A real option methodology to determine the optimal intervention windows for railway infrastructure

Transcription

1 Computers n Ralways XIV 437 A real opton methoology to etermne the optmal nterventon wnows for ralway nfrastructure N. Lethanh & B. T. Aey Insttute of Constructon an Infrastructure Management, ETH Zürch, Swtzerlan Abstract In ths paper, a real opton methoology s presente for the etermnaton of the optmal expecte tme n the future for a ralway nfrastructure manager to ece what types of nterventons, f any, are to be execute. Ths tme s heren referre to as the optmal tme to ece to execute an nterventon to emphasze that when the tme s fxe t s not yet known f nterventons wll actually be execute. Such a methoology s partcularly useful n the management of ralway nfrastructure that belongs to a multnatonal freght corror where multple ralway management organzatons are nvolve. Every tme one executes an nterventon t can affect the flow of trans on the part of the nfrastructure controlle by others. In the methoology an aaptaton of the moel use to value optons n fnancal engneerng usng the Black an Scholes fferental equaton s use. The moel enables the valuaton of the ablty to wat to ece to etermne whether or not an nterventon shoul be execute. The methoology s emonstrate by etermnng the optmal tme to ece to execute nterventons on a fctve ral corror for a ralway management organzaton. Use of the methoology s expecte to mprove the coornaton of the executon of nterventons on multple parts of the corror, an gve a pero of tme n whch t s relatvely certan that no nterventons wll be execute. Keywors: mantenance, ral nfrastructure, real optons. o: /cr140361

2 438 Computers n Ralways XIV 1 Introucton Management of ralway nfrastructure that belongs to a natonal track tran system or a multnatonal freght corror (e.g. the trans-europe transportaton network comprses sx freght corrors [1]) requres ralway managers to execute nterventons to ensure that an aequate level of servce s prove. As multple ralway management organzatons are nvolve, an every tme one executes an nterventon t can affect the flow of trans on the part of the corror controlle by others, t woul be esrable for all nvolve organzatons to agree on a tme to execute nterventons even f t s not known exactly whch nterventons are to be execute when ths tme s fxe. Ths woul mprove the coornaton of the executon of nterventons on multple parts of the corror, an gve a pero of tme n whch t s relatvely certan that no nterventons wll be execute. As the ecson to execute nterventons s base on the values of numerous varables wth whch there s uncertanty, e.g. ncreases n the number of trans, the eteroraton ue to the wearng of tracks an the changes n track geometry ue to floong, t s n the best nterest of each ralway management organzaton to etermne for tself the optmal tme to execute nterventons even f t s unclear as to the nterventons, f any, that wll be execute. Ths optmal tme s heren referre to as the optmal tme to ece to execute nterventons (OTD) to emphasze that when the tme s fxe t s not yet known f nterventons wll actually be execute. It may also be seen as the optmal wnow n whch nterventons coul be, but must not be, execute. Recent work has ncate that ths s possble usng a methoology bult on real optons (RO), such as propose by [2] n other cvl engneerng applcatons. A methoology bult on RO makes t possble to rectly take nto conseraton the probablty of obtanng new nformaton n the future an the fact that nterventons wll only be execute f certan contons are met. Most of the work focuse on nvestgatng the use of RO methoologes n the fel of cvl engneerng ecson makng has been focuse on ecson makng relate to the constructon of new nfrastructure, such as the constructon of a new arport [3], of a hgh-spee passenger tran system [4], of an electrcty strbuton network [5], an of a new aton to an exstng hghway network or n the evelopment of constructon projects n general [6]. Almost no work has been focuse on ecson makng relate to the mantenance of exstng nfrastructure, an none on ecson makng relate to the mantenance of ralway nfrastructure. The former nclues the example gven by [2] of how a RO methoology can be use n ecson makng relate to the mantenance of offshore structures. Although no work has been one on the latter, however, many researchers are nvestgatng the etermnaton of optmal nterventon strateges for ral nfrastructure. In the methoology propose n ths work for the etermnaton of the OTD, an aaptaton of the moel use to value European call optons n fnancal engneerng usng the Black an Scholes fferental equaton s use. The moel enables, as o all RO moels, the valuaton of the ablty to wat to ece to

