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Pricing perishable inventories by using marketing restrictions with applications to airlines Li, Michael Zhi-Feng 1994

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PRICING PERISHABLE INVENTORIES BY USING MARKETINGRESTRICTIONS WITH APPLICATIONS TO AIRLINESByMichael Zhi-Feng LiB. Sc. (Mathematics) Beijing Normal UniversityPh. D. (Mathematics) University of ReginaA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYillTHE FACULTY OF GRADUATE STUDIESFACULTY OF COMMERCE AND BUSINESS ADMINISTRATIONWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1994© Michael Zhi-Feng Li, 19944In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Faculty of Commerce and Business AdministrationThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1Z1Date: /,(iAbstractThis thesis addresses the problem of pricing perishable inventories such as airline seatsand hotel rooms. It also analyzes the airline seat allocation problem when two airlinescompete on a single-leg flight. Finally, several existing models for seat allocation withmultiple fares on a single-leg flight are compared.The pricing framework is consistent with modern yield management tools which utilize restrictions such as weekend stayover to segment the market. One model analyzedconsiders a restriction which is irrelevant to one set of consumers, but which the othersfind so onerous that they will not purchase a restricted ticket at any price. If the consumers who do not mind the restriction are less price sensitive than those who find therestriction onerous, then the thesis shows that there is an optimal policy for a monopolistwhich will sell fares at no more than three price levels.When two restrictions are allowed in the model, if one is more onerous than the otherin the sense that the set of consumers who would not buy a ticket with the first restrictionis a subset of those who would not buy it with the second restriction, then the restrictionsare said to be nested. If the sets of consumers who would not buy tickets with the firstrestriction is disjoint from those who would not buy with the second restriction, thenthe restrictions are said to be mutually exclusive. If two restrictions are either nested ormutually exclusive, then a monopolist needs at most four price levels with three types (i.e.combinations of restrictions) of product. With two general restrictions, the monopolistmay need five price levels with four types of product.The pricing model is applied to restrictions which are based on membership in aparticular organization. For example, employees of an airline are frequently eligible11for special fares. Some airlines provide special fares for government employees or foremployees of certain corporations. An analysis is given to help airlines understand thecosts and benefits of such arrangements.A model of two airlines competing on a single-leg flight is developed for the casewhere the airlines have fixed capacity and fixed price levels for two types of fares —full and discount. The airlines compete by controlling the number of discount fareswhich they sell. The split of the market between the airlines is modelled in two differentways. First, the airlines might share the market for a fare class proportionally to theirallocation of seats to that fare class. In this case, under certain conditions, there existsan equilibrium pair of booking limits for the discount fare such that each airline willprotect the same number of seats for the full fare customers, even when the demands arerandom and stochastically dependent. The second market sharing model assumes thatthe two airlines share the market demand equally. In this case, when the demands aredeterministic, then there is an equilibrium solution where each airline will protect enoughseats to split equally the market for the full fares.Finally, three existing seat allocation models for multi-fare single-leg flights withstochastically independent demands are compared. It is shown that the optimality conditions for each of these models are analytically equivalent, thus providing a unifiedapproach to this problem.111Table of ContentsAbstract iiList of Tables viiiAcknowledgement ix1 Introduction and Background 11.1 Introduction 11.2 Airline Seat Inventory Control 41.2.1 The Seat Allocation Problem for Single-leg Flights 41.2.2 The Seat Allocation Problem for Multi-leg Flights 51.2.3 The Seat Allocation Problem for Multiple Flights 91.2.4 The Overbooking Problem 91.2.5 The Seat Allocation Problem in the Presence of Competition . 111.3 Airline Fare Pricing 121.3.1 Introduction 121.3.2 Current Operating Environment 131.3.3 Modelling Practice 161.3.4 New Directions 211.4 Objectives and an Overview of This Thesis 222 Pricing Perishable Inventories by Using a Restriction 302.1 Introduction 30iv2.2 Rationing Rules and Wilson’s Pricing Model 322.2.1 Rationing Rules 322.2.2 Wilson’s Model 342.3 An Optimal Pricing Model by Use of Restrictions 422.3.1 The Model Settings 422.3.2 The Demand Constraints 462.3.3 The Formulation of the Optimal Pricing Model 472.3.4 Basic Properties of Optimal rn-Policies 482.4 Optimal Pricing Strategies 552.5 An Application to Airline Fare Pricing 692.6 Summary — Pricing by Using Restriction 753 General Optimality Results and Other Properties 763.1 Introduction 763.2 Auxiliary Results on General Pricing Policies 773.2.1 Notation and Definitions 773.2.2 A Lemma on the Property of Monotonicity 833.3 General Optimality Results 853.3.1 Three Fundamental Lemmas 853.3.2 The General Optimality Theorems 1023.4 Further Properties of Optimal Policies 1073.5 Summary — General Optimality Results 1134 Pricing Models with Two Types of Restrictions 1154.1 Introduction and Model Setting 1154.2 Pricing Problem by Using Two Nested Restrictions 1184.3 Pricing Problem by Using Two Mutually Exclusive Restrictions 125V4.4 Pricing Problem by Using Two General Restrictions 1314.5 Summary — Pricing by Using Two Types of Restrictions 1455 Airline Pricing by Using Membership and Product Restrictions 1475.1 Introduction 1475.2 Model Setting and Notation 1505.3 Cheaper Restricted Membership Fares Only 1515.4 Cheaper Unrestricted Membership Fares Only 1565.5 Cheaper Restricted Membership Fares and Cheaper Unrestricted Membership Fares 1595.6 Summary — Membership and Product Restrictions 1646 Seat Allocation Game on Flights with Two fares 1666.1 Introduction and Model Setting 1666.2 Seat Allocation Game Under Proportional Splitting Rule 1706.3 Seat Allocation Game Under Equal Splitting Rule with Deterministic Demand 1766.4 Summary — Seat Allocation Game 1807 A Note On Three Models for Multi-fare Seat Allocation Problem 1817.1 Introduction 1817.2 Wollmer’s Model 1827.3 Curry’s Model 1837.4 Brumelle-McGill’s Model 1857.5 Equivalence of Optimality Conditions 1877.6 Summary — On Three Model for Multi-fare Seat Allocation Problem . 191vi8 Summary of the Thesis and Future Directions 1928.1 Summary of the Thesis 1928.2 Future Directions 1948.2.1 On Seat Allocation Problems 1948.2.2 On Seat Allocation Games 1958.2.3 On the Pricing Problem by Using Restrictions 1968.2.4 On Further Applications 198Bibliography 200viiList of Tables1.1 An Example of Fare Classes for a Two-leg Flight 81.2 An Example of Virtual Nesting Classes for a Two-leg Flight 82.1 Effect of Fare Restrictions — Boeing Company (1982) 702.2 Solutions of the Tight Problem — A Numerical Example 74viiiAcknowledgementWith sincere appreciation I first acknowledge the advice and encouragement from themembers of my supervisory committee, Professors Shelby L. Brumelle, Tae H. Oum andMichael W. Tretheway. In particular, I want to thank my supervisor, Professor ShelbyL. Brumelle, for his tireless effort of helping me to finish this dissertation. I have alsobenefited a great deal from Professor Michael W. Tretheway for helping me to improvethe presentation of this thesis.I am deeply grateful to the Faculty of Commerce and Business Administration ofthe University of British Columbia for supporting my first two years of graduate studiesthrough a MacPhee Memorial Fellowship. I feel very fortunate that I had been awardeda University Graduate Fellowship by the Faculty of Graduate Studies of the Universityof British Columbia during the early stage of this dissertation. I am also indebted toProfessor Shelby L. Brumelle for supporting me in the last two years of my study.ixChapter 1Introduction and Background1.1 IntroductionSince the first important paper of Littlewood (1972,[131]) on the airline seat allocationproblem, the subject of yield management has become a main topic for revenue enhancement for most airlines in the world. For example, American Airlines, a leader in airlineyield management research and implementation, has experienced a tremendous success inthe last decade or so due at least in part to its yield management system. It is estimatedthat during the period of 1990 to 1992, the annual benefit to American Airlines from itsyield management programs was approximately 500 million dollars, or about a 2 percentimprovement on total passenger revenues.1Yield management is far from completely understood. For example, yield management is sometimes considered to be a pricing mechanism.2 But actually, yield management is a tool to assist firms in determining the maximum number of reservations toaccept at given price levels so as to maximize total expected revenues. This is not thesame as setting prices — prices are taken as given. The research so far on yield management shows no indication that it is capable of providing the optimal set of prices for thefirm. In fact, any seat inventory control system must have a set of pre-specified pricesso that demand can be forecasted at these price levels. For airlines, the actual pricing‘Refer to Smith et a! (1992, [221]) for a survey on yield management practice at American Airlines.2This view is often held by professionals from hotel industry, for example, Relihan (1989,[193]) andRanks et al (1992, [83]).1Chapter 1. Introduction and Background 2decisions are traditionally made separately, by a different department.3 The first goalof this thesis is to enhance seat management techniques by developing a tactical pricingmodel for perishable inventories like airline seats and hotel rooms. In fact, the main partof this thesis is dedicated to this goal. The key innovation is to explicitly incorporatethe use of artificial restrictions as marketing mechanisms. In particular, the questions Iwant to get answers include: (1) is it necessary to use the restrictions? (2) what are theoptimal pricing structures? (3) do we have a tractable characterization for the optimalpricing structures? (4) how can the model be used in industries, such as airlines?Another shortcoming of the existing yield management literature is that airlines donot operate in isolation of each other. The existing research in yield management still doesnot address the issue of competitive response to other airlines’ seat allocation and pricingdecisions. To analyze the strategic aspect of airline operations, we need to understandthe nature of the strategic interaction among airlines. An early view was that the airlineindustry is a contestable industry.4 But empirical studies show that the airline market isnot perfectly contestable, which implies that the strategic interaction among airlines isoligopolistic. Typical oligopolistic behaviour has three forms: (1) price-type competition,also known as Bertrand competition, where price is the primary strategic variable; (2)quantity-type competition, also known as Cournot competition, where quantity is theprimary strategic variable; and (3) collusion, where firms collude with each other toexploit their monopolistic power. A recent empirical study by Brander and Zhang (1990,[33]) shows that the competition among airlines is a quantity-type competition. Thisevidence suggests that it is important to develop a new generation of seat allocation3There is a trend that some airlines now use a single department to handle pricing and yield management together. This suggests that airlines are starting to realize the importance of coordinating thepricing decisions and inventory control decisions.4An industry is contestable when firms charge the same prices and supply the same quantities as acompetitive industry would, even there are only a few firms in the industry. A key argument that theairline industry is contestable is that airplanes — the key assets for airlines — are mobile.Chapter 1. Introduction and Background 3models which explicitly consider competitive quantity decision responses. The second goalof this thesis is to make the first step toward a full understanding of the seat allocationproblem in the presence of competition.It is well-known that modern yield management techniques critically depend on aproper understanding of the seat allocation problem for single-leg flights with multiplefares. The most recent development along this direction include three independentlydeveloped articles, Curry (1990, [52]), Woilmer (1992, [269]), and Brumelle and McGill(1993, [38]), which solve the seat allocation problem for single-leg flights with multiplefares when the random demands for different fare classes are independent. It is interesting to note that these three papers have utilized three different analytical tools todetermine optimal booking policies; and they have three different optimality rules forbooking policies. This raises the issue as to which approach is the best and which onehas a computational advantage over the other two. A later part of this thesis clarifies thisissue by showing that all three approaches are in fact analytically equivalent in the sensethat the optimal policy rules in fact are identical. The thesis provides a unified approachthat may shed useful light in dealing with the seat allocation problem for multi-faresingle-leg flights with dependent random demands.Before moving on to the main development of the thesis, brief overviews of airlineseat inventory control and airline fare pricing are presented. These overviews providethe necessary background information needed to evaluate the contributions of this thesisand serve as motivation for the general class of problems to be addressed.Chapter 1. Introduction and Background 41.2 Airline Seat Inventory Control1.2.1 The Seat Allocation Problem for Single-leg FlightsThis is the most studied area in yield management. For a flight with two fare classes withindependent demand, an optimal booking policy has been characterized by Littlewood(1972, [131]). He proves that the optimal protection level 7) for the high fare5 is givenby:*Mi{Offp(y> k—)}where fj and fh are the fare price for a low fare seat and a high fare seat respectively, Y isthe demand for the high fare class, k is the flight capacity, and P(.) denotes a probabilitydistribution. The formula gives a decision rule that tells an airline reservations controllerwhat to do when a booking request for the low fare arrives. Since the airline will notreject a high fare booking, the issue is when to stop booking the low fare tickets. Theformula simply says that the airline should accept bookings for the low fare until theexpected seat revenue from an uncertain high fare sale is greater than the certain low farerevenue f.6 If the demands are continuous random variables, then we have the followingclosed-form formula:fi = fhP(Y> k —This result was generalized by Brumelle et al (1990, [37]) to the case of two-fare flightswith dependent demands.The seat allocation problem for single-leg flights with multiple fares with independentdemands has been solved independently by Brumelle and McGill (1993, [38]), Curry5A proleclion level for the high fare is the total number of seats that will be proecied (i.e., reserved)for the high fare customers.6J is worthwhile to mention that the main reason why the Littlewood’s formula involves the surevalue of a low fare ticket—fi — is that each low fare booking, when it arrives, corresponds to a suresale. On the other hand, if there is no booking request for the low fare, then the decision of rejecting oraccepting a booking for the low fare never occurs. So by default, the formula is still valid.Chapter 1. Introduction and Background 5(1990, [52]), and Woilmer (1992, [269]). Belobaba (1987,[13], and 1989, [14]) introducedan heuristic method — the so-called Expected Marginal Seat Revenue (EMSR) method— to handle the seat allocation problem for a single-leg flight with multiple fare classes.The major advantage of this method is computational since it is a technique basedon Littlewood’s result. Consequently, if the random demands are independent, thenthe EMSR method gives optimal solutions for any two fare classes in isolation, butunfortunately this will not be optimal for the whole problem.7 The current challenge isto address the seat allocation problem for general multi-fare flights (that is, not limitedto two fares) with dependent demands.1.2.2 The Seat Allocation Problem for Multi-leg FlightsThis problem turns out to be a major headache for all airlines. If we ignore the randomness of the demands, then the seat allocation problem for multi-leg flights can beformulated as a network flow optimization problem. This had been a main focus ofresearch for one decade or so.8 But there are two main drawbacks to this approach:1. It cannot handle random demands;2. It is of little use for modern computer reservation systems.9To my knowledge, there is no major airline in the world that has actually implementedthis approach. What the airlines need is a framework that gives either optimal or heuristicsolutions and is compatible with their current reservations systems.7Simulation studies show that (1) the difference between the revenue derived from the EMSR methodand the optimal revenue is oniy 0.5 percent; and (2) the optimal booking policies may be drasticallydifferent from the policies derived from the EMSR heuristic. Refer to Brumelle et al (1990, [37]) for areason why revenue functions are insensitive to booking limits.8For example, refer to Ladany and Hersh (1977, [118]), Hersh and Ladany (1978, [89]), Glover et al(1982, [80]), and Dror, Trudeau and Ladany (1988, [63]).9This will become more clear from the discussions that follow.Chapter 1. Introduction and Background 6There are two other basic approaches in dealing with the seat allocation problem formulti-leg flights, which are the segment-based method’° and the revenue-based method.Segment-based methods allocate seats on a flight in an effort to maximize revenue oneach individual flight segment, independent of other flight segments. Currently, mostairlines manage seat inventories by flight segment only. A typical segment-based seatallocation model involves two stages:1. The airline first decides how many seats that will be made available for each segment; and2. With the given number of seats allocated to each segment, the airline specifiesbooking policies for fare classes on each specific segment.As claimed in Williamson and Belobaba (1988, [260]), several major airlines had implemented the EMSR method, which was originated by Littlewood for single-leg flights andwas later extended by Belobaba (1987, [12]) for multi-leg flights. The main idea from theEMSR model is to protect seats for a higher fare class as long as the expected marginalrevenue of the seats is greater than marginal revenues of the seats at a lower fare class.In the multi-leg EMSR approach, protection levels and booking limits are determined foreach fare or each nest” on each flight segment. The main drawback for the segment-basedapproach is the lack of consideration for the interaction of traffic across flight segments‘°A segment is an origin-destination (O-D) itinerary on the same flight number. For example, if legsA-H and H-B have the same flight number, then there are three separate segments: A-H, H-B and A-B.Refer to Curry (1990, [52]) for these definitions.“A nest is a group of similar fares on the same O-D on the flight. For example, for a two-leg flightA-B via H, the airline may pooi all fare classes on the segment H-B into a single nest. Clearly, on eachsegment, the maximum number of nests is the total number of fare classes on the segment. Traditionally,airlines offer as many as 15 fare classes on each flight segment, which implies that for a two-leg flight,there will be 45 different fare classes on the flight and the CRS must keep track of booking informationfor all these 45 fare classes regularly. Therefore, grouping fares on each segment into 4 or 5 differentnests will substantially reduce the amount of information that needs to be displayed on the CRS. Thisis precisely why the deterministic network approach is not implementable in the current reservationsystems.Chapter 1. Introduction and Background 7within the network in the determination of seat allocations and booking limits. On theother hand, the major advantage of a segment-based approach is its compatibility withcurrent airline reservations systems.12Revenue-based methods have been developed and implemented by American Airlines.The main ideas are:• Indexing all fare classes on a flight on the basis of the absolute ticket revenue ofeach fare class, where a lower index means a higher priority;’3and• Grouping all those fare classes with the same index, regardless of their O-D itineraries,into a bucket.In practice, only the buckets — not the actual inventory of seats by fare classes — aredisplayed on the reservations systems. This is why this method is often known as thevirtual nesting method. Consequently, a bucket is also known as a virtual inventory class.The optimization aspects of a revenue-based approach jnclude:’4• Development of a systematic method of indexing fare classes and a simple procedureof updating the indices during the booking period; and• Finding the protection levels, or equivalently the booking limits, for each bucket tomaximize total revenues of the flight.12There are several extensions for Belobaba’s EMSR methods, for example, refer to Wong (1990,[270]), Belobaba (1991, [15]), and most recently Wong et al (1992, [271]).‘3This is a critical component of any revenue-based method. Unfortunately, there is no public literaturethat addresses this issue. In the meantime, it is worthwhile to mention that this problem is also aninteresting theoretical issue. Clearly, the most efficient and consistent way of indexing fare classes on thewhole network is to have an index table that include all fare classes on all flights on the network, ratherthan indexing fare classes on a flight-by-flight basis. So the theoretical challenge here is to develop amodel for such an universal indexing table. Of course, the first question is whether or not such a tableis possible.14Relevant papers on this problem include Buhr (1982, [41]), Wang (1983, [253]), Simpson (1985,[217]), Simpson (1985, [218]), Smith and Penn (1988, [222]), Curry (1989, [51]), Curry (1990, [52]),Curry et al (1990, [54]), and Smith et al (1992, [221]).Chapter 1. Introduction and Background 8Fare Glass Segment A—B Segment A—H Segment H—BY $1000 $750 $700M $800 $500 $470Q $540 $300 $290B $330 $220 $200Table 1.1: An Example of Fare Classes for a Two-leg FlightBuckets (Virtual Classes) O-D ClassesV0 YABV1 YAH, MAB, YHBV2 MAH, BAB, MHBV3 BAH,QAB,BHBV4 QAH,QHBTable 1.2: An Example of Virtual Nesting Classes for a Two-leg FlightAccording to Smith et al (1992, [221]), American Airlines have also used the EMSRmethod in determining the booking limits for each bucket.To see how this approach works, let us consider Flight 101 from A to B via the hubcity H. This is a two-leg flight with three O-D itineraries, that is, A-B, A-H and H-B.Suppose that on each O-D itinerary, the airline offers four fare classes, Y, M, Q and B.Refer to Table 1.1 for the made-up revenues for each of the 16 fare classes for this flight.For the purpose of illustration, assume that the airline plans to group these fares intofive buckets, labeled as V0, V1, V2, V3, and V4. Refer to Table 1.2 for one specification ofthese five buckets. In the actual booking process, suppose there is a request for the fareclass M on segment H-B of Flight 101. When a travel agent looks at the system and ifthere still are some seats available for bucket V2, the agent can authorize this booking.But during the reservation process, the actual fare class MHB on Flight 101 never showsup.Chapter 1. Introduction and Background 91.2.3 The Seat Allocation Problem for Multiple FlightsEqually important, airlines need to address the seat allocation problem in the presenceof other flights. This becomes increasingly important because of the hub-and-spokenetworks airlines have developed especially since U.S. airline deregulation in 1978. Mostflights originating from a hub will pick up a substantial amount of connecting traffic fromother flights destined to the hub. Without explicitly incorporating the demand from theconnecting traffic, a yield management system cannot realize the full potential revenuefor each flight. For example, consider Flight 201 from city H (the hub) to city B andFlight 202 from city A to city H. There are two key questions that are important toairlines:• How much should the airline charge a traveller who flies from city A to city B usingflights 201 and 202 with a restricted ticket? And how much should be charged foran unrestricted ticket?• How many seats should be allocated on flight 202 for travellers from city A to cityB?These problems are closely related, but are not simple. Along this direction, except forthe network flow approach, there is no other theoretical work done in this area.151.2.4 The Overbooking ProblemSince airline seats cannot be consumed before the time of the flight, they are sold in theform of reservations. This causes an operating problem for airlines, since some customerswill either cancel their reservations at the last minute or simply not show up. Since the‘5Refer to Ladany (1977, [1161), Hersh and Ladany (1978, [89]), Glover et al (1982, [80]) and Dror etal (1988, [63]).Chapter 1. Introduction and Background 10airline has no time to resell these tickets, some seats will be unused. With overbooking,16the airline can generate additional revenue on these otherwise empty seats. The airlineoverbooking problem has two important components:• Determining the optimal overbooking levels, which are associated with the bookinglimits for fare classes or buckets; and• In the event that the flight is overbooked at the departure time, specifying a procedure for denying boardings to some passengers.The first part is an optimization problem. Along this direction, pioneering research canbe traced back to Beckmann (1958a, [9]), Beckmann (1958b, [10]), Thompson (1961,[235]) and Taylor (1962, [234]). Further analytical development has been studied inmany papers in the seventies.17 For most recent development, refer to Rothstein (1985,[203]), Brumelle et al (1990, [37]), and Bodily and Pfeifer (1992, [24]).18The second issue is an implementation problem. The fact is that when no-shows andcancellations are low, the flight becomes truly oversold, which means that the numberof passengers expecting to board the flight is larger than the flight seat capacity. Consequently, some passengers must be bumped, or denied boarding. This issue had been afocal point in the early stage of overbooking practice. The early practice of involuntarybumping created some serious consequences, including lawsuits against airlines. Eventually, the Simon-auction method, proposed by Simon (1968, [215]), was implementedby many airlines. The basic idea of a Simon-auction is: the airline asks passengers whodo not mind being bumped to submit a sealed bid for compensation and the airline will16Simply speaking, we say that an airline overbooks a flight if during the booking period, the totalnumber of accepted bookings for the flight is larger than the flight-capacity.‘7For example, refer to Rothstein (1971a, [199]), Rothstein (1971b, [200]), Simon (1972, [216]), Vickrey(1972, [250]), Rothstein (1975, [202]) and Shlifer and Vardi (1975, [214]).‘8For the overbooking problem in the context of hotels, refer to Rothstein (1974, [2011), and Libermanand Yechiali (1977, [127]; 1978, [128]).Chapter 1. Introduction and Background 11choose those passengers with the lowest bids to be bumped. Since the whole processis voluntary, it has no legal complications for the airline. Nowadays, most airlines usevariations of the original Simon-auction in their practice.1.2.5 The Seat Allocation Problem in the Presence of CompetitionSo far we know very little about how to manage the competition problem in the contextof yield management. A recent study by Brander and Zhang (1990, [33]) shows that theCournot model is more consistent for the data in their study than either the Bertrandor cartel models. This is a very interesting and important result for seat managementresearch, since the quantity competition problem in the airline industry is closely relatedto the seat allocation problem under competition.Consider that two airlines each have a scheduled flight from city A to city B, and thatthere are two fixed fares fh and f,, which both carriers charge. With random demandfor both of the two fare classes, each airline’s strategic decision variable is its protectionlevel for the high fare, or equivalently, its booking limit for the low fare, since one airline’sdecision on its protection level for the high fare will have revenue impact on the otherairline.’9 This seat allocation game can be analyzed in several ways. The most naturalapproach is to treat the game as a bimatrix game, since the protection level for the highfare is an integer bounded by the flight capacity.2°On the other hand, there are also many different theories of oligopoly in economics‘9The key issue here is how to split the demand when both airlines allocate a positive number of seatsfor a particular fare class.20A bimatrix game is a non-cooperative two-person non-zero sum game with a finite set of purestrategies for each player. There is a substantial amount of literature on bimatrix games. For example,refer to Bohnenblust et al (1950, [29], Nash (1950, [162]; 1951, [163j), Shapley and Snow (1950, [212]),Vorob’ev (1958, [252]), Kuhn (1961, [112]), Lemke and Howson (1964, [123]), Mangasarian (1964, [137]),Millham (1968, [147]; 1972, [148]; 1974, [149]), Raghavan (1970, [188]), Owen (1971, [175]), Parthasarathyand Raghavan (1971, [177]), Kreps (1974, [111]), Shapley (1974, [211]), Heuer (1975, [90]; 1979, [91]),Heuer and Millham (1976, [92]), Jansen (1981a, [93]; 1981b, [94]), and Okada (1984, [165]).Chapter 1. Introduction and Background 12and marketing that are useful for airlines to gain a general understanding of the strategicbehaviour of the firms. But they are not very useful at the operational level.2’ Therefore,I am not going to investigate the allocation game along this direction. For an excellentsurvey on theories of oligopoly, refer to Shapiro (1989, [2101).221.3 Airline Fare Pricing1.3.1 IntroductionSince U.S. airline deregulation in 1978, airlines have enjoyed complete freedom in settingfare levels. In fact, airline fare pricing has evolved from the pre-deregulation simpleand static practice to a more dynamic and complex structure. Marketing innovationshave been developed and used to segment consumers into groups with different demandelasticities and to stimulate market demand and create consumers loyalty. Two classicalexamples are the use of artificial restrictions on discount seats and the establishmentof frequent-fliers programs. On the other hand, even though the application of thesemarketing innovations is well-known in the airline industry, there is very little academicattention in economics and marketing literature to the theoretical implications of thesemarketing innovations.23 In fact, in the vast literature of pricing research in economics21j the context of homogeneous product with capacity constraints, refer to Levitan and Shubik (1972,[125]), Kreps and Scheinkman (1983, [110]), AlIen and Heliwig (1986, [2]), Davidson and Deneckere (1986,[59]), and Kiemperer and Meyer (1986, [107]). In the context of differentiated products, refer to Dixitand Stiglitz (1977, [61]), Hart (1979, [87]), Salop (1979, [205]), Shaked and Sutton (1982, [208]), andSingh and Vives (1984, [219]), Bulow et al (1985, [40]), Mas-Colell (1975, [138]), Moorthy (1985, [153]),Perloff and Salop (1985, [178]), Moorthy (1988, [154]), Anderson et al (1989, [5]), Economides (1989,[71]), and Cremer et al (1991, [47]).22Also, general discussions on the competition issue in the airline industry can be found in many otherpapers. For example, refer to Graham et al (1983, [82]), Levine (1987, [124]), Reiss and Spiller (1989,[192]), Tretheway (1989, [244]), Brueckner (1990, [36]), Hansen (1990, [85]), Morrison and Winston(1990, [157]), Sorenson (1990, [223]), and Strassmann (1990, [233]). These studies are policy-orientedand they are useful for us to have a better understanding of the airline industry, in particular, airlinederegulation.23A recent article of Gale and Homles (1993, [74]) studies the economic impact of using advance-purchasing in the context of airlines. The primary focus of this paper is on the mechanism design.Chapter 1. Introduction and Background 13and marketing, there is no analytical framework that is directly applicable to airline farepricing.24 In this section, we will briefly discuss several aspects of airline operations inconnection with marketing restrictions.1.3.2 Current Operating EnvironmentBefore moving on to the discussion of the modelling practice in the economics literaturethat may be relevant to airline fare pricing, let us first summarize some recent corporatechanges in the airline industry and some unique operating characteristics related to airlinefare pricing. First of all, as it is well-known, an airline fare class is characterized by a priceand a set of restrictions. The restrictions are intended to prevent business travellers frombuying deep discount fares targeted to leisure travellers. The main fare pricing decisionsfor an airline include:• Evaluating the impact of restrictions and choosing profit maximizing restrictions;and• Choosing profit maximizing prices and deciding how many fare classes to offer.One of the main consequences of U.S. airline deregulation is the emergence of complicatedfare structures. Before deregulation, airlines had limited freedom in setting fare pricesand selecting the type of fare classes on a particular route. After deregulation, the farestructure had become so complicated that on some flights, there are over 15 differentfare classes with all kind of restrictions or fences. This in turn means that a two-legflight will have 45 different fare classes. For carriers with a large operating network, themaintenance of such a complicated fare structure became a major operating challenge.24There are many empirical studies of airline demand. For example, refer to Oum and Gillen (1983,[171]) and Oum et al (1986, [172]). These studies, together with traditional economic theories, canprovide basic guidelines for airlines. But with the sheer number of ifights and the diverse nature ofdifferent city pairs, their usefulness is very limited for daily operations and pricing decisions.Chapter 1. Introduction and Background 14In a surprising move in April of 1992, American Airlines (AA) suggested a simplified fare structure and hoped that other major airlines would cooperate. Even thoughsuch an idea won a widespread praise from other airlines and the travel agency industry,this proposal was short-lived. By the end of April of 1992, none of U.S. major airlinesincluding AA decided to fully implement it. Such a resistance might be in part due tothe fact that airlines did not know how to price their products by explicitly incorporating the marketing restrictions they had been using for many years. In fact, there wasno theoretical justification for such a simplified fare structure, and AA did not provideany satisfactory explanation.25 There is another interesting development recently emerging from the airline industry. Again lead by American Airlines, many airlines in theworld started to merge the traditional separate functional departments for pricing andyield management into one department.26 This indicates that airlines now realize theimportance of coordinating the pricing decisions with yield management techniques.With these new developments in the marketplace, it is urgent for airlines to develop atactical pricing model on a flight-by-flight basis that is consistent with their existing yieldmanagement system. The main task of this thesis is to take this challenge. In particular,I will develop a general pricing model for perishable inventories, such as airline seats andhotel rooms, which explicitly incorporates the use of artificial restrictions.27Since the pricing model in this thesis is intended to be applicable for airlines, it isimportant to understand some basic, but unique, characteristics of the airline businessthat are related to fare pricing. There include:25Since AA had never explicitly revealed where the new simplified fare siructure come from: therationale behind the simplified fare structure is a secret.26Other examples include British Airways and Cathay Pacific. I have checked with additional majorairlines through telephone inquiries and have found that even though they have two different functionaldepartments, the interaction between the two is on a daily basis.27AS we will see later, the pricing model developed in this thesis will have optimal fare structures thatat least in part coincide with the new simplified fare structure suggested by AA, that is, offering threecoach fare classes with at least one unrestricted coach fare.Chapter 1. Introduction and Background 15• The airline product is perishable and consumers can neither store nor enjoy theproduct before a given time;• Capacity is fixed;• Costs are either fixed or sunk;• Airlines are operating in a network environment;• Market demand is easily segmented, for example, leisure travellers and businesstravellers;• Airlines practice price discrimination by use of artificial restrictions;• Matching competitors’ prices is the only way to survive because of the commoditynature of airline products; and• Sophisticated computer reservation systems are powerful tactical tools to allowairlines to make better decisions on allocating a limited number of seats amongmany different fare classes, which is very difficult for competitors to mimic becausethese are an airline’s private information.The traditional view of pricing perishable products, such as produce and dairy products,is that the price will decrease as the product approaches the end of its life. This is mainlydue to the fact that:• such a product can be consumed at any time during its life span; and• all consumers prefer to purchase and consume the product early.For airline seats, travellers will enjoy the pleasure and feel the value all in the sameperiod, that is, the period of the flight time. Because of this, consumers for airlineChapter 1. Introduction and Background 16seats can delay their purchasing decisions. The striking implication of this behaviour isthat airlines simply cannot sell down. If they do so, rational consumers may wait for thecheapest price.28 Therefore, it suggests that the traditional pricing theories for perishableproducts are not useful for airlines at all and new approaches must be called for.1.3.3 Modelling PracticeApparently the main reason that an airline uses marketing restrictions, such as Saturday-night stayover, advance booking, and corporate membership, is to effectively segmentmarket demand in order (1) to serve each market with a proper product and service, and(2) to improve the airline’s revenue by extracting the maximum possible revenue fromeach market segment. Therefore, a precise understanding of the impact of marketingrestrictions must be an integral part of airline fare pricing. This is also important toaircraft manufacturers since the seat configuration in an aircraft is the basis for airlinesto offer different products and services.On the other hand, modern yield management tools, in particular, the seat inventorycontrol models, require the airline to pre-specify a set of prices so that yield management specialists can conduct demand forecasting on this set of prices and make properdecisions on booking requests for different fare classes.29 Airline seats on a particularflight invariably are put on sale several months in advance of the actual departure time.Therefore airlines must make pricing decisions before the inventory control models arein effect. The traditional approach was to offer a large number of fare classes and let280f course, there are some consumers who just want to purchase the seats early in order to have theirreservations confirmed. The point here is that consumers no longer have any incentive to purchase early.29j the actual implementation of seat allocation models, demand forecasling is critical. Littlewood(1972, [131]) had some discussions on this issue. The latest development is the censored regression modelof McGill (1989, [144]).Chapter 1. Introduction and Background 17the yield management system take care of the rest. As a consequence, airline yield man—agement specialists face a very complicated fare structure for each flight. But the lackof progress on the theoretical work for the optimal seat inventory control problem formulti-fare flights with random dependent demands becomes a major setback for suchan approach since it is even difficult to evaluate the performance of heuristic methods.Furthermore, complicated fare structures require more manpower and other resources,which is again a cost burden to airlines.One of the main features of the modern airline fare pricing is that airlines offermultiple fare classes on each flight. Then it is natural that we should try to find answersfrom economic pricing theories that suggest the use of multiple prices. Let us start withthe argument, due to Lott and Roberts (1991, [132]), that airline fare pricing is a formof product differentiation. According to Lott and Roberts, the main reason why thosecustomers travelling on short notice are paying a substantially higher price than thosewho can book early is the cost of the service: providing the consumer with an “ability topurchase a ticket at the last minute”(p. 21). They argue thatIn the case of airline seats, the opportunity cost of keeping some seatsavailable until the last minute is that they may go unused. The limitednumber of seats that airlines make available for advance purchase discountsshows how easy it is for airline to sell advance tickets. If they could not sellmany of these discount tickets, the restriction on the number sold would besuperfluous. This also explains why airlines have penalties for cancellationsof these discount fares and not for reservations at the higher price.For airlines to be willing to hold seats for last minute travellers, they mustmake the same expected revenue from these seats as they do from those seatsChapter 1. Introduction and Background 18purchased iii advance. . ..