Finding second-best toll locations and levels by relaxing the set of first-best feasible toll vectors

Transcription

1 EJTIR Issue 14(1) ISSN: Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors Jokm 1 Lnköng Unversty Norrköng Sweden Ths er rovdes frmework for otmzng toll loctons nd levels n congeston rcng schemes for lrge urbn rod networks wth the objectve to mxmze the socl surlus. Ths otmzton roblem s referred to s the toll locton nd level settng roblem (TLLP) nd s both nonconvex nonsmooth nd nvolves bnry decson vrbles nd s therefore consdered s hrd roblem to solve. In ths er soluton roch s rovded whch nsted of drectly solvng the TLLP mkes use of the frstbest toll level soluton n whch no restrctons re mosed on toll loctons or levels. A frstbest rcng scheme cn be obtned by solvng convex rogrm nd t hs revously been shown tht for the used routes n the network the frstbest toll levels on route level re unque. By formultng n otmzton roblem whch nsted of mxmzng the socl surlus tres to fnd the lnk toll levels whch mnmze the devton from frstbest route tolls mxed nteger lner rogrm s obtned nd f the toll loctons re redetermned the resultng otmzton roblem s lner rogrm. The roch of mnmzng the devton from frstbest route tolls s led for two dfferent network models nd results re rovded to show the lcblty of the roch s well s to comre wth other roches. Also t s shown tht for the Stockholm network vrtully the frstbest level of socl surlus cn be obtned wth sgnfcntly reduced number of locted tolls. Keywords: congeston rcng; network desgn; user equlbrum 1. Introducton Mrgnl socl cost rcng (MSCP) n rod trffc network wll led to the most effcent usge of the rod nfrstructure (frst dscussed by Pgou 192) nd wll mxmze the socl surlus. One drwbck wth MSCP s however tht n rctce toll needs to be collected on every rod segment. Even though collectng toll on every rod segment s ossble wth the technology vlble tody t would result n rcng scheme n whch t would be vrtully mossble for the rod users to redct the cost ssocted wth choosng secfc route through the network. Also such rcng scheme my become exensve to mlement nd oerte. Therefore congeston rcng schemes re usully mlemented n rctce wth more lmted number of toll loctons nd n such wy tht the rod users esly cn understnd the system nd redct ther costs of trvellng. The rcng scheme cn be n the form of cordon rcng where the rod users y toll when crossng cordon n the form of re rcng where the rod users y toll for ccessng restrcted rt of the cty or s n the form of dstnce bsed rcng where the rod users re chrged fxed mount er klometer drven. In ll of these vrnts of rcng schemes restrctons re mosed on toll loctons nd/or toll levels but the m s stll to 1 A: Cmus Norrköng SE61 74 Norrköng Sweden T: F: E: jokm.ekstrom@lu.se

2 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors mxmze the socl surlus nd such rcng rncles re commonly referred to s secondbest ones s oosed to frstbest ones n whch there re no restrctons. When redctng effects of chnges n trnsortton nfrstructure or rcng of usng the nfrstructure t s commonly ssumed tht the trvelers re dstrbuted ccordng to Wrdron user equlbrum. In user equlbrum no trveler cn reduce hs/her trvel cost by lterng the choce of mode or route n the trnsortton network. Whle dstrbuton of rod users ccordng to user equlbrum ssumes tht the rod users hve erfect nformton bout trvel costs (both tme nd monetry costs) wthn the rod network nd tht they mke decsons whch mxmze ther ndvdul utlty the user equlbrum hs desrble mthemtcl roertes. Assumng tht the trffc condtons re sttc over the studed tme erod the roblem of determnng the user equlbrum cn be formulted s convex rogrm. The roblem of fndng otml toll loctons nd toll levels whch mxmze the socl surlus cn then be formulted s blevel otmzton rogrm. In whch the lower level rogrm s convex user equlbrum roblem nd on the uer level toll loctons nd levels re djusted to mxmze the socl surlus. A secl cse of the blevel rogrm rse when there re no restrctons on the toll loctons nd ther toll levels (frstbest rcng) nd for ths cse the roblem s convex. For the frstbest cse MSCP tolls lwys gve n otml soluton but there cn lso be lterntve otml solutons. The blevel rogrm whch s both nonconvex nd nonsmooth for the generl cse s smlr to other blevel rogrms rsng n trnsortton network desgn roblems. For revew on blevel rogrms wthn trnsortton lnnng see Mgdls (1995) nd for more recent revew on models nd methods the cse of congeston rcng see Tsekers nd Voß (29). For the cse when the toll loctons re redetermned the roblem s reduced to fndng otml toll levels whch s stll nonconvex nd nonsmooth roblem due to the ntrcte reltonsh between the uer nd lower level roblems. Ths wll be referred to the toll level settng roblem (TLP) nd t hs revously been solved wth both scent methods (Yng nd Lm 1996; Verhoef 22; Chen nd Bernsten 24; Lwhongnch nd Hern 24; et l. 29) nd metheurstc roches (Yn 2; Yng nd Zhng 23; Sheherd nd Sumlee 24; Zhng nd Yng 24). Introducng stochstc route choces scent methods re develoed n (Chen et l. 24; Yng nd Yng 25; Sumlee et l. 26; Connors et l. 27). Whle the scent roches cn only be used for fndng locl otml soluton nd need to del wth the nondfferentblty of the TLP metheurstc roches usully requre lrge number of toll level solutons to be evluted nd ech evluton of soluton to the TLP requres one user equlbrum roblem to be solved. On the other hnd metheurstc roches nclude mechnsm to vod gettng stuck n locl otml solutons lthough they cn nether gurntee locl or globl otmlty of the soluton. The toll locton nd level settng roblem (TLLP) hs lmost exclusvely been studed wth dfferent heurstc roches (Verhoef 22b; Zhng nd Yng 24; Sumlee 24; Sheherd nd Sumlee 24; et l. 29). Heurstc methods re however often bsed on herrchl decsons of toll loctons nd toll levels nd to evlute one secfc toll locton requre one TLP to be solved nd to cheve good results wth such heurstcs requre lrge number of toll loctons to be evluted. The roch n et l. (29) s nsted bsed on smoothenng of the dscrete rt of the objectve functon n order to smultneously determne the otml number of tolls to locte nd the corresondng toll loctons nd toll levels bsed on cost ssocted wth ech toll locton nd ths roch hs been demonstrted on network model of Stockholm n et l. (214). More recently globl otmzton roches hve been doted to solve the TLLP by Zhng nd vn Wee (212) nd et l. (212). These roches re bsed on ecewse lnerzton of the nonlner functons n the TLLP resultng n mxed nteger lner rogrm (MILP). So fr globl otmzton roches hve however only been demonstrted on smll network models due to the comuttonl burden of solvng the resultng MILPs.

