A Comparison of MCDA Techniques TOPSIS and MAROM in Evaluating Bus Alternative-fuel Modes

Transcription

1 A. Farkas A Comparson of MCDA Technques TOPSIS and MAROM n Evaluatng Bus Alternatve-fuel Modes A Comparson of MCDA Technques TOPSIS and MAROM n Evaluatng Bus Alternatve-fuel Modes András Farkas Óbuda Unversty, Faculty of Busness and Economcs 1084 Budapest, Tavaszmező út 17, Hungary e-mal: farkas.andras@kgk.un-obuda.hu Abstract. There are a great number of mult-crtera decson analyss (MCDA) methods used n practce. Most of them have been desgned to evaluate alternatves on one partcular scale of measurement only. But the alternatves are characterzed by many attrbutes whch, usually, correspond to dfferent types of measurement scale. In ths paper such a method called MAROM s compared to a classcal MCDA procedure called TOPSIS. The problem of alternatve-fuel modes of buses used for publc transportaton n urban areas s consdered. The rankng and the evaluaton procedures of the alternatve-fuel modes are shown for an emprcal study taken from the lterature. Comparson of the two methods s made on the results of the numercal computatons. The formal descrpton of the method MAROM s also presented n detal. Keywords: multple attrbute decson makng, alternatve-fuel modes of buses 1 Introducton Mult-crtera decson analyss (MCDA) and mult-attrbute decson makng (MADM) may apply to many complex decson makng processes. They are most applcable to solvng problems that are characterzed as a choce among alternatves. When used for group decson makng MADM allows the respondents to consder the values that each vews as beng mportant. In ths paper, two advanced MADM methods wll be used to produce a meanngful soluton to a transportaton problem of world-wde nterest. Typcal of the modern age s the phenomenon of urbanzaton,.e. the physcal growth of urban areas as a result of rural mgraton as well as the ever growng suburban concentraton nto many ctes, partcularly the very largest ones. The Unted Natons projected that half of the world s populaton lve n urban areas by the end of By 2050, t s predcted that 64.1% and 85.9% of the developng and developed world, respectvely, wll be urbansed [6]. Accordng to a revsed UN World Urbanzaton Prospects report, the growth wll be even faster, the total fgure s estmated to rse to 60% (4.9 bllon people) by 2030 [18]. 181

2 MEB th Internatonal Conference on Management, Enterprse and Benchmarkng 31 May - 1 June, 2013 Budapest, Hungary Under such crcumstances, ncreasng attenton has been gven to urban sustanablty over the past decades. The sustanable development of ctes largely depends upon a sound urban polcy. Transport sector s consdered one of the major contrbutors to ar and nose polluton n urban world. Emssons from motor vehcles affect strongly the health of people who are lvng n a town. Also, there s a drect relatonshp between a transport system and the ar polluton n a cty [4]. In ths context, sustanablty s defned as protecton of urban communtes from adverse development effects wth respect to human health and the envronment [8]. In partcular, ths paper focuses on modern technology and ts applcablty to mass transt systems as major contrbutors to sustaned urbanzaton. Of the varous optons, avalable for publc transportaton, effcent bus systems can be effectve and affordable. Ths paper wll buld on the excellent work of Tzeng et al. [17]. Ther research along wth a comprehensve emprcal example attempted to summarze the most promsng developments of alternatve-fuel buses sutable for the urban area and to explore ther favourable future drectons by comparng these alternatves to the characterstcs of the conventonal bus wth an nternal combuston desel engne. For ths purpose, they presented a comprehensve mult-attrbute nvestgaton of these alternatve-fuel modes wth a set of data provded by dfferent groups of Tawanese experts representng both engneerng bodes and academa. They used the method TOPSIS (Technque for Order Preference by Smlarty to Ideal Soluton) whch s one of the most popular compromse methods for evaluatng and rankng dfferent alternatves; see n [7] and [2]. TOPSIS defnes the best opton as the one that s closest to the deal opton and farthest away from the negatve deal pont. The authors of [17] clamed the assessment of the performances of the alternatve-fuel vehcles s based on experts judgements wthout usng mathematcal models for the evaluatons of crtera as ther contrbuton. Author of the present paper has consdered another concept n hs MCDA methodology called MultAttRbute Object Measurement (MAROM) [5]. Although MAROM was also desgned to be capable of nvolvng mxed data,.e. both tangble and ntangble attrbutes, however, t requres that the non-quantfable and quantfable attrbutes be treated n dfferent manners. In MAROM, the raw data, ether elcted from experts judgments or arsen from physcal measurements, are preserved for the computatons n forms of bnary varables, rank numbers and quanttatve data dependng upon ther correspondng scale of measurement,.e. nomnal, ordnal, nterval or rato. Ths way, they are not subject to any arbtrary transformaton onto a hgher or a lower order scale. Hence, to elmnate the dfferent unts of measurement by standardzaton or normalzaton, can only put through wthn the same type of scale of measurement. These features of MAROM are fundamentally dfferent to those of comprsed by the TOPSIS method. Our major objectve wll be the comparson of the prorty rankngs and the aggregate scores of the alternatve-fuel buses resulted from the use of the two methods MAROM and TOPSIS when they are appled to the same data set. 182

