Vladimir PAPI], Jovan POPOVI] 1. INTRODUCTION

Transcription

1 Yugoslav Joural of Operaos Research 200 umber 77-9 VEHICLE FLEET MAAGEMET: A BAYESIA APPROACH Vladmr PAPI] Jova POPOVI] Faculy of Traspor ad Traffc Egeerg Uversy of Belgrade Belgrade Yugoslava Absrac: Ths paper focuses o compaes ha have boh a flee o serve cusomers ad depos for vehcle maeace. Whe maageme of such a vehcle flee s cosdered oe of he mos mpora problems s he compuao esmao of he relably ad he avalably of he vehcles. Wha ofe maes hs compuao dffcul s he suffce daa for sascal ferece or oal lac of daa such a aggravag crcumsace arses for eample wheever he flee s reewed. Cocerag o such a case hs paper preses some aalycal formulae based o he Bayesa approach o uceray ha corbue o he soluo of he problem. Keywords: Vehcle flee maageme uceray Bayesa approach probably dsrbuos.. ITRODUCTIO The flees cosdered hs paper are maaed depos ha belog o he same compay as he flee. Whe he compay orgazes s curre acves ad plas fuure oes mus mae a log-erm geeral pla for employg he vehcles ad a eecuve wor pla EWP for he prese me ad ear fuure. Alog wh mag plas he compay mus esure ha he ecessary umber of approprae vehcles are worg codo whe eeded accordg o he EWP ad ha here are also sad-by vehcles avalable. Ths requres formao abou he sze ad srucure of he vehcle flee he codo of each vehcle whch s subjec o chages ad he maeace facles he depos. Iformao abou chages vehcle codos ad relably daa parcular cosue he bass for: deermg he probably of fulfllg he ass specfed by he EWP [6] defg he cocep of correcve maeace ad he sysem of preveve maeace defg he capacy of maeace facles ad he orgazao of he maeace sysem

2 78 V. Pap} J. Popov} /Vehcle Flee Maageme: A Bayesa Approach compug he avalably of he vehcles ad defg he ecessary raspor capaces he umber ad ype of vehcles whch correspod o he se of raspor demads. How s hs o be doe f here s a lac of approprae formao? I oher words how ca we deal wh he descrbed uceray? The objecve of hs paper s o help solve he problem of predcg he sae of he vehcle flee. The esece of uceray vehcle flee maageme ad maeace due o he lac of daa for sascal ferece s ae o cosderao. Sce o mehod has bee elaboraed o deerme he relably ad avalably of he vehcle flee uder ucera codos hs paper suggess a mehod based o he Bayesa reame of uceray whch ca clude all specfc feaures of he problem. 2. STATEMET OF THE PROBLEM A vehcle flee s usually a heerogeeous se comprsg vehcles of dffere srucure ad age. Vehcles of he same srucure ad age form homogeeous subses whch ca be called cosruco-operao groups or CO groups. Ofe a CO group comprses a small umber of vehcles. A heerogeeous vehcle flee havg more ha j oe CO group would have veory vehcles he j-h CO group. Our cosderao ca be cofed o oly oe CO group sce he procedure ca be repeaed for each CO group separaely. So hereafer de j wll be omed. Each CO group of vehcles has rasporao ass o fulfll a gve me perod he ass beg defed by he EWP. By preseg he varables releva o he EWP o a char me beg he abscssa for a CO group uder cosderao we ca vsualze he relao bewee he umber of vehcles ecessary o fulfll he rasporao ass ad he umber of avalable vehcles echcally f for operao. s a o-radom fuco of me. Ths fuco s ofe perodcal perods T beg oe day oe wee ec. A eample s gve Fg.. a T Fgure.

3 V. Pap} J. Popov} /Vehcle Flee Maageme: A Bayesa Approach 79 - The varables uder cosderao are: umber of vehcles a - umber of avalable vehcles a radom varable m -umber of vehcles o f for operao due o falure or eed of regular maeace a radom varable - umber of vehcles ecessary o fulfll he ass o - umber of vehcles operao a o-radom varable f o ad a radom varable f o a < s - umber of sad-by vehcles f for operao - vehcles reserve a radom varable - momes whe eher he umber of ecessary vehcles or he umber of avalable vehcles a chages. Three ypes of ervals ca be dsgushed o he char Fgure :. ervals such as ad whch < a ad hece o < a ad s > 0. These are he ervals whch all ass are fulflled ad s vehcles are reserve. 2. ervals such as 4 5 whch a ad hece o a whle s 0. I such ervals all ass are fulflled bu a a hgh rs sce here are o sad-by vehcles o vehcles reserve. 3. ervals such as 2 ad 3 4 whch > a ad hece o a < wh s 0. I such ervals some ass cao be fulflled. Sce all ass wll be fulflled oly whe a.e. whe m follows ha for gve ad lmaos are pu o he umber of vehcles o f for operao m.