3 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors Bsed on frstbest rcng schemes Yldrm nd Hern (25) formulte set of vld toll vectors whch fulfl the requrement of beng frstbest rcng schemes. Dfferent objectve functons cn then be formulted to choose between the dfferent frstbest toll vectors e.g. the toll vector wth mnmum number of locted tolls or whch mnmze the mxmum toll level. Lrsson nd Ptrksson (1998) nd Yn nd Lwhongnch (29) hve however shown tht whle the toll levels dffer between dfferent frstbest rcng schemes on lnklevel the sum of the collected tolls for ech used route s equl for ll frstbest rcng schemes under the ssumton tht the trffc demnd s deendng on the cost of trvellng. Thus s Lrsson nd Ptrksson (1998) onts out the set of frstbest toll vectors s of lmted nterest s the number of tollble lnks cn only be reduced f severl lnks cn be relced by sngle one. The study n et l. (214) suggests tht t s for the resented Stockholm study ossble to cheve 96% of the socl surlus ssocted wth frstbest rcng wth only 24% of the lnks beng tolled. To serch for frstbest rcng schemes wll result n rcng schemes whch ccount for 1% of the socl surlus but wll on the other hnd requre n ncresed number of lnks to be tolled. Thus close to frstbest rcng my be ossble usng sgnfcntly reduced number of tolled lnks. In the work resented n ths er the mxmzton of the socl surlus s relced by the mnmzton of the devton from frstbest rcng on route level. The resultng otmzton roblem s lner rogrm (LP) f the toll loctons re fxed nd MILP f the toll loctons re vrble. Whle the mnmzton of the devton from frstbest route tolls s not lkely to result n toll levels whch mxmze the socl surlus functon t s shown n the numercl results tht good toll levels nd loctons cn be obtned wthn resonble comuttonl tme for network model of Stockholm. The mn contrbuton of ths er s to rovde frmework for otmzng toll loctons nd toll levels whch s lcble to lrge urbn rod networks wthn resonble comuttonl tme. In the numercl results t s shown tht lthough the devton from the frstbest soluton s lrge toll level solutons whch mnmze ths devton results n toll levels close to wht hs been obtned wth other more comuttonl demndng methods. Ths er lso resented results whch show tht congeston rcng schemes resultng n level of socl surlus close to wht s reched wth frstbest rcng cn be cheved wth sgnfcntly reduced number of locted tolls. Whle Lrsson nd Ptrksson (1998) onts out tht the set of frstbest toll vectors s of lmted nterest the results resented n ths er shows tht for rctcl cse smll relxton of the frstbest toll set cn sgnfcntly reduce the number of tolled lnks wth vrtully the sme level of socl surlus. The resultng LP (for the cse of fxed toll loctons) ncludes constrnts whch rely on the comlete set of routes beng exlctly formulted for every ODr. In the otmum soluton t s however ssumed tht only subset of these constrnts re bndng. Therefore model wth only subset of constrnts ncluded s formulted nd n tertve soluton lgorthm s develoed for genertng ddtonl routes. For the cse of vrble toll loctons the resultng MILP s solved wth greedy heurstc. The remnder of the er s outlned s follows. In Secton 2 the set of frstbest toll vectors s formulted nd lter relxed n Secton 3 to llow for secondbest solutons. Together wth the relxed set of frstbest toll vectors n otmzton roblem s formulted n order to mnmze the devton from frstbest route tolls. To solve the resultng LP nd MILP soluton lgorthms re develoed n Secton 4 for both the cses of fxed nd vrble toll loctons. Numercl results re resented n Secton 5 for network models of Soux Flls nd of Stockholm nd n Secton 6 the results re summrzed nd further reserch drectons re dscussed.

4 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors 2. Vld toll vectors n the frstbest rcng roblem Consder rod trffc network wth set of orgndestnton (OD) rs I nd set of lnks A. For ech ODr I there s set of routes Π ech route Π wth f beng the number of trvelers usng route er hour. The number of crs er hour ν on lnk s gven by v f δ δ tkes on the vlue of 1 f route trverses lnk nd = where I Π otherwse. The lnk trvel cost s gven by c ( τ v ) = αt ( v ) + τ where t ( v ) s the trvel tme (n mnutes) on lnk t flow v τ the toll level on lnk nd α the vlue of tme whch trnsforms tme nto the sme monetry unt s the toll levels re gven n. The reltonsh between trvel cost nd demnd s exressed by the nverse trvel demnd functon whch for ODr I s gven by π 1 = D ( q) where π s the mnmum trvel cost nd q the trvel demnd n ODr n the unt of trvelers er hour. Note tht the lnk flow s gven n the unt of crs er hour nd the demnd nd route flows n the unt of trvelers er hour thus the men cr occuncy χ wll be used to rovde converson between trvelers nd crs. Throughout ths er the trvel tme functons re ssumed to be serble nd ncresng functons nd the nverse trvel demnd functons re ssumed to be serble nd decresng functons. The roblem of fndng the user equlbrum lnk flow nd demnd dstrbuton gven toll vector τ cn then be formulted s the comlementrty roblem (Sheff 1985) f > c( τ v) δ = π Π I (1) A f = c( τ v) δ π Π I (1b) A 1 ( ) 1 ( ) q > D q = π I (1c) q = D q π I (1d) f Π I (1e) q I (1f) v 1 f δ = A. (1g) χ I Π f = q I Π Constrnts (1) nd (1b) sttes tht ny used route wll hve cost equl to the mnmum cost π of mkng tr n ODr I nd ny unused route wll hve cost equl to or lrger thn π for ODr I. For ech ODr constrnts (1c) nd (1d) wll ensure tht otentl rod user only mkes tr f the ndvdul surlus wth mkng the tr exceeds the mnmum cost of trvellng n the ODr. Constrnts (1e) nd (1f) ensure nonnegtve route flows nd demnds resectvely (1g) gves the converson between route nd lnk flows nd (1h) ensure tht the sum of the route flows n ech ODr equls the demnd. For gven toll vector τ the user equlbrum lnk flows nd demnds cn be obtned by solvng the followng convex rogrm (Sheff 1985) qv v q 1 ( τ ) = χ ( ) ( ) τ A I mn G q v c u du D w dw subject to constrnts (1e)( 1h). (1h) (2)