3 A. Farkas A Comparson of MCDA Technques TOPSIS and MAROM n Evaluatng Bus Alternatve-fuel Modes 2 Choces and characterstcs of alternatve-fuel modes In the semnal paper of Tzeng et al. [17], 12 alternatve-fuel modes are consdered. Cted from the papers [17], [10] and [16], the man characterstcs of these twelve alternatve-fuel modes, denoted by AF_ k, where k =1,,12, are now descrbed: AF 1: Conventonal desel engne The conventonal desel engne bus s employed all over the world. It s the most effcent among all exstng nternal combuston engnes, makng t one of the major contenders as a power source n the 21st century. Its man advantages are the low purchasng costs, the flexblty to the speed of traffc and the low senstvty to road faclty. However, t has very hgh exhaust emsson rates (PM, NO x, CO x ). Ths vehcle s ntroduced n the set of alternatves n order to compare t wth the new fuel modes. We menton here that bo-desel fuels (e.g. Soy Desel) as possble substtutes for petroleum desel are not consdered here. AF 2: Compressed natural gas CNG Natural gas s used n several forms as vehcle fuel, stored on board,.e., compressed natural gas (CNG), lquefed natural gas (LNG), and attached natural gas (ANG). The CNG vehcle has already been commercalzed around the world and s matured n ts technology (there are about four mllon CNG vehcles n the world). Natural gas s a mxture of hydrocarbons, manly methane, produced ether from gas wells or together wth crude ol producton. Interest for natural gas as an alternatve fuel arses from ts clean burnng qualtes, ts wde resource base and ts commercal avalablty to users. The compressed natural gas vehcle s wdespread n countres wth ther own natural gas. Natural gas has numerous benefts n terms of economcs, pollutants, greenhouse gases, safety, general abundance and costs less than that of the desel ol. CNG vehcles emt only slght amounts of carbon doxde and carbon monoxde and they have hghoctane value; thus, they are sutable for utlzaton as publc transportaton vehcles. AF 3: Lquefed propane gas LPG There are countres that have used ths mode of fuel for publc transportaton. In Japan, Italy, and Canada, as much as 7% of the buses are powered by LPG, and several other European countres are also plannng to employ LPG vehcles, manly due to polluton consderatons. 183