(p.22)They then conclude thatIf this explanation is correct, we should observe similar pricing for the“perishable” goods with uncertain demand.Our explanation does not prove that pricing anomalies are due onlyto cost differences. .. . Our explanation is able to explain, however, why advanced reservation discounts exist even on highly competitive routes afterderegulation. (p.22)But in their footnote 10, they feel that the use of restrictions is still a puzzle in airlinefare pricing:Another puzzle is the requirement that consumers have to spend a Saturday night at their destination in order to receive the discount. This is typicallyexplained as an example of price discrimination against business travellers,but it may only be a form of peak-load pricing if those who stay over Saturdaynight travel on Sunday, the quietest time of the week. The puzzle remains asto why there is not an explicit discount for returning on Sunday, but this isalso a problem for the price discrimination explanation. (p.21)These qualitative arguments are indeed interesting. But I do not think that Lott andRoberts provide a convincing argument as to why the traditional product differentiationmodel is appropriate for airline fare pricing with marketing restrictions.Now, consider the argument that airlines engage in price discrimination. It is wellknown that airlines offer many different price levels with the identical product. Thissuggests that airlines are practising price discrimination. There are three types of priceChapter 1. Introduction and Background 19discrimination in the economics literature.30 First-degree price discrimination, also calledperfect price discrimination,3’“involves the sellers charging a different price to each unitof product in such way that the price charged for each unit is equal to the maximumwillingness to pay for the unit.” (p. 600) An intuitive way of describing this type of pricediscrimination is that the seller makes a single take-it-or-leave-it offer to each consumerthat extracts the maximum amount possible from the market. But the “leave-it” threatlacks credibility: it is not a rational way of bargaining since the firm will take the risk oflosing the consumer and incurring a permanent loss of revenue. On the other hand, evenif the seller had a way of making such a commitment, he typically lacks full informationabout the buyers’ preferences. He cannot determine for certain whether his offer will beactually accepted. The requirement of full information about buyers’ preference is themain drawback for first-degree price discrimination. For the sales of airline tickets, suchan informational requirement is simply impossible.Second-degree price discrimination, also called nonlinear pricing, occurs when pricesdiffer depending on the number of units of the good brought. In other words, consumers face the same price schedule, but the schedule involves different prices for differentamounts of the good purchased. This type of pricing critically relates to the existence ofmarket segments and use of self-selection constraints. But with unit demand for airlinetickets from most consumers, the quantity schedule is not meaningful in the context ofairline fare pricing.32Third-degree price discrimination means that different consumers are charged different prices, but each consumer pays a constant amount for each unit of the good brought.30This classification of the forms price discrimination is due to Pigou (1920, [182]).31Refer to Varian (1989, [249]).320ne exception here might be the pricing problem for group booking, where there are bulk sales oftickets to tour operators, sport teams, etc. The fact that bulk sales are also subject to the ordinaryrestrictions used on other tickets makes the direct application of second degree price discriminationquestionable.Chapter 1. Introduction and Background 20This pricing scheme requires that the market demand be segmented and different marketsegments be perfectly sealed. To seal different market segments, firms use certain mechanisms, such as discriminating by age. The determination of different groups of consumersis taken exogenously by the model. Even though most real life price discrimination fallsinto this category, the requirement of perfectly sealed market segments is too strong toallow us to use this type of price discrimination to explain the use of multiple prices inairline fare pricing.The idea of using a marketing mechanism to help firms to segment the market is toogood to throw away. Telephone companies effectively segment the market by offeringdiscounts for residential usage; theatres successfully attract additional consumers byoffering student discounts and senior discounts. All these cases are classic examplesof effective third-degree price discrimination. One striking feature is that telephonecompanies and theatres use artificial restrictions that are preventive to some consumers.This is of great importance to the understanding of airline fare pricing. The difficulty isthat third-degree price discrimination requires a perfect marketing mechanism in the sensethat it will perfectly seal the different consumer groups. As demonstrated by Gerstnerand Hoithausen (1986, [76]), the action of perfectly sealing the market segments may notbe in a monopolist’s best interests. On the other hand, the pricing model in Gerstnerand Hoithausen (1986, [76]) only addresses the case of two segments, where each segmentconsists of identical consumers. The problem for airlines is that the restrictions, suchas advance booking and Saturday-night stay, cannot effectively segment the market intoseveral consumer groups so that each group consists of identical travellers. Therefore,a more delicate analytical framework is required to accurately model the fare pricingproblem for modern airlines. This, in fact, is the main motivation for pricing modelin this thesis. More precisely, the model proposed in this thesis is a third-degree priceChapter 1. Introduction and Background 21discrimination model using restrictions which are imperfect marketing mechanisms.331.3.4 New DirectionsClearly, there are many important issues in airline fare pricing that need to be investigated. The most important task should be the development of a fare pricing model thatexplicitly incorporates the use of marketing restrictions. Such a model is a fundamentalstep toward a full understanding of the economic benefits of marketing restrictions. Thesecond important issue should be the design of marketing restrictions. From the theoretical point of view, it would be interesting to know what kind of marketing restrictionsare plausible. But in practice, an optimally designed restriction (in theory) may not beimplementable since the actual implementation requires the physical specification of amarketing restriction (such as Saturday stay condition). In this regard, the second issueshould be more empirically oriented. Another important issue is how to incorporate thenetwork structure, in particular, the hub-and-spoke system, into fare pricing. Again thisis a theoretically challenging question with direct impact on the market place. As longas airlines are operating on a network environment, different markets in the network willinteract with each other, which will be reflected in the demand structure over the network. Therefore, such an interaction should be explicitly considered in the correspondingpricing problem.In order to develop useful models for airlines, there are two basic criteria that shouldbe followed:33Recently, Gale and ilolmes (1993, [74]) investigate the use of advance-purchase as a mechanism ofdiscounting in the pricing problem for a monopoly airline. Their discussion focuses on how to find anoptimal direci-revelaiion mechanism that will lead each consumer to make a rational choice between apeak flight and an off-peak flight. Their model is based on the assumption of the existence of differenttime costs over the two flights for a continuum of risk-neutral consumers. They concludes that advancepurchase discount is an optimal selling mechanism. Their model also predicts that no seats on the peakflight are sold at a discount, which is not consistent with airline fare pricing practice. In reality, airlinessometimes do make some discount seats available on peak flights.Chapter 1. Introduction and Background 22• the model solution, either an optimal pricing structure or an optimal restriction,must be consistent with yield management practice;34 and• the model must consider the technological constraint of the computer reservationssystems.35With these two objectives in mind, I will dedicate most of this thesis to the development ofa fare pricing model that implicitly incorporates the use of marketing restrictions. Duringthe development, instead of limiting the discussion to airlines, I will cast the model asa general monopoly pricing model for perishable inventories that will be applicable toairline seats, hotel rooms, rental cars and other sectors, where the monopoly firm has todecide whether a marketing restriction should be used and how it should be used.1.4 Objectives and an Overview of This ThesisThis section provides a summary of the objectives and main results of the remainingchapters of this thesis.Summary of Chapter 2: Pricing Perishable Inventories Using A RestrictionThis chapter addresses the monopoly pricing problem for perishable inventories, suchas airline seats, using a single restriction to segment the market. The following assumptions are made• the demand function is decreasing and is a step function defined on a finite set ofprices;341n particular, with analytical models and specialized yield management software.35For example, most of existing computer reservations systems use the concept of nesting or virtualnesting to display the information on reservations.Chapter 1. Introduction and Background 23• the impact of the restriction is such that as price increases, the percentage ofcustomers who can accommodate the restriction in the market is decreasing;• the monopolist offers the product for sale from the lowest price to the highest price;and• the restricted units are sold first on some lower prices and unrestricted units aresold later at some higher prices.The chapter proposes to answer the following questions:• Is it necessary to use restrictions? What is the economic benefit?• What is the optimal pricing structure? With how many prices?The main contributions of this chapter are:• It gives a detailed analysis of Wilson’s pricing problem for a capacity-constrainedmonopolist in the context of unit demand. Additional results are derived so thatthey can be used to formulate the pricing problem when the monopolist uses restrictions.• It presents a formal analytical model for the pricing problem of perishable inventories by using a single restriction.• It further characterizes some optimal pricing polices that can be derived from aseries of linear programming problems, which have at most three active price levels.• It demonstrates that except for some trivial cases, the monopolist can do strictlybetter by offering some restricted units.Chapter 1. Introduction and Background 24• It illustrates how the pricing model developed in this chapter can be used in airlinefare pricing. It shows that the assumption of the impact of restriction is equivalentto saying, in the airline context, that leisure travellers are more price sensitive thanthe business travellers.Summary of Chapter 3: General Optimality Results and Other PropertiesThis chapter addresses two important issues that are assumed away in Chapter 2.Although the current pricing practice by airlines and hotels is consistent with the assumption that restricted prices are lower than unrestricted prices, it is important, from atheoretical point of view, to understand the issue of optimal pricing practice when usingan artificial restriction. The model in Chapter 2 does not explain why the monopolistshould limit itself to the policies that restricted units are sold first at some lower pricesand the unrestricted units are sold subsequently at some higher prices — a practice whichwill be called primary policies. Practically speaking, it might make sense to sell someunrestricted units at some very low price and then open sales of some restricted unitslater. The main goal of this chapter is to investigate whether such a pricing practice isnecessary. Another issue is that the model in Chapter 2 assumes that the monopolistopens sales at one price at a time. This is not consistent with current practice. Forexample, airlines make all offered prices together with allocated quantities available simultaneously. Technically speaking, the model in Chapter 2 is not affected by this issuesince it is a static model. But to airlines, this is very important simply because theoptimal pricing structures, if implemented, must be consistent with yield managementtechniques.The main contributions of this chapter are:• Any general pricing policy, which allows the monopolist to sell restricted unitsat any price level, is always weakly dominated by a primary policy, where theChapter 1. Introduction and Background 25dominance is defined in terms of realized revenue for the policy. This result impliesthat any optimal pricing policy in the class of primary policies is also optimal in theclass of general policies. Therefore, it resolves the issue of optimal pricing practicewhen a monopolist uses artificial restrictions; and• Under a behavioral assumption that if restricted units and unrestricted units areoffered at the same price level simultaneously every consumer in the market willtry to buy an unrestricted unit first, it is shown that there always exists an optimalpolicy in the class of general policies with the property that even all offered pricestogether with the allocated quantities are made available simultaneously, there willbe no negative impact on the realized revenue.Summary of Chapter 4: Pricing Models with Two Types of RestrictionsBy using one restriction, the monopolist can introduce two kinds of products to themarket. In this chapter, I will use the same techniques developed in Chapters 2 and 3to discuss the pricing problem when the monopolist can use two types of restrictions,which will allow the monopolist to introduce three to four different kinds of product toconsumers. This will enrich the techniques developed in previous two chapters to handlemore realistic pricing problems.The main findings of this chapter are:• For the cases of two nested restrictions and two mutually exclusive restrictions, themonopolist’s pricing problem can be formulated as a mathematical programmingproblem with four constraints. It is further shown that this formulation can berelaxed into a linear programming problem. The main result is that the monopolistneeds to offer at most four price levels with at most three types of product tomaximize its revenue; andChapter 1. Introduction and Background 26• For the general case of two restrictions, it is shown that under certain plausibleconditions, there exists optimal pricing polices that consist of at most four kindsof products with five price levels to maximize its revenue.Summary of Chapter 5: Airline Pricing by Using Membership and ProductRestrictionsThe purpose of this chapter is to present some interesting applications of the pricingmodels developed in Chapter 4 to airline fare pricing when a product restriction, suchas Saturday-night stay and advance booking, is incorporated in association with the useof membership fares. Note that besides the conventional product restrictions, there aremany other forms of restrictions in airline industry, which further segment the market demand. For example, airlines usually have special agreements with some corporate clientsand government clients for special fares; and airlines are under constant requests fromtour operators for additional discounts. Also, airlines must deal with internal travellers(i.e., employees) and travellers from a partner airline. Each of these traveller groupsconstitutes a certain form of membership. The traditional view of membership is thatit gives the members certain privileges, which are not available to the general public.Therefore, the presence of a membership can be considered as a form of restriction. Thischapter discusses the issue of membership together with a product restriction. Studiesin this chapter enable an airline to understand the operating environment for three commonly used membership privileges. It is further shown that the pricing models developedin previous chapters can be used to evaluate certain corporate commitments associatedwith membership deals.The main findings of this chapter are:• For the case that the membership privilege is limited to a restricted fare at a pricelevel that is lower than the public restricted fare (for example, tour operators andChapter 1. Introduction and Background 27corporate retreat programs), the airline has an optimal fare structure with thefollowing properties:— it consists of at most four fare classes; and—the demand for restricted membership fares will be exhausted right after thesales of these restricted fares targeted to members;• For the case that the membership privilege allows the members to purchase cheaperunrestricted fares only (for example, interval travellers and major service corporations), the pricing model for two mutually exclusive restrictions can be used, whichimplies that the airline needs at most four different fare classes to maximize theflight revenue.• It is well-known that most airlines have interval travel policies for employees, whichindicates that if an employee pays a small fraction of the (public) full fare, thenhe/she will have a confirmed seat on the flight. Usually, these interval fares arecheaper than the public restricted fares, which is in fact a corporate commitmentto employees. It is shown that unless an airline has a systematic method to monitorsome high-demand flights, such a commitment may cost the airline a lot of money;• If the membership privileges include cheaper restricted fares and cheaper unrestricted fares (for example, government employees and travellers from partner airlines), it is shown that the airline needs to use the pricing model for two generalrestrictions developed in Chapter 4. Sufficient conditions are given for airlines pursuing simple optimal fare structures. These models can be used to evaluate whetheror not certain corporate commitment is indeed consistent with the operating environment; andChapter 1. Introduction and Background 28The chapter concludes with the following insight: in order to fully exploit thesemarket segments derived from the existence of memberships, an airline must havea systematic approach that can be computerized so that:— given a particular membership, it can tag those flights that will be availableto these members; and—it is capable of identifying the most favorable membership group on the flight-by-flight basis.Chapter 6: Seat Allocation Game on Flights with Two FaresThe main purpose of this chapter is to propose a major initiative in modern airlineyield management research, that is, to investigate the seat allocation problem in thepresence of competition. In particular, I will focus on single-leg two-fare flights with random demands (not necessarily independent) shared by two airlines, where each airline’sstrategic variable is the booking limit for the low fare, or equivalently, the protectionlevel to the high fare. The discussion is based on how the two airlines split the marketdemand. If the splitting rule is proportional according to their respective allocations,two main results emerge:• At equilibrium, each airline will protect the same number of seats for the high fare;and• At equilibrium, the total number of seats that are available for low fare class fromtwo airlines is strictly smaller than the total number of seats that would be availablefor the low fare if two airlines cooperate.Under the equal splitting rule, the discussion of the seat allocation game is limited to thecase of deterministic demands. It is shown that allocating enough seats to capture halfChapter 1. Introduction and Background 29of the high fare demand is an equilibrium strategy for both airlines.Chapter 7: A Note on Three Models for Multi-fare Seat Allocation ProblemThe seat allocation problem for single-leg multi-fare flights with independent random demands has been solved independently by Curry (1988, [50]), McGill (1988, [143])and Woilmer (1988, [268]). On the other hand, there are some differences in the fieldon which model is more computationally efficient. The purpose of the this chapter istwo-fold. First, I want to clarify this issue by proving that the optimal conditions fromthree models are in fact analytically equivalent, which therefore implies that they areall computationally equivalent too. Second, by unifying the existing approaches, I provide further insight on the multi-fare seat allocation problem with dependent randomdemands.Chapter 2Pricing Perishable Inventories by Using a Restriction2.1 IntroductionThis chapter develops a model for monopoly pricing a perishable product with a capacityconstraint.’ Examples might include airline seats and hotel rooms. Firms, such as airlinesand hotels, typically offer multiple prices and impose artificial restrictions to segment themarket. Surprisingly, there is no useful framework in economics or marketing to deal withthis kind of problem. Models in the economics literature are driven by simplicity andelegance, but these typically do not provide operational rules for management to follow.Pricing models in the marketing literature are aimed at the managerial level, and areintended to be realistic and therefore possibly to be implemented in practice. But itis very hard to achieve these two objectives at a satisfactory level without some degreeof compromise. For example, Dobson and Kalish (1988, [62]) develop an operational,heuristic procedure to position and price a line of related, substitute products. Theirmathematical programming formulation, however, is not computationally tractable. Thegoal of this chapter is to develop a model that is analytically tractable and practicallyinteresting.Recently, there has been a growing interest in the dynamic monopoly pricing problemof inventories in the operations research (OR) literature. In particular, Rajan, Rakesh andSteinberg (1992, [189]) present a model of simultaneous pricing and inventory control for1While the model is developed for the case of a monopolist, it has some applicability for any firmwith some degree of market power.30Chapter 2. Pricing Perishable Inventories by Using a Restriction 31a monopolist retailer who orders, stocks and sells a single perishable, but storable productfacing a known demand function. The solution to the model gives the optimal dynamicprice2 and the optimal cycle length.3 Other related papers along this direction includeWernerfelt (1986, [258]), Stadie (1990, [226]), and Gallego and van Ryzin (1992, [75)).The main differences between these OR models and the model in this chapter becomevery clear after knowing the following aspects of the new model:• The market consists of at least two types of consumers with different willingnessto pay;• The product is perishable, but non-storable;• The firm imposes an artificial restriction on the product at lower prices;4 and• The availability of the product at lower prices is controlled.It will be useful to have a pricing framework that explicitly incorporates these characteristics. The purpose of this chapter is to take up this challenge. The model in thischapter fills an important theoretical gap in the pricing research.This chapter is organized as follows. Section 2.2 first motivates the notion of rationing, which is directly related to the specification of the residual demand when salesare rationed. I then discuss a pricing model developed by Wilson, which provided the inspiration for the development of this optimal pricing model. Section 2.3 formally presentsan optimal pricing model that explicitly incorporates the impact of using one restrictionon the product at some lower prices. Section 2.4 discusses optimal pricing strategies that2A dynamic price is a price functional of time.3A cycle length is the length of the time period when the retailer will not make another orderingdecision. It is interesting to mention that the optimal dynamic price is independent of the choice of cyclelength.4The reason I call the restriction artificial is because the action of imposing it does not cost the firmanything. In contrast when a firm segments the market by product differentiation, the firm must expendresources in order to achieve it.Chapter 2. Pricing Perishable Inventories by Using a Restriction 32have tractable characterizations. Section 2.5 discusses an application of the model to theairline industry, where the market is segmented into business travellers and leisure travellers. This section also provides a simple example to illustrate the application. Finally,the last section is a summary.2.2 Rationing Rules and Wilson’s Pricing Model2.2.1 Rationing RulesThe notion of rationing is commonly used in the duopoly model of price competitionwhen two firms are capacity-constrained. Historically, the use of capacity constraints ina pricing game, first suggested by Edgeworth (1897, [68]), was motivated to resolve thewell-known Bertrand Paradox.5 The key question related to rationing is: if two firmsoffer different prices, then what is the residual demand for the high-price firm after thesales of the low-price firm? If the low-price firm’s capacity is large enough to satisfy thewhole market demand, then there is no residual demand for the high-price firm. Thusthe main issue becomes specifying the residual demand when the low-price firm cannotexhaust total market demand at its price level. As a general rule, in models with capacityconstraints, firms make positive profit and the market price is greater than the marginalcost.The rationing issue also arises for a capacity-constrained monopolist. The innovationover the traditional model is that a monopolist does not need to allocate all its capacityat single price level. In fact, the monopolist can offer an allocation schedule at differentprice levels to maximize its total revenue.6 This problem was first studied by Wilson5Simply speaking, the Bertrand Paradox says that when two firms produce identical goods withidentical and constant marginal cost, then there exists a unique equilibrium such that both firms priceat marginal cost and make no profit. This result is considered to be a paradox by some because it ishard to believe that firms in industries with few firms never succeed in manipulating the market priceto make profits.6We should notice that from a consumer’s point of view, it is irrelevant who is offering a higher price.Chapter 2. Pricing Perishable Inventories by Using a Restriction 33(1988, [262]). As in the duopoly case, the solution to this problem is critically related toa more specific assumption concerning the manner in which sales are rationed.Two rationing rules have often been considered in the literature.7 Assume that amonopolist produces a homogeneous product and each consumer oniy needs one unit ofthe product. If we let D(p) be the demand function for the product defined in the interval[po, oo) with N D(po) < oo and po > 0, then for p p0, D(p) can be interpreted asthe size of the market that consists of those consumers who have reservation prices of por higher.8 Now suppose that the monopolist first offers qi units at price p Po. Theefficient-rationing ruic presupposes a residual-demand function given by:I D(p) — q if D(p) >d(p;pi) =( 0 otherwise.This residual demand function essentially assumes the most eager consumers buy theproduct at price p. This rationing is efficient because it maximizes consumer surplus.Therefore, it is the most undesirable rationing scheme for the monopolist. Also, theresidual demand function defined by the efficient rationing rule is the one that would beobtained if the consumers were able to costlessly resell the good to each other, that is,to engage in arbitrage.9In the duopoly case, the higher price is offered by another firm; and in the monopoly case, the higherprice is offered by the same firm. The point here is that if the rationing makes economic sense in theduopoly case, it will also be rationed in the monopoly case.7The following discussions follow from Chapter 5 of Tirole (1988, [236]), pp. 212—214.8This is a typical way of handling unit demand when a firm does not have perfect information aboutconsumers’ willingness to pay. Such an approach has been extensively used in the economics literaturein many different contexts, for example, refer to Harris and Raviv (1981, [86]), Wolinsky (1984, [264]),and Perloff and Salop (1985, [178]). On the other hand, if we define F(p) = 1 — D(p)/N, then wecan interpret F(p) as the probability distribution function of a consumer’s reservation price. Therefore1 — F(p) is the probability that a consumer has a reservation price of p or higher. Such an interpretationis also consistent with the notion of imperfect information in economics when consumers have inelasticdemand.9The use of the efficient-rationing rule can be found in many articles from economics literature. Forexample, refer to Beckmann (1965, [11]), Levitan and Shubik (1972, [125]), Kreps and Scheinkman (1983,[110]), and Perry (1984, [179]).Chapter 2. Pricing Perishable Inventories by Using a Restriction 34Another popular rationing rule is called the proportional-rationing rule. It assumesthat all consumers have the same probability of being rationed. Given that the monopolistoffers q units at price p P0, the probability of not being able to buy at price p isD(pi)— qD(pi)Hence, the residual demand function after the sales of q units at price Pi is given byD(pi)—qid(p;pi) = D(p) for p > p.D(p1)This rule is not efficient for consumers since some consumers with valuation p’ : P1 <p’ <p are able to buy the good at the bargaining price p’, while some consumers withthe higher reservation prices are unable to. However, the monopolist prefers this rule tothe efficient rule since the residual demand is higher at each price. Typical conditionsfor the proportional-rationing rule are:• consumers arrive in a random order and are served on a first-come-first-served basis;and• there is no reselling opportunity.Because of these conditions, the proportional-rationing rule is better than the efficient-rationing rule for some industries with perishable inventories because these conditionsare easily met, for example, by airline seats and hotel rooms.102.2.2 Wilson’s ModelWilson (1988, [262]) is the oniy published paper that deals with rationing sales in thecontext of a monopolist. His primary goal is to explain the existence of price dispersion‘°Papers that have used this rationing rule include Beckmann (1965, [11]), Allen and Hellwig (1986,[2]), Davidson and Deneckere (1986, [59]), and Wilson (1988, [262]).Chapter 2. Pricing Perishable Inventories by Using a Restriction 35by a monopolist, when consumers arrive in a random order and are served on the first-come-first-served basis. His paper uses the proportional-rationing rule.Consider that a monopolist produces a homogeneous good with a fixed capacity.Assume that the market demand consists of heterogeneous consumers with unit demand.Let D(p) be the market demand function, which is interpreted as the number of consumerswhose valuation for the product is at least p. To simplify his analysis, Wilson makes afurther assumption on the demand structure:• D(p) is a left-continuous, non-increasing step function on a finite set of prices{pi, P2, , p,-, } where 0 <P1 <P2 < <Pn <00.In other words, the demand function is specified by:D1 ifppiD(p) = D if p (pt-i, p] and i> 10 ifp>p,where cc > D1> D2 > ...> D> 0.Remark: Since the demand function is a step function, the set of prices that a rational firm will consider is finite, which in fact is the set {p1,p2,• ,p}. For any pricep (pj., pi), the firm will be better off charging p rather than p, since increasing fromp to pi does not result in any loss of the demand.Since the choice set of prices is finite, the firm only needs to decide how many unitsof the product to make available at the finite number of prices. This implies that thepricing decision becomes an allocation problem. In this regard, Wilson introduces thefollowing definition of a pricing policy.Chapter 2. Pricing Perishable Inventories by Using a Restriction 36Definition 2.2.1 A pricing policy for the monopolist is a vector {q,, q,...,q}, whereqj > 0 is the number of units for sale at price p.The monopolist’s decision problem is to find a pricing policy to maximize its totalrevenue pq,. The key is to characterize the set of feasible pricing policies. Thefeasibility of a pricing policy is guided by two market forces: the demand and the supply.The supply is bounded by the capacity, which implies that we must haveqi k,where k is the firm’s capacity limit. I call this the supply constraint.I now want to derive the feasibility condition for the demand side.” First, sincethe product is assumed to be homogeneous and non-storable, a rational firm will onlyconsider selling the product from the lowest price to the highest price, since otherwisethose consumers with higher reservation prices would prefer to wait to get the product ata lower price.’2 More specifically, for any given pricing policy {qi, , q}, the sellingprocess is as follows: the firm first puts q units of the products on sale at price p,; afterselling q units at p, the firm allocates another q units on sale at price P2; and so onand so forth. The key aspect of this selling process is that if after the sales at pricesp the demand at price p is not exhausted, the residual demands at higher pricesare positive, since we assume that the consumers arrive in random order.’3Given a pricing policy {qi, q2, . , qn}, for k i and i 1, let d,k be the residual demand at price pk after the sales of the product at prices ,p, according to q . , q.“The feasibility condition on the demand side is more involved since we allow the possibility of multiplep rices.‘2This issue is more subtle than it appears here. For many products, if the action of delaying thepurchase decision creates a substantial loss of productivity or a substantial amount of disutility, thenthe firm may consider selling the product from the highest price to the lowest price. Wilson’s modelhere and my model later do not address the pricing problem for this case.13This assumption allows us to use the proportional-rationing rule.Chapter 2. Pricing Perishable Inventories by Using a Restriction 37Then according to the proportional-rationing rule,d+l,k = d,k — d,k qi for k i + 1, (2.1)d,2+1where the second part in the right hand side is the leakage from the residual consumergroup consisting of these consumers with the reservation price Pk or higher who actuallypurchase the product at the price Pi+i• Also, for ease of presentation, we will denotedo,k E Dk for k 1.The following lemma gives a simple updating formula for the residual demand, which isin fact the key in Wilson’s development.Lemma 2.2.1 For every i 1 and all k i, we haved2,k = Dk(1 — (2.2)where can be interpreted as the total leakage ratio of the market demand dueto the sales of the product according to the partial plan {qi,• , q}.Proof: I prove the result by using induction on argument i. For i = 1, by definition,Dk qid,k = Dk — ——q1 = Dk(1—i—Il JJ1Therefore, the result is true for i = 1. Now assume that the result is true for i 1. Iwant to prove that the result is also true for i + 1. By (2.1) and (2.2), it follows thatd+l,k = di,,,—qi+iui,i+’qt Dk= Dk(1——)— qi-i-D D+i.t=1i+1= Dk(1—YZ-J-).Chapter 2. Pricing Perishable Inventories by Using a Restriction 38This proves that the lemma holds for i + 1. So by induction, the lemma is proved. LIThe demand side feasibility condition is the following: we must maintain a nonnegative residual demand at the end of the sales at each possible price level. This isequivalent to requiring that dk,k 0 for all k, that is,qjVkl,which is equivalent tosince qj’s are all non-negative and Di’s are all positive. I call this the demand constraint.Finally, putting the objective function, the supply constraint, and the demand constraint all together, Wilson concludes that the firm’s pricing problem can be formulatedas the following linear programming problem:Max 1pjqjs.t.<1 (2.3)q k (2.4)qi 0 Vi.From now on, we will call this formulation Wilson’s Pricing Model or simply the Wilson-model. The following result is an immediate consequence of this linear programmingformulation.Theorem 2.2.2 [Wilson (1988, [262])] To maximize revenue, the firm needs to chargeno more than two prices.Chapter 2. Pricing Perishable Inventories by Using a Restriction 39Proof: This follows immediately from the fact that the number of non-zero variablesin any basic optimal solution for a linear programming is no more than the number ofstructural constraints. DThe following result, not proved in Wilson’s paper, shows that any optimal solutionin the Wilson-model will make the demand constraint binding.Theorem 2.2.3 If k D, then for any optimal solution in the Wilson-model, thedemand constraint (2.3) is binding.Proof: Note that if k = D, it is clear that the Wilson-model has an unique optimalsolution {p, D}, which will automatically make the demand constraint (2.3) binding.Therefore, we only have to prove the theorem for the case that k > D.I will first prove that the theorem is true for any basic optimal solution. Note thatany basic optimal solution in the Wilson-model consists of a pair {q’, q’} (i <j). If bothquantities are positive, (2.3) and (2.4) must be binding. So the theorem holds for thiscase.Now suppose that an optimal solution consists of only one price, say p, with theoptimal quantity q. I want to show that(2.5)Suppose (2.5) is not true, which must imply that:q<D1.Therefore the constraint (2.3) is not binding. Since the given optimal solution is a basicsolution, then the constraint (2.4) must be binding, that isq = k.Chapter 2. Pricing Perishable Inventories by Using a Restriction 40Furthermore, it must be true thatk> D+,,since otherwise the firm can sell all k units at price level Pi+i that will realize a totalrevenues of which contradicts to the optimality of q’ = k at price level j. Also,since k> D, we must have k <n. In summary, we get:• q’ =• 0<DD+,<k<D.Define a new allocation plan only with qj > 0 and q+, > 0 such that--+ —D2 D2+,qi+qi-1-, = k.Solving this system, we obtaink — D2,qi =— L/j+1D-kqi+i—Now the revenue under is given byR = piqi + pi,qii > p(q + q+,) = pk,because q2, > 0. This contradicts the assumption that q,’ is an optimal solution, whichimplies that (2.5) is true. This proves that the theorem is true for any basic optimalsolution. For the general case, we note that any optimal solution in the Wilson-model isa convex combination of basic optimal solutions. Therefore, the theorem is true for anyoptimal solution since both (2.3) and (2.4) are convex constraints.’4 D14A constraint is said to be a convex constraint if for any two feasible vectors to the constraint, anyconvex combination of these two vectors remains to be feasible to the constraint. In fact, any linearconstraint is a convex constraint.Chapter 2. Pricing Perishable Inventories by Using a Restriction 41Remarks: We can make the following interesting observations from the above theorems.