5 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors The effcency of congeston rcng scheme s evluted by the chnge n socl surlus. The socl surlus mesure (SS) for gven toll vector τ conssts of the consumer surlus (CS) lus the oertor surlus (OS). The consumer surlus s exressed s q ( τ ) 1 τ CS( τ q( τ) v( τ)) = D ( ) ( ( )) ( ) w dw χ αt v τ + v τ (3) I A χ n whch the frst sum s the user benefts (gven by the Mrshlln mesure (Zerbe nd Dvely 1994) nd the second sum s the user costs. The oertor surlus (OS) s equl to the collected tolls OS( τ q( τ) v( τ)) = τ v ( τ). (4) A The socl surlus mesure cn then be exressed s SS( τ q( τ) v( τ)) = CS( τ q( τ) v( τ)) + OS( τ q( τ) v( τ)) q ( τ ) = D ( w) dw χ αt ( v ( τ)) v ( τ). I 1 A Let v nd q bet the lnk flow nd demnd vectors corresondng wth the nontolled soluton. The socl surlus for the nontolled soluton s then gven by q 1 = χ α I A SS( q v ) D ( w) dw t ( v ) v wth the chnge n socl surlus SS( τ ) nduced by the toll vector τ gven by SS( τ) = SS( τ q( τ) v( τ)) SS( q v ). (6) Consder the roblem of mxmzng the socl surlus by djustng the toll loctons nd toll levels. Let y be vrble for ech lnk A whch tkes on the vlue of 1 f lnk s tolled nd otherwse. The fesble combntons of toll loctons nd levels re then gven by the set V U T : ( τ y) y g τ τ y A y = k (7) A where k s the number of tolled lnks g s rmeter whch tkes on the vlue of 1 f lnk s U tollble nd otherwse nd τ s the mxmum toll level llowed to be chrged on lnk. For the cse when the toll loctons re fxed the fesble set of toll levels cn be formulted s T F U { τ τ τ g} : (8) The objectve n both the TLP nd TLLP s to mxmze (6) nd snce cn be used s objectve functon. The TLLP cn then be formulted s (5) SS( q v ) s constnt (5) q ( τ ) 1 mx F( τ) = SS( τ q( τ) v( τ)) = D ( ) ( ( )) ( ). w dw χ αt v τ v τ ( τ y) T V I A (9) V F For fxed toll loctons ( τ y) T s smly relced by τ T n (9). Note tht q( τ ) nd v( τ ) re gven mlctly by the soluton to the user equlbrum roblem. Ths mlct relton between the objectve functon nd toll levels gve rse to the nonconvex nture of the blevel rogrm. If g = 1 for every lnk nd k s equl to the number of lnks n the network (9) s convex rogrm wth the MSCP toll vector s one mong ossbly severl frstbest otml

6 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors solutons. The corresondng lnk flow nd demnd soluton s for ths cse commonly referred to s the system otml () lnk flow nd demnd soluton nd ny frstbest toll level soluton results n lnk flows nd demnds. Let π v nd f be the vectors of mnmum OD trvel costs lnk flows nd route flows resectvely corresondng to the lnk flow nd demnd soluton. Insertng f n (1) Yldrm (21) shows tht ny toll vector τ stsfyng ( ) A π v nd f > αt ( v ) + τ δ = π Π I (1) ( ) f = αt ( v ) + τ δ π Π I (1b) A wll mxmze the socl surlus. Note tht n order to formulte the set of frstbest toll vectors (1) t s ctully only necessry to know the lnk trvel tmes mnmum route trvel cost nd the set of used routes whch cn be comuted from v nd q. The set of route flows nd demnds stsfyng (1) s denoted s the set of fesble frstbest toll vectors. Lrsson nd Ptrksson (1998) show tht f the demnd s elstc the totl d toll for ech route through the network wth ostve flow s ndeendent of the choce of toll vector s long s the toll vector belong to the set of frstbest toll vectors. In Hern nd Rmn (1998) nd Yldrm (21) n equvlent lnk bsed formulton of (1) s used to mxmze or mnmze dfferent objectves over the set of fesble frstbest toll vectors e.g. to mnmze the number of locted tolls or the mnmum of the mxmum toll level. The resultng otmzton roblem s ether LP or MILP. Frstbest solutons re however usully not ossble n rctce when there re restrctons on toll loctons nd/or toll levels. In the next secton the set of fesble toll vectors (1) s therefore relxed to llow for secondbest solutons. 3. Relxng the frstbest toll set Let 1 Π nd Π be the set of routes n ODr wth flow f > nd wth flow f = 2 resectvely.e. the set of used nd unused routes resectvely. The set of frstbest fesble toll vectors (1) cn then be exressed s ( t v ) A ( t v ) A α τ δ π ( ) + = α + τ δ π ( ) Note tht the frstbest toll chrged for trvellng on route route exressed s τ π αt δ equvlent route toll A 1 Π I (11) Π 2 I (11b) Π n ODr I cn be =. Thus nsted of chrgng toll on ech lnk n τ route cn be chrged to ech user on route. Whle there my exst severl lnk toll vectors whch re vld n (11) Lrsson nd Ptrksson (1998) show tht under the ssumton of elstc demnd the route tolls cn be unquely determned nd ths route toll wll lwys be equl to the sum of the frstbest lnk tolls long route. For route wth zero flow t s only necessry for the route toll to be equl to or exceed π therefore be equl to zero f the route cost even wthout tolls exceeds. For unused routes τ cn π. The roblem of mxmzng the socl surlus (5) over the set of fesble toll vectors s nonlner rogrm f the set of fesble toll vectors re defned by (8) or mxed nteger nonlner rogrm f the set of fesble toll vectors re defned by (7). Both of these roblems belong to the route

