited in this manner. For example, an alternative nesting approach may be used, where, for example, both specific and flexible products are nested on each leg according to total fare with a flexible product closing entirely if it is closed on any of its constituent legs. Stated another way, nested fares are such that you always close one before the other, e.g., the "supersaver fare" is closed before other fares.

[00118] Moreover, it has been assumed in the above analysis that the airUne has complete discretion on which specific product to assign a flexible customer. However, one variation according to the invention is to allow flexible passenger to rank their alternatives. In this case, the airline may use one of a variety of schemes to match flexible customers with available flights in a way that best accommodates their stated preferences.

ANALYSIS 2: DETERMINISTIC ANALYSIS OF THE FLEXIBLE BOOKING PROBLEM

[00119] In this analysis, various concepts provided above are extended to cover full airline networks with arbitrary specifications of flexible products. As will be appreciated by persons versed in the art, these concepts can be used to resolve the deterministic flexible booking problem in a network.

[00120] The details of the model presented in this analysis are as follows. The network consists of m resources and the capacity of the resources is given by a vector c e R'ⁿ ₊. This network offers n specific products. The vector ? e R^m ₊ denotes the revenue derived from the specific products, i.e., pj denotes the revenue from the sale of one unit of the -th specific product, j = 1, ..., n. A e R"^{l xn} denotes the capacity utilization matrix, i.e., A,-₇-, i = 1, ..., m, j = 1, ..., n, denotes the amount of resource i required by one unit of the specific product j. The set of specific products is denoted by N.

[00121] The network also offers /flexible products. Each flexible product k = 1, ...,/, is described by a set N_t c N of specific products, and the network manager (as used herein,

"network manager" may be encompassed by a host server or system, or, for example, other computer-implemented methods used to provide revenue management) has the flexibility of assigning the customers for the flexible product k to any of specific products included in the set Nk. Let nk = |Nt| denote the number of specific products constituting the k-th flexible product, and let Cu denote the submatrix of A obtained by picking the columns of A corresponding to N_t. Moreover, the revenue from the flexible products is given by a vector r e R^f ₊, and the set of all flexible products is denoted by E.

[00122] The demand for the specific and flexible products is assumed to be given by a consumer choice model. Specifically, it is assumed that when a subset S c N E of products is offered by the network, the arrival rate for specific products is given by λ(S) e Rⁿ ₊, and the arrival rate for flexible products is given by γ(S) e R^₊. Specific choices for λ(S) and γ(S) are discussed below with reference to a column generation algorithm.

[00123] Network resources have to be sold over the time horizon [0, 7] and are worthless after T (the "time horizon"). It is assumed that the assignment of the demand for flexible products to particular specific products is made at time T. The goal of the network manager is to choose a sequence S_/ c N E, I = 1 , ... , L, of subset of products to offer and corresponding time intervals t(Sι), 1 = 1, ..., L, over which to offer these sets in order to highest revenue possible.

[00124] The optimization problem faced by the network manager is given by the following Unear program (LP): max∑_s( ?'2(S) + r'r(S))t(S)

A(∑_sλ(S)t(S)) + C_z ≤ c, ∑_sγ(S)t(S) -U_z = 0, subject to: ∑_st(S) ≤ T, (1) t(S) > 0,VS c N E, z > 0.

[00125] The decision variables in (1) are the times {t(S) : S c: N u E}, and the composition variables z (that is, the number of variables are 2^n+f+ ∑ =; tik-1). Because there are only m +f + 1 constraints in the problem, at most m +f+ 1 of the exponentially many variables in (1) can take a positive value. Moreover, suppose the k-t flexible product is offered for any period, then at least one of the components of the Z_k vector must be strictly positive. On the other hand, if the k-th flexible product is never offered, then the row corresponding to k in the constraint Σgγ(S)t(S) - U_z = 0 is effectively redundant and can be dropped without affecting the solution.

