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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 MARKETING RESTRICTIONS WITH APPLICATIONS TO AIRLINES By Michael Zhi-Feng Li B. Sc. (Mathematics) Beijing Normal University Ph. D. (Mathematics) University of Regina  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  ill  THE FACULTY OF GRADUATE STUDIES FACULTY OF COMMERCE AND BUSINESS ADMINISTRATION  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  April 1994  ©  Michael Zhi-Feng Li, 1994  4  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Faculty of Commerce and Business Administration The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1Z1  Date:  /,(i  Abstract  This thesis addresses the problem of pricing perishable inventories such as airline seats and hotel rooms. It also analyzes the airline seat allocation problem when two airlines compete on a single-leg flight. Finally, several existing models for seat allocation with multiple fares on a single-leg flight are compared. The pricing framework is consistent with modern yield management tools which uti lize restrictions such as weekend stayover to segment the market. One model analyzed considers a restriction which is irrelevant to one set of consumers, but which the others find so onerous that they will not purchase a restricted ticket at any price. If the con sumers who do not mind the restriction are less price sensitive than those who find the restriction onerous, then the thesis shows that there is an optimal policy for a monopolist which 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 other in the sense that the set of consumers who would not buy a ticket with the first restriction is a subset of those who would not buy it with the second restriction, then the restrictions are said to be nested. If the sets of consumers who would not buy tickets with the first restriction is disjoint from those who would not buy with the second restriction, then the restrictions are said to be mutually exclusive. If two restrictions are either nested or mutually 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 monopolist may need five price levels with four types of product. The pricing model is applied to restrictions which are based on membership in a particular organization.  For example, employees of an airline are frequently eligible 11  for special fares. Some airlines provide special fares for government employees or for employees of certain corporations. An analysis is given to help airlines understand the costs and benefits of such arrangements. A model of two airlines competing on a single-leg flight is developed for the case where 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 fares  which they sell. The split of the market between the airlines is modelled in two different ways. First, the airlines might share the market for a fare class proportionally to their allocation of seats to that fare class. In this case, under certain conditions, there exists an equilibrium pair of booking limits for the discount fare such that each airline will protect the same number of seats for the full fare customers, even when the demands are random and stochastically dependent. The second market sharing model assumes that the two airlines share the market demand equally. In this case, when the demands are deterministic, then there is an equilibrium solution where each airline will protect enough seats to split equally the market for the full fares. Finally, three existing seat allocation models for multi-fare single-leg flights with stochastically independent demands are compared. It is shown that the optimality con ditions for each of these models are analytically equivalent, thus providing a unified approach to this problem.  111  Table of Contents  ii  Abstract  viii  List of Tables  ix  Acknowledgement 1  1.1  Introduction  1  1.2  Airline Seat Inventory Control  4  1.3  1.4 2  1  Introduction and Background  1.2.1  The Seat Allocation Problem for Single-leg Flights  4  1.2.2  The Seat Allocation Problem for Multi-leg Flights  5  1.2.3  The Seat Allocation Problem for Multiple Flights  9  1.2.4  The Overbooking Problem  9  1.2.5  The Seat Allocation Problem in the Presence of Competition  11  Airline Fare Pricing  12  1.3.1  Introduction  12  1.3.2  Current Operating Environment  13  1.3.3  Modelling Practice  16  1.3.4  New Directions  21  Objectives and an Overview of This Thesis  Pricing Perishable Inventories by Using a Restriction 2.1  .  Introduction  22 30 30  iv  2.2  2.3  3  Rationing Rules and Wilson’s Pricing Model  32  2.2.1  Rationing Rules  32  2.2.2  Wilson’s Model  34  An Optimal Pricing Model by Use of Restrictions  42  2.3.1  The Model Settings  42  2.3.2  The Demand Constraints  46  2.3.3  The Formulation of the Optimal Pricing Model  47  2.3.4  Basic Properties of Optimal rn-Policies  48  2.4  Optimal Pricing Strategies  55  2.5  An Application to Airline Fare Pricing  69  2.6  Summary  —  Pricing by Using Restriction  General Optimality Results and Other Properties  75 76  3.1  Introduction  76  3.2  Auxiliary Results on General Pricing Policies  77  3.3  3.2.1  Notation and Definitions  77  3.2.2  A Lemma on the Property of Monotonicity  83 85  General Optimality Results 3.3.1  Three Fundamental Lemmas  3.3.2  The General Optimality Theorems  3.4  Further Properties of Optimal Policies  3.5  Summary  —  General Optimality Results  4 Pricing Models with Two Types of Restrictions  85 102 107 113 115  4.1  Introduction and Model Setting  115  4.2  Pricing Problem by Using Two Nested Restrictions  118  4.3  Pricing Problem by Using Two Mutually Exclusive Restrictions  125  V  5  4.4  Pricing Problem by Using Two General Restrictions  4.5  Summary  —  131 145  Pricing by Using Two Types of Restrictions  147  Airline Pricing by Using Membership and Product Restrictions 5.1  Introduction  147  5.2  Model Setting and Notation  150  5.3  Cheaper Restricted Membership Fares Only  151  5.4  Cheaper Unrestricted Membership Fares Only  156  5.5  Cheaper Restricted Membership Fares and Cheaper Unrestricted Mem 159  bership Fares Membership and Product Restrictions  164  Seat Allocation Game on Flights with Two fares  166  5.6 6  Summary  —  6.1  Introduction and Model Setting  166  6.2  Seat Allocation Game Under Proportional Splitting Rule  170  6.3  Seat Allocation Game Under Equal Splitting Rule with Deterministic De 176  mand 6.4 7  Summary  —  180  Seat Allocation Game  181  A Note On Three Models for Multi-fare Seat Allocation Problem 7.1  Introduction  181  7.2  Wollmer’s Model  182  7.3  Curry’s Model  183  7.4  Brumelle-McGill’s Model  185  7.5  Equivalence of Optimality Conditions  187  7.6  Summary  —  On Three Model for Multi-fare Seat Allocation Problem  vi  .  191  8  Summary of the Thesis and Future Directions  192  8.1  Summary of the Thesis  192  8.2  Future Directions  194  8.2.1  On Seat Allocation Problems  194  8.2.2  On Seat Allocation Games  195  8.2.3  On the Pricing Problem by Using Restrictions  196  8.2.4  On Further Applications  198 200  Bibliography  vii  List of Tables  1.1  An Example of Fare Classes for a Two-leg Flight  8  1.2  An Example of Virtual Nesting Classes for a Two-leg Flight  8  2.1  Effect of Fare Restrictions  2.2  Solutions of the Tight Problem  —  Boeing Company (1982) —  A Numerical Example  viii  70 74  Acknowledgement  With sincere appreciation I first acknowledge the advice and encouragement from the members of my supervisory committee, Professors Shelby L. Brumelle, Tae H. Oum and Michael W. Tretheway. In particular, I want to thank my supervisor, Professor Shelby L. Brumelle, for his tireless effort of helping me to finish this dissertation. I have also benefited a great deal from Professor Michael W. Tretheway for helping me to improve the presentation of this thesis. I am deeply grateful to the Faculty of Commerce and Business Administration of the University of British Columbia for supporting my first two years of graduate studies through a MacPhee Memorial Fellowship. I feel very fortunate that I had been awarded a University Graduate Fellowship by the Faculty of Graduate Studies of the University of British Columbia during the early stage of this dissertation. I am also indebted to Professor Shelby L. Brumelle for supporting me in the last two years of my study.  ix  Chapter 1  Introduction and Background  1.1  Introduction  Since the first important paper of Littlewood (1972,[131]) on the airline seat allocation problem, the subject of yield management has become a main topic for revenue enhance ment for most airlines in the world. For example, American Airlines, a leader in airline yield management research and implementation, has experienced a tremendous success in the last decade or so due at least in part to its yield management system. It is estimated that during the period of 1990 to 1992, the annual benefit to American Airlines from its yield management programs was approximately 500 million dollars, or about a 2 percent 1 improvement on total passenger revenues. Yield management is far from completely understood. For example, yield manage 2 But actually, yield manage ment is sometimes considered to be a pricing mechanism. ment is a tool to assist firms in determining the maximum number of reservations to accept at given price levels so as to maximize total expected revenues. This is not the same as setting prices  —  prices are taken as given. The research so far on yield manage  ment shows no indication that it is capable of providing the optimal set of prices for the firm. In fact, any seat inventory control system must have a set of pre-specified prices so 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. 2 T his view is often held by professionals from hotel industry, for example, Relihan (1989,[193]) and Ranks et al (1992, [83]).  1  Chapter 1. Introduction and Background  2  3 The first goal decisions are traditionally made separately, by a different department. of this thesis is to enhance seat management techniques by developing a tactical pricing model for perishable inventories like airline seats and hotel rooms. In fact, the main part of this thesis is dedicated to this goal. The key innovation is to explicitly incorporate the use of artificial restrictions as marketing mechanisms. In particular, the questions I want to get answers include: (1) is it necessary to use the restrictions? (2) what are the optimal pricing structures? (3) do we have a tractable characterization for the optimal pricing structures? (4) how can the model be used in industries, such as airlines? Another shortcoming of the existing yield management literature is that airlines do not operate in isolation of each other. The existing research in yield management still does not address the issue of competitive response to other airlines’ seat allocation and pricing decisions. To analyze the strategic aspect of airline operations, we need to understand the nature of the strategic interaction among airlines. An early view was that the airline 4 But empirical studies show that the airline market is industry is a contestable industry. not perfectly contestable, which implies that the strategic interaction among airlines is oligopolistic. 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 the primary strategic variable; and (3) collusion, where firms collude with each other to exploit their monopolistic power. A recent empirical study by Brander and Zhang (1990, [33]) shows that the competition among airlines is a quantity-type competition. This evidence suggests that it is important to develop a new generation of seat allocation There is a trend that some airlines now use a single department to handle pricing and yield man 3 agement together. This suggests that airlines are starting to realize the importance of coordinating the pricing decisions and inventory control decisions. An industry is contestable when firms charge the same prices and supply the same quantities as a 4 competitive industry would, even there are only a few firms in the industry. A key argument that the airline industry is contestable is that airplanes the key assets for airlines are mobile. —  —  Chapter 1. Introduction and Background  3  models which explicitly consider competitive quantity decision responses. The second goal of this thesis is to make the first step toward a full understanding of the seat allocation problem in the presence of competition. It is well-known that modern yield management techniques critically depend on a proper understanding of the seat allocation problem for single-leg flights with multiple fares. The most recent development along this direction include three independently developed articles, Curry (1990, [52]), Woilmer (1992, [269]), and Brumelle and McGill (1993, [38]), which solve the seat allocation problem for single-leg flights with multiple fares when the random demands for different fare classes are independent. It is inter esting to note that these three papers have utilized three different analytical tools to determine optimal booking policies; and they have three different optimality rules for booking policies. This raises the issue as to which approach is the best and which one has a computational advantage over the other two. A later part of this thesis clarifies this issue by showing that all three approaches are in fact analytically equivalent in the sense that the optimal policy rules in fact are identical. The thesis provides a unified approach that may shed useful light in dealing with the seat allocation problem for multi-fare single-leg flights with dependent random demands. Before moving on to the main development of the thesis, brief overviews of airline seat inventory control and airline fare pricing are presented. These overviews provide the necessary background information needed to evaluate the contributions of this thesis and serve as motivation for the general class of problems to be addressed.  4  Chapter 1. Introduction and Background  Airline Seat Inventory Control  1.2  The Seat Allocation Problem for Single-leg Flights  1.2.1  This is the most studied area in yield management. For a flight with two fare classes with independent demand, an optimal booking policy has been characterized by Littlewood (1972, [131]). He proves that the optimal protection level  7)  5 is given for the high fare  by: *Mi{Offp(y> k—)} where  fj  and fh are the fare price for a low fare seat and a high fare seat respectively, Y is  the demand for the high fare class, k is the flight capacity, and P(.) denotes a probability distribution. The formula gives a decision rule that tells an airline reservations controller what to do when a booking request for the low fare arrives. Since the airline will not reject a high fare booking, the issue is when to stop booking the low fare tickets. The formula simply says that the airline should accept bookings for the low fare until the expected seat revenue from an uncertain high fare sale is greater than the certain low fare revenue  6 f.  If the demands are continuous random variables, then we have the following  closed-form formula: fi  =  fhP(Y> k  —  This result was generalized by Brumelle et al (1990, [37]) to the case of two-fare flights with dependent demands. The seat allocation problem for single-leg flights with multiple fares with independent demands has been solved independently by Brumelle and McGill (1993, [38]), Curry A proleclion level for the high fare is the total number of seats that will be proecied (i.e., reserved) 5 for the high fare customers. J is worthwhile to mention that the main reason why the Littlewood’s formula involves the sure 6 is that each low fare booking, when it arrives, corresponds to a sure value of a low fare ticket fi sale. On the other hand, if there is no booking request for the low fare, then the decision of rejecting or accepting 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]) introduced an 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 based on Littlewood’s result. Consequently, if the random demands are independent, then the EMSR method gives optimal solutions for any two fare classes in isolation, but unfortunately this will not be optimal for the whole problem. 7 The current challenge is to address the seat allocation problem for general multi-fare flights (that is, not limited to two fares) with dependent demands. 1.2.2  The Seat Allocation Problem for Multi-leg Flights  This problem turns out to be a major headache for all airlines. If we ignore the ran domness of the demands, then the seat allocation problem for multi-leg flights can be formulated as a network flow optimization problem. This had been a main focus of 8 But there are two main drawbacks to this approach: research for one decade or so. 1. It cannot handle random demands; 9 2. It is of little use for modern computer reservation systems. To my knowledge, there is no major airline in the world that has actually implemented this approach. What the airlines need is a framework that gives either optimal or heuristic solutions and is compatible with their current reservations systems. Simulation studies show that (1) the difference between the revenue derived from the EMSR method 7 and the optimal revenue is oniy 0.5 percent; and (2) the optimal booking policies may be drastically different from the policies derived from the EMSR heuristic. Refer to Brumelle et al (1990, [37]) for a reason why revenue functions are insensitive to booking limits. For example, refer to Ladany and Hersh (1977, [118]), Hersh and Ladany (1978, [89]), Glover et al 8 (1982, [80]), and Dror, Trudeau and Ladany (1988, [63]). This will become more clear from the discussions that follow. 9  Chapter 1. Introduction and Background  6  There are two other basic approaches in dealing with the seat allocation problem for multi-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 on each individual flight segment, independent of other flight segments. Currently, most airlines manage seat inventories by flight segment only. A typical segment-based seat allocation model involves two stages: 1. The airline first decides how many seats that will be made available for each seg ment; and 2. With the given number of seats allocated to each segment, the airline specifies booking policies for fare classes on each specific segment. As claimed in Williamson and Belobaba (1988, [260]), several major airlines had imple mented the EMSR method, which was originated by Littlewood for single-leg flights and was later extended by Belobaba (1987, [12]) for multi-leg flights. The main idea from the EMSR model is to protect seats for a higher fare class as long as the expected marginal revenue 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 for each fare or each nest” on each flight segment. The main drawback for the segment-based approach 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 legs A-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 flight A-B via H, the airline may pooi all fare classes on the segment H-B into a single nest. Clearly, on each segment, 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 information for all these 45 fare classes regularly. Therefore, grouping fares on each segment into 4 or 5 different nests will substantially reduce the amount of information that needs to be displayed on the CRS. This is precisely why the deterministic network approach is not implementable in the current reservation systems.  Chapter 1. Introduction and Background  7  within the network in the determination of seat allocations and booking limits. On the other hand, the major advantage of a segment-based approach is its compatibility with current airline reservations systems. 12 Revenue-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 of each fare class, where a lower index means a higher priority;’ 3 and • 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  —  are  displayed on the reservations systems. This is why this method is often known as the virtual 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 procedure of updating the indices during the booking period; and • Finding the protection levels, or equivalently the booking limits, for each bucket to maximize total revenues of the flight. There are several extensions for Belobaba’s EMSR methods, for example, refer to Wong (1990, 12 [270]), Belobaba (1991, [15]), and most recently Wong et al (1992, [271]). This is a critical component of any revenue-based method. Unfortunately, there is no public literature 3 ‘ that addresses this issue. In the meantime, it is worthwhile to mention that this problem is also an interesting theoretical issue. Clearly, the most efficient and consistent way of indexing fare classes on the whole network is to have an index table that include all fare classes on all flights on the network, rather than indexing fare classes on a flight-by-flight basis. So the theoretical challenge here is to develop a model for such an universal indexing table. Of course, the first question is whether or not such a table is possible. Relevant papers on this problem include Buhr (1982, [41]), Wang (1983, [253]), Simpson (1985, 14 [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  Fare Glass Y M  Q B  Segment A—B $1000 $800 $540 $330  8  Segment A—H Segment H—B $700 $750 $470 $500 $290 $300 $200 $220  Table 1.1: An Example of Fare Classes for a Two-leg Flight  Buckets (Virtual Classes)  O-D Classes  0 V  YAB YAH, MAB, YHB MAH, BAB, MHB  V 1 2 V 3 V 4 V  BAH,QAB,BHB QAH,QHB  Table 1.2: An Example of Virtual Nesting Classes for a Two-leg Flight  According to Smith et al (1992, [221]), American Airlines have also used the EMSR method 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 hub city 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 into , V 1 , V 2 , and V 3 . Refer to Table 1.2 for one specification of 4 , V 0 five buckets, labeled as V these five buckets. In the actual booking process, suppose there is a request for the fare class M on segment H-B of Flight 101. When a travel agent looks at the system and if , the agent can authorize this booking. 2 there still are some seats available for bucket V But during the reservation process, the actual fare class up.  MHB  on Flight 101 never shows  Chapter 1. Introduction and Background  9  The Seat Allocation Problem for Multiple Flights  1.2.3  Equally important, airlines need to address the seat allocation problem in the presence of other flights.  This becomes increasingly important because of the hub-and-spoke  networks airlines have developed especially since U.S. airline deregulation in 1978. Most flights originating from a hub will pick up a substantial amount of connecting traffic from other flights destined to the hub. Without explicitly incorporating the demand from the connecting traffic, a yield management system cannot realize the full potential revenue for each flight. For example, consider Flight 201 from city H (the hub) to city B and Flight 202 from city A to city H. There are two key questions that are important to airlines: • How much should the airline charge a traveller who flies from city A to city B using flights 201 and 202 with a restricted ticket? And how much should be charged for an unrestricted ticket? • How many seats should be allocated on flight 202 for travellers from city A to city B? These problems are closely related, but are not simple. Along this direction, except for 15 the network flow approach, there is no other theoretical work done in this area. 1.2.4  The Overbooking Problem  Since airline seats cannot be consumed before the time of the flight, they are sold in the form of reservations. This causes an operating problem for airlines, since some customers will either cancel their reservations at the last minute or simply not show up. Since the Refer to Ladany (1977, [1161), Hersh and Ladany (1978, [89]), Glover et al (1982, [80]) and Dror et 5 ‘ al (1988, [63]).  Chapter 1. Introduction and Background  10  16 airline has no time to resell these tickets, some seats will be unused. With overbooking, the airline can generate additional revenue on these otherwise empty seats. The airline overbooking problem has two important components: • Determining the optimal overbooking levels, which are associated with the booking limits for fare classes or buckets; and • In the event that the flight is overbooked at the departure time, specifying a pro cedure for denying boardings to some passengers. The first part is an optimization problem. Along this direction, pioneering research can be traced back to Beckmann (1958a, [9]), Beckmann (1958b, [10]), Thompson (1961, [235]) and Taylor (1962, [234]). Further analytical development has been studied in 17 For most recent development, refer to Rothstein (1985, many papers in the seventies. [203]), Brumelle et al (1990, [37]), and Bodily and Pfeifer (1992,  [24]).18  The second issue is an implementation problem. The fact is that when no-shows and cancellations are low, the flight becomes truly oversold, which means that the number of passengers expecting to board the flight is larger than the flight seat capacity. Con sequently, some passengers must be bumped, or denied boarding. This issue had been a focal point in the early stage of overbooking practice. The early practice of involuntary bumping created some serious consequences, including lawsuits against airlines. Even tually, the Simon-auction method, proposed by Simon (1968, [215]), was implemented by many airlines. The basic idea of a Simon-auction is: the airline asks passengers who do not mind being bumped to submit a sealed bid for compensation and the airline will Simply speaking, we say that an airline overbooks a flight if during the booking period, the total 16 number of accepted bookings for the flight is larger than the flight-capacity. For example, refer to Rothstein (1971a, [199]), Rothstein (1971b, [200]), Simon (1972, [216]), Vickrey 7 ‘ (1972, [250]), Rothstein (1975, [202]) and Shlifer and Vardi (1975, [214]). For the overbooking problem in the context of hotels, refer to Rothstein (1974, [2011), and Liberman 8 ‘ and Yechiali (1977, [127]; 1978, [128]).  Chapter 1. Introduction and Background  11  choose those passengers with the lowest bids to be bumped. Since the whole process is voluntary, it has no legal complications for the airline. Nowadays, most airlines use variations of the original Simon-auction in their practice. 1.2.5  The Seat Allocation Problem in the Presence of Competition  So far we know very little about how to manage the competition problem in the context of yield management. A recent study by Brander and Zhang (1990, [33]) shows that the Cournot model is more consistent for the data in their study than either the Bertrand or cartel models. This is a very interesting and important result for seat management research, since the quantity competition problem in the airline industry is closely related to the seat allocation problem under competition. Consider that two airlines each have a scheduled flight from city A to city B, and that there are two fixed fares fh and  f,,  which both carriers charge. With random demand  for both of the two fare classes, each airline’s strategic decision variable is its protection level for the high fare, or equivalently, its booking limit for the low fare, since one airline’s decision on its protection level for the high fare will have revenue impact on the other 9 This seat allocation game can be analyzed in several ways. The most natural airline.’ approach is to treat the game as a bimatrix game, since the protection level for the high ° 2 fare is an integer bounded by the flight capacity. On the other hand, there are also many different theories of oligopoly in economics ‘ T 9 he key issue here is how to split the demand when both airlines allocate a positive number of seats for a particular fare class. A bimatrix game is a non-cooperative two-person non-zero sum game with a finite set of pure 20 strategies 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]), Parthasarathy and 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  12  and marketing that are useful for airlines to gain a general understanding of the strategic ’ Therefore, 2 behaviour of the firms. But they are not very useful at the operational level. I am not going to investigate the allocation game along this direction. For an excellent survey on theories of oligopoly, refer to Shapiro (1989,  1.3 1.3.1  [2101).22  Airline Fare Pricing Introduction  Since U.S. airline deregulation in 1978, airlines have enjoyed complete freedom in setting fare levels. In fact, airline fare pricing has evolved from the pre-deregulation simple and static practice to a more dynamic and complex structure. Marketing innovations have been developed and used to segment consumers into groups with different demand elasticities and to stimulate market demand and create consumers loyalty. Two classical examples are the use of artificial restrictions on discount seats and the establishment of frequent-fliers programs. On the other hand, even though the application of these marketing innovations is well-known in the airline industry, there is very little academic attention in economics and marketing literature to the theoretical implications of these 23 In fact, in the vast literature of pricing research in economics marketing innovations. 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 Dixit and Stiglitz (1977, [61]), Hart (1979, [87]), Salop (1979, [205]), Shaked and Sutton (1982, [208]), and Singh 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]). Also, general discussions on the competition issue in the airline industry can be found in many other 22 papers. 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-oriented and they are useful for us to have a better understanding of the airline industry, in particular, airline deregulation. A recent article of Gale and Homles (1993, [74]) studies the economic impact of using advance23 purchasing in the context of airlines. The primary focus of this paper is on the mechanism design. 21j  Chapter 1. Introduction and Background  13  and marketing, there is no analytical framework that is directly applicable to airline fare 24 In this section, we will briefly discuss several aspects of airline operations in pricing. connection with marketing restrictions. 1.3.2  Current Operating Environment  Before moving on to the discussion of the modelling practice in the economics literature that may be relevant to airline fare pricing, let us first summarize some recent corporate changes in the airline industry and some unique operating characteristics related to airline fare pricing. First of all, as it is well-known, an airline fare class is characterized by a price and a set of restrictions. The restrictions are intended to prevent business travellers from buying deep discount fares targeted to leisure travellers. The main fare pricing decisions for 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 complicated fare structures. Before deregulation, airlines had limited freedom in setting fare prices and selecting the type of fare classes on a particular route. After deregulation, the fare structure had become so complicated that on some flights, there are over 15 different fare classes with all kind of restrictions or fences. This in turn means that a two-leg flight will have 45 different fare classes. For carriers with a large operating network, the maintenance of such a complicated fare structure became a major operating challenge. There are many empirical studies of airline demand. For example, refer to Oum and Gillen (1983, 24 [171]) and Oum et al (1986, [172]). These studies, together with traditional economic theories, can provide basic guidelines for airlines. But with the sheer number of ifights and the diverse nature of different city pairs, their usefulness is very limited for daily operations and pricing decisions.  