Princing ATC/ATM services : Part I : the case of a public provider

Transcription

1 Priig ATM series : Part I : the ase of a publi proider Rabah Guettaf, Catherie Mael, Moussa Larbai, eli Atoio Claudio Mora-Camio To ite this ersio: Rabah Guettaf, Catherie Mael, Moussa Larbai, eli Atoio Claudio Mora-Camio. Priig ATM series : Part I : the ase of a publi proider. VIII SITRAERII RIITA 9, 8th Brazilia Air Trasportatio Researh Soiety Symposium ad d Ibero-Ameria Air Trasportatio Researh Soiety Symposium, No 9, Sao Paulo, Brazil. 9. <hal-93877> HAL Id: hal Submitted o May 14 HAL is a multi-disipliary ope aess arhie for the deposit ad dissemiatio of sietifi researh doumets, whether they are published or ot. The doumets may ome from teahig ad researh istitutios i rae or abroad, or from publi or priate researh eters. L arhie ouerte pluridisipliaire HAL, est destiée au dépôt et à la diffusio de doumets sietifiues de ieau reherhe, publiés ou o, émaat des établissemets d eseigemet et de reherhe fraçais ou étragers, des laboratoires publis ou priés.

2 PRICING O ATM SERVICES Part I: THE CASE O A PUBLIC PROVIER Rabah Guettaf 1, Catherie Mael 1, Moussa Larbai, éli Mora-Camio 1 1 LARA, ENAC, Toulouse, rae rabah.guettaf-et,atherie.mael,feli.mora@ea.fr, Kuala Lumpur Uiersity, Malaysia, moussa.larbai@gmail.om RESUMO Neste estudo uma abordagem multi ieis é proposta para estudar o problema da defiição das taas a serem apliadas por u proedor publio de seriços de otrole e gereiameto do trafego aéreo. O proedor de seriço proura, detro de toda seguraça, fometar a demada fial dos usuários do sistema de trasporte aéreo. Esta aalise lea a osiderar a resolução de um problema de programação matemátia de tipo bi iel ode o proedor do seriço é o líder e o setor das empresas aéreas é o seguidor. No aso moo dimesioal, uma solução aalítia é estabeleida e aalizada para este problema. Um esuema atural de egoiação etre o proedor de seriço e as empreasas aéreas é etão osiderado e odições sufiietes de oergêia do proesso para a solução ôtima são estabeleidas. ABSTRACT I this study a multileel framework is proposed to aalyze the harges defiitio problem for a publi ATM serie proider whose fial objetie is, while guarrateeig safety, to promote the leel of ed users demad. This leads to the formulatio of a bi-leel optimizatio program iolig ATM serie proider as the leader ad the whole airlie setor as the follower. A diret optimal solutio has bee obtaied i the simplisti osidered oe dimesioal ase. The a atural egotiatio proess betwee the ATM serie proider ad the airlie setor is itrodued. Suffiiet oditios are established so that the proposed egotiatio proess oerges towards the optimal solutio. Keywords: priig, bileel programmig, ATM, air trasportatio 1. INTROUCTION Alog the last deades, may studies i the fields of Operatios Researh, Systems Maagemet ad Applied Eoomis hae bee deoted to air trasportatio plaig, tariffs ad operatios related issues. I geeral the aalysis whih hae bee performed are limited to diret effets so that the sope of the adopted models are i geeral too limited. This implies that feedbak pheomea betwee the differet ators ad ioled air trasportatio atiities aot be fully take ito aout to perform a omprehesie aalysis ad to desig effiiet plas ad poliies. I this study a multileel approah is deelopped. The objetie assumed for airlies i this study is of a pure eoomi ature: profit maimizatio. The mai oer of this study beig with the defiitio of effiiet ATM harges, the whole airlie setor is take as a whole, so that market ompetitio betwee airlies is ot otemplated. This is a limitatio of the study, whih is doe i sake of limited ompleity, sie i fat, ATM harges may hae some ifluee o the euilibrium state of differet air trasportatio markets. Howeer, it is also worth to obsere that these airlies are i geeral represeted by a uiue etity durig egotiatios with ATM authorities.

