SCRAM: A Sharing Considered Route Assignment Mechanism for Fair Taxi Route Recommendations

Transcription

1 SCRAM: A Sharng Condered Route Agnment Mechanm for Far Tax Route Recommendaton Shyou Qan Department of Computer Scence and Engneerng, Shangha Jao Tong Unverty qhyou@jtu.edu.cn Iam Sahel Unverty of Lyon, ISA-Lyon, CITI-IRIA Lab am.ahel@na-lyon.fr Jan Cao Department of Computer Scence and Engneerng, Shangha Jao Tong Unverty cao-jan@jtu.edu.cn Mnglu L Department of Computer Scence and Engneerng, Shangha Jao Tong Unverty mll@jtu.edu.cn Frédérc Le Mouël Unverty of Lyon, ISA-Lyon, CITI-IRIA Lab frederc.le-mouel@nalyon.fr ABSTRACT Recommendng route for a group of competng tax drver almot untouched n mot route recommender ytem. For th knd of problem, recommendaton farne and drvng effcency are two fundamental apect. In the paper, we propoe SCRAM, a harng condered route agnment mechanm for far tax route recommendaton. SCRAM am to provde recommendaton farne for a group of competng tax drver, wthout acrfcng drvng effcency. By degnng a conce route agnment mechanm, SCRAM acheve better recommendaton farne for competng tax. By conderng the harng of road ecton to avod unneceary competton, SCRAM more effcent n term of drvng cot per cutomer (DCC). We tet SCRAM baed on a large number of htorcal tax trajectore and valdate the recommendaton farne and drvng effcency of SCRAM wth extenve evaluaton. Expermental reult how that SCRAM acheve better recommendaton farne and hgher drvng effcency than three compared approache. Categore and Subject Decrptor H.2.8 [Informaton Sytem]: Databae applcaton-data Mnng Keyword Recommender Sytem; Agnment Mechanm; Farne; Tax 1. ITRODUCTIO Tax drver need to fnd cutomer when ther tax are vacant. Meanwhle, cutomer expect to fnd a tax a quckly a poble. Wth the help of route recommender ytem, on one hand, tax Correpondng author. Permon to make dgtal or hard cope of all or part of th work for peronal or claroom ue granted wthout fee provded that cope are not made or dtrbuted for proft or commercal advantage and that cope bear th notce and the full ctaton on the frt page. Copyrght for component of th work owned by other than the author() mut be honored. Abtractng wth credt permtted. To copy otherwe, or republh, to pot on erver or to redtrbute to lt, requre pror pecfc permon and/or a fee. Requet permon from Permon@acm.org. KDD 15, Augut 1-13, 215, Sydney, SW, Autrala. Copyrght held by the owner/author(). Publcaton rght lcened to ACM. ACM /15/8...$15.. DOI: drver can more quckly fnd cutomer, ncreang ther revenue; on the other hand, cutomer can fnd a tax n a horter tme, avng watng tme. Therefore, route recommender ytem for tax drver are of great ocal and economc mportance becaue thee ytem am at provdng peronalzed and context-aware route recommendaton [12, 3]. Recommender ytem focued on publc tranportaton ervce have been tuded extenvely [6, 16, 11, 15, 17]. The common objectve of thee ytem to mnmze the drvng cot (n term of tme or dtance) of tax drver whle maxmzng the revenue whch are expreed a ucce probablte of pckng up cutomer. However, extng recommender ytem have everal lmtaton. Frt, thee ytem eldom addre the route recommendaton farne problem nvolvng a group of competng tax drver. Wthout a far route agnment mechanm, recommendaton farne not guaranteed for competng tax drver. Second, the harng of road ecton extng n the recommended route not condered when provdng route recommendaton, reultng n poor drvng effcency due to the unneceary competton of tax drver. Thrd, precely peakng, mot ytem, uch a [6, 11, 15], are coare-graned becaue a route recommended by thee ytem jut a drvng drecton n nature, rather than an actual drvng route compoed of conecutve road ecton. In the paper, we propoe SCRAM, a harng condered route agnment mechanm for far tax route recommendaton. SCRAM am to provde far and effcent route recommendaton for a group of competng tax drver. The prmary objectve of SCRAM to guarantee recommendaton farne for a group of competng tax drver. The by-product of SCRAM the mproved drvng effcency of tax drver by conderng the harng of road ecton extng n the recommended route. In addton, SCRAM capable of provdng tax drver wth actual drvng route, rather than rough drvng drecton. SCRAM dffer from mot extng route recommender ytem n four apect. Frt, SCRAM conder the harng degree of road ecton n the proce of computng recommended route. The conderaton of harng the ba on whch to provde recommendaton farne for competng tax drver. Second, the ucce probablty of pckng up cutomer ha prorty over drvng cot n the route evaluaton functon employed by SCRAM. Th prncple naturally conform to the realty that the goal of tax drver to mnmze drvng cot under the preme of fndng cutomer. Thrd, the ucce probablte of road ecton

2 contaned n a route are weghted decreangly wth the dtance from the tartng nterecton to the road ecton becaue potental cutomer at far road ecton are more lkely to be pcked up by other tax. Fourth, nether ucce probablty nor drvng cot uffcent to repreent the recommendaton farne and drvng effcency of tax drver. We preent a new metrc, called drvng cot per cutomer (DCC) whch combne both drvng cot and ucce probablty. We evaluate the recommendaton farne and drvng effcency of SCRAM baed on htorcal tax trajectore whch were collected n Shangha, Chna. The trajectore contan the GPS data of more than 4 tax drver operatng n a perod of 126 day. Expermental reult how that SCRAM outperform t counterpart to a large degree n term of both recommendaton farne and drvng effcency. The man contrbuton of the paper are a follow: By dentfyng three route evaluaton prncple, we defne an evaluaton functon whch comple wth the demand of tax drver. We degn a conce route agnment mechanm to guarantee the recommendaton farne of competng tax drver and prove the properte of the