Chapter 1. Introduction and Background  14  In a surprising move in April of 1992, American Airlines (AA) suggested a simpli fied fare structure and hoped that other major airlines would cooperate. Even though such 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 airlines including AA decided to fully implement it. Such a resistance might be in part due to the fact that airlines did not know how to price their products by explicitly incorporat ing the marketing restrictions they had been using for many years. In fact, there was no theoretical justification for such a simplified fare structure, and AA did not provide 25 There is another interesting development recently emerg any satisfactory explanation. ing from the airline industry. Again lead by American Airlines, many airlines in the world started to merge the traditional separate functional departments for pricing and 26 This indicates that airlines now realize the yield management into one department. importance of coordinating the pricing decisions with yield management techniques. With these new developments in the marketplace, it is urgent for airlines to develop a tactical pricing model on a flight-by-flight basis that is consistent with their existing yield management 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 and 27 hotel rooms, which explicitly incorporates the use of artificial restrictions. Since the pricing model in this thesis is intended to be applicable for airlines, it is important to understand some basic, but unique, characteristics of the airline business that are related to fare pricing. There include: 25 AA had never explicitly revealed where the new simplified fare siructure come from: the Since rationale behind the simplified fare structure is a secret. Other examples include British Airways and Cathay Pacific. I have checked with additional major 26 airlines through telephone inquiries and have found that even though they have two different functional departments, the interaction between the two is on a daily basis. AS we will see later, the pricing model developed in this thesis will have optimal fare structures that 27 at least in part coincide with the new simplified fare structure suggested by AA, that is, offering three coach 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 the product 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 business travellers; • Airlines practice price discrimination by use of artificial restrictions; • Matching competitors’ prices is the only way to survive because of the commodity nature of airline products; and • Sophisticated computer reservation systems are powerful tactical tools to allow airlines to make better decisions on allocating a limited number of seats among many different fare classes, which is very difficult for competitors to mimic because these 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 mainly due 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 same period, that is, the period of the flight time. Because of this, consumers for airline  Chapter 1. Introduction and Background  16  seats can delay their purchasing decisions. The striking implication of this behaviour is that airlines simply cannot sell down. If they do so, rational consumers may wait for the 28 Therefore, it suggests that the traditional pricing theories for perishable cheapest price. products are not useful for airlines at all and new approaches must be called for. 1.3.3  Modelling Practice  Apparently the main reason that an airline uses marketing restrictions, such as Saturdaynight stayover, advance booking, and corporate membership, is to effectively segment market 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 from each market segment. Therefore, a precise understanding of the impact of marketing restrictions must be an integral part of airline fare pricing. This is also important to aircraft manufacturers since the seat configuration in an aircraft is the basis for airlines to offer different products and services. On the other hand, modern yield management tools, in particular, the seat inventory control models, require the airline to pre-specify a set of prices so that yield manage ment specialists can conduct demand forecasting on this set of prices and make proper 29 Airline seats on a particular decisions on booking requests for different fare classes. flight 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 are in effect. The traditional approach was to offer a large number of fare classes and let 0f course, there are some consumers who just want to purchase the seats early in order to have their 28 reservations confirmed. The point here is that consumers no longer have any incentive to purchase early. j the actual implementation of seat allocation models, demand forecasling is critical. Littlewood 29 (1972, [131]) had some discussions on this issue. The latest development is the censored regression model of McGill (1989, [144]).  17  Chapter 1. Introduction and Background  the 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 lack of progress on the theoretical work for the optimal seat inventory control problem for multi-fare flights with random dependent demands becomes a major setback for such an 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 offer multiple fare classes on each flight. Then it is natural that we should try to find answers from economic pricing theories that suggest the use of multiple prices. Let us start with the argument, due to Lott and Roberts (1991, [132]), that airline fare pricing is a form of product differentiation. According to Lott and Roberts, the main reason why those customers travelling on short notice are paying a substantially higher price than those who can book early is the cost of the service: providing the consumer with an “ability to purchase a ticket at the last minute”(p. 21). They argue that In the case of airline seats, the opportunity cost of keeping some seats available until the last minute is that they may go unused.  The limited  number of seats that airlines make available for advance purchase discounts shows how easy it is for airline to sell advance tickets. If they could not sell many of these discount tickets, the restriction on the number sold would be superfluous. This also explains why airlines have penalties for cancellations of these discount fares and not for reservations at the higher price. For airlines to be willing to hold seats for last minute travellers, they must make the same expected revenue from these seats as they do from those seats  Chapter 1. Introduction and Background  purchased  iii  advance.  .  . .  18  (p.22)  They then conclude that If 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 only to cost differences.  ..  .  Our explanation is able to explain, however, why ad  vanced reservation discounts exist even on highly competitive routes after deregulation. (p. ) 22 But in their footnote 10, they feel that the use of restrictions is still a puzzle in airline fare pricing: Another puzzle is the requirement that consumers have to spend a Satur day night at their destination in order to receive the discount. This is typically explained 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 Saturday night travel on Sunday, the quietest time of the week. The puzzle remains as to why there is not an explicit discount for returning on Sunday, but this is also a problem for the price discrimination explanation. (p.21) These qualitative arguments are indeed interesting. But I do not think that Lott and Roberts provide a convincing argument as to why the traditional product differentiation model is appropriate for airline fare pricing with marketing restrictions.  Now, consider the argument that airlines engage in price discrimination. It is well known that airlines offer many different price levels with the identical product. This suggests that airlines are practising price discrimination. There are three types of price  Chapter 1. Introduction and Background  19  30 First-degree price discrimination, also called discrimination in the economics literature. n, “involves the sellers charging a different price to each unit 3 discriminatio perfect price ’ of product in such way that the price charged for each unit is equal to the maximum willingness to pay for the unit.” (p. 600) An intuitive way of describing this type of price discrimination is that the seller makes a single take-it-or-leave-it offer to each consumer that extracts the maximum amount possible from the market. But the “leave-it” threat lacks credibility: it is not a rational way of bargaining since the firm will take the risk of losing the consumer and incurring a permanent loss of revenue. On the other hand, even if the seller had a way of making such a commitment, he typically lacks full information about the buyers’ preferences. He cannot determine for certain whether his offer will be actually accepted. The requirement of full information about buyers’ preference is the main drawback for first-degree price discrimination. For the sales of airline tickets, such an informational requirement is simply impossible. Second-degree price discrimination, also called nonlinear pricing, occurs when prices differ depending on the number of units of the good brought.  In other words, con  sumers face the same price schedule, but the schedule involves different prices for different amounts of the good purchased. This type of pricing critically relates to the existence of market segments and use of self-selection constraints. But with unit demand for airline tickets from most consumers, the quantity schedule is not meaningful in the context of 32 airline fare pricing. Third-degree price discrimination means that different consumers are charged differ  ent prices, but each consumer pays a constant amount for each unit of the good brought. 30 classification of the forms price discrimination is due to Pigou (1920, [182]). This Refer to Varian (1989, [249]). 31 0ne exception here might be the pricing problem for group booking, where there are bulk sales of 32 tickets to tour operators, sport teams, etc. The fact that bulk sales are also subject to the ordinary restrictions used on other tickets makes the direct application of second degree price discrimination questionable.  20  Chapter 1. Introduction and Background  This pricing scheme requires that the market demand be segmented and different market segments be perfectly sealed. To seal different market segments, firms use certain mecha nisms, such as discriminating by age. The determination of different groups of consumers is taken exogenously by the model. Even though most real life price discrimination falls into this category, the requirement of perfectly sealed market segments is too strong to allow us to use this type of price discrimination to explain the use of multiple prices in airline fare pricing.  The idea of using a marketing mechanism to help firms to segment the market is too good to throw away. Telephone companies effectively segment the market by offering discounts for residential usage; theatres successfully attract additional consumers by offering student discounts and senior discounts. of effective third-degree price discrimination.  All these cases are classic examples  One striking feature is that telephone  companies 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 is that third-degree price discrimination requires a perfect marketing mechanism in the sense that it will perfectly seal the different consumer groups. As demonstrated by Gerstner and Hoithausen (1986, [76]), the action of perfectly sealing the market segments may not be in a monopolist’s best interests. On the other hand, the pricing model in Gerstner and Hoithausen (1986, [76]) only addresses the case of two segments, where each segment consists of identical consumers. The problem for airlines is that the restrictions, such as advance booking and Saturday-night stay, cannot effectively segment the market into several consumer groups so that each group consists of identical travellers. Therefore, a more delicate analytical framework is required to accurately model the fare pricing problem for modern airlines. This, in fact, is the main motivation for pricing model in this thesis. More precisely, the model proposed in this thesis is a third-degree price  Chapter 1. Introduction and Background  21  33 discrimination model using restrictions which are imperfect marketing mechanisms. 1.3.4  New Directions  Clearly, there are many important issues in airline fare pricing that need to be investi gated. The most important task should be the development of a fare pricing model that explicitly incorporates the use of marketing restrictions. Such a model is a fundamental step toward a full understanding of the economic benefits of marketing restrictions. The second important issue should be the design of marketing restrictions. From the theo retical point of view, it would be interesting to know what kind of marketing restrictions are plausible. But in practice, an optimally designed restriction (in theory) may not be implementable since the actual implementation requires the physical specification of a marketing restriction (such as Saturday stay condition). In this regard, the second issue should be more empirically oriented. Another important issue is how to incorporate the network structure, in particular, the hub-and-spoke system, into fare pricing. Again this is a theoretically challenging question with direct impact on the market place. As long as airlines are operating on a network environment, different markets in the network will interact with each other, which will be reflected in the demand structure over the net work. Therefore, such an interaction should be explicitly considered in the corresponding pricing problem. In order to develop useful models for airlines, there are two basic criteria that should be followed: Recently, Gale and ilolmes (1993, [74]) investigate the use of advance-purchase as a mechanism of 33 discounting in the pricing problem for a monopoly airline. Their discussion focuses on how to find an optimal direci-revelaiion mechanism that will lead each consumer to make a rational choice between a peak flight and an off-peak flight. Their model is based on the assumption of the existence of different time costs over the two flights for a continuum of risk-neutral consumers. They concludes that advance purchase discount is an optimal selling mechanism. Their model also predicts that no seats on the peak flight are sold at a discount, which is not consistent with airline fare pricing practice. In reality, airlines sometimes 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, 34 and must be consistent with yield management practice; • the model must consider the technological constraint of the computer reservations 35 systems. With these two objectives in mind, I will dedicate most of this thesis to the development of a fare pricing model that implicitly incorporates the use of marketing restrictions. During the development, instead of limiting the discussion to airlines, I will cast the model as a general monopoly pricing model for perishable inventories that will be applicable to airline seats, hotel rooms, rental cars and other sectors, where the monopoly firm has to decide whether a marketing restriction should be used and how it should be used.  1.4  Objectives and an Overview of This Thesis  This section provides a summary of the objectives and main results of the remaining chapters of this thesis.  Summary of Chapter 2: Pricing Perishable Inventories Using A Restriction This chapter addresses the monopoly pricing problem for perishable inventories, such as airline seats, using a single restriction to segment the market. The following assump tions are made • the demand function is decreasing and is a step function defined on a finite set of prices; 34 particular, with analytical models and specialized yield management software. 1n For example, most of existing computer reservations systems use the concept of nesting or virtual 35 nesting 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 of customers 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 are sold 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-constrained monopolist in the context of unit demand. Additional results are derived so that they can be used to formulate the pricing problem when the monopolist uses re strictions. • It presents a formal analytical model for the pricing problem of perishable inven tories by using a single restriction. • It further characterizes some optimal pricing polices that can be derived from a series of linear programming problems, which have at most three active price levels. • It demonstrates that except for some trivial cases, the monopolist can do strictly better by offering some restricted units.  24  Chapter 1. Introduction and Background  • It illustrates how the pricing model developed in this chapter can be used in airline fare pricing. It shows that the assumption of the impact of restriction is equivalent to saying, in the airline context, that leisure travellers are more price sensitive than the business travellers. Summary of Chapter 3: General Optimality Results and Other Properties This 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 as sumption that restricted prices are lower than unrestricted prices, it is important, from a theoretical point of view, to understand the issue of optimal pricing practice when using an artificial restriction. The model in Chapter 2 does not explain why the monopolist should limit itself to the policies that restricted units are sold first at some lower prices and the unrestricted units are sold subsequently at some higher prices  —  a practice which  will be called primary policies. Practically speaking, it might make sense to sell some unrestricted units at some very low price and then open sales of some restricted units later. The main goal of this chapter is to investigate whether such a pricing practice is necessary. Another issue is that the model in Chapter 2 assumes that the monopolist opens sales at one price at a time. This is not consistent with current practice. For example, airlines make all offered prices together with allocated quantities available si multaneously. Technically speaking, the model in Chapter 2 is not affected by this issue since it is a static model. But to airlines, this is very important simply because the optimal pricing structures, if implemented, must be consistent with yield management techniques. The main contributions of this chapter are: • Any general pricing policy, which allows the monopolist to sell restricted units at any price level, is always weakly dominated by a primary policy, where the  Chapter 1. Introduction and Background  25  dominance is defined in terms of realized revenue for the policy. This result implies that any optimal pricing policy in the class of primary policies is also optimal in the class of general policies. Therefore, it resolves the issue of optimal pricing practice when a monopolist uses artificial restrictions; and • Under a behavioral assumption that if restricted units and unrestricted units are offered at the same price level simultaneously every consumer in the market will try to buy an unrestricted unit first, it is shown that there always exists an optimal policy in the class of general policies with the property that even all offered prices together with the allocated quantities are made available simultaneously, there will be no negative impact on the realized revenue. Summary of Chapter 4: Pricing Models with Two Types of Restrictions By using one restriction, the monopolist can introduce two kinds of products to the market. In this chapter, I will use the same techniques developed in Chapters 2 and 3 to 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 to consumers. This will enrich the techniques developed in previous two chapters to handle more realistic pricing problems. The main findings of this chapter are: • For the cases of two nested restrictions and two mutually exclusive restrictions, the monopolist’s pricing problem can be formulated as a mathematical programming problem with four constraints. It is further shown that this formulation can be relaxed into a linear programming problem. The main result is that the monopolist needs to offer at most four price levels with at most three types of product to maximize its revenue; and  Chapter 1. Introduction and Background  26  • For the general case of two restrictions, it is shown that under certain plausible conditions, there exists optimal pricing polices that consist of at most four kinds of products with five price levels to maximize its revenue. Summary of Chapter 5: Airline Pricing by Using Membership and Product Restrictions The purpose of this chapter is to present some interesting applications of the pricing models developed in Chapter 4 to airline fare pricing when a product restriction, such as Saturday-night stay and advance booking, is incorporated in association with the use of membership fares. Note that besides the conventional product restrictions, there are many other forms of restrictions in airline industry, which further segment the market de mand. For example, airlines usually have special agreements with some corporate clients and government clients for special fares; and airlines are under constant requests from tour operators for additional discounts. Also, airlines must deal with internal travellers (i.e., employees) and travellers from a partner airline. Each of these traveller groups constitutes a certain form of membership. The traditional view of membership is that it 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. This chapter discusses the issue of membership together with a product restriction. Studies in this chapter enable an airline to understand the operating environment for three com monly used membership privileges. It is further shown that the pricing models developed in previous chapters can be used to evaluate certain corporate commitments associated with membership deals. The main findings of this chapter are: • For the case that the membership privilege is limited to a restricted fare at a price level that is lower than the public restricted fare (for example, tour operators and  Chapter 1. Introduction and Background  27  corporate retreat programs), the airline has an optimal fare structure with the following properties: —  —  it consists of at most four fare classes; and the demand for restricted membership fares will be exhausted right after the sales of these restricted fares targeted to members;  • For the case that the membership privilege allows the members to purchase cheaper unrestricted fares only (for example, interval travellers and major service corpora tions), the pricing model for two mutually exclusive restrictions can be used, which implies that the airline needs at most four different fare classes to maximize the flight revenue. • It is well-known that most airlines have interval travel policies for employees, which indicates that if an employee pays a small fraction of the (public) full fare, then he/she will have a confirmed seat on the flight. Usually, these interval fares are cheaper than the public restricted fares, which is in fact a corporate commitment to employees. It is shown that unless an airline has a systematic method to monitor some high-demand flights, such a commitment may cost the airline a lot of money; • If the membership privileges include cheaper restricted fares and cheaper unre stricted fares (for example, government employees and travellers from partner air lines), it is shown that the airline needs to use the pricing model for two general restrictions developed in Chapter 4. Sufficient conditions are given for airlines pur suing simple optimal fare structures. These models can be used to evaluate whether or not certain corporate commitment is indeed consistent with the operating envi ronment; and  Chapter 1. Introduction and Background  28  The chapter concludes with the following insight: in order to fully exploit these market segments derived from the existence of memberships, an airline must have a systematic approach that can be computerized so that: —  given a particular membership, it can tag those flights that will be available to these members; and  —  it is capable of identifying the most favorable membership group on the flightby-flight basis.  Chapter 6: Seat Allocation Game on Flights with Two Fares The main purpose of this chapter is to propose a major initiative in modern airline the yield management research, that is, to investigate the seat allocation problem in presence of competition. In particular, I will focus on single-leg two-fare flights with ran dom demands (not necessarily independent) shared by two airlines, where each airline’s strategic variable is the booking limit for the low fare, or equivalently, the protection level to the high fare. The discussion is based on how the two airlines split the market demand. 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 from two airlines is strictly smaller than the total number of seats that would be available for the low fare if two airlines cooperate. Under the equal splitting rule, the discussion of the seat allocation game is limited to the case of deterministic demands. It is shown that allocating enough seats to capture half  Chapter 1. Introduction and Background  29  of the high fare demand is an equilibrium strategy for both airlines.  Chapter 7: A Note on Three Models for Multi-fare Seat Allocation Problem The seat allocation problem for single-leg multi-fare flights with independent ran dom 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 field on which model is more computationally efficient. The purpose of the this chapter is two-fold. First, I want to clarify this issue by proving that the optimal conditions from three models are in fact analytically equivalent, which therefore implies that they are all computationally equivalent too. Second, by unifying the existing approaches, I pro vide further insight on the multi-fare seat allocation problem with dependent random demands.  Chapter 2  Pricing Perishable Inventories by Using a Restriction  2.1  Introduction  This chapter develops a model for monopoly pricing a perishable product with a capacity constraint.’ Examples might include airline seats and hotel rooms. Firms, such as airlines and hotels, typically offer multiple prices and impose artificial restrictions to segment the market. Surprisingly, there is no useful framework in economics or marketing to deal with this kind of problem. Models in the economics literature are driven by simplicity and elegance, 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 are intended to be realistic and therefore possibly to be implemented in practice. But it is very hard to achieve these two objectives at a satisfactory level without some degree of compromise. For example, Dobson and Kalish (1988, [62]) develop an operational, heuristic procedure to position and price a line of related, substitute products. Their mathematical programming formulation, however, is not computationally tractable. The goal of this chapter is to develop a model that is analytically tractable and practically interesting. Recently, there has been a growing interest in the dynamic monopoly pricing problem of inventories in the operations research (OR) literature. In particular, Rajan, Rakesh and Steinberg (1992, [189]) present a model of simultaneous pricing and inventory control for 1 W hile the model is developed for the case of a monopolist, it has some applicability for any firm with some degree of market power.  30  Chapter 2. Pricing Perishable Inventories by Using a Restriction  31  a monopolist retailer who orders, stocks and sells a single perishable, but storable product facing a known demand function. The solution to the model gives the optimal dynamic 3 Other related papers along this direction include 2 and the optimal cycle length. price Wernerfelt (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 become very clear after knowing the following aspects of the new model: • The market consists of at least two types of consumers with different willingness to 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 charac teristics. The purpose of this chapter is to take up this challenge. The model in this chapter fills an important theoretical gap in the pricing research. This chapter is organized as follows. Section 2.2 first motivates the notion of ra tioning, which is directly related to the specification of the residual demand when sales are rationed. I then discuss a pricing model developed by Wilson, which provided the in spiration for the development of this optimal pricing model. Section 2.3 formally presents an optimal pricing model that explicitly incorporates the impact of using one restriction on the product at some lower prices. Section 2.4 discusses optimal pricing strategies that A dynamic price is a price functional of time. 2 3 cycle length is the length of the time period when the retailer will not make another ordering A decision. It is interesting to mention that the optimal dynamic price is independent of the choice of cycle length. The reason I call the restriction artificial is because the action of imposing it does not cost the firm 4 anything. In contrast when a firm segments the market by product differentiation, the firm must expend resources in order to achieve it.  Chapter 2. Pricing Perishable Inventories by Using a Restriction  32  have tractable characterizations. Section 2.5 discusses an application of the model to the airline industry, where the market is segmented into business travellers and leisure trav ellers. This section also provides a simple example to illustrate the application. Finally, the last section is a summary. Rationing Rules and Wilson’s Pricing Model  2.2 2.2.1  Rationing Rules  The notion of rationing is commonly used in the duopoly model of price competition when two firms are capacity-constrained. Historically, the use of capacity constraints in a pricing game, first suggested by Edgeworth (1897, [68]), was motivated to resolve the 5 The key question related to rationing is: if two firms well-known Bertrand Paradox. offer different prices, then what is the residual demand for the high-price firm after the sales of the low-price firm? If the low-price firm’s capacity is large enough to satisfy the whole market demand, then there is no residual demand for the high-price firm. Thus the main issue becomes specifying the residual demand when the low-price firm cannot exhaust total market demand at its price level. As a general rule, in models with capacity constraints, firms make positive profit and the market price is greater than the marginal cost. The rationing issue also arises for a capacity-constrained monopolist. The innovation over the traditional model is that a monopolist does not need to allocate all its capacity at single price level. In fact, the monopolist can offer an allocation schedule at different price  6 This problem was first studied by Wilson levels to maximize its total revenue.  Simply speaking, the Bertrand Paradox says that when two firms produce identical goods with 5 identical and constant marginal cost, then there exists a unique equilibrium such that both firms price at marginal cost and make no profit. This result is considered to be a paradox by some because it is hard to believe that firms in industries with few firms never succeed in manipulating the market price to make profits. We should notice that from a consumer’s point of view, it is irrelevant who is offering a higher price. 6  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 to a more specific assumption concerning the manner in which sales are rationed. 7 Assume that a Two rationing rules have often been considered in the literature. monopolist produces a homogeneous product and each consumer oniy needs one unit of the 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 as  the size of the market that consists of those consumers who have reservation prices of p 8 Now suppose that the monopolist first offers qi units at price p or higher.  Po. The  efficient-rationing ruic presupposes a residual-demand function given by:  d(p;pi)  =  I  D(p)  (  0  —  q  if D(p) > otherwise.  This residual demand function essentially assumes the most eager consumers buy the product at price p. This rationing is efficient because it maximizes consumer surplus. Therefore, it is the most undesirable rationing scheme for the monopolist. Also, the residual demand function defined by the efficient rationing rule is the one that would be obtained if the consumers were able to costlessly resell the good to each other, that is, 9 to engage in arbitrage. In the duopoly case, the higher price is offered by another firm; and in the monopoly case, the higher price is offered by the same firm. The point here is that if the rationing makes economic sense in the duopoly case, it will also be rationed in the monopoly case. The following discussions follow from Chapter 5 of Tirole (1988, [236]), pp. 212—214. 7 This is a typical way of handling unit demand when a firm does not have perfect information about 8 consumers’ willingness to pay. Such an approach has been extensively used in the economics literature in many different contexts, for example, refer to Harris and Raviv (1981, [86]), Wolinsky (1984, [264]), D(p)/N, then we and Perloff and Salop (1985, [178]). On the other hand, if we define F(p) = 1 can interpret F(p) as the probability distribution function of a consumer’s reservation price. Therefore 1 F(p) is the probability that a consumer has a reservation price of p or higher. Such an interpretation is also consistent with the notion of imperfect information in economics when consumers have inelastic demand. The use of the efficient-rationing rule can be found in many articles from economics literature. For 9 example, 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  34  Another popular rationing rule is called the proportional-rationing rule. It assumes that all consumers have the same probability of being rationed. Given that the monopolist offers q units at price p  P0,  the probability of not being able to buy at price p is D(pi) q D(pi) —  Hence, the residual demand function after the sales of q units at price d(p;pi)  =  D(p)  D(pi) qi for p ) 1 D(p —  >  Pi  is given by  p.  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 with the higher reservation prices are unable to. However, the monopolist prefers this rule to the efficient rule since the residual demand is higher at each price. Typical conditions for 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 efficientrationing rule for some industries with perishable inventories because these conditions 10 are easily met, for example, by airline seats and hotel rooms. 2.2.2  Wilson’s Model  Wilson (1988, [262]) is the oniy published paper that deals with rationing sales in the context 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  35  by a monopolist, when consumers arrive in a random order and are served on the firstcome-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 consumers whose valuation for the product is at least p. To simplify his analysis, Wilson makes a further 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:  D(p)  where cc  >  > D 1 D 2  =  > ...