3 I the ase of a publi ATM serie proider, it is osidered that the mai objetie is to promote air trasportatio for ed users, i.e. the passegers freight is ot osidered epliitly i this study, through a safe ad effiiet trasportatio supply by airlies ad ATM authorities. Hee a passegers demad model, reatie to airlies tariffs is itrodued to take ito aout idiret ifluee of ATM harges o passegers demad leels oer differet air trasportatio markets. Aother objetie, whih is take ito aout through ieuality ostraits, is that the eoomi performae of the airlie setor is ot impaired by the retaied leels of ATM harges. Also, ATM osts related diretly with the urret traffi situatio iestmets osts leadig to elarged ATM apaity to fae future traffi situatios are ot osidered should be adeuately oered. This results i a bileel optimizatio problem whih is of the upper liear-lower liear lass. To illustrate the approah a oe dimesioal system is osidered. I that ase it is possible to establish diretly its optimal solutio while suffiiet oditios are established so that a egotiatio proess betwee ATM authorities ad the airlie setor leads to this optimal solutio. There, the problem is split i two depedet problems: oe where ATM authorities determie ATM harges for gie airlies tariffs ad oe where the airlie setor determies the tariffs ad seat apaity supplies oer the differet markets for gie ATM harge leels.. BASIC ASSUMPTIONS Here is osidered the ase of a elemetary air trasportatio system omposed of a sigle pair of airports liked by a sigle air route. There is a uiue ATM operator ad a uiue airlie operatig betwee these two airports. igure 1: The elemetary ATM ase The potetial demad is supposed to be omposed of roud trips ad to obey, for the sake of simpliity ad the ability to deelop lear aalytial ad graphial results, to the followig affie demad futio: 1 1 where is the effetie leel of potetial demad. Here it is supposed that there is a uiue lass of traelers ad a uiue apparet prie 1 is adopted for roud trips. is a absolute potetial demad ad is a ostat positie parameter harateristi of the respose of the market to prie hages. The parameter represets a ta ide applied to eah trip tiket. Other demad models suh as the epoetial oe with ostat prie elastiity :

4 e or either uspeified models suh as: 3 with adeuate assumptios suh as: ad 4 ould hae bee adopted. Beyod the harges, a additioal way to fud the series whih appears atural, is to assig a proportio,1 of the aboe ta, whih is paid by the fial users, to the serie proider. The trasport apaity of the airlie is gie by the maimum affordable freuey of serie f whih is related with the size of the fleet of the airlie. Here for simpliity ad ma osiderig that for the gie time period f a be a high umber, f will be take as real. Whe a freuey of serie f is adopted, the operatios osts are supposed to be gie by: here: f 5 C - is a positie parameter a mea ariable ost with respet to freuey. It is related with the prie of fuel, the ost of the rew ad the legth of the flights. - C is a fied ost related with the sizes of the fleet ad the rews of the airlie as well as with the harateristis of the operated etwork. - is the tariff applied to a roud flight, iludig e route otrol ad airport otrol. Here o distitio will be made betwee airport taes ad approah ad e route harges. The aailable seat apaity is gie by: f 6 where is the mea seat apaity of a airraft of the fleet of the airlie. The operatig osts of the supplier are gie by: here: f C 7 - is a positie parameter a mea ariable ost with respet to freuey. It is related maily with the legth of the flights. - C is a fied ost related with the harateristis of the otrolled airspae ad with the size of the staff. No saturatio effets with oseuees oer the ost futios of the serie proider ad the airlie are osidered i this study. 3. OPTIMIZING CHARGES OR A PUBLIC ATM SERVICE SUPPLIER Here it is supposed that the fial objetie of the publi ATM serie supplier is to maimize the satisfied demad while guarrateeig a miimum eoomi retur for the ATM series, R, ad a miimum eoomi retur R for the airlie. It is supposed also that the airlie tries to maimize her beefit takig ito aout her ost futio ad the ATM tariff.

5 3.1 A biliel program for publi ATM priig Aordig to the aboe assumptios, a biliel program a be established: with f ma f 9 C R 8 where is the ta rate applied to air traellers, ad the airlie s profit ostrait: f 1 C R where ad f are gie by the solutio of: with ad ma, f f C,mi f, 1 11 ma 1 f f ma Solig the airlies profit maimizatio problem To sole the airlie s profit maimizatio, two ases must be osidered with respet to the effetie leel of passegers demad. Either : f 14 or: a Case i whih effetie demad is determied by the seat apaity I this first ase, we hae: f 1 16 where 1 17 Now, osiderig airlie s profit leel ures, we get: p p C f 18 where p is a hose leel of profit for the airlie. Chagig the alue of p ad osiderig ostrait 13 i the, f plae, we get hyperbola ars for the profit leel ures.