agnment mechanm by three theorem. Baed on real trajectory data, extenve experment are conducted to evaluate the recommendaton farne and drvng effcency of SCRAM. The ret of the paper organzed a follow. Secton 2 dcue related work, hghlghtng the dfference between our propoed approach and the extng one. Secton 3 formulate the far route recommendaton problem. Secton 4 defne the route evaluaton functon whch ncorporate three evaluaton prncple. Secton 5 preent the route agnment mechanm and the proof of three theorem whch theoretcally decrbe the properte of SCRAM. Secton 6 llutrate expermental reult. We conclude the paper n ecton RELATED WORK Route recommendaton for tax drver ha drawn much attenton from reearcher [6, 16, 11, 8, 14, 15, 17, 4, 18, 19, 9]. Mot extng recommender ytem are degned baed on real world GPS trajectore whch were collected through a large number of probng tax. Thee recommender ytem can be roughly dvded nto two categore: macrocopc recommender ytem and mcrocopc one. For macrocopc recommender ytem, uch a [6, 16, 8, 15, 19], only drvng drecton are provded for tax drver rather than actual drvng route. Generally, cutomer locaton are extracted from GPS trajectore by thee ytem and the locaton are clutered nto multple repreentatve mall area, whch are the recommended drvng drecton for tax drver. For example, n the LCP approach propoed n [6], thee mall area are referred to a pck-up pont whch are learned from the trajectore of hgh-proft tax drver. A tax drver who requet a route recommendaton provded wth a equence of pck-up pont a drvng drecton. Compared wth macrocopc recommender ytem, mcrocopc one provde tax drver wth actual drvng route. T-Fnder propoed n [17] ue the ame method a LCP [6] to extract the repreentatve mall area from the trajectore, but thee mall area are termed a parkng place. Intead of jut tellng tax drver where to go, T-Fnder goe one tep further than LCP by provdng tax drver wth detaled drvng route connectng parkng place. SCRAM belong to the mcrocopc category. However, SCRAM dffer from extng mcrocopc recommender ytem, uch a T-Fnder, n two apect. Frt, SCRAM addree the route recommendaton problem for a group of competng tax drver, rather than a ngle one. We degn a conce route agnment mechanm to guarantee the recommendaton farne for tax drver. Second, we conder the harng of road ecton occurrng n the recommended route, whch ext n realty and gnored n mot recommender ytem. The combnaton of thee two apect reult n a far and effcent route recommender ytem for competng tax drver. To the bet of our knowledge, the problem of recommendng route for a group of competng tax drver eldom tuded n the extng route recommender ytem. In [6] and [17], the round-robn mechanm ued to agn recommended route to multple requetng tax drver, wthout guaranteeng the recommendaton farne. The route recommender ytem propoed n [11] manly erve a ngle novce by provdng an optmal drvng drecton connectng multple mall area. In eence, the core of recommendng route for multple competng tax drver can be regarded a a load balance problem. A for load balance, recommendaton farne and drvng effcency are two contradctng factor. Therefore, the challenge of degnng a load balance polcy to guarantee the recommendaton farne wthout acrfcng the drvng effcency of competng tax drver. In the paper, we propoe SCRAM to addre th challenge. 3. PROBLEM FORMULATIO In realty, the road network of a cty, uch a Shangha, characterzed by a et of nterecton and a et of road ecton. Thu, a road network can be repreented by a graph G = (I, R), where I = {I 1, I 2,..., I n } a fnte et of n nterecton and R = {R 1, R 2,..., R m } a fnte et of m road ecton. A road ecton determned by two nterecton, R = (I j, I k ). A road ecton R aocated wth fve properte, the drecton (oneway or bdrectonal) R.dr, the peed contrant R.peed, the length R.length, the tartng nterecton R., and the endng nterecton R.e. Furthermore, by mnng the real trajectore generated by a large number of tax, each road ecton ha a ucce probablty P (R ) whch repreent the chance of pckng up cutomer for tax drver, P (R ) [, 1.] for 1 m. A route a drected equence of L road ecton, W = {R 1, R 2,..., R L }, where W. = R 1., W.e = R L.e and R j.e = R j+1. for 1 j < L whch mean that conecutve road ecton contaned n a route hould hare an nterecton. Let D be a group of competng tax drver that requet route recommendaton, D = {D 1, D 2,..., D }. Baed on the above defnton and notaton, we can formally defne the problem of far tax route recommendaton a: DEFIITIO 1. The far tax route recommendaton problem. Gven: a road network G = (I, R), a probablty et P = {P (R 1 ), P (R 2 ),..., P (R m )}, a cot matrx T repreentng the drvng cot between two neghborng nterecton, a group of competng tax drver D = {D 1, D 2,..., D } located near an nterecton. Objectve: Fndng a et of route W = {W 1, W 2,..., W } from the canddate route and agnng a route to each requetng drver (D, W j ) whle atfyng the followng condton. CODITIO 1. When agnng route to drver, the agnment mechanm mut guarantee recommendaton farne for competng tax drver from a long-term perpectve. The drvng cot per cutomer (DCC) hould be a contant for all competng