4 80 V. Pap} J. Popov} /Vehcle Flee Maageme: A Bayesa Approach Vehcles are o f for operao eher due o regular maeace acves or due o falures. Cosderg he fluece of varous ds of falures o he process of rasporao hree groups of falures ca be dsgushed: A - B - C - "umpora" falures - hose whch do o drecly fluece he basc fucog of a vehcle ad safe drvg. The falure s defed afer he vehcle has compleed s as ad has reured o he depo. The umber of vehcles wh hs d of falure wll be I deoed by m. "delay provog" falures - hose whch obsruc he basc fucog of he vehcle or safe drvg bu ca be correced o he spo. The vehcle may be repared by he drver or by a group of reparme so ha he vehcle complees s as wh some delay. The umber of vehcles wh hs II d of falure wll be deoed by m. "crcal" falures - hose whch hamper he basc fucog of he vehcle or safe drvg ad are o be ae care of a he depo. III The umber of vehcles wh hs d of falure wll be deoed by m. By adopg hs classfcao of falures he sae of he vehcles for he me perod uder cosderao may be represeed by he dagram show Fgure 2. m MAITEACE I THE DEPOT m VEHICLE CHECK-I I THE DEPOT a m VEHICLES I OPERATIO m MAITEACE BY THE DRIVER OR BY A ITERVETIO GROUP OF WORKERS Fgure 2. The possbly of falures ad he ecessy of regular maeace requre he esece of sad-by vehcles s. The fluece of falures from group A o he sze of he sad-by flee ca usually be egleced sce s s rarely dmshed due o hese falures. The repar capaces a he depo are usually suffce o correc hese falures before he deparure me of he vehcle scheduled by EWP. The fluece of falures from group B s small sce such falures requre he use of a sad-by vehcle oly f he delay overlaps he begg of he e as assged o he vehcle.

5 V. Pap} J. Popov} /Vehcle Flee Maageme: A Bayesa Approach 8 Falures of group C have a srog fluece o he sze of he "sad-by" group of vehcles. Due o hese falures a vehcle from he sad-by flee mus be moblzed o coue he as. Ofe hese falures requre a ow vehcle o be moblzed as well. The mome he vehcles are roduced o operao s mpora o ow he regulary of appearace of varous ds of falures ad he dyamcs of her elmao. To our owledge aemps o solve hs problem whch corporae he above d of uceray are o foud he avalable leraure. The possbly of predcg chages he sae of he vehcle flee obag owledge abou he fucog of he maeace sysem ad havg approprae ools o quafy hese processes whch s of ulmae cocer here would eable he deermao of a sad-by flee ha sasfes real demads ad reduces coss. To reach hs goal we sared from he assumpo ha he me bewee cosecuve falures s dsrbued epoeally for each vehcle. As s ow from relably heory [348] ad llusraed usg epermeal daa ad he smulao mehod [2] hs assero s vald uder he followg codos: he sysem vehcle may be regarded as a comple sysem srucured o s compoes assembles ha are muually depede regard o possble falure; he umber of compoes.e. he umber of possble ypes of falures s s large a leas several doze ; each compoe has s ow dsrbuo of he me bewee falures; ay falure of ay of he compoes resuls he falure of he sysem.e. of he vehcle. Respecg he foregog codos oe ca smulae he falure of each compoe by assgg o each a cera dsrbuo. These dsrbuos ca he be combed o gve a superposed dsrbuo whch afer applyg a sascal es proves o be epoeal. The CO groups may dffer he umber of compoes ad hey usually dffer he dsrbuos assged o each compoe. Cosequely he CO groups usually dffer he parameers of he resula epoeal dsrbuos ad falure raes. Due o he assumpo ha he resula dsrbuo s a epoeal dsrbuo he uceray ca be corporaed hrough he uow falure rae. The codoal dsrbuo of he umber of falures me perod herefore has a Posso dsrbuo P ad he Posso dsrbuo has a aural cojugae [49]. ow all he codos ecessary o use he Bayesa approach o uceray are fulflled. I dealed aalyss hs umber ca amou o several housad.