4 MEB th Internatonal Conference on Management, Enterprse and Benchmarkng 31 May - 1 June, 2013 Budapest, Hungary AF 4: Fuel cell (hydrogen) The so called fuel cell battery can transform hydrogen and oxygen nto power for vehcles, but hydrogen s not sutable for onboard storage. The research on fuel cell hydrogen buses has already been concluded wth success, and test results wth the expermental vehcle operatng on hydrogen fuel ndcate that ths vehcle has a broad surface n the burnng chamber, low burnng temperature and the fuel s easly made nflammable. Damler Benz Company has already developed a prototype vehcle wth a fuel cell. To date, the only vehcles offered for sale wth fuel cell technology s the Zevco London tax whch was launched n London n July Due to the fact that the energy to operate ths vehcle comes from the chemcal reacton between hydrogen and oxygen, no detrmental substance s produced and only pure water, n the form of ar, s emtted. A fully loaded fuel tank can last as far as 250 km. Fuel cells generate electrcty n order to power the vehcle from an electrochemcal reacton between hydrogen and oxygen under controlled condtons. The only waste generated n ths process s water vapour. Hydrogen s energy densty s very low compared to that of the methanol and especally when compared to gasolne s. However, ths low densty requres very large and heavy tanks on board of the vehcle. Addtonally, t would be necessary to create an entre new nfrastructure,.e., to set up refuelng statons. AF 5: Methanol The fuel of methanol s related to vehcles wth gasolne engnes. The combnaton rate of methanol n the fuel s 85% (so called M85). The engne that can use ths fuel wth dfferent combnaton rates s termed as flexble fuel vehcle (FFV). The FFV engne can run smoothly wth any combnaton rate of gas wth methanol. Ths way, methanol wll act as an alternatve fuel and helps to reduce the emsson of black smoke and ntrous oxdes (NO 2 ). A great number of ntense experments are pursung wth methanol, especally n the US. Fuel statons provdng methanol are already avalable n Japan snce The thermal energy of methanol s lower than that of the gasolne, but the capablty of contnuous travelng by ths vehcle s nferor to conventonal vehcles. When methanol becomes usable as an alternatve fuel, t wll sgnfcantly reduce vehcle emssons of pollutants and greenhouse gases. AF 6: Electrc vehcle opportunty chargng The source of power for the opportunty chargng electrc vehcle (OCEV) s the combnaton of a loaded battery and fast opportunty chargng durng the tme the bus s dle when stopped. Whenever the bus starts from the depot, ts loaded battery wll be fully charged. Durng the seconds when the bus s stopped, the power recepton sensor on the electrc bus (nstalled under the bus) wll be lowered to the chargng supply plate nstalled n front of the bus stop to charge the battery. Wthn 10 seconds of a stop, the battery s charged wth 0.15 kwh power (dependng on the desgn of the power supply faclty), and the power suppled s adequate for t to move to the next bus stop. 184

5 A. Farkas A Comparson of MCDA Technques TOPSIS and MAROM n Evaluatng Bus Alternatve-fuel Modes AF 7: Drect electrc chargng A zero-emsson alternatve to petroleum that has been avalable for many years s electrcty, and s an opton currently used n many ctes wth electrc-cable buses. Recent technology, however, uses electrcty ndependently of a fxed electrc cable by usng fuel cells or battery storage. The bg appeal of electrcty s havng a clean and quet operatng system. Some ctes and countres have begun to use electrc buses, but ther future s unlkely because of the hgh costs. Ths type of electrc bus s n the prototype desgn stage. The power for ths vehcle comes manly from the loaded battery. Once the battery power s nsuffcent, the vehcle wll have to return to the plant to conduct rechargng. The development of a sutable battery s crtcal for ths mode of vehcle. If a greater amount of electrcty can be stored n the battery, the crusng dstance by ths vehcle wll be longer. AF 8: Electrc bus wth exchangeable batteres The objectve of an electrc bus wth an exchangeable battery s to affect a fast battery charge and to acheve a longer crusng dstance. The bus s modfed to create more on-board battery space and the number of on-board batteres s adjusted to meet the needs of dfferent routes. The fast exchangng faclty has to be ready to conduct a rapd battery exchange so that the vehcle moblty can be mantaned. AF 9: Hybrd electrc bus wth gasolne engne The electrc-gasolne vehcle has an electrc motor as ts major source of power and a small-szed gasolne engne. When electrc power fals, the gasolne engne can take over ts functon and contnue the trp. The knetc energy rendered durng the drve wll be turned nto electrc power to ncrease the vehcles crusng dstance. AF 10: Hybrd electrc bus wth desel engne The electrc desel vehcle has an electrc motor and a small-szed desel engne as ts major sources of power. When electrc power fals, the desel engne can take over and contnue the trp, whle the knetc energy rendered durng the drve wll be turned nto electrc power to ncrease the vehcles crusng dstance. AF 11: Hybrd electrc bus wth CNG engne The electrc-cng vehcle has an electrc motor and a small-szed CNG engne as ts major sources of power. When electrc power fals, the CNG engne takes over and provdes the power, wth the knetc energy produced s converted to electrc power to permt contnuous travel. AF 12: Hybrd electrc bus wth LPG engne The electrc-lpg vehcle has an electrc motor and a small-szed LPG engne as ts major sources of power. When electrc power fals, the LPG engne takes over and provdes the power, wth the knetc energy produced s converted to electrc power to permt contnuous travel. 185