7 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors clss of mthemtcl rogrms wth equlbrum constrnts (MPEC) nd re n the generl cse nonconvex nd nonsmooth roblems. Insted of drectly solvng the nonlner rogrm or the mxed nteger nonlner rogrm ths er rooses nother strtegy for obtnng toll levels nd loctons n order to mxmze the socl surlus. By serchng for lnk toll vectors whch mnmze the devton frstbest route tolls toll levels cn be obtned by solvng LP nd toll loctons nd levels by solvng MILP. The set (11) cn be relxed by ntroducng the vrbles ρ nd σ nto (11) for ech route Π 1 I nd the vrbles µ nto (11b) for ech ODr I. The relxed set of frstbest toll vectors cn then be exressed s ( αt v τ ) A ( ) A ( ) + δ + ρ σ = π αt v + τ δ + µ π ( ) Π I (12) 1 Π 2 I. (12b) For route wth f > ρ nd σ wll descrbe the mount negtve nd ostve resectvely by whch the current route toll gven by devtes from the frstbest route toll. For route wth τδ A f = the negtve devton from the frstbest route toll s nsted only gven by the mxmum devton n ODr µ. Ths wll mke t ossble to lter tertvely generte the constrnts for unused routes wthout ntroducng ddtonl vrbles. If ρ σ nd μ re zero the corresondng toll vector τ whch stsfes (12) nd (12b) wll be vld frstbest toll vector. By enlzng the devton from frstbest rcng.e. the vlues on ρ σ nd μ mnmzton roblem cn be formulted wth the otml soluton equl to the toll loctons nd levels whch mnmze the enlzed devton from frstbest route tolls. The roblem of mnmzng the enlzed devton () s formulted s z = β1 f ( ρ + σ ) + β 2 q µ (13) y I 1 Π I ( αt( v ) + τ) δ + ρ σ = π Π 1 I (13b) A ( αt( v ) + τ) δ + µ π Π 2 I (13c) mn ρσ µτ A ρ µ Π 1 I (13d) ( τ y) Τ V (13e) ρ σ Π 1 I (13f) µ I. (13g) Constrnt (13d) s ntroduced to mke t ossble to develo n effcent soluton lgorthm nd the rctcl nterretton of (13d) s tht the negtve devton from frstbest route tolls for route wth ostve flow cnnot exceed the mxmum negtve devton for the routes wth zero flow n the sme ODr. Ths s further dscussed n the next secton. 1 In the objectve functon (13) for ny route wth ostve flow Π the devton from frstbest route tolls s weghted by constnt β 1 nd the route flow f. For ech ODr I the mxmum devton from frstbest route tolls s weghted by the constnt β 2 nd the trvel demnd q. The weghts reflect tht for routes wth ostve flows t s resonble tht smll devton from frstbest route tolls s more mortnt for routes wth lrge flows comred wth routes wth smller flows. For routes wth zero flow n the soluton the sme

8 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors rgument s used when weghtng µ by q to reflect the mortnce of ODrs wth hgh demnd. If the frstbest tolls cn be comletely recreted on route level the resultng toll vector s frstbest soluton wth the otml objectve functon vlue z*=. For fxed toll loctons n equvlent mnmzton roblem cn be formulted s ρσ µτ subject to τ Τ ( ) mn z = β f ρ + σ + β qµ 1 2 I 1 Π I F nd constrnts (13b)(13g) (14) Note tht the lnk flows re ssumed to be fxed (to flows) nd the otmzton roblem ether tkes the form of n LP (f the toll loctons re fxed) or MILP (f the toll loctons re vrble). For otml solutons to (13) wth otml objectve functon vlues close to zero ths s resonble nd the toll level soluton wll be close to the globl mxmzer of (9). As the number of tollble lnks s reduced the dfference between the true equlbrum lnk flows nd the system otml ones wll ncrese nd the toll level soluton to (13) cn be exected to move further wy from the globl mxmzer of (9). The beneft on the other hnd s the ossblty to fnd good toll locton nd toll levels for lrge networks n resonble tme. Even f the set of used routes s known t s stll mtter of exressng (11b) for every unused route whch cn be exected to be lrge. Therefore cuttng constrnt lgorthm (CCA) s used both for solvng the LP nd the MILP. For the MILP cse the roblem however becomes too lrge to be solved to otmlty for lrger networks nd greedy heurstc s doted for the exmle of the Stockholm network resented n Secton 5. Whle (11) s formulted bsed on route flows corresondng lnk bsed model cn be formulted (Yldrm nd Hern 25). Snce the lnk bsed formulton wll not rely on exlctly formultng the set of used nd used routes t cn drectly be mlemented nd solved wth ny commerclly vlble solver. The mjor lmtton wth dotng lnk bsed formulton s the number of constrnts ntroduced. A lnk bsed formulton wll roxmtely hve the number of constrnts equl to the number of lnks multled wth the number of ODrs whle the route bsed verson doted n ths er wll hve the number of constrnts equl to the number of used routes lus the number of ddtonlly dded constrnts for unused routes. Genertng the unused routes tertvely wth CCA t s thus ossble to kee the number of constrnts consderbly smller comred wth the lnk bsed formulton. One otentl roblem s the nonunqueness of route flows. In contrst to the lnk flow soluton the route flow soluton s not unque ether n terms of used routes or flows on the routes. For the cse of (11) the set of frstbest toll vectors wll not deend on the route flow soluton. When formultng (13) the objectve functon wll however deend on the route flow soluton (through the weghtng rmeters). Thus the route flow soluton cn ffect the resultng objectve functon vlue s well s the comuted toll levels nd loctons. Whle ths s clerly otentl lmtton of the resented roch t hs for the exmle of the Stockholm network (used n Secton 5 for numercl results) been shown tht n rctce the choce of route flow solutons hs neglgble effect on the results. 4. Soluton roch for lrge networks Constrnt (13b) s formulted for every route wth ostve flow n the soluton nd constrnt (13c) s formulted for every route n the network wth zero flow n the soluton. The totl number of routes wth zero flow wll for rel world trffc networks be lrge nd to generte them ll ror s not rctcl ossble. Also t not exected tht every constrnt n