6 82 V. Pap} J. Popov} /Vehcle Flee Maageme: A Bayesa Approach 3. SOLUTIO OF THE PROBLEM I order o solve he problem he esseal queso o be aswered s he followg: how o oba codos of uceray he dsrbuo of he umber of falures per u me erval. U me ervals called sequeces are oos assocaed o sequeal plag ad record eepg. A he ed of he -h sequece order o pla for sequece respecg he umber of dspesable vehcles he eeded capacy of he maeace depo ad oher elemes of he logsc suppor s ecessary o predc he umber of falures sequece. The umber of falures recorded he -h sequece s o be used a he begg of sequece o correc he parameers of he falure rae dsrbuo. Le us ow cosder a homogeeous se of vehcles each havg he same * falure rae whch s uow. Falure rae s he average umber of falures a sequece. I accordace wh he Bayesa approach a uow rae s reaed as a radom varable. I order o have compable dsrbuos [ 4 9] hs case a gamma dsrbuo s chose: * Γ 0 < < A he begg of he Bayesa reame of uceray.e. a he begg of he frs sequece he a pror values of parameers ad he gamma dsrbuo mus be deermed. Ths s doe eher o he bass of avalable paral formao or subjecvely whe daa abou he falure rae do o es [4 7]. Due o he addvy of he gamma dsrbuo he whole se of vehcles.e. he whole CO group s characerzed by falure rae whch s reaed as a radom varable wh a gamma dsrbuo: Γ 2 Ths dsrbuo he a pror dsrbuo s characerzed by he desy fuco: e f Γ > 0 The margal dsrbuo of he umber of falures he whole group of vehcles durg he frs sequece u me erval deoed by ca be obaed from he followg equao: 3 P P 0 Γ! Γ f d 4

7 V. Pap} J. Popov} /Vehcle Flee Maageme: A Bayesa Approach 83 from whch follows ha has a egave bomal dsrbuo.b.:.b. 5 Whe compug dvdual probables accordg o 5 he followg recurre formula s used: P P P 6 Ths formula s used o predc he umber of falures a homogeeous group of vehcles durg he frs sequece. If durg he frs cosecuve sequeces u me ervals... 2 we have regsered... 2 falures for he whole CE group uder cosderao he a poseror dsrbuo of wll be: ] [... Γ e d f P f P f Ths meas ha he a poseror dsrbuo of s aga a gamma dsrbuo:... 2 Γ 8 A comparso of he a poseror dsrbuo 8 o he a pror dsrbuo 2 reveals how he umber of regsered falures... 2 ad he umber of pas sequeces are used o correc he parameers of he dsrbuo. Hece sequece u me erval he dsrbuo of he umber of falures whch wll serve for plag purposes s obaed from he followg equao: d f P P! Γ Γ 9

8 84 V. Pap} J. Popov} /Vehcle Flee Maageme: A Bayesa Approach Thus from 9 follows ha he dsrbuo s aga a egave bomal dsrbuo:.b. 0 The epeced umber of falures ad he correspodg varace respecvely are gve by: E ad 2 V The se of falures for whch he former calculaos were performed was he se comprsg all falures: "umpora" falures ype A "delay provog" falures ype B ad "crcal" falures ype C. ow calculaos wll be performed frs for he subse comprsg oly B ad C falures he uo of subses B ad C ad he for he subse comprsg oly C falures. Le us loo for he dsrbuo of he umber of falures of ype B or C. These are falures whch ule hose of ype A do fluece he fulfllme of he ass. I order o do hs we wll modfy equao 0 usg a suable heorem []. If he probably ha he falure s of ype A s deoed by p he he probably ha s of ype B or C s p q. Afer sequeces whch he umber of falures was regsered he umber of falures durg he sequece has a egave bomal dsrbuo 0 wh parameers ad he umber of falures of ype B or C gve by:. Accordg o he heorem [] follows ha Y also has a egave bomal dsrbuo Y.B. q 2 We shall ow proceed o calculaos for he subse comprsg C falures aloe. Falures of ype C are a subse of he se of all falures. These "crcal" falures always requre a sad-by vehcle o be moblzed. From he vewpo of vehcle flee maageme s covee ha hese falures occur rarely. I order o rea hese falures he followg model s roduced.