[00126] According to the principles of the present invention, the following column generation algorithm is provided as an efficient algorithm for solving equation (1). 1. Select an initial collection S ⁰⁾ of subsets of N E. Set q<— 0. 2. Solve the following restricted LP (3) corresponding to S ^q). Let (t^(q),z^(q)) denote the optimal solution of the restricted primal and (u^(q ,v ^q),β ^q)) denote the corresponding optimal dual vector. ∞∑_s≡sm (P'MS) + r'γ(S))t(S) subject to: A(∑_SeS(P) λ(S)t(S)) + C_z ≤ c,

∑_s≡sm γ(S)t(S) -U_z = 0, (3)

t(S) > 0,VS € S⁽⁰⁾ z > 0. 3. Compute the "minimum set" S ^q corresponding to (u^(q ,v^(q ,β ^q)). Let _o(_gκ (u ^q)yAλ(S^) + (v^yγ(S⁽"⁾) + β p'λ(S^(q)) + r'γ(S^iq)) denote "length" of set S^(q). 4. If / (S^(q))>l, the solution of the current restricted primal problem is optimal for the full primal LP(1). Stop. 5. Else, set q÷-q+1, S(q)<-S(q-1) u S(q-l). Return to step 2.

[00127] The results of the computational experiments with a subset of airUne flights from JFK, LGA, STL, ORD to SFO are now provided. The details of the flights in this model are shown in Table 5. Table 5

[00128] There were n = 26 specific products consisting of all of the one-hop flights shown in Table 5 and the two-hop flights from NYC to SFO that satisfy the time constraints. There were/ = 5 flexible products consisting of the origin-destination (O-D) pairs: NYC to SFO, NYC to STL, NYC to ORD, ORD to SFO, and STL to SFO. The set N_t for each flexible product was set equal to all the flights that served the corresponding O-D pair.

[00129] A demand model based on utility over a set of alternatives was used. This particular form makes the demand larger in terms of the basic utility w for the product, as the price increases and as it takes longer for the customer to arrive at the destination. The details of the demand model used are as follows. Each O-D pair was assigned equal utility w. For each specific product j = 1, ... , n, g_j = exp(w - c _j - δt_j ), g_j = -^ (14)

where p_j is the revenue from the flight j (shown in the last column of Table 5), tj is the duration of the flight, a = 0.004 and δ = 0.05 (the constants a and δwere chosen to make the terms in the exponent comparable). The fare r_& of the &-th flexible product was set to M^βmf-y^ {_m} _an(j _e choice model coefficients were set to: h h_k = exp( v-(xr_k - δmax{t_j}),h_k = -*-,£ = ...,/. (15) J^εFk ZU [00130] Let R f(T) denote the optimal revenue for time horizon T when the network does offer flexible products, and R _nf(T) denote the optimal revenue for time horizon T when the network does not offer flexible products. The gain Gf(T) that results from using flexible products over not using flexible products is defined as:

[00131] The effects of several model parameters, such the time horizon T, the utility u, the discount factor γ, etc, on the gain Gf(T) are now explained.

Effect Of Varying The Time Horizon

[00132] The effects of varying the time horizon T while holding all other parameters (particularly, capacity) are now considered. A typical plot for the gain Gf(T) as a function of the time horizon T (all other parameters held constant) is show in FIG. 6. The gain curve starts out flat, next, it enters a phase where it does not have any noticeable trend, and finally monotonically decreases to zero. It should be noted that, although in the plot shown in FIG. 6 the slope of the gain curve is negative immediately after the flat section, this will not always be the case.

[00133] The gain curve Gf(T) can be divided into three parts, depending upon the value of the time horizon T. For small T, the gain curve is flat because no capacity constraints are binding and the entire gain is due to demand induction. For sufficiently high values of T, all of the capacity can be sold to specific products and the gain is therefore 0. For intermediate values of T, the gain can go through both increasing and decreasing regions, although it ultimately decreases to 0. Therefore, an airline or other business should generally be wilUng to offer flexible products when it anticipates that there is a low probabiUty that it will sell out its capacity, in order to take advantage of the demand induction benefits. For the example represented by FIG. 6, it is generally even more advantageous to offer flexible products in the intermediate region, in which the airline benefits from both demand induction and improved capacity utiUzation.