3 Computers n Ralways XIV 439 etermne whether or not an nterventon shoul be execute. The methoology s emonstrate by etermnng the optmal tme to ece to execute nterventons on a fctve ral corror for a ralway management organzaton. The remaner of ths paper s structure as follows. The RO moel use n the methoology to etermne the OTD an the assumptons mae n establshng the moel parameters are explane n secton 2. The RO methoology s shown by conuctng an example to etermne the OTD for a fctve ral lnk for a ralway management organzaton n secton 3. Conclusons an scusson are gven n secton 4. 2 The moel The moel s evelope assumng that a ralway management organzaton s ntereste n etermnng the OTD urng a fnte tme pero T, for an exstng ralway lnk. Ths ecson strategy (DS) comprses only one pont n tme, z, n whch t can be ece whether an nterventon s to be execute or not, an the nterventon s execute. The organzaton s ntereste n etermnng the DS that wll maxmze the total expecte net benefts,.e. optmal z on 0 z T. In ths moel, there exsts two possbltes at z: (1) the global conton state (GCS) of the lnk oes not trgger the executon of an nterventon, or (2) the GCS of the lnk trggers the executon of an nterventon. If possblty (1) occurs, the probablty of the lnk enterng the GCS where an nterventon s requre ncreases. If possblty (2) occurs, then the lnk s restore to an as-new GCS an the eteroraton starts over, but not necessarly n the same way. In both cases, once the OTD has passe, t s assume that lmts on tran movements, e.g. spee restrctons, are mpose f the GCS, where an nterventon s requre, s reache. The tme pero (0,T) s ve nto three parts. The frst s from now to the start of the OTD (0,z), the secon s urng the OTD, an the thr s from the en of the OTD to the en of the nvestgate tme pero (z, T). It s assume here for mathematcal convenence that the OTD s nstantaneous. The mpacts ue to routne mantenance an operaton before an after z are represente as R (0,z) an R (z,t), an B (0,z)an B (z,t), respectvely. The superscrpt refers to nterventon type ( D), where to o nothng s also seen as an nterventon type an D s a set of nterventon types (The nterventon consere here s the nterventon on the lnk an nclues the nterventons to be execute on all objects of the lnk.) The negatve mpacts ncurre urng the executon of an nterventon, for example ue to nterrupte tran scheules, are represente as C z. The OTD s then gven as: z 0 T t z * t (1) S(0 : T) = S(0, t) e t e S ( z, T) e t z T z * Max e S ( z, T) S ( z, T),0 z

4 440 Computers n Ralways XIV where, St ()= Bt () Rt () Ct (): s the net beneft. Here, wthout loss of generalty, the superscrpt an subscrpt are gnore. The beneft of each DS s formulate as the subtracton of total revenue Bt (), routne mantenance Rt (), an cost of nterventon Ct (). * s use to enote the executon of a reference nterventon, e.g. o nothng, at z; s scount factor; Bt () nclues B(0, z ], B * (, z T ], an B (, z T ] an can be expresse as: I (2) Bt ()= N() t h() t O() t =1 where, N() t ( =1,, I ) s nex representng the quantty of transporte goos; s the nex use to ncate the type of goos; h () t s unt prce of nex, per ton of transport goos of type I; O () t s the operatonal cost on. It s mple n Equaton (1) that f the executon of nterventon s more benefcal than the executon of nterventon *, the nterventon wll be * execute. Otherwse, the nterventon wll be execute. The values of each of these parameters coul be moele probablstcally, e.g. the prces of electrcty an ol. In many cases, they can be moele as a geometrc Brownan moton as: h()= t h() t t h() t (3), h, h, t where, h, s the rft parameter of the geometrc Brownan moton; h, represents the stanar evaton of the change n the prce of ; t, s a parameter use to moel uncertan varables usng the Wener process. It has zero mean an stanar evaton of h, t. The expecte value of uncertan varables gven the value at tme t, e.g. the future prce of ol h( t u) gven the prce at tme t h() t can be escrbe as: u Eh [ ( tu)]= h( t) e (4) where, u s the length of tme between t to t u; whenever the lnk s n a GCS where an aequate level of servce s not prove, negatve mpact s ncurre, W, an the value of Bt () s gven by: I Bt ()= Ft () N() t h() t O() t (1 Ft ()) W (5) =1 where, F () t s the probablty of the ralway lnk beng n a GCS that oes not trgger the executon of an nterventon (ths s sometmes referre to as survval probablty), an W are the negatve mpacts ue to enterng a GCS before an