>  1 D  ifppi  D  if p  0  ifp>p,  (pt-i, p] and i> 1  D> 0.  Remark: Since the demand function is a step function, the set of prices that a ratio nal firm will consider is finite, which in fact is the set p  (pj., pi), the firm will be better off charging  p  ,p2,• 1 {p  rather than  p,  ,p}.  For any price  since increasing from  p 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 units of the product to make available at the finite number of prices. This implies that the pricing decision becomes an allocation problem. In this regard, Wilson introduces the following definition of a pricing policy.  Chapter 2. Pricing Perishable Inventories by Using a Restriction  36  Definition 2.2.1 A pricing policy for the monopolist is a vector {q,, q,...  ,  q},  where  qj > 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 total revenue  pq,. The key is to characterize the set of feasible pricing policies. The  feasibility 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 have qi  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, since the product is assumed to be homogeneous and non-storable, a rational firm will only consider selling the product from the lowest price to the highest price, since otherwise those consumers with higher reservation prices would prefer to wait to get the product at 2 More specifically, for any given pricing policy {qi, a lower price.’  ,  q},  the selling  process is as follows: the firm first puts q units of the products on sale at price p,; after selling q units at p, the firm allocates another q units on sale at price P2; and so on and so forth. The key aspect of this selling process is that if after the sales at prices p the demand at price p is not exhausted, the residual demands at higher prices 3 are positive, since we assume that the consumers arrive in random order.’ Given a pricing policy {qi, q2,  .  ,  qn},  for k  i and i  mand at price pk after the sales of the product at prices  1, let d,k be the residual de ,p, according to q  .  ,  q.  “The feasibility condition on the demand side is more involved since we allow the possibility of multiple p rices. This issue is more subtle than it appears here. For many products, if the action of delaying the 2 ‘ purchase decision creates a substantial loss of productivity or a substantial amount of disutility, then the firm may consider selling the product from the highest price to the lowest price. Wilson’s model here and my model later do not address the pricing problem for this case. This assumption allows us to use the proportional-rationing rule. 13  Chapter 2. Pricing Perishable Inventories by Using a Restriction  37  Then according to the proportional-rationing rule, d,k  —  d+l,k  =  d,k  1 + 2 d,  qi for k  (2.1)  i + 1,  where the second part in the right hand side is the leakage from the residual consumer group consisting of these consumers with the reservation price Pk or higher who actually purchase the product at the price  Also, for ease of presentation, we will denote  Pi+i•  do,k  Dk for  E  k  1.  The following lemma gives a simple updating formula for the residual demand, which is in fact the key in Wilson’s development.  Lemma 2.2.1 For every i  k ,d 2  where  i, we have  1 and all k =  Dk(1  (2.2)  —  can be interpreted as the total leakage ratio of the market demand due  to 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 d,k  =  Dk  Dk  —  1 ——q  =  Dk(1  =  1, by definition,  qi —  1 JJ  i—Il  Therefore, the result is true for i  =  1. Now assume that the result is true for i  1. I  want to prove that the result is also true for i + 1. By (2.1) and (2.2), it follows that d+l,k  =  di,,, —  =  ui,i+’  qi+i  Dk qt Dk(1——)— qi-iD D+i .t=1 i+1  =  Dk(1—YZ-J-).  Chapter 2. Pricing Perishable Inventories by Using a Restriction  This proves that the lemma holds for i + 1. So by induction, the lemma is proved.  38  LI  The demand side feasibility condition is the following: we must maintain a non negative residual demand at the end of the sales at each possible price level. This is equivalent to requiring that  dk,k  0 for all k, that is, qj  Vkl,  which is equivalent to  since 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 con straint all together, Wilson concludes that the firm’s pricing problem can be formulated as the following linear programming problem: Max  pjqj 1  s.t. <1  (2.3)  k  (2.4)  q qi  0 Vi.  From now on, we will call this formulation Wilson’s Pricing Model or simply the Wilsonmodel. The following result is an immediate consequence of this linear programming formulation.  Theorem 2.2.2 [Wilson (1988, [262])] To maximize revenue, the firm needs to charge no more than two prices.  Chapter 2. Pricing Perishable Inventories by Using a Restriction  39  Proof: This follows immediately from the fact that the number of non-zero variables in any basic optimal solution for a linear programming is no more than the number of structural constraints. D  The following result, not proved in Wilson’s paper, shows that any optimal solution in the Wilson-model will make the demand constraint binding.  D, then for any optimal solution in the Wilson-model, the  Theorem 2.2.3 If k  demand constraint (2.3) is binding. Proof: Note that if k solution  {p,  =  D, it is clear that the Wilson-model has an unique optimal  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 that any basic optimal solution in the Wilson-model consists of a pair {q’, q’} (i  <j).  If both  quantities are positive, (2.3) and (2.4) must be binding. So the theorem holds for this case. Now suppose that an optimal solution consists of only one price, say p, with the optimal quantity q. I want to show that (2.5) Suppose (2.5) is not true, which must imply that: . 1 q<D Therefore the constraint (2.3) is not binding. Since the given optimal solution is a basic solution, then the constraint (2.4) must be binding, that is q  =  k.  40  Chapter 2. Pricing Perishable Inventories by Using a Restriction  Furthermore, it must be true that k> D+,, since otherwise the firm can sell all k units at price level Pi+i that will realize a total revenues 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  --+ 2 D +, 2 D  q+,  > 0 such that  —  , qi+qi1  k.  =  Solving this system, we obtain qi  qi+i  =  k—D , 2 — L/j+1 D-k —  is given by  Now the revenue under R  = piqi  + pi,qii >  p(q  + q+,)  =  pk,  because q , > 0. This contradicts the assumption that q,’ is an optimal solution, which 2 implies that (2.5) is true. This proves that the theorem is true for any basic optimal solution. For the general case, we note that any optimal solution in the Wilson-model is a convex combination of basic optimal solutions. Therefore, the theorem is true for any 4 optimal solution since both (2.3) and (2.4) are convex constraints.’  A 14  D  constraint is said to be a convex constraint if for any two feasible vectors to the constraint, any  convex combination of these two vectors remains to be feasible to the constraint. In fact, any linear constraint is a convex constraint.  Chapter 2. Pricing Perishable Inventories by Using a Restriction  41  Remarks: 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 can actually sell all units at the highest positive price level  —  p,, the firm’s pricing  decision is primarily driven by the market demand in the sense that there will be no 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 two prices only when the firm cannot reach its best possible revenue at the price that exhausts all its capacity. This occurs when the single-price revenue function is not concave. It is worthwhile to notice that k < D says that the firm can sell all its inventories at the highest possible price level  p,  which consequently implies that there is no need to  ration 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 that kD. Now for a given demand structure, since the optimal revenue value in Wilson’s formu lation oniy depends upon the capacity level k, we can write this derived revenue function (k): R ” as R’(k). Wilson proves the following properties for 1  Theorem 2.2.4 [Wilson (1988, [262j)j It is true that 1. R”(k) is a concave and non-decreasing function of k; 2. RL(k) is piece-wise linear in k; (k), 3 (k) is piece-wise linear in a neighborhood of k, then Ru(k) > R 3 3. if RY  Chapter 2. Pricing Perishable Inventories by Using a Restriction  42  where R (k) is the best single-price revenue defined by: 3 (k) 3 R  =  max xD’(x).  O<x<k  The first property is true in general. The second property is a consequence of as sumption that the demand function is a step function. On the other hand, if the best single-price revenue function is concave, the firm always charges one price. The signif icance 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 best single-price revenue. An Optimal Pricing Model by Use of Restrictions  2.3 2.3.1  The Model Settings  Consider 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 hotel needs to sell a fixed number of rooms on each particular day; and a car rental company wishes to rent out a fixed number of cars on each day. The main characteristic of the products we are concerned with here is perishability, that is, if the firm cannot sell the product 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 the product can not be stored and consumed before a given time. Before moving on to the formal development of the model, it is necessary to highlight the 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 consumer group into two subgroups  —  those who do not mind the restriction and those who  do. 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 restriction is not attached. Then the total demand for the product at the unrestricted price p is 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 percentage of those consumers with reservation prices of p or higher who will be unable to buy the product because of the restriction. Then a(p) is given by: Dr(p) D(p) =1D(p) = D(p) (p) and 7 As in the Wilson-model, to further simplify the analysis, assume that both D D(p) are step functions defined on a set of prices {p1,p2,. ,p}, where P0 <n  <00.  More specifically speaking, for  Di(p) =  1 = r, u,  , 1 D  ifp  , 1 D  if p E (p,p] for 2  o  ifp>p,  _<pi  i  n  where D , is decreasing (not necessarily strictly decreasing) in i. Denoting 1 =  Dr,i  for i =  then the total market demand function for unrestricted product is given by: 1 D D(p)=  1 ifpp ,p]for2in 1 _ 2 ifpE(p  0  ifp>p.  P1 <P2 <  Chapter 2. Pricing Perishable Inventories by Using a Restriction  44  Consequently, a(p) is also a step function:  a(p) =  a,  ifpp,  cv,  if p E (p—i,p] for 2  1  ifp>p,  i 5n  where cv, a(pj) for i= 1,.•. , n. Before introducing the definition of a pricing policy, I need to highlight all the as sumptions used in this chapter: • D: is strictly decreasing in i; • D , is decreasing for 1 = r, u; 1 • a is strictly increasing in i, which implies that as the price for restricted prod uct increases, the percentage of potential consumers who can accommodate the restriction is decreasing; 5 • The monopolist will sell the restricted units first and the unrestricted units later;’ • The monopolist sells its product from the lowest price level to the highest price level; and • k is the capacity and is such that k > D, which rules out the trivial case of the 6 pricing problem.’ Sillce the monopolist sells the restricted product first, it is natural to choose a value of m such that (1) 1  m  n, and (2) the prices p,,— ,pm are attached with the  restriction. Of course, the value of m is controlled by the monopolist. Then the firm’s problem can be divided into two parts: Chapter 3, I will call this type of policy a primary policy. 16 will rule out the trivial case as pointed out in Theorem 2.2.3. This  Chapter 2. Pricing Perishable Inventories by Using a Restriction  45  • For any given value m, characterize the optimal pricing policies and the correspond ing 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 given m; 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 prob lem is in fact an allocation problem. More specifically, I give the following definition of a pricing 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  { q,.,i,”  ,  q,.,; qu,m,  ,  m  n, a rn-policy for the firm is a vector  where q is the number of units of the product allocated to  sell at the price p with restriction for 1  m, and q,, 3 is the number of units of the  product allocated to sell at the price p 3 without restriction for m  j  n.  Remark: For convenience, from now on, I call a price with the restriction a restricted price and a price without the restriction an unrestricted price. If a unit of the product is sold at a restricted price, I call it a restricted product. Similarly, if a unit of the product is 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 offer restricted products and unrestricted products at the same price Pm. This is a technical trick 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  2.3.2  46  The Demand Constraints  Let me first discuss market demand at the restricted prices spond to the market demands Dr,i, Dr,2,  ,  P1,P2,’”  ,  Pm, which corre  Dr,m. Since Dr,, is decreasing in i, then  Wilson’s technique of rationing demand works here. Thus we have the following demand constraint for the market of restricted products at prices  Pi,•  .  ,  Pm  1il.  To analyze the residual unrestricted market after the sales of restricted products according to the quantity schedule {qr,i,•  for  ,  j  rn, we introduce the following  notation: be the residual demand for the restricted product at price pj;  •  • dm,j be the total residual demand at the (unrestricted) price pj; .dum,j  ,Jr  =,J —  m,j  —  Evidently, the residual demand dm,j is a function depending upon {qr,i,—  For  ,  tunately, the following lemma shows that this functional relationship has a simple form.  , 1 Lemma 2.3.1 After the sales of the product with restrictions at prices {p according to {qr,i,  (j  ,  .  ,pm}  2 the residual demand for the unrestricted product at price p  m) is given by: dm,j =  —  Drj  )  -,  where the second term in the right hand side is the fulfilled part of the restricted demand after the sales of the restricted products. Proof: Define  to be the residual demand for the restricted product at price p 3 after  the sales of the restricted products according to {qr,i,  ,  qrm}. Since the demand market  Chapter 2. Pricing Perishable Inventories by Using a Restriction  47  for the restricted products is nested, then it follows from Lemma 2.2.1 (in Section 2.2) that  m dm,j T  —  D  “1 21  D,.  Therefore, dm,j=Du,j+d,j=Dj_Dr,j+dn,j=Dj_Dr,j>’. This proves the lemma.  D  From the proof of the above lemma it is clear that  is decreasing in  j.  Since  is also decreasing by assumption, it implies that the residual demand function dm,j is decreasing too. So the residual market for unrestricted product defined on the price set of {Pm,  ,p} is consistent with the settings in Wilson’s model as discussed in the  previous section. Therefore, the demand constraint for the unrestricted product in the residual market is given by: j=m m,j which, by Lemma 2.3.1, is equivalent to n  qu,j -m j=m D-D r,j L_dil  Z  q_ D, 1  As we can see here, this constraint is no longer a linear constraint.  2.3.3  The Formulation of the Optimal Pricing Model  Together with the objective function, the capacity constraint (or the supply constraint) and the two demand constraints, we have successfully formulated the pricing problem as a mathematical programming problem, which is summarized in the following theorem:  Chapter 2. Pricing Perishable Inventories by Using a Restriction  Theorem 2.3.2 For a given m: 1  48  m < n, a monopolist’s pricing problem can be  formulated as choosing a vector of quantities {qr,i,  ,  q; qu,m,  ,  to solve the  following mathematical programming problem: Max  pqi +  Z=m pjqu,j  s.t. r  >D 3= D 3 q,i  qu,j  Dr i  (2.6)  1 < 1  (2.7)  —  k  +  (2.8)  0, Vi and Vj.  Remark: For ease of reference, the above pricing model is called BL Pricing Model, or  simply, the BL-model. 17 The optimal value of the revenue derived from the BL-model will be denoted by R(m). We will also call any optimal solution to the BL-model with  given m as an optimal rn-policy. 2.3.4  Basic Properties of Optimal rn-Policies  Note that the formulation in Theorem 2.3.2 is computationally inconvenient because it  involves 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 is the main motivation for the discussions in the next section. Before moving on to the next section, we first need to establish a comparison result between R(m) and Rw  —  the  revenue derived from the Wilson-model. Recall that the Wilson-model solves the pricing Here “BL” stands for Brumelle-Li. 17  Chapter 2. Pricing Perishable Inventories by Using a Restriction  49  problem without using a restriction, which is given by: Max  pjqj 1  s.t. v,.n L...ii=1 D 1  —  q k 1 ZL 0 Vi.  qi  The following theorem establishes the following result: if the Wilson-model fails to generate an optimal pricing strategy consisting of only one price, then the firm has a strict incentive to use the restriction. As we will see later, this result plays a key role in my discussions in the next section.  Theorem 2.3.3 Suppose an optimal solution in the Wilson-model consists of two prices, say  j  and p 3 (i  <i).  Then:  R (z) So consequently, R(i) Proof: Let  —  {q,, qj}  —  Ru —  (p  —m)  (k  —  —  in I  n .fn.  L’,j  —  )  n. —  Ru > 0 since c is strictly increasing in i.  be an optimal solution in the Wilson-model with i  must have —  3 k—D j  L) -  qw,j  D:k  —  j  I-)  and Dj<k<D:. The corresponding revenue is given by: = piqw,i  +  p,qw,j.  <j.  Then we  Chapter 2. Pricing Perishable Inventories by Using a Restriction  50  Consider the following allocation plan: • The firm offers q,.,j units of the restricted product at p and qu,j units of the unre stricted product at pj; and •  and qu,j satisfy the following system: 3 = q, qu,j =  —  k  —  D,.,,--——;  q,j.  It is easy to check that the above policy is feasible, that is, the allocations satisfy all three constraints (2.6), (2.7) and (2.7) in the BL-model. Now solving the above system, we get k—D,  qr,:  D r,z,  — —  —  3 q,  Dr,iDj Dr,i  —  —  —  Dr,jk Dr,j  Denote the corresponding revenue to be R = piq,-,i + p,q, . Since 3 q + qw,j = k, it is true that R  = (pi  —  p)(qj  —  —  Now note that —  qu,,  —  —  — —  D k 2 3 D Dr,iDj Dr,jDi D•” ‘‘ ) 3 (Dr,i Dr,,)(Di D D (1 1 (1 c)D D 2) (Dr,i Dr,j)(Dj D) D 1 (a—ajD D’ (D,.,, Dr,,)(Dj D)  Dr,jD, Dr,j k  —  3 DD  Dr,jk  —  —  —  —  —  —  —  —  — —  k 1  —  —  —  —  —  —  — —  k 1  —  —  + qu,j  k and  Chapter 2. Pricing Perishable Inventories by Using a Restriction  51  This proves the theorem. 0  The above theorem presents a comparison result for the case that Wilson’s model has an optimal policy that consists of two prices. At the end of the next section, I will establish a similar result for the case that Wilson’s model has an optimal policy that consists of exactly one price.  Before concluding this section, I present two additional properties on optimal m policies.  Theorem 2.3.4 Any optimal rn-policy does not need more than two unrestricted prices. Proof: Let m be given and {q ,•.. q; 1 m /3  ,  p!-,  q} be any optimal rn-policy. Define m  *  and k,.  q.  To prove the theorem, it suffices to show that {q,” , q} is an optimal solution of .  the following linear programming problem: Max  pjqu,j  s.t. /_j=m  D,  q,j l3Drj  Zj=m qu,j qu,j  k  < 1 —  —  kr  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. Now  suppose that it is not an optimal solution to the reduced problem. Then it implies that  52  Chapter 2. Pricing Perishable Inventories by Using a Restriction  there is another set of quantities {q,m,”  ,  q} that constitutes a feasible solution to  the reduced problem and is such that PJqUJ  >  j=m  j—rn On the other hand, it is obvious that  q;  ,  q} is a feasible solution  to the BL-model associated with m. The corresponding revenue is given by n m n m p,q + 3q > p + p j=m 3m i=1  >  which contradicts the assumption that  q;  ,  q} is an optimal m  policy. Therefore, the {q,” , q} must be an optimal solution to the reduced prob lem. 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 monop olist needs at most two unrestricted prices. So the theorem is proved. D  As 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 {q , 2  .  q;  .  ,  q} is an optimal m  policy for the BL-model, then the demand constraint for unrestricted product (2.7) in the BL-model is always binding; that is,  3=m D 3 Proof: Let kr  =  q and k  q,j m  —  =  D r,j L.1=i  =m q• 1 ’ 3 YZD  q  = 1.  (2.9)  Suppose that (2.9) is not true. As  an immediate consequence of this, we know that the supply constraint (2.8) must be binding, that is, icr + k = k.  Chapter 2. Pricing Perishable Inventories by Using a Restriction  53  As before, denote  From the proof of Theorem 2.3.4, we know that given the allocation plan for the restricted product, {q , 1  q}, the reduced problem in the residual market for the unrestricted  product is a Wilson-type problem. Therefore, by Theorem 2.2.3, if (2.9) is not true, then it must be the case that k < D  —  I3Dr,n,  (2.10)  which obviously implies that the monopolist will sell all unrestricted units at price p, that is, *  q  fo  ifj<ri  (  ifj=n.  —  Define io=max{im:q>0}. Choose c> 0 such that k +  D,  —  0 /9DT, and q  0 <n Case 1: i For this case, define a new policy {q ,• 1  .  .  —€  j  .  ,  q} as follows:  if iio  q  q=  q;  ; 0 ifz=i  and fO  ifj<n  (  ifj=n.  —  k+e  I claim that this new policy is a feasible policy to the BL-model associated with m. In fact, it is easy to see that 1,  Chapter 2. Pricing Perishable Inventories by Using a Restriction  54  which implies that the plan is feasible for the restricted market. For the unrestricted market, first note that the residual demand for unrestricted product at price p, after the sales according to {q ,... q}, is given by 1 =  D  —  (3  —  0 r,2  )Dr,n> D  which indicates that the allocation plan {qrn,•  ,  /3Dr,n  —  ku +  f,  q} is also feasible in the residual  market for unrestricted product. Therefore, this new plan is indeed feasible for the BL model associated with m. Now note that the total revenue generated by this new plan is R’  +  =  jm  i1  + (p  +  = which contradicts to the fact that  .  .  q;  —  )f > R(m), 0 pj  .  ,  q} is an optimal rn-policy.  This contradiction leads to the conclusion that (2.9) must be true.  Case 2: i 0  =  n  Clearly under this case, we must have that m +k  =  n. It is clear that  D.  Then since kr + Ic,, = k > D, we must have some i < n such that q 1 > 0. Because of this, define =  Now choose  €  >  0 such that q 11  max{i < n : €  and Ic,, +  > €  0}. —  /3DT,. Define a new policy as  follows: q,j ,} 1 j min{q,,, d_  if i#i;  Chapter 2. Pricing Perishable Inventories by Using a Restriction  and q’  =  55  k + e. Similarly, by the same argument used in case 1, we know that this new  policy is feasible to the BL-model associated with m, which generates a total revenue of  =  0 since i’  <  R(m) +  + pq’  (pn  —  pj)e> R(m),  n. This again leads to a contradiction. Therefore (2.9) is also true for this  case. In summary, we show that (2.9) must be always true. L:i.  2.4  Optimal Pricing Strategies  The main purpose of this section is to demonstrate that the monopolist’s pricing problem can be reduced to solving n linear programming problems. First, note that the previous section solves the monopolist’s pricing problem for a given value of m. On the other hand, the choice of m is completely controlled by the monopolist. Therefore, it is the monopolist’s best interest to choose a value of R(ñi)  =  ni.  such that (2.11)  max R(m). 1<m<n  Consequently, an optimal solution to the initial monopolist’s pricing problem should be an 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 opti  mal 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) for  . {q* , restricted product is not binding for some optimal solution 1  .  .  ,  q;  .  ,  Chapter 2. Pricing Perishable Inventories by Using a Restriction  56  in the BL-model, that is, *  or equivalently, the sales of restricted products according to the partial plan {q ,• , 1 do not exhaust the restricted demand at price level p,, then this optimal policy contains exactly one unrestricted price. Proof: We know that after the sales of the restricted products, the residual demand for the product at the restricted price p 3  (j  ni)  is given by *  = (1  = Dr,j(1  —  13)Dr,j;  —  and by Lemma 2.3.1 the residual demand for the product at the unrestricted price pj is given by  -  m = D,  Since 3  <  1, it follows that  >  —  Dr,j  *  >  0 for all  =  j  rh.  —  BDr,j.  Define  Tm,j d 1—o=--——,forj=m,••.,n. —  Um,j  Then we have •-  It is easy to check that & is strictly increasing in and ,8  <  j because  is strictly increasing in  j  1.  On the other hand, as demonstrated in Theorem 2.3.4, {q,” , q} solves the following reduced problem:  Chapter 2. Pricing Perishable Inventories by Using a Restriction  Max  57  Z—,pjqu,j  S .t.  j=m d,,,,  EL q,, qu,j  where k,.  =  q,.,.  —  k  —  , 7 O,V  Clearly, this is just Wilson’s formulation for the residual market for  the unrestricted product. More specifically speaking, this subproblem can be restated as follows: the market demand at price p 3 is given by  :  j  =  ñ,.  ,  n, and the firm has  the option of imposing the restriction on the product, where the impact of the restriction is characterized by  a,, which is strictly increasing as proved above.  Therefore we can use  Theorem 2.3.3 to this subproblem, which says that if the above reduced problem (Wilson type) has an optimal solution consisting of two distinct prices, then the firm can do strictly better by properly using the restriction. This implies that the optimal solution to the above reduced problem must consist of exactly one unrestricted price since other wise it will lead to a contradiction to the assumption that  .  ,q;  .  ,  q} is  an optimal solution. Therefore, the theorem is proved. D  We notice that the main difficulty in the formulation of the pricing model in Theorem 2.3.2 is that the first demand constraint (2.6) may not be binding, which will force us to deal with a nonlinear constraint in the model. But if the first demand constraint (2.6) is in fact binding, then the BL-model leads to the following linear programming problem:  Chapter 2. Pricing Perishable Inventories by Using a Restriction  Max  m ii  58  n q, + Zj=m pjqu,j p 2  S .t.  c—sm  q,  L.j=i Dr,i  —  /_.dj=m D,—D 1 , 7  < 1 —  i + Ej=m qu,j  k  0, Vi and Vj.  qr,i,qu,j  I use (m) to denote the optimal revenue value derived from above linear programming problem. I call it the tight problem since the first demand constraint (2.6) in BL model is forced to be tight (or equivalently, binding). The implication of the fact that (2.6) is tight is that there will be no residual demand for restricted product at price p the sales of the restricted products according to the partial plan {qr,i,•  ,  p, after  q}.  Before presenting the main theorem of this section, I need to clarify a simple technical issue here: • R(m) will not be well-defined at all m such that Dr,m  >  k since there is no feasible  solution for the above tight linear programming problem • As a convention, I will let R(m)  0 at any m where the tight linear programming  problem has no feasible solution. The following theorem, the main result of this section, shows that a linear programming characterization for R(ñi) is possible by using R(m).  Theorem 2.4.2 It is always true that max (m)  1mn  =  max R(m).  1mn  Chapter 2. Pricing Perishable Inventories by Using a Restriction  Proof: Let  ñi  59  be such that max R(m).  R() I now first prove that there exists some  1<m<n ri  such that  R(ñ2) ? R(fn). Let {q,•  ,  (2.12)  q; q,••• , q} be any basic optimal ni-policy derived from the BL  model. If this optimal solution satisfies: *  then this solution is feasible to the tight problem associated with holds for th  ñi.  Therefore (2.12)  =  I now prove (2.12) when /3  <  the case that exactly one of q,,j  1. By Theorem 2.4.1, we can limit our discussion on =  ,  n is positive, say, q. By Theorem 2.3.5, the  second constraint (2.7) must be binding, so it follows that  q Now let th  =  3,  =  =  —  . /3D , 7  and define:  =  q’  if1<i<ni,  0  ifth+li3—1,  (1  /3)Drj  if i  =3;  and qu,j =  I  ffj3, 0  It is straightforward to check that the policy  otherwise. {r,j,  1  feasible solution to the tight problem associated with  ñi; ,j, ñi ñi.  Then since q  j =  n} is a —  /3Dr’,  Chapter 2. Pricing Perishable Inventories by Using a Restriction  60  we have m n >piqr,i + pjqu,j = >pqZ,. +p(qr, + q,)  R(m)  i1  i1  37  +p;(D;  =  —  pjq*.  /3Dr,)  +pq = R(ñi),  which implies that R(th).  R(ñi)  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. D  Let me say a few words here on the computational issue on th. The worst case is to solve n linear programming problems associated with R(m) for m = 1,•• ,n. The natural approach is to study the curvature property of R(m). A desirable property is that (m) is in fact quasi-concave, which will reduce the number of linear programming problems that are needed to obtain an optimal policy from the order of n to the order of ln(n). But at the this moment, I can not prove or disprove whether  m) is indeed  quasi-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 there  exists an optimal solution in Wilson’s model that consists of exactly one price. I first introduce the following notation: i = max{i : piDr,i = max pjDr,j}; 13n  (2.13)  Chapter 2. Pricing Perishable Inventories by Using a Restriction  min{i  =  max p D,}. 3  =  61  (2.14)  13n  I now first prove the following interesting lemma.  Lemma 2.4.3 For any i such that =  max  it is true that iT  Proof: I will prove the result by contradiction. First suppose that  By definition, it follows that (1  Pi Dr,j  —  ‘i )Pi; -D  pj  = (1  —  *D 2 , 2 cj )p  which is impossible since, by the monotonicity assumption on at’s and the definition of i (i.e., (2.13)), we have 1  —  aj <1  —  c. and p*D, <pD.  Therefore we must have i < i. Now suppose that L  <  By definition, we have =  =  Chapter 2. Pricing Perishable Inventories by Using a Restriction  62  which leads to a contradiction since, again by the monotonicity assumption on ai’s and the definition of i (i.e., (2.14)), we have aj <c and  Then we must have i’ <i. Hence the lemma is proved. D  I now prove the following interesting result on the structure of optimal policies in Wilson’s model that consist of only one price.  Lemma 2.4.4 Suppose that there exists an optimal policy in Wilson’s model that consists  of one price, say, {pj , qj 0 }, and such that the supply constraint (2.4) is not binding, then 0 20 p  =  max p D. 2  (2.15)  Proof: By Theorem 2.2.3, we know that the demand constraint (2.3) is binding, that is, . 0 D  (2.16)  0 < k. D  (2.17)  0 qj  =  By assumption, we also know that  I now prove the result by contradiction. Suppose that (2.15) is false, that is, D < max p 0 p D. 2  (2.18)  Define =  max{i”  =  max pD}.  i’. So we only need to consider the following two 0 Clearly, from (2.18), we know that i cases.  Chapter 2. Pricing Perishable Inventories by Using a Restriction  Case 1: io  63  <*  Under this case, it follows that 20 > D, D which implies that the firm can sell D. units at the price p. By (2.18), the allocation plan {p, D } is strictly preferred by the monopolist. This contradicts to the condition that 0 {pj . , q} is an optimal solution to the Wilson-model.  0 > Case 2: i 10 }, it follows that First, by the optimality of the plan {pj ,D 0 (2.19)  k<D. Now let  0 } satisfy the following system: q =1  0 D  D  =k.  0 q+q  Solving the above system leads to I  O  —  r  —  ,  * 2 .LJ —  10 q —  0 L/j  D—k  ,  -  —  which generates the total revenue of R =pq +p q 0  =  ,  .L.JjS —  Then by (2.17), (2.18) and (2.19), we know that , 0 R>p 0D  +  D—k , —  . D 0 p  Chapter 2. Pricing Perishable Inventories by Using a Restriction  64  0 } is an optimal solution. which contradicts to the fact that 0 {p D , In summary, both cases lead to a contradiction. Therefore, we conclude that (2.15) is true, as required. 0  The following theorem investigates whether the monopolist still prefers to use the restriction when there is an optimal solution in Wilson’s model that consists of only one price.  Theorem 2.4.5 Suppose that Wilson’s model has an optimal solution consisting of ex  0 }, with the optimal revenue value of Rv actly one price, say, {pj ,D 0  =  10 which is D 0, p  such that the supply constraint (2.4) is not binding. Then a necessary and sufficient condition that there exists an integer m such that R(m) > R° is: (2.20)  i < i,  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 that  io E {i’ :  max pD} 1<,<n  and i <io <it.  Sufficiency of (2.20): Here we need to show that (2.20) implies that R(m) > R”-’ for some m. I will prove this through two cases.  Case 1: i  =  io.  Clearly, under this case, we must have that  65  Chapter 2. Pricing Perishable Inventories by Using a Restriction  Now let m =  = i and an allocation plan {qr,i,  q,j  f  =  ,  qu,m,”  ,  as follows:  D,,,  0  otherwise;  3 D,  ifj=i  0  otherwise.  and qu,j =  I.  It is obvious that the above plan is feasible to the BL-model associated with m since k> Dm = Dr,m + Du,m  0 + Dr,  By definition of i, we know that 0 < pj* pj D 0 which implies that . 