Effect Of Changes In Utility _

[00134] This section provides the results of experiments where the utiUty w derived from the journey was varied. It will be understood that, as used herein, utility (or w) refers to an intrinsic measure of value derived from a product by a customer, subtracting the product's cost in terms of invested money and time. From equations (14) and (15) provided above, it follows that the arrival rate is an increasing function of the utility w. Because the total capacity c is fixed, the flexible products are Ukely to become less attractive as the arrival rate increases, or equivalently w increases — the network would not like to waste valuable resources on a low revenue alternative. On the other hand, when the utility is low, flexible products should become attractive because of the demand induction effect, and consequently, the gain G/(T) due to flexible products will be high. Moreover, because reducing rates is equivalent to scaling capacity up, it can be expected that for low values of w, flexible products will remain attractive over a larger time horizon T. [00135] The experimental results are consistent with the above analysis. From the plot of revenue vs. T shown in FIG. 7, it is clear that revenue is an increasing function of w. On the other hand, gain Gf(T) plotted in FIG. 8 is, on the whole, a decreasing function of w. The gain curves corresponding to different values of w tend to cross in the indeterminate range of the time horizon that was identified in the previous section. In this range, capacity utiUzation effects dominate, and the network offers flexible products (not to enhance demand, but to better manage capacity). Thus, the gains are high even for high value of the utility w. In fact, capacity utilization effect explains most of the gain for the case with w = 3. Table 6 below lists the number of flexible products in the optimal collection or set.

Table 6 [00136] For fixed utility w, the number of flexible products used decreases with the time horizon T (this is consistent with results in the previous section). For a fixed time horizon T, the number of flexible products used decreases with w. [00137] In summary, flexible products are especially attractive for low values of utility w and small time horizon T because of the demand induction effect (the network can improve the revenue by at least about 8% by offering flexible products). At intermediate values of T, the demand induction is not as important, and it is not as clear to what extent flexible products are beneficial. For large T, one can sell all the capacity to customers demanding specific products, and flexible products are thus completely unattractive.

Effect Of Outside Alternatives

[00138] This section discusses the effect of competition, or the outside alternative, on the gain Gf(T). The effect of competition is captured by altering the rates gj ,j = 1, ..., n, and h , k= l, ..., k, where both of these factors measure the attractiveness of products offered by competitors. Recall that, initially (see equations (14) and (15)), it was set that gjl gj = H , / hk = 20. In this section, iteration is used such that gjl gj = h~k, I hk = φ, where φ= {5, 10, 15, 20, 25}. It is clear that, for at least some choice models, increasing ^has the effect of increasing the rate of customers that leave the network without purchasing any product (that is, the outside alternative becomes more attractive).

[00139] FIG. 9 plots the gain G_f(T) vs. the φfor the utility w = 1, and FIG. 10 is the same plot for w = 2 (note that the gain curve corresponding to φ = 20 is the same as that in FIG. 6). As used herein, is an index measuring the relative attractiveness of products for a company relative to the products of its competitors. From these plots, it is clear that increased competition has the effect of shifting the gain curve up and to the left. For φ small (that is, when the outside alternative is not very attractive), the network manager has a captive customer pool and there is no incentive for offering flexible products. Thus, the gain Gf(T) is small. The gain in this scenario is explained by capacity utilization effects, rather than demand induction. On the other hand, for large φ, (that is, in highly competitive environments), the network manager attempts to capture customers by offering a cheaper flexible alternative. In this scenario, the manger is attempting to enhance demand by offering flexible products.

Effect Of Changing Discounts

[00140] In this section, where revenue ru = (1 - ε)

{pj}, the relation between ε and Gf(T) is investigated for ε e {0, 0.25, 0.5, 0.75}. It will be understood that ε represents a small number, and is used to bound the ratio of the heuristic to an upper bound on the revenues. As FIGS. 11-12 illustrate, increasing the discount on flexible products decreases the gain. Interestingly, even offering a 75% discount produces a positive gain for certain time horizons T. In particular, when w = 0.5 (see FIG. 11), a 75% discount produces positive gains over the same range as a 0% discount.

Demand Induction vs. Capacity Utilization

[00141] As previously noted, the revenue improvement from flexible products results from two sources: demand induction and capacity utilization. In order to aid in the understanding of the relative contribution of both of these sources to the total gain, the gain G_d(T) has been computed using the same set of specific products when no flexible products are offered and the demand rates are given by: _ ϊg_j + h_k , j = argmax_{[j≡Fk ]}{p_J }forsomek e F ¹ 1 g_j otherwise,

(i.e., all of the demand induced by a flexible product is added to the highest fare specific product).

[00142] FIG. 13 plots the gains G_f(T) and G_d(T) as a function of the time horizon T. At T = 150, the gain curve G_d(T) drops below Gf(T) and stays below for all T> 150. Therefore, it is possible to conclude that capacity utilization is the primary explanation for increase in revenue for moderate to large values of T. The gain G_d(T) ~ 0 for T> 180, and therefore, it follows that demand induction is important for small T (if at all).