5 Computers n Ralways XIV 441 nterventon s execute, e.g. ue to the automatc mposton of spee restrctons. Values of W can be etermne by usng hstorcal ata [7]. Ths type of evaluaton s smlar to the so calle European call opton n fnancal engneerng, where at a preetermne tme z, the holer of an opton s allowe to make a ecson on whether the opton wll be exercse, but not the oblgaton to o so. Wth such a moel t s possble to estmate the probablty of a GCS occurrng n whch an nterventon s to be execute usng a Webull functon [8]. One of the man avantages of usng the Webull functon s that t s not wthout memory [9]. Usng the Webull functon, the functonal form of the probablty of not enterng the trggerng GCS F () t n Equaton (5) s: m F ( )=exp( ) (6) where, s the so-calle arrval ensty (or falure rate), an m s the acceleraton or shape parameter. Values of parameters an m can be estmate usng regresson analyss wth avalable ata. The beneft S () t of executng nterventon f nterventon s execute are gven by: S ()= t B () t R () t C () t (7) If t = z, C () t C ( z), otherwse, t equals to 0. The total net beneft between the DS n whch there s the possblty at z to ece whether or not to execute nterventon an the DS n whch there s the possblty at z to ece whether or not to execute nterventon *, (enote as ) at tme 0 (analogous to the payoff n European call opton) s gven by: zs, ( ) where, t g Z T ( T z) z, S ( t) zszt, (, ) = e gz T (8) s the expecte value of: * gx ( )= MaxS ( zt, ) S ( zt, ),0 (9) The soluton for Equaton (8) has been extensvely escrbe n numerous references on applyng Black an Scholes formulaton. The explct formulas to estmate the total net beneft are efne n followng equatons: zszt, (, ) ( T z ) * 1 2 = SzT (, ) ( ) e S ( zt, ) ( ) (10)

6 442 Computers n Ralways XIV wth T z (11) ln S S (, z T) 1 2 * T z (, z T) 2 T z (12) where ( x) s the cumulatve strbuton functon for normal stanar strbuton. 2 2 x s 1 ( x)= e s 2 (13) In orer to fn the optmal z,.e. the OTD, t s necessary to solve the set of Equatons (8) to (13). These equatons, whch nvolve the ntegral of an embee stochastc process,.e. the geometrc Brownan moton, can be solve usng the analytcal an numercal approach suggeste by Black an Scholes [10]. Although the formulaton gven above s the one to compute the expecte total beneft of havng the opton to ece at tme z to execute an nterventon or not, whch s analogous to the etermnaton of the payoff value n a European call opton, t s possble to run smulatons wth fferent values of z to compute how the value of the opton to ece changes over tme. The use of ths RO moel to etermne the OTD s emonstrate wth a fctve example n the next secton. 3 Example 3.1 General In orer to emonstrate the RO methoology a fctve example s one, n whch t s magne that the ralway management organzatons of a European freght corror, e.g. Proral n the Netherlans, DB n Germany, SBB an BLS n Swtzerlan, an RFI n Italy are the ralway management organzatons of the Rhne-Alp Corror, nee to etermne when they wll ece that nterventons shoul be execute on the corror or not. Each operator shoul make the ecson at a tme when t coul be the most benefcal for them. The OTD for one organzaton over a 30-year tme pero s etermne for a fctve ralway lnk n the freght corror. The lnk s use to transport 200,000 tons of goos per year (N), an the annual growth s expecte to be 0.5%. The salvage value of the network s assume to be 0.