0 > Pr,io D  Pr,io Dr,jo + Pu,i.  Therefore for this m, we have that R(m) > RY.  Case 2: i 0 > i. i, Now define a policy such that the part of  . Then it is clear that m 0 Let m = i  the allocation plan for the unrestricted product {qu,m,  qu,j =  (  D,,  ifj =  0  otherwise.  ,  is given as follows:  By definition of i, we know that pjqu,j pmDu,m. j=m  (2.21)  Chapter 2. Pricing Perishable Inventories by Using a Restriction  Denote  k,.  k  66  Then by assumption, we know that  —  k> Dm = Dr,m + Du,m  Dr,m +  which leads to  ICr Now if, in addition, kr  >  Dr,m.  Dr,i, then define the allocation plan for the restricted product  as follows:  (  n  Ii  0  otherwise.  =  Then by definition of i and the fact that i <m, we obtain D,j 0 pjDr,j = pir*Dr,i > p  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. For this case, let  q  m; and  = 0 for  and qr,m be the solution of the following  system:  Dr,i  +  qr,m Dr,m  = 1  + qr,m = kr. It is easy to check that the above system has the following unique solution: —  —  —  qr,m  (2.23)  r r,m r,: —’r,ra 1 Dr,i;kr  —  —  r,m  D r,m  Chapter 2. Pricing Perishable Inventories by Using a Restriction  67  which, according to (2.22) and (2.23), leads to the following: + Pmqr,m  = —  kr  Dr,m  —  kr  —  Pi; Dr,i +  —  ,-  Pm Dr,m  —  Dr,i* Dr,m Pm Dr,m + r -Ir,m 1 Jr,i 1 = PmDr,m. >  kr  —  kr r-, Pm Dr,m  —  —  —  (2.24)  Therefore by (2.21) and (2.24), it follows that R(m)  m fl >piqr,i + pjqu,j > pmDr,m +PmDu,m = pmDm = Rw. i=1 j=m  This 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 that max R(m)  1<m<n  RW = p , 0 0D  which, according to Theorem 2.4.2, is equivalent to max R(m) = Ru.  1<m<n  (2.25)  Note that by Lemmas 2.4.3 and 2.4.4, i = i implies that . 0 = i: = i ) 0 It is evident that (i  (2.26)  R’. Therefore, to show (2.25), it suffices to prove that R(m) <Ru.  (2.27)  Note that for these rn’s such that (m) = 0, (2.27) is true by default. Also, for these rn’s such that R(m) > 0, we know that R(rn) must be the optimal objective value of  Chapter 2. Pricing Perishable Inventories by Using a Restriction  the tight problem associated with m. Let {q, ,” 1  q;  68  be an arbitrary  optimal solution to the tight problem associated with m. Define =  ,  for z = 1,... , m, or  —  j  —  m,  ,  ri.  Then we know that =1,  7i 1  i.  Now it is easy to see that by the definitions of i and i, (m)  =  pq + i=1  >  m =  pq  j=m n +  j=m n T)jpj Dr,i; + 7jPi i=1 j=m < pi Dr,i + i=1  m  10 = = pj D since i = i = io. This proves that (2.27) is true. Consequently, (2.25) is proved. This finishes the proof of the necessity of (2.20). D  Remark: We know that when Wilson’s model has an optimal solution that consists of only one price and there is excess capacity left, it means that the monopolist will choose the best value of pD . The significance of Theorem 2.4.5 is two-fold. First, it shows that 1 the monopolist can always sell some restricted units and obtain a strictly larger value  Chapter 2. Pricing Perishable Inventories by Using a Restriction  69  of revenues by further exploiting the two different market segments. Second, it gives plausible evidence that the monopolist will utilize the limited capacity more effectively if he uses the restriction as a marketing mechanism. It is also interesting to note that the only case which makes the use of the restriction unattractive is when  Z  =  i. This  simply says that when both markets reach the maximum revenue point at the same price level and the capacity is high, it makes no difference whether or not the monopolist uses the restriction.  2.5  An Application to Airline Fare Pricing  In this section, I will present a simple application of the pricing model developed in the last section to the airline fare pricing problem. It is well-known that it is a common prac tice for airlines to apply the restriction on some low-priced fares and that the availability of these restricted fares is controlled. Unfortunately, there has been no theoretical pricing model that can be used to justify such a widespread business practice. The purpose of this section is to show that the new pricing model developed in this chapter may be a useful framework that will fill that theoretical gap. It is commonly recognized that there are at least two distinct consumer groups for air travel industry: the leisure group and the business group (sometimes called must-go 18 I only consider the case that the market demand group). For the purpose of illustration, consists of two segments  —  leisure and business.  (p) and Db(p) be the demand from leisure segment and business segment for 1 Let D the unrestricted tickets at price p respectively. Suppose that the airline has an option of a restriction on some fare classes. To simplify the analysis, I make the following assumptions: The actual number of consumer segments is not crucial in applying the new model. 8 ‘  Chapter 2. Pricing Perishable Inventories by Using a Restriction  70  Percentage of market satisfying restrictions Business Travellers Leisure Travellers (1 7b) (1 71) 53% 82% 27% 68% 11% 49% 11% 53% 8% 46% 4% 35%  Restrictions Advance Bookings Minimum Stay (Days) (Days) None 7 None 14 None 30 7 7 14 7 7 30  —  —  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 segment from D’(p) to (1  —  D where )p), 1 7 (  71  —  a fixed constant  —  is the proportion of the  leisure segment travellers who cannot fly because of the restrictions. • The restriction on a ticket of price p will reduce the demand from the business seg ment from Db(p) to (1  —  p), D where 6 ) 7 (  7b  —  a fixed constant  —  is the proportion  of the business segment travellers who cannot fly because of the restriction. • 71 < 7b,  that is, a higher percentage of business segment travellers cannot accom  modate the restriction. The Boeing Company (1982) reported some empirical results on the effect of fare restric tions in the airline industry, which are given in Table 2.1. The table gives clear evidence that 71 <7b.  Now it is clear that the demand for the restricted tickets at price p is given by: Dr(p)  =  (1  —  p) + (1 71 ( 1 )D  —  p). 7b)D ( 6  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  71  Then c(p), the percentage of total demand at price p that will not buy the restricted ticket at price p, is specified by 1  —  —  —  Dr(p) D(p)  —  (1  —  —  b)Db(p) 71 ( 1 )D p) + (1 7 (p) + D’(p) 6 D —  In the development of my pricing model, I require that a(p) be a strictly increasing function. In the airline case considered here, I am going to show that this assumption is equivalent to requiring that the leisure segment travellers are more price sensitive than the 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 1 D ( p) and D”(p) are differentiable and decreasing in p. If the impact of the restriction is such that  j  <7h,  then a(p) is strictly  increasing if and only if >  where  and  i  (p)  for all p,  (2.28)  are the prices elasticities of the leisure segment demand and business  segment demand respectively, that is, dD ( 1 p) dp  p and (p)’ 1 D  dDb(p)  —  () = —  dp  X  p Db(p)  Pro of: Note that 1 Since (1  —  (1 =  —  7b)/(1  —  71) <  —  ‘y)D ( 1 p) + (1 7b)D”(P) Db(p) + D’(p)  +  —  — —  (1  —  71) D (p)  +1  1, then a(p) is strictly increasing if and only if the function  Dt(p)/Db(p) is strictly decreasing, which is equivalent to the condition that  dD ( t p) Db(p) dp  dD ( 6 p) dp  Chapter 2. Pricing Perishable Inventories by Using a Restriction  72  which leads to the condition (2.28) immediately since both D’ and Db are assumed to be decreasing. Therefore the lemma is proved.  D  Remark: It is not quite right to claim that the above lemma is consistent with the dis cussions in previous sections because of the fact that D 1 and D’ are assumed to be step functions in previous sections, rather than differentiable functions as required in Lemma 2.5.1. Nevertheless, it shows us that the monotonicity assumption on the c’s is plausible in the airline case.  One of the main reasons that airlines impose restrictions on low-price tickets is to discourage travellers who are from the business segment from buying the discounted, restricted fares. As demonstrated in the above lemma, the existence of consumer groups with different prices elasticities, together with the condition that the restriction has a non-uniform impact across different consumer groups will provide sufficient conditions for a 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 was 1.5 whereas for the must-go group it was only 1.15. Then according to the pricing model in the previous section, we may say that the practice of using restrictions in the airline passenger market is justifiable. Let us now go through the following simple example on airline fare pricing by using one type of restriction.  Chapter 2. Pricing Perishable Inventories by Using a Restriction  73  Example 2.5.1 (Airline Fare Pricing Problem) Consider that the demand function from leisure segment is given by  D(p)=  500  ifp100  300  ifpE(100,150]  100  if p E (150,250]  0  ifp>250;  and that the demand function from the business segment is given by  Db(p) =  Further assume that  ‘  = 0 and  7b  400  ifp100  300  if p E (100,150]  200  if p E (150,250]  50  if p E (250,400]  0  ifp>400.  = 0.5, that is, the restriction will result in 50 percent  of business segment travellers not willing to purchase a restricted ticket and have no impact on the leisure segment. Then the airline has four possible prices $15O,p3 = $250, and p4  pi  = ,P2 100 $  1 = 0.23,a 2 = 0.27, a 3 = 0.33, and a $400, with a 4 = 0.5,  which is strictly increasing. The corresponding demand for the restricted fares are: 73 = 200, Dr Dr,i = 700, Dr 2 = 450, D 4 = 25; and the total demand for unrestricted fares are: 1 = 900, 2 D D = 600, D 3 = 300, D 4 = 50. Table 2.2 summarizes the solutions to the Tight Problems for this example. It is clear from the table that ñz = 3 and there exists a unique optimal solution which is given by: 2 q  = 180,q 3 = 120, q 3 = 100,  Chapter 2. Pricing Perishable Inventories by Using a Restriction  Highest Restricted p = $100 (m = P2 = $150 (m = p3 = $250 (m = J34 = $400 (m =  Fare 1) 2) 3) 4)  qr2 qr2  An Optimal Solution no feasible solution no feasible solution = 180, = 120, q3 = 100 = 370.6, q,4 = 4.4, q4 = 25  Table 2.2: Solutions of the Tight Problem  with i(3)  =  —  74  Derived Revenue 0 0 82,000 67,350  A Numerical Example  82,000.’  On the other hand, there is another optimal solution to the original BL-model. Note that in the above optimal solution, the airline will sell 120 restricted tickets and 100 unrestricted tickets at the same price level p3 tickets at P2  =  =  $250 after the sales of 180 restricted  $150. We also know that at end of the sales at the price level p, there  will be no residual demand for both the restricted product and the unrestricted product at p  p3. This implies that after the sales of 180 restricted tickets at  P2 =  $150, the  airline can actually allocate the remaining 220 tickets all unrestricted at the price level 3.  Consequently, the following policy: qr2 =  180,  =  220  is also optimal. It is worthwhile to mention that that this new optimal policy seems to be more appealing to the airline since it does not involve selling some restricted tickets and unrestricted tickets at the same price level.  Remarks: In conclusion to this example, we can make the following two general com ments: • The original BL-model may have multiple optimal solutions, which might allow the Just for the sake of comparison, if the airline decides not to use the restriction on any fare, then the 9 ‘ optimal revenue derived from Wilson’s model is 80,000.  Chapter 2. Pricing Perishable Inventories by Using a Restriction  75  firm to avoid using an optimal policy that is less appealing; The Tight Problem can be used for two purposes: (i) to find the maximum revenue value; and (ii) to derive a simple optimal pricing structure.  2.6  Summary  —  Pricing by Using Restriction  In this chapter, I have investigated the monopoly pricing problem for a unique type of perishable product, such as airline seats and hotel rooms. The model explicitly incorpo rates the use of artificial restrictions. It is shown that the monopolist’s optimal pricing problem can be formulated as a mathematical programming problem. Furthermore, if the impact of the restriction on the demand is such that as price increases, the percentage of consumers who can accommodate the restriction is decreasing, then the monopolist will have a linear programming characterization on the optimal revenue and the optimal pricing policy, which substantially simplifies the original formulation. I also give a simple application of the model to airlines. I show that the model assumption that p) is strictly increasing, in the airline case, is equivalent to saying that leisure travellers are more price sensitive than business travellers, which is exactly what the airlines have in mind when they use restrictions.  Chapter 3  General Optimality Results and Other Properties  3.1  Introduction  Chapter 2 developed a simple pricing model for a monopolist who uses restrictions as a mean of segmenting the demand market. It showed that the monopolist’s pricing problem can be reduced to a series of linear programming problems. It demonstrated that by properly setting the level of the highest restricted price and rationing the sales at different prices, the monopolist needs to charge no more than three prices to maximize the revenue. On the other hand, there is a key assumption that the monopolist offers the restricted prices first, and then after all sales have been accommodated at the restricted prices, sales are offered at the unrestricted prices. I call this type of policy a primary policy. Even though a primary policy is very natural, it does limit the monopolist’s choice on possible pricing structures. Another implicit assumption was that the firm offers one kind of price at a time, which means that if the firm is selling restricted products, then unrestricted products are not available. In practice, for example airlines, both restricted fares and unrestricted fares are used and they are made available at the same time. I will call this the simultaneous availability issue. The purpose of this chapter is to resolve these two important issues. In response to the first concern, I will introduce a general pricing policy which allows the firm to put restrictions at any possible prices. I am going to show that any general pricing policy  76  Chapter 3. General Optimality Results and Other Properties  77  is weakly dominated by a primary policy, where the criterion for dominance is defined in terms of realized revenue. This consequently implies that any optimal policy in the class of primary policies remains optimal in the class of general policies. To address the simultaneous availability issue, I will demonstrate that based an op timal 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 avail able can still generate the same amount of the revenue realized by the primary optimal policy. In other words, the firm can always find an optimal policy that is consistent with existing pricing practice by use of artificial restrictions. To make the argument go through, a reasonable behavioral assumption is needed, that is, when some unrestricted units and some restricted units are offered at the same price level, consumers from the restricted market will first try to buy the unrestricted units. This chapter is organized as follows. Section 3.2 establishes some preliminary results on general pricing policies. Section 3.3 will prove the general optimality results. Then section 3.4 discusses simultaneous availability issue for optimal policies. The last section is a summary.  3.2 3.2.1  Auxiliary Results on General Pricing Policies Notation and Definitions  I will use the same model settings as in Chapter 2. That is, there is a monopoly firm which wants to sell a fixed number of units of a certain product. The product is perishable and 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 the demand for the product at price p only when the restriction is not attached. Denote the  Chapter 3. General Optimality Results and Other Properties  78  total demand at price p by D(p). Then D(p) = Dr(p) + D(p). Let a(p) be the percentage of those consumers with reservation prices of p or higher who will be unable to buy the product because of the restriction, which is given by: D(p) Dr(p) =1D(p) D(p) Further assume that Dr(p) and D(p) are step functions defined on a finite set of prices , hthatPj<P ...<PpJ-<oo’Thatis,forl=r,u 2 {Pl,.,PN}suc <  Dj(p) =  , 1 D  ifpPi;  , 1 D  if p E (P:_i,Pj] for 2  o  ifp>FN.  i  N;  Consequently D(p) and (p) are also step functions:  D(p)=  where D = Dr,t +  1 D  ifpPi  D,  i,F]for2in ifpE(P _ 2  o  ifp>P,  c  ifpPi  a  ifpE(P..i,P]for2in  1  ifp>P;  and  . As in Chapter 2, I will make the following assumptions throughout a(P ) where a. = 1 this chapter: D is strictly decreasing in i; 1 T here is a minor notational change here from Chapter 2, where the price set is specified by ,p}. The change here is purely presentational. {Pi,  Chapter 3. General Optimality Results and Other Properties  • D , is decreasing for 1 1 • cx:j  =  79  r, U;  is strictly increasing in J.  Since the monopolist has a choice of imposing restrictions on any possible prices and offering the product with and/or without restrictions at the same price, a general pricing policy 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 general 2 First, a general policy should allow the firm to offer restricted and unre pricing policy. stricted product at the same price level in an arbitrary order in the sense that the firm can sell unrestricted product first and the restricted product later at the same price, and vice versa. Because of this, it is possible there are more than two allocations on the same price, for example, the firm may offer some restricted product at price P1, then some unrestricted product at the same price, then again offering some additional restricted , 1 product at this price. Consequently, the number of initial prices, P  ..  ,  F, has to be  modified to capture this possibility. Therefore, by looking at the order of offered prices together with the option of attaching restrictions, we may have a sequence of prices Pi  P2  such that pj E {Pl,”.,PN}, for all i  =  1,... ,n. Furthermore, at each  of these offered prices, there is also a tag which indicates the nature of this price. For example, let S is the tag variable on the price pj, then S be a restricted price; and S  =  =  r implies that the price  j  u indicates that the price is unrestricted. I now introduce  the following definition of a general pricing policy.  The following discussion also helps us to understand the reason why I need the above minor notational 2 change in this chapter.  Chapter 3. General Optimality Results and Other Properties  80  Definition 3.2.1: A general pricing policy is a bundle {n;(p,q,,Sj),i  =  such that ,... ,FN}, for alli 1 • piE {P • S  =  1,..,n such that p  P2  r indicates that p is attached with restriction and S  =  =  u implies that p, is  unrestricted; and is the quantity available for sale at the price i associated with the tag variable  • Si.  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 quantities with the the same characteristics (that is, restricted or unrestricted) consecutively at the same price level. For example, if S  =  r and qr,i > 0, then there is no need  to consider any policy such that  —  —  —  S+i =  Pi+1 qr,i+i  r;  = pi;  >  0.  • For the rest of this chapter, when we refer to G-policy, we require that if there are positive quantities allocated at the same price level, then the characteristics of the units 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; 2 •p=P f 1 _ 3 ori=1,.,m,andp=P orj=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 re stricted units are sold before the sales of the allocated unrestricted units, the monopolist’s pricing problem can be formulated as the following two-stage process: 1. For any given 1  m  N, solve the following linear programming problem: Pqj +  Max s.t.  Lji=1 Dri çN qi L..ij=m DjDr,, m  +  q,-  —1 —  N Zj=m q,  q  O,Vi and Vj,  qri,quj  which leads to a realized revenue of .(m); 2. Find a value of th such that Ti) =  max f(m).  1<m<N  It is the purpose of this chapter to show that there exists an optimal policy in the C-class that has the above special structure. For presentational purpose, I give a special name for the type of polices discussed in Chapter 2:  Chapter 3.  General Optimality Results and Other Properties  Definition 3.2.2: We call a G—policy {n; (p, qs,j,  i  82  1,.-. , n} a primary rn-policy  =  if l.S=r,foriz=l,...,rn;and6=u,fori=m+1,...,n;and 2. Pm  Pm+i,  where m: 1 <m <n.  Remark: Because of the special structure of a primary policy, we can represent any  primary policy as follows: {(pi, q,i),  , (Pm, qr,m); (Pm, qu,m),” , (p,, qu,n)};  or simply, ,qu,n}.  {q,i,.  I will use this representation if it causes no confusion. We should also keep in mind that under 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 pricing policies 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, inates another feasible G-policy {n’; (p,  6), i  6), i 1,.  .  .  , n’}  =  1,..  .  , n} weakly dom  if  •1  If the above inequality is strict, we call it strict dominance. And if it holds with equal ity, we say that the two pricing policies are equivalent. Consequently, we say a feasible G-policy is optimal if it weakly dominates all other feasible policies.  General Optimality Results and Other Properties  Chapter 3.  Given any G-policy {n; (pt, • For i  =  • Let -y  1,  =  •  ,  ri let d  =  Si), i  =  Dr(pj), d  1,.  =  ,  83  we introduce the following notation:  D(p) and d  =  d + d.  be the percentage of consumers who will not purchase the product  —  at price Pj if restrictions are attached. • For t  =  1,.  ,  n and  j  t, let  be the total residual demand at the unrestricted  price P, after the sales according to the partial policy {(p,  —  =  1,... ,t}.  be the corresponding residual demand at the restricted price Pj• Define  And let _1u__1. 3 u,t  SJ, i  1r  —  ut,j. —  • Let  7t,j =  1  be the percentage of residual consumers who will not purchase  the product at the restricted price Pj after the sales according to {(pi, qs,i, Si), i 1,... ,t}. It is worthwhile to mention that 1. 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, ; and 7  4. {-y,} is increasing, but may not be strictly increasing as c is. 3.2.2  A Lemma on the Property of Monotonicity  Let us now go back to the discussion of the G-policy. For a G-policy as defined in Defini tion 3.2.1, if the sales are at an unrestricted price, then there is no impact on the values of the ‘s, which means that the portion of these consumers who can not accommodate the restriction remains unchanged due to the use of proportional-rationing rule to update  Chapter 3.  84  General Optimality Results and Other Properties  the residual demand. But if the sales are at a restricted price, the values of ‘s will be changed. The following lemma shows that the monotonicity of 7’s will be maintained.  i = 1,.  Lemma 3.2.1: For any given G-policy {n; (p , 2  ..  , n} and given t,  j  is  t.  increasing in j on j  1 = u, then the sales of Proof: I prove the result by induction. First, consider t = 1. If S  unrestricted products makes no impact on the  that is,  7’S,  7i,j  =  7j,  which is increasing  by the monotonicity assumption on cvi’s. Now if S = r, then  and 3 = d, , 1 d  —  Therefore,  dr  (l—)(l—7) l—(i— ) 7  Since 7 is increasing in that  71,j  j and  -  1, it follows immediately from the above expression  is also increasing. This proves the result for t = 1. is increasing. By the same argument,  Suppose that the result holds for t, that is, if S+ = u, then  7t+1,j  =  7j;  hence the result is true for t + 1 in this case. If 8+ = r,  since —  —  the monotonicity of that  qr,t+i  7t+1,j  3 7t+i,  —  , 1 d÷ , 1 d+  —  (1  (1 —  —  1— (1  —  7t,j)  qre+1  follows immediately from the monotonicity of  and the fact  This implies that the result also holds for t + 1 when  Therefore by induction the result is true for t = 1, 2,... and the lemma is proved.  = r. D  Chapter 3. General Optimality Results and Other Properties  85  The 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 statements  is true: 1. 7tj 2.  =  {yt,j}  1 for all  j  t;  has a subsequence that is strictly increasing in  j.  Furthermore, we have: 1. for a given t and any ji  <32  such that p , if 7t,j 32 1 <p 1 < 1, then 1 < 7t,j 7t,j ; 2  2. for all  j  t” > t’, 7t’,j  Remark: The above result shows us that as long as the residual demand for the restricted product is positive, the 7-sequence is strictly increasing in terms of price levels. Note that the monotonicity condition on the original a-sequence is also in reference to price levels. Therefore, the result in Corollary 3.2.2 will enable us to use the BL-model in Chapter 2 for the residual market if we limit to primary policies for the residual market.  3.3 3.3.1  General Optimality Results Three Fundamental Lemmas  Recall 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 than  the restricted products. Such a type of policy limits the firm’s choices on possible pricing  Chapter 3. General Optimality Results and Other Properties  86  structures. For example, it rules out the possibility that the firm may consider to sell some unrestricted units before the sales of some restricted units. The purpose of this section is to further investigate the firm’s pricing problem under the expanded set of pricing policies  —  the set of general pricing policies as defined by Definition 3.2.1, which  contains policies that allow the firm to impose the restrictions at any price level. In this section, I want to prove a very interesting general optimality result, which shows that any G-policy is in fact weakly dominated by a primary policy. Consequently, any optimal policy 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 of linear programming problems; and • It is very difficult, if not impossible, to discuss the nature of optimal polices in the G-class. For ease of presentation, I introduce the following definition:  Definition 3.3.1: For any G—policy {n; (pg, q,, , Sj, i 2 be active if qs,j =  >  0. If 6  =  =  1,.  .  .  ,n}, the price  i  is said to  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 that is strictly less than an active restricted price. The following lemma shows that I can focus on the policies such that there is no unrestricted product sold at the level of the last active restricted price prior to the sales of the restricted product at this price.  Chapter 3.  General Optimality Results and Other Properties  87  Lemma 3.3.1: There always exists an optimal policy in the G-class with the following properties: 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 restricted units.  In words, at the last active restricted price level, the firm sells the allocated restricted units 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 last active restricted price and io be the largest index for an active unrestricted price before the restricted price Pm, that is, io =max{i : i < m,.5—u and 0 It is evident that i  m  —  1 and p 0  > O}.  Pm.  Now if p 0 < Pm, then Lemma 3.3.1 is true by default since it means that there are no unrestricted units that are sold at the price level Pm before the sales of the allocated qr,m restricted units (also) at Pm. So we only need to consider the case that Pio = Pm  Without loss of generality, we can take i 0 = m  —  1, i.e., Pm—i = Pm. Hence Pm—i is the  largest unrestricted price that is precedent to Pm. By definition of Pm, we know that after price Pm, the firm only sells the unrestricted products. Therefore, again without loss of generality, we can assume that Pm <p  Pm+i <  We need to discuss two cases here.  Case 1: Pm <Pm+i. By Corollary 3.2.2, we know that strictly increasing in  j  7t_i,j  is strictly increasing in  j  m since p 3 is  m. Then by Theorem 2.4.2 and Lemma 3.2.1, I know that for  Chapter 3. General Optimality Results and Other Properties  the remaining market specified by {dm_i,j, d_ 1 : set {Pm,  ..  j  = m,  ,  88  n} together with the price  ,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),.  ,  u)}; and  (pa,  • there is no residual demand for restricted product after the sales at the last active restricted 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 > Pm Consequently, since Pm—i = Pm, qr,m  — —  Um_1,m  —  ,  ,.  — —  Um...2,m_1  q,—i  —  )m—2,m—i  .  )m—2,m  —  —  and the total residual demand at the (unrestricted) price p 3 after the sales up to the restricted price Pm is given by: dm,j =  = d_ , = (1 1  —  qu,m-i  , for 2 )d_  m—2,m—i  j  m.  (3.2)  By (3.1), it is easy to check that qu,m—i  +  qr,m  >  n—2,m =  ,m_i• 2 d_  Let us now define a new G-policy as follows: {n; (ps, q,., 6), i = 1,• S  ifl<i<m—2  r  ifi=m—1  it  ifim;  .  ,  n} such that  Chapter 3. General Optimality Results and Other Properties  89  arid iflim—2 =  ifi=m—1  ,m 2 dn_ qu,m—i + qr,m  —  if i  ,rn 2 d_  =  m  ifi>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 that Pm—i becomes the last active restricted price;  • furthermore, the new policy is equivalent to the old policy since it is easy to check that pqs,i; and • after the sales at the restricted price Pm—i, there is no residual demand for the restricted 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  = qu,m—i + qr,rn  —  m—2,m  = qu,m—i + (1  qu,rn—i —  Urn —2,m— i  =  1m—2,m— 1 d’  r )dm_ rn—i 2  —  r dm_ rn—i 2  (3.3)  qu,m—i.  Um_2 rn—i  Note that, after the sales of qm.i units at the restricted price Pm—i, there is no demand for restricted product at any price Pj for  j  m. As usual, let  be the total residual  demand at price P 2 after the sales according to the new partial G-policy {(p,qçi,S) : i = 1,”.,m—1}.  90  Chapter 3. General Optimality Results and Other Properties  Then it is clear that for  d’m_i,j =  the unrestricted price Pj m,j  >  m  1.  —  Pm, it follows from (3.3) that for the new policy the residual demand at  Since Pm—i  U  j  —  (i  m) after sales up to the unrestricted price Pm  qu,m  r—  —  —  )‘m—2,j  —  )Um2,j  —  Um_2,m  —  given by  q,—i ‘u  ,  )tlm_2,j’  —  —  Um2,mi  18  Um_2,m_i  which is identical to (3.2). This shows that the new policy is feasible. Therefore the new G-policy ,S,),i = 1,..• ,n} 1 {n;(p,q is also an optimal policy and is such that at the last active restricted price level, the unrestricted product, if allocated with a positive amount, will be sold after the sales of the restricted product at this price. Case 2: Pm = Pm+i  Under this case, we know that at the level of the last active restricted price, the firm will sell some unrestricted units first (qu,m._i), then some restricted units (qr,m), and then some unrestricted units again (qu,m+i). Note that after the sales according to the original policy 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  together with the set of prices  {Pm,Pm+2,”  91  ,pn}. By the same argument used in Case  1, without loss of generality, we can assume that  = 0. Therefore (3.1) and (3.2) are  still valid. Now define another new G-policy  Sfl, i  {n”; (p’, such that n” = u  —  = 1,.  ,  1 and 73  ifl<i<m  Pi+i  if m + 1 <i <n”;  6  ifi <i<m—2  r  ifi=m—1  u  ifmin”;  and if1im—2 =  ifi=m—1  ,m 2 d_ qu,m—i + qr,m  —  m + qu,m+i 2 d_  if i = m ifm+1 i<n”.  qu,i+i  Again, we only need to show that this new G-policy is feasible. First, by (3.3), we get —  qu,m  —  dLm—2,m—i qu,m—i Um_2,m_i  + qu,m+i.  (3.4)  , be the total residual demand at price Pj after the sales according to 1 Similarly, let d_ the partial new policy {(p’,  S’) : i = , = 1 d_  ,  for  m  j  —  1}. As in Case 1, we know that m  —  1.  Under the new policy, by (3.4) and the fact that Pm—i = Pm at the unrestricted price P 3  (i  Pm+i, the residual demand  m) after the sales up to the unrestricted price Pm  iS  92  Chapter 3. General Optimality Results and Other Properties  given by: I,  q,,,, —  —  m,j  — —  im—i,j  ,j,,  qu,m—i  11  Um_2,m_i  m—1,m  Pm+i, then according to (3.2), for  Now since Pm  ,‘  =  (Lm+1,j  —  qu,m+i ‘u —  —  j  ‘-‘m—2,m  m + 1,  \ju )Um,j dum ,m+ 1  dUm,j =  Um,j  —  qu,m+i  ,  ‘m,m+1 qu,m—i  (1  —  —  qu,m+i )m—2,j  —  m—2,m—1 11  —  m—2,m+1  qu,m—i  qu,m+i  Um_2,m_1  (Lm_2,m  I,J.  —  ‘m—2,j  —  \AU  which implies that 1 = dm+i,j, for d,_ Note that for  j  =  m + 2,•  .  ,  n, q,_ 1  =  j  =  m + 1,  ,  ii.  (3.6)  qu,i. So (3.6) implies that the new G-policy is  indeed feasible. Therefore the new G-policy is an optimal policy with the property that Pm—i  is the last active restricted price.  In summary, we have shown that there always exists another optimal G-policy such that Pm—i iS the last active restricted price. If there exists another active unrestricted price pj such that •  0 i’  < m  •  = 11,  • Pi  =  —  1,  Pm—i,  then we can repeat the above procedure. In the end, we will have an optimal policy such that pq, will be the active active restricted price. Consequently, by induction, this will lead an optimal policy with the following properties:  93  Chapter 3. General Optimality Results and Other Properties  • if we let p, be the active restricted price and define lo = min{l : pj then  10>  Pnt, Si = u, qu,i  >  O},  m + 1.  This implies that there are no unrestricted units at the price p, that are sold before the sales of the allocated restricted units also at the price level p,. Therefore the lemma is proved.  