6 MEB th Internatonal Conference on Management, Enterprse and Benchmarkng 31 May - 1 June, 2013 Budapest, Hungary A comprehensve lterature revew on alternatve-fuels for mass transt shows a varety of approaches to new energy technologes for buses [9]. Perhaps the most mportant source of nformaton on the development of alternatve-fuels can be found n the U.S. Department of Energy s Alternatve Fuels Data Center (AFDC) whch mantans an Alternatve Fuels Data Base [1]. Ths webste provdes a comprehensve source of nformaton on alternatve-fuels through collectng operaton nformaton from vehcles runnng on alternatve-fuels, analyzes the data, and makes them avalable to the publc. Insght nto operaton of alternatve-fuel buses s provded n a CUTR report of alternatve-fuels used by buses [11]. Ths report analyzed ar qualty data and dentfed the most pressng ar qualty problems that could be addressed by an alternatve-fuel program n the US. It consdered alternatve-fuel vehcles n transt and evaluated advantages and dsadvantages of each of eght fuel types: Reformulated gasolne and desel fuel (RFG, RFD), Propane - beng the man ngredent n lquefed petroleum gas (LPG), Compressed natural gas (CNG), Lquefed natural gas, Ethanol, Methanol, Bodesel, Electrc vehcles (ncludng EVs wth solar rechargng statons). 3 Weghtng of crtera and evaluaton of alternatvefuels In the paper of Tzeng et al. [17], 11 evaluaton crtera were establshed. Ths set of crtera (attrbutes) was revealed as a result of an nteractve group decson makng process n the course of whch bus manufacturng professonals, academc communty people, researchers and bus operatons experts have partcpated. These eleven sngle crtera (attrbutes), wth respect to whch each alternatve-fuel mode wll be evaluated are as follows [17]: C 1: Energy supply Yearly amount of costs of supply, storage and fuel C 2: Energy effcency Energy consumpton related to fuel heatng value C 3: Ar polluton Chemcal substance harmful to health C4: Nose polluton Nose effect produced durng operaton tme C5: Industral relatonshp Impact on other locomotve ndustry branches C6: Costs of mplementaton Costs of producton, purchase and mplementaton C 7: Costs of mantenance Yearly costs of mantenance C 8: Vehcle capablty Crusng dstance, gradeablty, speed of vehcle, etc. C 9: Road faclty Necessary features of road for the vehcle, e.g. pavement, slope C 10: Speed of traffc flow Conformty to traffc flow C 11: Sense of comfort Travelng comfort and aesthetc appeal 186

7 A. Farkas A Comparson of MCDA Technques TOPSIS and MAROM n Evaluatng Bus Alternatve-fuel Modes In Table 1, the normalzed average weghts,.e., the relatve mportance values for every crteron are ndcated [17, p.1377], whch were determned by groups of experts usng the AHP method [12]. In Table 1, the averages of the assessed values for each alternatve-fuel mode wth respect to every crteron are also presented. These values, denoted by u j, 0 u j 1, are taken from [17, p.1378]. The evaluaton results have been derved through conductng a survey by applyng a Delph procedure that was repeated twce. The experts have been selected on a well-grounded manner. They represented manufacturng ndustres, governmental departments, energy commttees, research nsttutes and academc staff from hgher educatonal nsttutons. C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 10 C 11 Weght AF AF AF AF AF AF AF AF AF AF AF AF Table 1 Value assessment for the alternatve-fuels and the crtera weghts [17] 187

8 MEB th Internatonal Conference on Management, Enterprse and Benchmarkng 31 May - 1 June, 2013 Budapest, Hungary 4 Comparson of the results from TOPSIS and MAROM The MAROM procedure (see n the Appendx) requres that nature of the data of each crteron be adequate to the propertes of the type of the scale of measurement to whch these data correspond to. Therefore, as ts frst step, each crteron s assgned to the approprate scale of measurement. Furthermore, we extended the number of crtera from 11 to 15, snce n the paper of Tzang et al. [17] some addtonal nformaton s also presented whch were not drectly captured n ther analyss. These data relate to some mportant engneerng/chemcal characterstcs of alternatve-fuels whch stem from relable sources (physcal measurements). They are gven n [17, p ] wth ther accompanyng unts of measurement. To preserve the unformty of the two data sets as much as possble whch were used by Tzang et al. [17] and the present author, see Table 1 and Table 2, respectvely, mnmal changes have been made only. Ths way, crtera C4, C5, C9, C10 and C11 of the orgnal data set have been retaned, but were assgned to ordnal scales so that ther orgnal performance values, u j gven n Table 1, were converted to rank numbers usng a nne-grade ordnal scale [1, 1.5, 2, 2.5,, 5], where an deally best performance, f there s any, would receve