• Theorem 2.2.3 tells us that if a firm’s capacity is not so small that the firm canactually sell all units at the highest positive price level—p,, the firm’s pricingdecision is primarily driven by the market demand in the sense that there will beno residual demand after the sales at the last positive allocation.• Theorem 2.2.2 and Theorem 2.2.3 together imply that a firm needs to offer twoprices only when the firm cannot reach its best possible revenue at the price thatexhausts all its capacity. This occurs when the single-price revenue function is notconcave.It is worthwhile to notice that k < D says that the firm can sell all its inventories atthe highest possible price level p, which consequently implies that there is no need toration the sales at all. Clearly, this is a trivial case. From now, unless explicitly stated,I will rule out this case. In other words, throughout this chapter I will assume thatkD.Now for a given demand structure, since the optimal revenue value in Wilson’s formulation oniy depends upon the capacity level k, we can write this derived revenue functionas R’(k). Wilson proves the following properties for R1”(k):Theorem 2.2.4 [Wilson (1988, [262j)j It is true that1. R”(k) is a concave and non-decreasing function of k;2. RL(k) is piece-wise linear in k;3. if RY3(k) is piece-wise linear in a neighborhood of k, then Ru(k) > R3(k),Chapter 2. Pricing Perishable Inventories by Using a Restriction 42where R3(k) is the best single-price revenue defined by:R3(k) = max xD’(x).O<x<kThe first property is true in general. The second property is a consequence of assumption that the demand function is a step function. On the other hand, if the bestsingle-price revenue function is concave, the firm always charges one price. The significance of Theorem 2.2.4 is that if the best single-price revenue function is not concave,then the firm can achieve a concave revenue function that is strictly better than the bestsingle-price revenue.2.3 An Optimal Pricing Model by Use of Restrictions2.3.1 The Model SettingsConsider that a firm plans to sell a fixed number of units of a certain product or service.For example, an airline wants to sell a fixed number of seats on a particular flight; a hotelneeds to sell a fixed number of rooms on each particular day; and a car rental companywishes to rent out a fixed number of cars on each day. The main characteristic of theproducts we are concerned with here is perishability, that is, if the firm cannot sell theproduct before a certain time, the product has no further salvage value. Because of this,the firm hopes to sell as many units as possible before the product totally losses its value.Also, I will assume that the product is not storable for consumers in the sense that theproduct can not be stored and consumed before a given time.Before moving on to the formal development of the model, it is necessary to highlightthe model’s assumptions:• Each consumer will purchase at most one unit of the product;• The firm has a fixed capacity k and its goal is to maximize revenue;Chapter 2. Pricing Perishable Inventories by Using a Restriction 43• The firm may choose to impose a restriction on the product to divide the consumergroup into two subgroups — those who do not mind the restriction and those whodo. For this, let D(p) be the demand for the product at price p with the restriction;and let D(p) be the demand for the product at price p only when the restrictionis not attached. Then the total demand for the product at the unrestricted price pis given by:D(p) = Dr(p) + D(p); and• The firm chooses a set of prices, each of which will have the restriction attached.To highlight the impact of the restriction on market demand, let a(p) be the percentageof those consumers with reservation prices of p or higher who will be unable to buy theproduct because of the restriction. Then a(p) is given by:D(p) Dr(p)= D(p) =1- D(p)As in the Wilson-model, to further simplify the analysis, assume that both D7(p) andD(p) are step functions defined on a set of prices {p1,p2,. ,p}, where P0 P1 <P2 <<n <00. More specifically speaking, for 1 = r, u,D1, ifp _<piDi(p) = D1, if p E (p,p] for 2 i no ifp>p,where D1, is decreasing (not necessarily strictly decreasing) in i. Denoting= Dr,i for i =then the total market demand function for unrestricted product is given by:D1 ifpp1D(p)= ifpE(p2_,p]for2in0 ifp>p.Chapter 2. Pricing Perishable Inventories by Using a Restriction 44Consequently, a(p) is also a step function:a, ifpp,a(p) = cv, if p E (p—i,p] for 2 i 5 n1 ifp>p,where cv, a(pj) for i= 1,.•. , n.Before introducing the definition of a pricing policy, I need to highlight all the assumptions used in this chapter:• D: is strictly decreasing in i;• D1, is decreasing for 1 = r, u;• a is strictly increasing in i, which implies that as the price for restricted product increases, the percentage of potential consumers who can accommodate therestriction is decreasing;• The monopolist will sell the restricted units first and the unrestricted units later;’5• The monopolist sells its product from the lowest price level to the highest pricelevel; and• k is the capacity and is such that k > D, which rules out the trivial case of thepricing problem.’6Sillce the monopolist sells the restricted product first, it is natural to choose a valueof m such that (1) 1 m n, and (2) the prices p,,— ,pm are attached with therestriction. Of course, the value of m is controlled by the monopolist. Then the firm’sproblem can be divided into two parts:Chapter 3, I will call this type of policy a primary policy.16This will rule out the trivial case as pointed out in Theorem 2.2.3.Chapter 2. Pricing Perishable Inventories by Using a Restriction 45• For any given value m, characterize the optimal pricing policies and the corresponding revenues, say R(m); and• Find the value of m that maximizes R(m).This section focuses on the formulation of the monopolist’s pricing problem for a givenm; and the next section will discuss the characterization of optimal choices on m.Note that for a given value m, as argued in Wilson’s model, the firm’s pricing problem is in fact an allocation problem. More specifically, I give the following definition of apricing policy, which is very similar to Definition 2.2.1 as in Wilson’s model.Definition 2.3.1 For a given integer m: 1 m n, a rn-policy for the firm is a vector{ q,.,i,” , q,.,; qu,m, , where q is the number of units of the product allocated tosell at the price p with restriction for 1 m, and q,,3 is the number of units of theproduct allocated to sell at the price p3 without restriction for m j n.Remark: For convenience, from now on, I call a price with the restriction a restrictedprice and a price without the restriction an unrestricted price. If a unit of the product issold at a restricted price, I call it a restricted product. Similarly, if a unit of the productis sold at an unrestricted price, I call it an unrestricted product.As we can see here, Definition 2.3.1 allows the possibility that the firm may offerrestricted products and unrestricted products at the same price Pm. This is a technicaltrick that will enable us to give a simple characterization of an optimal pricing strategy,as we will see in the next section.Chapter 2. Pricing Perishable Inventories by Using a Restriction 462.3.2 The Demand ConstraintsLet me first discuss market demand at the restricted prices P1,P2,’” , Pm, which correspond to the market demands Dr,i, Dr,2, , Dr,m. Since Dr,, is decreasing in i, thenWilson’s technique of rationing demand works here. Thus we have the following demandconstraint for the market of restricted products at prices Pi,• . , Pm1il.To analyze the residual unrestricted market after the sales of restricted productsaccording to the quantity schedule {qr,i,• , for j rn, we introduce the followingnotation:• be the residual demand for the restricted product at price pj;• dm,j be the total residual demand at the (unrestricted) price pj;.du =,J ,Jrm,j — m,j —Evidently, the residual demand dm,j is a function depending upon {qr,i,— , Fortunately, the following lemma shows that this functional relationship has a simple form.Lemma 2.3.1 After the sales of the product with restrictions at prices {p1, . ,pm}according to {qr,i, , the residual demand for the unrestricted product at price p2(j m) is given by:dm,j = — Drj ) -,where the second term in the right hand side is the fulfilled part of the restricted demandafter the sales of the restricted products.Proof: Define to be the residual demand for the restricted product at price p3 afterthe sales of the restricted products according to {qr,i, , qrm}. Since the demand marketChapter 2. Pricing Perishable Inventories by Using a Restriction 47for the restricted products is nested, then it follows from Lemma 2.2.1 (in Section 2.2)thatmdT D “1m,j — D,.21Therefore,dm,j=Du,j+d,j=Dj_Dr,j+dn,j=Dj_Dr,j>’.This proves the lemma. DFrom the proof of the above lemma it is clear that is decreasing in j. Sinceis also decreasing by assumption, it implies that the residual demand function dm,jis decreasing too. So the residual market for unrestricted product defined on the priceset of {Pm, ,p} is consistent with the settings in Wilson’s model as discussed in theprevious section. Therefore, the demand constraint for the unrestricted product in theresidual market is given by:j=m m,jwhich, by Lemma 2.3.1, is equivalent tonZ qu,jD-D -m q_j=m r,j L_dil D,1As we can see here, this constraint is no longer a linear constraint.2.3.3 The Formulation of the Optimal Pricing ModelTogether with the objective function, the capacity constraint (or the supply constraint)and the two demand constraints, we have successfully formulated the pricing problem asa mathematical programming problem, which is summarized in the following theorem:Chapter 2. Pricing Perishable Inventories by Using a Restriction 48Theorem 2.3.2 For a given m: 1 m < n, a monopolist’s pricing problem can beformulated as choosing a vector of quantities {qr,i, , q; qu,m, , to solve thefollowing mathematical programming problem:Max pqi + Z=m pjqu,js.t.>D r 1 (2.6)< 1 (2.7)3= D3 Dr i —q,i + k (2.8)qu,j 0, Vi and Vj.Remark: For ease of reference, the above pricing model is called BL Pricing Model, orsimply, the BL-model.17 The optimal value of the revenue derived from the BL-modelwill be denoted by R(m). We will also call any optimal solution to the BL-model withgiven m as an optimal rn-policy.2.3.4 Basic Properties of Optimal rn-PoliciesNote that the formulation in Theorem 2.3.2 is computationally inconvenient because itinvolves a non-linear constraint. On the other hand, it is evident that if the constraint(2.6) in the BL-model is binding, then it will lead to a simple linear programming problem,which substantially simplifies the process of finding an optimal pricing strategy. This isthe main motivation for the discussions in the next section. Before moving on to thenext section, we first need to establish a comparison result between R(m) and Rw — therevenue derived from the Wilson-model. Recall that the Wilson-model solves the pricing17Here “BL” stands for Brumelle-Li.Chapter 2. Pricing Perishable Inventories by Using a Restriction 49problem without using a restriction, which is given by:Max 1pjqjs.t.v,.nL...ii=1 D1 —ZL1qkqi 0 Vi.The following theorem establishes the following result: if the Wilson-model fails togenerate an optimal pricing strategy consisting of only one price, then the firm has astrict incentive to use the restriction. As we will see later, this result plays a key role inmy discussions in the next section.Theorem 2.3.3 Suppose an optimal solution in the Wilson-model consists of two prices,say j and p3 (i <i). Then:R Ru(k — —(z) ——(p—m) in n .fn. n.I— L’,j ) —So consequently, R(i) — Ru > 0 since c is strictly increasing in i.Proof: Let {q,, qj} be an optimal solution in the Wilson-model with i <j. Then wemust have— k—D3L) j- D:kqw,j —I-) jandDj<k<D:.The corresponding revenue is given by:= piqw,i + p,qw,j.Chapter 2. Pricing Perishable Inventories by Using a Restriction 50Consider the following allocation plan:• The firm offers q,.,j units of the restricted product at p and qu,j units of the unrestricted product at pj; and• and qu,j satisfy the following system:q,3 = — D,.,,--——;qu,j = k—q,j.It is easy to check that the above policy is feasible, that is, the allocations satisfy allthree constraints (2.6), (2.7) and (2.7) in the BL-model. Now solving the above system,we getk—D,Dqr,:—r,z,—— Dr,iDj — Dr,jkq,3— Dr,i — Dr,jDenote the corresponding revenue to be R = piq,-,i + p,q,3. Since + qu,j k andq + qw,j = k, it is true thatR — = (pi — p)(qj —Now note that— Dr,jD, — Dr,jk DD3 — D2kqu,, —— Dr,j — — — D3— k D•” Dr,iDj — Dr,jDi— ‘‘ (Dr,i — Dr,,)(Di — D3)— 1k D(1 — c)D1D — (1 ——2) (Dr,i — Dr,j)(Dj — D)— 1k D’(a—ajD1(D,.,, — Dr,,)(Dj — D)Chapter 2. Pricing Perishable Inventories by Using a Restriction 51This proves the theorem. 0The above theorem presents a comparison result for the case that Wilson’s modelhas an optimal policy that consists of two prices. At the end of the next section, I willestablish a similar result for the case that Wilson’s model has an optimal policy thatconsists of exactly one price.Before concluding this section, I present two additional properties on optimal mpolicies.Theorem 2.3.4 Any optimal rn-policy does not need more than two unrestricted prices.Proof: Let m be given and {q1,•.. q; , q} be any optimal rn-policy. Definem * m/3 p!-, and k,. q.To prove the theorem, it suffices to show that {q,” . , q} is an optimal solution ofthe following linear programming problem:Max pjqu,js.t.q,j < 1/_j=m D, l3Drj —Zj=m qu,j k — krqu,j O,Vj.I name the above linear programming problem as the Reduced Problem.First of all, it is clear that , q} is feasible to the reduced problem. Nowsuppose that it is not an optimal solution to the reduced problem. Then it implies thatChapter 2. Pricing Perishable Inventories by Using a Restriction 52there is another set of quantities {q,m,” , q} that constitutes a feasible solution tothe reduced problem and is such thatPJqUJ >j—rn j=mOn the other hand, it is obvious that q; , q} is a feasible solutionto the BL-model associated with m. The corresponding revenue is given bym n m n+ > p3q > p,q + pi=1 3m j=mwhich contradicts the assumption that q; , q} is an optimal mpolicy. Therefore, the {q,” , q} must be an optimal solution to the reduced problem.Finally, since the reduced problem is a Wilson-type formulation, by Theorem 2.2.2,the monopolist needs no more two of q’s to be positive. This implies that the monopolist needs at most two unrestricted prices. So the theorem is proved. DAs a consequence of Theorem 2.2.3 and Theorem 2.3.4, we get the following result.Theorem 2.3.5 Suppose that k > D. If {q2, . . q; , q} is an optimal mpolicy for the BL-model, then the demand constraint for unrestricted product (2.7) in theBL-model is always binding; that is,q,j= 1. (2.9)D D m q3=m 3 — r,j L.1=iProof: Let kr = q and k= YZD3’1=m q• Suppose that (2.9) is not true. Asan immediate consequence of this, we know that the supply constraint (2.8) must bebinding, that is,icr + k = k.Chapter 2. Pricing Perishable Inventories by Using a Restriction 53As before, denoteFrom the proof of Theorem 2.3.4, we know that given the allocation plan for the restrictedproduct, {q1, q}, the reduced problem in the residual market for the unrestrictedproduct is a Wilson-type problem. Therefore, by Theorem 2.2.3, if (2.9) is not true, thenit must be the case thatk < D— I3Dr,n, (2.10)which obviously implies that the monopolist will sell all unrestricted units at price p,that is,*fo ifj<riq —( ifj=n.Defineio=max{im:q>0}.Choose c> 0 such that k + D, — /9DT, and q0Case 1: i0 <nFor this case, define a new policy {q1,• . . q; . , q} as follows:q if iioq=j —€ ifz=i0;andfO ifj<n—( k+e ifj=n.I claim that this new policy is a feasible policy to the BL-model associated with m. Infact, it is easy to see that1,Chapter 2. Pricing Perishable Inventories by Using a Restriction 54which implies that the plan is feasible for the restricted market. For the unrestrictedmarket, first note that the residual demand for unrestricted product at price p, afterthe sales according to {q1,... q}, is given by= D — (3 — )Dr,n> D — /3Dr,n ku + f,r,20which indicates that the allocation plan {qrn,• , q} is also feasible in the residualmarket for unrestricted product. Therefore, this new plan is indeed feasible for the BLmodel associated with m. Now note that the total revenue generated by this new planisR’ = +i1 jm= + + (p — pj0)f > R(m),which contradicts to the fact that. . q; . , q} is an optimal rn-policy.This contradiction leads to the conclusion that (2.9) must be true.Case 2: i0 = nClearly under this case, we must have that m = n. It is clear that+ k D.Then since kr + Ic,, = k > D, we must have some i < n such that q1 > 0. Because ofthis, define= max{i < n : > 0}.Now choose € > 0 such that q11 € and Ic,, + € — /3DT,. Define a new policy asfollows:q,jj min{q,,, d_1,} if i#i;Chapter 2. Pricing Perishable Inventories by Using a Restriction 55and q’ = k + e. Similarly, by the same argument used in case 1, we know that this newpolicy is feasible to the BL-model associated with m, which generates a total revenue of= + pq’ R(m) + (pn — pj)e> R(m),since i’0 < n. This again leads to a contradiction. Therefore (2.9) is also true for thiscase.In summary, we show that (2.9) must be always true. L:i.2.4 Optimal Pricing StrategiesThe main purpose of this section is to demonstrate that the monopolist’s pricing problemcan be reduced to solving n linear programming problems. First, note that the previoussection solves the monopolist’s pricing problem for a given value of m. On the otherhand, the choice of m is completely controlled by the monopolist. Therefore, it is themonopolist’s best interest to choose a value of ni. such thatR(ñi) = max R(m). (2.11)1<m<nConsequently, an optimal solution to the initial monopolist’s pricing problem should bean optimal flu-policy. For ease of reference, we introduce the following definition:Definition 2.4.1 If flu satisfies (2.11), then. any optimal flu-policy is named as an optimal policy for the monopolist’s pricing problem.The following theorem follows from Theorem 2.3.3 and Theorem 2.4.1.Theorem 2.4.1 Suppose that flu satisfies (2.11). If the demand constraint (2.6) forrestricted product is not binding for some optimal solution {q*1,. . . , q; . ,Chapter 2. Pricing Perishable Inventories by Using a Restriction 56in the BL-model, that is,*or equivalently, the sales of restricted products according to the partial plan {q1,• ,do not exhaust the restricted demand at price level p,, then this optimal policy containsexactly one unrestricted price.Proof: We know that after the sales of the restricted products, the residual demand forthe product at the restricted price p3 (j ni) is given by*= Dr,j(1—= (1— 13)Dr,j;and by Lemma 2.3.1 the residual demand for the product at the unrestricted price pj isgiven by -m *= D,— Dr,j > = — BDr,j.Since 3 < 1, it follows that > 0 for all j rh. DefinedTm,j —1—o=--——,forj=m,••.,n.Um,jThen we have•-It is easy to check that & is strictly increasing in j because is strictly increasing in jand ,8 < 1.On the other hand, as demonstrated in Theorem 2.3.4, {q,” , q} solves thefollowing reduced problem:Chapter 2. Pricing Perishable Inventories by Using a Restriction 57Max Z—,pjqu,jS .t.j=m d,,,, —EL q,, k —qu,j O,V7,where k,. = q,.,. Clearly, this is just Wilson’s formulation for the residual market forthe unrestricted product. More specifically speaking, this subproblem can be restated asfollows: the market demand at price p3 is given by : j = ñ,. , n, and the firm hasthe option of imposing the restriction on the product, where the impact of the restrictionis characterized by a,, which is strictly increasing as proved above. Therefore we can useTheorem 2.3.3 to this subproblem, which says that if the above reduced problem (Wilsontype) has an optimal solution consisting of two distinct prices, then the firm can dostrictly better by properly using the restriction. This implies that the optimal solutionto the above reduced problem must consist of exactly one unrestricted price since otherwise it will lead to a contradiction to the assumption that. ,q; . , q} isan optimal solution. Therefore, the theorem is proved. DWe notice that the main difficulty in the formulation of the pricing model in Theorem2.3.2 is that the first demand constraint (2.6) may not be binding, which will force us todeal with a nonlinear constraint in the model. But if the first demand constraint (2.6) isin fact binding, then the BL-model leads to the following linear programming problem:Chapter 2. Pricing Perishable Inventories by Using a Restriction 58m nMax iip2q, + Zj=m pjqu,jS .t.c—sm q,L.j=i Dr,i —< 1/_.dj=m D,—D71 —i + Ej=m qu,j kqr,i,qu,j 0, Vi and Vj.I use (m) to denote the optimal revenue value derived from above linear programmingproblem. I call it the tight problem since the first demand constraint (2.6) in BL modelis forced to be tight (or equivalently, binding). The implication of the fact that (2.6) istight is that there will be no residual demand for restricted product at price p p, afterthe sales of the restricted products according to the partial plan {qr,i,• , q}.Before presenting the main theorem of this section, I need to clarify a simple technicalissue here:• R(m) will not be well-defined at all m such that Dr,m > k since there is no feasiblesolution for the above tight linear programming problem• As a convention, I will let R(m) 0 at any m where the tight linear programmingproblem has no feasible solution.The following theorem, the main result of this section, shows that a linear programmingcharacterization for R(ñi) is possible by using R(m).Theorem 2.4.2 It is always true thatmax (m) = max R(m).1mn 1mnChapter 2. Pricing Perishable Inventories by Using a Restriction 59Proof: Let ñi be such thatR() max R(m).1<m<nI now first prove that there exists some ri such thatR(ñ2) ? R(fn). (2.12)Let {q,• , q; q,••• , q} be any basic optimal ni-policy derived from the BLmodel. If this optimal solution satisfies:*then this solution is feasible to the tight problem associated with ñi. Therefore (2.12)holds for th =I now prove (2.12) when /3 < 1. By Theorem 2.4.1, we can limit our discussion onthe case that exactly one of q,,j = , n is positive, say, q. By Theorem 2.3.5, thesecond constraint (2.7) must be binding, so it follows thatq = =— /3D7,.Now let th = 3, and define:q’ if1<i<ni,= 0 ifth+li3—1,(1 /3)Drj if i =3;andI ffj3,qu,j =0 otherwise.It is straightforward to check that the policy {r,j, 1 ñi; ,j, ñi j n} is afeasible solution to the tight problem associated with ñi. Then since q = — /3Dr’,Chapter 2. Pricing Perishable Inventories by Using a Restriction 60we haven mR(m) >piqr,i + pjqu,j = >pqZ,. +p(qr, + q,)i1 37 i1= +p;(D;— /3Dr,) pjq*. +pq = R(ñi),which implies thatR(ñi) R(th).So (2.12) is true. On the other hand, since R(m) < R(th) for all m : 1 < m <Therefore it is shown that= R(th).Hence the theorem is proved as required. DLet me say a few words here on the computational issue on th. The worst case isto solve n linear programming problems associated with R(m) for m = 1,•• ,n. Thenatural approach is to study the curvature property of R(m). A desirable property isthat (m) is in fact quasi-concave, which will reduce the number of linear programmingproblems that are needed to obtain an optimal policy from the order of n to the orderof ln(n). But at the this moment, I can not prove or disprove whether m) is indeedquasi-concave.Before concluding this section, I present another comparison result between the Ru— the optimal revenue derived from Wilson model — and R(n) for the case that thereexists an optimal solution in Wilson’s model that consists of exactly one price. I firstintroduce the following notation:i = max{i : piDr,i = max pjDr,j}; (2.13)13nChapter 2. Pricing Perishable Inventories by Using a Restriction 61= min{i = maxp3D,}. (2.14)13nI now first prove the following interesting lemma.Lemma 2.4.3 For any i such that= maxit is true thatiTProof: I will prove the result by contradiction. First suppose thatBy definition, it follows thatPi Dr,j (1 — ‘i )Pi; -D pj = (1 — cj )p2*D2,which is impossible since, by the monotonicity assumption on at’s and the definition ofi (i.e., (2.13)), we have1 — aj <1 — c. and p*D, <pD.Therefore we must have i < i.Now suppose thatL <By definition, we have= =Chapter 2. Pricing Perishable Inventories by Using a Restriction 62which leads to a contradiction since, again by the monotonicity assumption on ai’s andthe definition of i (i.e., (2.14)), we haveaj <c andThen we must have i’ <i. Hence the lemma is proved. DI now prove the following interesting result on the structure of optimal policies inWilson’s model that consist of only one price.Lemma 2.4.4 Suppose that there exists an optimal policy in Wilson’s model that consistsof one price, say, {pj0,qj0}, and such that the supply constraint (2.4) is not binding, thenp20 = max p2D. (2.15)Proof: By Theorem 2.2.3, we know that the demand constraint (2.3) is binding, that is,qj0 = D0. (2.16)By assumption, we also know thatD0 < k. (2.17)I now prove the result by contradiction. Suppose that (2.15) is false, that is,p0D < maxp2D. (2.18)Define= max{i” = max pD}.Clearly, from (2.18), we know thati0’. So we only need to consider the following twocases.Chapter 2. Pricing Perishable Inventories by Using a Restriction 63Case 1: io <*Under this case, it follows thatD20 > D,which implies that the firm can sell D. units at the price p. By (2.18), the allocationplan {p, D } is strictly preferred by the monopolist. This contradicts to the conditionthat {pj0., q} is an optimal solution to the Wilson-model.Case 2: i0 >First, by the optimality of the plan {pj0,D10 }, it follows thatk<D. (2.19)Now let q0 } satisfy the following system:=1D D0q+q0 =k.Solving the above system leads toI — O r—,.LJ2* L/j0—D—k,q10— -—which generates the total revenue of_______D—kR =pq +p0q =,+,p0D..L.JjS——Then by (2.17), (2.18) and (2.19), we know thatR > p0D0,Chapter 2. Pricing Perishable Inventories by Using a Restriction 64which contradicts to the fact that {p0,D0 } is an optimal solution.In summary, both cases lead to a contradiction. Therefore, we conclude that (2.15)is true, as required. 0The following theorem investigates whether the monopolist still prefers to use therestriction when there is an optimal solution in Wilson’s model that consists of only oneprice.Theorem 2.4.5 Suppose that Wilson’s model has an optimal solution consisting of exactly one price, say, {pj0,D0 }, with the optimal revenue value of Rv = p0D10, which issuch that the supply constraint (2.4) is not binding. Then a necessary and sufficientcondition that there exists an integer m such that R(m) > R° is:i < i, (2.20)where i and i are given by (2.13) and (2.14) respectively.Proof: By Lemmas 2.4.3 and 2.4.4, we know thatio E {i’ : max pD}1<,<nand i <io <it.Sufficiency of (2.20): Here we need to show that (2.20) implies that R(m) > R”-’ for somem. I will prove this through two cases.Case 1: i = io.Clearly, under this case, we must have thatChapter 2. Pricing Perishable Inventories by Using a Restriction 65Now let m = = i and an allocation plan {qr,i, , qu,m,” , as follows:f D,,,q,j =0 otherwise;andD,3 ifj=iqu,j =I. 0 otherwise.It is obvious that the above plan is feasible to the BL-model associated with m sincek> Dm = Dr,m + Du,m Dr,0 +By definition of i, we know thatpj0D0 < pj*which implies thatPr,io Dr,jo + Pu,i. > Pr,io D0.Therefore for this m, we have that R(m) > RY.Case 2: i0 > i.Let m = i0. Then it is clear that m i, Now define a policy such that the part ofthe allocation plan for the unrestricted product {qu,m, , is given as follows:( D,, ifj =qu,j =0 otherwise.By definition of i, we know thatpjqu,j pmDu,m. (2.21)j=mChapter 2. Pricing Perishable Inventories by Using a Restriction 66Denote k,. k—Then by assumption, we know thatk> Dm = Dr,m + Du,m Dr,m +which leads toICr > Dr,m.Now if, in addition, kr Dr,i, then define the allocation plan for the restricted productas follows:( n Ii=0 otherwise.Then by definition of i and the fact that i <m, we obtainpjDr,j = pir*Dr,i >p0D,j PmDr,m, (2.22)which implies that R(m) > Rw. We still need to prove the result for the following case:Dr,ni < kr < Dr,t. (2.23)For this case, let q = 0 for m; and and qr,m be the solution of the followingsystem:+qr,m= 1Dr,i Dr,m+ qr,m = kr.It is easy to check that the above system has the following unique solution:— r r,m—r,: 1—’r,ra— Dr,i;krDqr,m — r,m— r,mChapter 2. Pricing Perishable Inventories by Using a Restriction 67which, according to (2.22) and (2.23), leads to the following:= + Pmqr,m— kr— Dr,m — kr—Pi; Dr,i +,-Pm Dr,m—kr — Dr,m Dr,i* — kr> Pm Dr,m + r r-, Pm Dr,m—-1Ir,m 1Jr,i —= PmDr,m. (2.24)Therefore by (2.21) and (2.24), it follows thatm flR(m) >piqr,i + pjqu,j > pmDr,m +PmDu,m = pmDm = Rw.i=1 j=mThis proves the result under (2.23). In summary, we prove the sufficiency of (2.20).Necessity of (2.20): We want to prove that i = i implies thatmax R(m) RW = p0D0,1<m<nwhich, according to Theorem 2.4.2, is equivalent tomax R(m) = Ru. (2.25)1<m<nNote that by Lemmas 2.4.3 and 2.4.4, i = i implies that= i: = i0. (2.26)It is evident that (i0) R’. Therefore, to show (2.25), it suffices to prove thatR(m) <Ru. (2.27)Note that for these rn’s such that (m) = 0, (2.27) is true by default. Also, for thesern’s such that R(m) > 0, we know that R(rn) must be the optimal objective value ofChapter 2. Pricing Perishable Inventories by Using a Restriction 68the tight problem associated with m. Let {q,1,” q; be an arbitraryoptimal solution to the tight problem associated with m. Define= , for z = 1,... , m,—or j — m, , ri.Then we know that=1,17i i.Now it is easy to see that by the definitions of i and i,(m) = pq + > pqi=1 j=mm n= +i=1 j=mm nT)jpj Dr,i; + 7jPii=1 j=m< pi Dr,i += pj D10 =since i = i = io. This proves that (2.27) is true. Consequently, (2.25) is proved. Thisfinishes the proof of the necessity of (2.20). DRemark: We know that when Wilson’s model has an optimal solution that consists ofonly one price and there is excess capacity left, it means that the monopolist will choosethe best value of pD1. The significance of Theorem 2.4.5 is two-fold. First, it shows thatthe monopolist can always sell some restricted units and obtain a strictly larger valueChapter 2. Pricing Perishable Inventories by Using a Restriction 69of revenues by further exploiting the two different market segments. Second, it givesplausible evidence that the monopolist will utilize the limited capacity more effectivelyif he uses the restriction as a marketing mechanism. It is also interesting to note thatthe only case which makes the use of the restriction unattractive is when Z = i. Thissimply says that when both markets reach the maximum revenue point at the same pricelevel and the capacity is high, it makes no difference whether or not the monopolist usesthe restriction.2.5 An Application to Airline Fare PricingIn this section, I will present a simple application of the pricing model developed in thelast section to the airline fare pricing problem. It is well-known that it is a common practice for airlines to apply the restriction on some low-priced fares and that the availabilityof these restricted fares is controlled. Unfortunately, there has been no theoretical pricingmodel that can be used to justify such a widespread business practice. The purpose ofthis section is to show that the new pricing model developed in this chapter may be auseful framework that will fill that theoretical gap.It is commonly recognized that there are at least two distinct consumer groups forair travel industry: the leisure group and the business group (sometimes called must-gogroup). For the purpose of illustration,18 I only consider the case that the market demandconsists of two segments — leisure and business.Let D1(p) and Db(p) be the demand from leisure segment and business segment forthe unrestricted tickets at price p respectively. Suppose that the airline has an optionof a restriction on some fare classes. To simplify the analysis, I make the followingassumptions:‘8The actual number of consumer segments is not crucial in applying the new model.Chapter 2. Pricing Perishable Inventories by Using a Restriction 70Restrictions Percentage of market satisfying restrictionsAdvance Bookings Minimum Stay Leisure Travellers Business Travellers(Days) (Days) (1— 71) (1 — 7b)7 None 82% 53%14 None 68% 27%30 None 49% 11%7 7 53% 11%14 7 46% 8%30 7 35% 4%Table 2.1: Effect of Fare Restrictions — Boeing Company (1982)• The restriction on a ticket of price p will reduce the demand from the leisure segmentfrom D’(p) to (1 —71)D(p), where 71 — a fixed constant — is the proportion of theleisure segment travellers who cannot fly because of the restrictions.• The restriction on a ticket of price p will reduce the demand from the business segment from Db(p) to (1—7)D6(p), where 7b — a fixed constant — is the proportionof the business segment travellers who cannot fly because of the restriction.• 71 < 7b, that is, a higher percentage of business segment travellers cannot accommodate the restriction.The Boeing Company (1982) reported some empirical results on the effect of fare restrictions in the airline industry, which are given in Table 2.1. The table gives clear evidencethat 71 <7b.Now it is clear that the demand for the restricted tickets at price p is given by:Dr(p) = (1 — 71)D1(p) + (1 — 7b)D6(p).The demand for unrestricted tickets at price p is given by:D(p) = Db(p) + D’(p).Chapter 2. Pricing Perishable Inventories by Using a Restriction 71Then c(p), the percentage of total demand at price p that will not buy the restrictedticket at price p, is specified by1 — Dr(p) — (1 — 71)D1(p) + (1 —7b)Db(p)—— D(p)— D6(p) + D’(p)In the development of my pricing model, I require that a(p) be a strictly increasingfunction. In the airline case considered here, I am going to show that this assumption isequivalent to requiring that the leisure segment travellers are more price sensitive thanthe business segment travellers, which is of course a very plausible assumption. Basically,this is just saying that the market demand is indeed segmented.Lemma 2.5.1 Suppose that the demand functions D1(p) and D”(p) are differentiable anddecreasing in p. If the impact of the restriction is such that j <7h, then a(p) is strictlyincreasing if and only if> (p) for all p, (2.28)where and i are the prices elasticities of the leisure segment demand and businesssegment demand respectively, that is,dD1(p) p — dDb(p) pdp D1(p)’ and () =— dpX Db(p)Pro of: Note that(1 — ‘y)D1(p) + (1 — 7b)D”(P)—+1—= Db(p) + D’(p) — (1 — 71) + 1D (p)Since (1 — 7b)/(1—71) < 1, then a(p) is strictly increasing if and only if the functionDt(p)/Db(p) is strictly decreasing, which is equivalent to the condition thatdDt(p) Db(p) dD6(p)dp dpChapter 2. Pricing Perishable Inventories by Using a Restriction 72which leads to the condition (2.28) immediately since both D’ and Db are assumed to bedecreasing. Therefore the lemma is proved. DRemark: It is not quite right to claim that the above lemma is consistent with the discussions in previous sections because of the fact that D1 and D’ are assumed to be stepfunctions in previous sections, rather than differentiable functions as required in Lemma2.5.1. Nevertheless, it shows us that the monotonicity assumption on the c’s is plausiblein the airline case.One of the main reasons that airlines impose restrictions on low-price tickets is todiscourage travellers who are from the business segment from buying the discounted,restricted fares. As demonstrated in the above lemma, the existence of consumer groupswith different prices elasticities, together with the condition that the restriction has anon-uniform impact across different consumer groups will provide sufficient conditions fora monopolist to consider using restrictions. Empirical results of Oum, Gillen and Nobel(1986, [172]) indicated that in U.S., the average price elasticity for the leisure group was1.5 whereas for the must-go group it was only 1.15. Then according to the pricing modelin the previous section, we may say that the practice of using restrictions in the airlinepassenger market is justifiable.Let us now go through the following simple example on airline fare pricing by usingone type of restriction.Chapter 2. Pricing Perishable Inventories by Using a Restriction 73Example 2.5.1 (Airline Fare Pricing Problem)Consider that the demand function from leisure segment is given by500 ifp100300 ifpE(100,150]D(p)=100 if p E (150,250]0 ifp>250;and that the demand function from the business segment is given by400 ifp100300 if p E (100,150]Db(p)= 200 if p E (150,250]50 if p E (250,400]0 ifp>400.Further assume that‘= 0 and 7b = 0.5, that is, the restriction will result in 50 percentof business segment travellers not willing to purchase a restricted ticket and have noimpact on the leisure segment. Then the airline has four possible prices pi =$100,P2$15O,p3 = $250, and p4 $400, with a1 = 0.23,a = 0.27, a3 = 0.33, and a4 = 0.5,which is strictly increasing. The corresponding demand for the restricted fares are:Dr,i = 700, Dr2 = 450, D73 = 200, Dr4 = 25;and the total demand for unrestricted fares are:D1 = 900, D2 = 600, D3 = 300, D4 = 50.Table 2.2 summarizes the solutions to the Tight Problems for this example. It is clearfrom the table that ñz = 3 and there exists a unique optimal solution which is given by:q2 = 180,q3 = 120, q3 = 100,Chapter 2. Pricing Perishable Inventories by Using a Restriction 74Highest Restricted Fare An Optimal Solution Derived Revenuep = $100 (m = 1) no feasible solution 0P2 = $150 (m = 2) no feasible solution 0p3 = $250 (m = 3) qr2 = 180, = 120, q3 = 100 82,000J34 = $400 (m = 4) qr2 = 370.6, q,4 = 4.4, q4 = 25 67,350Table 2.2: Solutions of the Tight Problem — A Numerical Examplewith i(3) = 82,000.’On the other hand, there is another optimal solution to the original BL-model. Notethat in the above optimal solution, the airline will sell 120 restricted tickets and 100unrestricted tickets at the same price level p3 = $250 after the sales of 180 restrictedtickets at P2 = $150. We also know that at end of the sales at the price level p, therewill be no residual demand for both the restricted product and the unrestricted productat p p3. This implies that after the sales of 180 restricted tickets at P2 = $150, theairline can actually allocate the remaining 220 tickets all unrestricted at the price level3. Consequently, the following policy:qr2 = 180, = 220is also optimal. It is worthwhile to mention that that this new optimal policy seems tobe more appealing to the airline since it does not involve selling some restricted ticketsand unrestricted tickets at the same price level.Remarks: In conclusion to this example, we can make the following two general comments:• The original BL-model may have multiple optimal solutions, which might allow the‘9Just for the sake of comparison, if the airline decides not to use the restriction on any fare, then theoptimal revenue derived from Wilson’s model is 80,000.Chapter 2. Pricing Perishable Inventories by Using a Restriction 75firm to avoid using an optimal policy that is less appealing;The Tight Problem can be used for two purposes: (i) to find the maximum revenuevalue; and (ii) to derive a simple optimal pricing structure.2.6 Summary — Pricing by Using RestrictionIn this chapter, I have investigated the monopoly pricing problem for a unique type ofperishable product, such as airline seats and hotel rooms. The model explicitly incorporates the use of artificial restrictions. It is shown that the monopolist’s optimal pricingproblem can be formulated as a mathematical programming problem. Furthermore, ifthe impact of the restriction on the demand is such that as price increases, the percentageof consumers who can accommodate the restriction is decreasing, then the monopolistwill have a linear programming characterization on the optimal revenue and the optimalpricing policy, which substantially simplifies the original formulation.I also give a simple application of the model to airlines. I show that the modelassumption that p) is strictly increasing, in the airline case, is equivalent to sayingthat leisure travellers are more price sensitive than business travellers, which is exactlywhat the airlines have in mind when they use restrictions.Chapter 3General Optimality Results and Other Properties3.1 IntroductionChapter 2 developed a simple pricing model for a monopolist who uses restrictions asa mean of segmenting the demand market. It showed that the monopolist’s pricingproblem can be reduced to a series of linear programming problems. It demonstratedthat by properly setting the level of the highest restricted price and rationing the sales atdifferent prices, the monopolist needs to charge no more than three prices to maximizethe revenue. On the other hand, there is a key assumption that the monopolist offers therestricted prices first, and then after all sales have been accommodated at the restrictedprices, sales are offered at the unrestricted prices. I call this type of policy a primarypolicy. Even though a primary policy is very natural, it does limit the monopolist’s choiceon possible pricing structures.Another implicit assumption was that the firm offers one kind of price at a time, whichmeans that if the firm is selling restricted products, then unrestricted products are notavailable. In practice, for example airlines, both restricted fares and unrestricted faresare used and they are made available at the same time. I will call this the simultaneousavailability issue.The purpose of this chapter is to resolve these two important issues. In response tothe first concern, I will introduce a general pricing policy which allows the firm to putrestrictions at any possible prices. I am going to show that any general pricing policy76Chapter 3. General Optimality Results and Other Properties 77is weakly dominated by a primary policy, where the criterion for dominance is defined interms of realized revenue. This consequently implies that any optimal policy in the classof primary policies remains optimal in the class of general policies.To address the simultaneous availability issue, I will demonstrate that based an optimal policy derived from the Tight Problem, it is possible to construct another policy,which may not be primary, that has the property that making all allocated units available can still generate the same amount of the revenue realized by the primary optimalpolicy. In other words, the firm can always find an optimal policy that is consistentwith existing pricing practice by use of artificial restrictions. To make the argument gothrough, a reasonable behavioral assumption is needed, that is, when some unrestrictedunits and some restricted units are offered at the same price level, consumers from therestricted market will first try to buy the unrestricted units.This chapter is organized as follows. Section 3.2 establishes some preliminary resultson general pricing policies. Section 3.3 will prove the general optimality results. Thensection 3.4 discusses simultaneous availability issue for optimal policies. The last sectionis a summary.3.2 Auxiliary Results on General Pricing Policies3.2.1 Notation and DefinitionsI will use the same model settings as in Chapter 2. That is, there is a monopoly firmwhich wants to sell a fixed number of units of a certain product. The product is perishableand not storable for consumers (for example, airline tickets and hotel rooms). Let Dr(p)be the demand for the product at price p with the restriction; and let D(p) be thedemand for the product at price p only when the restriction is not attached. Denote theChapter 3. General Optimality Results and Other Properties 78total demand at price p by D(p). ThenD(p) = Dr(p) + D(p).Let a(p) be the percentage of those consumers with reservation prices of p or higher whowill be unable to buy the product because of the restriction, which is given by:D(p) Dr(p)D(p) =1- D(p)Further assume that Dr(p) and D(p) are step functions defined on a finite set of prices{Pl,.,PN}suchthatPj<P2<...