D  Remark: Simply speaking, Lemma 3.3.1 says that if the firm plans to sell some un restricted units and some restricted units at the last active restricted price level, then without loss of generality the firm can limit its attention on these policies such that the restricted units are sold first.  The following two lemmas further explore this issue by considering cases where there are 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),..  .  ,  u)}  (pa,  is equivalent to a primary policy, where p  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 one active 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 restricted  Chapter 3. General Optimality Results and Other Properties  94  units at this active restricted price will exhaust all the residual demand for the restricted product, then the optimal policy is equivalent to a primary policy. Proof: Clearly, under the given policy, I know that after the sales of the restricted product at price for  j  P2,  there will be no residual demand for restricted product at price p,  2. Therefore, the total residual demand at price p 3 is given by , 2 d  =  =  (1  3.  —  (3.7)  Consequently, the feasibility condition for the remaining unrestricted market is  1  j=3  3  I now construct a new policy as follows: I 1 q-,  2 q,  (1  —  11 a , 1  I q, 1  =  qu,i  1 q ---)d, q,, 2  =  qr,2  =  q,j,Vj  = qu,i  A  = 0, where x  A  y  =  —  —  qr,i;  ; 2 q,.,  3;  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 price level p’, where the upper bound is determined by the existing demand d and the original allocation • if at  Pi,  qu,1  at the unrestricted price pi;  the allocation q 1 cannot exhaust all the restricted demand, then q 2 will  take care of the residual market for restricted product at p2; • the new policy leads to a primary policy since either (i) q 1 following primary 1-policy: ; q, 1 , qu,2, 1 { q,  ,  =  d, which leads to a  General Optimality Results and Other Properties  Chapter 3.  or (ii) q 2  =  (1  —  /d)d 1 q  1 0, which implies that q  >  =  95  0 and consequently leads  to a following primary 2-policy: , q,3, 2 , qr,2; q, 1 { q,  ,  q,};  • it is easy to check that the new derived primary policy is equivalent to the original policy. Let {q. , 1  “,  q; q, qu,t+i,  specification, where I  Case 1: t  =  1 or I  ,  q,n}  be the derived primary t-policy from the above  2.  =  1.  This implies that 1 = q  Let (p2,  d  q, = qu,i qu,i, 1  —  and  2 = qr,2, dq,  = qu,j,  be the total residual demand after the sales according to  V.)  ), 1 (p’, q  3. (p1,  ) and 1 q  ). Then it is easy to check that 2 q —  q  —  d  dL1  qr,2  du2  ,ju i•  Therefore, to prove the feasibility of the new primary policy, it suffices to show that, ,,Vj 2 d 3 which, by (3.7), is equivalent to .4  Jn what follows, the symbol  means “equivalent”.  3  96  Chapter 3. General Optimality Results and Other Properties  2 q,id U  d  U  1 U  1  ,Iu  ,.?  2 U  U1(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 that 1 q.  =  2 q,  =  qu,i  2 c1;q  =  q,i  (1—  1 a ..Ir /  2 = q,id(  1 U  1 —a;-)—  T d  —  d1  Again, let d’, be the total residual demand after the sales according to ) and 2 q  1  3.  =  (p2,  -  1 U  (p2,  (p1,  , q ) 1  Then =  (1  —  U2)U  (1  =  —  qu,i  1  Ui  ,1u ,1 2,ju 1 ..lr,1u) j• 2 1 U U  I hope to show that , d , 2 which, by (3.7), is equivalent to du dr  T d  d’  2 d —  <  1 d  2 U  d 1 d  ——  .,lu— 1u 1 U  dL < 47i  2 d  72,  which follows from the monotonicity condition on 7’s. This shows that the new policy is feasible. This proves Case 2. In summary, the lemma is proved.  D  97  Chapter 3. General Optimality Results and Other Properties  The following lemma considers the case that there are two active restricted prices between the first two active unrestricted prices.  Lemma 3.3.3: Any optimal G-policy of the following form:  { (pi,  u),  q,i,  (p2, qr,2,  r), (ps, q,3, r); (pa, q,4, u),.  .  .  ,  (pa,  u)}  is equivalent to a primary policy, where Pi  P2<  P3  4<  <p and  qu,i  > 0,qr,2>  0,qT,3  =  3 >0. d,  In words, it says that if there exists an optimal policy in the G-class such that (1) it has two consecutive active restricted prices, (2) there is one and only one active unrestricted price before these two active restricted prices, (3) there is no residual demand left for the restricted product at the end of sales of the restricted units at the second restricted price level, then this optimal policy is equivalent to a primary policy. Proof: First of all, it is easy to show that  which is equivalent to qu,i  qr,2 _L Jr  Ui  2 U,  qf,3 Jr  (43  —  —  This also implies that there is no residual demand for restricted product after the sales up to (ps, q-,3). Therefore, the total residual demand after the sales up to (ps, q,.,3) is given by (3.9)  =d=(1_)d7. d , 3 Define a new policy as follows 1 q,  = qu,i  A d, q, 1  =  q,i  —  ; 1 q.,  98  Chapter 3. General Optimality Results and Other Properties  qr,2 =  (1  1 q ——)dr A 2  —  2 = q,  qr,2  3 = q,,,  q,.,  —  ; 2 q,,  U,’ I  I  3 = q,.,  I  qr,2,  (1  q,,  qr,2  r ,  —  —  —  2 U  I  4.  =  Step 1: To show that the above new allocation plan corresponds to a primary policy. 1 First of all, at price p,, if q 0, and q 2  • q,  =  d. This implies that  d, it indicates that q,,  0,  3 = = q  2 1 < d and q which leads to a primary 1-policy. Secondly, if q  =  (1  —  -)d, then it  implies that • q, 1  = qu,i, q, =  3 0, and q  0, q 2  =  0,  2 < (1 which leads to a primary 2-policy. Finally, if q, < d, q. (1  —  —  1 • q  —  !!-)d and q 3  =  ?-)d, then it implies that  2 = = q  0,  0, and q 3  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 that q, =  2 = q•  3 q,  d  qu,i,q, = qu,i  —ct  Vj  4.  = q,3,  = qu,j,  99  Chapter 3. General Optimality Results and Other Properties  Some further observations can be made here: • the new plan is feasible for the restrictedpi.market; • there is no residual demand for restricted product at any price Pj  1) after the  (i  sales of q 1 units at the restricted price Let d, 3 be the total residual demand after the sales according to  ) 2 q. ) , (pi, q ), (p2, q 1 (p’, 1  and (ps, q ). Then I get 3 d’3,j  =  (1  —  —  1 d  —  2 d  =  3 d  (1  — —  d  —  ‘dU .1uI 3’ (3  —  2u  Au  U  “2  4.  Vj  Therefore to prove the feasibility of the new policy, it suffices to show that ,, 3 d which, by (3.9), is equivalent to requiring that 1 lu  3u  1 a  ,lu 3 U,  2 u  + 1 du1 d  Au  +  1  == qu,1(——)+  —  <  ,u  —  d dL  q, 1 qr,2 + + d d-’2d ‘1’  2 d q,-,3  dV du1 3d  +  3 d  d’ <_L —  1 U,  1.  (3.10)  On the other hand, by (3.8), I know that to show (3.10), it suffices to prove that dr >dandd- 4=  T1 d 2  dU  -  1 Ct  1 U,  d 1 d 3 d —and—>— Au Au Au  1 d —  u  —  1 d  2 U,  1 U  3 U  d  d”  T d  T d  a 1 d’ 1 d  2 ct  1 a  3 Ct  du  du  dU  2 d  1 d  —  —  3 d  which follows from our monotonicity assumption on 7’s. Therefore (3.10) is true. Hence the result is proved for Case 1. Case 2: t  =  2.  Under this case, I will have the following: 1 = qu,i < q  2 q,  = qr,2  —  =  =  (1— =  d  2’  ,Vj 3 q,  4.  Chapter 3. General Optimality Results and Other Properties  100  I also know that • P2  is the last active restricted price;  • there is no residual demand for restricted product after the sales at restricted price P2.  Again, let d, 3 be the total residual demand at price p 3 to  ), 1 (pi, q  (p2,  ), 2 q,  3 d,  4) after the sales according  ) and (ps, q 2 q ). Then 3  (p2,  =  U  (1—  c,2  —  U3)U  =  2 q,2—q,  (1—  —  I now want to show that 4,  ,,Vj 3 d 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 that id”  1  1  d  1  d1  d  d  => U U 2  1 U  U  2 U  2 U  1 d  ‘-2  2 d  3 d  1 U  2 U,  3 U,  dr  du  d’ d’ and —- < —p 2 d 2 d 3 d 1 7 a < 2 , 3 nd7 -y  —1 d  4=  2 d  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  101  Similarly, I know that under this case, q,., = q,i  qu,i  I  <  = qr,2  ,q 2 _-j_)d <(1— 3  =  qu,i  qr,2  ‘-“1  2  (1—————  1 tL  3 = q,3 q  Let d, 3 be the total residual demand at price Pi ), (p2, q 1 ), 2 (pi, q,  —  (  ,q 3 q 3  r  4.  =  4) after the sales according to  (p3, q. ) and (ps, q 3 ). Then 3  I need to show that d,,  d , 3 ,  which, by (3.9), is equivalent to (3.12) Again using the fact (3.8), to show (3.12), it suffices to prove 1u  1  I  3 U  1 I  J U3  1 (hi  47371, (Li  3 UU  1  U3  (hi  which 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 policy is indeed feasible. Finally, it is easy to see that +  = piqu,i  +  which implies that the new primary policy {q , 1  pjqr,j  +  q; q,  pjqu,j,  ,  q} is in fact equivalent  to the original policy. Therefore, we have shown that there exists a feasible primary policy that is equivalent to the original policy. So the lemma is proved.  D  Chapter 3. General Optimality Results and Other Properties  102  In summary, Lemma 3.3.1 resolves the issue on the order of sales of the unrestricted units and the restricted units at the last active restricted price level by showing that the firm can limit its attention on policies such that the restricted units are sold first at the level of the last active restricted price. Lemmas 3.3.2 and 3.3.3 demonstrate that any optimal policy with the following two properties: • the first active unrestricted price is followed by one restricted price or two consec utive restricted prices; and • there is no residual demand for the restricted product at the end of sales of the allocated restricted units at the last active restricted price, will be equivalent to a primary policy. 3.3.2  The General Optimality Theorems  We now use Lemmas 3.3.1, 3.3.2 and 3.3.3 to prove the following theorem, which deals with the situation where following the first active unrestricted price, there are many re stricted prices.  Theorem 3.3.4: For m: 2  m  { (pt, q,i, u); (p2, qr,2, r),.  n, any G-policy of the following form: .  ,  (pm, qr,m, r); (Pm+i, qu,m+i, u),.  .  .  ,  (pa,  u)}  is weakly dominated by a primary policy, where P1P2<<PrnPrn+1<”Pn.  Proof: Clearly, I only need to consider the case that qu,i > 0. Note that, given the value of qu,i, the total residual demand at price Pj is given by: —  (1  —  -)d,j  2;  General Optimality Results and Other Properties  Chapter 3.  103  and the residual demand for restricted product is given by: =(1—1’)d,j 2. 3 d, Given that after the sales of unrestricted product at price stricted products at prices  P2,  Pi  pj the firm plans to sell re  and unrestricted products at prices Pm--i,” ,Pn.  ,Pm  We also know that the sales of qu,i units at the unrestricted price so in  = 7j for all  j.  Tf Pm =  j  Pm+1,  does not change -y’s,  > 2. By Corollary 3.2.2, if Pm <Pm+1, then yj is strictly increasing  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 Chapter 2 to the residual market specified by {(d,, d ,) : 1 set {p2,  •  j  = 2,•  ,  n} together with the price  ,pn} By Theorem 2.4.2, we know that there will be a primary policy ), 2 , q, 2 {(p’  ,  (p’,, q,,’);  qL’),.  ,  (P’n’,  which is defined on the residual market such that = n if Pm <Pm+1 and u’ =  1.  2. the set of prices  {j4,.  —  1 if Pm  Pm+i  ,p’,} is the same as the set of prices {p2,••  3. dim i = 0, that is, there is no residual demand for restricted product at price 4 p  p’,’ after the sales according to  4. at most two of q , 2  ••  ,  {(p, qu,i), (p’,q, ),” 2  (p’,, q,,)};  are strictly positive;  5. it weakly dominates the partial policy:  {(p2,qr,2),”,(pm,qr,m);(pm+1,qu,m+1),”,(pn,qun)}.  General Optimality Results and Other Properties  Chapter 3.  104  This consequently leads to the new G-policy:  which in turn is weakly dominating the original policy. Now if all of q, , 2  q’ are  zeros, 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 positive allocations at restricted prices, then the theorem follows from Lemma 3.3.3. This proves the theorem.  D  I 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, q ,j, 82), i 1 tive restricted price and let Pui < {pi,  =  1,.  •  ,  n}. Let Pm be the last ac  < Pu 3 be the unrestricted prices in the price set  ,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: 5 , 3 {(pu ) ;(pi,qr,i),i qu,u = 8 u + 1,••• ,m;(pj,qu,),j = m+ 1,... ,n},  which will lead to a policy in the form of {(p,q),i = us,... ,m ; (p,qj),j = m 8 3 + 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  price that is below pm ; and 5  15  the largest unrestricted  Chapter 3. General Optimality Results and Other Properties  • there are oniy s  —  1 active unrestricted prices among in the price set  105  ,Pms },  which are Pui,” , Pus_i. As we can see here, every time we use Theorem 3.3.4 on the modified policy, we will eliminate one active unrestricted price that is strictly less than the last active restricted price. Therefore, after s times, we will end up with a policy that has no active unrestricted price that is less than the last active restricted price, or equivalently, we will get a primary policy, as required. Hence the theorem is proved. 0 To conclude this section, I here present an example that demonstrates that there are situations which have optimal policies that are not primary.  Example 3.3.1 Existence of Non-Primary Optimal Policy Consider the following simple demand structure: 1 = $100, P P 2 = $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: 2 = 50, 1 = 100,q 1 = lOO;q q which generates total revenue of $27, 000. Let’s consider the following policy in a form as 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 P 1 = $100, then sells 50 restricted units also at P 1 = $100, and finally 50 unrestricted units at P 2 = $140. It is clear that this policy also generates total revenue of $27,000. Let’s check that this policy 1 = 300 and D is also feasible. First, note that D 2 = 100. If the firm sells 150 unrestricted units at F , then we know the following: 1  Chapter 3. General Optimality Results and Other Properties  106  • among these 150 units sold, the portion that is originally from the restricted market is given by: 1 Dr  =  100  X  150  =  50,  o’J’)  which implies that the residual demand for the restricted product at the price level 1 is 50; P • there is residual demand for the restricted product at the price level P 2 since Dr,2  =  0; and • the residual demand for the unrestricted product at P 2 is =(1._-)x100=50. 2 (1_1)D, Therefore, the allocated 50 restricted units at the price level P 1 and the allocated 50 unrestricted units at the price level P 2 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 difficulty of some primary optimal policies where the firm has to sell some restricted units first and some unrestricted units later at the same price level. 4 These observations prove to be quite useful in the next section too. 1n the above example, it is not easy to sell 100 restricted units after knowing that there are still 4 another 100 unrestricted units available later. This is what I mean implemena1ion difficulty. The new non-primary policy does not have this difficulty.  Chapter 3. General Optimality Results and Other Properties  3.4  107  Further Properties of Optimal Policies  In the above section, I have demonstrated that there exists a primary optimal pricing policy in the G-class. As pointed out at the beginning of this chapter, there is another implicit assumption in the BL-model, which assumes that the firm offers one type of the product at a time. In other words, the allocated unrestricted units will not be put on sale until the restricted units are sold out according to the plan. On the other hand, such an assumption is not consistent with the existing pricing practice by airlines, which use multiple prices and make all offered prices available at the same time. The purpose of this section is to resolve this issue by arguing that there always exists an optimal policy with the property that all offered quantities  —  both restricted and unrestricted  —  at different  price levels can actually be made available at the same time. To finish the argument, I need the following behavioral assumption: 5 • Behavioral Assumption: If there are restricted units and unrestricted units al located at the same price level and all these units are made available at the same time, then the unrestricted units will be sold first, or equivalently, when restricted units 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 from Theorem 2.4.2. Clearly, we only need to focus on the basic primary optimal policies that are derived from the Tight Problem in Chapter 2. Recall that the Tight Problem is a linear programming problem with three structural constraints, which implies that any basic primary optimal policy consists of at most three positive quantities, of which at least one, but at most two, of them is at an unrestricted price level. Because of this, I need to discuss two scenarios. Even though I do make this assumption, but it is oniy used once. 5  Chapter 3. General Optimality Results and Other Properties  108  Scenario 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  • qu,k =  <k with active prices F, F 3 and Fk. We also know that D,k,  which implies that there will be no residual demand for the unrestricted  product at p • q, 3  =  (1  Fk  —  after the sales of the unrestricted units at Fk; which implies that there will be no residual demand for the re  stricted product at price level p to {(F, q,.,j),  3 after the sales of the restricted units according F  qj)}.  3 <Pk, it is easy to see that I now need to consider three cases. First, for the case that F if the two restricted prices P, F and the one unrestricted price Fk are all made available at the same time, then the firm still can sell  units restricted product at P,  q,.,j  units  , and qu,k units of unrestricted units at F,. due to the following 3 of restricted product at F fact: • Consumers from two market segments will only buy the product targeted to their respective segment because a consumer from the restricted market will not only buy the unrestricted product because the restricted units are cheaper and a consumer from the unrestricted market will not buy a restricted unit because he cannot accommodate the restriction. For the case that i  <j  =  k, consider the following new policy:  qr,t  q,,j  ift=i  0  otherwise;  =  1  Chapter 3. General Optimality Results and Other Properties  109  and  1I  u,k+-ff)  r,k  1  t—  otherwise.  0  Clearly, this new policy is also a primary optimal policy with only one active restricted price P 1 and one active unrestricted price Fk. We now argue that for this new optimal . are available at 1 policy, if both the restricted units at F, and the unrestricted units at P the same, the firm still can reach the projected sales of q. 1 and q 1 restricted units at P unrestricted 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, the firm will capture all the demand for the unrestricted product at price Pk, which is ; 1. D,  • for consumers from the restricted market, they will not buy the restricted product at price Pk until the allocated q 1 restricted units are sold out; • after the allocated q restricted units are sold out, according to the proportionalrationing 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 the  —  unrestricted product at price P 1. 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  is obvious that the firm just sells Dr,k + D, 1. 1. P  =  q,.,j  =  0, qr,j  =  Then it  Dk unrestricted units at the price level  without selling any restricted unit.  Scenario II: There are exactly two unrestricted prices.  Chapter 3. General Optimality Results and Other Properties  110  Under this scenario, we can write an primary optimal policy derived from the Tight Problem as follows:  { q;  }, 2 qu,k  with the following properties: • i < k 1 <Ic ; 2 • q • qu,k 2 =  (1  —  We now need to discuss two cases. First, consider the case that i < k , that is, P <Fk,. 1 Using the same the argument as for the first case under Scenario I, we know that even making the restricted price P 2 and the two unrestricted prices the same time, the firm still can reach the projected sales of qr,ki  unrestricted units at Pk 1 and  For the case that i  =  2 qu,k  Pk,  q  and Fk 2 available at  restricted units at F , 1  unrestricted units at Fk . 2  , additional care is needed if the firm makes all offered prices 1 k  available at the same time because of the situation that the firm now offers restricted units and unrestricted units at the same price level Fk . The dilemma is that if only 1 one of the potential Dr,ki consumers from the restricted market actually purchases an unrestricted unit at Pk , the firm will not be able to sell Dr,ki restricted units. In fact, 1 under the Behavioral Assumption, if  qu,ki  unrestricted units and  available at the same price level Fk, then the allocated  qu,ki  qr,k,  restricted units are  unrestricted units will be  sold first. Among these who have purchased the unrestricted product at price level Pk , 1 there is a positive fraction of them who are from the restricted market. This implies that 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 decides  Chapter 3.  General Optimality Results and Other Properties  111  to make all allocated units available at the same time. Therefore, we have to construct a new optimal policy. For this, consider the following policy: 6 • the firm first sells  q  unrestricted units at price level Fk ; 1  (1 +  =  • the firm then offers q 1  (1  =  —  • the firm finally allocates q 2  1 )Dr,k  2 = qu,k  (1  restricted units at price level Pk ; 1 —  2 ‘—)D,k  unrestricted units at price  level Fk . 2 To conclude our argument, we have to finish three steps: (1) this new policy is in fact feasible, (2) it is equivalent to the original primary optimal policy, and (3) if all offered prices in this new optimal policy are available at the same time, the firm still can reach its projected total revenue. Regarding the feasibility of this new policy, we observe that: • the allocation q 1 is feasible at Pk 1 since =  1 + Dr,k  =  qu,ki  , 1 + Dr,ki = Dk  1 + Dr,ki) < (D,k  which further implies that 1 q,k  —  qu,ki  —  • after the sales of the q 1 unrestricted units at the price Pk,, the residual demand for the restricted product is given by: =  (1  which implies that q 11  1)Dr,j = (1  —  =  1 Dk (l  —  —  qu,ki  1 D,k  )Dr,j  for  j  1 is indeed feasible; )Dr,ki restricted units at P 1  This construction of a non-primary optimal policy is motivated from the discussions in Example 6 3.3.1.  Chapter 3.  General Optimality Results and Other Properties  • after the sales of the  1 q  112  unrestricted units at the price level Fk , the residual 1  demand for the unrestricted product from the unrestricted market is specified by: —  (1  =  —  which indicates that q 2  1 Dk =  (1  (1  —  qu,ki  for  1 D,k  j  , 1 k  is also feasible.  —  So I have shown that the new policy is feasible. To see that it is also equivalent to the original primary policy optimal, note that 1 Pk  1 + (q,,ç  + Pk 2 ‘lu,k 2  =  ) + 1 1 (qu,ki + Dr,k Fk  =  (q,k, 1 Pk  2 qu,k Fk 2  + qr,ki) + 2 q,k Fk .  Now consider that for this new optimal policy, the firm makes q 1 unrestricted units at , 1 Fk  , and q,k 1 2 all available at the 1 restricted units at Fk qr,k 2 unrestricted units at Fk  same time. Then we have the following observations: • under the Behavioral Assumption, q 1 unrestricted units at the price level Fk 1 will be sold before the sales of  restricted units, which implies that at the beginning  of the sales, only the unrestricted Fk 1 is actually active; • after the sales of the unrestricted units at Fk , the restricted Fk 1 1 and the unrestricted 2 Pk  become active;  • since Pk , then by the same argument as used in the first case under Scenario 2 1 <Fk I, we know that the firm can reach the projected sales levels in both residual markets, that is, selling qk 2 unrestricted units. 1 restricted units and q,k In conclusion, we claim that all allocated quantities in the new optimal policy can be made available without any adverse impact to total projected revenue.  Chapter 3. General Optirnality Results and Other Properties  113  As a last comment of this section, it is interesting to note that in order to be able to maintain the projected value of the optimal revenue and to be able to make all allocated quantities available at the same time, the firm might have to implement a non-primary optimal policy.  3.5  Summary  —  General Optimality Results  This chapter addresses two unsettled issues in Chapter 2. The first issue is the limitation of only using the type of pricing policy that sell restricted product at lower first at some lower prices and later unrestricted product at some higher prices, namely, the primary policies. The other issue is whether all offered prices, restricted and unrestricted, can be made available simultaneously and still generate the same amount of revenue realized by a primary optimal policy. First of all, this chapter has completely resolved the first issue by showing that even allowing the firm to attach restrictions at any possible price and at any order, an optimal primary policy remains to be optimal in this general class of pricing policies. The main implication of this generality result is that if a firm decides to use a marketing restriction in its pricing decision, then the firm can limit its pricing practice to primary policies, that is, selling restricted units first at a set of prices that are lower than these prices for unrestricted units. This chapter also gives a satisfactory answer to the second issue by demonstrating that there always exists an optimal policy, which may be non-primary, with the property that making all allocated units  both restricted and unrestricted  —  available at the  same time will not have any negative impact on the optimal revenue value derived from a primary optimal policy. This is important because it shows that some optimal policies are consistent with the existing pricing practices by firms like airlines, which (1) offer multiple prices  —  restricted and unrestricted, and (2) make all allocated quantities at  Chapter 3. General Optimality Results and Other Properties  different price levels available the same time.  114  Such a consistency makes the pricing  model developed in the last chapter and this chapter much more useful to firms which use artificial restrictions in the process of pricing perishable inventories such as airline seats and hotel rooms.  Chapter 4  Pricing Models with Two Types of Restrictions  4.1  Introduction and Model Setting  Chapter 2 developed a simple pricing model for a monopolist who uses one restriction as a mean of segmenting the market demand. It showed that by properly setting the level of the highest restricted price and rationing the sales at lower prices, the monopolist needs to charge no more than three prices to maximize its revenue. Then Chapter 3 further analyzed the problem by considering a general class of pricing policies which allow firms to offer restricted units at any price level. It showed that the optimal policies characterized in Chapter 2 remain to be optimal in this general class of policies. Also in Chapter 3, I have addressed the issue of simultaneous availability problem for all prices that 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 these consumers who cannot accommodate the restriction is strictly increasing, the optimal pricing practice is to sell restricted units at some lower prices and unrestricted units at some subsequent higher prices, and the firm only needs at most three prices to maximize its revenue. On the other hand, the discussions in the previous chapters are limited to the use of one type of restrictions. As a consequence of this, the firm can only use two types of product  restricted product and unrestricted product. But any realistic situation  involves multiple restrictions and multiple types of product. Firms, such as airlines and  115  Chapter 4. Pricing Models with Two Types of Restrictions  116  hotels, must deal with several product restrictions together with several other corporate restrictions.’ It is important to note that each restriction is targeted to a unique market segment; and firms should be able to explore these market segments by dealing with all these restrictions at the same time. In this chapter, I will develop pricing models for perishable 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 different types of products. This will substantially increase the flexibility of pricing practice for perishable inventories, such as airline seats and hotel rooms. Consider that a monopoly firm has a limited quantity (k) of a certain perishable product, such as airline seats or hotel rooms. Let D(p) be the market demand function for the product at price p. We can interpret D(p) as the number of consumers in the market who are willing to buy one unit of the product at price p. The firm also considers to use two types of restrictions in its pricing decisions. Let 7t(p) be the percentage of consumers who can not purchase the product if type t restriction is attached to the product at price p, where t = 1,2. Therefore the demand for type t restricted product at price p is given by Dt(p) = (1  —  yt(p))D(p), t = 1,2.  As in Chapters 2 and 3, Twill assume that the demand functions D(p), are all step functions defined on the price set {pi,  D(p)=  .  .  ,  (p) 1 DT  (p) 2 and Dr  p,,,}, that is,  13,  ifpp,,  D  if p_ <ppfori=2,•••,ri,  0  ifp>p;  example, airlines offer special fares to major corporations and government, which are not avail able to the general public. The existence of these fares can be analyzed by treating them as corporate restrictions.  Chapter 4. Pricing Models with Two Types of Restrictions  and fort  =  1,2,  rrtI \ 1 j’j=  i  117  1,... ,n, and t  =  t D  ifppi,  D  if  0  ifp>p,  p_ <ppforz=2,-,n,  1,2, denote 7t,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 BL model to the case that there are two types of nested restrictions, which is a quite common 2 I show in this section that there exists optimal policies that can be practice for airlines. characterized by a linear programming problem, which indicates that the firm needs to offer at most four price levels to maximize its revenue by using three different types of product. Section 4.3 will extend the basic BL-model to the case of two type of restrictions that are mutually exclusive. I also show that in this case the firm needs to offer no more than four different price levels with three different types of product. And then in Section 4.4 I will use the results developed in Sections 4.2 and 4.3 to study the pricing problem when a firm uses two general types of restrictions. I demonstrate that under certain reasonable conditions, the firm can limit itself to the type of pricing policies that consist of at most five different price levels with four different types of product. Finally, the last section is a summary. For example, airlines offer discount fares with advance booking and Saturday night conditions. In 2 the meantime, the discount fares just with Saturday night condition are also available.  118  Chapter 4. Pricing Models with Two Types of Restrictions  4.2  Pricing Problem by Using Two Nested Restrictions  I first introduce the following definition:  Definition 4.2.1: We call type-s restriction is nested into type-i restriction if there exist aj(p), t 1. 1 2, 1  —  —  1,2 such that  =  yi(p)  72(p)  = 1 = (1  —  (p); and 1 c  —  cri(p))(i  —  (p)). 2 a  From the definition, it follows that if type-2 restriction is nested into type-i restriction, then I will have (p) = (1 2 DT  —  (4.1)  (p))D”(p), 2 a  which, in fact, is the main reason why I call them nested since it implies that the group of those who can accommodate type-2 restriction is a subset of of those who can accom modate type-i restriction. Hence, it is clear that type-2 restriction is more restrictive than 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 many units of the product available at each of these price are offered. Throughout this section, I will make the following assumptions: • both  {ai}  } are strictly increasing; 22 and {a  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 restric tions is quite natural:  Definition 4.2.2: Suppose that restriction 2 is nested into the restriction 1. Then a pricing policy for the firm is in a form of  { (p, qr2,i)  : i  =  1,.  .  ,  m; (ps, qri,j) :  j  =  ,•• 2 m  where Pmt is the highest type-t restricted price,  qrt,i  ,  ; (pk, q) : k 1 m  =  • m , 1  ,  is the quantity available for sale at  the type-t restricted price pj, q, is the quantity available for sale at the unrestricted price Pk,  in. andi<m m 2  Remarks: From the definition, I have the following observations:  2 • If m  =  1 and  qr2,1 =  0, then the above policy indicates that only type-2 restrictions  will be used; 2 • If m  =  1 m  =  rn and qri,m = 0, then it indicates that only type-i restriction will  be used; • If m 2  =  1 m  =  1 and qr2,j  = qri,i =  that is, no restriction is used.  0, then it means that all units are unrestricted,  120  Chapter 4. Pricing Models with Two Types of Restrictions  1 and m Also, since the values of m 2 are completely controlled by the firm, it is of the firm’s interest to find the best possible combination of these two values. But first, I need , mi). For this, I 2 to specify the set of all feasible pricing policies for any given pair (m will 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),.  2 for i with demands given by D  .  ,q,m ,(pm ) 2 }  1,.. ,m. Therefore, the feasibility condition for the  =  type-t restricted product is given by  After the sales of the type-2 restricted product, the residual demand for type-i restricted product is given by D’  =  —  , for D 2 j3  j  =  ,••• , n. 2 m  Consequently, the feasibility condition for the plan {(Pm , qi, 2 ), 2  , (pm , q, 1 )} 1  for the  type-i restricted product is 1 m  1 m  ,-1 q 3  /3_V  P1 =  • 3_rn 2  U  —  L..  rl q 3 flrl  2 3_rn  2  flr2  0  —  ‘  Finally, let us consider selling unrestricted product according to the plan ,. 1 m  .  ,  {(pk, q)  : k =  n}. It is easy to see that the residual demand for the unrestricted product after  the sales of the product with restrictions is given by ,k = Q1kDk + (1 1 drn  —  ,k = 2 /31)d  Dk  —  D’ 1 j3  —  (1  —  )/3 for k 1 /3 , D 2  . 1 m  Hence, the feasibility condition for the remaining unrestricted product is given by: n  fl  qk  k=mj  qk  —  drni,k —  1 Dk kzrn  —  —  (1  —  )3 1 3 D 2  <1 —  121  Chapter 4. Pricing Models with Two Types of Restrictions  Summarizing the above discussions, I obtain the following formulation of the firm’s pricing problem when using two nested restrictions: Max  2 m  V”fl  Pii +  +  ji  L.ak=mi Pkqk  s.t. sçrn2  (4.2)  < 1  L=i  —  qr1,  (4.3)  < 1  2 j=m  —  (4.4)  1 qr2,i  +  +  2 qri,j )j=m  ‘cn 1 q /_..jk=m  q  O,Vi,Vj, and Vk.  qr2,i,qrl,j,qk  I will call the above pricing model N-model, where N represents for nested.  Let  , mi) be the derived maximum revenue from the above N-model. The following the 2 R(m orem proves that there is a linear programming characterization for the optimal revenue: max  ,mi). 2 R(m  2 mi 1m  ,mi) be the optimal objective value of the following linear 2 Theorem 4.2.1: Let R(m programming problem (named as the Tight N-model): +  Max  p,qri,,  +  Z=mj pkqk  s.t. 2 ç-m  qr2,i —  L.di=i  .çmi  qrl,j  2 D’—D L..ej=m 2 gk  L.dkmi .Dk—D’  2.—i  qr2,i  +  2 qri,j YZ,=m  qr2,i, qri,j, qk  +  1 =  1  =1 YZk=mi q  O,Vi,Vj, and Vk.  q  122  Chapter 4. Pricing Models with Two Types of Restrictions  Then, max  ,mi) 2 R(m  max  =  <m 2 1m < 1 n  1<m2<mln  (4.5)  ,mi). 2 R(m  Proof: It is evident that max  n 1 m 2 1<m  , mi) 2 R(m  max  , mj). 2 k(m  max  ,mi). 2 .i(m  n 1 m 2 1m  Then to prove (4.5), it suffices to show that max  ,mi) 2 R(m  mj<n 2 1<m  <  <n 1 m 2 1m  To prove (4.6), I oniy need to show that for any pair: 1  n, there exists  Consider any pair 1  n and let  2 m  1,•••  (4.7)  . n 1 , 2 (th )  ,mj) 2 R(m  =  1 m  1 <n such that th  2 another pair: 1 <th  {  2 m  (4.6)  ,m2;(pj,qrl,j),j =  ,•• ,m 2 m ; (pk,qk),k 1  =  ,...,n} 1 m  be an optimal solution to the N-model. Denote 2 m  q2  which is the total sales for type-2 restricted units. Note that the residual demand for the type-i restricted product after the sales of the type-2 restricted product is given by: =  —  , for D 2 3  j  =  ,•• 2 m  ,  and the total residual demand for unrestricted product at price p, after the sales of the type-2 restricted product is given by: ,j 2 dm  =  D,  , for D 2 3  j  =  • m , 2  ,  123  Chapter 4. Pricing Models with Two Types of Restrictions  Therefore the residual percentage of consumers who can not accommodate the type-i restriction is given by: r’r1  jrl —  r’r2  i2  -‘-‘j  j Um , 2  —  —  3 P2’—’  1j  —  —  j = dm , 2  —  D  —  D, = 1 2 /3  —  132(1  —  i)(i  —  ) 2 a  2 are strictly increasing by assumption. which is strictly increasing since both c and cr Therefore, I can use Theorem 2.4.2 to the residual market for the type-i restricted product and the unrestricted product. This implies that considering q 2 I know that there exists an integer m  rñi <  —  q as the capacity limit,  n and a new optimal allocation plan in  the form of {(pj,qri,j),j =m2,.••,thl;(pk,ijk),k=thl,.•.,n},  such that 3 q, —  1 krn  .2  =  D,.  1  48  1  (4.9)  —  k  n qij  q  +  —  (4.10)  q; and  2 j=m n Pjqri,j +  n  +  pq 2 j=m  j=m2  (4.11)  pkqk.  1 k=m  We should notice that the relationship (4.9) is independent of the value of  /32.  In fact, as  long as (4.8) holds, that is, the feasibility constraint for the type-i restricted product is tight, (4.9) is automatically true! Now let  i 4  q. Therefore o is the total number of units of the product  allocated for unrestricted units. Then with o units protected, the firm needs to solve the following subproblem:  124  Chapter 4. Pricing Models with Two Types of Restrictions  +  Max  piqi,i  s.t. 2 m P2  q,.,j  —  —  qri,j  /_13m2  1  <  D 2 D—/3  1  < —  2 m  1 qr2,i + YZ j=m D 2 qri,j  —  q  —  qo  O,Vi, andVj.  qr2,i,qrl,j  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 that there exists another integer 1  th and another allocation plan  2 rn  {(pi,qr2,i),i =  1,  ,,th 2 th } 1 ,  ,2;(pj,r1,j),j =  such that 2 m  —  =  1,  (4.12)  =  1,  (4.13)  1 m 3  2 rn  3  1 rn  n  + i1  +  q;  qk  and  (4.14)  1 k=  3=7712  2 rn Pjq,-2, i=1  since it is easy to check that  +  )  Pjri,j  piqr2,i  2 =rn {qr2,i,  i  +  (4.15)  Pj’,-i,j,  i1 =  1,  ,  ; 2 m  =  ,• , ñ} is also a feasible 2 m  policy for the above subproblem. Finally, by (4.9), (4.12), (4.13) and (4.14), I know that the new allocation plan {(pi,r2,i)  : i  =  ,ql,) :j 3 l,.,rn2;(p  =  ñ2,  ,rhl;(pk,qk)  k  =  125  Chapter 4. Pricing Models with Two Types of Restrictions  constitutes a feasible policy for the Tight N-model with respect to  2  and  iii.  Further  more, by (4.11) and (4.15), it follows that 2 rn ,ñii) 2 (ñi  1 rn  fl  ir2,i+ >Pjqr1,j+ pkqk  2 =rn 2 m >ptq,.2, 2 m  n  ‘i  +  Pj’lri,j  +  3=m 2 1 m  + pjqri,j+ 2 pq2, i=1  pkqk=R(m2,m1). k=mi  3=m2  So (4.7) is true; and therefore (4.6) is proved.  pkqk k=ñ 1 n  C  As an immediate consequence of the above theorem, I have the following useful result on the optimal pricing policy by using two nested restrictions.  Corollary 4.2.2: If two type of restrictions are nested, then there exists an optimal pricing policy that consists of at most four different prices, which is characterized by a linear programming.  As a final remark of this section, the above discussion can be easily extended to the case of multiple nested restrictions.  4.3  Pricing Problem by Using Two Mutually Exclusive Restrictions  In 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 the product if both restrictions are attached to the product.  Then we say that these two  126  Chapter 4. Pricing Models with Two Types of Restrictions  restrictions are mutually exclusive if 7(p) = 1.  The mutual exclusiveness here means that those who can accommodate the type-i restriction cannot accommodate the type-2 restriction, and vice versa. Or intuitively speaking, if two types of restrictions are mutually exclusive, then they are targeting two distinct consumer groups. In other words, the set of those consumers who do not mind the type-i restriction is disjoint from the set of those consumers who are do not mind the type-2 restriction. This sounds very restrictive, but it is nevertheless an important case I need to discuss. In the next section, I will discuss the general case of two types of restrictions that are not mutually exclusive and not nested, by integrating the model developed in the last section and in this section. If the two restrictions are mutually exclusive and the units with restrictions are sold first, then the following lemma shows that the firm can limit itself to policies that are such that both restrictions share the same highest price.  Lemma 4.3.1: If two restrictions are mutually exclusive, then there exists an optimal policy of the following form: {m; (qrt,i,•  where Pm  ZS  ,  qrt,m), t = 1,2; (q,  ...  ,  qn)},  the common highest restricted price for both restrictions, q is the quantity  available at price p with restriction t and q, is the quantity available at the unrestricted price  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 that  for each restriction alone, the firm only needs to consider policies that sell restricted units first. Let  } be the 2 ,pmi} be the set of the type-i restricted prices and {p,... ,pm  1 < n-i . 2 set of the type-2 restricted prices. Without loss of generality, I assume that m  Chapter 4. Pricing Models with Two Types of Restrictions  127  So this leads to policies of the following form: : i  {  =  :j  1,...,m1;(p,qT2j,q)  = ml,...,m2;(pk,qk) :  k  =  ,...,n}, 2 m  where • for 1 < i < m , the firm allocates qri,j and 1  q,.,j  units at price pj with type-i  restriction and type-2 restriction respectively; • for m 1  <j 2 <m and at the price level p,, the firm sells qr2,j type-2 restricted units  first and the q 3 unrestricted units; and • for m 2  k  n, the firm sells unrestricted units only.  After the sales according to {(pj, q,.-,j, qr2,i) : i • d 1 I  =  =  1,••• , mi} up to price level Pmi, let  be the residual demand for type-I restricted units at price p 3 for  j  m and  1,2;  • dmi ,j be the total residual demand at the unrestricted price level p 3 for  j  ; 1 m  and • 7m 1 ,tj be the percentage of consumers in the residual market who can not accom modate type-I restriction at price Pj. Then it is straightforward to check that for I 1 d  =  =  1,2,  (1 —ii)D, with  So dmj,j  —  —  —  Note that —  1  —  ,2j 1 7m —  dmi,j  —  (1  —  —  is strictly decreasing if and only if  —  —  12 i 2 D  —  1 —(1  —  (1 )(1  —  —  772)(1  —  —(1  —  2)(1  —  72j)  Chapter 4. Pricing Models with Two Types of Restrictions  (1—71,+711  is strictly increasing,  which follows from the assumption that both 7mj,2j  128  is strictly increasing in  j;  713  and  723  are strictly increasing. Therefore  and similarly, 7mj,ij is also strictly increasing in  j.  Now, our pricing policy indicates that after the price level Pmi, the firm will stop selling type-i restricted units. In other words, the firm only sells type-2 restricted and unrestricted units in the residual market. But as illustrated above, for any given values of  771  and  772,  7m2  ,  is strictly increasing in  j.  Therefore in the residual market only with  type-2 restriction, we can use Theorem 3.3.5, which shows that there is another allocation plan of the following form: ,q 3 {(p ) 2 ,  :  j  —  ,••.,m’; 1 m  (p,,q)  : k  =  m’, •.,p},  which will perform as least as good as the original policy:  { (P3, q2,i, qj) Therefore, we can treat  Pm’  :j  ,• 1 m  ,  ; (pk, 2 m  q) : k  =  • m , 2  ,  n}.  3 This as the common highest price level for both restrictions.  proves the lemma. LI  With the help of Lemma 4.3.1, I know that the firm’s pricing problem can be formu lated as the following mathematical programming:  The main purpose of using the common price levels for both restrictions is to simplify the formulation 3 that leads to simple optimal pricing policies. This is purely for technical reasons. On the other hand, it is possible though that the two restrictions may not have a common last active price level.  Chapter 4. Pricing Models with Two Types of Restrictions  Max  p(qri,  +  qr2,i)  +  129  3 q 2 ZD=m p  s.t. m ‘  L_.ii=1 D’’ ç-m  qr2,i  /32 = L..jjl 2  i  (4.16)  <1  (4.17)  < —  —  Ej=m 2 — D 1 D ”—3 f3  Ei(qri,i  +  qr2,i)  +  (4.18)  1 q  3 E,=m q  O,Vi and Vj.  qrl,i,qr2,i,qj  I will call this formulation ME-Model, where “ME” stands for mutually exclusive. Again let 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 charac terization 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 pro gramming (named the Tight ME-model): Max  +  Zi  qr2,i)  +  YZj=m  pjqj  s.t. q-  m i=1 m  — —  qr2,i —  1 1  qj /_.j=m D—D’-D 21  i(qr1,i +  qr2,i)  +  Zrm q3  O,Vi and Vj.  q  Chapter 4. Pricing Models with Two Types of Restrictions  130  Then max R(m)  max R(m).  =  (4.19)  1<m<n  1<m<n  Proof: To prove (4.19), it suffices to show that for any m, there exists another th such  that R(m) <(th). Let  {(qrt,i,  ,  qt,,,j, t  =  1,2; (q,”  .  ,  (4.20)  q)} be an optimal solution to the ME-model with  respect 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 q , which, according to Theorem 2.3.5, must satisfy 3 q; Now take th  =  j  =  3 D  —  —  . D 2 /3  (4.21)  and define a new pricing policy as follows: for t  rt,i  =  q,j  ifl<i<m,  0  ifm+1i7—1,  (1  —  /3t)Dt;  if i  1,2,  =  and —  —  if j  2 D  =  3,  qj =  1  otherwise.  0  It is straightforward to check that the policy  {j,  i  1  a feasible solution to the Tight ME-model associated with  th, t ñi.  have n  J?(nz)  >pi(’r1,i i=L  +  r2,i)  +  =  =  1,2;  j,  ñì.  <j  < n} is  Furthermore, by (4.21) I  Chapter 4. Pricing Models with Two Types of Restrictions  131  therefore (4.20) is true. Hence the theorem is proved as required. 0  As an immediate consequence of the above theorem, I have  Corollary 4.3.3: If two types of restrictions are mutually exclusive, then there exists an optimal pricing policy that consists of at most four different prices, which is character ized by a series of linear programming problems.  Finally, it is straightforward to extend the above model to the case of multiple mutually exclusive restrictions.  4.4  Pricing Problem by Using Two General Restrictions  Recall that 7(p) is the percentage of consumers who cannot accommodate the type-f restriction, and y(p) is the percentage of consumers who cannot purchase the product if both restrictions are attached to the product. Then we should notice the followings: (p) or -y(p)  then it leads to the case of two nested restrictions;  • If y(p)  =  7i  • If 7(p)  =  1, then it becomes the case of two mutually exclusive restrictions.  =  72(P)),  On the other hand, it is clear that 7(p)  7t(p) for t  =  1, 2. Therefore, it is always true  that max(-yl(p),-y2(p))  (p)  1.  In the previous two sections I have discussed two extreme cases of two types of specific restrictions. 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  132  This implies that there are some consumers who can purchase the product even if both restrictions are attached. In this section, I will oniy consider the following type of pricing policies: • 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-S restriction; p) = (1 Dr ( • let 3  —  p) for i Dr ( -y(p))D(p) and denote D 3 = 3  i,... , n; and  • define 3 D’ 1—cj=—-,fori=i,•..,nandt——i,2, which measures the percentage of consumers in the market demand for type-t re stricted 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  —  ) = 1 2 c  —  7(p) = 1  —  7,  (4.23)  Chapter 4. Pricing Models with Two Types of Restrictions  133  which 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, where 1  : i  1,... , 1 m be the allocation plan for the type-3 restricted units, }  n. Denote d,  1 m  price P, for all  =  j  ,  as the residual demand for type-3 restricted product at  m after the sales according to this allocation. Then I know that 1 m  = (1  for all  j  . 1 m  —  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 restricted  product, each consumer can at most accommodate one type of restriction. Or equivalently speaking, after the sales of type-3 restricted units, all these who can accommodate both restrictions are satisfied. Therefore, type-i restriction and type-2 restriction are mutually exclusive in the residual demand market. By the discussions in the previous section, I can specify the remaining part of a pricing policy as follows:  { (pi, qi,i, qr2,j)  :  =  .  ,  m;  (pk, q)  : k  =  m,  .  ,  1 <m < n. where 1 <m In order to use Theorem 4.3.2 in Section 4.3, I need to check the monotonicity condi tions on the impact of type-i and type-2 restrictions on the residual market. Let  ,tj 1 7m  be the percentage of consumers in the residual demand market who can purchase type-t restricted product at price p 3 /33  =  (j  mi) after the sales of type-3 restricted units. Since  1, it follows that by (4.23) flrt 1 ,tj 1 17m  —  —  rr3 3 11 \ 3) 7 flr3” —  U, —  7i  7i  Yti  —  -  1  7tj Yj  Chapter 4. Pricing Models with Two Types of Restrictions  Therefore, for t  =  1, 2,  7m , 1 tj  is strictly increasing in  j  134  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 consumers who cannot buy type-t restricted product to the percentage of consumers who can not buy type-3 restricted product is strictly increasing. A sufficient condition is that yj  —  j,  yj is strictly decreasing in  or, equivalently,  DTl ‘  3 Dr ‘  is strictly decreasing in  j,  which implies that the percentage  of consumers who can accommodate type-t restricted product, but not type-3 restricted product, is strictly decreasing. This sufficient condition is stronger than required; but it has 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  Further assume that fort  =  =  1.  1,2,  is strictly increasing in j, then the firm’s pricing problem can be formulated as the following linear programming (I will call it Tight-G Model, where “C” stands for “general”): Max  +  Lmj  (qri,j 3 p  +  qr2,j)  s.t. ‘ç-mi  qr3,i  L.=i D 3  —  2 qi,  Ldj=ml  ” 3 D’D  —  —  qr2,j  1 D Lj=m -D 2  —  —  1 1  +  Z=m  Pkqk  Chapter 4. Pricing Models with Two Types of Restrictions  gk  i_..k=m 3 )—(D Dk—(D’—D — 2 ) D  ‘1r3,i +  135  —1 —  7Th  Zj=m (qri,j + qr2,j) + Zk=m qi qrt,i  0 and  Jk  q  0.  Proof: It follows from the above discussion that if ti/t is strictly increasing, then 7m ,tj 1 is strictly increasing for t  =  1,2. On the other hand, since /33  =  1, I know that in the  residual demand market, type-i and type-2 restrictions are mutually exclusive. Hence, the rest of the proof follows immediately from Theorem 4.3.2. D Therefore, if /33  1, then the firm needs to offer at most five prices to maximize its  revenue. The Tight-G model provides a benchmark on the optimal revenue value for any other general models since it at least provides a lower bound for the optimal revenue value. In Chapter 2 and Chapter 3, I have demonstrated that for the case of one type of restriction, an optimal solution for a tight model remains to be optimal in the general class of pricing policies. It is not clear at this moment whether or not Tight-G model in fact provides an optimal solution in the general context. The main issue is whether we can find an optimal solution such that /33  =  1. This is  not an easy task since if /3 < 1, then it is not even clear how to specify the remaining part of pricing policy because • /33 <  1 implies that if both type-i and type-2 restricted units are offered at the  same price level, some consumers can buy either one of them; • On the other hand, the impact on the residual demand market of selling type-i restricted units first and type-2 restricted units thereafter at the same price level is different from the impact on residual demand market of selling type-2 restricted units first and type-i restricted units later at the same price, which will complicate the process of updating the residual market.  136  Chapter 4. Pricing Models with Two Types of Restrictions  So the key question is: which type of restricted units will be offered first if both restricted units are offered at the same price? One way to get around this difficulty is to consider policies of the form: {(pj,qi,,qr2,j,qi,j)  where 1  m  1 m  :j  —  mi,.”,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 re stricted units and finally some type-i restricted units, all the same price level; • if at price p 3 the firm wants to sell type-2 restricted units first, it can do so by letting q’ 1  =  0; and  • it is possible possible for the firm to sell type-i restricted units first on a price set and 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 general specific restrictions:  Definition 4.4.1: For two general restrictions, a pricing policy is specified by  {(pi,qr3,i)  :i  =  ) j,qr2,j,q 1 l,...,m1;(pj,q’ j  :j  =  ml,...,m;(pk,qk) : k  =  m,.”,n},  1 <m <n. where 1 <m  Let us now analyze a special case that leads to Tight-G model.  Theorem 4.4.2: Assume that 7ij/7j and the following two conditions holds: (i)  is strictly increasing in  j,  72j/7j  are strictly increasing in  j.  If one of  137  Chapter 4. Pricing Models with Two Types of Restrictions  (ii)  ‘“ 7jY2j  is strictly increasing in  j,  there 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 optimal policy 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 the pricing problem into a mathematically programming formulation. The main technique here is to show that I need only to consider a small class of policies which contains an optimal policy. I will use the main results in Chapters 2 and 3. And the second step is to further simplify the model formulation so that I can have a linear programming formulation, which will lead to a simple optimal pricing structure. Technically speaking, I can start with policies specified by Definition 4.4.1.  But  because with the additional assumption in the theorem, I can actually focus on a smaller class of policies that makes the model formulation much simpler and easier. First of all, 4 Let the it is clear that any policy will start with the sales of type-3 restricted units. part of allocation for type-3 restricted units in an optimal policy be given by: :  { where m 1 is an integer such that 1  1 m  ,mi},  =  n. The key is to analyze the demand structure  in the residual market after the sales of type-3 restricted units. As above, let /)3 2=1  2  I will use the following notation in the residual market: let 1t 4  j  >  1 m  is possible though that the firm may not sell any type-3 restricted units.  138  Chapter 4. Pricing Models with Two Types of Restrictions  is the residual demand for type-t restricted units at price p 2 for t  •  =  1,2,3,  which are given by: 3 , 1 d • d 1  (1  —  )D and d, ,6 ; 3 3  =  Dt  —  D,r for t /3 , 3  =  1,2.  is the total residual demand for restricted product at price Pj (including the  demand type-i restricted product and the demand for type-2 restricted product), which is given by: mj,j  —  =  ,p2  ,lrl mi,j  1 (D  —  —  mj,j  —  mi,j  ) + d, 3 D  —  =  2 (flr  j  ‘S  1 n 3 D  —  flr 3 j )  2 +D  —  ,7r1 ,j 1 m  ‘  (1 + 3 )D /3 .  • dm, ,j is the total residual demand for unrestricted product at price Pj, which is given by: 3 dmi,j = (D  D+ ) 2+3 D  D’  =  —  . D 3  3 d  • Let l7mjj= -. 2 dm,j dlt  • Let 1—7m ,tj=-.fort=1,2. 1 • Let 1  —  ,tj = 1 m  1 m  Now if /33 = 1, then the rest of the proof follows from Theorem 4.3.2. So I only need to focus on the optimal policies such that /33 < 1. Let  7m,j 7  be the percentage of  consumers in the residual market for restricted product who cannot accommodate type-i restriction. Then, flrl  3r1  Q nr3  flrl  P3L)  Umi,j —  mi,1j  1 d  =  1 + DJ DJ 2  —  Q  nr3  /J3LI —  —  (1 + 3 )D /3  =  (Di’  —  ) + (D D 3 /3 2  —  Similarly, let ?)mi,j be the percentage of consumer in the residual demand market for unrestricted product who cannot accommodate any restrictions, then —  1mi,j  —  —  ,j 1 Urn  2 D’ + D  —  —  JJS, —  (1 + 3 )D /3 flr3  Claim 1:  139  Pricing Models with Two Types of Restrictions  Chapter 4.  Both 7 lmi,ij and  7mi,j  Proof of Claim 1: Clearly, 7mi 2 D’ D 3  ,  are strictly increasing in j.  is strictly increasing if and only if  is strictly decreasing.  On the other hand, note that flri  3 U  0  flr3  3 /J3UJ  —  —  (1  3 D  —  + -D 2 D 3  -  —  —  2 Dç  jj,r 2 j,r 3  3 D  +  3 D  73 7j  —  3 D” (1—/33)2  3 D D’’ ———--iD D D,  —  2 D  —  “+(i—33) —  72j  j 2 1—cy 2j  which is strictly decreasing since  7j72j  1  For  1mi  is strictly decreasing by assumption; and is strictly decreasing because  is strictly increasing.  23 cr  ,j, note that 1 + DJ  (1 + 3 )D /3 —8 D 3  2 DJ  2 D  D’ D,  —  —  3) D 3  D, 3 D’  —  —  (1  —  7i)+(l 723)(1+/33)(1 1—/33(1—73)  —  1—/33(1—73)  —  which is strictly decreasing if and only if (73 71j 1 /33(1 —  —  72j)  —  —  is strictly decreasing. But (-y  —  713  —  (  723)  1—/33(1—73)  —  —  71j  723)  (1+/33)73—/33  -y)  140  Chapter 4. Pricing Models with Two Types of Restrictions  =  —  —  7 (i’ (l+3)-—/33 73 1 / + 1 + (1 + 3)7j  723)  73 -“  —  —  7j  —  7j)  which is indeed strictly decreasing since • -yj is strictly increasing; and • both  --  and  -7,  are strictly increasing.  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-i restricted units, if allocated, should be sold before the sales of type-2 restricted units. Therefore, the firm can limit itself on the following type of policies in the residual market:  j  =  • m , 1  2 < m 1 < m where m  n.  {(p, qri,j) :  ,  ; (pi, qr2,l) : 1 2 m  ,  m; (pk, q) : k  =  m,•  ,  By using this type of policies, I know that the feasibility condition for type-i restricted allocations is  2 m  2 m  = —  qri,j  —  d1  — 3—rn 1  qri,j  < i  —  —  After the sales of type-i restricted units, the residual demand for type-i restricted product is i = dm , 2  (1  —  )(D’ 1 B  —  ,1 i3 ) D 3  ; 2 m  and the residual demand for type-3 restricted units is fl2,1 =  (1  —  ,8)(1  —  . 2 )D 1 > m 3 ,  Chapter 4. Pricing Models with Two Types of Restrictions  141  Consequently, the residual demand for type2 restricted product is given by r3  rr2 ‘‘1  —  2 ,1 m  ‘  r3  )  —  2 m  1  — 2 D + — / 1 ) . 3 D 3 /3 (/3 /3  =  Therefore the feasibility condition for type-2 restricted units is characterized by: qr2,1  k—’  <1.  — 2 D — ) 3 DT (/3i+/3 /3i/3  —  Now after the sales of type-2 restricted units, the residual demand for type-3 restricted product is d,k =  (1  —/32)(1  m; and  —/3)(1 3 )D —/3 , k  the residual demand for type-2 restricted product is jr2 Um,k  11  =  —  (1  =  —  i2 jr2 P2)Um , 2 k  )(D /3 2  (/3k + /33  —  /D 1 /3 ) ) 3 3 for k> rn.  —  Thus the total residual demand for type-l and type-2 restricted units is given by: —  (Lm,k  —  ,k 2 m  —  jr3 Um , 2 k)  i  t  jr2  Um,k  )+(1 2 3 )(D’ —D 1 )(D —i3 = (1 —,B  —  —D (/3i+/3 ) ) 3 /3i/3 ;  and the total residual demand for unrestricted product is: dm,k  =  (Dk  =  Dk  =  Dk  —  —  —  (D’ 1 /3  -  2+D D ) + d,k 3 —  -  ) 3 D  —  D+ 2 /3  —  (/31/32-  (1  —  /32)C81 + /33  (1- /3)(1  -  —  /D 1 /3 ) 3 3  2 /3 ) . 3 /3 D  So the feasibility condition for the final unrestricted units is n qk  k=m  Dk  —  —  D+ 2 /3  (/31/32  —  (1  —  i3)(1  —  2 ,8 ) 3 /3 D  <1 —  Summarizing these discussions, I get the following formulation for the firm’s pricing problem:  142  Chapter 4. Pricing Models with Two Types of Restrictions  +  Max  pjqri,j  j Z m 2  +  2 pjqr2,l Zm  +  Zk=m pkqk  s.t.  /3 =  (4.25)  1  Z2m, D—3 D 3  —  qr2,z  —  P2  (4.24)  1  3  /33  2..1=m2 3 (pi+/3 _thpa)Dr 2 Dr _  —  2 m  n  m  1 m q,-3, + Z j=mi qri,j + Z1=m2 qr2,1 + k=m ‘m ZD = 1 qrt,i  0 for t  =  1, 2, 3, and  (4.27)  1  D )/3 2 )(i—p ) D +U31132—(1—131 3 2 D— 1 Dk —3  q  0.  q  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,) (pl,qr2,z)  : i  =  : 1  =  1,...  ,  ; 1 m  (pj, qri,j)  •• ,m; m , 2 .  (pk,qk)  j  =  . m , 1  : k  =  m,•  :  .  .  ,  m;  ,n}  be an arbitrary optimal solution to the above formulation. If /33 = 1, then I know that our 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 the  pricing problem. On the surface, these properties should also lead to the model of two nested 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 subset of 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-3  restricted product. Most importantly, observe that • type-S restriction is nested into type-2 restriction; • both  amj,2j  and  ,2j 1 7m  are still strictly increasing; and  • during the sales of type-i restricted units, the residual demand for type-S restricted product is proportionally reduced. Therefore, in the residual market in regard of type-3 and type-2 restrictions I can use the results in Theorem 4.2.1, which says that after the sales of type-i restricted units that have included the residual demand for type-3 restricted product, there is no residual demand for type-3 restricted product. This implies that jr3 tLm , 2 1  which leads to  =  —  o  1  i-i  —  —  I_’1)11  —  ‘  -‘r3  3 /J3)L1  ,  = U,  1.  After the sales of type-i restricted units with /3 consists of exactly two types of consumers units and those who can. Let  ,21 2 7m  —  =  1, I know that the residual market  theoe who cannot purchase type-2 restricted  be the percentage of consumers in the residual market  2 restriction after the sales of type-3 and type-i restricted who cannot accommodate typeunits according to the allocation plan. Then since ,jr2 1 1  —  —  ,21 2 7m  flr2  ’rn 1 , 2 I  flr3 —  — —  ,1 2 m  —  j  =  1  —  (1  —  flrl  /31  =  1, it follows that  144  Chapter 4. Pricing Models with Two Types of Restrictions  which is strictly decreasing since Therefore  m2,2l  2!2-  is assumed to be strictly increasing for I  1,2.  =  it  is strictly increasing. Then by Theorem 3.3.5 and Theorem 2.4.2  follows that the firm only needs to consider policies such that  /32  1. In summary, I  have shown that the firm can focus on the class of policies such that i3  =  132  =  1. This  proves Claim 2. Using the results in Claim 2, I know that the pricing problem can be reduced to the following form:  Max  +  qri, + Zm P 2 Pjqr2,1 + YZk=m pkqk >T= m 2 13  s.t. —  P3 =  1 -‘m  q,j  L.=i  iZT  —  q,-i,  1 /_dj=m  2  1  —  1  4 29  —1  431  —  —  2 L.j=m  -D” 2 D  —  \fl  /_ak=m 1 m  Z—  +  Z  qrt,  D 3 + 2 Dk-D’-D  m2 1 j=m  qri,j  0 fort  +  =  —  n  m  2 qr2,1 + k=m YZ1=m  1,2,3, and q,  q  0.  Note that constraints (4.30) and (4.31) are no longer related to /3, the only parameter that 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 according to the plan: {(pj,qr3) : i  =  ,mi;(pj,qri,j) :  j  =  ; mi,...,m } 2  • o, the percentage of consumers in the market demand for type-i restricted product who can also accommodate type-2 restriction, is strictly increasing; and • The total sales for type-3 and type-i restricted units is bounded by n  m qr1 = q  qr2,l  —  2 1=m  qk,  —  k=m  Chapter 4. Pricing Models with Two Types of Restrictions  145  which can be considered as the capacity for type-3 and type-i restricted units since it 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 problem addressed in Chapter 2. Then using Theorem 2.4.2 and Theorem 3.3.5, the firm again only needs to focus on policies having the property of /33  =  1. Therefore I prove the  theorem, since the rest follows from Theorem 4.4.1. D  Note that the above theorem is aimed at the case that it is advantageous to delay the sales for one type of restricted units. There may be other cases that also lead to Tight-G formulation.  4.5  Summary  —  Pricing by Using Two Types of Restrictions  This chapter addresses the issue of pricing perishable inventories by using two types of restrictions. After studying two extreme cases, which are two nested restrictions and two mutually exclusive restrictions, I have also discussed the general case. I here provide the following 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 handling four different types of prices. • Extra demand structures are needed in order to obtain simplified optimal pricing structures. • Simplified optimal pricing structures can be characterized in a similar manner as in the basic BL-model.  Chapter 4. Pricing Models with Two Types of Restrictions  146  I have shown that for the two extreme cases, there exist optimal pricing structures that consist of at most four prices. And for the general case, I present two situations where there exist optimal pricing structures that consist of at most five prices. All of these optimal pricing structures are characterized by linear programming problems, which make these models tractable in application.  Chapter 5  Airline Pricing by Using Membership and Product Restrictions  5.1  Introduction  Chapter 2 and Chapter 4 developed a series of pricing models by using artificial restric tions. These models gave us conceptual tools for analyzing the use of restrictions as a mechanism. In this chapter, I will present an application, in a convincing way, in the context of airlines. It is well-known that all airlines have special arrangements with certain clientele-specific rates. It is important to note that the fares for these special consumer groups are not available to the general public through travel agents. In most cases, these rates are internally controlled by the airlines. These opportunities represent revenue potentials which the airline may be able to exploit. It is important to notice that any special clientele represents some kind of membership which covers only a small of portion of the general population. Because of this, from the pricing point of view, these memberships are in fact restrictions because their existence will prevent some consumers from being able to purchase the product at a price that is available to members only. On the other hand, the availability of the seats for these clientele needs to be controlled and a good yield management system must be capable of handling the presence of these special clientele. 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;  147  Chapter 5. Airline Pricing by Using Membership and Product Restrictions  148  • Restricted fares and unrestricted fares are understood as fares that are available to the general public, which means that all consumers are allowed to purchase these fares; and • These fares that are specifically available to members are called either restricted membership fares or unrestricted membership fares. Note that there is clearly an asymmetry between members and non-members on the access to the reservation information. A member can purchase a ticket either at a membership fare or at a public fare, whichever is less; but non-members can only purchase the public fares since the membership fares are not available to them. In this sense, we should treat the 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 airline faces two types of restrictions. The purpose of this chapter is to use the results developed in the previous chapters to illustrate how they can be applied in this context and what the implications are. In particular, I will discuss several different situations where a special membership deal makes sense to the airline. As a result of this analysis, it will improve 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 the airline 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 membership price 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 the first-come-first-serve basis.  