9 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors (13c) wll be bndng n the otml soluton. Thus constrnts n (13c) cn be generted tertvely when solvng both (13) nd (14). 4.1 Fxed toll loctons Let us frst consder the cse wth fxed toll loctons. To solve (14) reduced set of constrnts for the routes wth zero flow s formulted Π whch s (ossbly emty) subset of Π. (14) cn then be formulted s mn ρσ µτ y 2 ( ) z = β f ρ + σ + β q µ ( αt v τ ) A ( ) A 1 2 I 1 Π I ( ) + δ + ρ σ = π ( ) + + αt v τ δ µ π Π I 1 Π I 2 2 (15) (15b) (15c) ρ µ Π 1 I (15d) ( y) τ Τ V (15e) ρ σ Π 1 I (15f) µ I. (15g) * * * * Let ( ) (14) for ODr cn be formulted s ρ σ µ τ be the otml soluton to (15). Then the serch for volted constrnt n ( ) w = αt v + τ δ + µ * * mn ( ). 2 Π A To only serch for routes n Π 2 (16) whch mnmze (16) s not trvl snce t wll requre comlete enumerton of the routes wth zero flow n ech ODr. Snce constrnt (13d) s ncluded n (14) ny route n αt ( v ) + τ δ + µ π. Thus 1 Π wll lso stsfy A( ) * (16) cn be solved by fndng the shortest th n ech ODr (snce µ s constnt) wth lnk costs gven by αt ( v ) + τ. Let w be the otml objectve functon vlue to (16). If w * * < π * then there exsts route wth zero flow whch mkes the current soluton nfesble f t would be ncluded n Π. 2 The CCA for solvng (15) cn now be formulted s: Ste. For ech ODr I ntte the set of routes wth zero flow 2 Π. Ste 1. Solve (15) wth otml soluton ( τ µρσ ) nd objectve functon vlue z. Ste 2. Solve (16) for ech ODr I wth otml objectve functon vlue * w nd corresondng otml soluton *. If * 2 2 w < π set Π : = Π. If * w π for every ODr I termnte the lgorthm otherwse contnue wth Ste 1. In ech terton of the CCA t lest one route s dded or t s concluded tht the soluton to (15) lso solves (14). Snce every lnk hs ostve cost there wll be no routes whch nclude cycles nd thus the lgorthm must termnte n fnte number of tertons.

10 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors 4.2 Vrble toll loctons For vrble toll loctons the otmzton roblem (13) tkes the form of MILP. Usng the CCA resented n the revous secton would requre one MILP to be solved n ech terton whch s not rctcl for lrger networks. Also for the numercl results resented for the Stockholm network n the next secton t hs not been ossble to solve the MILP to otmlty even when β 2 s set to zero nd no unused routes need to be generted. Thus to show the lcblty of usng (13) to fnd toll loctons nd levels whch rovde good solutons to (9) greedy heurstc hs been develoed. The urose of the greedy heurstc s to rovde good but not necessrly otml solutons to (13) n order to evlute the roch of mnmzng the devton from frstbest route tolls. U Consder (8) nd let λ be the dul vrble corresondng wth the constrnt τ τ g n (7). The vlue of the dul vrble gves n estmte on how much the objectve functon vlue would mrove by unt chnge of the rght hnd sde. Thus for lnk wth g = λτ wll gve n estmte on the otentl mrovement of the objectve functon vlue from ntroducng toll U equl to τ on lnk. Whle λ s not lkely to be vld for the whole rnge from to τ nd t s not known wht the ctul vlue on τ would be f lnk s ctully tolled λ cn stll be used s n estmte on the mortnce of tollng lnk. In the greedy heurstc lnks re chosen tertvely to be ncluded n the soluton bsed on the λ vlues nd fter lnk s dded (14) s resolved n order to udte the λ vlues bsed on the currently selected toll loctons. The greedy lgorthm cn more formlly be wrtten s Ste. Intte by settng g : = for every lnk A Ste 1. Solve TLP n (14) to obtn λ b= rgmn λ (1 g ) nd set g : = 1. Ste 2. Add toll locton Fnd toll locton ( ) Ste 3. Termnte lgorthm f k = otherwse contnues wth Ste 1. g A A low toll level my ndcte less mortnt toll loctons nd to further mrove the soluton qulty the greedy lgorthm s rerun wth the toll loctons wth toll level τ below the threshold κ removed. Ths rovdes mechnsm to remove toll loctons whch seemed mortnt n the erly tertons but whch n the end turned out to gve smll contrbuton to reducng the objectve functon vlue. Removng toll loctons wth τ < κ cn be reeted severl tmes wth κ reduced by rmeter ψ ech tme. 4.3 Boundng the mnmum toll booth soluton For comrson t s of nterest to obtn good estmte on the mnmum number of toll loctons requred to rech lnk flows nd demnds. Drong (11b) from (11) results n the relxed set of frstbest toll vectors 1 ( αt + τ) δ = π Π I. (17) A A relxton of the mnmum toll booth roblem (Hern nd Rmn 1998) cn then be formulted s mn y A y 1 ( αt + τ) δ = π I Π subject to. A b (18)

11 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors The soluton to (18) wll gve n underestmton of the number of tolls requred to cheve frstbest rcng. Usng the greedy heurstc but contnung to dd toll loctons untl the otml objectve functon vlue of (14) s below some threshold close to zero 2 wll result n n uer bound on the number of tolls requred for chevng frstbest rcng. 5. Numercl results The roch for fndng good toll loctons nd toll levels n congeston rcng scheme bsed on the roch hs been led to network models of Soux Flls nd Stockholm. The Soux Flls model s well used n reserch ers ddressng otml network desgn nd otml rcng schemes. The verson of the Soux Flls network doted n ths er s the elstc demnd model frst resented n Yldrm (21) nd lter used n et l. (213) for evlutng globl otmlty roch when otmzng toll loctons nd toll levels. The Stockholm network hs revously been used n et l. (214) to study otml toll loctons nd corresondng toll levels wth heurstc roch bsed on smoothenng technque. To solve both (13) nd (14) requre number of LPs to be solved (roblems (15) nd (16)) nd for the numercl results resented here the commerclly vlble solver CPLEX verson 12.2 (IBM 21) hs been used. 5.1 The Soux Flls network The verson of the Soux Flls network model used n ths er hs 79 lnks nd 3 ODrs. The lnk trvel tme functons re on the BPRform (Bureu of Publc Rods 1964) nd the comlete network dt s gven n Yldrm (21). For the Soux Flls network ll costs nd tolls re gven n the unt of mnutes nd lyng MSCP tolls results n n mrovement of the socl surlus by 2722 mnutes. In et l. (213) globl otmzton roch s led to fnd the number of otml toll loctons nd ther locton nd corresondng otml toll levels gven cost for loctng ech toll. Whle the number of toll loctons re vrble n et l. (213) the number of tolls to locte s fxed n (9). To be ble to comre the results the number of tolls to locte wll therefore be gven by the resultng number of toll loctons from et l. (213). For the Soux Flls network model (13) cn be solved to otmlty by lyng the CCA drectly.e. by lettng the toll loctons be vrble n (15). Ths s only ossble for smll network model for whch the resultng MILP s ffordble to solve to otmlty n ech terton of the CCA. For ll exerments β 1 s set equl to one nd β 2 s vred between nd 1.5 n stes of.5. In Tble (1) the resultng mrovement n socl surlus s resented for 7 11 nd 14 number of locted tolls (k) nd the results re comred wth the best found soluton from usng the globl otmzton roch n et l. (213). From the results t s cler tht for ths lmted nlyss the roch resented n ths er erform well n comrson to the results from usng the globl otmzton roch. The globl otmzton roch should n theory be ble to fnd the globl otml soluton. In rctce the comuttonl tme requred to rove otmlty of soluton s lrge nd therefore the globl otmzton roch s led wth tme lmt. Thus globl otmlty s n rctse usully not roven for the solutons form ths roch. For 11 locted tolls ths s clerly cse when the globl otmzton roch erform worse comred wth the roch resented n ths er. Also excet for β 2 = the soluton s robust n terms of how the weghtng rmeters re set. 2 For lrger networks there wll be some convergence error when solvng the user equlbrum roblem nd thus t s not rorte to use zero s termnton crter.

