Contract optimization with front-end fare discounts for airline corporate deals

Transcription

1 Transportaton Research Part E xxx (2006) xxx xxx Contract optmzaton wth front-end fare dscounts for arlne corporate deals Julan Pachon a, Murat Erkoc b, *, Eleftheros Iakovou b a Navtare, Inc., 9130 Jollyvlle Road, Sute 100, Austn, TX 78759, Unted States b Department of Industral Engneerng, Unversty of Mam, P.O. Box , Coral Gables, FL 33124, Unted States Receved 28 June 2005; receved n revsed form 29 November 2005; accepted 5 December 2005 Abstract Ths paper develops a non-lnear programmng model to desgn optmal corporate contracts for arlnes stpulatng front-end dscounts for all nets, whch are defned by combnaton of routes, cabn types, and fare classes. The arlne s proft s modeled usng a multnomal logt functon that captures the clent s choce behavor n a compettve market. Alternatve formulatons are employed to nvestgate the mpact of prce elastcty, demand, and competton on optmal dscountng polces. A case study nvolvng a major carrer s presented to demonstrate the model. The results ndcate that arlnes can ncrease revenues sgnfcantly by optmzng corporate contracts usng the suggested model. Ó 2006 Elsever Ltd. All rghts reserved. Keywords: Arlne corporate contracts; Revenue management; Busness travel; Non-lnear programmng; Lagrangan relaxaton; Multnomal logt functon 1. Introducton Large corporatons wth hgh levels of travelng needs often contract wth one or more arlnes to receve travel dscounts and other ncentves. On the other hand, arlnes are constantly lookng to ncrease ther corporate travel as t accounts for the majorty of ther operatng earnngs. Whle corporate travel accounts for approxmately 55% of total ar travel passengers (Granados et al., 2005), accordng to they are responsble for more than two-thrds of arlne revenues. In the past, corporate deals were a prvlege only enjoyed by the major arlnes; today however, the entry of low-cost carrers nto more busness markets and ther ncreased efforts to penetrate the corporate travel market s, more than ever, promptng companes to reconsder ther suppler portfolos. Though such arlnes prmarly are utlzed through spotbuys, a growng number of companes are consderng them as secondary preferred carrers. Top executves at low-cost carrers happly pont out that more corporatons proactvely are seekng ways to nclude dscount * Correspondng author. Tel.: ; fax: E-mal addresses: [email protected] (J. Pachon), [email protected] (M. Erkoc), [email protected] (E. Iakovou) /$ - see front matter Ó 2006 Elsever Ltd. All rghts reserved. do: /j.tre

2 2 J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx arlnes n ther programs, especally as these carrers grow at several tmes the rate of legacy arlnes and nvade major carrer hubs and busness markets around the US. Ths trend s further exacerbated by the emergence of e-commerce that has enabled companes to have access to a larger number of low-cost smaller arlnes wth lower overhead and transacton costs. In ths new era of ncreased competton and strngent customer servce requrements arlnes are compelled to provde compettve yet cost effcent deals to ther busness customers. In ths envronment where the Internet has been movng from smply an nformaton medum to a medum of transactons and managng busness travel (Smth et al., 2001), effcent decson support systems and tools that can help develop optmal polces are n hgh demand. Arlnes that recognze the new challenges for the corporate travel sector must be wllng to develop creatve programs or offer entcements so as to at least hold onto ther busness. Consequently, robust decson support systems and tools are needed to fully capture the emergng busness models n the corporate ar travel sector. Corporate dealng s generally a functon wthn the sales dvson of an arlne. Account managers are assgned ether to a sngle major corporate account or to multple medum sze and small accounts. The account manager s n charge of settng and negotatng the corporate contract, trackng the contract performance, and managng the relatonshp wth the clent s travel management team. A contract between the arlne and a corporate customer s defned by the set of dscounts/ncentves that the arlne offers to the customer for each net (combnaton of route, cabn, and fare class) that the corporaton fles. The major challenges faced by account managers nclude the lack of data vsblty needed for makng ntellgent decsons, the lack of decson support systems that would allow them to evaluate multple negotaton strateges n a short perod of tme, and the dffculty n convncng corporate travelers to book on the preferred carrer when a compettor offers an equal or superor product at an equal or lower prce. Furthermore, today, n the arlne ndustry, not enough emphass s placed on the ntegraton of the systems used to support corporate contract dealng. Many account managers stll rely on computer prntouts and manually generated charts to make corporate contract decsons. The ndustry lacks ntegrated decson support systems that cover the entre spectrum of corporate dealng ncludng customer value segmentaton, contract desgn, contract optmzaton, and performance management. Such ntegrated systems should address the followng busness opportuntes for arlnes: () understand the value of each corporate customer s busness to the arlne, () ncrease contract complance to drve more predctable revenue, and () mprove margns through decson support at the pont of contract desgn. An adequate ntegrated decson support system would further help an arlne s sales force by optmzng complex contracts nvolvng multple regons, maxmzng margn contrbutons, and montorng current contract performance. It should use forecasts of corporate passenger demand to determne optmal dscount polces that maxmze expected contract performance whle mnmzng rsk. Such a system must consder three basc modules: customer segmentaton, contract optmzaton, and performance management (see Fg. 1). The customer segmentaton module dentfes customers who provde predctable and rreplaceable busnesses to ensure maxmum return on sales efforts consderng customers travel polces and the compettve envronment. On the other hand, the contract optmzaton module takes nto account the hstorcal travel for each corporaton, t forecasts future demand, and then bulds market response functons to estmate the tradeoff Customer Segmentaton Contract Optmzaton Performance Management -Identfes predctable & rreplaceable busness -Segregate customers n dfferent value buckets -Generates optmal dscounts for each net -Comply wth RM, sales and contractual gudelnes -Track contract complance -Identfy underperformng and over-performng markets Fg. 1. The Arlne Contract Soluton contans three man modules to handle contract desgn and trackng.