<PpJ-<oo’Thatis,forl=r,u,D1, ifpPi;Dj(p) = D1, if p E (P:_i,Pj] for 2 i N;o ifp>FN.Consequently D(p) and (p) are also step functions:D1 ifpPiD(p)= D, ifpE(P2_i,F]for2ino ifp>P,where D = Dr,t + andc ifpPia ifpE(P..i,P]for2in1 ifp>P;where a. = a(P1). As in Chapter 2, I will make the following assumptions throughoutthis chapter:D is strictly decreasing in i;1There is a minor notational change here from Chapter 2, where the price set is specified by{Pi, ,p}. The change here is purely presentational.Chapter 3. General Optimality Results and Other Properties 79• D1, is decreasing for 1 = r, U;• cx:j is strictly increasing in J.Since the monopolist has a choice of imposing restrictions on any possible prices andoffering the product with and/or without restrictions at the same price, a general pricingpolicy must include• the choice of a set of prices which will be attached with the restriction;• the allocation plan.Some observations are needed here before I formally introduce the definition of a generalpricing policy.2 First, a general policy should allow the firm to offer restricted and unrestricted product at the same price level in an arbitrary order in the sense that the firmcan sell unrestricted product first and the restricted product later at the same price, andvice versa. Because of this, it is possible there are more than two allocations on the sameprice, for example, the firm may offer some restricted product at price P1, then someunrestricted product at the same price, then again offering some additional restrictedproduct at this price. Consequently, the number of initial prices, P1, .. , F, has to bemodified to capture this possibility. Therefore, by looking at the order of offered pricestogether with the option of attaching restrictions, we may have a sequence of pricesPi P2 such that pj E {Pl,”.,PN}, for all i = 1,... ,n. Furthermore, at eachof these offered prices, there is also a tag which indicates the nature of this price. Forexample, let S is the tag variable on the price pj, then S = r implies that the price jbe a restricted price; and S = u indicates that the price is unrestricted. I now introducethe following definition of a general pricing policy.2The following discussion also helps us to understand the reason why I need the above minor notationalchange in this chapter.Chapter 3. General Optimality Results and Other Properties 80Definition 3.2.1: A general pricing policy is a bundle {n;(p,q,,Sj),i =such that• piE {P1,... ,FN}, for alli = 1,..,n such that p P2• S = r indicates that p is attached with restriction and S = u implies that p, isunrestricted; and• is the quantity available for sale at the price i associated with the tag variableSi.Remarks:• For ease of presentation, we will call the class of general pricing polices G-class;and any general pricing policy will be called a G-policy.• It is clear that there is no point for the firm to offer several positive quantitieswith the the same characteristics (that is, restricted or unrestricted) consecutivelyat the same price level. For example, if S = r and qr,i > 0, then there is no needto consider any policy such that— S+i = r;— Pi+1 = pi;— qr,i+i > 0.• For the rest of this chapter, when we refer to G-policy, we require that if there arepositive quantities allocated at the same price level, then the characteristics of theunits must be alternating.Recall that Chapter 2 limited the discussion to the following type of policies:Chapter 3. General Optimality Results and Other Properties 81• n=N+l;•p=P2fori=1,.,m,andp=P_j m+1 ”,n;and•=rfori=1,••,mand6=uforj=m+1,•-•,n.I have shown that under the assumptions: a, is strictly increasing and the allocated restricted units are sold before the sales of the allocated unrestricted units, the monopolist’spricing problem can be formulated as the following two-stage process:1. For any given 1 m N, solve the following linear programming problem:Max Pqj +s.t.Lji=1 DriçN qi—1L..ij=m DjDr,, —m Nq,- + Zj=m q, qqri,quj O,Vi and Vj,which leads to a realized revenue of .(m);2. Find a value of th such thatTi) = max f(m).1<m<NIt is the purpose of this chapter to show that there exists an optimal policy in the C-classthat has the above special structure. For presentational purpose, I give a special namefor the type of polices discussed in Chapter 2:Chapter 3. General Optimality Results and Other Properties 82Definition 3.2.2: We call a G—policy {n; (p, qs,j, i = 1,.-. , n} a primary rn-policyifl.S=r,foriz=l,...,rn;and6=u,fori=m+1,...,n;and2. Pm Pm+i,where m: 1 <m <n.Remark: Because of the special structure of a primary policy, we can represent anyprimary policy as follows:{(pi, q,i), , (Pm, qr,m); (Pm, qu,m),” , (p,, qu,n)};or simply,{q,i,. ,qu,n}.I will use this representation if it causes no confusion. We should also keep in mind thatunder this notation, at the price level Pm, the restricted units are sold first.Since the monopolist’s goal is to maximize its revenue, it is natural to rank the pricingpolicies in term of the corresponding revenues. For this, I introduce the following concept:Definition 3.2.3: We say a feasible G-policy {n; (ps, 6), i = 1,.. . , n} weakly dominates another feasible G-policy {n’; (p, 6), i 1,. . . , n’} if•1If the above inequality is strict, we call it strict dominance. And if it holds with equality, we say that the two pricing policies are equivalent. Consequently, we say a feasibleG-policy is optimal if it weakly dominates all other feasible policies.Chapter 3. General Optimality Results and Other Properties 83Given any G-policy {n; (pt, Si), i = 1,. , we introduce the following notation:• For i = 1, • , ri let d = Dr(pj), d = D(p) and d = d + d.• Let-y = — be the percentage of consumers who will not purchase the productat price Pj if restrictions are attached.• For t = 1,. , n and j t, let be the total residual demand at the unrestrictedprice P, after the sales according to the partial policy {(p, SJ, i = 1,... ,t}.And let be the corresponding residual demand at the restricted price Pj• Define_1u__1. 1r— u,t3— ut,j.• Let 7t,j = 1—be the percentage of residual consumers who will not purchasethe product at the restricted price Pj after the sales according to {(pi, qs,i, Si), i1,... ,t}.It is worthwhile to mention that1. d is in general not equal to D;2. {d} is decreasing, but may not be strictly decreasing as D is;3. yj is in general not equal to a,7; and4. {-y,} is increasing, but may not be strictly increasing as c is.3.2.2 A Lemma on the Property of MonotonicityLet us now go back to the discussion of the G-policy. For a G-policy as defined in Definition 3.2.1, if the sales are at an unrestricted price, then there is no impact on the valuesof the ‘s, which means that the portion of these consumers who can not accommodatethe restriction remains unchanged due to the use of proportional-rationing rule to updateChapter 3. General Optimality Results and Other Properties 84the residual demand. But if the sales are at a restricted price, the values of ‘s will bechanged. The following lemma shows that the monotonicity of 7’s will be maintained.Lemma 3.2.1: For any given G-policy {n; (p2, i = 1,. .. , n} and given t, j isincreasing in j on j t.Proof: I prove the result by induction. First, consider t = 1. If S1 = u, then the sales ofunrestricted products makes no impact on the 7’S, that is, 7i,j = 7j, which is increasingby the monotonicity assumption on cvi’s. Now if S = r, thenandd1,3 = d, —Therefore,dr (l—)(l—7)l—(i—7)-Since 7 is increasing in j and - 1, it follows immediately from the above expressionthat 71,j is also increasing. This proves the result for t = 1.Suppose that the result holds for t, that is, is increasing. By the same argument,if S+ = u, then 7t+1,j = 7j; hence the result is true for t + 1 in this case. If 8+ = r,since— d÷1, — (1 — (1 — 7t,j)— 7t+i,3 —— qre+1d+1, 1— (1the monotonicity of 7t+1,j follows immediately from the monotonicity of and the factthat qr,t+i This implies that the result also holds for t + 1 when = r.Therefore by induction the result is true for t = 1, 2,... and the lemma is proved. DChapter 3. General Optimality Results and Other Properties 85The following is an immediate consequence of Lemma 3.2.1.Corollary 3.2.2: For anyt = 1,••• ,n, one and only one of the following two statementsis true:1. 7tj = 1 for all j t;2. {yt,j} has a subsequence that is strictly increasing in j.Furthermore, we have:1. for a given t and any ji <32 such that p1 <p32, if 7t,j1 < 1, then7t,j1 < 7t,j2;2. for all j t” > t’,7t’,jRemark: The above result shows us that as long as the residual demand for the restrictedproduct is positive, the 7-sequence is strictly increasing in terms of price levels. Notethat the monotonicity condition on the original a-sequence is also in reference to pricelevels. Therefore, the result in Corollary 3.2.2 will enable us to use the BL-model inChapter 2 for the residual market if we limit to primary policies for the residual market.3.3 General Optimality Results3.3.1 Three Fundamental LemmasRecall that the BL-model in Chapter 2 only considered a special type of pricing policy— one which only allows the firm to sell the unrestricted products at higher prices thanthe restricted products. Such a type of policy limits the firm’s choices on possible pricingChapter 3. General Optimality Results and Other Properties 86structures. For example, it rules out the possibility that the firm may consider to sellsome unrestricted units before the sales of some restricted units. The purpose of thissection is to further investigate the firm’s pricing problem under the expanded set ofpricing policies — the set of general pricing policies as defined by Definition 3.2.1, whichcontains policies that allow the firm to impose the restrictions at any price level. In thissection, I want to prove a very interesting general optimality result, which shows thatany G-policy is in fact weakly dominated by a primary policy. Consequently, any optimalpolicy determined in the BL-model, which is thus primary, is also optimal in the G-class.This is a very important and useful result because:• There exists an optimal pricing policy in the BL-model characterized by a series oflinear programming problems; and• It is very difficult, if not impossible, to discuss the nature of optimal polices in theG-class.For ease of presentation, I introduce the following definition:Definition 3.3.1: For any G—policy {n; (pg, q,,2 Sj, i = 1,. . . ,n}, the price i is said tobe active if qs,j > 0. If 6 = r, I call it an active restricted price; and similarly, if= u, I call it an active unrestricted price.By definition, a primary policy will not sell any unrestricted product at a price thatis strictly less than an active restricted price. The following lemma shows that I canfocus on the policies such that there is no unrestricted product sold at the level of thelast active restricted price prior to the sales of the restricted product at this price.Chapter 3. General Optimality Results and Other Properties 87Lemma 3.3.1: There always exists an optimal policy in the G-class with the followingproperties:1. At the level of the last active restricted price, there are at most two active prices;and. The first one of these two active prices is associated with the sales of restrictedunits.In words, at the last active restricted price level, the firm sells the allocated restrictedunits before the sales of allocated unrestricted units at this price level, if any.Proof: Consider any optimal G-policy {n; (pg, q5,,j, SJ, i = 1,.• , ri}. Let Pm be the lastactive restricted price and io be the largest index for an active unrestricted price beforethe restricted price Pm, that is,io =max{i : i < m,.5—u and > O}.It is evident that i0 m — 1 and p0 Pm.Now if p0 < Pm, then Lemma 3.3.1 is true by default since it means that there areno unrestricted units that are sold at the price level Pm before the sales of the allocatedqr,m restricted units (also) at Pm. So we only need to consider the case that Pio = PmWithout loss of generality, we can take i0 = m — 1, i.e., Pm—i = Pm. Hence Pm—i is thelargest unrestricted price that is precedent to Pm.By definition of Pm, we know that after price Pm, the firm only sells the unrestrictedproducts. Therefore, again without loss of generality, we can assume that Pm Pm+i <<p We need to discuss two cases here.Case 1: Pm <Pm+i.By Corollary 3.2.2, we know that 7t_i,j is strictly increasing in j m since p3 isstrictly increasing in j m. Then by Theorem 2.4.2 and Lemma 3.2.1, I know that forChapter 3. General Optimality Results and Other Properties 88the remaining market specified by {dm_i,j, d_1 : j = m, , n} together with the priceset {Pm, .. ,p}, there is a new primary policy for the residual market such that• it is at least as good as the original partial policy{ (Pm, qr,m, r), (Pm+i, qu,m+i, u),. , (pa, u)}; and• there is no residual demand for restricted product after the sales at the last activerestricted price under this new primary policy for the residual market.Because of this, without loss of generality, we will assume that = 0 for all j m,which implies that after the sales at the last active restricted price in the original policy,there is indeed no residual demand left for the restricted product at price level p > PmConsequently, since Pm—i = Pm,— — ,_________,. —q,—i .qr,m — Um_1,m ——)m—2,m—i ——)m—2,mUm...2,m_1and the total residual demand at the (unrestricted) price p3 after the sales up to therestricted price Pm is given by:dm,j = = d_1, = (1 — qu,m-i )d_2, for j m. (3.2)m—2,m—iBy (3.1), it is easy to check thatqu,m—i + qr,m > n—2,m = d_2,m i•Let us now define a new G-policy as follows: {n; (ps, q,., 6), i = 1,• . , n} such thatS ifl<i<m—2r ifi=m—1it ifim;Chapter 3. General Optimality Results and Other Properties 89aridiflim—2dn_2,m ifi=m—1=qu,m—i + qr,m — d_2,rn if i = mifi>m.Intuitively speaking, the new G-policy has the following properties:• it has the same price set as the original G-policy;• we switched the tag variables on prices Pm—i and Pm in the original G-policy so thatPm—i becomes the last active restricted price;• furthermore, the new policy is equivalent to the old policy since it is easy to checkthatpqs,i; and• after the sales at the restricted price Pm—i, there is no residual demand for therestricted product at price Pm—i or higher.Now we need to show that the new policy is in fact feasible. By (3.1),qu,rn—i r rqu,rn = qu,m—i + qr,rn— m—2,m = qu,m—i + (1—)dm_2 rn—i — dm_2 rn—iUrn—2,m— id’1m—2,m— 1= qu,m—i. (3.3)Um_2 rn—iNote that, after the sales of qm.i units at the restricted price Pm—i, there is no demandfor restricted product at any price Pj for j m. As usual, let be the total residualdemand at price P2 after the sales according to the new partial G-policy{(p,qçi,S) : i = 1,”.,m—1}.Chapter 3. General Optimality Results and Other Properties 90Then it is clear thatd’m_i,j = for j > m — 1.Since Pm—i Pm, it follows from (3.3) that for the new policy the residual demand atthe unrestricted price Pj (i m) after sales up to the unrestricted price Pm 18 given by— r- qu,m —_________— ,q,—i ‘uU m,j ——)‘m—2,j ——)Um2,j ——)tlm_2,j’Um_2,m Um2,mi Um_2,m_iwhich is identical to (3.2). This shows that the new policy is feasible. Therefore the newG-policy{n;(p,q1,S,) i = 1,..• ,n}is also an optimal policy and is such that at the last active restricted price level, theunrestricted product, if allocated with a positive amount, will be sold after the sales ofthe restricted product at this price.Case 2: Pm = Pm+iUnder this case, we know that at the level of the last active restricted price, the firmwill sell some unrestricted units first (qu,m._i), then some restricted units (qr,m), and thensome unrestricted units again (qu,m+i). Note that after the sales according to the originalpolicy up to Pm—i, the relevant part of 7-sequence is{7m—i,m, 7m—i,m+2, , 7m—i,n},which, by Corollary 3.2.2, is strictly increasing. And the partial policy{ (pm,qr,m),(pm+i,qu,m+i),is a primary 1-policy for the residual market demand{ (d_i,m, dm_i ,m) , (d,im+2, dmi,m+2), . , dmi,n)},Chapter 3. General Optimality Results and Other Properties 91together with the set of prices {Pm,Pm+2,” ,pn}. By the same argument used in Case1, without loss of generality, we can assume that = 0. Therefore (3.1) and (3.2) arestill valid. Now define another new G-policy{n”; (p’, Sfl, i = 1,. ,such that n” = u — 1 and73 ifl<i<mPi+i if m + 1 <i <n”;6 ifi <i<m—2r ifi=m—1u ifmin”;andif1im—2d_2,m ifi=m—1=qu,m—i + qr,m — d_2m + qu,m+i if i = mqu,i+i ifm+1 i<n”.Again, we only need to show that this new G-policy is feasible. First, by (3.3), we getdL— m—2,m—iqu,m—qu,m—i + qu,m+i. (3.4)Um_2,m_iSimilarly, let d_1, be the total residual demand at price Pj after the sales according tothe partial new policy {(p’, S’) : i = , m — 1}. As in Case 1, we know thatd_1, = for j m — 1.Under the new policy, by (3.4) and the fact that Pm—i = Pm Pm+i, the residual demandat the unrestricted price P3 (i m) after the sales up to the unrestricted price Pm iSChapter 3. General Optimality Results and Other Properties 92given by:I,—q,,,,— 11 qu,m—i qu,m+i ‘um,j ——,j,, im—i,j — —________m—1,m Um_2,m_i ‘-‘m—2,mNow since Pm Pm+i, then according to (3.2), for j m + 1,,‘______\ju(Lm+1,j =— du )Um,jm ,m+ 1dUm,j= Um,j— qu,m+i,‘m,m+1— (1 qu,m—i qu,m+i——)m—2,j—‘m—2,jm—2,m—1 m—2,m+1— 11 qu,m—i qu,m+i \AU— I,J.Um_2,m_1 (Lm_2,mwhich implies thatd,_1 = dm+i,j, for j = m + 1, , ii. (3.6)Note that for j = m + 2,• . , n, q,_1 = qu,i. So (3.6) implies that the new G-policy isindeed feasible. Therefore the new G-policy is an optimal policy with the property thatPm—i is the last active restricted price.In summary, we have shown that there always exists another optimal G-policy suchthat Pm—i iS the last active restricted price. If there exists another active unrestrictedprice pj such that• i’0 < m — 1,•= 11,• Pi = Pm—i,then we can repeat the above procedure. In the end, we will have an optimal policy suchthat pq, will be the active active restricted price. Consequently, by induction, this willlead an optimal policy with the following properties:Chapter 3. General Optimality Results and Other Properties 93• if we let p, be the active restricted price and definelo = min{l : pj Pnt, Si = u, qu,i > O},then 10> m + 1.This implies that there are no unrestricted units at the price p, that are sold before thesales of the allocated restricted units also at the price level p,. Therefore the lemma isproved. DRemark: Simply speaking, Lemma 3.3.1 says that if the firm plans to sell some unrestricted units and some restricted units at the last active restricted price level, thenwithout loss of generality the firm can limit its attention on these policies such that therestricted units are sold first.The following two lemmas further explore this issue by considering cases where thereare many (at least one) restricted prices between two active unrestricted prices.Lemma 3.3.2: Any optimal G-policy of the following form:{ (pa, qu,i, u), (p2, qr,2, r); (ps, q,3, u),.. . , (pa, u)}is equivalent to a primary policy, wherep P2 3 <4 < <Pn and qu,i > 0, qr,2 =In words, it says that if an optimal policy in the G-class is such that (1) there is only oneactive restricted price; (2) there is exactly one active unrestricted price that is smaller(not necessarily strict) than the only active restricted price; and (3) the allocated restrictedChapter 3. General Optimality Results and Other Properties 94units at this active restricted price will exhaust all the residual demand for the restrictedproduct, then the optimal policy is equivalent to a primary policy.Proof: Clearly, under the given policy, I know that after the sales of the restrictedproduct at price P2, there will be no residual demand for restricted product at price p,for j 2. Therefore, the total residual demand at price p3 is given byd2, = = (1 — 3. (3.7)Consequently, the feasibility condition for the remaining unrestricted market is1 j=3 3I now construct a new policy as follows:I 11 Iq-,1 = qu,i A a1, q,1 = qu,i— qr,i;q1q,2 (1 — ---)d, q,,2 = qr,2—q,.,2;= 0, = q,j,Vj 3;where x A y = min(x, y). Some observations here will be helpful:• the basic idea here is to sell the restricted product as much as possible at the pricelevel p’, where the upper bound is determined by the existing demand d and theoriginal allocation qu,1 at the unrestricted price pi;• if at Pi, the allocation q1 cannot exhaust all the restricted demand, then q2 willtake care of the residual market for restricted product at p2;• the new policy leads to a primary policy since either (i) q1 = d, which leads to afollowing primary 1-policy:{q,1;q,1,qu,2, ,Chapter 3. General Optimality Results and Other Properties 95or (ii) q2 = (1 —q1/d)d > 0, which implies that q1 = 0 and consequently leadsto a following primary 2-policy:{q,1, qr,2; q,2,q,3, , q,};• it is easy to check that the new derived primary policy is equivalent to the originalpolicy.Let {q.1, “, q; q, qu,t+i, , q,n} be the derived primary t-policy from the abovespecification, where I = 1 or I = 2.Case 1: t 1.This implies thatq1 = d qu,i, q,1 = qu,i — dq,2 = qr,2, and = qu,j, V.) 3.Let be the total residual demand after the sales according to (p’,q1), (p1, q1) and(p2,q2). Then it is easy to check that—q — d qr,2 ,judL du i•1 2Therefore, to prove the feasibility of the new primary policy, it suffices to show that,d2,,Vj 3which, by (3.7), is equivalent to3.4Jn what follows, the symbol means “equivalent”.Chapter 3. General Optimality Results and Other Propertiesq,id2,.? ,Iu96q,iq.1 = qu,i c1;q2 = (1—a1q,2 =..Ir / 2U1= q,id(dT1——a;-)— d - 11 U1= 3.Again, let d’, be the total residual demand after the sales according to (p1,q1),(p2, q2) and (p2, Then= (1 — U2)U = (1 —,1u ,1qu,i 1 2,ju1 ..lr,1u) j•Ui U1 U2I hope to show thatwhich, by (3.7), is equivalent tod2,,d2 d1 d dL— < —— < 47i 72,.,lu— 1u d1 d2U2 U1which follows from the monotonicity condition on 7’s. This shows that the new policy isU d1 U1 U U2U1(dh1d2)<did2d1d2==‘di,which follows from the feasibility condition for the original policy. This proves Case 1.Case 2: t = 2.Under this case, we have thatdu dr dT d’feasible. This proves Case 2. In summary, the lemma is proved. DChapter 3. General Optimality Results and Other Properties 97The following lemma considers the case that there are two active restricted pricesbetween the first two active unrestricted prices.Lemma 3.3.3: Any optimal G-policy of the following form:{ (pi, q,i, u), (p2, qr,2, r), (ps, q,3, r); (pa, q,4, u),. . . , (pa, u)}is equivalent to a primary policy, wherePi P2< P3 4< <p and qu,i > 0,qr,2> 0,qT,3 = d,3 >0.In words, it says that if there exists an optimal policy in the G-class such that (1) it hastwo consecutive active restricted prices, (2) there is one and only one active unrestrictedprice before these two active restricted prices, (3) there is no residual demand left for therestricted product at the end of sales of the restricted units at the second restricted pricelevel, then this optimal policy is equivalent to a primary policy.Proof: First of all, it is easy to show thatwhich is equivalent toqu,i qr,2_Lqf,3 —Jr Jr —Ui U,2 (43This also implies that there is no residual demand for restricted product after the salesup to (ps, q-,3). Therefore, the total residual demand after the sales up to (ps, q,.,3) isgiven byd3,=d=(1_)d7. (3.9)Define a new policy as followsq,1= qu,i A d, q,1 = q,i — q.,1;Chapter 3. General Optimality Results and Other Properties 98q1 r Iqr,2 = (1 — ——)d2 A qr,2, q,2 = qr,2 — q,,2;U,’I Iq,, qr,2 rq,.,3 = (1 — — , q,,,3 = q,., —I U2= 4.Step 1: To show that the above new allocation plan corresponds to a primary policy.First of all, at price p,, if q1 = d, it indicates that q,, d. This implies that• q, 0, and q2 = q3 = 0,which leads to a primary 1-policy. Secondly, if q1 < d and q2 = (1 — -)d, then itimplies that• q,1 = qu,i, q, = 0, q2 0, and q3 = 0,which leads to a primary 2-policy. Finally, if q, < d, q.2 < (1 — !!-)d and q3 =(1 — — ?-)d, then it implies that• q1 = q2 = 0, and q3 0,which leads to a primary 3-policy. In summary, the new plan leads to a primary policy.Step : To show that the new derived primary policy is feasible.Let the derived primary policy be in the form of{(pi, q,,), . . . , (ps, q); (pi+i, q,t÷i),. , (Pn,where 1 <t <3. I will finish this step by studying three cases.Case 1: t = 1.Under this case, I have thatq, = d qu,i,q, = qu,i —ctq•2 = q,3 = q,3, = qu,j, Vj 4.Chapter 3. General Optimality Results and Other Properties 99Some further observations can be made here:• the new plan is feasible for the restricted market;• there is no residual demand for restricted product at any price Pj (i 1) after thesales of q1 units at the restricted pricepi.Let d,3 be the total residual demand after the sales according to (p’, q.1), (pi,q1), (p2, q2)and (ps,q3). Then I getd’ = (1 — — — = (1 — — d — — ‘dU Vj 4.Au 2u .1uI 3’3,j d1 d2 d3 U“2 (3Therefore to prove the feasibility of the new policy, it suffices to show thatd3,,which, by (3.9), is equivalent to requiring that1 1 d’<_L==qu,1(——)++—lu 3u ,lu — d2 d3 U,1a1 u2 U,3+ + <d q,1 qr,2 q,-,3d1 du Au ,u — dL d + ‘1’ + dV1. (3.10)d-’- du11 2d 3dOn the other hand, by (3.8), I know that to show (3.10), it suffices to prove thatdT dr d d” dT dTdU 1 >dandd- 4=2 -U,1 Ct1 a1 ct2 a1 Ct3d1 d d1 d3 d’ du du dU— —and—>—u Au Au — Au d1 — d2 d1 — d3d1 U,2 U1 U3which follows from our monotonicity assumption on 7’s. Therefore (3.10) is true. Hencethe result is proved for Case 1.Case 2: t = 2.Under this case, I will have the following:q1 = qu,i < = (1— d 2’q,2 = qr,2 — = = q,3,Vj 4.Chapter 3. General Optimality Results and Other Properties 100I also know that• P2 is the last active restricted price;• there is no residual demand for restricted product after the sales at restricted priceP2.Again, let d,3 be the total residual demand at price p3 U 4) after the sales accordingto (pi,q1), (p2, q,2) (p2, q2) and (ps,q3). Thend,3 = (1— c,2 — U3)U = (1— q,2—q, —I now want to show thatd3,,Vj 4,which, by (3.9), is equivalent to(—-- + )qu,i + qr,2+1. (3.11)Again, using (3.8), we know that to show (3.11), it suffices to show thatid” 1 1 d 1 d1 d d=>U U2 U1 U U2 U2‘-2d1 d2 d2 d3U1 U,2 U,3dr d’ du d’—-and —- < —pd1 d2 d2 d34=7<-yand73,which again follows from the monotonicity assumption on v’s. Therefore (3.11) holds.So this proves result for Case 2.Case 3: t = 3.Chapter 3. General Optimality Results and Other Properties 101Similarly, I know that under this case,I qu,i qu,i qr,2 rq,., = q,i < = qr,2 <(1— _-j_)d2,q3= (1—————tL1‘-“1 2q3 = q,3 — q3,q3 = 4.Let d,3 be the total residual demand at price Pi ( 4) after the sales according to(pi, q,1) (p2,q2), (p3, q.3) and (ps,q3). ThenI need to show thatd,, d3,,which, by (3.9), is equivalent to(3.12)Again using the fact (3.8), to show (3.12), it suffices to prove1 1u 1 J 1I U3 I U3 (hi47371,(Li UU3 1 U3 (hiwhich is again assumed. This proves (3.12). Then the result is proved for this case too.Summarizing the above three cases, I have shown that the new derived primary policyis indeed feasible. Finally, it is easy to see that+ = piqu,i + pjqr,j + pjqu,j,which implies that the new primary policy {q1, q; q, , q} is in fact equivalentto the original policy.Therefore, we have shown that there exists a feasible primary policy that is equivalentto the original policy. So the lemma is proved. DChapter 3. General Optimality Results and Other Properties 102In summary, Lemma 3.3.1 resolves the issue on the order of sales of the unrestrictedunits and the restricted units at the last active restricted price level by showing that thefirm can limit its attention on policies such that the restricted units are sold first at thelevel of the last active restricted price. Lemmas 3.3.2 and 3.3.3 demonstrate that anyoptimal policy with the following two properties:• the first active unrestricted price is followed by one restricted price or two consecutive restricted prices; and• there is no residual demand for the restricted product at the end of sales of theallocated restricted units at the last active restricted price,will be equivalent to a primary policy.3.3.2 The General Optimality TheoremsWe now use Lemmas 3.3.1, 3.3.2 and 3.3.3 to prove the following theorem, which dealswith the situation where following the first active unrestricted price, there are many restricted prices.Theorem 3.3.4: For m: 2 m n, any G-policy of the following form:{ (pt, q,i, u); (p2, qr,2, r),. . , (pm, qr,m, r); (Pm+i, qu,m+i, u),. . . , (pa, u)}is weakly dominated by a primary policy, whereP1P2<<PrnPrn+1<”Pn.Proof: Clearly, I only need to consider the case that qu,i > 0. Note that, given the valueof qu,i, the total residual demand at price Pj is given by:— (1— -)d,j 2;Chapter 3. General Optimality Results and Other Properties 103and the residual demand for restricted product is given by:d,3=(1—1’)d,j 2.Given that after the sales of unrestricted product at price Pi the firm plans to sell restricted products at prices P2, ,Pm and unrestricted products at prices Pm--i,” ,Pn.We also know that the sales of qu,i units at the unrestricted pricepjdoes not change -y’s,so= 7j for all j > 2. By Corollary 3.2.2, if Pm <Pm+1, then yj is strictly increasingin j. Tf Pm = Pm+1, then 71,m = 71,m+i and{71,2,”,71,m,71,m+2, ••,7n}constitutes a strictly increasing sequence. Therefore, we can use the results in Chapter2 to the residual market specified by {(d,,d1,) : j = 2,• , n} together with the priceset {p2, • ,pn} By Theorem 2.4.2, we know that there will be a primary policy{(p’2,q,2) , (p’,, q,,’); qL’),. , (P’n’,which is defined on the residual market such that1. = n if Pm <Pm+1 and u’ = — 1 if Pm Pm+i2. the set of prices {j4,. ,p’,} is the same as the set of prices {p2,••3. dim4 = 0, that is, there is no residual demand for restricted product at pricep p’,’ after the sales according to {(p, qu,i), (p’,q,2),” (p’,, q,,)};4. at most two of q2, ••,are strictly positive;5. it weakly dominates the partial policy:{(p2,qr,2),”,(pm,qr,m);(pm+1,qu,m+1),”,(pn,qun)}.Chapter 3. General Optimality Results and Other Properties 104This consequently leads to the new G-policy:which in turn is weakly dominating the original policy. Now if all of q,2, q’ arezeros, then the theorem follows immediately. If there is exactly one of them is positive,then the theorem follows from Lemma 3.3.2. And finally, if there are exactly two positiveallocations at restricted prices, then the theorem follows from Lemma 3.3.3. This provesthe theorem. DI now present the key result of this section.Theorem 3.3.5: Any G-policy is weakly dominated by a primary policy.Proof: Consider any G-policy {n; (ps,q1,j, 82), i = 1,. • , n}. Let Pm be the last active restricted price and let Pui < < Pu3 be the unrestricted prices in the price set{pi, ,Pm}. By Lemma 3.3.1, without loss of generality, we may assume that pu <Pm.Now for the given partial G-policy{ (p, qs,i, 6), i = 1,..., u — 1},we can use Theorem 3.3.4 on the remaining policy:{(pu3,qu,5);(pi,qr,i),i = u8 + 1,••• ,m;(pj,qu,),j = m+ 1,... ,n},which will lead to a policy in the form of{(p,q),i = us,... ,m8; (p,qj),j = m3 + 1,” ,n}.Therefore, after this application of Theorem 3.3.4, we know the followings:• Pms becomes the last active restricted price and Pus_i 15 the largest unrestrictedprice that is below pm5; andChapter 3. General Optimality Results and Other Properties 105• there are oniy s — 1 active unrestricted prices among in the price set ,Pms },which are Pui,” , Pus_i.As we can see here, every time we use Theorem 3.3.4 on the modified policy, we willeliminate one active unrestricted price that is strictly less than the last active restrictedprice. Therefore, after s times, we will end up with a policy that has no active unrestrictedprice that is less than the last active restricted price, or equivalently, we will get a primarypolicy, as required. Hence the theorem is proved. 0To conclude this section, I here present an example that demonstrates that there aresituations which have optimal policies that are not primary.Example 3.3.1 Existence of Non-Primary Optimal PolicyConsider the following simple demand structure:P1 = $100, P2 = $140; D,1 = 200, D,2 = 100; D,1 = 100, Dr,2 = 0; k = 250.It is very easy to check that there exists a unique primary optimal policy given by:q1 = lOO;q1 = 100,q2 = 50,which generates total revenue of $27, 000. Let’s consider the following policy in a formas given by Definition 3.2.1:{ (p = = 150, u), (p2 = P1,qr,2 = 50, r), (p3 = F2,q,3 = 50,u)},which says that the firm first sells 150 unrestricted units at P1 = $100, then sells 50restricted units also at P1 = $100, and finally 50 unrestricted units at P2 = $140. It isclear that this policy also generates total revenue of $27,000. Let’s check that this policyis also feasible. First, note that D1 = 300 and D2 = 100. If the firm sells 150 unrestrictedunits at F1, then we know the following:Chapter 3. General Optimality Results and Other Properties 106• among these 150 units sold, the portion that is originally from the restricted marketis given by:Dr1 100= X 150 = 50,o’J’)which implies that the residual demand for the restricted product at the price levelP1 is 50;• there is residual demand for the restricted product at the price level P2 since Dr,2 =0; and• the residual demand for the unrestricted product at P2 is(1_1)D,2=(1. -)x100=50.Therefore, the allocated 50 restricted units at the price level P1 and the allocated 50unrestricted units at the price level P2 are indeed feasible.Remark: The above example has some very interesting implications:• it is possible to have optimal policies that are not primary;• the use of the non-primary optimal policies will resolve the implementation difficultyof some primary optimal policies where the firm has to sell some restricted unitsfirst and some unrestricted units later at the same price level.4These observations prove to be quite useful in the next section too.41n the above example, it is not easy to sell 100 restricted units after knowing that there are stillanother 100 unrestricted units available later. This is what I mean implemena1ion difficulty. The newnon-primary policy does not have this difficulty.Chapter 3. General Optimality Results and Other Properties 1073.4 Further Properties of Optimal PoliciesIn the above section, I have demonstrated that there exists a primary optimal pricingpolicy in the G-class. As pointed out at the beginning of this chapter, there is anotherimplicit assumption in the BL-model, which assumes that the firm offers one type of theproduct at a time. In other words, the allocated unrestricted units will not be put onsale until the restricted units are sold out according to the plan. On the other hand, suchan assumption is not consistent with the existing pricing practice by airlines, which usemultiple prices and make all offered prices available at the same time. The purpose of thissection is to resolve this issue by arguing that there always exists an optimal policy withthe property that all offered quantities — both restricted and unrestricted — at differentprice levels can actually be made available at the same time. To finish the argument, Ineed the following behavioral assumption:5• Behavioral Assumption: If there are restricted units and unrestricted units allocated at the same price level and all these units are made available at the sametime, then the unrestricted units will be sold first, or equivalently, when restrictedunits and unrestricted units are offered at the same price time at the same time,consumers from the restricted market will try to buy the unrestricted units first.Let us now start with the characterization results on primary optimal policies fromTheorem 2.4.2. Clearly, we only need to focus on the basic primary optimal policies thatare derived from the Tight Problem in Chapter 2. Recall that the Tight Problem is alinear programming problem with three structural constraints, which implies that anybasic primary optimal policy consists of at most three positive quantities, of which atleast one, but at most two, of them is at an unrestricted price level. Because of this, Ineed to discuss two scenarios.5Even though I do make this assumption, but it is oniy used once.Chapter 3. General Optimality Results and Other Properties 108Scenario I: There is exactly one unrestricted price.Under this scenario, we can write any basic primary optimal policy as follows:qr,j; qu,k},where i <j <k with active prices F, F3 and Fk. We also know that• qu,k = D,k, which implies that there will be no residual demand for the unrestrictedproduct at p Fk after the sales of the unrestricted units at Fk;• q,3 = (1 — which implies that there will be no residual demand for the restricted product at price level p F3 after the sales of the restricted units accordingto {(F, q,.,j), qj)}.I now need to consider three cases. First, for the case that F3 <Pk, it is easy to see thatif the two restricted prices P, F and the one unrestricted price Fk are all made availableat the same time, then the firm still can sell units restricted product at P, q,.,j unitsof restricted product at F3, and qu,k units of unrestricted units at F,. due to the followingfact:• Consumers from two market segments will only buy the product targeted to theirrespective segment because a consumer from the restricted market will not only buythe unrestricted product because the restricted units are cheaper and a consumerfrom the unrestricted market will not buy a restricted unit because he cannotaccommodate the restriction.For the case that i <j = k, consider the following new policy:q,,j ift=iqr,t =1 0 otherwise;Chapter 3. General Optimality Results and Other Properties 109and1I u,k+-ff) r,k 1 t—0 otherwise.Clearly, this new policy is also a primary optimal policy with only one active restrictedprice P1 and one active unrestricted price Fk. We now argue that for this new optimalpolicy, if both the restricted units at F, and the unrestricted units at P1. are available atthe same, the firm still can reach the projected sales of q.1 restricted units at P1 and qunrestricted units. Note that:• at the beginning of the sales in the both markets, Dr,j and D,1. are exposed;• since consumers from the unrestricted market will buy the restricted product, thefirm will capture all the demand for the unrestricted product at price Pk, which isD,1.;• for consumers from the restricted market, they will not buy the restricted productat price Pk until the allocated q1 restricted units are sold out;• after the allocated q restricted units are sold out, according to the proportional-rationing rule, the residual demand from the restricted market at price Pk is (1 —which implies that the firm can sell up to (1—units of theunrestricted product at price P1. to the restricted market.Therefore, the firm will reach its projected sales in both markets.At last, for the case