9 A. Farkas A Comparson of MCDA Technques TOPSIS and MAROM n Evaluatng Bus Alternatve-fuel Modes Table 2 Input data of the alternatve fuel-modes for MAROM AF1 AF2 AF3 AF4 AF5 AF6 AF7 AF8 AF9 AF10 AF11 AF12 Nomnal Aggregated weght of nomnal scale scale Crteron weght Best value on nomnal scale for crteron C1 1 1.Depot Ordnal Aggregated weght of ordnal scale scale Crtera weghts Best values on ordnal scale for crtera C2-C Road faclty Nose polluton Indust. rel.shp Speed of traffc Sense of comfort Interval Aggregated weght of nterval and rato scales scale Crteron weght Best value on nterval scale for crteron C Worst value on nterval scale for crteron C Energy effcency [dm.less] Rato Crtera weghts scale Best values on scale for crtera C8-C Worst values on scale for crtera C8-C Fuel costs [1000 NT$] 9.Exhaust emsson (PM+NO x+hc+co x) [%] 10.Crusng dstance [km] 11.Number of passengers [No] 12..Max speed [km/h] 13.Recharge tme [mn] 14.Costs of mplementaton [1000 NT$] 15.Costs of mantenance [1000 NT$] 189

10 MEB th Internatonal Conference on Management, Enterprse and Benchmarkng 31 May - 1 June, 2013 Budapest, Hungary Utlzng the techncal data collected by [17], several new crtera are now ntroduced. As seen n Table 2, they are: Depot, whch can be small or large characterzng the depostary needs of the buses, s a nomnal varable [0 or 1], whle Crusng dstance, Number of passengers, Maxmum speed for urban/suburban servces and Recharge tme are rato scale varables wth partcular unts of measurements. They consttute the extended form of the old crteron Vehcle capablty whose weght has been unformly allocated to them. The old crteron Energy effcency s a dmensonless varable, snce t gves the rato of the alternatve-fuel effcency/fuel heatng value related to that of the desel bus, and hence, t s reasonable to assgn t to an nterval scale. We hope that establshng ths new data base for the same problem wll provde us more powerful and relable evaluaton outcomes. Table 2 presents ths reformulaton of the orgnal data set that meets the requrements of the theory of measurement. Here, the characterstc values for the 12 alternatve-fuel buses, the 15 partly modfed crtera weghts and the aggregated weghts for the scales of measurement are gven. Table 3 Comparson of the rankngs and the evaluaton scores for TOPSIS [17] and MAROM TOPSIS MAROM Rank Score Rank Score Rank Score Rank Score Basc Compr. Indv. Aggreg. Electrc bus wth exchangeable batteres Electrc bus wth opportunty chargng Electrc bus wth drect chargng Hybrd electrc wth gasolne engne Hybrd electrc wth CNG engne Hybrd electrc wth LPG engne Hybrd electrc wth desel engne Fuel cell (hydrogen) Methanol Compressed natural gas engne (CNG) Lqudate propane gas engne (LPG) Conventonal desel engne bus The results of the mult-crtera evaluaton of the 12 alternatve-fuel buses are shown n Table 3. Here, both the ranks and the evaluaton ndces called relatve standngs or scores yelded by TOPSIS (basc and compromse solutons) and MAROM (for the 190