3 between prce, market share, and demand. Once these functons are developed, an optmzaton engne would generate optmal dscounts for each route cabn class combnaton n each corporate contract based on revenue management and sales gudelnes. Lastly, the performance management module mproves contract complance aganst targets and objectves by dentfyng deals that are under-performng or over-performng. Ths module provdes root cause analyses of ndvdual and team performance wthn the sales organzaton. In ths paper we focus our analyss on the contract optmzaton module. The purpose of ths paper s to hghlght the mportance of a decson support system n desgnng arlne corporate contracts and demonstrate the economcal benefts the arlnes are leavng on the table by not applyng optmzaton n ths feld. Our study examnes optmal contract desgn polces that nclude offerng effcent front-end travel dscount deals to corporate customers at each net. We ntroduce a proft maxmzng non-lnear mathematcal programmng model that determnes optmal prce dscounts for a corporate account over all servced nets gven predetermned corporate requrements/restrctons and arlne revenue management and sales gudelnes. The non-lnearty of the objectve functon s due to market share mappng that s modeled by a multnomal logt (MNL) functon. The MNL functon captures successfully the corporate choce behavor for all nets takng nto account the mpact of the competton. The proposed model can also be used to nvestgate the mpact of prce elastctes, demand structure, costs, servce qualty, and arlne competton on an arlne s optmal corporate dscount levels. Our results reveal that the nterplay between optmal dscount polces at dfferent nets s more sgnfcant n the context of corporate travel n that optmal dscount decsons at all nets are senstve to changes n demand and cost parameters n one net. The rest of the paper s organzed as follows. Next secton revews the related lterature. In Secton 3, we provde a descrpton of the bascs of arlne corporate contract deals. Next, we ntroduce a market response model that captures the corporate response to the arlne s offers n Secton 4. In Secton 5 we propose an optmzaton-based model wth four dfferent nstances: frst we nvestgate the unconstraned problem and then three dfferent nstances wth three dstnct corporate restrctons as practced today n ndustry. For each of these nstances we present the detals of the soluton methodology and dscuss extensvely the obtaned manageral nsghts. In Secton 6, we present a case study nvolvng a major nternatonal carrer. Secton 7 concludes the paper wth summary and potental areas of future research. 2. Lterature revew J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx 3 Most of the research n compettve arlne prce optmzaton has focused on defnng optmal prces for B2C markets (busness to customer) and consstently most revenue management systems are desgned n ths context. Revenue Management, the process by whch the dscount fares are allocated to scheduled flghts for the purpose of balancng demand and ncreasng revenues, has been adopted by most arlnes snce the deregulaton of the arlne ndustry n 1978 (Pfefer, 1989). Ths B2C prcng strategy has developed n part as a response by major carrers to prce competton from low-cost arlnes. By offerng a lmted number of seats at dscount fares, arlnes can be compettve n prce wth low-cost carrers and mght be able to fll otherwse empty seats (Belobaba, 1987). A comprehensve taxonomy of revenue management problems s presented by Weatherford and Bodly (1992). For a recent overvew of prcng models for revenue management we refer the reader to Barnhart et al. (2003) and Btran and Caldentey (2003). Several papers that study compettve prcng n the arlne ndustry ncorporate an explct modelng of customer choce behavor as a functon of prce and/or servce qualty. Andersson (1998) dscusses the applcaton of logt choce models to estmate buy-up (buyng a hgher fare after lower fares are no more avalable) behavors of passengers at one of Scandnavan Arlne Systems hubs. Algers and Besser (2001) study customer choce probabltes for flghts and fare classes usng revealed and stated passenger preference data. A related study on ths subject s provded by Tallur and van Ryzn (2004) where the authors estmate multnomal logt (MNL) choce probabltes for fare classes n a sngle-leg flght va the maxmum lkelhood method. The authors nvestgate optmal polces for offerng fare products at each pont n tme by solvng a control problem. Suzuk et al. (2004) also employ MNL probabltes to model passenger choce behavors to evaluate the mpact of arfares n a regon on arport leakages. They nvestgate optmal arfare polces for an arlne at a certan arport va smulaton. Other related papers that study arlne prcng strateges nclude Toh et al. (1986), Bennett and Boyer (1990), Dresner and Tretheway (1992), Botmer (1996), Zhang and Cooper (2004) and Netessne and Shumsky (2005).

4 4 J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx All the aforementoned papers consder a general B2C framework. Although some of the results from these research efforts are stll vald for B2B (busness to busness) settngs, n most part, they fall short of ncorporatng unque characterstcs and requrements of corporate travelng n ther models. Surprsngly, there are only a few papers n the lterature that study corporate deals and servces n the arlne ndustry even though busness travelers form the largest part of the most proftable sectors of the ar travel market. To our knowledge all of the papers n ths venue analyze the busness travel problem from the perspectve of the busness traveler (buyer) rather than the arlne company (suppler). In a recent paper, Suzuk and Walter (2001) study the effectve use of frequent flyer programs offered by almost all arlners that can enable sgnfcant ar travel cost savngs for a frm. The authors propose an nteger programmng model that maxmzes the cost savngs for the frm under a determnstc settng where the frm s all future travelng plans are known a pror. Later, Suzuk (2002) extends ths model to the case where the busness travelers future trp plans are stochastc. He proposes a heurstc approach that can be used to determne whch arlne to choose for each trp and when to redeem the earned mles. The arlne selecton problem s also addressed by Degraeve et al. (2004) from the perspectve of the frm. Instead of frequent fler programs, ther work consders regmes such as front-end dscounts, absolute-volume dscounts, and market share dscounts as the base of the frm s arlne selecton problem. In ths approach, the authors develop a mxed nteger programmng model that ncorporates the actvty based costng herarchy allowng for varable costs at dfferent levels n the organzaton. They present a case study conducted at Alcatel Bell. In contrast to the B2B papers dscussed above, our study consders a contract optmzaton problem from the perspectve of the arlne company that offers front-end dscounts for ts corporate accounts across multple legs. The proposed approach s novel n that t ntegrates both the model of the corporate customer and the competton nto the optmzaton problem through revenue management and sales gudelnes, corporate specfc restrctons, and market response mappng. The proposed model combnes the dscountng decsons at all nets served by the arlne n a monolthcal framework. 3. Bascs of corporate contracts In the contractng process, corporatons submt RFPs (request for proposal) to the arlne, descrbng the specfc routes, cabn, and classes they generally fly, and the arlne n return offers ncentves to try to wn ther busness. These ncentves take the form of front-end fare dscounts (dscounts that are offered at the bookng tme), back-end fare dscounts (dscounts that are offered at the end of the lfe of the contract provdng the customer ht a target volume), or addtonal benefts (VIP lodges, lmos, upgrades, etc). For small and mdszed corporatons, arlnes may offer dfferent programs that allow the corporatons to earn ponts n proporton to travel mles n addton to the personal frequent flyer rewards gven to the travelng company employees. These ponts can then be redeemed wth round trps tckets worldwde n frst class, busness or coach, VIP lodges passes and membershps, and upgrades. Once a contract between the arlne and the corporaton s n place, t s generally adjusted and renewed on an annual bass. In the current corporate dealng process at most major arlnes, account managers create dfferent scenaros by predctng the behavor of the corporate customer. Rules of thumb, market competton prces, and customer requests generally gude the dscounts that they offer. Currently, ths process does not account for seasonal travelng behavors, specfc route cost, and contrbuton margn for each flght segment. Once the account managers determne what they consder the rght deal, the draft contract s sent for approval to hgher echelon decson makers n the arlne. Once t s approved, the sales team negotates the fnal detals wth the corporate travel managers. When the contract s n place, the sales team s responsble for montorng the performance of the deal; ths ncludes measurng contract complance aganst targets and objectves and dentfyng deals that are under-performng or over-performng. For the arlne to gan and keep the corporaton as a clent, t should propose a deal that satsfes both ts own and the corporaton s targets and objectves. From the arlne s pont of vew, deals are generally evaluated usng the metrcs that nclude volume, cost of deal, fnal revenue, and contrbuton margn volume measures the total number of passengers flyng on nets. Cost of deal ncludes the dscounts and commsson/bookng costs and t s expressed as a percentage of the contract gross revenue. Fnal revenue represents the total revenues that the arlne receves from a corporate contract. It s computed by subtractng dscount and the bookng payments/commsson costs from the gross