12 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors Tble 1. Resultng mrovement n socl surlus for the Soux Flls network. k β 2 =. β 2 =.5 β 2 =1. β 2 =1.5 Globl otmlty roch In order to evlute the greedy lgorthm t hs been led wth β 1 =β 2 =1. In Fgure 1 comrson of the mrovement n socl surlus s gven for both the 7 11 nd 14 locted tolls resented n Tble 1 nd ddtonlly for nd 25 locted tolls. Frst of ll the results resented n Fgure 1 show tht the beneft from ctully solvng (13) to otmlty s lrge comred wth usng the greedy lgorthm. It s lso cler tht the dfferences between the two soluton methods dmnsh when the number of locted tolls ncreses. It s eseclly nterestng to see tht the soluton s reched wth 26 locted tolls nd tht the greedy lgorthm requres 27 locted tolls n order the rech the soluton. In terms of comutng the mnmum toll booth soluton the greedy heurstc erforms well. These results cn lso be comred wth the mnmum toll booth soluton from Yldrm (21) whch rovde mnmum toll booth soluton wth 28 locted tolls bsed on solvng the (MILP) lnk bsed mnmum toll booth roblem. In Yldrm (21) the soluton lgorthm s termnted before n otml nteger soluton hs been verfed due to excessve comuttonl tme lthough soluton wth 26 locted tolls s found by dhoc mens. Formultng the set of fesble frstbest toll vectors bsed on route flows rther thn lnk flows nd solvng the mnmum toll booth roblem bsed on the sme CCA s descrbed here results n the otml soluton of 26 locted tolls wthn seconds whch suggests tht n terms of comuttonl effcency the route bsed verson s sueror. Fgure 1. Comrson between otml nd greedy soluton of (13) bsed on the mrovement n socl surlus. 5.2 The Stockholm network model The Stockholm network used n ths er hs 392 lnks (312 f the connectors to orgn nd destnton zones re excluded) nd 4 zones resultng n 156 ODrs. Some rs of lnks re only used to gve relstc grhcl reresentton of the network nd cn be relced wth one sngle lnk whch reduce the number of tollble lnks requred for MSCP to 291 lnks. The lnk trvel tme functons re gven on the form 4 v v t( v) = L 1 n 3 n

13 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors where L s the lnk length n the number of lnes nd 1 4 lnk tye secfc rmeters. The rctcl rod ccty s gven by 3n. The demnd model descrbes the choce between ublc trnsort nd cr for the trvellers wth ccess to cr durng the mornng rush hour. The choce between ublc trnsort nd cr s gven by bnoml logt model. Assumng tht the trvel cost for ublc trnsort s not deendng on the number of trvellers usng t the vot ont verson of the bnoml logt model Kumr (198) cn be used whch for ODr tkes the form q A = D ( π ) = T A Ke ηπ ( π ) + Where T s the totl trvel demnd n ODr nd A nd K re the demnd for cr nd ublc trnsort n the nontolled scenro. The mnmum trvel cost n ODr n the nontolled scenro s gven by π nd η> s the dserson rmeter. The number of trvellers by cr n the nontolled scenro s lwys less thn T nd when π s ncresed q wll decrese. The Stockholm network reresents n ggregted trffc network of the Stockholm regon (Fgure 2). The demnd model (19) s bsed on dt from the demnd forecst model T/RIM (Engelson nd Svlgård 1995). The T/RIM model s however clbrted for full Stockholm network wth bout 11 lnks nd 125 zones. In ths er n ggregted verson of the T/RIM model s used nd t s ossble tht usng the ggregted trffc network together wth T/RIM dt wthout further clbrtons wll result n hgher lnk flows comred wth results from other models for the Stockholm regon. In Trnsek (23) severl lterntve models re comred for Stockholm nd for the nontolled scenro the flow cross the current cordon n Stockholm vry between nd vehcles er hour whle the ggregted model used n ths er results n vehcles er hour. For the urose of evlutng the roch resented n ths er the ggregted Stockholm network s consdered to be good exmle of rel network model. The cr occuncy χ =1.13 trvelers er cr nd the dserson rmeter η =.7 re lso rovded from the T/RIM model nd the vlue of tme s set to 1.2 SEK er mnute. (19)

14 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors Fgure 2. The Stockholm network. The currently mlemented congeston rcng scheme n Stockholm s resented n Fgure (3) for cutout of the centrl rts of Stockholm. In the ggregted network there re totl of 2 toll fcltes locted for the currently mlemented scheme to be comred wth the 37 ctul locted toll fcltes. For ech locted toll 2 SEK s chrged ech cr ssng the toll fclty durng ek hour. For the lnk flow nd demnd dstrbuton whch cn be cheved by the MSCP MSCP tolls τ the mrovement n socl surlus s SEK er rush hour whch cn be comred wth the mrovement n socl surlus cheved by the currently mlemented cordon whch s SEK er rush hour. When the demnd s gven by (19) the user equlbrum roblem wth elstc demnd (2) cn be solved by usng the rtl lnerzton lgorthm resented n Evns (1976) n whch seres of fxed demnd user equlbrum roblems re solved tertvely. In ths er the rtl lnerzton lgorthm s used together wth the Dsggregted Smlcl Decomoston (DSD) lgorthm (Lrsson nd Ptrksson 1992) for solvng ech fxed demnd roblem. The beneft from usng the DSD lgorthm s the vlblty of route nformton whch s needed when formultng (13). Also the DSD lgorthm hs reotmzton cblty whch s useful feture when solvng seres of smlr fxed demnd user equlbrum roblems. Any lgorthm rovdng exlct route nformton cn however be used. For lrge trffc network the convergence when solvng the user equlbrum roblem wll not be erfect nd there my exst routes wth ostve flow but wth the route trvel cost dfferng from the mnmum ODtrvel cost. In ths er ths s hndled by removng ny route wth f <.1 from the set of used routes. Whle the otmzton roblem solved n et l. (214) ncludes the cost of loctng the toll collecton fcltes nd the number of toll loctons s vrble n the roblem the resultng toll loctons from et l. (214) cn stll be used s comrson to the ones comuted wth the roch resented n ths er.