3 Frequency (x1) Dtance (km) (a) CDF Dtance (km) (b) Fgure 1: Dtrbuton and CDF of drvng dtance tax drver. In other word, the tandard devaton of DCC hould be mnmzed. The objectve of SCRAM to addre the recommendaton farne problem for a group of competng tax. If tax are far away from each other wthout competton, route recommendaton can be provded ndependently. Thu, n the paper, t aumed that a group of competng tax drver located near an nterecton whch ued a the tartng pont to generate canddate route. Th tuaton obervable n realty. For example, tax tend to wat for cutomer at certan place, uch a ubway ext or hoppng center, but ometme the queue too long, o ome tax are reluctant to contnue watng. The aumpton can be releaed to a more common cae where tax are cattered n a mall area. In th cae, the nearet nterecton from the center of the area could be elected a the tartng pont. Snce all recommended route are generated from the tartng pont, ome tax need to go there to follow the recommended route. Superfcally, th may affect the recommendaton farne of competng tax drver. Due to the harng of road ecton extng n the recommended route, n fact, the mpact neglgble. We dcu th cae n detal n Secton 5. When recommendng drvng route for tax drver, the length of route uually retrcted. ote the length of a route the number of road ecton that are contaned n the route. Let λ be the maxmum number of road ecton that are connected to nterecton n a road network. In a general cae wth no retrcton on the length of recommended route, the number of canddate route n λ(λ 1) 1, where n the number of nterecton appearng n the road network. A proved n [6], the complexty of computng recommended route O(n!). When the length of recommended route lmted to a contant number L, the number of canddate route reduced to λ(λ 1) L 1. For the far tax route recommendaton problem, a length-contraned cae more realtc, whch comple wth the traffc pattern extracted from the real trajectore generated by a large number of tax. By analyzng the tax trajectore collected n Shangha, Chna, we fnd that tax drver uually travel along a mall number of road ecton to pck up cutomer. The dtrbuton and cumulatve dtrbuton functon (CDF) of drvng dtance are depcted n Fgure 1. A hown n the fgure, more than 4% of tax drver pck up cutomer n le than 1 km. Th obervaton called localty drvng behavor of tax drver, whch ued to determne the length of recommended route. The dtrbuton and CDF of drvng tme taken by tax drver are hown n Fgure 2, whch alo confrm the localty drvng behavor of tax drver. It oberved that almot 4% of tax drver fnd a cutomer n le than 5 mnute. Further detal on the trajectory data are gven n Secton EVALUATIO FUCTIO The way to evaluate route crtcal to electng recommended route from the canddate one. Generally, a route can be meaured Frequency(x1) Tme (mn) (a) CDF Tme (mn) (b) Fgure 2: Dtrbuton and CDF of drvng tme by t ucce probablty of pckng up cutomer or drvng cot. When recommendng route to tax, ucce probablty and drvng cot hould be condered multaneouly. The defnton of an evaluaton functon hould comply wth the demand of tax drver. 4.1 Evaluaton Prncple Prorty Prncple The mot mportant apect of defnng a route evaluaton functon to determne the goal of tax drver. If tax drver olely chooe energy avng a ther purpoe, the bet choce for them jut to tay n the ame place. However, each tax drver need to pay a certan amount of money a a management fee even f he/he doe nothng each day. Therefore, the goal of tax drver to mnmze drvng cot under the preme of fndng cutomer. aturally, the frt prncple of evaluaton that ucce probablty hould have hgher prorty than drvng cot Decayng Prncple The ucce probablte of road ecton contaned n a route hould not be weghted equally. The drvng dtance from the tartng pont to a road ecton crucal for tax drver to ucceed n fndng cutomer. Generally, tax drver more ealy fnd potental cutomer at near road ecton, whle potental cutomer at far road ecton are more lkely to be pcked up by other tax drver. Therefore, the econd prncple of evaluaton that the ucce probablte of road ecton contaned n a route hould be weghted decreangly wth the drvng dtance from the tartng nterecton to the road ecton Sharng Prncple Gven a road ecton wth a hgh ucce probablty, f only one tax crue on t, the tax drver more lkely to fnd a cutomer. If there are multple tax traverng the road ecton, the chance of pckng up cutomer ubtantally decreaed. To guarantee recommendaton farne and drvng effcency, tax hould be prevented from all drvng toward a route whch ha hgh ucce probablty. Thu, the thrd prncple of evaluaton that when multple route contan a road ecton, no matter t place n each route, t reaonable to hare the ucce probablty of the road ecton. To effectvely evaluate route, we propoe the expected drvng cot (EDC) functon whch ncorporate the three evaluaton prncple mentoned above. 4.2 EDC wthout Sharng To mplfy the dcuon, we llutrate the EDC functon va an example. Fgure 3 how a route compoed of L road ecton, R 1 R 2... R L 1 R L. P (R ) and T (R ) are the ucce probablty and drvng cot of road ecton R (1 L), repectvely. Let E be the event that a tax drver pck up a cutomer at road ecton R. When a tax drver follow th

4 Fgure 3: An example of a route T (R 1 ) T (R 2 ) P (R 1 ) P (R 2 ) EDC W W W W W W route, he/he may pck up a cutomer at road ecton R wth a probablty P (E ). Let E be the event that a pck-up event never happen. Ω = {E } [1,L] E defne the unvere of event and P (Ω) = L P (E) + P (E ) = 1. We defne the EDC functon a: EDC = L T (E ) (2P (E ) 1) where the event probablte are: + T (E ) (2P (E ) 1), (1) L + 1 P (R 1) = 1 1 P (R ) (1 P (R j )) [2, L] P (E ) = j=1 L (1 P (R j)) = j=1 and the drvng cot are: T (R 1 )(1 P (R 1 )) = 1 1 T (R j ) + T (R )(1 P (R )) [2, L] T (E ) = j=1 L T (R j )/P (R j ) = j=1 (3) In Equaton (1), the exponental proceng of P (E ) realze the prorty prncple, whch hghlght the prorty of ucce probablty over drvng cot. An llutraton gven n Table 1. Route W 1 and W 2 are compoed of two road ecton. The frt road ecton of W 1 ha a lower ucce probablty and drvng cot, whle the frt road ecton of W 2 ha a hgher ucce probablty and drvng cot. The econd road ecton of W 1 and W 2 have the ame ucce probablty and drvng cot. In th cae, accordng to the prorty prncple, W 2 hould be uperor to W 1 n term of EDC. Due to the exponental proceng, the prorty of ucce probablty over drvng cot exacerbated wth hgher ucce probablty value. For example, a lted n Table 1, due to the