5 revenue (publshed fare tmes volume). Fnally, contrbuton margn s the money earned by the arlne after subtractng the route cost (.e., fnal revenue mnus route cost). The route cost s comprsed of the ncremental and dedcated costs. The ncremental costs are a functon of volume ncludng dealng costs (e.g. sales force salares), food, and a porton of fuel expenses. The dedcated route costs are the fxed costs that encompass: () the cost to fly the arplane; () the termnal costs such as check-n, baggage handlng, gate fees, etc.; () the overhead costs such as corporate costs and other support staff (mantenance, jantors, admn, etc.), and (v) the general uplft costs such as FFPs, call centers, etc. On the other hand, from the corporaton s pont of vew, an attractve deal promses protecton of top routes (routes wth hgh volume), substantal savngs n travel cost, and hgh average dscounts. Bascally, top routes are those wth large volume, where the customer perceves the greatest value of a corporate deal. Generally, prospectve clents defne a mnmum dscount n the top routes and the arlne plans around them. Optmal desgn of corporate contracts can be developed through an optmzaton-based ntegrated corporate contract decson support system. The ntegrated framework components nclude three modules: () demand forecastng, () market response model, and () prce optmzaton model. The demand forecastng module apples seasonal forecastng algorthms (e.g. Holt Wnters) to acheve a hghly accurate corporaton seasonal flyng behavor forecast on whch real-tme contract performance mght be predcted. Improvng contract performance clearly begns wth understandng future demand. The travel behavor of a corporaton n each route s drven by the dfferent busnesses they have, the geographes where those busnesses take place, the number of employees, number of ther subsdares, the fnancal status of the corporaton, etc. In few organzatons an ndvdual or a department s n charge of bookng the travel for the entre company, whle n others ndvdual employees book ther own travel. Travel polces n some corporatons constran the bookng choces to ether the lowest rate, or to any rate that falls wthn a range determned by the corporaton s management. In determnng optmal dscounts, the arlne must frst understand the travel pattern of the corporaton n each contracted net. Wth the help of hstorcal data, and standard forecastng methods, hghly accurate seasonal corporaton flyng behavor can be obtaned. A wde varety of forecastng methods are avalable to arlne management (see Makrdaks et al., 1998). Account managers generally mantan a close busness relatonshp wth the travel department at the corporaton, allowng them to get addtonal nformaton and nsghts about crtcal travel behavor changes that may not be captured wth hstorcal data. In ths paper, the man focus of our analyss s rather on the market response and optmzaton modules. 4. The market response model J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx 5 Ths model generates corporate-specfc arlne market share functons that capture the arlne market share response to changes n dscount levels based on the state of the competton, customer preferences, and hstorcal prce data. Any market share model must capture the prce elastcty of the demand, whch s drectly related to the possbltes of substtuton for that product. A relatvely large number of substtutes wll mply a hgh prce elastcty, whereas a lack of substtutes wll lkely force demand to become more rgd so that demand for the product may become relatvely nelastc. Qute often, many carrers compete wth each other on the same route, provdng a case for ntra-modal substtuton. Although travelers also face nter-modal substtuton, n corporate travel, usually ntra-modal substtuton s consdered much more sgnfcant n affectng the demand for a partcular arlne. Predctably, the market share of a gven arlne for a corporaton s determned by the dscount and qualty of servce that the arlne offers, as well as the dscount and qualty of servce offered by the competng arlnes. A typcal form of a market response curve s shown n Fg. 2. When the dscount offered by the arlne exceeds the dscounts offered by the competton, the arlne expects to get a hgher market share. There s a mnmum and maxmum market share that the arlne expects as n many corporatons the employees choose the arlne at the tme of bookng. Provded that the prce comples wth the travel polcy of the corporaton, the employee usually selects hs/her favorte arlne or the one wth whch he/she has a frequent flyer program. To model the market share functon, we employ the multnomal logt (MNL) model. The MNL model s wdely used n travel demand marketng and forecastng n both lterature and practce. Snce the ntra-modal substtuton approach n general satsfes the ndependence from rrelevant alternatves property (Ben-Akva

6 6 J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx Market Share Max. Market Share Mn. Market Share Comp 1 Comp. 2 % Dscount Dscount dscount Dscount Max dscount Fg. 2. Corporate travel market response functon. and Lerman, 1985), the MNL model s commonly used n modelng the market share among arlnes as a functon of prce and servce qualty. MNL models requre that each arlne s market share s non-negatve and the sum of market shares of all arlnes on a gven net s unty. As such they satsfy the logcal-consstency requrements and can effcently capture the S-shape market share curve. The accuracy of the model depends on the avalable data and estmaton technques employed. The parameters of the MNL model are generally computed based on hstorcal data. For new contracts, the hstorcal data s replaced by data of corporatons of smlar sze, ndustry, and busness orentaton. Several estmaton approaches such as maxmum lkelhood, least squares, regresson, and ntegral calculus are dscussed n ths context by Ben-Akva and Lerman (1985) and Hsu and Wlcox (2000). An excellent example n the context of arlne operatons s provded by Suzuk et al. (2001), where the authors use the standard least squares method to estmate the MNL parameters to model an arlne company s market share. Tran (2003) dscusses the estmaton of the choce probabltes for varous dscrete-choce models usng smulaton-based methods. Although the accuracy of the estmates s mportant, the man beneft of the MNL models s that they provde meanngful manageral nsghts regardng market-share elastctes and sales-volume elastctes (Basuroy and Nguyen, 1998). In ths study, as we focus our analyss on the mpact of arfares dscounts on arlne corporate contract metrcs, our model assumes that all other parameters related to customer servce qualty are exogenous. In ths context, dscount s defned as the percentage reducton from the publshed fare. Let d k denote the dscount offered by arlne k for net. Then, by defnton, 0 6 d k < 1. Now, let A represent the set of arlnes that have a contract wth the corporaton for net. Gven dscount levels of all arlnes n A, the market share for arlne k on net s determned by the market response functon, f k as follows: b k f k ¼ e þc ln 1 1 d k ð1þ P b j þc ln 1 1 d j2a e j In the foregong functon, b k s used to model all non-arfare factors such as customer servce qualty, passenger trp length, servce frequency, safety records, and ntangble non-monetary parameters. Certanly, the non-arfare factors can dffer for each arlne/net/corporaton combnaton. Snce the market response functon ncludes the attrbutes of all arlnes, any arlne usng ths functon needs hstorcal data or expert opnon to estmate not only ts own parameters but also attrbutes of other arlnes operaton on the same nets. In the absence of suffcent hstorcal data and expert opnon, the other arlnes market share parameters can be aggregated usng the clent s total demand nformaton and the arlne s own performance data. 1 1 We note that n ths case, the MNL model would be reduced to a bnary logt model where flyng wth other arlnes s aggregated to a sngle choce for each net.