7 Computers n Ralways XIV Global conton states The GCSs are gven n Table 1. The conton state of each object n the lnk (e.g. track, brges, tunnels, an the elements of whch the objects are comprse such as ballast, sleepers, an rals) are scretely escrbe n a range of 1 to 5, whch s convenent for nspecton an often use n practces. Table 1: Global conton states (GCSs). GCS 0 1 Descrpton All elements of all objects are n a new or as-new conton state (CS1). The probablty of the network provng an aequate level of servce n the upcomng year s 100%. All elements of all objects are n a goo conton state (CS2) or better. The probablty of the network provng an aequate level of servce n the upcomng year s at least 90%. Example Network level Element CS Label Track Base layer Ballast CS % % % % Track Base layer Ballast % % % % Note: The GCSs are efne n ths example purely for llustraton purposes. The efnton of such GCSs so that they are useful n practce requres conserable work [11]. 3.3 Decson strateges The possble DSs,.e. the GCS that trggers a ecson an the optons for the organzaton at that tme, are gven n Table 2. For example, DS 1 refers to the strategy when the organzaton s to make a ecson once the GCS1 s reache an the organzaton can ece to ether execute an nterventon or to o nothng. Table 2: Decson strateges. DS Trggerng Interventon to be execute rather than o nothng nterventon GCSs No. Descrpton Example 1 1 I-2 Mnor nterventon Improve the base layer an geometrc level 1 conton of some but not all of the track 2 1 I-3 Major nterventon Improve the base layers an geometrc level 1 conton of all of the track

8 444 Computers n Ralways XIV 3.4 Deteroraton an mprovement The values of the parameters of the eteroraton moel at the start of the nvestgate tme pero an followng the executon of each nterventon are gven n Table 3. For example, f the GCS1 s reache an nterventon 2 s execute then the eteroraton moel to be use followng executon of the nterventon wll have the parameter values α = an m = The effectveness of the nterventons are represente through the values of the eteroraton moel parameters (the parameter values for the o-nothng nterventon are gven as t=0 values corresponng to each GCS an can also be seen n Table 3). Table 3: Values of eteroraton moel parameters followng the executon of nterventons. Parameters GCS1 t = 0 I-1 I m It s mportant to note that the restoraton of nfrastructure to an as-new conton oes not necessarly mean that the eteroraton of the nfrastructure followng the executon of the nterventon eterorates s the same as the newly bult nfrastructure. The three cases are: The eteroraton may be the same, e.g. all objects are replace wth objects that are entcal to the orgnal objects. The eteroraton may be faster, e.g. some objects are replace wth objects entcal to the orgnal objects but others are not replace an, therefore, have a hgher falure rate than they at t = 0. The eteroraton may be slower, e.g. all objects are replace wth objects that eterorate more slowly than the orgnal objects, whch s somethng that may happen ue to the conseraton of nformaton gathere over the tme pero snce the constructon of the orgnal objects n the esgn of the new objects an the ntegraton of new technologes. The nterventons assume n ths example correspon to the thr case. The changes n the probabltes of not reachng GCS are shown n Fgure 2. These changes are shown assumng no nterventons are execute (the lower curve), an for one possble future scenaro, assumng each DS was followe (the two upper curves) an ether nterventon 1 or nterventon 2 was execute. In Fgure 2 t can be seen, for example, that f a ecson s to be mae n year 13 as to whch nterventon s to be execute there s a 30% chance that the lnk wll not yet have reache GCS1. At that pont n tme the organzaton wll be able to ece to execute the o-nothng nterventon, or to execute a mnor nterventon or major nterventon,.e. to follow DS0, DS1 or DS2.