Effect Of Customer Uncertainty

[00143] The choice model coefficients that have been used for flexible products from equation

(15) assume that the prospective customer evaluates a flexible product based on the longest trip time of any specific product within the flexible product. This is a highly "risk-averse" model because it assumes that consumers evaluate flexible products on a "worst-case scenario" assumption. In this section, an alternate model is considered in which customers evaluate flexible products based on their subjective probabilities that they will be assigned to each constituent specific product. However, a term that penalizes the uncertainty in the flexible products is also needed, because a customer choosing a flexible product would be less able to finaUze his travel plans until closer to the flight time. Therefore, a model is considered where the demand rate for a flexible product is calculated as: K = (u -cr_k - βt_k ) -H(p^k)) ,

where H(p^k) = -∑ ,_eF p^k log( ?*) is the entropy of the probabiUty mass function p^k.

[00144] In this model, the entropy term H(p) penaUzes uncertainty. That is, if there is an equal probability of being assigned to all the specific products (in which case the buyer of the specific product has no idea of what he is getting), the rate will be lower than if there is a high probability of being assigned to a particular specific product (in which case the buyer of the flexible product can be fairly certain of what he is getting). The rates gj,j = l, ..., n, for specific products are still given by (14).

[00145] FIG. 14 shows the gain curves for the entropy model. Reasonable gains were produced when the buyer of a flexible product can be fairly certain of which product he is buying (that is, when the probability of a flexible product being assigned to a certain specific product is high). When the assignment probabilities were uniform (that is, when the customer has no idea what product he is ultimately buying), the maximum gain is less than 1%. When the customer can be 99% certain of buying a certain product, the maximum gain is slightly less than 4%. This indicates that the value of flexible products is less if customers are highly uncertain about their ultimate assignment, and, they strongly penaUze uncertainty.

[00146] In any real network, the demand for the flexible and specific products is Ukely to be random. Below, the performance of a feasible Unear problem-based poUcy in a stochastic network is explained.

[00147] It is assumed that when the network manager offers a set S N u E requests for a specific product j e S n N arrive according to a Poisson process with rate λ/S) and requests for flexible products k e S n E arrive according to a Poisson process with rate Jj(S), where λ(S) and γ(S) are defined by (5) and (6) respectively.

[00148] It can be established that the optimal value of the LP (1) is an upper bound on the expected revenue achievable in the stochastic network. On the other hand, optimal solution {(S ι, t^* _/), / = !, ..., L}of the LP yields a feasible policy defined as follows: (a) Select a permutation π of (1, ..., L) and open sets in the order {S _/,: 1= 1, ..., L} for a period {t _/,: / = 1, ..., L} and

(b) At the beginning of the time interval corresponding to the set S _/ open all products included in the set S /. At any time t e [0, T , let x(i) e Rⁿ and y e R^f denote respectively the number of specific and flexible products sold.

[00149] When a booking request is received, update the state (x, y) (that is, if the arriving request is for a specific product, increment the appropriate component of x by 1, else increment the appropriate component of y by 1). If there exists an integer composition vector z such that y = Uz and Ax + Cz ≤ c, accept the arriving request. Otherwise, reject the request, reset the state (x, y) to the original value and close the product corresponding to the current request. Moreover, it should be noted that, in order to implement the policy, an integer program needs to be solved at every booking request.

[00150] For the computational experiments using the above feasible policy, a network with m = 5 resources, n = 10 specific products, and/= 4 flexible products was considered. The remaining data, including the matrix A, the sets Nk, k = l, ...,f, and the revenue vectors r and p, were randomly generated. For each random instance of the underlying network, the performance of the linear program-based policy was simulated as the time horizon T and the total capacity c are simultaneously scaled up by a factor α = 1, 10, 100, 1000. For each scale factor the revenue generated by the above policy was averaged over 500 trials. In every random instance of the network, it was found that the permuting sets {S _/} did not improve the performance of the optimal poUcy. Therefore, the results for the identity permutation are reported.