7 J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx 7 Fg. 3. Market response curve for dfferent values of c. Clearly, when all the arlnes servng the net select the same dscount level, ther market share wll be manly determned by the non-arfare factors. The mpact of the dscount n the market response functon s captured by c. We refer to ths parameter as the prce elastcty factor, whch s assumed to be unform across all nets. Ths assumpton s justfed as we are dealng wth busness and not lesure travel. Hgher values for c mply more drastc customer reactons to prce dscounts as llustrated n Fg. 3. When c 1, the market response functon s convex n the dscount level. The S-shape becomes more apparent as c ncreases beyond 1. Fg. 3 exemplfes the market share functon for an arlne wth two compettors employng dscount levels of 25% and 30%. Observe that below the compettors dscount levels, the arlne loses more market share as c ncreases. On the other sde, for any dscount rate above the compettors rates, the arlne s market share ncreases n c. The stuaton c!1mples that the customer s merely a prce-taker and thus, by outbddng the hghest competton dscount the arlne can get all the demand. In contrast to B2C settngs, both c 1 and c!1cases are unlkely under B2B contracts. In general, c s strctly greater than one and ts value s usually hgher for small to mdsze busnesses compared to large corporatons. As our analyss tackles the contract desgn problem for a sngle arlne, for brevty, we denote the other arlnes dscount weghts as ¼ X j2a =k b j þc ln 1 1 d e j We refer to as the collectve appeal of other arlnes on net. Clearly, as ncreases, the arlne s market share decreases. Addtonally, we drop the arlne ndex from the functon and re-wrte the market share functon for net as follows: þc ln 1 f ¼ eb 1 d ð3þ þ e b þc ln 1 1 d ð2þ 5. Contract dscount optmzaton The contract dscount optmzaton model determnes the optmal dscount rate for each net. Ths nvolves analyzng hundreds of thousands of combnatons to select those that are optmally suted for a gven cus-

8 8 J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx tomer s contract at a specfc pont n tme. The objectve of the corporate contract optmzaton problem s to set an optmal dscount level for each net based on the forecasted demand, the market response mappng, revenue management gudelnes, and corporate specfc restrctons. The arlne company s goal s to maxmze total profts by choosng the approprate dscounts for each net. The proft for the arlne from net can be computed usng the followng functon: P ¼ D f ðp ð1 d Þ ðcc þ RC ÞÞ ð4þ In the proft functon, D and f denote the forecasted demand of the corporaton and the market share of the arlne on net, respectvely. P, CC,andRC are the publshed arfare, the route cost per passenger, and the commsson and bookng cost per tcket on net respectvely. Let N denote the set of all nets ncluded n the contract. Then the arlne ams to maxmze P 2N P. Note that n ths model the total travel demand on any net by the corporate s assumed to be constant and not a functon of the offered dscounts. Ths assumpton s justfed by the fact that n corporate travel the demand s created by the corporaton travel needs (e.g. the projects and sales that they are actvely workng on dfferent regons) not prce. As opposed to lesure travel where prce motvate more customer to fly, n the busness world for a typcal medum to large sze corporaton the total travel does not present hgh varatons based on prce n that the corporaton wll not usually send addtonal employees for sales and projects just because prce dscounts are offered. The prce manly affects the choce of carrer The unconstraned case We frst analyze the maxmzaton problem for each net separately gnorng the constrants mposed by revenue management gudelnes and corporate restrctons. Solutons to the unconstraned problems can be used as a benchmark aganst the general case where dscountng decsons must be made jontly due to constrants. In ths case, the problem can be easly decomposed nto nets snce the proft functons for all nets are ndependent from each other. To determne the optmal dscountng polcy we frst establsh that the proft functon for each net has a unque maxmzer by showng that the proft functon s unmodal n d 2 [0,1). 2 Proposton 1. Gven c > 1, the proft functon P s unmodal for d 2 [0,1). Furthermore, 1. P monotoncally decreases n [0,1) and thus, the optmal dscount s zero (.e., d ¼ 0) f and only f ½P ðc 1Þ cðrc þ CC ÞŠ P e b < 0 ð5þ 2. Otherwse, the unque maxmzer, d, satsfes the followng frst order optmalty condton: f 0 P ð1 d f Þ ðrc þ CC Þ ¼ P. ð6þ The most mportant contrbuton of ths result s that the optmal dscount level can be determned usng the frst order optmalty condtons. Although we do not have a closed form soluton for the optmal amount, ths can be easly found through a smple lne search usng (6). In addton, Proposton 1 provdes us wth other useful nsghts and observatons regardng optmal dscountng polces; these are summarzed below: If the prce elastcty s not suffcently large (e.g. c 6 1) then t s never preferable for the arlne to offer any dscount to the corporaton. The arlne company should not offer any dscount f the margnal proft on a net s negatve or too small and/or ts non-arfare related servces and qualty (b ) are suffcently hgh. Under an unconstraned settng, the optmal (front-end) dscount s ndependent of the total demand for the net. It rather depends on the market share, whch s modeled n terms of percentages, publshed arfare, and costs. The optmal dscount rate on net ncreases n P and, and decreases n CC, RC, and b. 2 All proofs are provded n Appendx A.