11 A. Farkas A Comparson of MCDA Technques TOPSIS and MAROM n Evaluatng Bus Alternatve-fuel Modes ndvdual and the aggregate weghtng cases) are ndcated on nterval scales. As t does not come as a surprse, the two methods have produced rather dfferent rankngs and scores. Comparsons on the fndngs, however, should be made very carefully. As a remarkable outcome, observe the bg dfferences n the ranks of the conventonal desel engne bus. The last poston of the desel engne n the TOPSIS rankngs seems to be very strange regardng the fact that Tzeng et al. s nvestgatons refer to the year It s also strkng that there are rather sgnfcant dfferences n the prorty scores of the alternatve-fuel modes produced by the two methods. We ntend not to go nto detaled techncal explanatons, only to menton that author, as beng a vehcle engneer, strongly beleves that the MAROM rankng reflects better the at that tme exstng stuatons than that of TOPSIS. The relatve hgh postons of the conventonal desel engne bus n the MAROM rankngs as opposed to those of the alternatve-fuel modes follows manly from the tardness of the requred engneerng developments and the bus manufacturng capabltes as well as the weak achevements of the cvl ntatves concernng envronmental protecton ssues. However, there s no doubt as urban mass transt technology wll get stronger and mprove n the future, then more buses wll be powered by alternatve means n the search for more effcent energy use, cleaner ar, queter operatons and more travelng convenence, especally, when they wll be able to effcently serve n suburban areas as well. Conclusons In ths paper, two MADM methods the TOPSIS and the MAROM have been compared based on numercal experence. These methods have been appled to the same emprcal example, the selecton of alternatve-fuel buses operatng n urban areas. Although the results produced by the two methods were qute dfferent, however, both approaches have shown that the use of alternatve-fuel modes to mprovng human health and the envronment offers huge opportuntes for ensurng sustanable urban developments. Appendx The formal descrpton of the method MAROM s presented below: Consder the followng data matrx: A = [ a ], = 12,.., m; k = 1,..., n, ( 1) k nvolvng n optons (alternatves). The n columns gve for every opton the values of m varables (row vectors) denotng varous characterstcs (attrbutes, crtera) of these alternatves. In (1), a value (crsp number) s assgned to each entry a k whch s ether elcted from respondents judgments or arsen from physcal measurements. Thereby, the nature of a partcular data may be of a subjectve type (qualtatve) and/or an objectve type (quanttatve). A column vector a k of matrx A represents a composte vector a k =(a k (N), a k (O), a k (I), a k (R) ) whch s parttoned nto four blocks. Thus, A conssts of varables of mxed type, where the superscrpt N refers to nomnal (usually bnary), O to ordnal, I to nterval and R to rato varables. Of course, n a concrete real-world case, varables of any type may be mssng. 191

12 MEB th Internatonal Conference on Management, Enterprse and Benchmarkng 31 May - 1 June, 2013 Budapest, Hungary An addtonal column vector, denoted by b, and called a reference vector, s to be constructed whch represents an deal (hypothetcal) opton, entres of whch are composed of the best values of the set of alternatves wth respect to each attrbute. It has the same element-wse structure as that of vector a k. Numercal scales are: [0,1] on a nomnal; [1,,5] on an ordnal; [0,,1] or [actual values,.e. row data emergng from measurements] on an nterval and/or on a rato scale, respectvely. Because of rato scale (and sometmes nterval) varables have usually dfferent unts of measurements the row vectors a (R)T (and a (I)T ) are standardzed so that ther means are equal to 0 and ther standard devatons are equal to 1. E.g., for the rato varables, these standard devatons can be obtaned as (R) k (R) s a a,,.., ; k,...,. k (R) m(r) 2 (R) (R) (R) = 1 n n 2 1 m n k (R) (R) = 1 = 1 ( 2) 1 = 1 = 1 Wth (2), the standardzed elements are 1 '( R) ( R) ( R) ak = ( R) ( ak a. ), = 1,.., m; k = 1,..., n. ( 3) s A representatve group of respondents (experts, customers, users, etc.) s then formed. Every commttee member should evaluate each alternatve by supplyng hs judgments on each qualtatve varable wth respect to the nomnal and ordnal scaled crtera. It s recommended that the number of voters l, l =1,,q, to be at least 10 persons. The mult-attrbute decson makng model for preference measurng s as follows l l l l d = w d + ε, k = 1,.., n; l = 1,..., q, ( 4) k m = 1 k k where k l s the overall dstance of alternatve k from the deal alternatve for the lth voter; w l s the weght of attrbute ; d k s the dstance of the kth alternatve (object) from the reference pont on attrbute ; ε k s the value of an error random varable whch may nclude model msspecfcaton, measurement errors and respondents uncertantes. To determne the weghts of the attrbutes, w l, =1,,m, the analytc herarchy process (AHP) method s proposed [12]. These weghts are then usually normalzed, so that n k=1w l =1. The dstance measure d k n Eq. (4) takes on dfferent functonal forms for alternatve k: (a) For the nomnal vectors, d k (N) (a k (N),b (N) ), denotng them smply as x,y 0 N, the dstance measure s the Tanmoto (also called Jaccard) coeffcent [14]: where ( N d ) α β + γ ( x, y) = 1 =, k α + β + γ α + β + γ 192