that i= j = k, it must be true that q,.,j = 0, qr,j = Then itis obvious that the firm just sells Dr,k + D,1. = Dk unrestricted units at the price levelP1. without selling any restricted unit.Scenario II: There are exactly two unrestricted prices.Chapter 3. General Optimality Results and Other Properties 110Under this scenario, we can write an primary optimal policy derived from the TightProblem as follows:{q; qu,k2}with the following properties:• i < k1 <Ic2;• q• qu,k2 = (1 —We now need to discuss two cases. First, consider the case that i < k1, that is, P <Fk,.Using the same the argument as for the first case under Scenario I, we know that evenmaking the restricted price P2 and the two unrestricted prices Pk, and Fk2 available atthe same time, the firm still can reach the projected sales of q restricted units at F1,qr,ki unrestricted units at Pk1 and qu,k2 unrestricted units at Fk2.For the case that i = k1, additional care is needed if the firm makes all offered pricesavailable at the same time because of the situation that the firm now offers restrictedunits and unrestricted units at the same price level Fk1. The dilemma is that if onlyone of the potential Dr,ki consumers from the restricted market actually purchases anunrestricted unit at Pk1, the firm will not be able to sell Dr,ki restricted units. In fact,under the Behavioral Assumption, if qu,ki unrestricted units and qr,k, restricted units areavailable at the same price level Fk, then the allocated qu,ki unrestricted units will besold first. Among these who have purchased the unrestricted product at price level Pk1,there is a positive fraction of them who are from the restricted market. This impliesthat the firm definitely cannot sell additional Dr,ki restricted units at the price level Fk,.Consequently the firm cannot directly use the given primary optimal policy if it decidesChapter 3. General Optimality Results and Other Properties 111to make all allocated units available at the same time. Therefore, we have to constructa new optimal policy. For this, consider the following policy:6• the firm first sells q = (1 + unrestricted units at price level Fk1;• the firm then offers q1 = (1 — )Dr,k1 restricted units at price level Pk1;• the firm finally allocates q2 qu,k2 = (1 — ‘—)D,k2 unrestricted units at pricelevel Fk2.To conclude our argument, we have to finish three steps: (1) this new policy is in factfeasible, (2) it is equivalent to the original primary optimal policy, and (3) if all offeredprices in this new optimal policy are available at the same time, the firm still can reachits projected total revenue. Regarding the feasibility of this new policy, we observe that:• the allocation q1 is feasible at Pk1 since= + Dr,k1 = qu,ki (D,k1 + Dr,ki) < + Dr,ki = Dk1,which further implies thatq,k1— qu,ki—• after the sales of the q1 unrestricted units at the price Pk,, the residual demandfor the restricted product is given by:= (1 — 1)Dr,j = (1 — qu,ki )Dr,j for jDk1 D,k1which implies that q11 = (l —1)Dr,ki restricted units at P1 is indeed feasible;6This construction of a non-primary optimal policy is motivated from the discussions in Example3.3.1.Chapter 3. General Optimality Results and Other Properties 112• after the sales of the q1 unrestricted units at the price level Fk1, the residualdemand for the unrestricted product from the unrestricted market is specified by:— (1—= (1 — qu,ki for j k1,Dk1 D,k1which indicates that q2 = (1 — is also feasible.So I have shown that the new policy is feasible. To see that it is also equivalent to theoriginal primary policy optimal, note thatPk1 (q,,ç1 + + Pk2 ‘lu,k2 = Fk1 (qu,ki + Dr,k1)+ Fk2qu,k2= Pk1(q, , + qr,ki) + Fk2q,.Now consider that for this new optimal policy, the firm makes q1 unrestricted units atFk1, qr,k1 restricted units at Fk1, and q,k2 unrestricted units at Fk2 all available at thesame time. Then we have the following observations:• under the Behavioral Assumption, q1 unrestricted units at the price level Fk1 willbe sold before the sales of restricted units, which implies that at the beginningof the sales, only the unrestricted Fk1 is actually active;• after the sales of the unrestricted units at Fk1, the restricted Fk1 and the unrestrictedPk2 become active;• since Pk1 <Fk2,then by the same argument as used in the first case under ScenarioI, we know that the firm can reach the projected sales levels in both residualmarkets, that is, selling qk1 restricted units and q,k2 unrestricted units.In conclusion, we claim that all allocated quantities in the new optimal policy can bemade available without any adverse impact to total projected revenue.Chapter 3. General Optirnality Results and Other Properties 113As a last comment of this section, it is interesting to note that in order to be able tomaintain the projected value of the optimal revenue and to be able to make all allocatedquantities available at the same time, the firm might have to implement a non-primaryoptimal policy.3.5 Summary — General Optimality ResultsThis chapter addresses two unsettled issues in Chapter 2. The first issue is the limitationof only using the type of pricing policy that sell restricted product at lower first at somelower prices and later unrestricted product at some higher prices, namely, the primarypolicies. The other issue is whether all offered prices, restricted and unrestricted, can bemade available simultaneously and still generate the same amount of revenue realized bya primary optimal policy. First of all, this chapter has completely resolved the first issueby showing that even allowing the firm to attach restrictions at any possible price andat any order, an optimal primary policy remains to be optimal in this general class ofpricing policies. The main implication of this generality result is that if a firm decidesto use a marketing restriction in its pricing decision, then the firm can limit its pricingpractice to primary policies, that is, selling restricted units first at a set of prices thatare lower than these prices for unrestricted units.This chapter also gives a satisfactory answer to the second issue by demonstratingthat there always exists an optimal policy, which may be non-primary, with the propertythat making all allocated units both restricted and unrestricted— available at thesame time will not have any negative impact on the optimal revenue value derived froma primary optimal policy. This is important because it shows that some optimal policiesare consistent with the existing pricing practices by firms like airlines, which (1) offermultiple prices— restricted and unrestricted, and (2) make all allocated quantities atChapter 3. General Optimality Results and Other Properties 114different price levels available the same time. Such a consistency makes the pricingmodel developed in the last chapter and this chapter much more useful to firms whichuse artificial restrictions in the process of pricing perishable inventories such as airlineseats and hotel rooms.Chapter 4Pricing Models with Two Types of Restrictions4.1 Introduction and Model SettingChapter 2 developed a simple pricing model for a monopolist who uses one restriction asa mean of segmenting the market demand. It showed that by properly setting the levelof the highest restricted price and rationing the sales at lower prices, the monopolistneeds to charge no more than three prices to maximize its revenue. Then Chapter 3further analyzed the problem by considering a general class of pricing policies whichallow firms to offer restricted units at any price level. It showed that the optimal policiescharacterized in Chapter 2 remain to be optimal in this general class of policies. Also inChapter 3, I have addressed the issue of simultaneous availability problem for all pricesthat are planned to be offered. These two chapters together prove a very powerful result:as long as the restriction is effective, that is, as price increases, the percentage of theseconsumers who cannot accommodate the restriction is strictly increasing, the optimalpricing practice is to sell restricted units at some lower prices and unrestricted units atsome subsequent higher prices, and the firm only needs at most three prices to maximizeits revenue.On the other hand, the discussions in the previous chapters are limited to the useof one type of restrictions. As a consequence of this, the firm can only use two typesof product restricted product and unrestricted product. But any realistic situationinvolves multiple restrictions and multiple types of product. Firms, such as airlines and115Chapter 4. Pricing Models with Two Types of Restrictions 116hotels, must deal with several product restrictions together with several other corporaterestrictions.’ It is important to note that each restriction is targeted to a unique marketsegment; and firms should be able to explore these market segments by dealing with allthese restrictions at the same time. In this chapter, I will develop pricing models forperishable inventories when a firm uses two types of restrictions. As we will see later,the use of two types of restrictions will allow the firm to introduce three or four differenttypes of products. This will substantially increase the flexibility of pricing practice forperishable inventories, such as airline seats and hotel rooms.Consider that a monopoly firm has a limited quantity (k) of a certain perishableproduct, such as airline seats or hotel rooms. Let D(p) be the market demand functionfor the product at price p. We can interpret D(p) as the number of consumers in themarket who are willing to buy one unit of the product at price p. The firm also considersto use two types of restrictions in its pricing decisions. Let 7t(p) be the percentage ofconsumers who can not purchase the product if type t restriction is attached to theproduct at price p, where t = 1,2. Therefore the demand for type t restricted productat price p is given byDt(p) = (1— yt(p))D(p), t = 1,2.As in Chapters 2 and 3, Twill assume that the demand functions D(p), DT1(p) and Dr2(p)are all step functions defined on the price set {pi, . . , p,,,}, that is,13, ifpp,,D(p)= D if p_ <ppfori=2,•••,ri,0 ifp>p;example, airlines offer special fares to major corporations and government, which are not available to the general public. The existence of these fares can be analyzed by treating them as corporaterestrictions.Chapter 4. Pricing Models with Two Types of Restrictions 117and fort = 1,2,Dt ifppi,rrtI \1 j’j= D if p_ <ppforz=2,-,n,0 ifp>p,i 1,... ,n, and t = 1,2, denote7t,i = 1—Then it is easy to see that ‘y(p) is also a step function.This chapter is organized as follows. Section 4.2 presents an extension of the BLmodel to the case that there are two types of nested restrictions, which is a quite commonpractice for airlines.2 I show in this section that there exists optimal policies that can becharacterized by a linear programming problem, which indicates that the firm needs tooffer at most four price levels to maximize its revenue by using three different types ofproduct. Section 4.3 will extend the basic BL-model to the case of two type of restrictionsthat are mutually exclusive. I also show that in this case the firm needs to offer no morethan four different price levels with three different types of product. And then in Section4.4 I will use the results developed in Sections 4.2 and 4.3 to study the pricing problemwhen a firm uses two general types of restrictions. I demonstrate that under certainreasonable conditions, the firm can limit itself to the type of pricing policies that consistof at most five different price levels with four different types of product. Finally, the lastsection is a summary.2For example, airlines offer discount fares with advance booking and Saturday night conditions. Inthe meantime, the discount fares just with Saturday night condition are also available.Chapter 4. Pricing Models with Two Types of Restrictions 1184.2 Pricing Problem by Using Two Nested RestrictionsI first introduce the following definition:Definition 4.2.1: We call type-s restriction is nested into type-i restriction if thereexist aj(p), t = 1,2 such that1. 1— yi(p) = 1 —c1(p); and2, 1— 72(p) = (1 — cri(p))(i —a2(p)).From the definition, it follows that if type-2 restriction is nested into type-i restriction,then I will haveDT2(p) = (1 —a2(p))D”(p), (4.1)which, in fact, is the main reason why I call them nested since it implies that the groupof those who can accommodate type-2 restriction is a subset of of those who can accommodate type-i restriction. Hence, it is clear that type-2 restriction is more restrictivethan type-i restriction. Denote = at(pi) for t 1,2 and j = 1,’ .. , i-i.The firm needs to address the following fundamental questions:• Is it necessary to use restrictions? Two or just one?• What are the optimal pricing policies?Therefore a pricing policy should tell the firm which prices are restricted and how manyunits of the product available at each of these price are offered. Throughout this section,I will make the following assumptions:• both {ai} and {a22} are strictly increasing;Chapter 4. Pricing Models with Two Types of Restrictions 119• type-2 restricted units are sold first, then type-i, and finally the unrestricted;• if type-2 restricted units and type-i restricted units are sold at the same price level,then type-2 restricted units are sold first;• if type-i restricted units and unrestricted units are sold at the same price level,then type-i restricted units are sold first; and• the product is sold at prices in the order of price levels Pi,• ,p.With these in mind, the following definition of a pricing policy under two nested restrictions is quite natural:Definition 4.2.2: Suppose that restriction 2 is nested into the restriction 1. Then apricing policy for the firm is in a form of{ (p, qr2,i) : i = 1,. . , m; (ps, qri,j) : j = m2,•• , m1; (pk, q) : k = m1,• ,where Pmt is the highest type-t restricted price, qrt,i is the quantity available for sale atthe type-t restricted price pj, q, is the quantity available for sale at the unrestricted pricePk, andi<m2m n.Remarks: From the definition, I have the following observations:• If m2 = 1 and qr2,1 = 0, then the above policy indicates that only type-2 restrictionswill be used;• If m2 = m1 = rn and qri,m = 0, then it indicates that only type-i restriction willbe used;• If m2 = m1 = 1 and qr2,j = qri,i = 0, then it means that all units are unrestricted,that is, no restriction is used.Chapter 4. Pricing Models with Two Types of Restrictions 120Also, since the values of m1 and m2 are completely controlled by the firm, it is of thefirm’s interest to find the best possible combination of these two values. But first, I needto specify the set of all feasible pricing policies for any given pair (m2,mi). For this, Iwill use the same techniques used in Chapter 2 to derive these feasibility conditions.First, consider that type-2 restricted units are put on sale according to{(pi,qi),. . ,(pm2q,)}with demands given by D2 for i = 1,.. ,m. Therefore, the feasibility condition for thetype-t restricted product is given byAfter the sales of the type-2 restricted product, the residual demand for type-i restrictedproduct is given by= D’ — j32D, for j = m2,••• , n.Consequently, the feasibility condition for the plan {(Pm2,qi,2), , (pm1q,1)} for thetype-i restricted product ism1 ,-1 m1 rl/3_V q3 q3P1 =— L.. flrl 0 flr2 —• U ‘3_rn2 2 3_rn2Finally, let us consider selling unrestricted product according to the plan {(pk, q) : k =m1,. . , n}. It is easy to see that the residual demand for the unrestricted product afterthe sales of the product with restrictions is given bydrn1,k = Q1kDk + (1— /31)d2,k = Dk — j31D’ — (1 — /31)/32D,for k m1.Hence, the feasibility condition for the remaining unrestricted product is given by:n flqk — qk <1k=mj drni,k — kzrn1 Dk — — (1 —31)32D —Chapter 4. Pricing Models with Two Types of Restrictions 121Summarizing the above discussions, I obtain the following formulation of the firm’spricing problem when using two nested restrictions:m2 V”flMax ji + Pii + L.ak=mi Pkqks.t.sçrn2 < 1 (4.2)L=i—qr1, < 1 (4.3)j=m2 —1 (4.4)‘cnqr2,i + )j=m2qri,j + /_..jk=m1q qqr2,i,qrl,j,qk O,Vi,Vj, and Vk.I will call the above pricing model N-model, where N represents for nested. LetR(m2,mi) be the derived maximum revenue from the above N-model. The following theorem proves that there is a linear programming characterization for the optimal revenue:max R(m2,mi).1m2miTheorem 4.2.1: Let R(m2,mi) be the optimal objective value of the following linearprogramming problem (named as the Tight N-model):Max + p,qri,, + Z=mj pkqks.t.ç-m2 qr2,i—1L.di=i.çmi qrl,j= 1L..ej=m2 D’—D2gk=1L.dkmi.Dk—D’2.—i qr2,i + YZ,=m2 qri,j + YZk=mi q qqr2,i, qri,j, qk O,Vi,Vj, and Vk.Chapter 4. Pricing Models with Two Types of Restrictions 122Then,max R(m2,mi) = max R(m2,mi). (4.5)1<m2<mln 12< nProof: It is evident thatmax R(m2,mi) max k(m2,mj).1<m2n 12nThen to prove (4.5), it suffices to show thatmax R(m2,mi) < max .i(m2,mi). (4.6)1<m2mj<n 12<nTo prove (4.6), I oniy need to show that for any pair: 1 m2 m1 n, there existsanother pair: 1 <th2 th1 <n such thatR(m2,mj) (th2,n1). (4.7)Consider any pair 1 m2 n and let{ = 1,••• ,m2;(pj,qrl,j),j = m2,•• ,m1; (pk,qk),k =m1,...,n}be an optimal solution to the N-model. Denotem2q2which is the total sales for type-2 restricted units. Note that the residual demand for thetype-i restricted product after the sales of the type-2 restricted product is given by:= —32D, for j = m2,•• ,and the total residual demand for unrestricted product at price p, after the sales of thetype-2 restricted product is given by:dm2,j = D, 32D, for j = m2,• ,Chapter 4. Pricing Models with Two Types of Restrictions 123Therefore the residual percentage of consumers who can not accommodate the type-irestriction is given by:jrl r’r1 i2 r’r2—— Um2,j-‘-‘j — P2’—’3 1j—— dm2,j = — D — /32D, = 1 — 132(1 — i)(i — a2)which is strictly increasing since both c and cr2 are strictly increasing by assumption.Therefore, I can use Theorem 2.4.2 to the residual market for the type-i restricted productand the unrestricted product. This implies that considering q—q as the capacity limit,I know that there exists an integer m2 rñi < n and a new optimal allocation plan inthe form of{(pj,qri,j),j =m2,.••,thl;(pk,ijk),k=thl,.•.,n},such thatq,3 1 48— —.2D,.= 1 (4.9)krn1 knqij + q — q; and (4.10)j=m2n nPjqri,j + pq + pkqk. (4.11)j=m2 j=m2 k=m1We should notice that the relationship (4.9) is independent of the value of /32. In fact, aslong as (4.8) holds, that is, the feasibility constraint for the type-i restricted product istight, (4.9) is automatically true!Now let 4i q. Therefore o is the total number of units of the productallocated for unrestricted units. Then with o units protected, the firm needs to solve thefollowing subproblem:Chapter 4. Pricing Models with Two Types of Restrictions 124Max + piqi,is.t.m2 q,.,j< 1P2— —qri,j < 1/_13m2 D—/3 —m2 —D1 qr2,i + YZj=m2 qri,j q — qoqr2,i,qrl,j O,Vi, andVj.This again leads to the basic model in Chapter 2. By Theorem 2.3.5, I know that (4.8)holds at any optimal solution. Furthermore, again using Theorem 2.4.2 , I conclude thatthere exists another integer 1 rn2 th and another allocation plan{(pi,qr2,i),i = 1, ,2;(pj,r1,j),j = th2,,th1},such thatm2 —= 1, (4.12)m1= 1, (4.13)3 3rn2 rn1 n+ + qk q; and (4.14)i1 3=7712 k=1rn2Pjq,-2, + ) Pjri,j piqr2,i + Pj’,-i,j, (4.15)i=1 =rn2 i1since it is easy to check that {qr2,i, i = 1, , m2; = m2,• , ñ} is also a feasiblepolicy for the above subproblem. Finally, by (4.9), (4.12), (4.13) and (4.14), I know thatthe new allocation plan{(pi,r2,i) : i = l,.,rn2;(p3,ql,) :j = ñ2, ,rhl;(pk,qk) k =Chapter 4. Pricing Models with Two Types of Restrictions 125constitutes a feasible policy for the Tight N-model with respect to 2 and iii. Furthermore, by (4.11) and (4.15), it follows thatrn2 rn1 fl(ñi2,ñii) ir2,i+ >Pjqr1,j+ pkqk=rn2m2 ‘i n>ptq,.2, + Pj’lri,j + pkqk3=m2 k=ñ1m2 m1 npq2,+pjqri,j+ pkqk=R(m2,m1).i=1 3=m2 k=miSo (4.7) is true; and therefore (4.6) is proved. CAs an immediate consequence of the above theorem, I have the following useful resulton the optimal pricing policy by using two nested restrictions.Corollary 4.2.2: If two type of restrictions are nested, then there exists an optimalpricing policy that consists of at most four different prices, which is characterized by alinear programming.As a final remark of this section, the above discussion can be easily extended to thecase of multiple nested restrictions.4.3 Pricing Problem by Using Two Mutually Exclusive RestrictionsIn this section, I will discuss another case of two types of restrictions. Before moving on,I need to introduce the following definition of mutually exclusive restrictions:Definition 4.3.1: Let (p) be the percentage of consumers who cannot purchase theproduct if both restrictions are attached to the product. Then we say that these twoChapter 4. Pricing Models with Two Types of Restrictions 126restrictions are mutually exclusive if 7(p) = 1.The mutual exclusiveness here means that those who can accommodate the type-irestriction cannot accommodate the type-2 restriction, and vice versa. Or intuitivelyspeaking, if two types of restrictions are mutually exclusive, then they are targeting twodistinct consumer groups. In other words, the set of those consumers who do not mindthe type-i restriction is disjoint from the set of those consumers who are do not mindthe type-2 restriction. This sounds very restrictive, but it is nevertheless an importantcase I need to discuss. In the next section, I will discuss the general case of two typesof restrictions that are not mutually exclusive and not nested, by integrating the modeldeveloped in the last section and in this section.If the two restrictions are mutually exclusive and the units with restrictions are soldfirst, then the following lemma shows that the firm can limit itself to policies that aresuch that both restrictions share the same highest price.Lemma 4.3.1: If two restrictions are mutually exclusive, then there exists an optimalpolicy of the following form:{m; (qrt,i,• , qrt,m), t = 1,2; (q, . . . , qn)},where Pm ZS the common highest restricted price for both restrictions, q is the quantityavailable at price p with restriction t and q, is the quantity available at the unrestrictedprice Pk after the sales of the product with restrictions.Proof: Since 7tj is strictly increasing, then by Theorem 3.3.5 of Chapter 3, I know thatfor each restriction alone, the firm only needs to consider policies that sell restricted unitsfirst. Let ,pmi} be the set of the type-i restricted prices and {p,... ,pm2} be theset of the type-2 restricted prices. Without loss of generality, I assume that m1 < n-i2.Chapter 4. Pricing Models with Two Types of Restrictions 127So this leads to policies of the following form:{ : i = 1,...,m1;(p,qT2j,q) :j = ml,...,m2;(pk,qk) : k =m2,...,n},where• for 1 < i < m1, the firm allocates qri,j and q,.,j units at price pj with type-irestriction and type-2 restriction respectively;• for m1 <j <m2 and at the price level p,, the firm sells qr2,j type-2 restricted unitsfirst and the q3 unrestricted units; and• for m2 k n, the firm sells unrestricted units only.After the sales according to {(pj, q,.-,j, qr2,i) : i = 1,••• , mi} up to price level Pmi, let• d1 be the residual demand for type-I restricted units at price p3 for j m andI = 1,2;• dmi ,j be the total residual demand at the unrestricted price level p3 for j m1;and• 7m1 ,tj be the percentage of consumers in the residual market who can not accommodate type-I restriction at price Pj.Then it is straightforward to check that for I = 1,2,Sod1 = (1—ii)D, withdmj,j — — —Note that— (1 —— (1— 772)(1—1— 7m1,2j— dmi,j — — —i12D2 — 1 —(1 — )(1——(1— 2)(1— 72j)is strictly decreasing if and only ifChapter 4. Pricing Models with Two Types of Restrictions 128(1—71,+711 is strictly increasing,which follows from the assumption that both 713 and 723 are strictly increasing. Therefore7mj,2j is strictly increasing in j; and similarly, 7mj,ij is also strictly increasing in j.Now, our pricing policy indicates that after the price level Pmi, the firm will stopselling type-i restricted units. In other words, the firm only sells type-2 restricted andunrestricted units in the residual market. But as illustrated above, for any given valuesof 771 and 772, 7m2,is strictly increasing in j. Therefore in the residual market only withtype-2 restriction, we can use Theorem 3.3.5, which shows that there is another allocationplan of the following form:{(p3,q2): j —m1,••.,m’; (p,,q) : k = m’, •.,p},which will perform as least as good as the original policy:{ (P3, q2,i, qj) : j m1,• , m2; (pk, q) : k = m2,• , n}.Therefore, we can treat Pm’ as the common highest price level for both restrictions.3Thisproves the lemma. LIWith the help of Lemma 4.3.1, I know that the firm’s pricing problem can be formulated as the following mathematical programming:3The main purpose of using the common price levels for both restrictions is to simplify the formulationthat leads to simple optimal pricing policies. This is purely for technical reasons. On the other hand, itis possible though that the two restrictions may not have a common last active price level.Chapter 4. Pricing Models with Two Types of Restrictions 129Max p(qri, + qr2,i) + ZD=mp2q3s.t.‘m < i (4.16)L_.ii=1 D’’ —ç-m qr2,i <1 (4.17)/32 = L..jjl 2 —Ej=mD1—f3”—32 1 (4.18)Ei(qri,i + qr2,i) + E,=m q3 qqrl,i,qr2,i,qj O,Vi and Vj.I will call this formulation ME-Model, where “ME” stands for mutually exclusive. Againlet R(m; -y), or R(m) in brief, be the optimal objective value of the above ME-Model.The following theorem demonstrates that there also exists a linear programming characterization for the optimal revenue value— maxl<m<n R(m).Theorem 4.3.2: Let (m) be the optimal objective value of the following linear programming (named the Tight ME-model):Max Zi + qr2,i) + YZj=m pjqjs.t.mq-— 1i=1 —m qr2,i— 1qj 21/_.j=m D—D’-Di(qr1,i + qr2,i) + Zrm q3 qO,Vi and Vj.Chapter 4. Pricing Models with Two Types of Restrictions 130Thenmax R(m) = max R(m). (4.19)1<m<n 1<m<nProof: To prove (4.19), it suffices to show that for any m, there exists another th suchthatR(m) <(th). (4.20)Let {(qrt,i, , qt,,,j, t = 1,2; (q,” . , q)} be an optimal solution to the ME-model withrespect to m. If at this solution, constraints (4.16), (4.17), (4.18) are all binding, then(4.20) clearly holds for = m.By Theorem 2.4.1 of Chapter 2, I know that if either one of the constraints (4.16)and (4.17) is not binding, then exactly one of these, qn can be strictly positive.Let us call it q3, which, according to Theorem 2.3.5, must satisfyq; = D3 — — /32D. (4.21)Now take th= j and define a new pricing policy as follows: for t = 1,2,q,j ifl<i<m,rt,i 0 ifm+1i7—1,(1— /3t)Dt; if i =and—— D2 if j = 3,qj =1 0 otherwise.It is straightforward to check that the policy {j, 1 i th, t = 1,2; j, ñì. <j < n} isa feasible solution to the Tight ME-model associated with ñi. Furthermore, by (4.21) IhavenJ?(nz) >pi(’r1,i + r2,i) + =i=LChapter 4. Pricing Models with Two Types of Restrictions 131therefore (4.20) is true. Hence the theorem is proved as required. 0As an immediate consequence of the above theorem, I haveCorollary 4.3.3: If two types of restrictions are mutually exclusive, then there exists anoptimal pricing policy that consists of at most four different prices, which is characterized by a series of linear programming problems.Finally, it is straightforward to extend the above model to the case of multiple mutuallyexclusive restrictions.4.4 Pricing Problem by Using Two General RestrictionsRecall that 7(p) is the percentage of consumers who cannot accommodate the type-frestriction, and y(p) is the percentage of consumers who cannot purchase the product ifboth restrictions are attached to the product. Then we should notice the followings:• If y(p) = 7i (p) or -y(p) = 72(P)), then it leads to the case of two nested restrictions;• If 7(p) = 1, then it becomes the case of two mutually exclusive restrictions.On the other hand, it is clear that 7(p) 7t(p) for t = 1, 2. Therefore, it is always truethatmax(-yl(p),-y2(p)) (p) 1.In the previous two sections I have discussed two extreme cases of two types of specificrestrictions. In this section, I will discuss the case of two general product restrictions,that is,0 < 1—7(p) <min(1— i(p), 1 — 72(p)). (4.22)Chapter 4. Pricing Models with Two Types of Restrictions 132This implies that there are some consumers who can purchase the product even if bothrestrictions are attached. In this section, I will oniy consider the following type of pricingpolicies:• the firm first sells some units with two restrictions attached;• the firm then offers type-i and type-2 restricted units at certain orders; and• the firm finally offers the rest of units without any restrictions.I need to introduce additional notation here:• I will call the restriction consisting of type-i and type-2 restrictions the type-Srestriction;• let Dr3(p) = (1 — -y(p))D(p) and denote D3 = Dr3(p) for i i,... , n; and• defineD’31—cj=—-,fori=i,•..,nandt——i,2,which measures the percentage of consumers in the market demand for type-t restricted product who can accommodate both restrictions.Throughout this section, I will assume the following monotonicity conditions:•-y is strictly increasing in i for t = 1, 2; and• is strictly increasing in i for t = 1, 2.On the other hand, it is easy to check that(1-x)(l — = (1 — 72i)(i — c2) = 1 — 7(p) = 1 — 7, (4.23)Chapter 4. Pricing Models with Two Types of Restrictions 133which implies that only four of these five sets of parameters can be independently given.As a consequence of this, I know that-y is also strictly increasing.Let {(p, : i = 1,... , m1} be the allocation plan for the type-3 restricted units,where 1 m1 n. Denote d,,as the residual demand for type-3 restricted product atprice P, for all j m after the sales according to this allocation. Then I know thatm1= (1—for all j m1.And it is clear that the feasibility condition for type-3 restricted units is given by/332>31.Suppose that /33 = 1. It implies that in the residual demand market for restrictedproduct, each consumer can at most accommodate one type of restriction. Or equivalentlyspeaking, after the sales of type-3 restricted units, all these who can accommodate bothrestrictions are satisfied. Therefore, type-i restriction and type-2 restriction are mutuallyexclusive in the residual demand market. By the discussions in the previous section, Ican specify the remaining part of a pricing policy as follows:{ (pi, qi,i, qr2,j) : = . , m; (pk, q) : k = m, . ,where 1 <m1 <m < n.In order to use Theorem 4.3.2 in Section 4.3, I need to check the monotonicity conditions on the impact of type-i and type-2 restrictions on the residual market. Let 7m1,tjbe the percentage of consumers in the residual demand market who can purchase type-trestricted product at price p3 (j mi) after the sales of type-3 restricted units. Since/33 = 1, it follows that by (4.23)flrt rr31 — — 3 — 11 \ 7i 7i — Yti 1 7tj17m,tj flr3”7) -—U, YjChapter 4. Pricing Models with Two Types of Restrictions 134Therefore, for t = 1, 2, 7m1,tj is strictly increasing in j if and oniy if-yj/-y,j is strictly increasing in j,which means that as the price increases, the relative ratio of the percentage of consumerswho cannot buy type-t restricted product to the percentage of consumers who can notbuy type-3 restricted product is strictly increasing. A sufficient condition is thatyj—yj is strictly decreasing in j,DTl Dr3or, equivalently, ‘ ‘ is strictly decreasing in j, which implies that the percentageof consumers who can accommodate type-t restricted product, but not type-3 restrictedproduct, is strictly decreasing. This sufficient condition is stronger than required; but ithas an intuitive interpretation.Summarizing the above discussions, I have the following theorem.Theorem 4.4.1: Suppose that the firm only consider the type of policies such that /33 = 1.Further assume that fort = 1,2,is strictly increasing in j,then the firm’s pricing problem can be formulated as the following linear programming (Iwill call it Tight-G Model, where “C” stands for “general”):Max + Lmjp3(qri,j + qr2,j) + Z=m Pkqks.t.‘ç-mi qr3,iL.=i D3 —qi,2— 1Ldj=ml D’D3”—qr2,j— 1Lj=m1D2-D —Chapter 4. Pricing Models with Two Types of Restrictions 135gk—1i_..k=m Dk—(D’—D3) (D2— —7Th‘1r3,i + Zj=m (qri,j + qr2,j) + Zk=m qi qqrt,i 0 and Jk 0.Proof: It follows from the above discussion that if ti/t is strictly increasing, then 7m1,tjis strictly increasing for t = 1,2. On the other hand, since /33 = 1, I know that in theresidual demand market, type-i and type-2 restrictions are mutually exclusive. Hence,the rest of the proof follows immediately from Theorem 4.3.2. DTherefore, if /33 1, then the firm needs to offer at most five prices to maximize itsrevenue. The Tight-G model provides a benchmark on the optimal revenue value forany other general models since it at least provides a lower bound for the optimal revenuevalue. In Chapter 2 and Chapter 3, I have demonstrated that for the case of one typeof restriction, an optimal solution for a tight model remains to be optimal in the generalclass of pricing policies. It is not clear at this moment whether or not Tight-G model infact provides an optimal solution in the general context.The main issue is whether we can find an optimal solution such that /33 = 1. This isnot an easy task since if /3 < 1, then it is not even clear how to specify the remainingpart of pricing policy because• /33 < 1 implies that if both type-i and type-2 restricted units are offered at thesame price level, some consumers can buy either one of them;• On the other hand, the impact on the residual demand market of selling type-irestricted units first and type-2 restricted units thereafter at the same price levelis different from the impact on residual demand market of selling type-2 restrictedunits first and type-i restricted units later at the same price, which will complicatethe process of updating the residual market.Chapter 4. Pricing Models with Two Types of Restrictions 136So the key question is: which type of restricted units will be offered first if both restrictedunits are offered at the same price? One way to get around this difficulty is to considerpolicies of the form:{(pj,qi,,qr2,j,qi,j) :j — mi,.”,m},where 1 m1 m n. This type of policy has the following technical properties:• it allows the firm to sell some type-i restricted units first, then some type-2 restricted units and finally some type-i restricted units, all the same price level;• if at price p3 the firm wants to sell type-2 restricted units first, it can do so byletting q’1 = 0; and• it is possible possible for the firm to sell type-i restricted units first on a price setand type-2 restricted units on the remaining price set, or vice versa.Consequently, I have the following definition of a pricing policy when using two generalspecific restrictions:Definition 4.4.1: For two general restrictions, a pricing policy is specified by{(pi,qr3,i) : i = l,...,m1;(pj,q’j,qr2,j,):j = ml,...,m;(pk,qk) : k = m,.”,n},where 1 <m1 <m <n.Let us now analyze a special case that leads to Tight-G model.Theorem 4.4.2: Assume that 7ij/7j and 72j/7j are strictly increasing in j. If one ofthe following two conditions holds:(i) is strictly increasing in j,Chapter 4. Pricing Models with Two Types of Restrictions 137(ii) ‘“ is strictly increasing in j,7jY2jthere exists an optimal policy that is characterized by the Tight-C model.Proof: By Theorem 4.4.1, I know that it suffices to show that there exists an optimalpolicy such that /33 = 1. It is evident that conditions (i) and (ii) are perfectly symmetric.I will prove the result by assuming (i).I will prove the theorem through two steps. In the first step, I will formulate thepricing problem into a mathematically programming formulation. The main techniquehere is to show that I need only to consider a small class of policies which contains anoptimal policy. I will use the main results in Chapters 2 and 3. And the second stepis to further simplify the model formulation so that I can have a linear programmingformulation, which will lead to a simple optimal pricing structure.Technically speaking, I can start with policies specified by Definition 4.4.1. Butbecause with the additional assumption in the theorem, I can actually focus on a smallerclass of policies that makes the model formulation much simpler and easier. First of all,it is clear that any policy will start with the sales of type-3 restricted units.4 Let thepart of allocation for type-3 restricted units in an optimal policy be given by:{ : = ,mi},where m1 is an integer such that 1 m1 n. The key is to analyze the demand structurein the residual market after the sales of type-3 restricted units. As above, let/)32=1 2I will use the following notation in the residual market: let j > m141t is possible though that the firm may not sell any type-3 restricted units.Chapter 4. Pricing Models with Two Types of Restrictions 138• is the residual demand for type-t restricted units at price p2 for t = 1,2,3,which are given by:d1,3 (1 — ,63)D; and d,3 = Dt — /33D,r, for t = 1,2.• d1 is the total residual demand for restricted product at price Pj (including thedemand type-i restricted product and the demand for type-2 restricted product),which is given by:,lrl ,p2—— (flr2 — flr3 ,7r1mj,j — mi,j mj,j mi,j — ‘S j j ) ‘ m1,j= (D1 — D3) + d, = D3n1 + D2 — (1 + /33)D.• dm, ,j is the total residual demand for unrestricted product at price Pj, which isgiven by:dmi,j = (D3 D’ D2 + D3)+ = —3D.d3• Let l7mjj=2-.dm,jdlt• Let 1—7m,tj=-.fort=1,2.• Let 1— m1,tj =m1Now if /33 = 1, then the rest of the proof follows from Theorem 4.3.2. So I onlyneed to focus on the optimal policies such that /33 < 1. Let 7m,j be the percentage ofconsumers in the residual market for restricted product who cannot accommodate type-irestriction. Then,3r1 flrl Q nr3 flrl Q nr3Umi,j—P3L)—/J3LI— mi,1j d1 = DJ1 + DJ2 — (1 + /33)D = (Di’ — /33D)+ (D2 —Similarly, let ?)mi,j be the percentage of consumer in the residual demand market forunrestricted product who cannot accommodate any restrictions, then—— D’ + D2 — (1 + /33)D1mi,j ——flr3Urn1,j JJS,—Chapter 4. Pricing Models with Two Types of Restrictions 139— D3 (1 —Dç2 —+D2 — D3D’’ D3 D”3———--i- (1—/33)2— D D________________+— j,r2j,3D, D3— 73“+(i—3 1—cy2j3)7j — 72j 2jif(73 — 71j — 72j)1 — /33(1—Claim 1: Both 7lmi,ij and 7mi,j are strictly increasing in j.Proof of Claim 1: Clearly, 7mi,is strictly increasing if and only ifD2D’3 is strictly decreasing.On the other hand, note thatflri 0 flr3U3 /J3UJ —______D2-D -which is strictly decreasing sinceis strictly decreasing by assumption; and7j72j1 is strictly decreasing because cr23 is strictly increasing.For 1mi ,j, note thatD’ D2DJ1 + DJ2 — (1 + /33)D— D, D, 3) D3D’33—8—— (1 7i)+(l 723)(1+/33)(1 -y)—1—/33(1—73)—__ ______—1—/33(1—73)which is strictly decreasing if and onlyis strictly decreasing. But(-y— 713— 723) ( — 71j 723)1—/33(1—73)— (1+/33)73—/33Chapter 4. Pricing Models with Two Types of Restrictions 140= 7 (i’ 723)(l+3)-—/33 73 73— 1 / -“— 1 ++ (1 + 3)7j ——7j — 7j)which is indeed strictly decreasing since• -yj is strictly increasing; and• both -- and -- are strictly increasing.7,This proves Claim 1.Now by Theorem 3.3.5, the monotonicity properties of 7lmj,1j and 7mj,j indicate,• in the residual market, the firm only needs to consider policies such that type-irestricted units, if allocated, should be sold before the sales of type-2 restrictedunits.Therefore, the firm can limit itself on the following type of policies in the residual market:{(p, qri,j) : j = m1,• , m2; (pi, qr2,l) : 1 , m; (pk, q) : k = m,• ,where m1 < m2 < m n.By using this type of policies, I know that the feasibility condition for type-i restrictedallocations ism2 m2= qri,j — qri,j < i—d—— —1 3—rn1After the sales of type-i restricted units, the residual demand for type-i restricted productisdm2,i = (1 — B1)(D’ — i33D),1 m2;and the residual demand for type-3 restricted units isfl2,1 = (1 — ,8)(1 —3)D,1 > m2.Chapter 4. Pricing Models with Two Types of Restrictions 141Consequently, the residual demand for type-2 restricted product is given by— ‘rr2 r3 r3m2,1 ‘‘1 — ) m2 1=D—(/3+/33//3)D.Therefore the feasibility condition for type-2 restricted units is characterized by:qr2,1 <1.k—’D2—(/3i+/33i/)DT —Now after the sales of type-2 restricted units, the residual demand for type-3 restrictedproduct isd,k = (1 —/32)(1 —/3)(1 —/33)D,k m; andthe residual demand for type-2 restricted product isjr2 11 i2 jr2Um,k = — P2)Um,k= (1 — /32)(D—(/3k + /33 — /313)D for k> rn.Thus the total residual demand for type-l and type-2 restricted units is given by:— jr3 i jr2(Lm,k— m2,k — Um2,k) t Um,k= (1 —,B)(D’ —D3)+(1 —i32)(D — (/3i+/3/3i)D;and the total residual demand for unrestricted product is:dm,k = (Dk — — D2 + D3) + d,k= Dk — /31(D’ — D3) — — (1— /32)C81 + /33 — /313)D= Dk - - /32D + (/31/32- (1- /3)(1 - /32)/33D.So the feasibility condition for the final unrestricted units isnqk <1k=m Dk — — /32D + (/31/32 — (1 — i3)(1 — ,82)/33D —Summarizing these discussions, I get the following formulation for the firm’s pricingproblem:Chapter 4. Pricing Models with Two Types of Restrictions 142Max + Z2mj pjqri,j + Zm2pjqr2,l + Zk=m pkqks.t./33 3 1 (4.24)/3 — Z2m, D—33 1 (4.25)— qr2,zP2= 2..1=m2Dr_(pi+/3thpa)Dr —Dk —31D—2D+U31132—(1—131)(i—p2/3D 1 (4.27)m1 m2 m nZD1=q,-3, + Zj=mi qri,j + Z1=m2 qr2,1 + k=m ‘m qqrt,i 0 for t = 1, 2, 3, and q 0.I now show that there exists an optimal solution to the above formulation such that= /3i = /32 = 1,which will reduce the above formulation into a linear programming.Claim 2: There exists an optimal policy such that /3 = /32 = 1.Proof of Claim 2: Let{ (pg, q3,) : i = 1,... , m1; (pj, qri,j) : j = m1,. . , m;(pl,qr2,z) : 1 = m2,•• . ,m; (pk,qk) : k = m,• . ,n}be an arbitrary optimal solution to the above formulation. If /33 = 1, then I know thatour problem leads to the case studied in Theorem 4.4.1. I now consider that /33 < 1.First of all, the monotonicity properties of ?7m,1j and TImi,j help us to formulate thepricing problem. On the surface, these properties should also lead to the model of twonested restrictions. This is not so since• the firm sells type-2 restricted units after the sales of type-i restricted units;Chapter 4. Pricing Models with Two Types of Restrictions 143• the set of those consumers who can buy type-i restricted product is not a subsetof the set of these consumers who can buy type-2 restricted product.On the other hand, if ,83 < 1, then there is a positive residual demand market for type-3restricted product. Most importantly, observe that• type-S restriction is nested into type-2 restriction;• both amj,2j and 7m1,2j are still strictly increasing; and• during the sales of type-i restricted units, the residual demand for type-S restrictedproduct is proportionally reduced.Therefore, in the residual market in regard of type-3 and type-2 restrictions I can usethe results in Theorem 4.2.1, which says that after the sales of type-i restricted unitsthat have included the residual demand for type-3 restricted product, there is no residualdemand for type-3 restricted product. This implies thatjr3— i-i 1 o ‘ -‘r3 ,tLm2,1 — — I_’1)11 — /J3)L1 = U,which leads to = 1.After the sales of type-i restricted units with /3 = 1, I know that the residual marketconsists of exactly two types of consumers — theoe who cannot purchase type-2 restrictedunits and those who can. Let 7m2,21 be the percentage of consumers in the residual marketwho cannot accommodate type-2 restriction after the sales of type-3 and type-i restrictedunits according to the allocation plan. Then since /31 = 1, it follows that,jr2 flr2 flr31 — 1’rn2,I —1— 7m2,21 ——— flrlm2,1 j=1 — (1—Chapter 4. Pricing Models with Two Types of Restrictions 144which is strictly decreasing since 2!2- is assumed to be strictly increasing for I = 1,2.Therefore m2,2l is strictly increasing. Then by Theorem 3.3.5 and Theorem 2.4.2 itfollows that the firm only needs to consider policies such that /32 1. In summary, Ihave shown that the firm can focus on the class of policies such that i3 = 132 = 1. Thisproves Claim 2.Using the results in Claim 2, I know that the pricing problem can be reduced to thefollowing form:Max + >T=2m1P3qri, + Zm2Pjqr2,1 + YZk=m pkqks.t.