9 Computers n Ralways XIV 445 Fgure 1: Probabltes of beng n each GCS f each IS s followe. 3.5 Impacts It s assume that mpacts are ncurre n three general tme peros from the start of the nvestgate tme pero to the executon of the frst planne nterventon, urng the executon of planne nterventons, an after the executon of planne nterventons. The mpacts that are ncurre are groupe as follows: urng the executon of unplanne nterventons, W, whch have an ncreasng probablty of happenng as the tme between nterventons ncreases, an urng the executon of planne nterventons, C, between nterventons ue to routne mantenance, R, between nterventons ue to normal operaton, O. The parameters of the moels of the mpacts are gven n Table 4. For example, f an nterventon s execute when the lnk enters GCS1, t s assume that an mpact of mu ue to scheule restrctons (W), an mpact of mu ue to the executon of the nterventon (C), an mpact of mu/year on average ue to routne mantenance (R) untl the next nterventon an an mpact of mu/year ue to normal operaton (O) snce the last nterventon. Followng the executon of an nterventon, R an O are assume to ncrease etermnstcally annually by 1% an 1.5%, respectvely. Table 4: Impact moel parameters. Parameters Unt GCS1 t=0 I-2 I-3 W 10 6 mu C 10 6 mu R 10 6 mu/year O 10 6 mu/year

10 446 Computers n Ralways XIV For further example, n year 13 a ecson wll be mae to o nothng or to execute nterventon 1 or 2, epenng on the conton of the nfrastructure n year 13 an the expecte conton state of the nfrastructure from year 13 to T. If ether nterventon 1 or 2 s execute the lnk wll be restore to an as-new conton an mpacts C wll be ncurre, where C s larger for nterventon 2 as t s more extensve (1.6 x 10 6 mu vs x 10 6 mu). In both cases O an W, whch are the same as they epen on the tme to arrval n the trggerng GCS an are nepenent of the type of nterventon execute, wll be ncurre. In both cases R wll be ncurre but the value of R wll be hgher f nterventon 1 s execute than f nterventon 2 s execute. Ths s because f a mnor nterventon s execute the routne mantenance costs wll be hgher followng the nterventon than f a major nterventon s execute. The value of each mpact type ht () n Equaton (5) s moele as a geometrc Brownan moton (Equaton (2)), wth a rft parameter = an stanar evaton =0.2. A scount rate of =2% s use. The nput was then use wth the moel escrbe n secton 3 to obtan the results n secton Results The expecte beneft of watng to etermne f an nterventon shoul be execute f each of the DSs are followe,.e. the opton value (Equaton (8)), s gven n Fgure 2 an Table 5. The opton value s the fference between the benefts f the DS s followe an the benefts f the o-nothng reference strategy s followe. Each pont n each curve represents the opton value of ecng to execute or not execute an nterventon at tme z n the nvestgate pero of 30 years. For example, f DS2 s followe an the ecson s mae n year 7 whether to execute the nterventon or not, the expecte beneft wll be mu (pont C n Fgure 2). If DS2 s followe an the ecson s mae n year 15 whether to execute the nterventon or not, the expecte beneft s mu (pont A n Fgure 2). If DS2 s followe an the ecson s mae n year 20 whether to execute the nterventon or not, the expecte beneft s mu (Pont B n Fgure 2). Fgure 2: Opton values of the nvestgate ecson strateges.