[00151] FIG. 15 displays the results for generic random network instance. In the plot, μ denotes the average expected revenue, i.e., ll T E[R(aT)], where R(αT) is the random return of the poUcy over [0, ctT], the upper and lower bounds, labeled μ± 2σ respectively, are given by (l/α7 *E[R(αT)] + (2l( T)²) Var[R( T)], and the "lp opt" denotes the optimal value of the LP. From the plot it appears that as the scale factor α T ∞, the average revenue appears to asymptotically reach the LP optimal. Thus, for large time horizons Tthe LP solution is a good approximation for the achievable revenue rate. This result is consistent with similar scaling results for networks with only specific products. ANALYSIS 3: STRATEGIC PRICING IN CONSTRAINED MARKETS WITH REPEATED TRANSACTIONS [00152] In this section, a class of models are considered with the following four properties. First, assume that sellers have fixed capacity or a fixed inventory of a good for sale. Examples of fixed capacity include airline selUng seats on a flight, hotels renting rooms, and manufacturers selling capacity slots in a pure order-to-delivery environment. Examples of fixed inventory of goods for sale include fashion retailing where inventory is ordered once and cannot be replenished because lead times are longer than the sale horizon.

[00153] Second, assume that buyers interact with sellers over a number of time periods to transact the good. The model is developed in terms of a "capacity sale," in which buyers are purchasing reservations for capacity to be consumed at a future time (as in airline reservations). The model is also applicable to "inventory sales," in which buyers receive the product at the time of purchase (as in fashion goods).

[00154] Third, assume that the value of unsold goods drops to zero at the end of the final period. In other words, there is no salvage value for the unsold goods. Positive salvage values can be incorporated by adjusting prices without loss of generality.

[00155] Fourth, assume that buyers and sellers will interact more than once, and that buyer expectations and competitive actions by sellers will be influenced by observed past behavior.

[00156] The general model provided below relating to the above four properties can be used to manage revenue, and covers a wide number of real-world markets including airUne seats, hotels, fashion goods, high technology goods, gas pipeline capacity sales, automobiles and many others. This model provides pricing and availability policies when there are multiple transactions and the behavior of buyers depends upon the availability that they have seen in the past.

[00157] Special cases of perishable, constrained capacity models that will be expanded upon include "revenue and yield management" and "markdown management." The revenue and yield management problem is faced by airlines, hotels, rental cars, and other industries for which prices are generally assumed to be fixed, and the seller's problem is to determine how much capacity to reserve for higher price, late-booking customers.

[00158] In markdown management, on the other hand, the problem facing the seller is when and by how much to markdown unsold goods over a finite season. The number of times that goods can be marked down is usually Umited, and it is often assumed that the price of a good cannot be raised once it has been marked down.

[00159] For the basic model, assume that a seller has a fixed, perishable capacity C, and that sales of this capacity occur over two periods. During period , the supplier sells capacity at a price of . It is assumed that r; and r₂ are announced ahead of time and known to both the seller and all buyers. At the end of period 2, the value of unsold capacity drops immediately to 0 (that is, there is no salvage value). However, it should be noted that the problem can be reformulated by setting n <— - r3 for i e [1, 2, 3} if there is a residual value r < r₂. The case in which rj > r is denoted the markdown management problem, and the case in which r i < ∑ is denoted the revenue management problem.

[00160] It is assumed that there is a population of potential customers D. Each customer within this population can be characterized by a wilUngness to pay vector (Wi, Wi), where j denotes his wilUngness to pay in period i for capacity. The following three possibilities exist.

[00161] First, Wi = W₂. The willingness to pay of a buyer is the same in the first period as the second. This buyer's behavior will be driven entirely by the prices being offered in the two periods and his belief on capacity availability in period 2.

[00162] Second, Wi ≥Wz. This buyer is wilUng to pay more in period 1. This would characterize markdown markets in which the value of the object purchased declines over time - such as fashion goods as well as some industrial markets where the buyer is willing to pay a premium to lock-in future capacity and eUminate supply uncertainty.

[00163] Third, Wi < W₂. This buyer is wilUng to pay more in period 2. This is considered to be a characteristic of some airline passengers who may be uncertain during period 1 about their need to travel, but have that uncertainty resolved in period 2.

[00164] It is assumed that the willingness to pay in the two periods has a joint density function/ with the support on R² ₊. Note that this model is general. For example, as a special case, D could consists of two independent populations (one of which has Wi > 0 and W₂ = 0, and the other of which has Wi = 0 and W₂ > 0).

[00165] The unconstrained demand in period i is denoted by Z>,. In each period, the seller can choose to accommodate demand up to his level of unsold capacity, or he may choose to set a sales Umit, 0 < bi < C on the amount of capacity that he allows to be sold. Then, expected total revenue is given by: TR(bj, b₂) = rjE[min(Dι, bi)] + nEminlD , b_% max(C-Dj, C- bj)].