9 The mpact of prce elastcty on the optmal dscount rate s ambguous for c > 1. In general, the optmal dscount rate ncreases n c f the collectve appeal of other arlnes s hgh. It may decrease when the opposte s true. The frst two of the above observatons follow drectly from (5). If the prce elastcty s low the dscounts wll play a small role n the arlne s market share. Specfcally, when the elastcty s less than or equal to one, the margnal ncrease n market share cannot acheve to offset the margnal loss on revenues. Subsequently, the dscounts cannot be used to mprove the arlne s earnngs. Even for relatvely hgher values of c, dscounts may not be preferable due to small revenue margns. If the arlne can persuade travelers manly through ts non-arfare related efforts, dscounts may also become unnecessary. The last three of the observatons are the drect byproduct of the frst order optmalty condton provded by (6). Frst, observe that snce total demand for a certan net does not change as a functon of the dscount rates, the arlne s decsons are manly determned by the market share. Second, an ntutve observaton s that hgher proft margns ncrease the arlne s market power and hence allow for hgher dscount rates. On the other hand, hgher can also motvate hgher dscount rates. In ths case the arlne wll be compelled to ncrease ts dscount rate to protect ts share n the market. Naturally, there wll be a negatve correlaton between the arlne s non-arfare related efforts and the dscount rates t s wllng to offer. Thrd, we observe from (6) the mpact of the prce elastcty on the optmal dscount rate s context specfc. If the competton offers relatvely hgher dscount rates and/or better non-arfare related servces, the market share of the arlne wll declne as the prce elastcty ncreases. Ths pattern s a straghtforward consequence of the multnomal logt functon. To offset the market share loss, the arlne needs to ncrease ts dscount offer under hgher values of c Customer servce constrants and revenue management gudelnes Although the unconstraned dscount contract model provdes valuable nsghts for optmal arlne dscountng strateges, n the real-world dscount decsons have constrants mposed by revenue management gudelnes and/or corporate restrctons. Revenue management gudelnes are bascally the route level dscount restrctons. Contract gudelnes bound the mnmum contrbuton margn and maxmum cost of dealng of the entre contract based on the customer segment. Corporate specfc restrctons may nclude dscount lmts for the top routes, maxmum devaton from current levels of dscount, mnmum average dscount, mnmum weghted average, and mnmum total corporate savngs. These restrctons are modeled through the so-called partcpaton constrant(s) n the optmzaton problem. Any vable contract must satsfy these constrants Total-demand-averaged (TDA) dscount constrant Below, a mathematcal formulaton for the constraned optmzaton problem s provded: X z ¼ Max P. d s.t. d l 6 d 6 d u 8 2 N ð8þ P P 2ND d P ^d ð9þ 2N D J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx 9 The objectve functon (7) maxmzes the sum of contrbutons for all nets n the contract. Constrants (8) bound the dscounts for each net based on revenue management gudelnes, contract gudelnes, and/or corporate gudelnes. Eq. (9) tes all nets n the optmzaton model by restrctng the weghted average of all new dscounts to be greater than a certan threshold denoted by ^d. Bascally, the rght-hand sde of the nequalty gves the mnmum average dscount per corporate flght acceptable by the customer. The threshold value can be determned by the weghted average of the dscounts n the current contract wth the corporaton or as a result of negotatons wth the corporaton f the contract s new. Clearly, the foregong problem s a non-lnear programmng (NLP) model wth a non-lnear objectve functon and lnear constrants. Wthout constrant (9), the model can be decomposed to nets easly and solved ð7þ

10 10 J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx usng (6) and bounds gven n (8). If ths soluton satsfes constrant (9) then most of the conclusons for the unconstraned case stay vald. Otherwse, t s straghtforward to conclude that constrant (9) must be bndng at optmalty. In ths case, the optmal soluton for small to mdsze problems can be obtaned by commercally avalable NLP solvers. We note that the optmzaton model s a separable non-lnear program and as such solutons to large sze problems can be approxmated usng Separable Programmng. In a separable non-lnear program, the objectve functon and constrants can be expressed as the sum of functons, each nvolvng only one varable. Clearly, the objectve functon n (7) has ths property. Bascally, n Separable Programmng the non-lnear objectve functon s approxmated to a pecewse lnear functon by parttonng each net proft functon nto small ntervals that are determned by grd ponts (see Bazaraa et al., 1993 for detals). Snce the proft functon for each net s unmodal, the new model can be then solved usng the smplex method. Close approxmatons can be acheved f the number of grd ponts (and thus, the number of varables n the LP) s suffcently hgh. Alternatvely, the constrant n (9) can be moved to the objectve functon and then the problem could be solved va Lagrangan relaxaton/decomposton. Through Lagrangan relaxaton, we can obtan valuable manageral nsghts as dscussed below. For the followng proposton, recall that d denotes the optmal dscount amount on net under the unconstraned case and let d o represent the optmal dscount amount for net n the constraned model. Proposton 2. Followng are true for the optmzaton model gven n (7) () For any net, f d < d u then d o P d. Otherwse, do ¼ d u. () At any optmal soluton, for any gven net, f d o < ^d then optmal dscount amounts on all nets are nondecreasng n total demand on net (D ). However, they are non-ncreasng n D f d o > ^d. () For any gven net, where P ð1 d o Þ P CC þ RC ; f d o > ^d then the total optmal arlne profts are nondecreasng n D. On the other hand, ncreased demand on net may degrade arlne s profts f d o < ^d. The frst observaton n Proposton 2 states that the optmal dscount rates n the constraned model wll be n general above the optmal dscount rates for the unconstraned case. In contrast to the unconstraned case, we observe that optmal dscount rates can be nfluenced by demand on nets n the constraned model. Clearly, at optmalty the arlne wll offer dscount rates below the corporate threshold on some nets and balance the average by offerng hgher rates on other nets. We call nets wth optmal rates below the threshold as the low dscount rate (LDR) nets and above the threshold as the hgh dscount rate (HDR) nets. If demand on a LDR net ncreases the current average dscount rate wll declne. As such, the arlne wll need to ncrease ts dscount rates on other nets so as to brng the average back to the requred level. On the other hand, f the demand ncrease takes place on a HDR net, the average dscount rate wll ncrease gvng room for dscount reductons on other nets. Typcally, a net becomes a LDR net f the proft margn of the arlne on ths net s relatvely small and/or the non-arfare related weght of the arlne s suffcently hgh. Both of these factors create ncentves to lower the dscount rates. Hgh dscount rates can be afforded when the proft margn s hgh or the non-arfare related weght s low. To further exemplfy these results consder an arlne offerng a dscount contract to a potental corporate clent for three nets. It s gven that the collectve appeal of other arlnes ( ) on nets 1, 2, and 3 are 12, 9, and 12, respectvely. On Net 1 total demand s 30 unts, the publshed fare s $100, and unt overall cost s $50. These values are 25, $100, and $80 for Net 2 and 20, $100, and $20 for Net 3, respectvely. For the example, the prce elastcty factor s assumed to be 3 and the threshold dscount level 30%. Moreover, we let b 1 = 0.7, b 2 =1,andb 3 = 0.5. For brevty we do not consder any lower and upper bounds on dscount rates. We note that n the gven example Net 2 wll lkely be a LDR net snce the proft margn on ths net s relatvely small and the non-arfare weght s relatvely hgh. On the other hand, Net 3 s more proftable yet the arlne s not as strong on ts non-arfare related qualty on ths net. Therefore, t s a HDR net. Fg. 4a depcts the mpact of changes n total demand at Net 2 on optmal dscount rates. Observe that optmal dscount rate for Net 2 s always below the threshold and thus, from Proposton 2, optmal dscount rates on all nets must ncrease n demand on Net 2. Ths stuaton s llustrated n the fgure. On the other hand, Net 3 dscount rates are always above the threshold and thus, any ncrease on total demand on ths net wll lead to lower optmal dscount rates at all nets as llustrated n Fg. 4b.