6 It appears learly i figure that the maimum profit is obtaied whe ostrait 13 redues to euality. This result is also alid see figure 3 whe the tagey of a profit leel ure ad the demad lige proides a freuey aboe f ma. I the ase of o atie fleet ostraits, we get a double solutio for the followig euatio: whe p reahes the alue: p C 1 19 The: p 1 ma C f ma,mi f ma, * 1 igure : Optimizatio of airlie profit, o atie fleet ostrait, full apaity igure 3: Optimizatio of airlie profit with a atie fleet ostrait at full apaity

7 ad either o atiatio of the fleet ostrait: * or atiatio of the fleet ostrait: * f ma 3 while the optimal airlie profit is gie either by p relatio 16 or by: * ma p p * f ma f ma C 4 3..b Case i whih effetie demad is determied by the prie leel I this seod ase, we hae: f 5 Now, osiderig airlie s profit leel ures p, we get: p C f 1 6 where p is a hose leel of profit for the airlie. Chagig the alue of p ad osiderig ostrait 5 i the, f plae, we get parabola ars for the profit leel ures. igure 4: Optimizatio of airlie profit with pried ostraited demad It appears that here agai the optimal solutio is gie by relatios:

8 * f 7 with agai, ad : 8 Relatio reeals the ifluee of ATM tolls oer air trasportatio tariff ad oseuetly oer satisfied demad relatio 8. I the followig, it will be assumed that the fleet ostrait remais iatie. 3.3 Solig the publi ATM serie proider problem Cosiderig the solutio of the airlie profit maimizatio problem, problem 8, 9, 1 with 11, 1 ad 13 beomes: ma 9 uder the ostraits: C R 3 C R 31 ad 3 This last oditio isures that there is some satisfied demad. This problem a be rewritte as: mi 33 uder the ostraits: C R 34 C R 35 ad ma mi 36 Costraits 34 ad 35 a be rewritte as: b a 37

9 with C R b a 1 38 ad 4 4 C R 39 The euatio: b a 4 presets real roots whih are the positie, whe: a b 41 or R C 4 Whe =, this oditio redues to: R C 43 Let 1 be the miimum real root of 4 whe it eists: 1 1 with 41 C R 44 Whe =, we get: 1 45 with 4 C R 46 ad 1 is the less tha ma if: 47 It will be supposed i the followig that, ad R are hose so that 1 is less tha ma.

10 The roots of euatio: 4 4 C R 48 are always real, oe of them beig always positie: The smallest of these roots, Cosiderig 36, is suh as: C R, is suh as: C R ma Here also,it will be supposed i the followig that ad R are hose so that is less tha ma. By ispetio of all the possiblities, oly two situatios lead to a solutio for problem 33, 34, 35 ad 36. They are represeted graphially bellow: - I the ase i whih is positie superior to, to hae a solutio, 1 must be iferior or eual to ad superior to, the the solutio * is eual to 1. - I the ase i whih is less tha, there is a solutio gie by * ma 1, whe: ma ma 1, 5 The ordiates i figure 5 ad figure 6 bellow represet the profits aboe the guarrateed alues R ad R ANL.

11 igure 5: Solutio of the publi ATM supplier problem ase i whih is superior to igure 6: Solutio of the publi ATM supplier problem ase i whih is iferior to 4. GLOBAL SOLUTION OR PUBLIC ATM SERVICE PROVIER It will be supposed for the followig that ad R are hose so that figure 5: 4 C 1 R C R C R 53 the, oly the first ase is to be osidered ad the solutio of the whole problem 8 to 13 is gie by: * 1 The leel of the satisfied demad is gie by: 54 1 * * 55 The profit of the publi supplier is eual see figure 5 to the miimum guarrateed leel R while the profit of the airlie is gie by: 1 * 1 p* * 1 whih is superior or eual see figure 4 to the miimum guarrated leel R. C 56