9 V. Pap} J. Popov} /Vehcle Flee Maageme: A Bayesa Approach 85 Le be he umber of vehcles he CO group uder cosderao. Le us suppose ha he umber per sequece of ype C falures s a radom varable deoed by Z wh a codoal bomal dsrbuo B p where parameer p s uow. Thus: P Z p p p Parameer p represes he probably per sequece of a ype C falure. The Bayesa learg algorhm reas p as a radom varable wh a bea dsrbuo. The desy fuco g p wh he a pror deermed values of parameers a ad b s gve by: a b p p g p 0 < p < a > 0 b > 0. B a b 4 Afer sequeces whch z... z2 z falures of ype C are regsered usg he Bayes formula we ca oba he a poseror dsrbuo whch s defed by he desy fuco: g p z z... z 2 a z b z p b z p B a z I follows ha p has a a poseror dsrbuo whch s aga a bea dsrbuo: 5 z z2 z B a z b z p... 6 If we deoe by Z he umber of falures sequece he he dsrbuo of Z s deermed by he equao: P Z B a z b z B a z b z 7 Ths dsrbuo s used o forecas he umber of ype C falures he sequece. Wh hese forecass a had he decso-maer should be able o do beer plag. To summarze: The dsrbuo of all falures ypes A B ad C s gve by 0 he dsrbuo of "crcal" falures ype C by 7 ad he dsrbuo of "flueal" falures falures of ype B or C by 2. Thereby we have mplcly cosdered falures of ype A B ad C.

10 86 V. Pap} J. Popov} /Vehcle Flee Maageme: A Bayesa Approach 4. UMERICAL EAMPLE I order o llusrae he valdy of he derved formulae ad show wha based o hem ca be furher compued ad ulzed he decso process we wll ow prese a real-lfe eample [5]. The daa were colleced for a cosrucooperao group CO of 3 cser vehcles model FAP34 durg he perod from July 987 ll 3 December 990. All hese vehcles were pu o operao he Publc Compay "Gradsa ^so}a" Belgrade as brad ew a he begg of ha perod. Commo characerscs for he whole group are: early decal dffcul worg codos for all vehcles vehcles are used o wash he srees maually herefore hey wor he s gear durg her operao me ad appromaely he same mleage durg he correspodg me perods. Falures were recorded durg he July December 990 me perod ad classfed accordg o ype A B ad C falures. Table shows he umber of falures he defed me perods. Table : umber of falures Tme perod Type of falure July Dec. 987 Jauary Dec. 988 Jauary Dec. 989 Jauary Dec. 990 A B C B C C Le us cosder ha we are a he begg of he observao perod July 987 ad ha he u me erval s a caledar moh. Sce he vehcles are brad * ew ad we do o have ay record of falures of our ow we ca forecas he falure rae per vehcle whch s he same for each of he 3 observed vehcles eher usg oher cusomers' daa or he maufacurer's forecass. I order o show how he choce of a pror forecas of he average umber of falures durg oe moh of vehcle operao affecs he soluo we made a sesvy aalyss of oucomes gve he choce of a pror values of dsrbuo parameers observg he obaed resuls four me pos whch cocde wh he ed of he caledar year ad hree dffere aleraves. The frs wo of hese aleraves are smlar ha he a pror esmae of he average umber of falures per vehcle per moh regardless of he falure ype herefore belogg o class ABC s 5. The dfferece bewee hem s he degree of our cofdece ha value whch s dffere so ha he frs more opmsc alerave we rus ha esmae more whle he secod oe we rus less. I he frs alerave ha assumpo led o he smaller varace of esmaes of gamma dsrbuo parameers formula. I he hrd alerave we assumed ha o average here are 4 falures regardless of he falure ype per vehcle per moh ad ha we have a opmsc aude regardg he rao bewee varace ad mea value. Dsrbuos ad correspodg values of dsrbuo parameers