hgh ucce probablty of W 4 frt road ecton, even the drvng cot of W 4 frt road ecton four tme that of W 3, and W 4 tll better than W 3 n term of EDC. The denomnator n Equaton (1), and L+1, fulfl the decayng prncple. For example, route W 5 and W 6 lted n Table 1 both contan two road ecton wth the ame drvng cot. W 5 frt road ecton ha a larger ucce probablty, and the econd road ecton ha a maller ucce probablty. On the contrary, W 6 frt road ecton ha a maller ucce probablty, whle the econd road ecton ha a larger ucce probablty. In th cae, due to the decayng prncple, W 6 uperor to W 5 n term of EDC. 4.3 EDC wth Sharng The harng of road ecton occurrng n route make the evaluaton of route more complcated and cotly. Accordng to (2) Table 1: Example route ued to explan the evaluaton prncple Fgure 4: An llutraton of route harng a common road ecton ID Road Secton W 1 R 1 R 2 R 3 W 2 R 1 R 4 R 5 W 3 R 1 R 6 R 7 W 4 R 8 R 9 R 1 W 5 R 11 R 12 R 13 W 6 R 14 R 15 R 16 Table 2: Lt of canddate route the harng prncple, f a road ecton contaned n multple route, the ucce probablty of the road ecton hould be hared by thee route. The problem of mplementng the harng prncple that when a road ecton hared by multple route, t non-trval to compute the recommended route from the canddate one nce the hared tme of the road ecton condtonal. Therefore, condtonal EDC hould be computed for route contanng common road ecton. An example gven to llutrate the defnton of condtonal EDC. A hown n Fgure 4, there are x canddate route lted n Table 2. Of thee route, W 4, W 5 and W 6 are ndependent route, whle W 1, W 2 and W 3 hare a common road ecton R 1. Three tax drver located at nterecton I 8 requet route recommendaton. Therefore, three recommended route hould be elected from the et of canddate route {W 1,..., W 6} n term of EDC. Concernng the harng tme of R 1, three tuaton may appear, whch are lted n the followng. 1. The harng prncple appled by mply dvdng the ucce probablty of R 1 by the number of route that contan R 1. In th example, the ucce probablty of R 1 dvded by 3. Due to the dvon of R 1 ucce probablty, W 4, W 5 and W 6 may be elected a the three optmal route. 2. R 1 aumed to be hared by only two route. Snce the ucce probablty of R 1 jut dvded by two, the poblty of W 1 and W 2 beng optmal promoted. So, W 1, W 2 and W 5 may be elected a the three optmal route. 3. If the prncple of harng ucce probablty not appled n the computaton of EDC value, W 1, W 2 and W 3 may be elected a the three optmal route for recommendaton. However, tatcally determnng the harng tme, uch a the above three polce, not correct, becaue the above three tuaton maybe occur multaneouly. The hared tme of R 1

5 hould be determned condtonally n a combnaton that elect three route from the canddate et. For the example hown n Fgure 4, there are C(6, 3) = 2 combnaton to elect three route from the x canddate one. In each combnaton, the harng tme of a road ecton determntc. Therefore, condtonal EDC hould be computed for each route accordng to the actual hared tme of R 1 n the combnaton. The combnaton wth the mnmum ummed condtonal EDC value the oluton whch contan the three optmal route. Let S(R, c) be the number of route that hare the road ecton R n a combnaton c, the condtonal event probablte are defned a: P (R 1 ) S(R 1 = 1,c) 1 P (R ) (1 P (Rj) ) [2, L] P (E, c) = S(R, c) S(R j=1 j, c) (4) L (1 P (R j) S(R j, c) ) = j=1 The condtonal EDC of a route n the combnaton c defned a: L EDC = T (E ) (2P (E,c) 1) + T (E ) (2P (E ) 1) L + 1 (5) The dvon of P (R ) by S(R, c) realze the harng prncple etablhed for route evaluaton, preventng tax from all drvng toward a route wth hgh ucce probablty. The ucce probablty of a road ecton an expectaton ued n the evaluaton functon. Actually, the arrval tme of tax may affect the harng of ucce probablte, but the arrval order cannot be taken nto account n advance. It would be too complcated to model the arrval order n the evaluaton functon. To mplfy the model, the tme relaton are not condered n the evaluaton functon. 4.4 Computaton Accordng to the defnton of EDC, optmal route computaton a combnatoral optmzaton problem n nature. Even though the number of canddate route mall, the number of combnaton very large. Therefore, the computaton performance hould be condered erouly. We deal wth th problem from three apect. Frt, the computaton of canddate route can be done offlne. Second, to reduce computaton cot, t acceptable to obtan an approxmate oluton rather than the exact one. Thrd, the dvdeand-conquer dea can be utlzed to decreae the computaton complexty by dvdng a map nto multple overlappng area. Actually, the computaton of canddate route can be done offlne. The nformaton that needed to compute canddate route cont of the road network, the route length and the traffc pattern (ucce probablte and drvng cot of road ecton), whch are all avalable n advance. Wth the avalablty of the road network and the route length, all canddate route can be generated by ung the depth frt earch algorthm whch conder the retrcton n the road network, uch a one-way treet or noturn-rght nterecton. It not neceary to generate canddate route for all nterecton, jut for part of the popular nterecton whch can be extracted from the trajectore. Snce computng optmal route a combnatoral optmzaton problem n nature, multple optmzaton algorthm can be utlzed to olve the problem. In the experment, we ue Genetc Algorthm (GA) [7] to compute recommended route by obtanng an approxmate oluton. In the ntalzaton tage, route are elected from the canddate et, where alo the number of tax drver requetng route recommendaton. The EDC functon ued a Fgure 5: Overlappng dvon of a map the ftne functon to evaluate the oluton doman. By teratng a certan number of round, GA produce an approxmate oluton whch contan the recommended route. For bg cte, uch a