11 J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx 11 Optmal Dscount Rates (a) Impact of Total Demand for Net 2 on Dscount Rates Net 3 Net 1 Net Net 2 Demand Optmal Dscount Rates (b) Impact of Total Demand for Net 3 on Dscount Rates Net 3 Net 1 Net Net 3 Demand Fg. 4. Demand for flghts vs. dscount rates. Arlne Proft Impact of Total Demand for Net 2 on Proft Total Demand on Net 2 Fg. 5. Impact of total demand at a low demand rate (LDR) net on profts. Another result contrary to the unconstraned case s that total demand on a net may nfluence the arlne profts on other nets. We note that ths occurs only when constrant n (9) s strctly bndng at optmalty (.e., when the customer restrctons are tght). In such a case, Proposton 2 establshes that arlne profts wll ncrease wth total demand on any HDR net. We know that ncreased demand on a HDR net wll result n lower optmal dscount rates, whch wll lead to lower market share. However, ncrease n the total HDR net demand and savngs from lower dscount rates wll more than compensate the losses n market share. Ths result cannot be generalzed to LDR net demands n that an ncrease n total demand on a LDR net does not necessarly lead to hgher profts as llustrated n Fg. 5. In ths example, when total demand on Net 2 s suffcently small (.e., <20), proft margns on ths net are stll hgh enough and thus the arlne s payoff ncreases n the total demand. The loss n proft margns due to hgher dscount rates s compensated by ncreased sales. However, as the total demand contnues to grow the proft margn on Net 2 further dmnshes Effectve-demand-averaged (EDA) dscount constrant The constrant gven n (9) stpulates a mnmum average dscount per corporate bookng. However, the corporate clent may not be nterested to work wth such threshold as often tmes ts employees wll not prefer ths arlne for all ther busness travels (.e., the market shares are strctly less than one at most nets). It may be more appealng to the customer to receve an average dscount cap for each bookng wth the arlne offerng the contract. Consequently, t may prefer to mpose the followng constrant nstead of (or n addton to) constrant (9): P P 2Nf D d 2N f D P ^d f ð10þ 3 In fact, f demand gets too hgh on LDR nets, optmal arlne profts can become negatve. In such cases, the arlne company should ether re-negotate the threshold dscount level wth the corporate clent or remove LDR nets wth excessve total demand from the contract.

12 12 J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx In order for ths constrant to work, the contractng arlne and the corporate travel agent must have a consensus on the demand estmates and the arlne s market share wth the corporate customer. The arlne can acheve ths through collaboratng wth the clent on estmatng the demand and the market shares. We note that the relaton among optmal dscount rates, demand, and profts under constrant (10) s smlar to Proposton 2 due to the fact that f d ncreases n d. That s, the dscount rates wll ncrease n demand at LDR nets and decrease at HDR nets. The arlne s profts ncrease n demand at HDR nets and the opposte may occur at LDR rates. Snce market shares are below one, the mpact of demand changes wll be relatvely smaller wth constrant (10). In the aforementoned example, the arlne s profts start to drop n Net 2 wth demand above 160 flghts wth constrant (10) n contrast to 20 flghts wth constrant (9) under the same threshold values. We also pont that n the same example, when net demands are 30, 20, and 25, the optmal profts for the arlne are $ and $ under constrants (9) and (10), respectvely for the same threshold value of 30%. Clearly, for ths nstance, the latter constrant leads to hgher profts for the arlne. However, total corporate savngs wll be $ and $741.83, respectvely mplyng that constrant (9) s more preferable to the corporate. It should be underlned that ths result cannot be generalzed to all cases as explaned below. Note that n constrant (10) the dscount rates are weghed based on the effectve demand rather than on total demand. At any gven soluton, the dfference between the TDA and EDA profts decreases as market shares of the arlne grow n LDR nets or dmnsh n HDR nets under fxed demand and dscount rates. Ths s due to the fact that n both cases the value of the left-hand sde n constrant (10) ncreases whle constrant (9) remans the same. Consequently, we should expect hgher payoff for the arlne wth constrant (9) when market shares at LDR (HDR) nets are relatvely low (hgh). Lower (hgher) market shares are expected when the collectve appeal of others s hgher (lower). For nstance, n the example when the collectve appeal of others s lowered from 9 to 6, the arlne profts under constrants (9) and (10) wll be $857.3 and $798.8, respectvely, ndcatng that the arlne s better off wth constrant (9) ths tme. On the other hand, the expected savngs for the corporate are $ and $ mplyng that the latter constrant wll perform better for the corporate customer Mnmum total (MINT) dscount constrant Another constrant that can be appended to the optmzaton problem s due to a threshold value on corporate savngs. The corporate clent could demand a mnmum total savngs of, say s, n the dscount contract. Ths threshold value can be determned based on hstorcal data or the performance of the currently employed contract. In ths case the new constrant s gven by X f D P d P s. ð11þ 2N Imposng the new constrant leads to the followng observatons: Proposton 3. For the optmzaton model wth constrant (11): () At any optmal soluton, for any gven net, optmal arlne profts ncrease wth D and P, and decrease wth. () Optmal dscount rates are non-ncreasng functons of demand and prce, and non-decreasng functons of at all nets. 6. Case study at Brtsh Arways In ths secton we demonstrate the mplementaton of our model on a real-world case study nvolvng renewal of the corporate contract that Brtsh Arways negotated wth CITI Group n Numbers have been scaled to protect confdentalty. Brtsh Arways (BA) has more than 200 account managers and sales executves who manage over 100,000 contract dscounts n more than 600 corporate contracts, representng 2.7 bllon USD n revenue. The CITI Group contract s negotated based on 419 dfferent nets encompassng