— -‘m1 q,j— 1 2P3 = L.=i iZTq,-i,— 1 4 29/_dj=m1 —__________—L.j=m2D2-D” —\fl—1 431/_ak=m Dk-D’-D2+3—m1 m2 m nZ— + Zj=m1 qri,j + YZ1=m2 qr2,1 + k=m qqrt, 0 fort = 1,2,3, and q, 0.Note that constraints (4.30) and (4.31) are no longer related to /3, the only parameterthat causes a non-linear constraint (4.29). Furthermore, I have the following observations:• The demand for type-i restricted product will be exhausted after the sales accordingto the plan: {(pj,qr3) : i = ,mi;(pj,qri,j) : j = mi,...,m2};• o, the percentage of consumers in the market demand for type-i restricted productwho can also accommodate type-2 restriction, is strictly increasing; and• The total sales for type-3 and type-i restricted units is bounded bym nqr1 = q — qr2,l — qk,1=m2 k=mChapter 4. Pricing Models with Two Types of Restrictions 145which can be considered as the capacity for type-3 and type-i restricted units sinceit is independent of the allocation plan for type-3 and type-i restricted units.Hence this leads to a well-defined subproblem that has the same structure as the problemaddressed in Chapter 2. Then using Theorem 2.4.2 and Theorem 3.3.5, the firm againonly needs to focus on policies having the property of /33 = 1. Therefore I prove thetheorem, since the rest follows from Theorem 4.4.1. DNote that the above theorem is aimed at the case that it is advantageous to delay thesales for one type of restricted units. There may be other cases that also lead to Tight-Gformulation.4.5 Summary — Pricing by Using Two Types of RestrictionsThis chapter addresses the issue of pricing perishable inventories by using two types ofrestrictions. After studying two extreme cases, which are two nested restrictions and twomutually exclusive restrictions, I have also discussed the general case. I here provide thefollowing additional insights:• Extensions from the basic BL-model to multiple restrictions are not straightforward.• A pricing model by using two general types of restrictions is capable of handlingfour different types of prices.• Extra demand structures are needed in order to obtain simplified optimal pricingstructures.• Simplified optimal pricing structures can be characterized in a similar manner asin the basic BL-model.Chapter 4. Pricing Models with Two Types of Restrictions 146I have shown that for the two extreme cases, there exist optimal pricing structuresthat consist of at most four prices. And for the general case, I present two situationswhere there exist optimal pricing structures that consist of at most five prices. All ofthese optimal pricing structures are characterized by linear programming problems, whichmake these models tractable in application.Chapter 5Airline Pricing by Using Membership and Product Restrictions5.1 IntroductionChapter 2 and Chapter 4 developed a series of pricing models by using artificial restrictions. These models gave us conceptual tools for analyzing the use of restrictions asa mechanism. In this chapter, I will present an application, in a convincing way, inthe context of airlines. It is well-known that all airlines have special arrangements withcertain clientele-specific rates. It is important to note that the fares for these specialconsumer groups are not available to the general public through travel agents. In mostcases, these rates are internally controlled by the airlines. These opportunities representrevenue potentials which the airline may be able to exploit. It is important to notice thatany special clientele represents some kind of membership which covers only a small ofportion of the general population. Because of this, from the pricing point of view, thesememberships are in fact restrictions because their existence will prevent some consumersfrom being able to purchase the product at a price that is available to members only. Onthe other hand, the availability of the seats for these clientele needs to be controlled and agood yield management system must be capable of handling the presence of these specialclientele. For ease of presentation, I will use the following convention on terminologies:• Any consumer in a special clientele is called a member-i• I use fares and tickets interchangeably with prices and units;147Chapter 5. Airline Pricing by Using Membership and Product Restrictions 148• Restricted fares and unrestricted fares are understood as fares that are available tothe general public, which means that all consumers are allowed to purchase thesefares; and• These fares that are specifically available to members are called either restrictedmembership fares or unrestricted membership fares.Note that there is clearly an asymmetry between members and non-members on the accessto the reservation information. A member can purchase a ticket either at a membershipfare or at a public fare, whichever is less; but non-members can only purchase the publicfares since the membership fares are not available to them. In this sense, we should treatthe existence of membership as another restriction in addition to the product restrictions,such as, Advance Booking, Saturday Night Stay, and No Refund. As a result, the airlinefaces two types of restrictions. The purpose of this chapter is to use the results developedin the previous chapters to illustrate how they can be applied in this context and whatthe implications are. In particular, I will discuss several different situations where aspecial membership deal makes sense to the airline. As a result of this analysis, it willimprove our understanding about the operating environment for each of these situations.Consider that an airline has a scheduled flight with a fixed capacity. Assume that theairline has a private agreement with a special group under the following simple terms:• As a member of the group, the traveller can purchase a ticket at a membershipprice that will be lower than the comparable price offered to the general public;and• The availability of these special fares are limited and members are served on thefirst-come-first-serve basis.Chapter 5. Airline Pricing by Using Membership and Product Restrictions 149When an airline decides to use both membership restriction and product restriction,the airline can offer at most four different types of fares: restricted membership fares,restricted fares, unrestricted membership fares and unrestricted fares. Because of themembership agreement, we know that:• Restricted membership fares, if offered, should riot be higher than the planned restricted fares; and• Unrestricted membership fares, if offered, should not be higher than the plannedunrestricted fares.In this section I will consider three common arrangements in terms of membership privileges: (1) Cheaper restricted fares only; (2) Cheaper unrestricted fares only; and (3)Cheaper restricted fares and cheaper unrestricted fares. The aim of this chapter is tofind out conditions that lead to a proper choice among these arrangements. This is ofgreat importance since the airline needs to know under what kind of operating environment, a particular membership deal is worthwhile at the first place.1This chapter is organized as follows. Section 5.2 presents the basic model settingand notation. Section 5.3 addresses the airline fare pricing problem when members arepromised that they can by restricted tickets for less. I will give two examples that areconsistent with this scenario. Section 5.4 discusses the airline fare pricing problem whenmembers are offered cheaper unrestricted fares. In particular, I will examine the impactof corporate policy for interval travellers and conclude that such a corporate commitmentcan cost an airline a lot of money. Then Section 5.5 deals with the case that an airlinegives members both cheaper restricted fares and cheaper unrestricted fares. Sufficient1The readers should be warned that our model is purely tactical. When an airline negotiates a specialdeal with an interest group, there are many other issues involved which can not be directly addressedin our model. Hopefully, our model will be helpful for airline managers to understand the impact ofdifferent types of membership deals.Chapter 5. Airline Pricing by Using Membership and Product Restrictions 150conditions are identified for simple optimal fare structures. Examples are also used togive further support. The last section is a summary.5.2 Model Setting and NotationLet D(p) be the total market demand function at price p (unrestricted), among whichDM(p) of them have the membership. Denote DN(p) to be the demand from the non-member group. Then,Dv(p) = D(p)— DM(p).Naturally, since membership is a form of restriction, we let 7M(p) be the percentage ofconsumers in the demand market at price p who are non-members. Then clearly,DM(p)1M(p)tD(p)On the impact of product restriction, we note that membership restricted fares are morerestrictive that the general restricted fares. Therefore I use the following notation:• 7Mr(P) is the percentage of consumers in the total demand market who cannotpurchase the product at the restricted membership fare p, that isDMr ()17Mr(p)_ D(p)where DMT(p) is the demand for the product at the restricted membership fare p;• 7r(p) be the percentage of consumers in the total demand market who cannotaccommodate the product restriction at price p, that isDr(p)1—7r(p)where D’ (p) is the demand for product at price p with product-restriction oniy.Chapter 5. Airline Pricing by Using Membership and Product Restrictions 151Further assume that D(p) and DM(p) are non-decreasing step function defined on theset of fares {pi, ,p,, }. As usual, for i = 1,... , n, denoteYM,i = yM(pi);yMr,i = yMr(pi);yr,i =and= D(p); D = D’(p); D = DM(pj); DrT = DMT(p1).Throughout this chapter, I will assume thatYMr,i arid Yr,i are strictly increasing in j.Additional conditions may be assumed in each of the following three sections in order toobtain simple optimal fare structures.5.3 Cheaper Restricted Membership Fares OnlyLet us first analyze the case where an airline only offers the members lower restricted fares.As a consequence of this, as a member, the traveller has three types of fares to choosefrom: restricted membership fares, (public) restricted fares and (public) unrestrictedfares. On the other hand, any non-member can only purchase the public restricted faresand the public unrestricted fares.Now if I call the restriction consisting of membership restriction and product restriction as the type-S restriction and the product restriction as the type-i restriction, thenit is clear that type-S restriction is nested into type-i restriction. This allows me to usethe model discussed in Section 4.2 of Chapter 4 to handle the problem here. But in thepresentation that follows I will avoid to use the notion of type-i and type-3 restrictionshere after. I put them here for the purpose of illustration only.Chapter 5. Airline Pricing by Using Membership and Product Restrictions 152Let Mr,i be the percentage of consumers in the demand market at restricted price Piwho are non-members. Then 1—aMr,i= for i > 1. Hence 1 7MT,i = (i 7r,i)(1 —aMr,i), which indicates that type-3 restriction is indeed nested into type-i restriction.On the other hand, since restricted membership fares are supposed not to be higherthan the general restricted fares, I can define a pricing policy for the airline as follows:{ (pi, : = 1,... , m2; (pj, qj) : j = m2,••• , mi; (pk, q) : k m1,• .. ,where is the allocation to members at the restricted price Pi for 1 i m2, qr,jis the allocation to general public at the restricted price Pj for m2 j m1, q isthe allocation to the general public at the unrestricted price Pk for m1 k n, and1 m2 m1 <n. Furthermore, I also know from my discussions in Section 4.2 that theairline’s pricing problem can be formulated as the following N-model:Max +=m2 P” + k=m1pkqks.t.— ‘ç.m2 Jj4,-P2 = —— v-”mj q,-,, < 1P1 = L..’j=m2 D-f32 —v-’n qk <1/_jk=m1 Dk—3lD—(1—/3l)fl —m2 m1 nzi=1 + Zj=m2 + YZk=m1 q qqMr,i,qr,j,qk O,Vi,Vj, andVk.Rephrasing Theorem 4.2.1 in Section 4.2, I get the following proposition:Proposition 5.3.1: Suppose that the airline only offers lower restricted membershipfares, and that both and aMr,i are strictly increasing, then• The airline needs to offer at most four fares to the market; andChapter 5. Airline Pricing by Using Membership and Product Restrictions 153• The optimal fare structures are designed in such a way that the demand for restrictedmembership fares will be exhausted right after the sales of these restricted farestargeted to members.I now discuss the implications of the assumptions in the above proposition. Clearlythe assumption that is strictly increasing is a basic requirement for an effective productrestriction. For the assumption that ‘Mr,i is strictly increasing, it simply means that asthe price increases, among these who can purchase restricted product, the percentage ofthe members is strictly decreasing. As demonstrated in Theorem 4.2.1, this assumptionplays a key role in the process of obtaining a simple pricing structure, or a simple farestructure in our context here. Basically, it allows us to focus on the type of policies withthe property that the demand for restricted membership fares will be exhausted rightafter the sales of these restricted fares targeted to members.Offering lower restricted fares to special clienteles is a very common approach forairlines dealing with membership issues. I here present two examples that are consistentwith this particular approach.Example 5.3.1: Tour OperatorsConsider vacation operators which sell tour packages or vacation packages with manypredetermined destinations. One of the key ingredients in these packages is the air fare.Most packages announce that they obtain much cheaper fares than the general fares,which implies that the vacation operators must have reached an agreement with a particular airline. From an airline point of view, these vacation operators create substantialrevenue opportunities since:• A successful operator usually offers different packages all year round, which areplanned many months in advance;Chapter 5. Airline Pricing by Using Membership and Product Restrictions 154• It is relatively predictable on the size of each package;• An airline needs not to deal each individual traveller in each package;• In most cases, air fares are the main part of the cost for each package, which impliesthat these operators who can strike good deals with some airlines will more likelybe the successful ones;• An airline can play a crucial role on flight schedule arrangement of the package ifthe airline indeed makes concessions on air fares; and• Almost all vacationers who arrange their vacations through operators will not flyat an unrestricted fare, which is usually much higher than an ordinary restrictedfare.Therefore, for these airlines which are willing to take that extra step to negotiate withvacation operators and are willing to give additional discounts on air fares, additionalrevenue can be generated. But the airline must address the following issues:• Given the time condition by the vacation operator, the airline needs to tell themwhich flight may be available;• If there are many flights that satisfy the time constraint and capacity constraint,then the airline can either give its own best choice on the basis of most revenue ornegotiate with the vacation operator on a common choice; and• On the other hand, if the airline cannot find an appropriate flight, then it is possiblefor the airline to suggest an alternative flight.All these issues involve delicate work on pricing and inventory control. Our above modelwill be helpful in this regard. First, since the airline only deals with the vacation operatorChapter 5. Airline Pricing by Using Membership and Product Restrictions 155with a clearly defined capacity requirement, our model at least gives a bottom-line valuefor the total revenue for this particular part of the capacity if these seats are offered tothe operator. It will also give the airline a base for any negotiation with the operator.Example 5.3.2: Corporate Retreat ProgramsMany corporations have annual retreat programs to reward their high performanceemployees. Every year, the management will select a group of employees whose contributions are better than most of the other employees. The company will send this groupto a special place for a couple of days of retreat. Typical examples include companiesfrom the insurance, lodging and real estate industries. The main characteristics for thesetypes of programs are:• The company will take care of all expenses and make all the arrangements;• Air fare, if the program decides to use air transportation, is a major part of thetotal expense, which is tax deductible;• These programs are usually arranged for weekends; and• Some top executives in the company usually need to show up at the program; andthese executives are in fact captives of the unrestricted fare or the first class faresince they simply cannot make an early commitment.Many large corporations use charter service for their retreat programs because it is flexibleand convenient. But major airlines can also capture a large market share of this business.Airline managers who understand the impact of these programs on the capacity andrevenue should be able to explore this market segment.Chapter 5. Airline Pricing by Using Membership and Product Restrictions 1565.4 Cheaper Unrestricted Membership Fares OnlyI now discuss the case where an airline considers oniy to offer lower unrestricted membership fares. There are many situations that lead to this type of practice. For example,an airline’s employees can fly wherever and whenever they want to for free subject to theavaibility of seats. Most airlines also have an internal policy that will give an employeea guaranteed seat and all the flexibility if the employee pays a small fraction of the unrestricted fare. This is of course a result of its internal corporate policy. But nevertheless,it has a direct revenue impact. Another example may be a big consulting corporationwhich constantly has its consultants on different assignments all the time. These travellers require the maximum flexibility. As a result of this, most of them must pay thesubstantially higher unrestricted fares. On the other hand, as modern telecominunication technology becomes more advanced and much cheaper, many companies have cuttheir travel budget to cope with the high cost of air travel. From an airline’s point ofview, these travellers are part of the frequent travellers. And increasing these travellers’frequency will create substantial revenue impact for the airline. Technically speaking,an airline may consider negotiating a special travel deal with such a corporation undercertain provisions, such as a minimum commitment on the total number of trips. Thegeneral feature for this type of membership demand structure is that there is no demandfor restricted fares from members. Therefore, I can directly use the ME-model developedin Section 4.3.But the case for internal travellers is slightly different and in fact quite interesting.It is well-known that some internal travellers, even those who can fly free, are willing tobuy an unrestricted ticket at a price only lower that the (public) restricted fare in orderto get a guaranteed seat.2 In other words, we have the following:course, the majority of internal travellers are free-riders. But technically speaking, these travellershave no impact in airline’s revenue. They also have no negative impact on airline’s operation since theseChapter 5. Airline Pricing by Using Membership and Product Restrictions 157• = 0, or equivalently, 7Mr,i = 1 for all i; that is, there are no internal travellerswho are willing to pay an restricted fare even if it is very cheap; and• A pricing policy is in the form of{ (p, : = l,• . . , mi; (pd, : j = m1,• . , m; (pk, q) : k = m,• ,n},where q is the allocation to members at the (unrestricted) price level p for1 <i m1, qr,j is the allocation to the general public at the restricted price pj form1 <j <m, and qj is the allocation to the general public at the unrestricted pricepk for m < k < n.In summary of the above discussions, we have the following result.Proposition 5.4.1: If there is no demand for restricted tickets from the membershipgroup and the airline is committed to offer unrestricted membership fares that are lowerthan restricted fares, then the airline ‘s pricing problem can be formulated as follows:Max + pjqr,j + YZm pkqks.t.\fl <1/_.ikm Dk—fllD-/32—m1qM, + Zj=mj qr,j + YZk=m q qqM,i, q, q > 0,Vi,Vj, and Vk.Proof: Straightforward. The details are omitted here. Eltraveller are called stand-bys, which means that their seats are not guaranteed. In other words, theymay be dumped during their trip. I here only focus on paying internal travellers.Chapter 5. Airline Pricing by Using Membership and Product Restrictions 158Even though the membership restriction and the product restriction are mutuallyexclusive in this context, the airline must address an additional constraint that requiresthe airline to offer the members unrestricted fares that are lower than the restricted fares.In the above formulation, it is easy to make j3 = 1. A natural condition for this is thatis strictly decreasing, which says that for non-members, as the price increases, thepercentage of those who are willing to purchase restricted tickets is strictly decreasing.Since the restriction is targeted to non-members, this condition is just a condition forthe restriction to be effective, as argued in Chapter 2.In order to obtain a simple optimal fare structure, the key is to make /32 equal toone. But the problem is that there is no natural conditions that generically guarantee/32 = 1. On the other hand, 132 = 1 is in fact a corporate policy since the airline willgive any employee a guaranteed unrestricted ticket if the employee pays a small fractionof the public unrestricted fare. This is good news for the above formulation, which canaccomplish a simple optimal fare structure because it reduces to a linear programmingproblem. But it may be bad news to the airline since it may cause a revenue loss. Forthese flights with low demand from internal travellers, the above formulation will likelylead to an optimal policy that is consistent with an optimal solution derived directlyfrom ME-model. But for some high demand flights or flights at busy travelling seasonssuch as Christmas, the commitment that /32 = 1 may prove to be costly. Therefore, theabove analysis calls for close monitoring on these flights and some modification may benecessary. One possible modification is to satisfy a fixed percentage of those who requesta booking on a flight. There are several benefits for this provision. First, it will stillreduce the above formulation to a linear programming, since /32 is a fixed constant and/31 can always be taken as one. Second, it is not hard to implement because it involvesthe airline’s own employees. For example, it can be implemented through a lottery drawby a computer among those employees who wish to book the same flight. Or it can beChapter 5. Airline Pricing by Using Membership and Product Restrictions 159implemented according to the seniority of employees.3 Another possible modificationis to ask internal travellers to pay for an unrestricted ticket at a restricted price. Thebest possible modification is to implement an optimal fare structure derived from theME-model, which may force some internal travellers to pay for an unrestricted ticket ata price that is higher than the restricted fares. In summary, the main point here is thatstrictly committing 2 = 1 may cost an airline a lot of money.5.5 Cheaper Restricted Membership Fares and Cheaper Unrestricted Membership FaresIn the above two sections, I have discussed several cases which indicate that it is sensible for an airline to offer either lower restricted membership fares or lower unrestrictedmembership fares. But in some situations, with enough information about the demandbehaviour from the members, the airline may consider offering both lower restricted membership fares and lower unrestricted membership fares. This leads to the pricing problemwith two general restrictions — one membership restriction and one product restriction.To see that we can obtain the same results as in Section 4.4, we might use the followingnotation:• we can call the product restriction a type-i restriction, which implies that a type-irestricted ticket is a (public) restricted ticket;• we can call the membership restriction a type-2 restriction, which means a type-.2restricted ticket is an unrestricted membership ticket;3Some airlines practice the rationing for the internal travelers in term of service seniority, whichmeans that these who are with the airline longer will have the first priority of obtaining confirmed seats.4But, in fact, I will avoid these generic notation in our discussions that follow. I present them herejust for the sake of consistency with the discussions in Section 4.4.Chapter 5. Airline Pricing by Using Membership and Product Restrictions 160• we can call a restricted membership ticket a type-S restricted unit since if it isattached with both type-i and type-2 restrictions.It is clear that for the case here, the airline has the flexibility of offering four typesof fares: restricted membership fares, (public) restricted fares, (unrestricted) membership fares, and (public) unrestricted fares. As argued at the beginning of this section,restricted membership fares, if offered, should not be higher than the public restrictedfares; and similarly, the unrestricted membership fares, if offered, also should not behigher than the public unrestricted fares. Clearly, the airline should always considerselling restricted membership fares first, since they are most restricted among these fourproposed fares. From Theorem 4.4.1 of Section 4.4 it follows that if the membership dealbetween the airline and the members requires that an airline must give a guaranteed booking for a restricted ticket from any member who requests for it, then the airline’s pricingproblem can be formulated as a simple linear programming, which shows that the airlineneeds at most five fare levels with four different types of ticket. Rephrasing Theorem4.4.1 in our context, we have the following proposition:Proposition 5.5.1: Suppose that an airline plans to use a membership restriction anda product restriction. Assume that• both -- and --- are strictly increasing in jYMr,j 7Mr,j• the airline will satisfy all bookings for restricted membership fares.Then the airline ‘s pricing problem can be formulated as the following l.p.:Max pjqMr,j + mj p, (qr,j + qir,j) + k—_m PkqkS. t.ç-mi— 1L.ij1 —Chapter 5. Airline Pricing by Using Membership and Product Restrictions 161_________— i/_ij=m1 D—D —___ __— 1J_Jj=mi D-D—v_’n gk—1Lak=m Dk_(D_D)_(Dj&D)—m1 n:i=i qMr,i + Zj=m1(qr,j + qMj) + YZkzm q qqM,j 0 and q 0where q,j is the allocation to the members at the restricted price j, qr,j is the allocationto the general public at the restricted price Pj, qM,j is the allocation to the members atthe unrestricted price P3 and q is the allocation to the general public at the unrestrictedprice pk.Let us analyze the two assumptions used in the above proposition. First, the conditionthat .2_ and —-- are strictly increasing in j is equivalent to requiring that7Mr,j YMr,jDM_DM’• both and D:_Dr are strictly decreasing,which implies that in the residual market where no members will buy restricted tickets,as price increases, the percentage of members is strictly decreasing and the percentageof consumers who can buy restricted tickets is also strictly decreasing. In essence, theseconditions are conditions of effectiveness for both membership restriction and productrestriction. They together provide sufficient information for an airline to decide whetheror not to offer cheaper membership (unrestricted) fares and (public) restricted fares.The second assumption is in fact a corporate commitment from the airline. It mayinvolve issues other than pricing. A good example is a bilateral agreement between twopartner airlines,5 where each airline gives a special treatment to its partner’s employeeswho use its passenger service. If an airline’s partner has such a commitment, this airline5Partner airlines nsually involve agreements on code-sharing and traffic-feeding.Chapter 5. Airline Pricing by Using Membership and Product Restrictions 62may have no choice but to give its partner’s employees the same treatment. Our modelshows that the airline will have a very simple optimal fare structure. On other hand,purely from revenue maximization point of view, such a commitment may have a negativeimpact on revenue. So it is really important for airline managers to understand that thebest result is such that the commitment is consistent with market behaviour. Theorem4.4.2 provides some additional conditions which guarantee that such a commitment willnot cause any revenue loss. For this, I have the following proposition:Proposition 5.5.2: Suppose that an airline plans to use a membership restriction anda product restriction. Assume that• both -- and -— are strictly increasing in j.YMr,j YMr,jThen1. if is strictly increasing in j, or equivalently, in the residual market wherethere is no members who can still buy restricted tickets, the relative size of themarket for unrestricted tickets from members and the market for restricted ticketsfrom general public is strictly increasing, the airline has an optimal fare structurehaving the following properties:• the airline ‘s commitment of satisfying all bookings from members for restrictedtickets is consistent with revenue maximizing;• the (unrestricted) membership fares will not be lower than the (public) restricted fares;2. if M,,,7,, is strictly increasing in j, or equivalently, in the residual market where7M,-,j 7M,)there is no members who can still buy restricted tickets, the relative size of themarket for restricted tickets and the market for unrestricted tickets from membersChapter 5. Airline Pricing by Using Membership and Product Restrictions 163is strictly increasing, the airline has an optimal fare structure having the followingproperties:• the airline’s commitment of satisfying all bookings from members for restrictedtickets is consistent with revenue maximizing;• the (unrestricted) membership fares will not be higher than the (public)stricted fares.Furthermore, in both cases, the airline needs to offer no more than five different fares.Proof: It follows from Theorem 4.4.2 and its proof. UAgain, let us look into the implications of the assumptions used in each of the twocases. Note thatrM riMr7Mr,3— 7M,3 “j —fMr,j — 7r,3 — —So intuitively speaking, if ignoring the group that consists of members who can buyrestricted tickets, the first case says that the market demand for restricted tickets isdecreasing faster than the market demand for unrestricted tickets from members; andthe second case is vice versa.6 With this in mind, it should not be surprising to see thatin the first case, the airline will offer (unrestricted) membership fares between the (public)restricted fares and the (public) unrestricted fares. Similar argument can be made for thesecond case too. This is clearly consistent with the traditional idea of economic pricingdiscrimination: the firm should charge a higher price to the segment that is less sensitiveto prices than to other segments.I now give some examples that fit into the model. First, consider the market thatconsists of travellers from a partner airline. These travellers must pay for their tickets if6lnterestiiigly, I can easily state these assumptions in term of price elasticities as used in Section 5of BL (1993a).Chapter 5. Airline Pricing by Using Membership and Product Restrictions 164they want to fly on a flight from the partner airline. On the other hand, because of theiraffiliation with an airline and its internal benefit of free flying, these travellers are muchmore price sensitive than the general public for both restricted fares and unrestrictedfares. I feel that this situation fits nicely into the second case in the above proposition.Another example are government. It is well-known that there is an enormous amountof travelling activities from employees of government or government agencies. Some ofthis travelling can be planned in advance, but there is a substantial portion of it thatcannot be planned ahead. In this case, an airline can give these clients a break onboth restricted fares and unrestricted fares. It is evident that the government will bepleased to let its employees fly with unrestricted fare at a price that is between the publicrestricted fares and public unrestricted fares. It is a special treatment that may inducemore travelling from government employees. In my view, the first case in the aboveproposition will be helpful for airlines to negotiate acceptable and rational fare priceswith the government.5.6 Summary — Membership and Product RestrictionsIn the above three sections I have illustrated how the pricing models for two types ofrestrictions can be used by airlines when product restriction and membership restrictionsare used at the same time. Unlike product restrictions, which are fairly difficult todesign, membership restrictions are relatively easy to plan and execute. In my view,the use of membership restrictions will further enhance an airline’s revenues. On theother hand, addressing all possible membership restrictions at the same time could becounter-productive because it is hard to monitor the interactions among many differentmember groups, in addition to the product restriction. A simple solution to this problemis to develop a heuristic procedure that has the following features:Chapter 5. Airline Pricing by Using Membership and Product Restrictions 165• it can model the impact of each individual membership restriction on each flight;• the airline can place a flag on a flight showing which member group(s) can bookthis flight; and• the computer reservation system should be automated so that ordinary travel agentscan authorize bookings for members.The key ingredient here is the tagging of flights. As an example, for internal travelers,it may be prudent for airlines to limit employees on personal trips to off-peak flightsonly. To tag a flight several months ahead is not an easy task, but with more advancedforecasting tools, these pricing models in Chapter 4 will be helpful since• they give optimal allocation plans in the short-run, which can help the airline tagthese flights that are available for bookings from certain members;• they specify the operating environment for an effective and rational fare structure;and• they are consistent with inventory control tools used in yield management systems.In summary, to enable an airline to take the full advantage of these revenue opportunities,the airline needs to have a much broader approach than what they do at present, sinceeach of these opportunities represents a market niche. The airline must have a systematicmanagement tool that is useful to specifically explore this potential source of revenues.A critical part of such a management system is the information system. The computerreservation system needs to be enhanced so that it is capable of providing real timerecommendations to managers.Chapter 6Seat Allocation Game on Flights with Two fares6.1 Introduction and Model SettingConsider international flights between two countries which have a bilateral agreement,and where each carrier offers a few flights per week, such as a daily flight. Such a lowfrequency of flights between two destinations will make the issue of competition in theallocation decisions very important for the sake of market share and profit.For example, Canadian Airlines International (CAT) and Japan Airlines (JAL) aresole carriers offering direct service between Vancouver and Tokyo. And both CAT andJAL offers daily flights. The schedules of both airlines are very close and their fares areactually identical. In particular, both airlines have scheduled flights on Saturdays andSundays, which indicates that both airlines are competing heavily in the business market.The strategic interaction between two airlines involves the following decisions:1. how many discounts seats will be available?2. what should an airline do when the other carrier stops selling discount tickets?In this chapter, I will discuss the seat allocation problem when each of two airlines isoperating a single-leg flight with two given fares — a full fare and a discount fare. I willuse the following notation:’• The discount fare is PB and the full fare is py;1The notation in this chapter is self-contained.166Chapter 6. Seat Allocation Game on Flights with Two fares 167• Airline k’s flight has a capacity of Ck, for k = 1,2;• The market demand for discount fare is given by B and the market demand for fullfare is given by Y. B and Y are not assumed to be independent.