Chapter 5. Airline Pricing by Using Membership and Product Restrictions  149  When 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 the membership agreement, we know that: • Restricted membership fares, if offered, should riot be higher than the planned re stricted fares; and • Unrestricted membership fares, if offered, should not be higher than the planned unrestricted fares. In this section I will consider three common arrangements in terms of membership priv ileges: (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 to find out conditions that lead to a proper choice among these arrangements. This is of great importance since the airline needs to know under what kind of operating environ 1 ment, a particular membership deal is worthwhile at the first place. This chapter is organized as follows. Section 5.2 presents the basic model setting and notation. Section 5.3 addresses the airline fare pricing problem when members are promised that they can by restricted tickets for less. I will give two examples that are consistent with this scenario. Section 5.4 discusses the airline fare pricing problem when members are offered cheaper unrestricted fares. In particular, I will examine the impact of corporate policy for interval travellers and conclude that such a corporate commitment can cost an airline a lot of money. Then Section 5.5 deals with the case that an airline gives members both cheaper restricted fares and cheaper unrestricted fares. Sufficient The readers should be warned that our model is purely tactical. When an airline negotiates a special 1 deal with an interest group, there are many other issues involved which can not be directly addressed in our model. Hopefully, our model will be helpful for airline managers to understand the impact of different types of membership deals.  Chapter 5. Airline Pricing by Using Membership and Product Restrictions  150  conditions are identified for simple optimal fare structures. Examples are also used to give further support. The last section is a summary.  5.2  Model Setting and Notation  Let D(p) be the total market demand function at price p (unrestricted), among which DM(p) of them have the membership. Denote DN(p) to be the demand from the nonmember group. Then, Dv(p) = D(p)  —  (p). M D  Naturally, since membership is a form of restriction, we let  7M(p)  be the percentage of  consumers in the demand market at price p who are non-members. Then clearly, 1M(p)t  p) D ( M D(p)  On the impact of product restriction, we note that membership restricted fares are more restrictive that the general restricted fares. Therefore I use the following notation: • 7Mr(P) is the percentage of consumers in the total demand market who cannot  purchase the product at the restricted membership fare p, that is DMr  17Mr(p)_ where  DMT(p)  ()  D(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 cannot  accommodate the product restriction at price p, that is Dr(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  Further assume that D(p) and  DM(p)  ,p,, }. As usual, for i  set of fares {pi,  151  are non-decreasing step function defined on the =  1,... , n, denote  YM,i = yM(pi);yMr,i = yMr(pi);yr,i  =  and =  D(p); D  =  D’(p); D  =  DM(pj); DrT  . = 1 DMT(p )  Throughout this chapter, I will assume that YMr,i  arid  Yr,i  are strictly increasing in  j.  Additional conditions may be assumed in each of the following three sections in order to obtain simple optimal fare structures.  5.3  Cheaper Restricted Membership Fares Only  Let 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 choose from: restricted membership fares, (public) restricted fares and (public) unrestricted fares. On the other hand, any non-member can only purchase the public restricted fares and the public unrestricted fares. Now if I call the restriction consisting of membership restriction and product restric tion as the type-S restriction and the product restriction as the type-i restriction, then it is clear that type-S restriction is nested into type-i restriction. This allows me to use the model discussed in Section 4.2 of Chapter 4 to handle the problem here. But in the presentation that follows I will avoid to use the notion of type-i and type-3 restrictions here after. I put them here for the purpose of illustration only.  Chapter 5. Airline Pricing by Using Membership and Product Restrictions  Let  Mr,i  be the percentage of consumers in the demand market at restricted price Pi  who are non-members. Then 1 aMr,i),  152  =  for i > 1. Hence 1  —aMr,i  7MT,i =  (i  7r,i)(1  —  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 higher than the general restricted fares, I can define a pricing policy for the airline as follows:  { (pi, where  :  =  m (pj, qj) : ; 1,... , 2  j  =  ,••• , mi; (pk, q) : k 2 m  ,• 1 m  is the allocation to members at the restricted price Pi for 1  2 is the allocation to general public at the restricted price Pj for m  ..  i  1  2 m  , 2 m  qr,j  , q is 1 m  j  1 the allocation to the general public at the unrestricted price Pk for m  ,  k  n, and  1 <n. Furthermore, I also know from my discussions in Section 4.2 that the m  airline’s pricing problem can be formulated as the following N-model: +  Max  =m2  P” +  1 pkqk k=m  s.t. —  2 ‘ç.m  Jj4,-  P2 =  —  — v-”mj q,-,, P1 = L..’j=m 2 2 D-f3 D  < 1 —  qk v-’n 1 2 /_jk=m Dk—3lD—(1—/3l)fl D  2 m zi= 1  +  1 m  2 Zj=m  qMr,i,qr,j,qk  +  <1 —  n YZk=m q 1  q  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 membership fares, and that both  and  aMr,i  are strictly increasing, then  • The airline needs to offer at most four fares to the market; and  Chapter 5. Airline Pricing by Using Membership and Product Restrictions  153  • The optimal fare structures are designed in such a way that the demand for restricted membership fares will be exhausted right after the sales of these restricted fares targeted to members. I now discuss the implications of the assumptions in the above proposition. Clearly the assumption that  is strictly increasing is a basic requirement for an effective product  restriction. For the assumption that  ‘Mr,i  is strictly increasing, it simply means that as  the price increases, among these who can purchase restricted product, the percentage of the members is strictly decreasing. As demonstrated in Theorem 4.2.1, this assumption plays a key role in the process of obtaining a simple pricing structure, or a simple fare structure in our context here. Basically, it allows us to focus on the type of policies with the property that the demand for restricted membership fares will be exhausted right after the sales of these restricted fares targeted to members. Offering lower restricted fares to special clienteles is a very common approach for airlines dealing with membership issues. I here present two examples that are consistent with this particular approach.  Example 5.3.1: Tour Operators Consider vacation operators which sell tour packages or vacation packages with many predetermined 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 par ticular airline. From an airline point of view, these vacation operators create substantial revenue opportunities since: • A successful operator usually offers different packages all year round, which are planned 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 implies that these operators who can strike good deals with some airlines will more likely be the successful ones; • An airline can play a crucial role on flight schedule arrangement of the package if the airline indeed makes concessions on air fares; and • Almost all vacationers who arrange their vacations through operators will not fly at an unrestricted fare, which is usually much higher than an ordinary restricted fare. Therefore, for these airlines which are willing to take that extra step to negotiate with vacation operators and are willing to give additional discounts on air fares, additional revenue can be generated. But the airline must address the following issues: • Given the time condition by the vacation operator, the airline needs to tell them which 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 or negotiate with the vacation operator on a common choice; and • On the other hand, if the airline cannot find an appropriate flight, then it is possible for the airline to suggest an alternative flight. All these issues involve delicate work on pricing and inventory control. Our above model will be helpful in this regard. First, since the airline only deals with the vacation operator  Chapter 5. Airline Pricing by Using Membership and Product Restrictions  155  with a clearly defined capacity requirement, our model at least gives a bottom-line value for the total revenue for this particular part of the capacity if these seats are offered to the operator. It will also give the airline a base for any negotiation with the operator.  Example 5.3.2: Corporate Retreat Programs  Many corporations have annual retreat programs to reward their high performance employees. Every year, the management will select a group of employees whose contri butions are better than most of the other employees. The company will send this group to a special place for a couple of days of retreat. Typical examples include companies from the insurance, lodging and real estate industries. The main characteristics for these types 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 the total 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; and these executives are in fact captives of the unrestricted fare or the first class fare since they simply cannot make an early commitment. Many large corporations use charter service for their retreat programs because it is flexible and 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 and revenue should be able to explore this market segment.  Chapter 5. Airline Pricing by Using Membership and Product Restrictions  5.4  156  Cheaper Unrestricted Membership Fares Only  I now discuss the case where an airline considers oniy to offer lower unrestricted mem bership 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 the avaibility of seats. Most airlines also have an internal policy that will give an employee a guaranteed seat and all the flexibility if the employee pays a small fraction of the unre stricted 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 corporation which constantly has its consultants on different assignments all the time. These trav ellers require the maximum flexibility. As a result of this, most of them must pay the substantially higher unrestricted fares. On the other hand, as modern telecominunica tion technology becomes more advanced and much cheaper, many companies have cut their travel budget to cope with the high cost of air travel. From an airline’s point of view, 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 under certain provisions, such as a minimum commitment on the total number of trips. The general feature for this type of membership demand structure is that there is no demand for restricted fares from members. Therefore, I can directly use the ME-model developed in 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 to buy an unrestricted ticket at a price only lower that the (public) restricted fare in order 2 In other words, we have the following: to get a guaranteed seat. course, the majority of internal travellers are free-riders. But technically speaking, these travellers have no impact in airline’s revenue. They also have no negative impact on airline’s operation since these  Chapter 5. Airline Pricing by Using Membership and Product Restrictions  •  =  0, or equivalently,  7Mr,i =  157  1 for all i; that is, there are no internal travellers  who are willing to pay an restricted fare even if it is very cheap; and • A pricing policy is in the form of  { (p, where 1 <i 1 m pk  <j  q  :  =  l,•  .  .  ,  :  mi; (pd,  ,• 1 j=m  .  ,  m; (pk, q) : k  =  m,• ,n},  is the allocation to members at the (unrestricted) price level p for  , qr,j is the allocation to the general public at the restricted price pj for 1 m <m, and qj is the allocation to the general public at the unrestricted price  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 membership  group and the airline is committed to offer unrestricted membership fares that are lower than restricted fares, then the airline ‘s pricing problem can be formulated as follows: Max  +  pjqr,j  +  YZm pkqk  s.t.  \fl  /_.ikm Dk—fllD-/3 D 2 1 m qM,  +  <1 —  Z j=mj qr,j + YZk=m q  qM,i, q, q  >  q  0,Vi,Vj, and Vk.  Proof: Straightforward. The details are omitted here. El  traveller are called stand-bys, which means that their seats are not guaranteed. In other words, they may be dumped during their trip. I here only focus on paying internal travellers.  Chapter 5. Airline Pricing by Using Membership and Product Restrictions  158  Even though the membership restriction and the product restriction are mutually exclusive in this context, the airline must address an additional constraint that requires the 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 that  is strictly decreasing, which says that for non-members, as the price increases, the percentage 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 for the 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 to  one. 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 will  give any employee a guaranteed unrestricted ticket if the employee pays a small fraction of the public unrestricted fare. This is good news for the above formulation, which can accomplish a simple optimal fare structure because it reduces to a linear programming problem. But it may be bad news to the airline since it may cause a revenue loss. For these flights with low demand from internal travellers, the above formulation will likely lead to an optimal policy that is consistent with an optimal solution derived directly from ME-model. But for some high demand flights or flights at busy travelling seasons such as Christmas, the commitment that /32  =  1 may prove to be costly. Therefore, the  above analysis calls for close monitoring on these flights and some modification may be necessary. One possible modification is to satisfy a fixed percentage of those who request a booking on a flight. There are several benefits for this provision. First, it will still reduce the above formulation to a linear programming, since /31  /32  is a fixed constant and  can always be taken as one. Second, it is not hard to implement because it involves  the airline’s own employees. For example, it can be implemented through a lottery draw by a computer among those employees who wish to book the same flight. Or it can be  Chapter 5. Airline Pricing by Using Membership and Product Restrictions  159  3 Another possible modification implemented according to the seniority of employees. is to ask internal travellers to pay for an unrestricted ticket at a restricted price. The best possible modification is to implement an optimal fare structure derived from the ME-model, which may a  price  internal travellers to pay for an unrestricted ticket at  that is higher than the restricted fares. In summary, the main point here is that  strictly committing  5.5  force some  2 =  1 may cost an airline a lot of money.  Cheaper Restricted Membership Fares and Cheaper Unrestricted Mem bership Fares  In the above two sections, I have discussed several cases which indicate that it is sensi ble for an airline to offer either lower restricted membership fares  or  lower unrestricted  membership fares. But in some situations, with enough information about the demand behaviour from the members, the airline may consider offering both lower restricted mem bership fares and lower unrestricted membership fares. This leads to the pricing problem with 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 following notation: • we can call the product restriction a type-i restriction, which implies that a type-i restricted ticket is a (public) restricted ticket; • we can call the membership restriction a type-2 restriction, which means a type-.2 restricted ticket is an unrestricted membership ticket; Some airlines practice the rationing for the internal travelers in term of service seniority, which 3 means that these who are with the airline longer will have the first priority of obtaining confirmed seats. But, in fact, I will avoid these generic notation in our discussions that follow. I present them here 4 just 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 is attached with both type-i and type-2 restrictions. It is clear that for the case here, the airline has the flexibility of offering four types of fares: restricted membership fares, (public) restricted fares, (unrestricted) member ship 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 restricted fares; and similarly, the unrestricted membership fares, if offered, also should not be higher than the public unrestricted fares. Clearly, the airline should always consider selling restricted membership fares first, since they are most restricted among these four proposed fares. From Theorem 4.4.1 of Section 4.4 it follows that if the membership deal between the airline and the members requires that an airline must give a guaranteed book ing for a restricted ticket from any member who requests for it, then the airline’s pricing problem can be formulated as a simple linear programming, which shows that the airline needs at most five fare levels with four different types of ticket. Rephrasing Theorem 4.4.1 in our context, we have the following proposition:  Proposition 5.5.1: Suppose that an airline plans to use a membership restriction and a product restriction. Assume that • both  --  YMr,j  and  ---  7Mr,j  are strictly increasing in  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) +  S. t. ç-mi L.ij1  — —  1  k—_m Pkqk  Chapter 5. Airline Pricing by Using Membership and Product Restrictions  1 D—D /_ij=m J_Jj=mi D-D  v_’n  —  i  —  1  —  —  gk  Lak=m Dk_(D_D)_(Dj&D)  1 m :i=i qMr,i  +  1 (qr,j Zj=m qM,j  where  q,j  +  —1 —  n  qMj)  0 and  + YZkzm q  q  qM,j  q  0  is the allocation to the members at the restricted price j,  to the general public at the restricted price Pj,  161  qr,j  is the allocation  is the allocation to the members at  the unrestricted price P 3 and q is the allocation to the general public at the unrestricted price  pk.  Let us analyze the two assumptions used in the above proposition. First, the condition that .2_ and 7Mr,j • both  —--  YMr,j  DM_DM’  are strictly increasing in and  D:_Dr  j  is equivalent to requiring that  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 percentage of consumers who can buy restricted tickets is also strictly decreasing. In essence, these conditions are conditions of effectiveness for both membership restriction and product restriction. They together provide sufficient information for an airline to decide whether or not to offer cheaper membership (unrestricted) fares and (public) restricted fares. The second assumption is in fact a corporate commitment from the airline. It may involve issues other than pricing. A good example is a bilateral agreement between two partner airlines, 5 where each airline gives a special treatment to its partner’s employees who use its passenger service. If an airline’s partner has such a commitment, this airline Partner airlines nsually involve agreements on code-sharing and traffic-feeding. 5  Chapter 5. Airline Pricing by Using Membership and Product Restrictions  62  may have no choice but to give its partner’s employees the same treatment. Our model shows 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 negative impact on revenue. So it is really important for airline managers to understand that the best result is such that the commitment is consistent with market behaviour. Theorem 4.4.2 provides some additional conditions which guarantee that such a commitment will not 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 and  a product restriction. Assume that • both  --  YMr,j  and  -—  YMr,j  are strictly increasing in  j.  Then 1. if  is strictly increasing in j, or equivalently, in the residual market where  there is no members who can still buy restricted tickets, the relative size of the market for unrestricted tickets from members and the market for restricted tickets from general public is strictly increasing, the airline has an optimal fare structure having the following properties: • the airline ‘s commitment of satisfying all bookings from members for restricted tickets is consistent with revenue maximizing; • the (unrestricted) membership fares will not be lower than the (public) re stricted fares; M,,,7,, is strictly increasing in j, or equivalently, in the residual market where 2. if 7M,-,j 7M,)  there is no members who can still buy restricted tickets, the relative size of the market for restricted tickets and the market for unrestricted tickets from members  Chapter 5. Airline Pricing by Using Membership and Product Restrictions  163  is strictly increasing, the airline has an optimal fare structure having the following properties: • the airline’s commitment of satisfying all bookings from members for restricted tickets 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.  U  Again, let us look into the implications of the assumptions used in each of the two cases. Note that 7Mr,3 fMr,j  —  —  rM  3 7M, 7r,3  “j  riMr —  — —  So intuitively speaking, if ignoring the group that consists of members who can buy restricted tickets, the first case says that the market demand for restricted tickets is decreasing faster than the market demand for unrestricted tickets from members; and 6 With this in mind, it should not be surprising to see that the second case is vice versa. in 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 the second case too. This is clearly consistent with the traditional idea of economic pricing discrimination: the firm should charge a higher price to the segment that is less sensitive to prices than to other segments. I now give some examples that fit into the model. First, consider the market that consists of travellers from a partner airline. These travellers must pay for their tickets if lnterestiiigly, I can easily state these assumptions in term of 6 of BL (1993a).  price elasticities as used in Section 5  Chapter 5. Airline Pricing by Using Membership and Product Restrictions  164  they want to fly on a flight from the partner airline. On the other hand, because of their affiliation with an airline and its internal benefit of free flying, these travellers are much more price sensitive than the general public for both restricted fares and unrestricted fares. 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 amount of travelling activities from employees of government or government agencies. Some of this travelling can be planned in advance, but there is a substantial portion of it that cannot be planned ahead.  In this case, an airline can give these clients a break on  both restricted fares and unrestricted fares. It is evident that the government will be pleased to let its employees fly with unrestricted fare at a price that is between the public restricted fares and public unrestricted fares. It is a special treatment that may induce more travelling from government employees. In my view, the first case in the above proposition will be helpful for airlines to negotiate acceptable and rational fare prices with the government.  5.6  Summary  —  Membership and Product Restrictions  In the above three sections I have illustrated how the pricing models for two types of restrictions can be used by airlines when product restriction and membership restrictions are used at the same time. Unlike product restrictions, which are fairly difficult to design, 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 the other hand, addressing all possible membership restrictions at the same time could be counter-productive because it is hard to monitor the interactions among many different member groups, in addition to the product restriction. A simple solution to this problem is 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 book this flight; and • the computer reservation system should be automated so that ordinary travel agents can 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 flights only. To tag a flight several months ahead is not an easy task, but with more advanced forecasting 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 tag these 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, since each of these opportunities represents a market niche. The airline must have a systematic management 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 computer reservation system needs to be enhanced so that it is capable of providing real time recommendations to managers.  Chapter 6  Seat Allocation Game on Flights with Two fares  6.1  Introduction and Model Setting  Consider 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 low frequency of flights between two destinations will make the issue of competition in the allocation decisions very important for the sake of market share and profit. For example, Canadian Airlines International (CAT) and Japan Airlines (JAL) are sole carriers offering direct service between Vancouver and Tokyo. And both CAT and JAL offers daily flights. The schedules of both airlines are very close and their fares are actually identical. In particular, both airlines have scheduled flights on Saturdays and Sundays, 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 is operating a single-leg flight with two given fares  —  a full fare and a discount fare. I will  use the following notation:’ • The discount fare is PB and the full fare is The notation in this chapter is self-contained. 1  166  py;  Chapter 6. Seat Allocation Game on Flights with Two fares  • Airline k’s flight has a capacity of Ck, for k  =  167  1,2;  • The market demand for discount fare is given by B and the market demand for full fare 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 variable and its objective is to achieve the highest revenue possible. Let  k 1  be the booking limit for the discount fare set by airline k, k  k’s expected revenue is determined by the joint decision expected revenue function as rk(lk, l) for k Bk(lk,  (ik,  =  1,2.2  Since airline  ii), we can write airline k’s  1,2. More specifically speaking, if we let  i) be the airline k’s demand share for the discount fare and Yk(ik, ij) be the airline  k’s demand share for the full fare, then the expected revenue function rk(lk, ij) is given by: 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 derived from the pair of booking limits (ik, 13). Therefore, this leads to a simple two-person game with payoff functions r . With this in mind, we give the following definition of an 2 1 and r equilibrium pair of booking limits.  Definition 6.1.1: A pair of booking limits (l, 1) is said to be an. equilibrium pair of booking limits if fork  =  1,2  rk(lk, l,) From now on, I 2  rk(lk,  l,) for all lk  =  0, 1,... ,Ck.  (6.2)  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  168  Note that there is no guarantee of the existence of an equilibrium pair of booking limits since it is of the pure strategy form. To extend this concept, I introduce the notion of mixed strategy. First of all, airline k’s pure strategy space is given by Sk={O,1,—.,Ck}, fork=1,2. And a booking strategy for airline k is a probability distribution function on Sk. Let Fk be the collection of all possible booking strategies for airline k. By definition, for any E Fk, we have that  ak  booking strategy pair  EES  (ak,  1, where of k(l)  ak(l)  O,Vl e Sk. Now for any given  cs), define  Rk(ak,a,)  ofk(lk)aJ(l)rk(lk,lJ).  =  IkESk IJESJ  I will call the game characterized by (R ,F 1 ;R 1 ) as the seat allocation game. I now 2 ,F 2 give the following definition of an equilibrium booking strategy, which provides a solution for our seat allocation game.  Definition 6.1.2: An equilibrium booking strategy is a pair (c4,c4): k  =  4  for  1,2, such that R(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. By definition, a bimatrix game is a two-person non-zero sum game where each player has a finite number of pure strategies. For this type of games, the existence of an equilibrium booking 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 admit mixed strategy equilibrium as well as pure strategy equilibrium.  Chapter 6. Seat Allocation Game on Flights with Two fares  169  It follows immediately from Definition 6.1.2 that our equilibrium booking strategies are Nash equilibria. Therefore, by the Nash Theorem, we always have at least one equilibrium booking strategy. In this section, my primary interest is to find an equilibrium pair of booking limits, that is, I want to characterize the pure strategy equilibria for the seat allocation game if they indeed exist. So far, I have not discussed how to obtain the derived revenue functions r 1 and r . 2 Without properly specifying these two functions, the seat allocation game is not even well defined. Since the two airlines face the same market demands (both of the full fare market and discount fare market), the specification of the revenue functions are critically related to the way two airlines share market demands. In this chapter, I will explore the seat allocation game under two types of market splitting rules: the proportional splitting rule and the equal splitting rule. By the proportional splitting rule, I mean that in the event that the total commitment for a certain fare class from two airlines exceeds the market demand, then the two airlines will split the market demand according to their proportions of the total commitment. 3 By the equal splitting rule, I mean that in the event that the total commitment for a certain fare class from two airlines exceeds the market demand, each airline will get a half of the market demand or reach its commitment level, whichever is less. 4 This chapter is organized as follows.  Section 6.2 investigates the seat allocation  game when the proportional splitting rule is used to define each airline’s derived revenue function. Section 6.3 studies the seat allocation game under the equal splitting rule with deterministic demands. And Section 6.4 is a summary. 1n fact, the intuition behind the proportional splitting rule comes from the capacity share or the 3 market share consideration. 1n some sense, the equal splitting rule is build upon the premise that consumers are absolutely 4 indifferent between the two carriers.  Chapter 6.  Seat Allocation Game on Flights with Two fares  170  Seat Allocation Game Under Proportional Splitting Rule  6.2  Under 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 by =  Bk(lk,l)  ik+lj  B, for k  =  1,2.  (6.3)  And the actual sales for airline k’s discount fare should be the minimum of its booking limit and its demand share, that is, Sk(lk,  i)  A  Bk(lk, i)  =  k 1  =  lk(1 A  B lk+ij  (6.4)  After the booking for the discount fare is closed, airline k has a residual capacity of Lk  =  Ck  —  Sk(lk,  13).  (6.5)  Consequently, airline k’s demand share for the full fare, again according to the propor tional splitting rule, now becomes Yk(lk,l)  =  Lk Y. k + j  (6.6)  T  Consequently, airline k’s expected revenue function rk is given by Tk(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;  • 3 Lk+L = (C—(1AB). For clarity of presentation, I 5  will use  Bk  and Yk to denote BkQk, l) and Yk(lk,lj) respectively.  Chapter 6.  Seat Allocation Game on Flights with Two fares  171  We are now going to use the technique of Brumelle et al (1990, [37]) to look for any equilibrium pair of booking limits. Given that airline j’s booking limit is fixed at 13 and that airline k has accepted  k 1  —  1 requests for its discount fare and there is an  additional request for its discount fare, then airline k must decide to accept or to reject this particular request for the discount ticket. If airline k decides to reject it, its expected revenue is given by E(rk(lk  —  1, l)Bk  ik);  and if airline k decides to accept the request, its expected revenue becomes E(rk(lk,l)IBk  ik).  The key is the revenue difference between these two decisions. For this, define the direct incremental gain function Gk(lk, 13) for airline k as follows: Gk(lk, 3 1 )  =  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 Gk(lk, ii)  =  =  1,2, PB  —  pyP(Y> C  —  1IB> 1)  —  Y (C 1)(C 1+1) X  E(YIY<C—l,B>l)P(Y<C—1IB>l), 2 . 1 whereCC +C andlli+l . 2 Proof: First of all, it is easy to check that  Therefore, Gk(lk,l)  E[YA(Cl)IBl]+ 01 pB+pY  (6.8)  Chapter 6. Seat Allocation Game on Flights with Two fares  172  x E[YA(C—l+1)IBlj PB+PY  Ck—lk  c—i  {E[Y A (C  —  X  l)Y> C—i, B> l]Pr(Y> C  —  ljB  1) +  +E[YA(C—i)IYC—l,Bl}Pr(YC—iBl)}+ Ck—ik+l  C—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.  D  On the other hand, it is important to notice that if airline k declines a request for the discount fare, the passenger will make the same request to airline j. Airline j may or may not accept the request. The above direct incremental gain function is calculated under the assumption that airline j’s decision is fixed at 1,. If airline j accepts the request, that is, airline  j  changes its initial booking limit for the discount fare from 1, to 13 + 1, then  airline 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  173  Proof: By the definition of indirect incremental gain function, it follows that g(l,  l)  =  E[i(lk, l)Bk  =  [ikpB  ik] lk  —[(1k  —  —  PY  C  l)pB  —  E[r(lk  E(Y A (C  C  +  —  —  1,13 +  l)IB  1)IB  k] 1  1)] +  E(Y A (C  —  l)IB> 1)]  E(YA(C_l)IBl), 1 PB_PYC  =  which proves the lemma.  