11 V. Pap} J. Popov} /Vehcle Flee Maageme: A Bayesa Approach 87 accordace wh formulae 2 ad 5 as well as dsrbuos of he forecas umber of falures for he followg moh ad correspodg values of meas ad varaces of he umber of falures accordace wh formulae 0 ad are preseed Table 2. The resuls are gve for four me pos ed of caledar year ad for hree aleraves descrbed above. Table 2. Dsrbuo of he * falure rae Dsrbuo of he falure rae of he whole CE group Margal dsrb. of he umber of all ypes ABC of falures he Aleraves I II III * Γ0 2 * Γ 0.2 * Γ 8 2 Γ30 2 Γ3 0.2 Γ04 2.B. 30 2/3 E 65.B. 3 /6 E 65.B. 04 2/3 E 52 frs moh V 97.5 V 390 V 78 ad umercal characerscs Afer 6 u me ervals mohs have passed 3 Dec. 987 umber of regsered falures ABC 6 32 umber of regsered falures B. 45 8/9 E V B E V Afer 8 mohs have passed 3 Dec. 988.B /2 9 E V B E V Afer 30 mohs have passed 3 Dec. 989.B /33 3 E V B E V B /9 E V B. 8020/2 9 E V 6.95.B. 9732/33 3 E V Afer 42 mohs have passed 3 Dec. 990.B /45 43.B B /45 43 E E E V V V

12 88 V. Pap} J. Popov} /Vehcle Flee Maageme: A Bayesa Approach Comparg he resuls of dffere aleraves some eresg facs mgh be oced. Sce he frs wo aleraves deal wh a well chose esmae of he average umber of falures per vehcle per moh * 5 ca be see ha afer a small umber of perods he dffereces bewee forecas epeced values are mmal ad he dffereces bewee aleraves are small. The mos mpora fac s ha egave bomal dsrbuo whch s used o descrbe he forecas umber of falures coverges faser o he Posso dsrbuo whch s he basc assumpo of he model. Ths pheomeo could be proved easly observg he rao bewee varace ad mea value whch compared o he sarg value.5 he s vara ad 6 he 2 d decreased o he value of jus above. I s also eresg o oe ha he much worse esmae of average umber of falures per vehcle per moh * 4 he 3 rd alerave despe our prey large a pror fah esmao capably epressed hrough he relao V. 5E qucly adaped hrough he corporao of already recorded daa o parameers of he a poseror dsrbuo. Ths ca be observed hrough he fac ha he epeced values of he umber of forecas falures regardless of he falure ype hs alerave dffer by less ha % ad sadard devaos by less ha 0.5% from values oher aleraves afer 42 u me ervals mohs ad oly slghly more afer jus 8 mohs of daa collecg. We poed ou earler ha some ypes of falures BC lead o dsurbaces he eecuo of he EWP eecuve wor pla herefore we wll dscuss hem separaely. We propose o mae forecass usg formula 2. I he hree aleraves descrbed earler we aga observe me pos a he ed of he caledar year. I fac we could have preseed he resuls for each moh f he complee daa were dsplayed. I formula 2 e o he a pror values of parameers ad - umber of pas me us - ordal umber of observed me po ad - oal umber of recorded falures of ypes ABC we roduced q - probably ha he falure s of ype B or C. A he begg of he calculag process show Table 3 he value q 0. was ae. The was adjused a frs o q 0. 5 ad he o q 0.6 because he creasg edecy of hese ypes of falures was observed. The correco from q 0. o q 0. 5 was made afer 30 mohs of vehcle flee operao because was observed ha he epeced umber of falures of ypes B or C for he whole flee per year he frs case was 75 ad he secod case depedg o he alerave 3 2 ad respecvely. Followg he crease he umber of falures of ype B or C afer 42 mohs of operao he probably creased o q 0. 6 whch correspods o he level of appromaely 22 falures per year per ere flee of 3 vehcles.

13 V. Pap} J. Popov} /Vehcle Flee Maageme: A Bayesa Approach 89 Table 3. Margal dsrb. of he umber of falures ype B or C Y he frs moh ad umercal characers. q Aleraves I II III Y.B E Y 6.5 V Y Y.B E Y 6.5 V Y 9.75 Y.B E Y 5.2 V Y 5.46 Afer 6 u me ervals mohs have passed 3 Dec q q q 0. q q 0.6 Y 7.B E Y V Y Y 7.B E Y V Y Afer 8 mohs have passed 3 Dec. 988 Y 9.B E Y V Y Y 9.B E Y V Y Y 7.B E Y V Y Y 9.B E Y V Y Afer 30 mohs have passed 3 Dec. 989 Y 3.B Y 3.B Y 3.B E Y * E Y * E Y * V Y V Y V Y Y 3.B E Y 9.36** 3 V Y Y 3.B E Y 9.338** 3 V Y Y 3 Afer 42 mohs have passed 3 Dec. 990 Y.B E Y V Y Y 43.B E Y V Y B E Y 9.239** 3 V Y Y.B E Y V Y Sce ype C falures "crcal falures" are he wors regardg fulfllme of he EWP because he egageme of a replaceme vehcle s eeded forecasg her umber s of grea mporace from he plaer's po of vew. Sce Z he codoal dsrbuo of he umber of falures cludes he uow probably of a ype C falure appearg ay of he vehcles Table 4 preses he chages caused by recordg ype C falures he me pos ha cocde o he ed of he caledar year. Aga we observe hree aleraves accordg o a pror chose parameer