• Each airline chooses a booking limit for the discount fare as its decision variableand its objective is to achieve the highest revenue possible.Let 1k be the booking limit for the discount fare set by airline k, k = 1,2.2 Since airlinek’s expected revenue is determined by the joint decision (ik, ii), we can write airline k’sexpected revenue function as rk(lk, l) for k 1,2. More specifically speaking, if we letBk(lk, i) be the airline k’s demand share for the discount fare and Yk(ik, ij) be the airlinek’s demand share for the full fare, then the expected revenue function rk(lk, ij) is givenby:rk(lk, ij) = PBE(Bk(lk, i) A ik) + pyE(Yk(lk, 13) A [Ck — (Bk(lk, i) A ik)]) (6.1)where E represents the expectation and rk(lk, i) is the airline k’s random revenue derivedfrom the pair of booking limits (ik, 13). Therefore, this leads to a simple two-person gamewith payoff functions r1 and r2. With this in mind, we give the following definition of anequilibrium pair of booking limits.Definition 6.1.1: A pair of booking limits (l, 1) is said to be an. equilibrium pair ofbooking limits if fork = 1,2rk(lk, l,) rk(lk, l,) for all lk = 0, 1,... ,Ck. (6.2)2From now on, I will use k and j to identify the airlines. By convention, I always assume that k j.Chapter 6. Seat Allocation Game on Flights with Two fares 168Note that there is no guarantee of the existence of an equilibrium pair of bookinglimits since it is of the pure strategy form. To extend this concept, I introduce the notionof mixed strategy. First of all, airline k’s pure strategy space is given bySk={O,1,—.,Ck}, fork=1,2.And a booking strategy for airline k is a probability distribution function on Sk. Let Fkbe the collection of all possible booking strategies for airline k. By definition, for anyak E Fk, we have that EES ak(l) 1, where ofk(l) O,Vl e Sk. Now for any givenbooking strategy pair (ak, cs), defineRk(ak,a,) = ofk(lk)aJ(l)rk(lk,lJ).IkESk IJESJI will call the game characterized by (R1,F1;R2,F2) as the seat allocation game. I nowgive the following definition of an equilibrium booking strategy, which provides a solutionfor our seat allocation game.Definition 6.1.2: An equilibrium booking strategy is a pair (c4,c4): 4 fork = 1,2, such thatR(cq,c) for all k E Fk.Since the pure strategy spaces for both airlines (the players) are discrete and finite,our two-person game belongs to a special class of game, the so-called bimatrix games. Bydefinition, a bimatrix game is a two-person non-zero sum game where each player has afinite number of pure strategies. For this type of games, the existence of an equilibriumbooking strategy is guaranteed by the following famous theorem of Nash (1951, [163]):Nash Theorem: Every bimatrix game has at least one Nash equilibrium if we admitmixed strategy equilibrium as well as pure strategy equilibrium.Chapter 6. Seat Allocation Game on Flights with Two fares 169It follows immediately from Definition 6.1.2 that our equilibrium booking strategies areNash equilibria. Therefore, by the Nash Theorem, we always have at least one equilibriumbooking strategy. In this section, my primary interest is to find an equilibrium pair ofbooking limits, that is, I want to characterize the pure strategy equilibria for the seatallocation game if they indeed exist.So far, I have not discussed how to obtain the derived revenue functions r1 and r2.Without properly specifying these two functions, the seat allocation game is not evenwell defined. Since the two airlines face the same market demands (both of the full faremarket and discount fare market), the specification of the revenue functions are criticallyrelated to the way two airlines share market demands. In this chapter, I will explore theseat allocation game under two types of market splitting rules: the proportional splittingrule and the equal splitting rule. By the proportional splitting rule, I mean that in theevent that the total commitment for a certain fare class from two airlines exceeds themarket demand, then the two airlines will split the market demand according to theirproportions of the total commitment.3 By the equal splitting rule, I mean that in theevent that the total commitment for a certain fare class from two airlines exceeds themarket demand, each airline will get a half of the market demand or reach its commitmentlevel, whichever is less.4This chapter is organized as follows. Section 6.2 investigates the seat allocationgame when the proportional splitting rule is used to define each airline’s derived revenuefunction. Section 6.3 studies the seat allocation game under the equal splitting rule withdeterministic demands. And Section 6.4 is a summary.31n fact, the intuition behind the proportional splitting rule comes from the capacity share or themarket share consideration.41n some sense, the equal splitting rule is build upon the premise that consumers are absolutelyindifferent between the two carriers.Chapter 6. Seat Allocation Game on Flights with Two fares 1706.2 Seat Allocation Game Under Proportional Splitting RuleUnder the proportional splitting rule, we know that given a pair of booking limits (ik, ii),airline k’s demand share for the discount fare is given byBk(lk,l) = B, for k = 1,2. (6.3)ik+ljAnd the actual sales for airline k’s discount fare should be the minimum of its bookinglimit and its demand share, that is,Sk(lk, i) = Bk(lk, i) A 1k = lk(1 AB (6.4)lk+ijAfter the booking for the discount fare is closed, airline k has a residual capacity ofLk = Ck— Sk(lk, 13). (6.5)Consequently, airline k’s demand share for the full fare, again according to the proportional splitting rule, now becomesLkYk(lk,l)= T Y. (6.6)k + jConsequently, airline k’s expected revenue function rk is given byTk(ik, i) = PBE(Bk A ik) + pyE(Yk A Lk) E(rk(lk, ii)), (6.7)where r(lk, i) is the random revenue associated with the pair of booking limits (ik, ii).By (6.3), (6.4) and (6.5), it is straightforward to verify the following facts:• = lk(1 A (B/i));• Sk + 53 = B Al;• Lk+L=(C—(1AB).5For clarity of presentation, I will use Bk and Yk to denote BkQk, l) and Yk(lk,lj) respectively.Chapter 6. Seat Allocation Game on Flights with Two fares 171We are now going to use the technique of Brumelle et al (1990, [37]) to look forany equilibrium pair of booking limits. Given that airline j’s booking limit is fixed at13 and that airline k has accepted 1k — 1 requests for its discount fare and there is anadditional request for its discount fare, then airline k must decide to accept or to rejectthis particular request for the discount ticket. If airline k decides to reject it, its expectedrevenue is given byE(rk(lk — 1, l)Bk ik);and if airline k decides to accept the request, its expected revenue becomesE(rk(lk,l)IBk ik).The key is the revenue difference between these two decisions. For this, define the directincremental gain function Gk(lk, 13) for airline k as follows:Gk(lk,13) = E[rk(lk,l)IBk ik] — E[rk(lk — 1,lj)IBk ik].The following lemma gives a simple expression for the direct incremental gain functions.Lemma 6.2.1: For/c = 1,2,Gk(lk, ii) = PB — pyP(Y> C — 1IB> 1) — Y (C 1)(C 1+1) XE(YIY<C—l,B>l)P(Y<C—1IB>l), (6.8)whereCC1.+2andlli+l2.Proof: First of all, it is easy to check thatTherefore,Gk(lk,l) pB+pY01E[YA(Cl)IBl]+Chapter 6. Seat Allocation Game on Flights with Two fares 172x E[YA(C—l+1)IBljCk—lkPB+PY c—i X{E[Y A (C— l)Y> C—i, B> l]Pr(Y> C — ljB 1) ++E[YA(C—i)IYC—l,Bl}Pr(YC—iBl)}+Ck—ik+lC—i+1{E[YA(C—l+1)Y>C—l,Bl]Pr(Y>C—lIBl)++E[YA(C—l+1)YC—l,Bl}Pr(YC—lIBl)}= PBPYPT(Y>C11B>l)+C—l,B l]Pr[Y_<C—1IBl].This proves the lemma. DOn the other hand, it is important to notice that if airline k declines a request for thediscount fare, the passenger will make the same request to airline j. Airline j may or maynot accept the request. The above direct incremental gain function is calculated underthe assumption that airline j’s decision is fixed at 1,. If airline j accepts the request, thatis, airline j changes its initial booking limit for the discount fare from 1, to 13 + 1, thenairline k must take into account the impact of this action by airline j on its revenue.Because of this, we introduce the notion of indirect incremental gain function:gk(ik, i) = E[k(lk, hf)IBk ik] — E[ik(ik — 1, i + 1)IBk ik].The following lemma gives us a simple formula for the indirect incremental gain function.Lemma 6.2.2: Fork 1,2,gk(lk,lj) = PB— PYE(c A 1IB 1).Chapter 6. Seat Allocation Game on Flights with Two fares 173Proof: By the definition of indirect incremental gain function, it follows thatg(l, l) = E[i(lk, l)Bk ik] — E[r(lk — 1,13 + 1)IB 1k]= [ikpB— PY Clk E(Y A (C— l)IB 1)] +—[(1k— l)pB +C E(Y A (C— l)IB> 1)]= PB_PYCE(YA(C_l)IBl),which proves the lemma. LiIt is interesting to see that Lemma 6.2.2 leads to the following facts:• gl(l1,12) =g2(i2,il);• gk(lk, 13) is decreasing in 1 = 11 + 12, and consequently, is decreasing in 1k for anygiven ij, and vice versa.On the other hand, it is easy to check that thatgk(lk, i) = PB — pyP(Y> C — lB 1)C—l,B l)P(Y C—lIB 1),which, together with (6.8), leads toGk(lk,l) > gk(lk,ij),because < 1. Now because of this inequality, each airline will only focus on itsdirect incremental gain function. But it is not clear whether or not the direct incremental gain function Gk is decreasing in 1k for any given 1,. If Gk is indeed decreasing in1k for any given ij, then we can use each airline’s response function to characterize anequilibrium pair of booking limits. In fact, we have the following proposition:Chapter 6. Seat Allocation Game on Flights with Two fares 174Proposition 6.2.3 If for k = 1,2, the gain function Gk(lk, i) is deceasing in 1k for anygiven ij, there exists an equilibrium pair of booking limits (l, 1) such that= max{lj : Gk(lk, 1) 0}. (6.9)Proof: Because Gk(lk, i) is decreasing in ik for any given ij, I can define airline k’sresponse function, to airline j’s choice i as follows:‘lk(lj) = max{lk E : Gk(lk,l3) 0}= max{lkESk:P(Y>C—1IB>l)+— 0(0—i +1)E(YIY <C — l,B l)P(Y C — 1IB> 1)First, it is clear that Tik is well-defined. To finish the proof, it suffices to show that, forl and l given by (6.9), I will haver1k(i) l, for k = 1,2. (6.10)Note that (6.8) and (6.9) imply that each airline will protect the same number of seatsfor the high fare, that is,1* 1*= 1—t 2—t.Upon agreeing on this, both airlines will face the exactly same decision on how to choosetheir booking limit for the discount fare so that there is no further possible revenue gainby allocating additional seats for the discount fare. This is in fact captured by the response function. Since in the end neither airline can unilaterally improve its revenue, thepair of booking limits (li, 1) must be an equilibrium. This proves the result. DAs an immediate consequence of the above proposition, I get the following interestingcorollary:Chapter 6. Seat Allocation Game on Flights with Two fares 175Corollary 6.2.4: Under the same assumption as in Proposition 6.2.3, each airline willprotect the same number of seats for the high fare at equilibrium.Proof: It follows directly from (6.8) and (6.9). LIThe following corollary indicates that competition will collectively reduce the total number of seats that will be made available for the low fare.Corollary 6.2.5: Under the same assumption as in Proposition 6.2.3, at the equilibrium,the total number of seats available for the low fare class is less than the total number ofseats that will be available for the low fare if the airlines completely cooperate.Proof: If the airlines completely cooperate, then they together will act as a monopoly.Then it follows from Brumelle et al (1990, [37]) that the optimal booking limit n” for thelow fare is given by:71=max{0<C:P(Y>C—IB)<}.pySo I need to show that1* l + 1 < 71*. (6.11)Again, this is obvious since by Proposition 6.2.3 I know that= Max{lkESk:P(Y>C_lk_lIBlk+l)+(GllCll*+l)XE(YIY < C — — l, B 1k + 1)P(Y C — — lIB 1k + 1)which implies thatP(Y > C_l*IB>1*) + (C lj(Cl*+1) xpyChapter 6. Seat Allocation Game on Flights with Two fares 176Therefore, I must haveP(Y > C — l*IB> j*) <PBpyThen by definition of ‘‘ (6.11) must be true. This proves the corollary. DAs a last comment for this section, I should say that the seat allocation game underthe proportional splitting rule is not completely solved yet in the sense that we still needa complete characterization of the whole set of equilibrium booking strategies, ratherthan one equilibrium pair of booking limits only. On the other hand, the above resultsare indeed quite encouraging.6.3 Seat Allocation Game Under Equal Splitting Rule with DeterministicDemandIn this section I will look at the seat allocation game from a different perspective. Aftermany unsuccessful attempts to find equilibrium solutions to the seat allocation gameunder the equal splitting rule with uncertain demands, I am here only able to reportsome preliminary analysis for the case that both B and Y are in fact deterministic, thatis, they are fixed constants. So the main purpose of this section is to see how the use ofthe equal splitting rule changes the way we handle the seat allocation game.The main question faced by each airline is: how many seats should each airline protectfor its full fare By definition, a protection level for full fare by each airline must be acommitment in the following sense: if Pk is the protection level, then airline k will onlysell at most Ck— Pk discount tickets even knowing that it can not sell pk full fare tickets.Let me now motivate the idea of the equal splitting rule. Given a pair of protectionlevels (pi, P2), airline k will only sell Ck — Pk low fare tickets. Once the discount fare is6j deliberately switch from booking limit to protection level. They are technically equivalent.Chapter 6. Seat Allocation Game on Flights with Two fares 177closed, both airlines start to sell the full fare. First of all, if Pi +p2 < Y, then each airlineis able to sell the full fare tickets up to its respective protection level. Now consider thatPi + P2 > Y, that is, the joint commitment for full fare is too high. If p > Y/2 andP2 > Y/2, then we should expect that each airline will be able to sell Y/2 full fare ticketsif travellers have no preference over airlines and travellers arrive at a random order. Asa consequence of this, neither airline can sell enough seats to exhaust its commitment.If Pi Y/2 and p’ + P2 > Y, then airline 1 will sell P’ discount tickets and airline 2will sell Y— P1 discount fares. Note that in this case airline 2 is unable to sell up to itsprotections level since P2 > Y — p’. The essence of this type of demand sharing is thatboth airlines will share the market with a certain degree of equality, or equivalently, theywill split the market evenly whenever it is possible.Summarizing the above discussion, Ihave the following:• If an airline’s protection level is not greater than half of the full market, it will sellup to its protection level regardless of the other airline’s commitment;• If an airline’s protection level is greater than the half of the full fare market, it willcapture at least half of the market.Now let Yk(pk,p3) be the airline k’s share of the full fare demand, where k = 1,2. Thenthe above motivation leads to the following formal specification: ‘Y—p ifpk+p >Y,pk>Y/2,p <Y/2;Yk(pk,p) = if pk > Y/2, pj > Y/2;Pk otherwise.7Again, I will use convention that k,j = 1,2 and kj.Chapter 6. Seat Allocation Game on Flights with Two fares 178For the demand for low fare, denote 1k = Ck — pk for k = 1, 2. Then similarly, I will haveB—i3 iflk+1>B,lk>B/2,lj<B/2;Bk(pk,p) = if > B/2, l > B/2;otherwise.Since under the above splitting rule, the the number of sales for each fare class for eachairline is in fact equal to the actual share of the demand, airline k’s revenue functionTk(Pk,Pj) is given byTk(Pk,Pj) = pBBk(pk,p) + pyYk(pk,p,).Since the goal is to find an equilibrium strategy for each airline, for completeness Igive the following definition:Definition 6.3.1: A pair of protection levels (p,p) is said to be an equilibrium for theseat allocation game ifrk(pk,p) > rk(pk,p), for all 73k = 0,1, ,Ck and k = 1,2.The following proposition gives a characterization for an equilibrium strategy for airlines when the demands are deterministic.Proposition 6.3.1: Under the deterministic demands, (Y/2, Y/2) is always an equilibrium pair of protection levels if Ck > Y/2 fork = 1,2..Proof: To prove that (Y/2, Y/2) is an equilibrium, we need to prove that neither airlinehas any strict incentive to deviate.Case 1:C+C2>B+Y.Chapter 6. Seat Allocation Game on Flights with Two fares 179In this case, the joint capacity exceeds the total demand of the discount fare and the fullfare. Given pk = Y/2, airline k is guaranteed sales of Y/2 units of the full fare ticketsregardless what is the airline j ‘s commitment for the full fare tickets. Even though theremay have many equally good responses from airline j, p, = Y/2 is always one of theseresponses. As a consequence of this, each airline will also sell B/2 discount fare tickets.8Case 2:C1+C_<B+Y.Since Ck Y/2, each airline can sell at least Y/2 at the full fare, which implies that atthe equilibrium, for k = 1,2,p Y/2.Once p = Y/2, the airline k’s best possible result on the sales of the full fare is alsoY/2. Note the if pk > Y/2, then two things will happen: (1) airline k can still oniysell Y/2 full fare tickets; and (2) since a protection level is a commitment, airline k onlyallocate Ck— pk number of seats for the discount fare, which may reduce the chance toobtain its fair demand share for the discount fare. Therefore p = Y/2 must be one of thebest choices. This proves that no airline can strictly improve its revenue by unilaterallydeviating from (Y/2, Y/2). This proves that the proposed pair of protection levels mustbe an equilibrium. C.Remark: Technically speaking, the above proposition is not very interesting. One plausible aspect is the fact that both airlines will make the same commitment for the full fareclass, which is consistent with the findings in the previous section. It remains to be seenwhether this property is still true when the demands are actually random.8This implies that when total capacity level is too high, the larger carrier has no clear advantage interms of sales.Chapter 6. Seat Allocation Game on Flights with Two fares 1806.4 Summary — Seat Allocation GameThis chapter discusses the seat allocation problem in the presence of another airline. Itis shown that under the proportional splitting rule, there exists an equilibrium bookingpolicy such that each airline will protect the same number of seats for the full fare. Ialso demonstrate that at equilibrium the total number of seats that are available for thediscount fare is smaller than the total number of seats that would be available if the twoairlines cooperate. Under the equal splitting rule, it is shown that if the demands aredeterministic, there is an equilibrium such that each airline will protect enough seats forhigh fare so that each airline will split the market demand for the high fare equally.Chapter 7A Note On Three Models for Multi-fare Seat Allocation Problem7.1 IntroductionAs we can see from the discussions in Chapter 1, there are many interesting problems inpricing and yield management that still need answers. The purpose of this chapter is toaddress the seat allocation problem for single-leg flights with multiple fares with independent random demands. There are three different models to solve the seat allocationproblem for multi-fare flights, done independently by Curry (1990, [52]), Wollmer (1992,[269]), and Brumelle and McGill (1993, [38]).’ After briefly reviewing these three models, I will present a proposition showing that their optimality conditions are analyticallyequivalent, which implies that none of them has computational advantage over others,as misleadingly claimed by Curry and Woilmer.Throughout this chapter, I will use the following standard notation: 2• n is the total number of fare classes for a single-leg flight;• fk is the average revenue of a fare class k ticket, where f, > f2> ... > f,;• Xk is the random demand for fare class k;• Pk is the protection level for fare classes k, k — 1, . . , 1;‘The time difference on publication for three papers is not important since all three papers have beencirculating for several years before they are published. In fact, all of them announced their findings in1988, refer to Wollmer (1988, [268]), McGill (1988, [143]) and Curry (1988, [50]).2A11 notation in this chapter is also self-contained, which hears no relationship with any other notationused in other part of this thesis.181Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem 182• C is the capacity of the flight.The following assumptions are used in three models:• Low fares are booked first;• Those fares, once closed, will not reopen;• There are no cancellations and no no-shows; and consequently, there is no overbooking; and• Random variables X1,X2,.. , X are independent.This chapter is organized as follows. Section 7.2 presents the seat allocation modeldue to Wollmer (1992, [269]), where the demands are assumed to be independent discrete random variables. Section 7.3 discusses the model due to Curry (1990, [52]) whichaddresses the problem for independent continuous random demands. Section 7.4 analyzes the model due to Brumelle and McGill (1993, [38]) which solves the seat allocationproblem for any independent random demands. Section 7.5 proves an equivalence resultfor the optimality conditions from three models. And the last section is a summary.7.2 Wollmer’s ModelWolimer (1988, [268]; 1992, [269]) studied the seat allocation problem for multi-fareflights by considering that the demands are discrete random variables. By formulatingthe seat allocation problem into a Markov decision problem, Wollmer derives optimalityconditions that are natural extensions of Littlewood’s optimal condition for two-fareflights with independent demands.Here is a sketch of Wolimer’s model. Let Rk(.s) be the expected revenue under anoptimal policy if s seats are available for booking when fare class k is the lowest fare thatChapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem 183is open.3 Define,ARk(s) = Rk(s) — Rk(s — 1),which is the incremental value of an additional seat. By using simple facts on conditionalprobabilities, it is easy to derive the following recursive relationships:Ri(s) =f1E(X A s), AR1(s) =f1P(X > s);andRk(pk_1 + i) = Rk_1(pk_1) + [ARk_l(pk_l + j — i)P(X i) + fkP(Xk + 1)]i—iARk(pk_1 + i)=ARk_l(pk_1 + j — i)P(X = i) + fkP(Xk j), (7.1)where E(.) represents the “expectation” or expected value, P(.) is the probability andxAy min(x, y). By proving that ARk(s) is decreasing in s, Woilmer gives the followingcharacterization for optimal booking policies:p*k= max{sI A Rk(s) > fk+1}, for k 1. (7.2)Based on this characterization, Wolimer also presented algorithms to find the optimalbooking policies.7.3 Curry’s ModelCurry (1988, [50]; 1990, [52]) studied the seat allocation problem for multi-fare single-legflight by assuming that the demands are continuous random variables. His approach is inthe spirit of marginal seat value since his analysis is based upon the concept of the revenueslope, or equivalently, the marginal seat revenue curve. Let Rk(pk_1, s) be the expected31n other words, Rk(S) is the total expected revenue for the remaining s seats for fare classes i =k, k — 1, . .. , 1 only. It simply means that fare classes i = n, .. , k + 1 are closed.Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem 184revenue for the remaining s seats that are available for fare classes i = k, k— 1,... , 1 ifthe protection level for fare classes i = k — 1,. , 1 is given by Pk—1• Define the slopefunction of the expected revenue function as follows:aRk(Pk_l, s)Sk(pk_1,s)= OsThe recursive relationship on Rk ‘s isfSpk_1Rk(pk_1,s)= j (f,x + Rk_j(pk_2,s — x))dFk(x) +0+((s— pk—1)fk + Rk_1(pk_2,pk_1))F(Xk > ‘— Pk—1),where Fk is the probability distribution of the random variable Xk. And the recursiverelationship on Sk’S takes the following form:ORk(pk_i,s)= (_fk +k_;2_1))F(Xk >3 pk-1)andfSpk_1Sk(pk_1, s) fkP(Xk >— Pk-1) + J Sk_1(pk_1, — x)dFk(x). (7.3)0After showing that the revenue function Rk(pk_1, s) is concave in s, Curry uses firstorder conditions to characterize the optimal booking policies. More specifically speaking,he proves that optimal booking policies in terms of a vector of protection levels p(p1*2,..,p) is given recursively by:0, and Sk(p_1,p)= fk+i fork = 1,•,n— 1, (7.4)which says that the optimal protection level for fare classes k, k — 1,. , 1 is chosen insuch a way that marginal value of an additional seat is equal to the value of a fare classk + 1 ticket, that is, fk+1. This is clearly consistent with Littlewood’s condition for caseof two-fare classes with independent demand.Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem 1857.4 Brumelle-McGill’s ModelBrumelle and McGill (1993, [38]) investigated the seat allocation problem for multi-faresingle-leg flights with independent demands. The key difference is that their analysis doesnot depend upon whether the demand is continuous or discrete. In fact, by using non-smooth optimization techniques and optimal stopping rules, they establish two importantresults:• there is a closed-form solution for the optimal booking policies; and• the best policy in the class of booking policies in terms of protection levels is infact also optimal in the class of all possible booking policies.I briefly sketch their model here. Let rk(s; p, x) be the revenue function generated by k highest fare classes, given that (a) there are s seats available, (b) the vector p = ‘pm) is the protection level booking policy, and (c) the vector x =(x1,x2, , x,) is a realization of the vector of random demands (X1,X2,. . . , X,). Intheir model, the recursive relationships are built directly on the revenue function ratherthan on the expected revenue function:I fis, ifO<s<x1,ri(s;p,x)=f1xi, if x1 < s;andrk(s;p,x) if 0 s pk,rk+1(s;p,x) = (s—pk)fk+1 +rk(pk,p,x) if Plc <Pk +Xk+1,Xk+lfk+1 + rk(S — xk+1;p,x) ifs Plc + Xk+1.To use the non-standard optimization method, we have to deal with the notion of leftderivative and right derivative. Let S_ and be the operators for left derivative andChapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem 186right derivative of the revenue function with respect to s. Then it follows thatI fi ifs<x1Sr1(s;p,x) =0 ifs>x1I fi ifs<x1S_ri(s;p,x) =1. 0 ifs>x1.In general,61rk(s;p,x) if 0 s <73k6+rk(s;p,x)= fk+1 if 73k <73k + Xk+1SJrk(s—xk+1;p,x) ifpk+xk+1 sS_rk(s;p,x) if 0 <s pkS_rk(s;p,x)= fk+1 ifpk<s<pk+xk+1S_rk(s—xk+1;p,x) ifpk+xk+1 <5.The main analytical tools for non-standard optimization are the following two facts: forany continuous, piecewise-linear function f(s),1. f(s) is concave on s > 0 if and only is 6f(s) 6_f(s) for all s; and2. A concave f(s) can be maximized at any point .s such that 0 E Sf(s*), whereSf(s*) is the subdiffereritial defined as the closed interval determined by 6+f(s*)and S_f(s*).Define the expected revenue function to beRk(s;p,X) = E(rk(s,p,X)),where X = (X1,2.. ,X) is the vector of random demands. It can be interpreted asthe expected revenue for the remaining s seats if the sales are available to fare classes k, k—Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem 1871,... , 1 and the booking policy p (pl,p2, ,p,) is used. By proving some concavityproperties of Rk, they derive the following condition for optimal booking policies:fk+i eSRk(p;(p,p,”•,p_l),X). (7.5)After some analytical simplifications, they further prove that if demand distribution functions are continuous, then there is a closed-form characterization for optimal bookingpolicies:fk+l=flP(Xl>p,Xl+X2>p,,Xl++Xk>p),fork=1,_. n—1.(7.6)Clearly, when n = 2, it reduces to Littlewood’s formula.7.5 Equivalence of Optimality ConditionsI am now going to show that the optimality conditions for all three models are in factanalytically equivalent, which therefore implies that they are also computationally equivalent.First of all, it is clear that Wolimer’s model is just a discrete version of Curry’s model.This becomes more evident if we compare Wollmer’s dynamic condition (7.2) with Curry’sdynamic condition (7.3). Because of this, I only need to prove the equivalence of (7.4)and (7.6).Proposition 7.4.1: Conditions (7.4) and (7.6) are equivalent.Proof: (i) (7.4) = (7.6)For simplicity, let p = (pl,p2, ..) be an optimal policy derived from condition (7.4).First note that by (7.4),Si(po,s) =f1P(X > s),Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem 188which implies thatf2 =f1P(X > pa).So (7.6) holds for k = 1. Now suppose that based on (7.4), we already have shown that(7.6) holds for i = 1,..., k, that is,f+i =f1P(X >p1,X + X2 > P2, ,X1 +• + X > lii), (7.7)for i = 1,.. . , k. We now need to show that (7.6) also holds for i = k + 1. To show this,I first show the following claim:Claim 6.1: If(7.6) holds fori = 1,.,k, then fori 1,,k+1 and ails 0,S(p_i,s)=fiP(Xi>p1,•,X+ ••+X_>p_,X+••+X_ >s). (7.8)Proof of Claim 6.1: I will prove it by induction. For k = 1, I need to show that (7.8) holdsfor i = 1 and i = 2. Clearly, it holds for i = 1. 1 now show that if f2 =f1P(X > p’),then (7.8) also holds for i = 2. By (7.4),ps—p1S2(pi,s) = f2P(X >s_i)+J Si(po,s—x)dF2(x0I—P1= f2P(X >s—i)+fiJ P(X1 > s—x)dF2( )0= f1P(X >P1)P(X2 > s—pr) + flE(I(xx5)I x<_)= f1P(X >p1,X2 >8—p1) + flE(I(xx2>s,x< _p)= f1P(X >pi,X2 > s—pi)+fiP(Xi +X2 > s,X <s—pr)= f1P(X >pi,Xi +X2 > s),where ‘A represents the indicator function defined on the set A, that IA(x) = 1 if x € A;and 0 if x A. This implies that (7.8) holds for i = 2. Therefore, the claim is provedfor k = 1.Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem 189Now suppose that the claim is true for k = 1,.•• , m, and that I want to show that itis also true fork = m + 1. First, I have shown that for i = 1, ,m + 1;S(p_j,s)=f1P(Xi>p•,X+•••+ _ >p_i,Xi+.+X_i+X>s); (7.9)and the inductive assumption implies thatf+1=fP(X>pi,Xi+X2p,•••,Xi+•••+X>p), fori=1,”•,m+1. (7.10)We only need to show that (7.8) holds for i = m+2. By (7.4), (7.9) and (7.10), it followsthatSm+2(p +i, s)fS_Pm+1= fm+2P(Xm+ > pm+i) + J Sm+i(pm,S — X)dFm+2(X)0= f1P(X > pi,Xi +X2 > P2, •,Xl + +Xm+i > pm+i)P(Xm+2> SPm+i) +fSPm+l+fiJ P(Xi>pi,,Xi++Xm>pm,Xi+”+Xm+i>S)dFm+0= fiP(Xi>pi,,Xi+.+Xm+i>pm+i,Xm+>Spm j)++f1P(X > Pi, ,X1 + + Xm > Pm,Xi +• + Xm+2 > S,Xm+2 S pm+i)= f1P(X >p1,•..,X + +Xm+i > Pm+i,Xi + Xm+l +Xm+2 > s),This proves (7.8) for i = m + 2. Hence the claim is true for k = m + 1. Therefore, byinduction, the claim is proved.I now go back to the original proof. In fact, I want to show that (7.6) holds for k + 1.Note that by inductive assumption, (7.7) is true for i = 1, . , k, which, by Claim 6.1,will lead to (7.8), that is, for i = 1,. , k + 1 and all s > 0,f1P(X >p1,•..,X ++X_1> p_1,X + +X_1 +X > s).Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem 190In particular, taking i = k + 1, I obtainSk+l(pk,s)=flP(Xl>pl,...,Xl+.-•+Xk>pk,Xl+.••+Xk+Xk+l>s).But (7.4) implies that Sk+1(pk,pk+1) = fk+2, it follows thatf2=flP(Xl>pl,,Xl++Xk>pk,X1++Xk+Xk+1>pk+1),which is exactly (7.6) for i = k + 1. This shows that (7.6) is true for k + 1. Then byinduction, (7.6) is always implied by (7.4). This finishes the first of the proofs.(ii) (7.6) (7.4)Again, I can prove this by induction. In fact, this part follows immediately from Claim6.1 since it only uses the facts (7.4) and (7.6). For k = 1, (7.4) follows directly from (7.6).Suppose that (7.4) folds for i = 1,.•• , k. I will show that (7.4) also holds for i = k + 1.Since (7.6) is true for all i = 1, . , n, then claim 6.1 implies thatSk+1(pk, s) =f1P(X > Pi,” , Xi +..• + Xk > Pk, X1 + ... + X + Xk+1 > s).Taking s= Pk+1 leads toSk+l (pk, pk+1)= flP(Xl>pl,...,Xl+...+Xk>pk,Xl+••.+Xk+l>pk+l)=fk+2which means that (7.4) holds for i = k + 1. Then by induction, (7.4) holds for all k.Therefore, the proposition is proved. DRemark: The significance of the above proposition is that it gives a closed-form representation for the dynamic equation in Curry’s formulation. This new representationmakes it clear on that the basic idea behind Littlewood’s result remains valid for themulti-fare case when demands are independent. It may also shed new light on a furtherextension to the case of certain dependent demands.Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem 1917.6 Summary On Three Model for Multi-fare Seat Allocation ProblemThis chapter provides a note on the seat allocation problem for multi-fare single-leg flightswith independent demands. It proves that three existing models for the seat allocationproblem for multi-fare flights with independent demands all have equivalent optimalityconditions, which clarifies the issue on which model has the computational advantageover the others. This equivalency result may also shed new light on the seat allocationfor multi-fare flight with dependent demands.Chapter 8Summary of the Thesis and Future Directions8.1 Summary of the ThesisThis dissertation studies the pricing problem for perishable inventories for a capacityconstrained monopolist. The main innovation is to explicitly incorporate the use ofartificial restrictions. And the main goal is to develop useful pricing models for firms likeairlines and hotels.Chapter 2 begins with a detailed discussion on a pricing model due to Wilson, whichleads to the notion of rationing sales at lower prices. I then present a monopoly pricingmodel for perishable inventories by using one type of restriction. A pricing policy iscalled a primary policy if the prices for restricted units are not larger than the prices forunrestricted units. Under the following conditions: (1) the impact of restriction on thedemand market has the property that as price increases, the percentage of consumerswho can accommodate the restriction is decreasing; (2) the demand function is a non-increasing step function; (3) the firm only uses primary policies; and (4) the firm sellsthe product at prices in an increasing order, then I prove that the monopolist’s pricingproblem can be formulated as a nonlinear mathematical programming problem with threeconstraints. I further show that the monopolist will have incentives to use the restrictionand can maximize its revenue by using no more than three prices. I also apply the modelto airline fare pricing. I demonstrate that if the demand market is divided into leisuretravellers and business travellers, then the condition imposed on the impact of restriction192Chapter 8. Summary of the Thesis and Future Directions 193on demand market is equivalent to saying that leisure travellers are more price sensitivethan the business travellers.Chapter 3 further discusses the same pricing problem in a more general context. Ifirst extend the class of pricing policies so that it is possible for the firm to place therestriction at any price level. I prove a general optimality theorem, which shows thatany optimal pricing policies in the class of primary policies remain optimal in the generalclass of policies. This implies that, for perishable inventories, if a monopolist decides touse an artificial restriction, an optimal pricing practice is to sell restricted units at pricesthat are lower than the prices for unrestricted units. Motivated by the current practice ofairlines, this chapter also investigates the issue of whether all active prices in an optimalprimary policy can be made available at the same time. It is shown that: (1) as longas active restricted prices are different from active unrestricted prices, then the optimalpricing primary will sustain even if all active prices— both restricted and unrestricted— are offered at the same time; and (2) if there exists an active restricted price that isequal to an active unrestricted price, then the firm will be better off by selling restrictedunits first at this price rather than making both restricted units and unrestricted unitsavailable at the same time.Chapter 4 extends the discussion to the case of using two types of restrictions. Threecases are analyzed. For two nested restrictions and two mutually exclusive restrictions,I show that there exist optimal pricing policies that consist of at most four prices byoffering three types of product. For the case of two general restrictions, I prove thatunder certain additional conditions, there exists optimal policies that consist of no morethan five prices by utilizing four types of product.Chapter 5 presents an application to the airline fare pricing problem in the presenceof membership and product restriction. In this chapter, I demonstrate that these pricingChapter 8. Summary of the Thesis and Future Directions 194models developed in previous chapters are useful tools for airlines to identify the operating environment for each one of three common membership privileges: (1) cheaperrestricted fares only; (2) cheaper unrestricted fares only; and (3) cheaper restricted faresand cheaper unrestricted fares. Examples are used to provide further support.Chapter 6 explores the seat allocation problem in the presence of another airline. Thediscussion is limited to two fare flights. It is shown that under the proportional splittingrule, there exists an equilibrium pair of booking limits for the discount fare such thateach airline will protect the same number of seats for the full fare regardless of theirrespective capacities. Under the equal splitting rule, it is shown that, for deterministicdemands, there exists an equilibrium such that each airline will protect enough seats tocapture half of the demand market for the full fare.Chapter 7 discusses three models for the multi-fare seat allocation problem withindependent random demands. We have shown that the optimality conditions from threemodels are in fact equivalent. This clarifies an issue of which method has a computationaladvantage over the other two. It also provides an integrated approach to the multi-fareseat allocation problem, which could be an important step toward the development of aseat allocation model for multi-fare flights with dependent demands.8.2 Future Directions8.2.1 On Seat Allocation ProblemsA key theoretical work that needs to be done along this direction is to solve the seat allocation problem for single-leg flights with dependent demands. There are many possibleapproaches. A natural approach is to extend the model of Brumelle and McGill (1993,[38]) to the case of dependent demands. In my view, a generic treatment of dependentdemands could be very difficult because a general dependency structure among multipleChapter 8. Summary of the Thesis and Future Directions 195random variables is too complicated. A more delicate approach is to introduce morestructural dependency among these random demands for fare classes. Recall that theoptimality conditions in Brumelle-McGiIl model (refer to Section 6.2) actually can berewritten as follows:fk+l=fkP(Xl++Xk>pkIXl++Xk_l>pk_l),fork=1,.”,n—1. (8.1)This is closely related to the original Littlewood formula and the result due to Brumelle etal. (1990, [37]) for two-fare flights with dependent demands. Note that Wk X1+. .should be interpreted as the total demand for fare class k. As long as {Xk}1 is asequence of independent non-negative random variables, the resulting sequence {Wk}1is a Markovian sequence. We should notice that working directly with the stochasticsequence {Wk}1 is consistent with the traditional view of economic demand functionwith uncertainties. It is also possible to introduce a more dedicate dependency structureso that the impact of restrictions can also be incorporated.As mentioned in Chapter 1, there are many other important seat allocation problems.Two of them are highlighted here again. First, I feel that the seat allocation problem inthe presence of connecting passengers is workable. A joint optimization model should bepossible. Another key problem is to develop optimal seat allocation models for multi-legflights. A concrete theoretical model that handles random demands and multiple faresremains to be a very major challenge in yield management research.8.2.2 On Seat Allocation GamesThe discussion in Section 6.3 is just a beginning of research along this direction. Inmy view, this should be a very fruitful area of research in the near future. It is a veryimportant step to push the field of yield management to a more realistic situation. Themain issues are:Chapter 8. Summary of the Thesis and Future Directions 196• Under the proportional splitting rule, (a) find the necessary and sufficient conditionsthat lead to the existence of equilibrium booking limits; and (b) find the conditionsthat lead to equilibria of mixed strategies.• Under the equal splitting rule, (a) solves the seat allocation problem under randomdemands; and (b) characterizes the equilibrium protection levels or the equilibriumbooking strategies.It is also important to discuss the seat allocation game for multiple fares. Note thatfor the case of two fares, each airline has oniy one strategic variable, either the bookinglimit for low fare, or the protection level for high fare. But when each airline operates aflight with n fares, then each airline will have n — 1 strategic variables. This gain opensup two possible ways of dealing with this game, which will be the same when n = 2. Oneway is from the static point of view — treating all these n— 1 variables as one strategicvector. This is equivalent to a one-shot game. Another is from the dynamic point ofview — treating these n — 1 strategic variables sequentially, which sounds more natural.But it is not clear at this moment which way will lead to more interesting results.8.2.3 On the Pricing Problem by Using RestrictionsThis thesis explores the pricing problem by using restrictions oniy for a special kind ofproduct, that is, it is perishable, and not storable for consumers with fixed supply. Froma different perspective, we may say that the results on this thesis show that, at least fora non-storable perishable products, a carefully designed restriction can be a very effectivemethod to practice price discrimination. So this leads to the following general question:• What are the product characteristics required to induce firms to intoduce restrictionsas a mechanism of price discriminationqChapter 8. Summary of the Thesis and Future Directions 197On the other hand, firms have complete freedom to use or not to use restrictionsduring the pricing process. This leads to a very interesting issue: suppose a firm candesign the restriction, so the question is:• Given the market demand structure for the unrestricted product, what is the monopolist’s most desirable restrictionTo put the above question in a more analytical way, recall that in Section 2.4, the aim isto characterize R(TYi, of), where— argmaxsR(m, a).So the above question is equivalent to asking to find an optimal a-function: äf(p) suchthat= argmaxR(ni,a),where F is the set of all increasing functions from the interval [pr, p,} to unit interval[0, 1]. So this is an optimization problem on a functional space. In marketing terms, thismay be termed the problem of optimal design of restrictions.Recall that the pricing models developed in this thesis have a common assumptionthat the monopolist offers one price at a time. Even though Section 3.4 argues that therealways exists an optimal policy that will realize the projected optimal revenue value fromthe BL-model, but the question still remains when the firm makes all allocated quantitiesavailable at the same time, but we do not have a formal pricing model that makes all pricesboth restricted and unrestricted — available at the same time. Clearly, we must dealwith a class of pricing policies that is different from the class of general policies discussedin Chapter 3. And questions related to the corresponding optimal pricing problem thatare of great interests include:Chapter 8. Summary of the Thesis and Future Directions 198• What is the formulation for the monopolist’s pricing problem?