Li  It is interesting to see that Lemma 6.2.2 leads to the following facts: • gl(l1,12) =g2(i2,il); • gk(lk, 13) given  is decreasing in 1  =  11  +  12,  and consequently, is decreasing in  k 1  for any  ij, and vice versa.  On the other hand, it is easy to check that that gk(lk, i)  =  PB  —  pyP(Y> C  —  lB  1)  C—l,B l)P(Y C—lIB 1), which, together with (6.8), leads to Gk(lk,l) >  because  <  gk(lk,ij),  1. Now because of this inequality, each airline will only focus on its  direct incremental gain function. But it is not clear whether or not the direct incremen tal gain function Gk is decreasing in k 1  k 1  for any given 1,. If Gk is indeed decreasing in  for any given ij, then we can use each airline’s response function to characterize an  equilibrium pair of booking limits. In fact, we have the following proposition:  Chapter 6. Seat Allocation Game on Flights with Two fares  Proposition 6.2.3 If for k  =  174  1,2, the gain function Gk(lk, i) is deceasing in  k 1  for any  given 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’s  response function, to airline j’s choice i as follows: ‘lk(lj)  : Gk(lk,l ) 3  0}  =  max{lk E  =  max{lkESk:P(Y>C—1IB>l)+ —  First, it is clear that  0(0—i + ik T  )E(YIY <C 1  —  l,B  l)P(Y  C  —  1IB> 1)  is well-defined. To finish the proof, it suffices to show that, for  l and l given by (6.9), I will have r1k(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 seats for the high fare, that is, =  1*  —t 1  1*  —t . 2  Upon agreeing on this, both airlines will face the exactly same decision on how to choose their booking limit for the discount fare so that there is no further possible revenue gain by allocating additional seats for the discount fare. This is in fact captured by the re sponse function. Since in the end neither airline can unilaterally improve its revenue, the pair of booking limits (li, 1) must be an equilibrium. This proves the result.  D  As an immediate consequence of the above proposition, I get the following interesting corollary:  Chapter 6. Seat Allocation Game on Flights with Two fares  175  Corollary 6.2.4: Under the same assumption as in Proposition 6.2.3, each airline will protect the same number of seats for the high fare at equilibrium. Proof: It follows directly from (6.8) and (6.9). LI  The following corollary indicates that competition will collectively reduce the total num ber 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 of seats 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 the  low fare is given by:  )<}. IB> 71 =max{0<C:P(Y>C— py So I need to show that 1*  (6.11)  l + 1 < 71*.  Again, this is obvious since by Proposition 6.2.3 I know that  =  Max{lkESk:P(Y>C_lk_lIBlk+l)+(GllCll*+l)X  E(YIY < C  —  —  l, B  k 1  + 1)P(Y  C  —  —  lIB  k 1  + 1)  which implies that  P(Y > C_l*IB>1*)  +  (C  lj(Cl*+1)  x  py  Chapter 6. Seat Allocation Game on Flights with Two fares  176  Therefore, I must have P(Y Then by definition of  ‘‘  >  C  —  l*IB> j*)  <PB  py  (6.11) must be true. This proves the corollary.  D  As a last comment for this section, I should say that the seat allocation game under the proportional splitting rule is not completely solved yet in the sense that we still need a complete characterization of the whole set of equilibrium booking strategies, rather than one equilibrium pair of booking limits only. On the other hand, the above results are indeed quite encouraging.  6.3  Seat Allocation Game Under Equal Splitting Rule with Deterministic Demand  In this section I will look at the seat allocation game from a different perspective. After many unsuccessful attempts to find equilibrium solutions to the seat allocation game under the equal splitting rule with uncertain demands, I am here only able to report some preliminary analysis for the case that both B and Y are in fact deterministic, that is, they are fixed constants. So the main purpose of this section is to see how the use of the 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 protect for its full fare By definition, a protection level for full fare by each airline must be a commitment in the following sense: if Pk is the protection level, then airline k will only sell 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 protection levels (pi, P2), airline k will only sell Ck 6j  —  Pk low fare tickets. Once the discount fare is  deliberately switch from booking limit to protection level. They are technically equivalent.  Chapter 6. Seat Allocation Game on Flights with Two fares  closed, both airlines start to sell the full fare. First of all, if Pi  177  +p2  <  Y, then each airline  is able to sell the full fare tickets up to its respective protection level. Now consider that Pi  +  P2 >  P2 >  Y, that is, the joint commitment for full fare is too high. If p  >  Y/2 and  Y/2, then we should expect that each airline will be able to sell Y/2 full fare tickets  if travellers have no preference over airlines and travellers arrive at a random order. As a consequence of this, neither airline can sell enough seats to exhaust its commitment. Y/2 and p’ +  If Pi  will sell Y  —  P2 >  Y, then airline 1 will sell P’ discount tickets and airline 2  P1 discount fares. Note that in this case airline 2 is unable to sell up to its  protections level since  P2 >  Y  —  p’. The essence of this type of demand sharing is that  both airlines will share the market with a certain degree of equality, or equivalently, they will split the market evenly whenever it is possible.Summarizing the above discussion, I have the following: • If an airline’s protection level is not greater than half of the full market, it will sell up 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 will capture at least half of the market. Now let  Yk(pk,p3)  be the airline k’s share of the full fare demand, where k  the above motivation leads to the following formal specification:  ifpk+p >Y,pk>Y/2,p <Y/2;  Y—p Yk(pk,p)  if  =  pk  >  Y/2, pj  otherwise.  Pk  Again, I will use convention that k,j 7  ‘  =  1,2 and kj.  >  Y/2;  =  1,2. Then  Chapter 6. Seat Allocation Game on Flights with Two fares  For the demand for low fare, denote 3 B—i Bk(pk,p)  k = 1  Ck  for k  pk  =  1, 2. Then similarly, I will have  >B,lk>B/2,lj<B/2; 3 iflk+1  if  =  —  178  >  B/2, l > B/2;  otherwise.  Since under the above splitting rule, the the number of sales for each fare class for each airline is in fact equal to the actual share of the demand, airline k’s revenue function Tk(Pk,Pj)  is given by Tk(Pk,Pj)  =  + pyYk(pk,p,).  pBBk(pk,p)  Since the goal is to find an equilibrium strategy for each airline, for completeness I give the following definition:  Definition 6.3.1: A pair of protection levels (p,p) is said to be an equilibrium for the seat allocation game if rk(pk,p) >  rk(pk,p),  for all  k 3 7  =  0,1, ,Ck and k  =  1,2.  The following proposition gives a characterization for an equilibrium strategy for air lines when the demands are deterministic.  Proposition 6.3.1: Under the deterministic demands, (Y/2, Y/2) is always an equilib rium 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 airline has any strict incentive to deviate.  Case 1: 2 B+Y. C + 1 C >  Chapter 6. Seat Allocation Game on Flights with Two fares  179  In this case, the joint capacity exceeds the total demand of the discount fare and the full fare. Given pk  =  Y/2, airline k is guaranteed sales of Y/2 units of the full fare tickets  regardless what is the airline  j ‘s commitment for the full fare tickets. Even though there  may have many equally good responses from airline  j, p,  =  Y/2 is always one of these  responses. As a consequence of this, each airline will also sell B/2 discount fare tickets. 8  Case 2: 2 + 1 C _ <B+Y. C Y/2, each airline can sell at least Y/2 at the full fare, which implies that at  Since Ck  the equilibrium, for k  =  1,2, p  Once p  =  Y/2.  Y/2, the airline k’s best possible result on the sales of the full fare is also  Y/2. Note the if pk > Y/2, then two things will happen: (1) airline k can still oniy sell Y/2 full fare tickets; and (2) since a protection level is a commitment, airline k only allocate Ck  —  pk  number of seats for the discount fare, which may reduce the chance to  obtain its fair demand share for the discount fare. Therefore p  =  Y/2 must be one of the  best choices. This proves that no airline can strictly improve its revenue by unilaterally deviating from (Y/2, Y/2). This proves that the proposed pair of protection levels must be an equilibrium. C.  Remark: Technically speaking, the above proposition is not very interesting. One plau sible aspect is the fact that both airlines will make the same commitment for the full fare class, which is consistent with the findings in the previous section. It remains to be seen whether this property is still true when the demands are actually random. This implies that when total capacity level is too high, the larger carrier has no clear advantage in 8 terms of sales.  Chapter 6. Seat Allocation Game on Flights with Two fares  6.4  Summary  —  180  Seat Allocation Game  This chapter discusses the seat allocation problem in the presence of another airline. It is shown that under the proportional splitting rule, there exists an equilibrium booking policy such that each airline will protect the same number of seats for the full fare. I also demonstrate that at equilibrium the total number of seats that are available for the discount fare is smaller than the total number of seats that would be available if the two airlines cooperate. Under the equal splitting rule, it is shown that if the demands are deterministic, there is an equilibrium such that each airline will protect enough seats for high fare so that each airline will split the market demand for the high fare equally.  Chapter 7  A Note On Three Models for Multi-fare Seat Allocation Problem  7.1  Introduction  As we can see from the discussions in Chapter 1, there are many interesting problems in pricing and yield management that still need answers. The purpose of this chapter is to address the seat allocation problem for single-leg flights with multiple fares with inde pendent random demands. There are three different models to solve the seat allocation problem for multi-fare flights, done independently by Curry (1990, [52]), Wollmer (1992, [269]), and Brumelle and McGill (1993, [38]).’ After briefly reviewing these three mod els, I will present a proposition showing that their optimality conditions are analytically equivalent, 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 been circulating for several years before they are published. In fact, all of them announced their findings in 1988, refer to Wollmer (1988, [268]), McGill (1988, [143]) and Curry (1988, [50]). A11 notation in this chapter is also self-contained, which hears no relationship with any other notation 2 used in other part of this thesis.  181  Chapter 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 over booking; and • Random variables X ,.. 2 ,X 1  ,  X are independent.  This chapter is organized as follows. Section 7.2 presents the seat allocation model due to Wollmer (1992, [269]), where the demands are assumed to be independent dis crete random variables. Section 7.3 discusses the model due to Curry (1990, [52]) which addresses the problem for independent continuous random demands. Section 7.4 ana lyzes the model due to Brumelle and McGill (1993, [38]) which solves the seat allocation problem for any independent random demands. Section 7.5 proves an equivalence result for the optimality conditions from three models. And the last section is a summary.  7.2  Wollmer’s Model  Wolimer (1988, [268]; 1992, [269]) studied the seat allocation problem for multi-fare flights by considering that the demands are discrete random variables. By formulating the seat allocation problem into a Markov decision problem, Wollmer derives optimality conditions that are natural extensions of Littlewood’s optimal condition for two-fare flights with independent demands. Here is a sketch of Wolimer’s model. Let Rk(.s) be the expected revenue under an optimal policy if s seats are available for booking when fare class k is the lowest fare that  Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem  183  is open. 3 Define, ARk(s)  =  Rk(s)  —  Rk(s  —  1),  which is the incremental value of an additional seat. By using simple facts on conditional probabilities, it is easy to derive the following recursive relationships: Ri(s)  =  E(X A s), AR f 1 (s) 1  P(X > s); f 1  =  and Rk(pk_1  + i)  =  Rk_1(pk_1)  + [ARk_l(pk_l + j  —  i)P(X  i) + fkP(Xk  + 1)]  i—i  ARk(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 and xAy  min(x, y). By proving that ARk(s) is decreasing in s, Woilmer gives the following  characterization 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 optimal booking policies.  Curry’s Model  7.3  Curry (1988, [50]; 1990, [52]) studied the seat allocation problem for multi-fare single-leg flight by assuming that the demands are continuous random variables. His approach is in the spirit of marginal seat value since his analysis is based upon the concept of the revenue slope, or equivalently, the marginal seat revenue curve. Let Rk(pk_1, s) be the expected 1n other words, Rk(S) is the total expected revenue for the remaining s seats for fare classes i 3 , k + 1 are closed. , 1 only. It simply means that fare classes i = n, k, k 1, —  .  ..  ..  =  Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem  revenue for the remaining s seats that are available for fare classes i the protection level for fare classes i  k  =  1,.  —  ,  1 is given by  =  Pk—1•  k, k  —  184  1,... , 1 if  Define the slope  function of the expected revenue function as follows: s)  aRk(Pk_l,  Sk(pk_1,s)  Os  =  The recursive relationship on Rk ‘s is fSpk_1  Rk(pk_1,s) =  (f,x + Rk_j(pk_2,s  j  —  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 recursive relationship on Sk’S takes the following form: ORk(pk_i,s) =  (_fk  +  k_;2_1))  F(Xk >3  pk-1)  and fSpk_1  Sk(pk_1, s)  fkP(Xk >  —  Pk-1)  +  J  Sk_1(pk_1,  —  x)dFk(x).  (7.3)  0  After showing that the revenue function Rk(pk_1, s) is concave in s, Curry uses first order 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 *,.. 1 (p * 2 p ,p) is given recursively by: 0, and 1 Sk(p_ , p)  =  fk+i  fork  =  1,•,n— 1,  which says that the optimal protection level for fare classes k, k  —  1,.  (7.4) ,  1 is chosen in  such a way that marginal value of an additional seat is equal to the value of a fare class k + 1 ticket, that is,  fk+1.  This is clearly consistent with Littlewood’s condition for case  of two-fare classes with independent demand.  Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem  185  Brumelle-McGill’s Model  7.4  Brumelle and McGill (1993, [38]) investigated the seat allocation problem for multi-fare single-leg flights with independent demands. The key difference is that their analysis does not depend upon whether the demand is continuous or discrete. In fact, by using  non-  smooth optimization techniques and optimal stopping rules, they establish two important results: • 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 in fact also optimal in the class of all possible booking policies. Let  I briefly sketch their model here.  rk(s; p,  x) be the revenue function gener  ated by k highest fare classes, given that (a) there are s seats available, (b) the vec tor p  ‘pm) is the protection level booking policy, and (c) the vector x  =  , 2 ,x 1 (x  ,  x,) is a  realization  ,X 1 ,. 2 of the vector of random demands (X  .  .  ,  =  X,). In  their model, the recursive relationships are built directly on the revenue function rather than on the expected revenue function:  ri(s;p,x)  I  fis,  , 1 ifO<s<x  xi, 1 f  if x 1 < s;  =  and  rk+1(s;p,x)  =  rk(s;p,x)  if 0  (s —pk)fk+1 +rk(pk,p,x)  if Plc  Xk+lfk+1  +  rk(S  —  xk+1;p,x)  ifs  s  pk,  <Pk  Plc +  +Xk+1,  Xk+1.  To use the non-standard optimization method, we have to deal with the notion of left derivative  and  right derivative.  Let S_ and  be the operators for left derivative and  Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem  186  right derivative of the revenue function with respect to s. Then it follows that  (s;p,x) 1 Sr  S_ri(s;p,x)  I  =  =  fi  1 ifs<x  0  1 ifs>x  I  fi  1 ifs<x  1.  0  . 1 ifs>x  In general,  +rk(s;p,x) 6  =  S_rk(s;p,x)  =  rk(s;p,x) 1 6  if 0  fk+1  if 73k  SJrk(s—xk+1;p,x)  ifpk+xk+1 s  S_rk(s;p,x)  if 0  fk+1  ifpk<s<pk+xk+1  S_rk(s—xk+1;p,x)  ifpk+xk+1 <5.  s <73k  <73k  +  Xk+1  <s pk  The main analytical tools for non-standard optimization are the following two facts: for any continuous, piecewise-linear function 1.  f(s)  is concave on s  >  f(s),  0 if and only is 6f(s)  6_f(s) for all  s;  and  2. A concave f(s) can be maximized at any point .s such that 0 E Sf(s*), where Sf(s*) is the subdiffereritial defined as the closed interval determined by 6+f(s*)  and S_f(s*). Define the expected revenue function to be Rk(s;p,X)  where X  =  =  E(rk(s,p,X)),  ,X 1 (X , 2 .. ,X) is the vector of random demands. It can be interpreted as  the 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  1,... , 1 and the booking policy p  (pl,p2,  ,p,)  187  is used. By proving some concavity  properties 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 func tions are continuous, then there is a closed-form characterization for optimal booking policies: >p,,Xl++Xk>p),fork=1,_.,n—1.(7.6) 2 fk+l=flP(Xl>p,Xl+X Clearly, when n  =  2, it reduces to Littlewood’s formula.  Equivalence of Optimality Conditions  7.5  I am now going to show that the optimality conditions for all three models are in fact analytically equivalent, which therefore implies that they are also computationally equiv alent. 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’s dynamic 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)  =  P(X > s), f 1  Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem  188  which implies that f2 =  So (7.6) holds for k (7.6) holds for i  > pa).  1. Now suppose that based on (7.4), we already have shown that  1,..., k, that is,  =  f+i for i  =  P(X f 1  =  P(X 1 f  ,X + X p 1 2  >  1 +• + X ,X  > P2,  1,.. . , k. We now need to show that (7.6) also holds for i  =  =  (7.7)  > lii),  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 1 ,•,X >p + •••+X_ 1 , >p_ + + ••+X_ X X >s). (7.8) Proof of Claim 6.1: I will prove it by induction. For k for i  1 and i  =  =  2. Clearly, it holds for i  then (7.8) also holds for i  =  =  1, I need to show that (7.8) holds  1. 1 now show that if f2  =  =  P(X f 1  >  p’),  2. By (7.4), 1 ps—p  (pi,s) 2 S  =  P(X >s_i)+J f 2  (x) 2 Si(po,s—x)dF  0  I —P1 =  P(X >s —i)+fiJ f 2  1 P(X  >  (x) 2 s—x)dF  0  where  ‘A  =  1.  P(X f 1  >  )P(X 1 P 2  =  P(X f 1  >  ,X 1 p 2  =  P(X >pi,X f 1 2  =  P(X >pi,Xi +X f 1 2  represents the  and 0 if x for k  =  > s  >8  >  —pr) +  —p1) +  + flE(I(x ) > ) 1 _ 5 < 2 I(x ) x  + flE(I(x > < 2 ) 1 s_p s,x ) x  s—pi)+fiP(Xi +X 2  indicator function  >  s,X <s—pr)  > s),  defined on the set A, that IA(x)  A. This implies that (7.8) holds for i  =  =  1 if x € A;  2. Therefore, the claim is proved  Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem  Now suppose that the claim is true for k  =  189  1,.•• , m, and that I want to show that it  is also true fork = m + 1. First, I have shown that for i = 1,  ,m + 1;  P(Xi 1 1 S(p_j,s)=f +•••+X_ >p_i,Xi+.+X_i+X>s); (7.9) >p•,X and the inductive assumption implies that = f+ P 1 (X f 2 > >pi,Xi+X , •••,Xi+•••+X>p), 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 follows that (pm+i, s) 2 Sm+ fS_Pm+1  P(Xm+ > fm+ = 2  pm+i) +  J  Sm+i(pm,S  —  (X) 2 X)dFm+  0  =  P(X > pi,Xi +X f 1 2 > P2, •,Xl +  +Xm+i > pm+i)P(Xm+ 2 > SPm+i) +  fSPm+l  +fiJ  () 2 P(Xi>pi,,Xi++Xm>pm,Xi+”+Xm+i>S)dFm+  0 =  >Spm+j)+ 2 fiP(Xi>pi,,Xi+.+Xm+i>pm+i,Xm+ P(X > Pi, +f 1  =  1 + ,X  2 > S,Xm+ + Xm > Pm,Xi +• + Xm+ 2  S  pm+i)  +Xm+i > Pm+i,Xi + Xm+l +Xm+ 2 > s),  P(X > 1 f 1 ,•..,X + p  This proves (7.8) for i = m + 2. Hence the claim is true for k = m + 1. Therefore, by induction, 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, will lead to (7.8), that is, for i  =  1,.  ,  . , k, which, by Claim 6.1,  k + 1 and all s > 0,  P(X > p f 1 ,•..,X 1 1 ,X + 1 ++X_ > p_  1 +X > s). +X_  Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem  In particular, taking i  =  190  k + 1, I obtain  Sk+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 that  =flP(Xl>pl,,Xl++Xk>pk,X1++Xk+Xk+1>pk+1), 2 fk+  which is exactly (7.6) for i  =  k + 1. This shows that (7.6) is true for k + 1. Then by  induction, (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 Claim 6.1 since it only uses the facts (7.4) and (7.6). For k Suppose that (7.4) folds for i Since (7.6) is true for all i Sk+1(pk, s)  Taking s  =  =  Pk+1  P(X f 1  =  =  1,  =  1, (7.4) follows directly from (7.6).  1,.•• , k. I will show that (7.4) also holds for i .  > Pi,”  k + 1.  =  n, then claim 6.1 implies that  ,  ,  1 + Xi +..• + Xk > Pk, X  ...  +X+  Xk+1 >  s).  leads to  Sk+l (pk, pk+1) =  , 2 flP(Xl>pl,...,Xl+...+Xk>pk,Xl+••.+Xk+l>pk+l)=fk+  which means that (7.4) holds for i  =  Therefore, the proposition is proved.  k + 1. Then by induction, (7.4) holds for all k.  D  Remark: The significance of the above proposition is that it gives a closed-form rep resentation for the dynamic equation in Curry’s formulation. This new representation makes it clear on that the basic idea behind Littlewood’s result remains valid for the multi-fare case when demands are independent. It may also shed new light on a further extension to the case of certain dependent demands.  Chapter 7. A Note On Three Models for Multi-fare Seat Allocation Problem  7.6  Summary  191  On Three Model for Multi-fare Seat Allocation Problem  This chapter provides a note on the seat allocation problem for multi-fare single-leg flights with independent demands. It proves that three existing models for the seat allocation problem for multi-fare flights with independent demands all have equivalent optimality conditions, which clarifies the issue on which model has the computational advantage over the others. This equivalency result may also shed new light on the seat allocation for multi-fare flight with dependent demands.  Chapter 8  Summary of the Thesis and Future Directions  8.1  Summary of the Thesis  This dissertation studies the pricing problem for perishable inventories for a capacity constrained monopolist. The main innovation is to explicitly incorporate the use of artificial restrictions. And the main goal is to develop useful pricing models for firms like airlines and hotels. Chapter 2 begins with a detailed discussion on a pricing model due to Wilson, which leads to the notion of rationing sales at lower prices. I then present a monopoly pricing model for perishable inventories by using one type of restriction. A pricing policy is called a primary policy if the prices for restricted units are not larger than the prices for unrestricted units. Under the following conditions: (1) the impact of restriction on the demand market has the property that as price increases, the percentage of consumers who can accommodate the restriction is decreasing; (2) the demand function is a nonincreasing step function; (3) the firm only uses primary policies; and (4) the firm sells the product at prices in an increasing order, then I prove that the monopolist’s pricing problem can be formulated as a nonlinear mathematical programming problem with three constraints. I further show that the monopolist will have incentives to use the restriction and can maximize its revenue by using no more than three prices. I also apply the model to airline fare pricing. I demonstrate that if the demand market is divided into leisure travellers and business travellers, then the condition imposed on the impact of restriction  192  Chapter 8. Summary of the Thesis and Future Directions  193  on demand market is equivalent to saying that leisure travellers are more price sensitive than the business travellers. Chapter 3 further discusses the same pricing problem in a more general context. I first extend the class of pricing policies so that it is possible for the firm to place the restriction at any price level. I prove a general optimality theorem, which shows that any optimal pricing policies in the class of primary policies remain optimal in the general class of policies. This implies that, for perishable inventories, if a monopolist decides to use an artificial restriction, an optimal pricing practice is to sell restricted units at prices that are lower than the prices for unrestricted units. Motivated by the current practice of airlines, this chapter also investigates the issue of whether all active prices in an optimal primary policy can be made available at the same time. It is shown that: (1) as long as active restricted prices are different from active unrestricted prices, then the optimal pricing 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 is  equal to an active unrestricted price, then the firm will be better off by selling restricted units first at this price rather than making both restricted units and unrestricted units available at the same time. Chapter 4 extends the discussion to the case of using two types of restrictions. Three cases 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 by offering three types of product. For the case of two general restrictions, I prove that under certain additional conditions, there exists optimal policies that consist of no more than five prices by utilizing four types of product. Chapter 5 presents an application to the airline fare pricing problem in the presence of membership and product restriction. In this chapter, I demonstrate that these pricing  Chapter 8. Summary of the Thesis and Future Directions  194  models developed in previous chapters are useful tools for airlines to identify the op erating environment for each one of three common membership privileges: (1) cheaper restricted fares only; (2) cheaper unrestricted fares only; and (3) cheaper restricted fares and cheaper unrestricted fares. Examples are used to provide further support. Chapter 6 explores the seat allocation problem in the presence of another airline. The discussion is limited to two fare flights. It is shown that under the proportional splitting rule, there exists an equilibrium pair of booking limits for the discount fare such that each airline will protect the same number of seats for the full fare regardless of their respective capacities. Under the equal splitting rule, it is shown that, for deterministic demands, there exists an equilibrium such that each airline will protect enough seats to capture half of the demand market for the full fare. Chapter 7 discusses three models for the multi-fare seat allocation problem with independent random demands. We have shown that the optimality conditions from three models are in fact equivalent. This clarifies an issue of which method has a computational advantage over the other two. It also provides an integrated approach to the multi-fare seat allocation problem, which could be an important step toward the development of a seat allocation model for multi-fare flights with dependent demands.  8.2 8.2.1  Future Directions On Seat Allocation Problems  A key theoretical work that needs to be done along this direction is to solve the seat al location problem for single-leg flights with dependent demands. There are many possible approaches. 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 dependent demands could be very difficult because a general dependency structure among multiple  Chapter 8. Summary of the Thesis and Future Directions  195  random variables is too complicated. A more delicate approach is to introduce more structural dependency among these random demands for fare classes. Recall that the optimality conditions in Brumelle-McGiIl model (refer to Section 6.2) actually can be rewritten as follows: (8.1)  fk+l=fkP(Xl++Xk>pkIXl++Xk_l>pk_l),fork=1,.”,n—1.  This is closely related to the original Littlewood formula and the result due to Brumelle et al. (1990, [37]) for two-fare flights with dependent demands. Note that Wk  X + 1 .  .  should be interpreted as the total demand for fare class k. As long as {Xk} 1 is a sequence of independent non-negative random variables, the resulting sequence {Wk} 1 is a Markovian sequence. We should notice that working directly with the stochastic sequence {Wk} 1 is consistent with the traditional view of economic demand function with uncertainties. It is also possible to introduce a more dedicate dependency structure so 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 in the presence of connecting passengers is workable. A joint optimization model should be possible. Another key problem is to develop optimal seat allocation models for multi-leg flights. A concrete theoretical model that handles random demands and multiple fares remains to be a very major challenge in yield management research. 8.2.2  On Seat Allocation Games  The discussion in Section 6.3 is just a beginning of research along this direction. In my view, this should be a very fruitful area of research in the near future. It is a very important step to push the field of yield management to a more realistic situation. The main issues are:  Chapter 8. Summary of the Thesis and Future Directions  196  • Under the proportional splitting rule, (a) find the necessary and sufficient conditions that lead to the existence of equilibrium booking limits; and (b) find the conditions that lead to equilibria of mixed strategies. • Under the equal splitting rule, (a) solves the seat allocation problem under random demands; and (b) characterizes the equilibrium protection levels or the equilibrium booking strategies. It is also important to discuss the seat allocation game for multiple fares. Note that for the case of two fares, each airline has oniy one strategic variable, either the booking limit for low fare, or the protection level for high fare. But when each airline operates a flight with n fares, then each airline will have n  —  1 strategic variables. This gain opens  up two possible ways of dealing with this game, which will be the same when n way is from the static point of view  —  treating all these n  —  =  2. One  1 variables as one strategic  vector. This is equivalent to a one-shot game. Another is from the dynamic point of view  —  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 Restrictions  This thesis explores the pricing problem by using restrictions oniy for a special kind of product, that is, it is perishable, and not storable for consumers with fixed supply. From a different perspective, we may say that the results on this thesis show that, at least for a non-storable perishable products, a carefully designed restriction can be a very effective method to practice price discrimination. So this leads to the following general question: • What are the product characteristics required to induce firms to intoduce restrictions as a mechanism of price discriminationq  Chapter 8. Summary of the Thesis and Future Directions  197  On the other hand, firms have complete freedom to use or not to use restrictions during the pricing process. This leads to a very interesting issue: suppose a firm can design the restriction, so the question is: • Given the market demand structure for the unrestricted product, what is the mo nopolist’s most desirable restriction To put the above question in a more analytical way, recall that in Section 2.4, the aim is to characterize R(TYi, of), where  —  argmaxsR(m, a).  So the above question is equivalent to asking to find an optimal a-function: äf(p) such that  = 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, this may be termed the problem of optimal design of restrictions.  Recall that the pricing models developed in this thesis have a common assumption that the monopolist offers one price at a time. Even though Section 3.4 argues that there always exists an optimal policy that will realize the projected optimal revenue value from the BL-model, but the question still remains when the firm makes all allocated quantities available at the same time, but we do not have a formal pricing model that makes all prices both restricted and unrestricted  —  available at the same time. Clearly, we must deal  with a class of pricing policies that is different from the class of general policies discussed in Chapter 3. And questions related to the corresponding optimal pricing problem that are of great interests include:  Chapter 8. Summary of the Thesis and Future Directions  198  • What is the formulation for the monopolist’s pricing problem? • l4 hat are the optimal policies? 7 • What can we say about any relationship between the optimal policies and the optimal polices 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 that needs to be addressed in the near future. Since the pricing problem in the context of a 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 allocation game. It is possible to approach this problem by treating it as a static game, since in this case each firm will throw out an allocation plan at the table and the let the market decide the outcome. Some key questions are: • What are the equilibrium pricing strategies? • What is the role of capacity? 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