13 J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx 13 Objectve Current Deal: Projected Scenaro One: Scenaro Two: Scenaro Three: Scenaro Four: Maxmze BA Proftablty Maxmze BA Proftablty Maxmze BA Proftablty Maxmze BA Proftablty Constrant TDA EDA MINT Arlne Demand Net Dscounts Fnal Revenue TDA (%) EDA (%) 38,520 37,076 39,165 45,477 7,524K 2,938K 3,315K 7,524K 16,726K 21% 23% 10,331K 17,468K 11% 11% 12,084K 39,668 3,512K 17,429K 21% 23% 11,704K 17,362K 20% 23% 11,722K 17,298K 30% 34% 10,540K Proft Margn Improvement (relatve to orgnal deal): 1,753K (+17%) 1,373K (+13%) 1,391K (+13.5%) 209K (2%) Fg. 6. Scenaro comparson based on total-demand-averaged (TDA) dscounts, effectve-demand-averaged (EDA) dscounts, mnmum total (MINT) dscounts, and net profts. destnatons n all contnents. Based on the sze and mportance of ths clent to BA, t s expected that the contrbuton margn n a deal wth ths customer wll be more than 60% whle the cost of dealng wll be less than 34% on average. Top routes n ths contract nclude LHR-JFK, LHR-EWR, LHR-ORD, LHR-FRA, LHR-ZRH, LHR-JNB, LHR-HKG, and LHR-NRT, for whch the current level of dscount (based on the 2003 contract) cannot be decreased. 4 Hence, the current dscount levels for these nets were treated as lower bounds. Based on market compettve condtons, the account manager has determned that there s no reason to ncrease the dscount for few of the routes such as LHR-SVO, LHR-DEL, LHR-FCO, LHR-MAD, LHR- BCN, LHR-DFW, and LHR-TPA. Therefore, dscount rates on these nets were fxed at the current levels and thus, were not ncluded n the model as decson varables. However, they were used n corporate partcpaton constrants for calculatons of the average dscount levels. The parameters of the MNL model are estmated and calbrated based on combnaton of hstorcal data, current market shares wth the clent, and expert opnon. Four dfferent scenaros were run before makng a fnal offer to CITI Group. In Scenaro 1, the dscounts are optmzed and only revenue management and contract gudelnes are consdered (upper and lower bounds on ndvdual dscount rates). Scenaro 2 employs constrant (9) wth a threshold value of 21% computed based on the prevous year s dscount rates. In ths scenaro, BA guarantees a total-demand-averaged dscount rate that s no less than what the current contract (2003 contract) would provde under the current demand forecast. Scenaro 3 ncludes constrant gven n (10) and the threshold value for the effectve-demand-averaged s also matched wth what the current contract can acheve (.e., 23%). In Scenaro 4, the constrant (11) s added to the problem so as to guarantee a mnmum total savngs for CITI Group. The optmal profts for Scenaros 3 5 are obtaned by solvng the non-lnear maxmzaton models gven n Secton 4 usng Lagrangan decomposton. The results are compared wth the 2003 contract s performance and are summarzed n Fg. 6. The frst column n Fg. 6 presents relevant statstcs of the contract assumng that the 2003 contract dscounts are not modfed for The outcome of Scenaro 1 results n ncreasng the dscounts for 129 nets and decreasng them n 184 nets wth a weghted average decrease of dscounts by 12% compared to the prevous year s contract. Ths contract provdes a 17% ncrease ( 1.75 mllon) n profts for BA compared to the prevous year. However, sellng ths contract to the customer would be very dffcult for two reasons. Frst, ther top routes have been modfed and second, the dscount per ar travel tcket has decreased from 23% to 11%. 4 In ths case study, each net s ndcated by a par of arport codes. We refer the reader to for explanaton of all arport codes worldwde. We also note that most of Brtsh Arways flghts subject to the deal are from LHR (London Heathrow Arport).

14 14 J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx 14 Arlne Profts (n mllon) EDA Rate (%) Fg. 7. Effectve-demand-averaged (EDA) dscount threshold vs. optmal arlne profts. The outcome of Scenaro 2 results n ncreased dscounts at 269 nets and decreased rates at 62 nets. Overall, for 79% of all nets the dscount rates were altered. Ths contract s expected to generate a 3% ncrease n demand and 11.7 mllon n profts, a 13% growth from the prevous year. The outcome of Scenaro 3 results n an expected contrbuton margn ncrease of 13.5% (.e., 1.4 mllon). Dscount rates are to be ncreased at 232 nets and decreased at 81 nets compared to last year s contract. In ths case at 75% of all the nets a change n dscount rates has occurred and the TDA dscount rate s decreased to 20%. In the last scenaro, a new contract that guarantees a total dscount of at least 7.5 mllon s desgned. In ths case, although the market share s ncreased by more than 10%, the consderable ncrease n dscount rates (30% TDA rate and 34% EDA rate) lmt the growth n optmal profts at 209 K, whch s well below what the other contracts can acheve. The results of ths analyss were valdated wth the account manager of the CITI Group account and the changes based on Scenaro 3 were mplemented for the 2004 contract renewal. Under ths contract, 50% of the total profts come from 15 nets (3.5% of the total number) ncludng: LHR-JFK(1), LHR-EWR, LHR-SVO, LHR-JNB, LHR-MAD, LHR-NRT, LHR-BOM, LHR-DXB, LHR-CDG, LHR-ARN, LHR-LIN, LHR- BOS, LHR-MIA, LHR-JFK(2), and LHR-JFK(3). Whle the dscount rates at the frst sx nets are not altered, they are ncreased at LHR-BOM, LHR-DXB, LHR-CDG, and LHR-ARN, and decreased at the last fve nets. At 24 nets (5.7% of the total), BA expects loss when sellng tckets to CITI Group employees under ths contract. The mpact of the EDA threshold on BA profts s further analyzed by computng the optmal profts under varyng values and results are depcted n Fg. 7. It s evdent from the analyss of Scenaro 1 that up to 11% threshold the optmal profts wll reman unchanged at mllon, a 17% ncrease n profts wth respect to the prevous contract. From ths pont on, the optmal profts decrease n EDA threshold n concave fashon. We note that up to the 43% level the arlne can desgn a contract through an optmzaton that generates hgher profts than the current contract wth a 23% EDA dscount rate. 7. Conclusons and future research In ths paper, we propose a non-lnear mathematcal model to determne the optmal desgn of corporate contracts that offer front-end dscount offers to corporate travelers. The contrbuton of ths paper to the lterature s that ths s the frst paper that nvestgates the corporate travel deals from the perspectve of the arlnes and presents a comprehensve vew of how optmzaton technques can be appled to ad the arlne desgnng the most cost effectve contracts. We model the arlne s proft functon usng a multnomal logt functon that captures the corporate clent s choce behavor under the mpact of the competton. The dscount decsons for all nets are nterdependent by the varous corporate restrctons that are ntegrated nto the model as constrants. The model confrms the strong correlaton between optmal dscount polces at dfferent nets n corporate contractng. It s often the case that hgh dscount rates at some legs that are crucal for the arlne mply lower or no dscounts at others. In general, we show that prce elastcty, total travel demand, costs, servce qualty, and arlne competton play sgnfcant roles n the arlne s optmal corporate dscountng polces.