[00166] It is assumed that sellers are seeking to determine to values of b,- to enhance as much as possible total revenue given their capacities, prices, and uncertain knowledge of demand. For simplicity, it is also assumed throughout that variable costs for the seller are zero, which is an appropriate assumption for sellers seeking to dispose of existing inventory, such as a retailer who has already purchased fashion goods or a hotel selling rooms. Moreover, b is denoted as "tactically optimal" if it increases total revenue assuming a single instance of the problem. Because capacity is perishable, b' = C. In addition:

n < r₂ → b i = (C -

- r,/r₂))⁺, (2) where G is the cumulative distribution function (c.d.f .) on D₂, here assumed to be continuous and strictly increasing on R⁺ (hence invertible).

[00167] As mentioned above, the analysis of this section focuses on the case of multiple instances where buyers and seller interact repeatedly over time. Each instance may correspond to capacity availability in each period, and hence, the unconstrained demand in each period will change for each instance. Let in) indicate unconstrained demand in period i during instance n ≥O, pι(n) indicate buyers' probability of availability during instance n, and b(n) the amount of capacity that the seller will offer for sale at price r,- during instance n. Let x n) denote sales and y_t(n) = x_t(n)/Di(n) the fraction of unconstrained demand in period i that is satisfied in instance n. Then, it is reasonable to expect that a buyer's posterior probability that capr2 - acity will be available in period n will be a function of the fraction of total demand that was satisfied in past interactions:

Pi( ) =_Pi (4)

[00168] ExpUcitly considering this process of expectation formation leads to pricing and availabiUty poUcies that can differ significantly from those that are tactically optimal. [00169] Situations are also considered in which there is more than one supplier. In these cases, the supplier is denoted by a superscript capital letter. Thus, x^A(n) indicates sales from supplier A during period i on interaction n. Absence of a superscript will indicate that there is only a single supplier.

[00170] The strategic pricing of distressed inventory is now considered. Marking down prices on obsolete or distressed inventory is a time-honored in many industries. Inventory may become distressed because it is physically perishable ~ as in airUne seats or goods such as fish or meat. Alternatively, inventory may become distressed because it becomes unseasonable or obsolete as new models are released - as in fashion goods or high tech electronic goods. In this connection with this analysis, the invention provides the optimal pricing of distressed inventory in the presence of buyers whose expectation of future discounts may influence current behavior.

[00171] In the distressed inventory model, prices are lower in the second period than the first, r₂ < r , and inventory perishes at the end of the second period. The optimal tactical policy is for sellers to allow all of their unsold capacity to be sold at the price r , that is, b'₂ = C (given the intuitive notion that making the distressed available for sale at any price r > 0 is more profitable than allowing it to "perish" unsold). However, the solution can be considerably different when future interactions between buyers and sellers are considered.

[00172] In this case, a buyer will either purchase the capacity at the full price in the initial period or choose to "wait for the sale" and hope to purchase the capacity at the discount price in the second period. For a particular buyer, the decision to purchase in period 1 or wait until period 2 will depend upon the prices in the two periods, his wilUngness to pay for capacity, the cost of -waiting nd his estimate of the likelihood that the capacity will be -available in period 2. If buyers believe that sufficient capacity will be available in period 2, and their wilUngness to pay does not change, then they will all wait for the sale and purchase at the lower price. If, on the other hand, buyers believe that there is a low chance that capacity will be available in period 2, then some buyers "lock in" their purchases in period 1 while others will wait until period 2 in hopes of getting the lower price. In a repeated interaction between buyers and sellers, it is reasonable that buyers will form their expectations about the availabiUty of distressed inventory based on their past interactions. This will provide an incentive for sellers to Umit the amount of distressed inventory that they offer. [00173] Let /?; be a buyer' s subjective probabiUty of capacity being available in period I with p_\ >/?2- It is assumed that buyers seeking to maximize their expected utiUty (as measured by consumer surplus, or the difference between the consumer's reservation price and the selUng price). A buyer with wilUngness to pay (Wi, Wi) would prefer to purchase in period 1 if (Wi-ri) > (W₂-r₂)⁺p₂, or equivalently, if W\ > w^*(W₂), where w*(W₂) = rι+p2(W₂-r₂)⁺. Consequently, the buyer's behavior will be determined by: Wι≥w*(W₂) → purchase in period 1 W₁<w*(W₂),W₂≥r₂ → wait to purchase in period 2 Wι<w*(W₂),W₂<r₂ → do not purchase in either period.