ew York or Shangha, there are a large number of nterecton and road ecton. It challengng to compute recommended route by ung only one erver. Th challenge can be addreed by dvdng a road network nto multple overlappng area. An area correpond to a erver whch reponble for proceng requet from the area. Snce a route may pan more than one area, overlappng dvon ued to deal wth th tuaton. The degree of overlappng ncreae wth route length. An example hown n Fgure 5 gven to llutrate the overlappng dvon. In the example, a map dvded nto four area, denoted by A1, A2, A3 and A4. The blue and green dahed lne repreent the overlappng area of A1 and A2, repectvely. Four erver are deployed to provde the route recommendaton ervce, each ervng an area. Through the dvde-and-conquer method, the computaton performance can be gnfcantly mproved, fully atfyng the performance requrement of real applcaton. The computatonal reource can be provded by cloud computng platform, uch a EC2. 5. ROUTE ASSIGMET MECHAISM The goal of degnng a route agnment mechanm to propoe an agnment functon f : D W j that agn each tax drver a route. The agnment problem ha been addreed largely by mechanm degner [5] [2]. Agnment mechanm can be clafed nto two famle. Money mechanm [1] allow the exchange of currency durng the agnment and non-money mechanm [13] do not allow t. The route agnment problem belong to the famly of money mechanm. A far route agnment mechanm for tax drver hould meet the followng condton. Frt, t hould guarantee recommendaton farne from a long-term perpectve. Some tax drver may eldom requet route recommendaton whle other may frequently do o. The agnment mechanm hould be able to guarantee recommendaton farne for tax drver who frequently requet route recommendaton. Second, the agned route cannot be predcted by tax drver, even gven the et of recommended route. For example, when a drver agned the wort route at the prevou tage, t hould not be guaranteed that the bet route wll be agned to hm n the current tage. Therefore, the mechanm hould make the agnment decon baed on multple htorcal agnment record, not jut one. Thrd, the agnment mechanm hould prevent volent vbraton when conecutvely

6 agnng route to a tax drver. By conderng all thee apect, we degn a far route agnment mechanm (SCRAM) whch work through the followng tep: 1. At the begnnng of the agnment, the balance of each drver D D et to b =. 2. At tage k, the route are orted on ther EDC value n acendng order and the drver are orted on ther balance n decendng order. Then, the j th drver n the drver lt agned the j th route n the route lt. 3. After the agnment at tage k, the balance are recomputed kj=k 1 a b k+1 b = j + c k, where c k = V k m k the amount charged to drver D. m k the mean value of the route EDC value and V k the EDC value of the route agned to drver D at tage k. In SCRAM, the drver wth the hghet balance receve the bet route. Th mechanm make the tandard devaton of drver balance at each tage reach t mnmum. Expermental reult alo verfy that SCRAM capable of guaranteeng recommendaton farne for a group of competng tax drver. A mentoned n Secton 3, a group of competng tax may be cattered n a mall area, and the nearet nterecton from the center of the area elected a the tartng pont to generate canddate route. So, ome tax that are not located at the nterecton need to get there to follow the recommend route. Th tuaton may affect the performance of SCRAM, but the mpact neglgble due to two fact. Frt, nce the road ecton near the tartng nterecton are orgnally ntenely hared, the gatherng of tax would not greatly exacerbate the competton. Second, n addton to the ntra-group competton arng from the tax drver who requet route recommendaton, tax drver alo face ntergroup competton comng from tax drver who do not requet route recommendaton. The harng prncple conder the ntragroup competton, whle the decayng prncple take nto account the nter-group competton. The combnaton of the two prncple well model realty. 5.1 Properte of SCRAM THEOREM 1. In SCRAM, the expected balance of drver at each tage zero, bk =, k +. PROOF. We wll prove that the agnment problem a zeroum game whch mean that the um of drver balance b k for k +. Th mean that b k =, k +. (6) We can how by nducton that for k +, the followng propoton true b k =, k +. (7) The game tart wth zero balance for all drver, b [1, ], therefore =, b =, (8) o the propoton (7) true for k =. Suppoe that the propoton (7) true for k, then for tage k +1 ( k ) b k+1 j=k 1 = bj + c k ( b k 1 + b k b k = + V k m ), k (9) where m k the mean of the route EDC value, So b k+1 = = + m k = b k 1 + V k m k b k 1 + V k + m k Snce bk 1 = bk V k m k =, o V k. (1) b k b k b k b k (11) =... = bk = and b k+1 =. (12) Therefore, the propoton (7) true for k + 1. By the prncple of nducton, the propoton (7) true for k +. THEOREM 2. In SCRAM, the drver wth the hghet balance receve the bet route. Th mechanm make the tandard devaton of drver balance at each tage reach t mnmum, mn( (b k )2 )1/2, k +. PROOF. A proved n Theorem 1, the mean of drver balance at each tage zero. The tandard devaton σ of drver balance at tage k + 1 ( (b k+1 ) 2 ) 1/2 σ =, (13) kj=k 1 where b k+1 b = j + c k. For preentaton mplcty, let kj=k 1 b x denote j and y repreent c k at tage k, then ( (x + y ) 2 ) 1/2 ( ) x 2 + 2x y + y 2 1/2 σ = =. (14) A mentoned above, the problem of route agnment to propoe a functon f : D W j for 1 and 1 j. Gven the order of drver, route agnment to get a permutaton of {1, 2,..., } whch realze the map from drver to route. Let S be the et of all permutaton of {1, 2,..., } and θ a permutaton n S. θ() denote the th element n θ. The problem of mnmzng σ equvalent to θ = arg mn θ S x y θ() (15)