14 90 V. Pap} J. Popov} /Vehcle Flee Maageme: A Bayesa Approach values. These chages fluece he dsrbuo parameers formulae 6 ad he epeced value of p. Table 4. cumulave o. of observed ype C falures ALTERATIVES Ip B.5 0 IIp B5 00 IIIp B4 20 p B.5 78 p B25 68 p B4 88 E p 0.28 E p 0.30 E p 0.37 p B p B p B E p 0.32 E p 0.32 E p 0.36 p B p B p B69345 E p 0.66 E p 0.55 E p 0.67 p B p B03560 p B92478 E p 0.6 E p 0.55 E p 0.6 A he ed we wll llusrae how formula 7 ca be cocreely ulzed a he me po of 3 December 990 o forecas he umber of ype C falures for he ere flee for he followg moh. For alerave II a5 b00 we calculaed probables P Z43 for dffere values ag o accou ha Z 88. The followg resuls were obaed: PZ PZ PZ PZ PZ PZ PZ PZ PZ PZ PZ Usg hese resuls he plaer ca forecas ha wh a probably greaer ha 0.95 he umber of ype C falures durg he 43 rd moh sce he begg of observaos wll be a mos COCLUSIO Accordg o he prese sae of he ar he feld of vehcle flee maageme we are whou aalycal plag ools case we lac daa o vehcle falures or he avalable daa are o suffce for sascal ferece. Uder he codos of such uceray he dsrbuos of he umber of falures are o

15 V. Pap} J. Popov} /Vehcle Flee Maageme: A Bayesa Approach 9 ow. As a alerave o omg aalycal plag whle wag for he daa o accumulae for sascal ferece we proposed a procedure for deermg he dsrbuos whch are subjeced o cosecuve correcos as he recordg of falures cosecuve sequeces advaces. The Bayesa learg algorhm proved o be a useful ool o oba hese dsrbuos accordg o whch plag ca be made. Plag here volves plag he umber of vehcles as well as plag he maeace facles sce he compaes cosdered hs paper have a maeace depo alog wh her vehcle flee. We fd he problem cosdered hs paper mpora eough o be acled by more ha oe approach. We epec he echque we proposed as well as oher moder echques o be used he fuure. Acowledgeme. We would le o epress our graude o oe of he referees for very useful suggesos. The addo of he umercal eample Seco 4 ogeher wh oher proposals led o sgfca mproveme of he paper as a whole. REFERECES [] Brow G.F. ad Rogers W.F. "A Bayesa approach o demad esmao ad veory provsog" RLQ [2] Bu~} S. "Deermao of he resource maageme mehodology of he subsysem of echcal eploao of he VTC auomaed daa processg codos" Ph.D. Thess Uversy of Belgrade 992 Serba. [3] Jorgeso D.W. McCall J.J. ad Rader R. Opmal Replaceme Polcy orh-hollad Amserdam 967. [4] Kapur K.C. ad Lamberso L.R. Relably Egeerg Desg Joh Wley.Y [5] Ku{} D. "Aalyss of he formao subsysem 'Vehcle operao ad maeace' he Publc Compay 'Gradsa ~so}a' Belgrade" graduao paper Faculy of Traspor ad Traffc Egeerg Belgrade 99 Serba. [6] Pap} V. "The deermao of he effecveess of a heerogeeous vehcle flee" Coferece "Scece ad Moor Vehcles" JUMV Belgrade 987 Serba. [7] Popov} J. ad Teodorov} D. "A adapve mehod for geerag demad pus o arle sea veory corol models" Trasp.Res. B [8] Todorov} J. "Egeerg he maeace of echcal sysems" JUMV Belgrade 993 Serba. [9] Zacs S. "Bayes sequeal desg of soc level" RLQ