15 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors Fgure 3. The current congeston rcng cordon n Stockholm. Fxed toll loctons For the exerment resented n ths secton β 1 s set equl to 1 for ll exerments nd β 2 s vred between nd 2. For β 2 = no tertons re needed wth the CCA snce routes wth zero flow wll not ffect the otml soluton for β 2 > the CCA generte between 542 nd 943 ddtonl routes durng 2 to 4 tertons. The soluton tme s between 22 CPUseconds usng one Intel P86 2.4GHz rocessor. Usng the senstvty nlyss bsed scent method resented n et l. (29) otml toll levels were comuted n et l. (214) for the currently mlemented cordon n Stockholm s well s for n extended Stockholm cordon n whch the byss hghwy Essngeleden s lso tolled (resultng n totl of 22 tolled lnks). The toll level soluton obtned from the scent method s denoted τ AS wth corresondng chnge n socl surlus ΔSS(τ AS ) nd for comrson these results re resented n Tble 2. The toll level soluton whch solves (14) s denotedτ nd led s toll level soluton n (9) results n n mrovement of the socl surlus by SS( τ ). The mrovement n socl surlus ssocted wth the toll levels obtned roch re resented n Tble 3 nd 4 for the AS current nd extended cordon resectvely. For comrson SS( τ )/ SSτ s lso gven n these tbles. Tble 2. Results (n SEK) for the Stockholm network from et l. (214). Scheme ΔSS AS τ Current cordon Current cordon otmzed Extd. current cordon Extd. current cordon otmzed

16 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors Tble 3. Results obtned when solvng (14) for the current Stockholm cordon. β 2 SS τ ( )(n SEK) SS SS ( τ ) AS ( τ ) Tble 4. Results obtned when solvng (14) for the extended Stockholm cordon. β 2 SS τ ( )(n SEK) SS SS ( τ ) AS ( τ ) For β 2 =1.5 the toll levels obtned by mnmzng the devton from frstbest route tolls reches AS 99% of SS( τ ). Whle the scent method requres severl hours n comuttonl tme the tme requred to solve (14) s between 22 seconds deendng on the number of tertons wth the CCA. For ll evluted choces of β 2 t s only β 2 = whch erforms oorly nd for ll other choces the results re close to wht s cheved by the scent method. Frst of ll ths suggests tht mnmzng the devton from frstbest route tolls my result n toll levels close to locl otml soluton nd secondly tht the roch s rctcl useful roch for mnmzng (9). The results lso suggest tht the roch s robust n terms of vlues on the βrmeters nd the reltve smll number of generted routes nd the low comuttonl tme suggests tht the roch wll be lcble for even lrger network models. To evlute how the choce of route flow soluton used for defnng the set of fesble frstbest toll vectors cn ffect the erformnce of the roch three ddtonl route flow solutons wth dfferent roertes hve been used when lyng the roch to the current Stockholm cordon. A lner rogrm cn be formulted wth the fesble regon defnng the route flows whch relse the lnk flow nd demnd soluton. Dfferent objectves cn then be led n order to comute route flow solutons wth dfferent roertes. The comlete set of unused routes hs not been ncluded n ths nlyss but subset of the unused routes (extrcted from the DSD lgorthm) hs been ncluded. Route set 1 s the ntl set of route from the DSD lgorthm. Set 2 s obtned by mnmzng the sum of the route flows on the ten routes wth lrgest flow n the ntl soluton. By mxmzng the sum of the route flows on the unused route (wth trvel cost equl to the mnmum OD trvel cost) n the ntl soluton route set 3 nd 4 re obtned wth the dfference tht n route set 4 no used route n the ntl soluton s llowed to be reduced to less thn 5% of the flow n the ntl soluton. The four dfferent route flow solutons re resented n Tble 5 together wth the otml objectve functon vlue (z*) of (14) nd resultng mrovement n socl surlus (ΔSS) for the exmle of β 1 = β 2 =1. Let P be the set of consdered equlbrum routes (for both used nd unused nt new routes) nd the ntl nd new route flow soluton denoted f nd f resectvely. The dstnce between the ntl nd new route flow soluton s then exressed s new nt D= f f nd s lso resented n Tble 5. P

17 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors Tble 5. Comrson of results from usng lterntve route flow solutons for defnng the set of frstbest toll vectors Route set Number of routes wth ostve flow z* SS (n SEK) D From Tble 5 t s cler tht for ths exmle the choce of route flow soluton hs neglgble effect on the toll level soluton. The concluson s the sme for other choces of vlues on β 1 nd β 2 nd for the extended Stockholm cordon s well. These results re therefore excluded from ths resentton. Vrble toll loctons The greedy heurstc resented n Secton 4.2 hs been led to solve (13). Whle the greedy heurstc s not exected to solve (13) to otmlty t rovdes solutons whch cn be used for evlutng the roch of mnmzng the devton from frstbest route tolls n order to obtn good toll loctons nd toll levels to (9). Results re resented for choosng nd 2 tollble lnks out of 291 ossble toll loctons. The κ rmeter n the greedy heurstc s scenro secfc nd set to nd 1 resectvely nd the reducton fctor hs been set to ψ = κ /1 resultng n totl of 1 reruns wth the greedy heurstc for ech scenro. The frst three scenros (24 43 nd 69 tollble lnks) corresond wth solutons from usng the smoothenng heurstc resented n et l. (29). Whle the smoothenng heurstc s used for vrble number of toll loctons n order to mxmze the socl surlus mnus the cost of settngu nd oertng the toll collecton fcltes the resultng toll loctons wll lso rovde good solutons for the cse when the number of toll loctons re fxed to the otml number of locted tolls from the smoothenng heurstc. In ths er ech of these scenros wll be used s comrson when serchng for the otml toll loctons gven the number of tolls to locte from the smoothenng heurstc. The scenro wth 69 locted tolls s obtned from et l. (214) n whch the setu nd oertonl cost s estmted to 5 SEK er lne of ech lnk. To obtn 24 nd 43 locted tolls the setu nd oertonl cost s set to 1 SEK nd 5 SEK resectvely for ech lnk. Note tht for the urose of ths er the cost ssocted wth toll locton s not relevnt nd s just set to vlue whch results n n rorte number of tolls beng locted. For nd 2 locted tolls there exst no comrson from the smoothenng heurstc nd results re resented for these scenros to show the erformnce of the roch when the number of tolled lnks s ncresed. Lower nd uer bound estmtons on the number of toll loctons requred for frstbest rcng s obtned by the roch resented n Secton 4.3 whch results n lower bound of 211 tolls 3 nd n uer bound of 219 tolls 4. Comutng the mrovement n socl surlus wth the 219 toll loctons results n the sme mrovement of the socl surlus s s reched wth MSCP tolls. In Fgure 4 the fnl objectve functon vlue of (13) from the greedy heurstc s gven s functon of the number of locted tolls for ech scenro. Let τ SH nd τ denote toll level solutons from the roch nd the smoothenng heurstc resectvely. In Tble 6 SH MSCP SS( τ ) s resented together wth SS( τ )/ SS( τ ) nd SS( τ )/ SS( τ ) for comrson. As n lterntve roch for determnng good toll loctons the k number of 3 The soluton to (18) s rovded by runnng CPLEX verson 12.2 for 8 hours 4 The uer bound s obtned by ddng toll loctons untl z ˆ = whch s the vlue on ẑ when every lnk s tollble.