[00174] Assume that total demand D is a Poisson random variable with parameter λ. If all buyers share the same values for/?ι and/?₂, then £>,^■ is also Poisson with parameters λj, i = 1, 2 given by: λ_l = λ?r{W_ϊ ≥ w* ⁽W₂)} λ₂ = λPr{W₁ < w* (W₂),W₂ ≥ r₂}.

[00175] Notice that λi is a non-increasing function of p_%. This means that buyers' expectation of availability of distressed inventory at a lower price will tend to cannibalize full-price sales.

[00176] The expected amount of full-price demand that is cannibalized is λVr{r_x < W, < w* W₂)} . Managing this quantity is the motivation for sellers to attempt to influence buyers' expectations. Defining w (Wι(n)) analogously with the definition of w^*(Wi) provided above, the full dynamic pricing system and Poisson demands D, rι) with parameters λ,, i( ) = 1, 2 can be specified as:

P_t (n) = g[y_t (n - 1), y, (n - 2), y. (n - 3),..., y, (0)] w* (W₂(n)) = r_{l +} p₂(n)(W₂(n)- r₂γ λ (n) = λ Pr { Wi ( ) ≥ w * (W₂ (n)) } λ₂( ) = tPr{W, (n) < w* (W₂(n)),W₂(n) ≥ r₂)} , ( ) = τmnlD_j^ (n), C] x₂ (n) = min[D₂ (n), C -x_x (n), b₂ (n)] y.(rø) = Xι( )lDi(n) fori = 1,2

[00177] To see the impact of customer expectation formation, consider the following example, where D follows a Poisson distribution with parameter λ: C = 100; =150; λ = 250; pι(0) = 1; P2(0) = 0; pt(n + 1) = in) + oi in) - pin)) for all n ≥l. In addition, it will be assumed that the wilUngness to pay is the same in the two periods and is given by an exponential distribution with rate 1/100 (mean = 100), that α = 1, and that bj(n) = C. For comparison purposes, the expected revenue that would be achieved from offering a single fare at $150 (that is, b₂(n) = 0) would be $8,367 with X (n) = 55.8 for all n.

[00178] Table 7 provided below shows the results of a sample path for the case r = 50 and biin) = 100. As expected, the tactical policy of making all remaining capacity available for sale at r results initially in a significant revenue increase. However, as buyers learn to anticipate that distressed inventory will be available and adjust their behavior accordingly, cannibaUzation of full-fare business begins to occur. Averaging the results of a simulation over 1000 sample paths results in an estimated expected equiUbrium revenue equal to $5,968.40 with a standard error of $16.22. This average represents 71.3% of the benchmark with b (n) = 0. 5CΘSffi3 35S ij _l i [^mλι fe ∞lϊftj

Table 7

[00179] The example in Table 7 illustrates that the tactically optimal solution of selUng all distressed inventory at a discount price can lead to a suboptimal solution when the impact on customer expectations is taken into account. Now, consider the case where the seller sets a higher distressed inventory price r₂ = 100 and sets a booking Umit b₂(«) = 50.

[00180] Table 8 provided below shows a sample path for this case. Again, the seller initially reaUzes a substantially increased revenue. As buyers learn to anticipate that distressed inventory will be available and adjust their behavior accordingly, cannibaUzation of full-fare business begins to occur. In this case, however, the demands and sales oscillate between two states during the first few instances (one in which total demand is split between the two classes, and one in which virtually all demand is for the discounted capacity). The simulation of over 1000 sample paths indicates that the system converges to an expected equiUbrium revenue of $7,789 (93.1% of the benchmark revenue), with a standard error of $59.50. Again, the seller would be worse off from selling distressed inventory, even at the higher fare and with the Umit in place.

Table 8

[00181] The above examples assume that consumers are totally myopic and base their expectations on the last realization. However, it has been observed that the limiting behavior of the system seems to be the same under exponential smoothing updates of probabilities for all positive values of . It is also interesting to observe that while the fluid Umit approximation can display oscillatory behavior, its limiting average behavior is very close to that of the corresponding stochastic system. The present invention also encompasses a system to determine the extent to which demands seen by the suppliers in each period are driven by the non-linear dynamics of the system as opposed to the intrinsic uncertainty.