7 Wthout lo of generalty, we uppoe that x orted n acendng order and y orted n decendng order. Th enough to fnd a permutaton θ S whch mnmze x y θ() (16) In the followng, the notaton x y θ() mplfed by (θ). We wll prove by nducton that for 2 the followng formula true (Id) (θ), θ S, (17) where Id the dentty permutaton appled n SCRAM. For = 2, S 2 = {θ 1, θ 2} where θ 1 = Id and θ 2 defned a θ 2(1) = 2 and θ 2(2) = 1. So (θ 2) (Id) = x 1y 2 + x 2 y 1 x 1 y 1 x 2 y 2 = (y 2 y 1 )(x 1 x 2 ). (18) Snce x n acendng order and y n decendng order, (Id) (θ2). Formula (17) true for = 2. Suppoe that Formula (17) true for > 2. Let x, y R +1, uch that x orted n acendng order and y orted n decendng order. We need to prove +1(Id) (θ), θ S +1. (19) +1 For all θ S +1, (θ) = x y θ() = x ȳ θ() + x +1 y j, (2) where j = θ( + 1), ȳ = (y 1,..., y j 1, y j+1,..., y +1 ) and θ S defned a θ() = { θ() f θ() {1,..., j 1} θ() 1 f θ() {j + 1,..., + 1} (21) Snce x n acendng order and ȳ n decendng order, the nducton aumpton mple (θ) x ȳ + x +1 y j = A j. (22) +1 Let compute the dfference A j A +1 for j {1,..., }, ( ) A j A +1 = x ȳ + x +1 y j x y + x +1 y +1 = j 1 x y + x y +1 + x +1y j =j x y x +1 y +1 = x y + x y +1 + x +1 (y j y +1 ) = =j =j x (y +1 y ) + x +1(y j y +1) =j x +1 (y +1 y ) + x +1 (y j y +1 ) =j = x +1(y +1 y j) + x +1(y j y +1) = (23) where A +1 = +1 (Id). We have A j +1(Id) (θ), θ S +1 (24) +1 Therefore, Formula (17) true for + 1. By the prncple of nducton, Formula (17) true for all 2. THEOREM 3. In SCRAM, the um of each drver balance converged to a contant C, lm k=1 bk = C for all D D. PROOF. A degned n SCRAM, the balance of drver D at kj=k 1 tage k + 1 b k+1 b = j + c k, where c k = V k m k. For the ake of mplcty, et =1. Then b k+1 = b k + c k, we get the ere of b a b = b 1 = b + c 1 = c 1 b 2 = b 1 + c 2 = c 1 + c 2... b = k=1 ck = k=1 (V k m k ) The um of D balance k=1 k=1 j=1 (25) k b k = (V j m j ) (26) When = 1, the prorty at tage k + 1 determned by the prorty at tage k. If the drver D get a hgher prorty at tage k, then he wll receve a relatvely lower prorty at tage k + 1. Th relatonhp can be expreed by a functon f (V k+1 m k+1 ) = f(v k m k ). (27) Therefore, the value of c k not monotoncally ncreaed wth the tage, but fluctuant, ether potve or negatve. So, lm k=1 b k = C. (28) 6. SIMULATIO RESULTS We evaluate the recommendaton farne and drvng effcency of SCRAM through mulaton whch are conducted baed on real world trajectore. Two parameter are ued n the experment, where the number of competng drver and L the length of recommended route. 6.1 Settng Road etwork The mulaton baed on the road network of Shangha whch contan about 22,42 nterecton and 32,919 road ecton. The average length of road ecton.45 km Trajectore The tax trajectore were collected n Shangha, Chna from approxmately 4 probe-tax operatng over a perod of 126 day [1]. A trajectory cont of a equence of pont. Each pont contan even feld: ID, tmetamp, longtude, lattude, peed, angle, and tatu. The meanng of the frt x feld well undertood. The lat feld the current tatu of a tax, ndcatng vacant and 1 for occuped.

8 6.1.3 Traffc Pattern Traffc pattern, uch a ucce probablte of road ecton, are extracted from the trajectore. Thee pattern are tme dependent. A common way to compute tme-dependent pattern knowledge to partton a day nto a certan number of fxed tme lot (e.g., 3 mnute a lot), a utlzed n [6]. In the experment, we alo employ th approach. Gven a road ecton, the number of vacant tax pang the road ecton counted, and the number of tax that ucceed n fndng cutomer at the road ecton alo ummed n each tme lot. The two number are ued to compute the ucce probablty of the road ecton. Traffc pattern are uually updated perodcally (weekly or monthly) wth new tax trajectore. The ucce probablte of road ecton and the dtrbuton of cutomer are extracted n the tme lot 8:AM - 8:3AM. 6.2 Compared Method SCRAM compared wth three approache. The frt one RA whch ued a a baelne. In RA, recommended route are randomly elected from the canddate et and then route are randomly agned to drver. The econd approach LCP, whch wa propoed n [6]. The route length of LCP et to three n the experment. Pleae note the route length of LCP the number of pck-up pont, rather than the number of road ecton. In the experment, we utlze the hortet path algorthm to determne the actual route connectng two pck-up pont for LCP. The roundrobn mechanm ued to agn route to tax n LCP. The thrd approach a veron of SCRAM wthout conderng the harng of road ecton, denoted by SCRAM-W/O. By comparng SCRAM wth SCRAM-W/O, we hghlght the effect of road ecton harng on recommendaton farne and drvng effcency. To provde a far comparon, the number of road ecton contaned n recommended route hould be the ame for the four approache. For RA, SCRAM-W/O, and SCRAM, the length of recommended route equal. Snce LCP recommend longdtance route and ue the round-robn mechanm to agn route, the number of recommended route hould be determned by the number of road ecton contaned n the route of SCRAM. For example, there are 1 tax drver requetng route recommendaton, and the route length of SCRAM and LCP 8 and 16, repectvely. For SCRAM, 1 route are recommended and agned to tax drver. For LCP, only 5 route are elected a the recommended route, whch are agned to tax drver by the round-robn mechanm. Both LCP and T-Fnder [17] extract mall popular area from the trajectore of hgh-proft tax drver. Thee area are called pck-up pont n LCP, whle they are termed a parkng place n T-Fnder. A route recommended by LCP mply a equence of pck-up pont wthout actual route to connect thee pont. T- Fnder goe one tep further by gvng the actual route connectng two parkng place. The hortet path algorthm ued by T- Fnder to generate actual route. In the experment, we adopt the ame algorthm to generate the actual route connectng two pck-up pont for LCP. Therefore, LCP extremely mlar to T-Fnder, whch not compared n the experment to avod duplcaton. 