13 A. Farkas A Comparson of MCDA Technques TOPSIS and MAROM n Evaluatng Bus Alternatve-fuel Modes α = mn( x, y ), β = x α, γ = y α, N. (b) For the ordnal vectors d k (O) (a k (O),b (O) ), denotng them as x,y 0 O, the dstance measure s the Soergel measure [13]: x + y 2 mn( x, y ) ( O d ) k ( x, y) = O x + y mn( x, y ),. (c) For the nterval vectors and the rato vectors, d k (I,R) (a k (I,R),b (I,R) ), denotng them as ether x,y 0 I, or x,y 0 R and ntroducng the L 2 norm of a vector x, 2 x = x = x T x, I, or R, 2 the dstance measure s the well-known Eucldean-metrc: ( I, O) T d ( x, y) = x y = ( x y) ( x y). k 2 Snce the metrc propertes hold for the above dstance functons used n the model (4) (see the proofs n [15]), n whch lnear scales are appled, therefore, the addtve type composte vector k l s also metrc. Furthermore, t s unque and for each of the partal vectors 0 d k (a k ( )T,b T ) 1. The dstance between any two composte vectors s proportonal to the degree of ntensty. The proportonalty unt s taken to be 1. Once the overall parwse dstances between each composte vector and the reference vector have been determned, the (column) vector of the relatve standngs or scores, s=(s k ), k=1,2,...,n, and thus the prorty rankng of the alternatves smply yelds as s=1 d k, respectvely. To establsh ether a [0 1] or a [1 100] nterval scale, a smple normalzaton procedure should be performed. To aggregate the ndvdual preferences nto a compromse rankng the mnmum varance method [3] s proposed. References [1] Alternatve Fuels Data Base. U.S. Department of Energy s Alternatve Fuels Data Center (AFDC). ( [2] Chen,S.J. and Hwang,C.L., Fuzzy Multple Attrbute Decson Makng: Methods and Applcatons. Sprnger Verlag. Berln, [3] Cook,W.D. and Seford,L.M., On the Borda-Kendall consensus method for prorty rankng problems, Management Scence. 28, (1982), [4] Faz,A.,Weaver,C.S. and Walsh,M.P., Ar Polluton from Motor Vehcles. Standards and Technologes for Controllng Emssons. The World Bank. Washngton DC,

14 MEB th Internatonal Conference on Management, Enterprse and Benchmarkng 31 May - 1 June, 2013 Budapest, Hungary [5] Farkas,A., Prorty rankng methods: A survey and an extenson, n: Busness Research and Management; Challenges n Central and Eastern Europe. (ed. Peter.S), IMC, Budapest, (1994), [6] [7] Hwang,C.L. and Yoon,K, Multple Attrbute Decson Makng Methods and Applcatons. Sprnger. New York, [8] Kazm,C. Evaluatng the envronmental mpact of alternatve-fuel vehcles. Journal of Envronmental Economcs and Management. 33, (1997), [9] Lynch,T.A, Elason L. and Dzurk,A., Energy and Envronmental Performance of Exstng Publc Transportaton Technologes. USDOT Research Report. DTRS93-G-0019 NUT14-FSU4. Florda State Unversty. College of Engneerng, Tallahassee, FL, [10] Morta,K., Automotve power source n 21st century. Journal of Socety of Automotve Engneers of Japan. 24, (2003), 3-7. [11] Reports of Alternatve-fuels Used by Buses. Center for Urban Transportaton Research (CUTR). Unversty of South Florda, FL, USA. ( [12] Saaty,T.L., A scalng method for prortes n herarchcal structures. Journal of Mathematcal Psychology. 15, (1977), [13] Soergel,D., Mathematcal analyss of documentaton systems. Informaton Storage Retreval. 3, (1967), [14] Sokal,R.R. and Sneath,P.H.A., Prncples of Numercal Taxonomy. Freeman and Co, San Francsco, [15] Späth,H., Cluster Analyss Algorthms. Wley, New York, [16] Sperlng,D., Future Drve Electrc Vehcles and Sustanable Transportaton. Island Press. Washngton DC., [17] Tzeng,G-H, Ln,C-W and Oprcovc,S., Mult-crtera analyss of alternatvefuel buses for publc transportaton. Energy Polcy. 33, (2005), [18] World Urbanzaton Prospects, the 2011 Revson. Unted Natons, Department of Economc and Socal Affars. Populaton Dvson. Workng Paper