11 Computers n Ralways XIV 447 Table 5: Optmal tmes to ece an opton values of ecson strateges. Strateges Ref 1 2 Maxmum expecte benefts (10 6 mu) Optmal tme to ece (years) NA Maxmum opton value per strategy (10 6 mu) NA Expecte benefts f ecson s to be mae n year 15 NA (10 6 mu) Opton value f ecson s to be mae n year 15 (10 6 mu) NA In ths example, both DSs yel more beneft than the o-nothng reference strategy. The OTD epens on the type of nterventon to be execute an the probablty of the lnk beng n the GCS n whch t woul be ece to execute an nterventon. In ths example, the hghest opton value s obtane from DS2 when the ecson to execute an nterventon s mae n year 15. The general ncrease n opton values that occurs followng tme 0 s ue to the ncrease nformaton that the ralway management organzaton receves an, therefore, upon whch the ecson to ntervene can be base. The man reasons for the ecrease n opton values that occurs towars the en of the 30-year pero are ue to the ecrease number of years n whch benefts can be obtane f an nterventon s execute an the ncrease spee of eteroraton, whch greatly ncreases the costs of unplanne nterventons (W). Thus, the expecte beneft at t = 30 s 0. It s note that, n general, DS2 yels hgher opton values than DS1. For example, f the ecson s to be mae n year 15, DS2 yels a hgher opton value ( mu) than DS1 ( mu). The expecte beneft at t 27 of DS1 s 0, whch nfers that no beneft can be obtane from executng an nterventon after year Conclusons In ths paper, a real opton methoology s presente for the etermnaton of the optmal expecte tme n the future for a ralway nfrastructure manager to ece what types of nterventons, f any, are to be execute. Such a methoology s partcularly useful n the management of ralway nfrastructure that belongs to a multnatonal freght corror, where multple ralway management organzatons are nvolve, an every tme one executes an nterventon t can affect the flow of trans on the part of the nfrastructure controlle by others. The real opton methoology can be use n aton to the conventonal lfe cycle cost estmaton technques that have been so far use n practce. Ths wll help managers to have better eas an optons so he/she can make goo management ecsons. The methoology s emonstrate by etermnng the optmal tme to

12 448 Computers n Ralways XIV ece to execute nterventons on a fctve ral corror for a ralway management organzaton. The atonal benefts that managers can gan s the opton values compare to the o-nothng strateges or compare to other strateges that managers can ece to execute. References [1] O. Ivanova, B. V. Zeebroeck an K. Karel Sptaels, Optmal nfrastructure chargng n a mult country ralway corror, n European Transport Conference, [2] S. Santa-Cruz an E. Herea-Zavon, Mantenance an ecommssonng real optons moels for lfe-cycle cost-beneft analyss of offshore platforms, Struct. Infrastruct. Eng., vol. 7, no. 10, pp , Oct [3] H. T. J. Smt, Infrastructure Investment as a Real Optons Game: The Case of European Arport Expanson, Fnanc. Manag., vol. 32, no. 4, p. 27, [4] P. M. Pmentel, J. Azeveo-Perera, an G. Couto, Hgh-spee ral transport valuaton, Eur. J. Fnanc., vol. 18, no. 2, pp , [5] B. Agusnata, Exploratory analyss to support real optons analyss: an example from electrcty nfrastructure nvestment, n Systems, Man an Cybernetcs, 2005 IEEE Internatonal Conference on, 2005, vol. 4, pp Vol. 4. [6] D. N. For, D. M. Laner, an J. J. Voyer, A Real Optons Approach to Valung Strategc Flexblty n Uncertan Constructon Projects, Constr. Manag. Econ., vol. 20, pp , [7] B. W. Schlake, C. P. L. Barkan, an J. R. Ewars, Tran Delay an Economc Impact of In-Servce Falures of Ralroa Rollng Stock, Transp. Res. Rec. J. Transp. Res. Boar, no. 2261, [8] K. Kobayash, K. Kato, an N. Lethanh, Deteroraton Forecastng Moel wth Multstage Webull Hazar Functons, J. Infrastruct. Syst., vol. 16, no. 4, pp , [9] B. Doson, The Webull Analyss Hanbook. Qualty Press, Amercan Socety for Qualty, [10] F. Black an M. Scholes, The Prcng of Optons an Corporate Labltes, J. Polt. Econ., vol. 81, no. 3, pp , [11] M. G. H. Bell an C. Cassr, Relablty of Transport Networks. Balock, Hertforshre, Englan: Research Stues Press, 2001.