6.3 Evaluaton Reult When a tax drver provded wth a recommended route, f there are cutomer dtrbuted on the route and the tax drver reache a cutomer earler than other competng tax by followng the route, the tax drver ucceed n pckng up a cutomer. In our mulaton, each experment conducted for ten round wth dfferent tartng nterecton that are randomly elected from the um of Cutomer RA LCP SCRAM W/O SCRAM Drver Fgure 6: Cutomer pcked up by tax drver. Mn Max Avg Std RA LCP SCRAM-W/O SCRAM Table 3: Stattc of pcked up cutomer et of all nterecton. In each round, a tax drver recommended a route and marked a to whether the tax drver pck up a cutomer or not. If the tax drver ucceed n pckng up a cutomer, he top crung on the route, and the dtance from the departng locaton to the place of the cutomer calculated a the drvng cot. Otherwe, the dtance from the departng locaton to the end of the recommended route computed. We ue the peed contraned on road ecton a the drvng peed n the experment The umber of Pcked up Cutomer In the experment, the number of competng tax drver ten and the length of recommended route L et to L = 8. The number of pcked up cutomer ummed for each tax drver. The reult are depcted n Fgure 6. Two obervaton can be made from the fgure. Frt, LCP able to fnd more cutomer than RA, SCRAM-W/O and SCRAM, by, on average, 27%, 24% and 11%, repectvely. Th attrbutable to the fact that the recommended route of LCP have longer dtance than the one of the other three approache. Therefore, the harng degree of road ecton correpondngly le ntene for LCP. Second, although the et of canddate route the ame for RA, SCRAM- W/O and SCRAM, SCRAM fnd 14% and 12% more cutomer than RA and SCRAM-W/O, repectvely. The reaon that SCRAM conder the harng of road ecton when computng recommended route, allevatng unneceary competton. The mnmum, maxmum, average and tandard devaton are lted n Table 3. It oberved that there no obvou dfference n the tandard devaton Drvng Dtance The ettng of the experment the ame a the prevou one. For each tax drver, the drvng dtance ummed for ten round. The reult are plotted n Fgure 7. LCP fnd the larget number of cutomer at the hghet drvng cot. On average, LCP ncur 137%, 134% and 223% more drvng cot than RA, SCRAM- W/O and SCRAM, repectvely. SCRAM ncur the leat drvng cot to fnd cutomer and the drvng dtance of tax drver are more table. The mnmum, maxmum, average and tandard devaton of drvng dtance are lted n Table 4. It oberved that the tandard devaton ncreae wth drvng dtance. LCP fluctuate more ntenely than the other three approache.

9 Drvng dtance (km) RA LCP SCRAM W/O SCRAM Drver Fgure 7: Drvng dtance of tax drver. Mn Max Avg Std RA LCP SCRAM-W/O SCRAM Table 4: Stattc of drvng dtance Drvng Cot per Cutomer ether the number of cutomer nor the drvng dtance uffcent to repreent the recommendaton farne and drvng effcency of tax drver. It meanngle to conder only one metrc. In order to compare the recommendaton farne and drvng effcency of tax drver, we propoe a new metrc called drvng cot per cutomer (DCC), whch defned a: DCC = drvng cot number of cutomer. (29) DCC repreent the average drvng cot to fnd a cutomer, reflectng the drvng effcency of tax drver. Meanwhle, the tandard devaton of DCC ndcate the recommendaton farne of competng tax drver. The DCC value of the four approache are hown n Fgure 8. On average, SCRAM 56%, 189% and 55% more effcent than RA, LCP, and SCRAM-W/O, repectvely. The mnmum, maxmum, average, and tandard devaton of DCC are lted n Table 5. It obvou that the tandard devaton of DCC ncreae wth the value of DCC. RA, LCP and SCRAM-W/O fluctuate ntenely. On the contrary, SCRAM much more table due to the conderaton of road ecton harng n the computaton of optmal route and the degn of an agnment mechanm to guarantee recommendaton farne. SCRAM acheve 322%, 278% and 77% better recommendaton farne than RA, LCP and SCRAM- W/O, repectvely. Overall, LCP provde tax wth long-dtance route to travere, whle SCRAM ugget tax crue n the local area. DCC RA LCP SCRAM W/O SCRAM Drver Fgure 8: DCC of tax drver. DCC Mn Max Avg Std RA LCP SCRAM-W/O SCRAM Table 5: Stattc of DCC =5 =1 =15 =2 =25 RA LCP SCRAM W/O SCRAM Method Fgure 9: The effect of tax drver. =5 =1 =15 =2 =25 RA LCP SCRAM-W/O SCRAM Table 6: Standard devaton of DCC Therefore, the area covered by the route of SCRAM maller than that of LCP. A route recommended by LCP longer and ha more chance of fndng cutomer, compared wth SCRAM. However, maller not necearly bad. A demontrated by the expermental reult, SCRAM farer and more effcent than LCP n term of drvng cot per cutomer. 6.4 Scalablty More nght of calablty are preented for the four approache n th ecton. The recommendaton farne and drvng effcency of SCRAM are evaluated by changng the number of tax drver and the length of recommended route The umber of Tax Drver Under the ame condton, ncreang the number of tax aggravate the competton, whch correpondngly lead to the growth of DCC. A hown n Fgure 9, DCC ncreae wth the number of tax drver for the four approache. For RA and LCP, the uperorty of SCRAM gradually dmnhe wth the ncreae of tax due to the aggravated competton. For example, when = 5, SCRAM 164% and 24% more effcent than RA and LCP, repectvely. When = 25, SCRAM only 19% and 77% more effcent than RA and LCP, repectvely. However, the uperorty of SCRAM over SCRAM-W/O ncreae wth tax. For example, when = 5, SCRAM 44% more effcent than SCRAM-W/O, but when = 25, SCRAM 74% more effcent than SCRAM-W/O. Th demontrate that SCRAM-W/O more affected by the number of tax than other approache. The tandard devaton of DCC are lted n Table 6. Overall, the tandard devaton of DCC ncreae wth the number of tax drver. When = 25, SCRAM tll acheve 127%, 56% and 16% better recommendaton farne than RA, LCP and SCRAM-W/O, repectvely, even under ntene competton.