12 rom relatios 54 ad 55, it is easy to show * that for a gie alue of, the alue of whih maimizes is * 1. The, a further step towards the optimizatio of the setor would be to hoose effiietly the rate of the ta applied to the trips. I that ase, osiderig 54 with =1, we should defie * by : 1 * 8 C R 57 ad the best alue of would be solutio of: with ma 1 * 58 ad 1 * 1 1 * C R 59 4 C R 6 Howeer, the alue of parameter is the result of a eogeous hoie proess where oerall eoomi as well as politial osideratios are take ito aout. igure 7: iaial flows ad atiity leels with publi ATM serie supplier 5. OPTIMALITY O A PRICING NEGOTIATION PROCESS

13 Sie the osidered priig problem ioles two major eoomi agets, the ATM serie proider ad the airlie setor, a solutio of the osidered bileel problem resultig from the hypothesis with respet to the ature, publi or priate, ad the goals of the ATM serie proider, through a egoiatio proess a be of iterest. ollowig the geeral formulatio 5 to 1, a atural egoiatio proess based o the objeties of the ioled eoomi agets ould be the followig: The publi ATM serie supplier soles at iteratio +1 the followig problem with respet to, gie ad f : 1 arg ma 61 with f ad the airlie s profit ostrait: f C R 6 f C R 63 where ad f are proided by the airlie setor whih soles the followig problem gie : ma mi f, f C 64, f with f f ma 65 This proess is represeted i figure 8: igure 8: Negotiaio proess betwee publi ATM ad airlies Sie, the ATM serie supplier has to satisfy the two budget ostraits with the miimum alue for, sie the profit of the airlie is a dereasig

14 futio of, the solutio of the problem orrespods to the saturatio of his ow budget ostrait, so that: f f C R 1 66 Here we osider the ase i whih the solutio of the airlie problem is gie by: f 67 with 68 the we get a reuret formula for : 1 C R 69 or simpliity we osider here oly the ase i whih =, the: 1 C R C R 7 or C R 1 71 Here we osider that the followig oditio is satisfied: 7 The i figure 9, the oergee of the egoiatio proess is aalyzed graphially. It appears that if:

15 4 C R 73 igure 9: Coergee of the publi ATM- Airlie egoiatio proess or R C 8 74 the there are two euilibrium poits: poit A whih is a stable euilibrium solutio ad poit B whih is a ustable euilibrium solutio. It appears that poit A orrespods to the optimal solutio of the bileel problem treated i setio III.3. I that ase the proposed egoiatio shemes leads to the optimal solutio. 6. CONCLUSION I this study a multileel framework has bee proposed to aalyze the ATM harges defiitio problem i the ase of a publi ATM serie proider i a deregulated market. Here the oe dimesioal ase has bee osidered. The objetie whih has bee assumed for the ATM publi serie proider is to promote air trasportatio for ed users while the objetie of the airlies setor is profit maimizatio. This has led to the formulatio of a bi-leel optimizatio program iolig ATM serie proider as the leader ad the whole airlie setor as the follower. A diret optimal solutio has bee obtaied i the simplisti osidered oe dimesioal ase. The a atural egotiatio proess betwee the ATM serie proider ad the airlie setor has bee itrodued, splittig this problem i two depedet problems: oe where ATM authorities determie ATM harges for gie airlies tariffs ad oe where the airlie setor determies the tariffs ad seat apaity supplies oer the differet markets for a gie ATM harge leel. Suffiiet oditios hae bee established so that the proposed egotiatio proess oerges towards the optimal solutio. The it appears of iterest to the serie proider to adopt a similar egotiatio proess whe dealig with the full sale etworked priig problem.

16 7. REERENCES Assad, A.A.1978 Multiommodity etwork flows-a surey, Networks, 8, pp Bard, J Pratial bi leel optimizatio: algorithms ad appliatios, Kluwer Book Series: No oe optimizatio ad its appliatios, 3. Berge, C. 1 The theory of graphs, oer. Brotore L., M. Labbé, P. Marotte ad G. Saard A bileel model for toll optimizatio o a multiommodity trasportatio etwork. Trasportatio Siee 35, empe S. oudatios of bileel programmig. Kluwer Aademie Publishers, ordreht. ord L.R. ad.r. ulkerso, 196, lows i etworks, Prieto, Prieto uiersity Press. Labbé M., P. Marotte ad G. Saard 1998 A bileel model of taatio ad its appliatio to optimal highway priig, Maagemet Siee 44,