• l47hat are the optimal policies?• What can we say about any relationship between the optimal policies and the optimalpolices from the BL-model in Chapter 2?This issue remains a very interesting but open question.The oligopoly pricing problem by using restrictions is another important issue thatneeds to be addressed in the near future. Since the pricing problem in the context ofa step-wise demand function is, in fact, a problem of quantity allocations by rationing,when there are several firms, the pricing game becomes a dynamic quantity allocationgame. It is possible to approach this problem by treating it as a static game, since inthis case each firm will throw out an allocation plan at the table and the let the marketdecide the outcome. Some key questions are:• What are the equilibrium pricing strategies?• What is the role of capacity?We should notice that the oligopoly pricing model is an important step toward a formalanalysis of the seat allocation game for multi-fare flights with random demands.8.2.4 On Further ApplicationsThis thesis exclusively focuses on applications to airlines. It is well-known that manyother industries, such as hotels, cruise lines, car rentals, and TV stations, also offerproducts that are characteristically similar to airline seats. The pricing models in thisthesis should be useful to these industries too. I here identify several problems in hotelsroom pricing.Bibliography[1] Abraham, M and T.E. Keeler, “Market structure, pricing and service quality in theairline industry under deregulation”, In W. Sichel and T. Gies (eds) Applicationof Economic Principles in Public Utility Industries Michigan Business Studies Vol.11(2), University of Michigan Press, pp. 103—120 (1981).[2] Allen, B. and M. Hellwig, “Bertrand—Edgeworth oligopoly in large markets”, Reviewof Economic Studies 53:175—204 (1986).[3] Aistrup, J., S. Boas, O.B.G. Madsen, and R.V.V. Vidal, “Booking policy for flightswith two types of passengers”, European Journal of Operational Research 27:274—288(1986).[4]. Anderson, S.E., “Operational planning in airline business — Can science improveefficiency? Experienced from SAS”, European Journal of Operational Research 43:3—12 (1989).[5] Anderson, S.P., A. de Palma and J. Thisse, “Demand for differentiated products,discrete choice models, and the characteristic approaches”, Review of EconomicStudies 56:21—35 (1989).[6] Bailey, E.E., “Peak laod pricing under regulatory constraint”, Journal of PoliticalEconomy 80:662—679 (1972).[7] Bailey, E.E., D.R. Graham and D.P Kaplan, Deregulating the Airlines MIT Press,Cambridge, MA (1985).[8] Bailey, E.E. and L.J. White, “Reversals in peak and off—peak prices”, Bell Journalof Economics and Management Science 5:75—92 (1974).[9] Beckmann, M.J., “Decision and team problems in airline reservations”, Econometrica 26:134—145 (1958).[10] Beckmann, M.J. and F. Bobkowski, “Airline demand: an analysis of some frequencydistributions”, Naval Logistics Research Quarterly 5:43—51 (1958).[11] Beckmann, M.J., “Edgeworth-Bertrand duopoly revisited”, in R. Henn, ed., Operations Research- Verfahren, III, Meisenheim: Verlag Anton Rein, pp. 55—68 (1965).200Bibliography 201[12] Belobaba, P.P., Air Travel Demand and Airline Seat Inventory Management. Unpublished Doctoral Dissertation, MIT; Cambridge, MA (1987).[13] Belobaba, P.P., “Airline yield management: an review of seat inventory control”,Transportation Science 21:63—73 (1987).[14] Belobaba, P.P., “Application of a probabilistic decision model to airline seat inventory”, Operations Research 37:183—197 (1989).[15] Belobaba, P.P., “Heuristic methods for airline O—D seat inventory control”, WorkingPaper, Flight Transportation Lab, MIT (1991).[16] Belobaba, P.P., “Optimal vs. heuristic methods for nested seat allocation”, Presentedat ORSA/TIMS Annual Meeting, San Francisco, CA (1992).[17] Ben—Akiva, M. and S.R. Lerman, Discrete Choice Analysis: Theory and Applicationto Travel Demand The MIT Press, Cambridge, MA (1985).[18] Benoit, J.P. and V. Krishna, “Dynamic duopoly: prices and quantities”, Review ofEconomic Studies 54:23—35 (1987).[19] Berchman, J. and 0. Shy, “Airline deregulation and the choice of networks”, Working Paper No.2—91 , the Sackler Institute of Economic Studies, Tel—Aviv University,Ramat Aviv, Israel (1991).[20] Bergstrom, T. and J.K. MacKie—Mason, “Some simple analytics of peak load pricing”, RAND Journal of Economics 22:241—249 (1991).[21] Bertrand, J., “Theorie Mathematique de la Richesse Sociale”, Journal des Savantspp. 499-508 (1883).[22] Bhatia, A.V. and S. Parekh, “Optimal allocation of seats by fare”, In AGIFORSSymp. Proc. vol.13, (1973).[23] Bitran, G.R. and S.M. Gilbert, “Managing hotel reservations with uncertain arrivals”, Working Paper, Sloan School, MIT (1992)[24] Bodily, S.E. and P.E. Pfeifer, “Overbooking decision rule.” OMEGA 20:129—133(1992).[25] Bodily, S.E. and L.R. Weatherford, “Yield management: achieving full profit potential from your market”, Darden School Working Paper No. 89—30, University ofVirginia (1989).Bibliography 202[26] Bodily, S.E. and L.R. Weatherford, “Perishable asset revenue management: yieldmanagement and pricing”, Darden School Working Paper, University of Virginia(1991).[27] Boeing Company, “Service Quality — Discount Fare Survey”, Company Report, Boeing Co. April, 1982.[28] Bohm, V., E. Maskin, H. Polemarchakis and A. Postlewaite, “Monopolistic quantityrationing”, Quarterly Journal of Economics 98:189—197 (1983).[29] Bohnenblust, H.F., S. Karlin and L.S. Shapley, “Solutions of discrete, two—persongames”, Annals of Mathematics Studies 24:51—72 (1950).[30] Boiteux, M., “Peak load pricing”, Journal of Business 33:157—179 (1960).[31] Borenstein, S., “Hubs and high fares: dominance and market power in the U.S.airline industry”, Rand Journal of Economics 20:344—365 (1989).[32] Borenstein, S. and N.L Rose, “Price discrimination in the US airline industry”,Discussion paper 306, Institute of Public Policy Studies, University of Michigan(1989).[33] Brander, J.A. and A. Zhang, “Market conduct in the airline industry: an empiricalinvestigation.” Rand Journal of Economics 21:567—583 (1990).[34] Brown, J.H., “An economic model of airline hubbing—and—spoking”, The Logisticsand Transportation Review 27(3) :225—239 (1991).[35] Brueckner, J.K. and P.T. Spiller, “Competition and mergers in airline networks”,International Journal of Industrial Organization 9:323—342 (1991).[36] Brueckner, J.K., N.J. Dyer and P.T. Spiller, “ Fare determination in airline hub—and—spoke networks”, Unpublished Working Paper, University of Illinois (1990).[37] Brumelle, S.L., J.I. McGill, T.H. Oum, K. Sawaki, and M.W. Tretheway, “Allocationof airline seat between stochastically dependent demands”, Transportation Science24:183—192 (1990).[38] Brumelle, S.L. and J.I. McGill, “Airline seat allocation with multiple nested fareclasses” Operations Research 41(1):127—137 (1993).[39] Bulow, J., “Durable—goods monopolies”, Journal of Political Economy 90:314—332(1982).[40] Bulow, J.I., J.D. Geanakoplos and P.D. Kiemperer, “Multimarket oligopoly: strategic substitutes and complements”, Journal of Political Economy 93:488—511 (1985).Bibliography 203[41] Buhr, J., “Optimal sales limits for two—sector flights”, AGIFORS Symp. Proc.22:291—304 (1982).[42] Call, G.D. and T.E. Keeler, “Airline deregulation, fares, and market behaviour”, inAnalytic Studies in Transportation Economics 221, edited by A. Daughety (1985).[43] Caplin, A. and B. Nalebuff, “Aggregation and imperfect competition: on the existence of equilibrium”, Econometrica 59:25—59 (1991).[44] Caplin, A. and B. Nalebuff, “Aggregation and social choice: a mean voter theorem”,Econometrica 59:60— (1991).[45] Chin, H., T. Parthasarathy and T.E.S. Raghavan, “Structure of equilibria in N—person non—cooperative games”, International Journal of Game Theory 3: 1—19(1974).[46] Chow, Y.S., H. Robbins, and D. Siegmund, Great Expectation: The Theory of optimal Stopping. Houghton Muffin Company, Boston (1971).[47] Cremer, H., M. Marchand, and J.F. Thisse, “Mixed ologopoly with differentiatedproducts”, International Journal of Industrial Organization 9(1):43—53 (1991).[48] Crew, M.A. and P.R. Kleindorfer, “Marshall and Turvey on peak load or joint product pricing”, Journal of Political Economy 79:1369—1377 (1971).[49] Crew, M.A. and P.R. Kleindorfer, “Peak load pricing with diverse technology”, BellJournal of Economics 7:207—231 (1976).[50] Curry, R.E., “Optimum seat allocation with fare classes nested on segments andlegs”, Technical Report Note 88—1, Aeronomics Incorporated, Fayetteville, Georgia(1988).[51] Curry, R.E., “A dynamic—systems approach to the optimal allocation of seat inventory”, A GIFORS Symp.Proc. 29:299—318 (1989).[52] Curry, R.E., “Optimal airline seat allocation with fare classes nested by origins anddestinations”, Transportation Science 24:193—204 (1990).[53] Curry, R.E., “Parallel nesting and overallOcation”, Presented at ORSA/TIMS Annual Meeting, San Francisco (1992).[54] Curry, R.E., M. Jaul, A.Storey, P.S. Swope Jr., and L.J. Whale, “E.R.O.S.: puttingpleasure into yield management”, AGIFORS Symp. Proc. 30:41—60 (1990).[55] Dansby, R.E., “Capacity constrained peak load pricing”, Quarterly Journal of Economics 92:387—398 (1978).Bibliography 204[56] Dasgupta, P. and E. Maskin, “The existence of equilibrium in discontirious economicgames, I: theory”, Review of Economic Studies 53:1—26 (1986).[57] Dasgupta, P. and E. Maskin, “The existence of equilibrium in discontinous economicgames, II: applications”, Review of Economic Studies 53:27—41 (1986).[58] d’Aspremont, C., J.J. Gabszewicz and J.K. Thisse, “On Hotelling’s stability in competition”, Econometrica 47:1145—1150 (1979).[59] Davidson, C. and R. Deneckere, “Long—run competition on capacity, short—run competition in price, and the Cournot model”, Rand Journal of Economics 17:404—415(1986).[60] DeSalvia, D.N., “An application of peak load pricing”, Journal of Business 42:458—476 (1969).[61] Dixit, A.K. and J.E. Stiglitz, “Monopolistic competition and optimum product diversity”, American Economic Review 67:297—308 (1977).[62] Dobson, G. and S. Kalish, “Positioning and pricing a product line”, MarketingScience 7:107—124 (1988).[63] Dror, M., P. Trudeau and S.P. Ladany, “Network models for seat allocation onflights”, Transportation Research B 22:239—250 (1988).[64] Dubey, P., “Price—quantity strategic market game”, Econometrica 50:111—126(1982).[65] Dunn, K.D. and D.E. Brooks, “Profit analysis: beyond yield management”, CornellHotel and Restaurant Administration Quarterly 31:80—90 (1990).[66] Eaton, B.C. and R.G. Lipsey, “Product Differentiation”, Chapter 12 of Handbook ofIndustrial Organization Vol.1 edited by R. Schmalensee and R.D. Willig. ElsenvierScience Publisher, B.V. (1989).[67] Economides, N., “Symmetric equilibrium existence and optimality if differentiatedproduct market”, Journal of Economic Theory 47:178—194 (1989).[68] Edgeworth, F., “La Theoria Pura del Monopolio”, Giornale degli Economisti 40:13-31 (1897). In English: “The Pure Theory of Monopoly”, in Papers Relating toPolitical Economy, volume 1, ed. F. Edgeworth, London: MacMillan (1925).[69] Ekelund, R.B., “Price discrimination and product differentiation ill economic theory:An early analysis”, Quarterly Journal of Economics 84:268—278 (1970).Bibliography 205[70] Etschmaier, M. and M. Rothstein, “Operations research in the management of airlines”, OMEGA 2:157—179 (1972).[71] Economides, N., “Symmetric equilibrium, existence and optimality in differentiatedproduct market”, Journal of Economic Theory 47:178—194 (1989).[72] Gale, D. and S. Sherman, “Solutions of finite two—person games”, Annals of Mathematics Studies 24:37—49 (1950).[73] Gale, I.L. and T.J Holmes, “The efficiency of advance-purchase discounts in thepresence of aggregate demand uncertainty”, International Journal of Industrial Organization 10(4):413-437 (1992).[74] Gale, I.L. and T.J. Holmes, “Advance-purchase discounts and monopoly allocationof capacity”, American Economic Review 83:135—146 (1993).[75] Gallego, G. and G. van Ryzin, “Optimal dynamic pricing of inventories with stochastic demand over finite horizons”, Working Paper, Department of JE/OR, ColumbiaUniversity, NY (1992).[76] Gerstner, E., “Peak load pricing in competitive markets”, Economic Inquiry 24:349—361 (1986).[77] Gerstner, E. and D. Holthausen, “Profitable pricing when market segments overlap”,Marketing Science 5:55—69 (1986).[78] Gerchak, Y.; M. Parlar and T.K.M. Yee, “Optimal rationing policies and productionquantities for products with several demand classes”, Canadian Journal of Administrative Sciences 2:161—176 (1985).[79] Gillen, D.W., T.H. Oum and M.W. Tretheway, Airline Cost and Performance: Implications for Public and Industry Policies Centre for transportation Studies, University of British Columbia, Vancouver, Canada (1985).[80] Glover, F., R. Glover, J. Lorenzo, and C. McMillian, “The passenger mix problemin the scheduled airlines”, Interfaces 12:73—79 (1982).[81] Goldman, M., H. Leland and D. Sibley, “Optimal nonuniform pricing”, Review ofEconomic Review 51:305—320 (1984).[82] Graham, D.R., D.P. Kaplan and D.S. Sibley, “Efficiency and competition in airlineindustry”, Bell Journal of Economics 14:118—138 (1983).[83] Hanks, R.D., R.G. Cross and R.P. Noland, “Discounting in the Hotel industry: anew approach.”, Cornell Hotel and Restaurant Administration Quarterly 32:15—23(1992).Bibliography 206[84] Hansen, Mark M., A Model of Airline Hub Competition. Unpublished Doctoral Dissertation, University of California, Berkeley, CA (1988).[85] Hansen, Mark M., “Airline competition in a hub—dominated environment: an application of noncooperative game theory”, Transportation Research B. 24(1):27—43(1990).[86] Harris, M. and A. Raviv, “A theory of monopoly pricing schemes with demanduncertainty”, American Economic Review 71:347—365 (1981).[87] Hart, O.D. , “Monopolistic competition in a large exoncomy with differentiatedcommodities”, Review of Economic Studies 42:353—382 (1979).[88] Heal, G., “Rational rationing and microeconomic references”, Economic Letters8:18—27 (1981).[89] Hersh, M. and S.P Ladany, “Optimal seat allocation for flights with intermediatestops”, Computers and Operations Research 5:31—37 (1978).[90] Heuer, G.A., “On completely mixed strategies in bimatrix games”, Journal of London Mathematical Society 11: 17—20 (1975).[91] Heuer, G.A., “Uniqueness of equilibrium points in bimatrix games”, InternationalJournal of Game Theory 8:13—25 (1979).[92] Heuer, G.A. and C.B. Miliham, “On Nash subsets and mobility chains in bimatrixgames”, Naval Logistics Research Quarterly 23: 311—319 (1976).[93] Jansen, M.J.M., “Regularity and stability of equilibrium points of bimatrix games”,Mathematics of Operations Research 6: 530—550 (1981).[94] Jansen, M.J.M., “Maximal Nash subsets for bimatrix games”, Naval Logistics Research Quarterly28: 147—152 (1981).[95J Johnson, R., (1985) “Networking and market entry in the airline industry”, Journalof Transport Economics and Policy 19:299—304 (1985).[96] Jones, P. and D. Hamilton, “Yield management: putting people in the big picture”,Cornell Hotel and Restaurant Administration Quarterly 32:89—95 (1992).[97] Jordan, W.J., “Heterogeneous users and the peak load pricing model”, QuarterlyJournal of Economics 97:127—138 (1983).[98] Joskow, P.J., “Contributions to the theory of marginal cost pricing”, Bell Journalof Economics 7:197—206 (1976).Bibliography 207[99] Kalish, S., “Monopolistic pricing with dynamic and production cost”, MarketingScience 2(2):135—159 (1983).[100] Kanafani, A. and A.A. Ghobrial, “Airline hubbing — some implications for airporteconomics”, Transportation Research A 19:15—27 (1985).[101] Kaplan, A., “Stock rationing”, Management Science 15:260—267 (1969).[102] Katz, M., “Firm—specific differentiation and competition among multiproductfirms”, Journal of Business 57: S149—S166 (1984).[103] Kendall, W.R. and C. Jordan, “Fare and income elasticities in the North Atlanticair travel market: economic and policy implications”, Transportation Journal, Summer (1989).[104] Kimes, S.E., “Yield management: a tool for capacity—constrained service firms”,Journal of Operations Management 4:348—363 (1989).[105] Kimes, S.E., “The basics of yield management”, Cornell Hotel and RestaurantAdministration Quarterly 30:14—19 (1989).[106] Kinberg, Y. and A.G. Rao, “Stochastic models of a price promotion”, ManagementScience 21(8):897—907 (1975).[107] Kiemperer, P. and M. Meyer, “Price competition vs. quantity competition: the roleof uncertainty”, Rand Journal of Economics 17:618—638 (1986).[108] Kling, J.A., C.M. Grimm and T.M. Corsi, “Hub—dominated airports: an empirical assessment of challenger strategies”, The Logistics and Transportation Review27(3):203—223 (1991).[109] Kraft, D.G.H., T.H. Oum and M.W. Tretheway, “Airlines seat management”, Logistics and Transportation Review 22:115—130 (1986).[110] Kreps, D.M. and J.A. Scheinkman, “Quantity precommitment and Bertrand competition yield Cournot outcome”, Bell Journal of Economics 14:326—337 (1983).[111] Kreps, V.L., “Bimatrix games with unique equilibrium points”, International Journal of Game Theory 3:115—118 (1974).[112] Kuhn, H.W., “An algorithm for equilibrium points in bimatrix games”, Proceedings,National Academy of Science U.S.A. 47:1656—1662 (1961).[113] Kunreuther, H. and J.F. Richard, “Optimal pricing and inventory decisions fornon—seasonal items”, Econometrica 39:173—175, 1971.Bibliography 208[114] Kunreuther, H. and L. Schrage, “Joint pricing and inventory decisions for constantpriced items”, Management Science 19:732—738, 1973.[115] Ladany, S.P., “Dynamic operating rules for motel reservations”, Decision Sciences7:827—840 (1976).[116] Ladany, S.P., “Bayesian dynamic operating rules for optimal hotel reservations”,Zeitschrift Opns. Res. 21: B165—B176 (1977).[117] Ladany, S.P and D.N. Bedi, “Dynamic booking rules for flight with intermediatestop”, OMEGA 5:721—730 (1977).[118] Ladany, S.P and M. Hersh, “Non—stop versus one—stop flights”, TransportationResearch 11:151—159 (1977).[119] Lambert, C.U. and J.M. Lambert and T. P. Cullen, “The overbooking question:a simulation”, Cornell Hotel and Restaurant Administration Quarterly 30(2):14—20(1990).[120] Lee, C.H. and E.H. Warren, “Rationing by seller’s preference and racial price discrimination”, Economic Inquiry 14:36—44 (1976).[121] Leland, H.E., “Theory of the firm facing uncertain demand”, American EconomicReview 62:278—291 (1971).[122] Leland, H.E. and R.A. Meyer, “Monopoly pricing structures with imperfect information”, Bell Journal of Economics 7(2):449—462 (1976).[123] Lemke, C.E. and J.T. Howson, Jr., “Equilibrium points of bimatrix games”, Journalof the Society of Industrial and Applied Mathematics 12:413—423 (1964).[124] Levine, M.E., “Airline competition in deregulated markets: theory, firm strategy,and public policy”, Yale Journal of Regulation 4:393—494 (1987).[125] Levitan, R. and M. Shubik, “Price duopoly and capacity constraints”, InternationalEconomics Review 13:111—122 (1972).[126] Lewis, R.C., “Consumer—based Hotel pricing”, Cornell Hotel and Restaurant Administration Quarterly 26(3): 18—21(1986).[127] Liberman, V. and U. Yechiali, “Hotel overbooking problem — inventory system withstochastic cancellations”, Advances in Applied Probability 9:220—??? (1977).[128] Liberman, V. and U. Yechiali, “On the hotel overbooking problem — an inventorysystem with stochastic cancellations”, Management Science 24:1117—1126 (1978).Bibliography 209[129] Lieberman, W., “Revenue enhancement opportunities through yield management”,Discussion Document, Arthur D. Little, Inc., San Francisco, CA (1992).[130] Lieberman, W., “Implementing yield management”, Presented at ORSA/TIMSAnnual Meeting, San Francisco (1992).[131] Littlewood, K., “Forecasting and control of passenger bookings”, AGIFORS Symp.Proc. 12:95—117 (1972).[132] Lott, J.R. Jr. and R.D. Roberts, “A guide to the pitfalls of identifying price discrimination”, Economic Inquiry 29:14—23 (1991).[133] Lovell, C.A. and K.L. Wertz, “Price discrimination in related markets”, EconomicInquiry 19:488—494 (1981).[134] Lovell, C.A. and K.L. Wertz, “Third degree price discrimination in imperfectlysealed markets”, Atlantic Economic Journal 13:1—11 (1985).[135] Luce, R.D. and H. Raiffa, Games and Decisons John Wiley and Sons, NY (1957).[136] Luenburger, D.G., Linear and Nonlinear Programming Second Edition, Addison-Wesley Publishing Company; Reading, MA (1984).[137] Mangasarian, O.L., “Equilibrium points of bimatrix games”, Journal of the Societyof Industrial and Applied Mathematics 12:778—780 (1964).[138] Mas-Colell, A., “A model of equilibrium with differentiated commodities”, Journalof Mathematical Economics 2:263—295 (1975).[139] Maskin, E. and J. Riley, “Monopoly with imperfect information”, Rand Journal ofEconomics 15:171—196 (1984).[140] Mayer, lvi., “Seat allocation, or a simple model of seat allocation vis sophisticatedones”, AGIFORS Symp. Proc. 16:103—135 (1976).[141] McCall, J.J., “Probabilistic microeconomics”, Bell Journal of Economics 2(2):403—433 (1971).[142] McCleary, K.W. and P.A. Weaver, “Are frequent—guest programs effective?” Cornell Hotel and Restaurant Administration Quarterly 31(3):39—45 (1991).[143] McGill, J., “Airline multiple fare class seat allocation”, Presented at FallORSA/TIMS Joint National Conference, Denver, Colorado (1988).Bibliography 210[144] McGill, J.I., Optimization and Estimation Problems in Airline Yield Management.Unpublished Doctoral Dissertation, Faculty of Commerce and Business Administration, the University of British Columbia, Vancouver, Canada (1989).[145] Mendelson, T., J.S. Pliskin and U. Yechiali, “A stochastic allocation problem”,Operations Research 28:687—693 (1980).[146] Meyer, J.R. and J.S. Strong, “From closed set to open set deregulation: an assessment of the U.S. airline industry”, The Logistics and Transportation Review28(1):1—21 (1992).[147] Miliham, C.B., “On the structure of equilibrium points in bimatrix games”, SIAMReview 10:447—449 (1968).[148] Millham, C.B., “Constructing bimatrix games with special properties”, Naval Research Logistics Quarterly 19:709—714 (1972).[149] Millham, C.B., “On Nash subsets of bimatrix games”, Naval Logistics ResearchQuarterly 21:307—317 (1974).[150] Mirman, L.J. and D. Sibley, “Optimal nonlinear prices for multiproduct monopolies”, Bell Journal of Economics 11:659—670 (1980).[151] Monroe, K.B. and A.J. Della Bitta, “Models for pricing decisions”, Journal ofMarketing Research 15:413—428 (1978).[152] Moorthy, K.S., “Market segmentation, self—selection, and production line design”,Marketing Science 3:288—307 (1984).[153] Moorthy, K.S., “Cournot competition in a differentiated oligopoly”, Journal ofEconomic Theory 36:86—109 (1985).[154] Moorthy, K.S., “Product and price competition in a duopoly”, Marketing Science7:141—168 (1988).[155] Morrison, S.A. and C. Winston, “Intercity transportation route structures underderegulation”, American Economic Review 75:57—61 (1985).[156] Morrison, S.A. and C. Winston, The Economic Effects of Airline Deregulation.Brookings Institute, Washington, D.C. (1987).[157] Morrison, S.A. and C. Winston, “The Dynamics of airline pricing and competition”,American Economics Review 80(2):389—393 (1990).[158] Mussa, M. and S. Rosen, “Monopoly and product quality”, Journal of EconomicTheory 18:301—317 (1978).Bibliography 211[159] Nagle, T., “Economic foundations for pricing”, Journal of Business 57:S3—S35(1984).[160] Nahmias, S., “Perishable inventory theory: a review”, Operations Research30(4):680—708 (1982).[161] Nahmias, S. and W.S. Demmy, “Operating characteristics of an inventory systemwith rationing”, Management Science 11:1236—1245 (1981).[162] Nash, J.F., “Equilibrium points in n—person games”, Proceedings, NationalAcademy of Science U.S.A. 36:48—49 (1950).[163] Nash, J.F., “Non—cooperative games”, Annals of Mathematics 54:286—295 (1951).[164] Nguyen, D.J., “The problems of peak loads and inventories”, Bell Journal of Economics 7:242—248 (1976).[165] Okada, -A., “Strictly perfect equilibrium points of bimatrix games”, InternationalJournal of Game Theory 13:45—154 (1984).[166] Oren, S.S., S. Smith, and R.B. Wilson, “Pricing a product line”, Journal of Business 57:S73—S100 (1984).[167] Orkin, E.B., “Boosting your bottom line with yield management”, Cornell Hoteland Restaurant Administration Quarterly 29(2):52—55 (1988).[168] Orkin, E.B., “Strategies for managing transient rates”, Cornell Hotel and Restaurant Administration Quarterly 30:34—39 (1990).[169] Osborne, W. and C. Pitchik, “Price competition in a capacity contrained duopoly”,Journal of Economic Theory 38:238—260 (1986).[170] Oum, T.H. and D.W. Gillen, “A study of inter—fareclass competition in airlinemarket”, Transportation Research Forum Proceedings, 599—609 (1980).[171] Oum, T.H. and D.W. Gillen, “Structure of intercity demands in Canada: theory,tests, and empirical result”, Transportation Research B 17:175—191 (1983).[172] Oum, T.H., D. Gillen and D. Nobel, “Demand for fareclasses and pricing in airlinemarkets”, Logistics and Transportation Review 23:195—222 (1986).[173] Oum, T.H. and M.W. Tretheway, Airline Economics. Published by Institute ofTransportation Studies, University of British Columbia, Vancouver, Canada (1992).Bibliography 212[174] Oum, T.H., A. Zhang and Y. Zhang, “Inter—firm rivalry and firm—specific demandin the deregulated airline markets”, Working Paper 91—TRA—008, Faculty of Commerce, University of British Columbia, Vancouver, Canada (1991).[175] Owen, G., “Optimal threat strategies of bimatrix games”, International Journal ofGame Theory 1:3—9 (1971).[176] Panzar, J., “Equilibrium and welfare in unregulated airline markets”, AmericanEconomic Review (Papers and Proceedings) 69:92—95 (1979).[177] Parthasarathy, T. and T.E.S. Raghavan, Some Topics in Two—person Games.American Elsevier Publishing Company, Inc., NY (1971).[178] Perloff, J.M. and S.C Salop, “Equilibrium with product differentiation”, Review ofEconomic Studies LII:107—120 (1985).[179] Perry, M., “Sustainable positive profit multiple-price strategies in contestable markets”, Journal of Economic Theory 32:246-265 (1984).[180] Pfeifer, P.E., “The airline discount fare allocation problem”, Decision Sciences20:149—157 (1989).[181] Phillps, L., The Economics of Price Discrimination Cambrudge University Press,Cambridge (1983).[182] Pigou, A.C., Economics of Welfare MacMillan, London (1920).[183] P’ng, I.P.L., “Reservations: customers insurance in marketing of capacity”, Marketing Science 8:248—264 (1989).[184] P’ng, I.P.L and D. Hirshleifer, “Price discrimination through offers to match price”,Journal of Business 60:365—383 (1987).[185] Pressman, I., “A mathematical formulation of the peak load problem”, Bell Journalof Economics and Management Science 1:304—326 (1970).[186] Pustay, M.W., “Toward a global airline industry: prospects and impediments”,The Logistics and transportation Review 28(1):103—128 (1992).[187] Pustay, M.W., “Six myths of contemporary airline industry”, Transportation Research Forum 32(2):286—291 (1992).[188] Raghavan, T.E.S., “Completely mixed strategies in bimatrix games”, Journal ofLondon Mathematics Society 2:709—712 (1970).Bibliography 213[189] Rajan, A.R. and R. Steinberg, “Dynamic pricing and ordering decisons by a monopolist”, Management Science 38(2):240—262 (1992).[190] Rao, V.R., “Pricing research in marketing: the state of the art”, Journal of Business57:S38—S64 (1984).[191] Reagan, P.B., “Inventory and price behaviour”, Review of Economic Studies47:137—142 (1982).[192] Reiss, P.C. and P.T. Spiller, “Competition and entry in small airline markets”,Journal of Law and Economics 32:S179—S202 (1989).[193] Relihan, W.J.III., “The yield management approach to hotel—room pricing”, Cornell Hotel and Restaurant Administration Quarterly 30(1):40—45 (1989).[194] Richter, H., “The Differential revenue method to determine optimal seat allotmentsby fare type”, AGIFORS Symp. Proc. 22:339—362 (1982).[195] Robinson, L.W., “Optimal and approximate control policies for airline booking withsequential fare classes”, Working Paper, Johnson Graduate School of Management,Cornell University (1991).[196] Rockafellar, R.T., Convex Analysis Princeton University Press (1970).[197] Ross, S.M., Stochastic Process John Wiley and Sons, New York (1982).[198] Ross, S.M., Introduction to Stochastic Dynamic Programming Academic Press, NewYork (1983).[199] Rothstein, M., “An airline overbooking model”, Transportation Science 5:180—192(1971).[200] Rothstein, M., “Airlines overbooking: the state of art”, Journal of Transport Economics and Policy 5:96—99 (1971).[201] Rothstein, M., “Hotel overbooking as a markovian sequential decision process”,Decision Sciences 5:389—404 (1974).[202] Rothstein, M., “Airlines overbooking: fresh approaches are needed”, TransportationScience 9:169—173 (1975).[203] Rothstein, M., “O.R. and the airline overbooking problem”, Operations Research33:392—435 (1985).[204] Salop, S., “The noisy monopolist: imperfect information, price disperson and pricediscrimination”, Review of Economic Studies 44:393—406 (1977).Bibliography 214[205] Salop, S., “Monopolistic competition with outside goods”, Bell Journal of Eco—nomics 10:141—156 (1979).[206] Salop, S. and J.E. Stiglitz, “Bargains and ripoffs: A model of monopolisticallycompetitive price disperson”, Review of Economic Studies 44:493—510 (1977).[207] Salop, S. and J.E. Stiglitz, “The theory of sales: A simple model of equilibriumprice diperson with identical agents”, American Economic Review 72(5):1121—1130(1982).[208] Shaked, A. and J. Sutton, “Relaxing price competition through product differentiation”, Review of Economic Studies 49:3—13 (182).[209] Shaked, A. and J. Sutton, “Natural oligopolies”, Econometrica 51(5): 1469—1483(1983).[210] Shapiro, C., “Theories of Oligopoly Behaviour”, Chapter 6 of Handbook of Industrial Organization Vol.1, 329—414; edited by R. Schmalensee and R.D. Willig.Elsenvier Science Publisher, B.V. (1989).[211] Shapley, L.S., “A note on the Lemke—Howson algorithm”, Mathematical Programming Study 1:175—189 (1974).[212] Shapley, L.S. and R.N. Snow, “Basic solutions of discrete games”, Annals of Mathematics Studies 24:27—35 (1950).[213] Shaw, S., Airline Marketing and Management 3rd Edition, Pitman Publishing,London (1990).[214] Shlifer, R. and Y. Vardi, “An airline overbooking policy”, Transportation Science9:101—114 (1975).[215] Simon, J., “An almost practical solution to airline overbooking”, Journal of Transport Economics and Policy 2:201—202 (1968).[216] Simon, J., “Airline overbooking: the state of art — a reply”, Journal of TransportEconomics and Policy 6:254—256 (1972).[217] Simpson, R.W., “Setting optimal booking levels for flight segments with multi—class, multi—market traffic”, AGIFORS Symp. Proc. 25:263—279 (1985).[218] Simpson, R.W., “Theoretical concepts for capacity/yield management”, AGIFORSSymp. Proc. 25:281—293 (1985).[219] Singh, N. and X. Vives, “Price and quantity competition in a differentiatedduopoly”, Rand Journal of Economics 15:546—554 (1984).Bibliography 215[220] Smith, B.C., “A group decision support model”, AGIFORS Symp. Proc. 30:27—39(1990).[221] Smith, B.C., J.F. Leimkuhler and R.M. Darrow, “Yield management at AmericanAirlines”, Interface 22:8—31 (1992).[222] Smith, B.C. and C.W. Penn, “Analysis of alternate origin—destination controlstrategies”, AGIFORS Symp. Proc. 28:123—143 (1988).[223] Sorenson, N., Airline Competitive Strategy: A Spatial Perspective. UnpublishedDoctoral Dissertation, Department of Geography, University of Washington, Seattle,WA. (1990).[224] Spence, A.M., “Nonlinear prices and welfare”, Journal of Public Economics 8:1—18(1977).[225] Spiller, P.T., “Pricing of hub—and—spoke networks”, Economics Letters 30:165—169(1989).[226] Stadie, W., “A full information pricing problem for sales of serveral identical commodities”, Z. Operations Research 34:161—169 (1990).[227] Stein, A.A., “Pricing and differentiation strategies”, Planning Review 17(5):30—34(1989).[228] Steiner, P., “Peak loads and efficient pricing”, Quarterly Journal of Economics71:585—610 (1957).[229] Stephenson, F.J. and R.J. Fox, “Corporate attitudes toward frequent—flier programs”, Transportation Journal 27:10—22 (1987).[230] Stokey, N., “Intertemporal price discrimination”, Quarterly Jounarl of Economics93:355—371 (1979).[231] Stokey, N., “Rational expectations and durable goods pricing”, Bell Journal ofEconomics 12:112—128 (1981).[232] Stone, R. and M. Diamond, “Optimal inventory control for a single flight leg”, StaffPaper, Operations Research Division, Northwest Airlines (1992).[233] Strassmann, D.L., “Potential competition in the deregulated airlines”, Review ofEconomics and Statistics (Netherlands) 72(4):696—702 (1990).[234] Taylor, C.J., “The determination of passenger booking levels”, AGIFORSSymp.Proc. 2:93—116 (1962).Bibliography 216[2351 Thompson, H.R., “Statistical problems in airline reservations control”, OperationalResearch Quarterly 12:167—185 (1961).[236] Tirole, J., The Theory of Industrial Organization The MIT Press, Cambridge, MA(1988).[237] Titze, B. and R. Griesshaber, “Realistic passenger booking behaviours and simplelow-fare/high-fare seat allotment model”, AGIFORS Symp. Proc. 23:197—223 (1983).[238] Toh, R., “Airline revenue yield protection: Joint reservation control over full andlow fare sales”, Transportation Journal Winter 74—80 (1979).[239] Toh, R., “An inventory depletion overbooking for the hotel industry”, Journal ofTravel Research 24:24—30 (1985).[240] Toh, R.S. and M.Y. Hu, “Frequent—flier programs: passenger attributes and attitudes”, Transportation Journal 29:11—22 (1988).[241] Toh, R.S., M.K. Kelley and M. Y. Hu, “An approach to the determination ofoptimal airline fares: some useful insights on price elasticities, monopoly power andcompetitive factors in the airline industry”, Journal of Travel Research 25(l):26—33(1986).[242] Toh, R.S. and M.J. Rivers, “Frequent—guest programs: do they fly?” Cornell Hoteland Restaurant Administration Quarterly 3l(3):46—52 (1991).[243] Topkis, D.M., “Optimal ordering and rationing policies in a non—stationary dynamic inventory model with n demand classes”, Management Science 15:160—176(1968).[244] Tretheway, M.W., “Frequent flyer programs: marketing bonanza or anti—competitive tool”, Proceedings of Canadian Transportation Research Forum, University of Saskatchewan Printing Service (1989).[245] Tretheway, M.W., “The characteristics of modern post—deregulation air transport”,Working Paper, Faculty of Commerce and Business Administration, University ofBritish Columbia; Vancouver, Canada (1991).[246] Tretheway, M.W. and T.H. Oum, Airline Economics: Foundations for Strategy andPolicy Centre for transportation Studies, University of British Columbia, Vancouver,Canada (1992).[247] Turvey, R., “Peak load pricing” Journal of Political Economy 76:101—113 (1968).Bibliography 216[235] Thompson, H.R., “Statistical problems in airline reservations control”, OperationalResearch Quarterly 12:167—185 (1961).[236] Tirole, J., The Theory of Industrial Organization The MIT Press, Cambridge, MA(1988).[237] Titze, B. and R. Griesshaber, “Realistic passenger booking behaviours and simplelow-fare/high-fare seat allotment model”, AGIFORS Symp. Proc. 23:197—223 (1983).[238] Toh, R., “Airline revenue yield protection: Joint reservation control over full andlow fare sales”, Transportation Journal Winter 74—80 (1979).[239] Toh, R., “An inventory depletion overbooking for the hotel industry”, Journal ofTravel Research 24:24—30 (1985).[240] Toh, R.S. and M.Y. Hu, “Frequent—flier programs: passenger attributes and attitudes”, Transportation Journal 29:11—22 (1988).[241] Toh, R.S., M.K. Kelley and M. Y. Ru, “An approach to the determination ofoptimal airline fares: some useful insights on price elasticities, monopoly power andcompetitive factors in the airline industry”, Journal of Travel Research 25(l):26—33(1986).[242] Toh, R.S. and M.J. Rivers, “Frequent—guest programs: do they fly?” Cornell Hoteland Restaurant Administration Quarterly 31(3):46—52 (1991).[243] Topkis, D.M., “Optimal ordering and rationing policies in a non—stationary dynamic inventory model with n demand classes”, Management Science 15:160—176(1968).[244] Tretheway, M.W., “Frequent flyer programs: marketing bonanza or anti—competitive tool”, Proceedings of Canadian Transportation Research Forum, University of Saskatchewan Printing Service (1989).[245] Tretheway, M.W., “The characteristics of modern post—deregulation air transport”,Working Paper, Faculty of Commerce and Business Administration, University ofBritish Columbia; Vancouver, Canada (1991).[246] Tretheway, M.W. and T.H. Oum, Airline Economics: Foundations for Strategy andPolicy Centre for transportation Studies, University of British Columbia, Vancouver,Canada (1992).[247] Turvey, R., “Peak load pricing” Journal of Political Economy 76:101—113 (1968).Bibliography 217[248] Varian, H.R., Microeconomic Analysis Second Edition, W.W. Norton and Company, New York (1984).[249] Varian, R., “Price Discrimination”, Chapter 11 of Handbook of Industrial Organization Vol.1 edited by R. Schmalensee and R.D. Willig. Elsenvier Science Publisher,B.V. (1989).[250] Vickrey, W., “Airline overbooking: some further solutions”, Journal of TransportEconomics and Policy 6:257—270 (1972).[2511 von Neumann, J. and 0. Morganstern, Theory of Games and Economic Behavior,3rd Edition Princeton University Press, Princeton, N.J. (1953).[252] Vorob’ev, N.N., “Equilibrium points in bimatrix games”, Theory of Probability andits Applications 3:297—309 (1958).[253] Wang, K., “Optimum seat allocation for multi—leg flights with multiple fare types”,AGIFORS Symp. Proc. 23:225—246 (1983).[254] Weatherford, L.R., Perishable Asset Revenue Management in General BusinessSituations Unpublished Doctoral Dissertation, Darden Graduate School of Business,University of Virginia; (1991).[255] Weatherford, L.R. and S.E. Bodily, “A taxonomy and research overview ofperishable—asset revenue management: yield management, overbooking, and pricing”, Operations Research ??:???—??? (1992).[256] Weatherford, L.R., S.E. Bodily and P.E. Pfeifer, “Modelling the customer arrivalprocess and comparing decision rules in perishable asset revenue management situations”, to appear in Transportation Science, (1993).[257] Wenders, J.T., “Peak load pricing in Electric Utility Industry”, Bell Journal ofEconomics 7:232—241 (1976).[258] Wernerfelt, B., “A special case of dynamic pricing policy”, Managament Science32:651—659 (1986).[259] William, F.E., “Decision theory and innkeeper: an approach for setting hotel reservation policy”, Interfaces 7:18—30 (1977).[260] Williamson, E.L. and P.P. Belobaba, “Optimization techniques for seat inventorycontrol”, AGIFORS Symp. Proc. 28:153—170 (1988).[261] Williamson, O.E., “Peak load prk*ig and optimal capacity under indivisibilityconstraint”, American Economic Review 56:810—827 (1966).Bibliography 218[262] Wilson, C.A., “On the optimal pricing of a monopolist”, Journal of Political Economy 96:164—176 (1988).[263] Wilson, R., “Efficient and competitive rationing”, Econometrica 57:1—40 (1989).[264] Wolinsky, A., “Product differentiation with imperfect information”, Review of Economic Studies 51:53—61 (1984).[265] Wollmer, R.D., “A hub—spoke seal management model”, Unpublished companyreport, Douglas Aircraft Company, McDonnell Douglas Corporation, Long Beach,CA (1986a).[266] Wollmer, R.D., “An airline reservation model for opening and closing fare classes”,Unpublished Company Report, Douglas Aircraft Company, Long Beach, CA(1986b).[267] Wolimer, R.D., “A seat management model for a single leg route”, UnpublishedCompany Report, Douglas Aircraft Company, Long Beach, CA (1987).[268] Wollmer, R.D., “A seat management model for a single leg route when lower fareclasses book first”, Presented at Fall ORSA/TIMS Joint National Conference. Denver, Colorado (1988).[269] Wollmer, R.D., “An airline seat management model for single leg route when lowfare classes book first”, Operations Research 40:26—37 (1992).[270] Wong, J.T., Airline Network Seat Allocation. Unpublished Doctoral Dissertation,Northwestern University, Evanston, IL (1990).[271] Wong, J.T., F.S. Koppleman and M.S. Daskin, “Flexible assignment approach toitinerary seat allocation”, Transportation Research Ser.B ??:??—?? (1992).[272] Zhang, A., Essays on Strategic and Contractual Relationships in Oligopoly Unpublished Doctoral Dissertation, Faculty of Commerce, the University of BritishColumbia; Vancouver, Canada (1990).

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