[00182] The above description of the present invention shows that there can be significant differences in profitabiUty depending upon the booking control mechanism used, where revenue management can comprise booking control mechanisms. The present invention also shows that, under reasonable assumptions, the benefits from offering flexible products can make it worthwhile to consider them as part of the overall market offering.

[00183] Although the invention has been described and illustrated in the foregoing illustrative embodiments, it is understood that the present disclosure has been made only by way of example, and that numerous changes in the details of implementation of the invention can be made without departing from the spirit and scope of the invention. For example, although the invention has been described in the paragraphs above with particular reference to the airline industry, it should be noted that the use of the methods and systems for managing flexible products as described herein is contemplated in a number of other industries as well (including the other industries described above). Moreover, it will be understood that certain features which are well known in the art have not been described in order to avoid complication of the subject matter of the present invention. The present invention is limited only by the claims which follow.

Claims

WE CLAIM:

1. A computer-implemented method for managing revenue from selUng a product comprising: (a) selUng a flexible product comprising at least two alternative products to a purchaser, wherein one of the alternative products is assigned to the purchaser after purchase of the flexible product and within a specified period of time that is estabUshed as a component of the flexible product prior to or upon purchase of the flexible product; (b) determining by the seller of the flexible product which alternative product is to be assigned to the purchaser; and (c) informing the purchaser which alternative product has been assigned within the specified period of time after the sale of the flexible product.

2. The method of claim 1 , further comprising receiving a request for the flexible product by the purchaser prior to step (a).

3. The method of claim 1 , further comprising offering the flexible product for sale prior to step (a).

4. The method of claim 1, wherein the determining comprises observing specific demand for at least one of the alternative products before the purchaser is assigned an alternative product.

5. The method of claim 1, wherein the the determining comprises waiting until uncertainty on specific product demand is substantially resolved before the purchaser is assigned an alternative product.

6. The method of claim 1, wherein the determining comprises using a linear program and/or a column generation approach.

7. The method of claim 1, wherein the determining comprises seeking to maximize quantity of products sold.

8. The method of claim 1, wherein the flexible product comprises a set of alternative products that have a perishable/constrained capacity and/or constrained inventory.

9. The method of claim 8, wherein the length of the specified period of time is based at least in part on the perishable nature of at least one of the alternative products.

10. The method of claim 1, wherein the flexible product comprises at least one of an airplane ticket reservation, a hotel room reservation, a concert ticket reservation, an internet Web page advertising space reservation, an air cargo reservation, and a vacation tour reservation.

11. The method of claim 1 , wherein the flexible product is an airplane ticket reservation, and the specific period of time is one week prior to the earliest departure date of the alternative products.

12. The method of claim 1, wherein the flexible product is an airplane ticket reservation, and the specific period of time is one week prior to the latest departure date of the alternative products.

13. The method of claim 1, wherein the flexible product is an airplane ticket reservation, and the specific period of time is 24 hours prior to the earliest departure time of the alternative products.

14. The method of claim 1, wherein the flexible product is an airplane ticket reservation, and the specific period of time is one day prior to the latest departure time of the alternative products.

15. The method of claim 1, wherein the flexible product is an airplane ticket reservation, and wherein the selling occurs no later than one week prior to the earUest depature date of the alternative products.

16. The method of claim 1, further comprising modifying the specified period of time is modified after the sale of the flexible product.

17. The method of claim 16, wherein the modification of the flexible product is the result of a negotiation between the seller and purchaser of the flexible product.

18. The method of claim 16, wherein the modification of the flexible product is a unilateral modification by one of the seller and the purchaser of the flexible product.

19. The method of claim 16, wherein the modification of the flexible product is caused by the seller of the flexible product, the method further comprising providing compensation from the seller to the purchaser.

20. The method of claim 16, wherein the modification of the flexible product is caused by the purchaser of the flexible product, the method further comprising providing compensation from the purchaser to the seller.

21. A computer-implemented method for managing revenue from selUng a product comprising: (a) selUng a flexible product comprising at least two alternative products to a purchaser, wherein one of the alternative products is assigned to the purchaser after purchase of the flexible product; (b) determining by the purchaser which alternative product is to be assigned to the purchaser; and (c) informing the seller of the flexible product which alternative product has been assigned.

22. The method of claim 21, wherein one of the alternative products is assigned to the purchaser within a specified period of time that is estabUshed as a component of the flexible product prior to or upon purchase of the flexible product.