18 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors lnks wth the lrgest MSCP toll levels re chosen to be tolled wth toll level equl to the MSCP lnk toll. The results from ths roch s denoted s the kmscp soluton nd the kmscp corresondng chnge n socl surlus ( SS( τ )) s gven n Tble 6 for comrson. Fgure 4. Objectve functon vlue when solvng (13) wth the greedy heurstc. From Fgure 4 t s cler tht ẑ s reduced when the number of locted tolls s ncresed. In Tble 6 t cn be seen tht the hghest vlue on SS( τ ) s obtned wth β 2 =1 for k= 24 wth β 2 =1.5 for k=43 nd wth β 2 =.5 for the remnng scenros. For k=16 nd k=2 the choce of β 2 seems to be less mortnt nd β 2 = n generl erforms oorly. It s however dffcult to mke ny defntve sttement bsed on the solutons from the greedy heurstc snce otmlty s not gurnteed. Comrng the best toll loctons for ech scenro bsed on SS( τ ) wth the SH soluton obtned by the smoothenng heurstc t s cler tht SS( τ ) < SS( τ ) for ll scenros. It should however be noted tht the comuttonl tme for the roch s between 1 nd 796 seconds deendng on the number of locted tolls whle the comuttonl tme requred by the smoothenng heurstc to rovde the solutons s counted n dys. For the cses wth nd 69 tollble lnks the mxmum comuttonl tme requred by the roch s 1156 seconds nd the resultng mrovement of the socl surlus s wthn the rnge 87% to 94% of wht s cheved wth the smoothenng heurstc. Comrng the roch wth the kmscp roch shows tht when the number of locted tolls s ncresed the dfference between the two roches dmnshes. For u to 69 locted tolls the roch however clerly outerforms the kmscp roch. For the best found soluton for ech number of locted tolls the scent method s led to AS further olsh the soluton. The resultng toll levels re denoted τ wth the chnge n socl AS surlus gven by SS( τ ) nd these results re resented n Tble 7 together wth AS SH AS MSCP SS( τ )/ SS( τ ) nd SS( τ )/ SS( τ ) for comrson.

19 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors The results n Tble 7 show tht wth 24 locted tolls (roxmtely 11% of the number of tolls MSCP requred to cheve frstbest rcng) t s ossble to rech 75% of SS( τ ) nd the best obtned toll locton soluton for 24 locted tolls s resented n Fgure 5. Allowng 12 toll MSCP loctons mkes t ossble to rech 96% of SS( τ ). Wth 16 nd 2 locted tolls the MSCP devton from SS( τ ) s only mnor. In the lght of these results t s rent tht by otmzng toll loctons nd toll levels ttrctve congeston rcng schemes cn be desgned wth sgnfcntly reduced number of locted tolls comred wth frstbest rcng. The results lso show tht for relstc trffc network severl of the toll loctons requred to cheve frstbest rcng wll hve neglgble contrbuton to the mrovement n the socl surlus. Tble 6. Results from solvng (13) wth the greedy heurstc nd comrson wth kmscp. k β SS( τ ) SS SS( τ ) ( τ ) SH SS( τ ) MSCP SS( τ ) SS SS kmscp ( τ ) MSCP ( τ ) Tble 7. Results obtned by usng the scent method for olshng τ. k β 2 AS SS( τ ) SS SS( τ ) AS ( τ ) SH SS SS( τ ) AS ( τ ) MSCP

20 EJTIR 14(1) Fndng secondbest toll loctons nd levels by relxng the set of frstbest fesble toll vectors Fgure 5. Best obtned toll locton soluton for 24 locted tolls. Tolled lnks re mrked wth grey br. 6. Dscusson nd further reserch In ths er n roch bsed on mnmzng the devton from frstbest route tolls s led for fndng toll loctons nd levels whch mxmze the socl surlus. In the numercl results t s shown for relstc trffc network tht results cn be obtned for fxed toll loctons wth smll dfferences from known locl otml solutons n very short comuttonl tme. For vrble toll loctons t s shown tht the roch s ble to fnd good solutons wthn resonble comuttonl tme. One lmtton of the roosed roch s tht the qulty of the soluton deends on the vlues of the rmeters β 1 nd β 2. Whle t my not be ossble to know wht vlues to use on these rmeters n dvnce the numercl results suggests tht the soluton roch s not very senstve on the selecton of the rmeter vlues. Gvng the unused routes wegh close to zero or double tht of the used routes (β 2 2β 1 ) clerly show worse erformnce of the roch. For ll other evluted βvlues the dfferences n term of soluton qulty re smll. Another otentl roblem s the set of used routes s not unque ether n terms of route flows or used routes. For the Stockholm network ths hs not shown to be n ctul roblem nd the soluton when otmzng the toll levels n the current cordon re not senstve to the choce of used routes nd ther flows. Ths er lso rovdes results whch gve some nsght nto close to frstbest rcng schemes. Whle t cn be determned tht between 211 nd 219 tolls re requred to cheve frstbest rcng t s ossble to fnd rcng schemes wth 16 locted tolls whch ccount for 99% of the mrovement n socl surlus ssocted wth frstbest rcng. Even more nterestng re the results for 24 nd 43 numbers of tolls to locte whch corresond to 11% nd 2%