10 DCC L=6 L=8 L=1 L=12 L=14 RA LCP SCRAM W/O SCRAM Method Fgure 1: The effect of route length. L=6 L=8 L=1 L=12 L=14 RA LCP SCRAM-W/O SCRAM Table 7: Standard devaton of DCC The Length of Recommended Route The effect of route length on DCC depcted n Fgure 1. Accordng to the localty drvng behavor dcovered n Fgure 1, t not neceary to tet large number of road ecton n the experment. For RA, route length ha almot no effect on DCC. The DCC of LCP decreae wth route length. A mentoned above, the number of road ecton contaned n the recommended route of the four approache hould be equal for the ake of far comparon. Increang the route length of SCRAM caue more route to be elected for the round-robn agnment n LCP, whch ndrectly allevate the competton. Increang the route length of SCRAM ndeed reult n fndng more cutomer but at the cot of drvng longer dtance. The ncreang growth of drvng dtance far larger than the ncreang growth of cutomer, caung an ncreae of DCC. Wthout conderng the harng of road ecton, the DCC of SCRAM-W/O more ealy affected by the length of route than SCRAM. The tandard devaton of DCC are lted n Table 7. On average, SCRAM acheve almot 1% better recommendaton farne than other three approache. 7. COCLUSIO Route recommendaton for tax drver of great economc and ocal mportance. On one hand, cutomer can quckly fnd tax, avng watng tme; on the other hand, tax drver can fnd cutomer n a horter tme by followng the recommended route, ncreang ther revenue. In th paper, we propoe SCRAM, a harng condered route agnment mechanm for far tax route recommendaton. In SCRAM, the harng of road ecton extng n the recommended route condered n the proce of computng optmal route. Then, recommended route are agned to tax drver from a long-term perpectve to guarantee recommendaton farne. In comparon wth three approache, SCRAM capable of provdng tax drver wth more effcent route that have the leat DCC. Furthermore, SCRAM acheve better recommendaton farne for competng tax drver, whch verfed by the tandard devaton of DCC. 8. ACKOWLEDGMETS Th work upported by atonal 863 Program (215AA1A22). Th work alo partally upported by Chna atonal Scence Foundaton ( , , ), Reearch Fund of Scence and Technology Common of Shangha Muncpalty ( ). Frédérc Le Mouël work funded by a grant from Rhone-Alpe Regon, France. 9. REFERECES [1] Suvnet-trace data. [2] A. Abdulkadroglu and T. Sonmez. School choce: A mechanm degn approach. The Amercan Economc Revew, 93(3): , 23. [3] G. Adomavcu and A. Tuzhln. Toward the next generaton of recommender ytem: A urvey of the tate-of-the-art and poble extenon. Knowledge and Data Engneerng, IEEE Tranacton on, 17(6): , 25. [4] Y. Dng, S. Lu, J. Pu, and L. M.. Hunt: A trajectory recommendaton ytem for effectve and effcent huntng of tax paenger. In MDM, page IEEE, 213. [5] A. G. Erdman and G.. Sandor. Mechanm degn: analy and ynthe (Vol. 1). Prentce-Hall, Inc., [6] Y. Ge, H. Xong, A. Tuzhln, K. Xao, M. Gruteer, and M. Pazzan. An energy-effcent moble recommender ytem. In SIGKDD, page ACM, 21. [7] D. E. Goldberg et al. Genetc algorthm n earch, optmzaton, and machne learnng, volume 412. Addon-weley Readng Menlo Park, [8] H. Hu, Z. Wu, B. Mao, Y. Zhuang, J. Cao, and J. Pan. Pck-up tree baed route recommendaton from tax trajectore. In Web-Age Informaton Management, page Sprnger, 212. [9] J. Lee, I. Shn, and G.-L. Park. Analy of the paenger pck-up pattern for tax locaton recommendaton. In CM, page IEEE, 28. [1]. an and A. Ronen. Algorthmc mechanm degn. In Proceedng of the thrty-frt annual ACM ympoum on Theory of computng, page ACM, [11] S. Qan, Y. Zhu, and M. L. Smart recommendaton by mnng large-cale gp trace. In WCC, page IEEE, 212. [12] P. Renck and H. R. Varan. Recommender ytem. Communcaton of the ACM, 4(3):56 58, [13] J. Schummer and R. V. Vohra. Mechanm degn wthout money. Algorthmc Game Theory, 1: , 27. [14] L.-Y. We, Y. Zheng, and W.-C. Peng. Contructng popular route from uncertan trajectore. In SIGKDD, page ACM, 212. [15] J. Yuan, Y. Zheng, C. Zhang, W. Xe, X. Xe, G. Sun, and Y. Huang. T-drve: drvng drecton baed on tax trajectore. In SIGSPATIAL, page ACM, 21. [16] J. Yuan, Y. Zheng, L. Zhang, X. Xe, and G. Sun. Where to fnd my next paenger. In UbComp, page ACM, 211. [17]. J. Yuan, Y. Zheng, L. Zhang, and X. Xe. T-fnder: A recommender ytem for fndng paenger and vacant tax. Knowledge and Data Engneerng, IEEE Tranacton on, 25(1): , 213. [18] D. Zhang, T. He, Y. Lu, and J. A. Stankovc. Callcab: A unfed recommendaton ytem for carpoolng and regular taxcab ervce. In Bg Data, Internatonal Conference on, page IEEE, 213. [19] M. Zhang, J. Lu, Y. Lu, Z. Hu, and L. Y. Recommendng pck-up pont for tax-drver baed on pato-temporal cluterng. In CGC), page IEEE, 212.