15 Corporate dealng decson support tools such as the one proposed n ths study can provde arlnes wth the means needed to create and manage successful corporate arlne contracts. Wth these tools, arlnes have the ablty to develop strateges that create wn wn stuatons for both the arlne and the corporate clents. A plot study at Brtsh Arways proved that by usng such a tool n optmzng corporate contracts, the arlne can ncrease the annual contrbuton margn from 3% to 5%; that s over 50,000,000. We note that n our study, the corporate contract s lmted to arfare dscount offers. In many stuatons, contracts may nclude non-arfare related offers such as cabn class upgrades, lmousne servce, hotel gft certfcates, resort certfcates, and merchandse rewards. Wth the use of the proposed MNL functon one could ncorporate non-arfare related decsons nto the model as well. However, a more generc model should consder jontly other dscount schemes such as frequent fler programs and back-end dscounts n addton to front-end dscounts. We note that deals wth large companes may lead to overcrowdng of some flght segments and thus shortage of capacty for proftable flyers. Future study should also address such dluton caused by multple routes wthn a large contract that flow through a common staton (hub). To tackle the problem related to the rsk of dluton a more comprehensve model that ntegrates multple contracts should be developed. Appendx A Proof of Proposton 1. Frst we need to show that P s unmodal n d n nterval [0,1). Unmodularty mples that the proft functon has at most one statonary pont n the gven nterval. Clearly, at any statonary pont, the equaton gven n (6) must be satsfed. We prove that the proft functon s unmodal and the statonary pont, f exsts, gves the maxmum value n [0,1) by showng that at any statonary pont the second dervatve must be strctly negatve when c > 1. Frst observe that the frst dervatve of the proft functon wth respect to d s P 0 ¼ D f 0 ðp ð1 d Þ ðcc þ RC ÞÞ D f P At any statonary pont, the foregong functon must return zero. We note that at ths pont the equaton n (6) holds. Next, we can wrte the second dervatve as follows: P 00 ¼ D f 00 ðp ð1 d Þ ðcc þ RC ÞÞ 2D f 0 P From (6), at any statonary pont the value of the second dervatve wll be P 00 f f 00 2f 0 ðc 1Þf ¼ D P ¼ D P ð1 d Þ f 0 Clearly, the foregong functon returns a strctly negatve value for d 2 [0,1) when c > 1, mplyng that any statonary pont must be a maxmzer and thus, P must be unmodal. It s straghtforward to see that P s decreasng at d = 0 f nequalty gven n (5) holds. From unmodularty, ths mples that P must be decreasng every where n [0,1) n that case. h Proof of Proposton 2. We use the Lagrange multpler to prove the frst part. Wth Lagrange multpler, k, constrant (9) can be removed and the objectve functon of the model can be rewrtten as follows: X z ¼ Max L ¼ Max P k ^d X D X! D d d d 2N 2N The frst order optmalty condtons requre that at optmalty P ð1 d o Þ ðrc þ CC Þ f P þ k ¼ 0 for all. f 0 J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx 15 From Proposton 1, we know that P s unmodal and t s straghtforward to see that k P 0 f the unconstraned soluton does not satsfy constrant (9). In such case, the objectve functon z must be ncreasng at d mplyng that d o P d must be true f d s below the upper bound gven n constrant (8).Whend s greater

16 16 J. Pachon et al. / Transportaton Research Part E xxx (2006) xxx xxx than or equal to the upper bound, the optmal dscount cannot be less than d u as ths would clearly degrade the objectve functon value. To prove the second part, frst observe that n the Lagrange model P j2n D j ^d d o j ¼ 0. For some net j f d o < ^d and D s ncreased, then the left-hand sde n ths equaton becomes strctly postve. To pull the lefthand sde down to zero agan, the dscount amount for at least one net (say net k) must be ncreased. Snce each net s proft functon s unmodal wth a maxmzer and k P 0, the left-hand sde n the frst dervatve evaluated at ths new dscount value wll be negatve. Then, to satsfy the frst order optmalty condton for ths net k must be ncreased. However, ths wll propagate postve dervatve values for other nets evaluated at the orgnal optmal soluton. Consequently, dscount rates at other nets wth d u j < do j < du j need to be ncreased as well, so as to satsfy the frst order optmalty condtons. Usng smlar analogy, one can see that ncreasng demand on any net wth d o > ^d wll result n decreased dscount rates at any net j wth d u j < do j < du j. For the proof of the thrd part we use the Envelop Theorem whch mples that the dervatve of the optmal Lagrange functon wth respect to D s ol ¼ f Pð1 d o od Þ ðcc þ RC Þ k ^d d o Clearly, the foregong dervatve s postve for d o > ^d suggestng that the optmal profts wll ncrease n D n such case. Observe from the same functon that the dervatve may return a negatve value mplyng downward sloppng of profts n demand for d o < ^d f the proft margn or market share of the arlne s suffcently small on net n the current optmal soluton. h Proof of Proposton 3. Under constrant (11), the Lagrange functon wll be L s ¼ X P k s s X! f D P d 2N From the Envelop Theorem, the frst dervatve of the optmal Lagrange functon wth respect to both D and P are postve mplyng that the optmal profts ncrease n both parameters. It s straghtforward to observe the opposte for. For the second part observe that as demand or prce at any net ncreases at any optmal soluton the value of the left-hand sde grows allowng for room to reduce the dscount rates. The opposte s true for values. h References Algers, S., Besser, M., Modelng choce of flght and bookng class a study usng stated preference and revealed preference data. Internatonal Journal of Servces Technology and Management 2, Andersson, S.E., Passenger choce analyss for seat capacty control: a plot project n Scandnavan arlnes. Internatonal Transactons n Operatonal Research 5 (6), Barnhart, C., Belobaba, P., Odon, A.R., Applcatons of operatons research n the ar transport ndustry. Transportaton Scence 37 (4), Basuroy, S., Nguyen, D., Multnomal logt market share models: equlbrum characterstcs and strategc mplcatons. Management Scence 44 (10), Bazaraa, M.S., Sheral, H.D., Shetty, C.M., Nonlnear Programmng: Theory and Algorthms. John Wley & Sons, NY. Belobaba, P., Arlne yeld management: an overvew of seat nventory control. Transportaton Scence 21 (2), Ben-Akva, M., Lerman, S.R., Dscrete Choce Analyss. MIT Press, Cambrdge, MA. Bennett, R.W., Boyer, K.D., Inverse prce/qualty tradeoffs n the regulated arlne ndustry. Journal of Transport Economcs and Polcy 24 (1), Btran, G., Caldentey, R., An overvew of prcng models for revenue management. Manufacturng and Servce Operatons Management 5 (3), Botmer, T.C., Effcency consderatons n arlne prcng and yeld management. Transportaton Research Part A 30 (4), Degraeve, Z., Labo, E., Roodhooft, F., Total cost of ownershp purchasng of a servce: the case of arlne selecton at Alcatel Bell. European Journal of Operatonal Research 156 (1), Dresner, M., Tretheway, M.W., Modelng and testng the effect of market structure on prce: the case of nternatonal transport. Journal of Transport Economcs and Polcy 26 (2),

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