Research Article A Low-Carbon-Based Bilevel Optimization Model for Public Transit Network

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1 Mathematcal Problems n Engneerng Volume 2013, Artcle ID , 6 pages Research Artcle A Low-Carbon-Based Blevel Optmzaton Model for Publc Transt Network Xu Sun, 1 Hua-pu Lu, 1 and Wen-un Chu 2 1 Insttute of Transportaton Engneerng, Tsnghua Unversty, Beng , Chna 2 School of Traffc and Transportaton, Beng Jaotong Unversty, Beng , Chna Correspondence should be addressed to Xu Sun; qngkong0113@126.com Receved 14 Aprl 2013; Revsed 6 June 2013; Accepted 6 June 2013 Academc Edtor: Valentna E. Balas Copyrght 2013 Xu Sun et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. To satsfy the demand of low-carbon transportaton, ths paper studes the optmzaton of publc transt network based on the concept of low carbon. Takng travel tme, operaton cost, energy consumpton, pollutant emsson, and traffc effcency as the optmzaton obectves, a blevel model s proposed n order to maxmze the benefts of both travelers and operators and mnmze the envronmental cost. Then the model s solved wth the dfferental evoluton (DE) algorthm and appled to a real network of Bao cty. The results show that the model can not only ensure the benefts of travelers and operators, but can also reduce pollutant emsson and energy consumpton caused by the operatons of buses, whch reflects the concept of low carbon. 1. Introducton Wth the rapd developments n urbanzaton and growth of car ownershp, the polluton and energy consumpton caused by traffc have become ncreasngly serous [1]. The low-carbon transportaton system, whch s characterzed by low emsson, low polluton, and low energy consumpton, s an effectve way to solve ths problem [2 4]. Publc transt, duetohavngahgherpassengercapactythancars[5], has been wdely recognzed as an mportant traffc mode n the low-carbon transportaton system. The operaton of transt vehcles can be affected by the transt network structures [6], so how to optmze the transt network s a key problem of the low-carbon system. In order to better meet the requrements of the low-carbon transport, t s necessary to optmze the transtnetworkbasedontheconceptoflowcarbon,whch can make publc transt become a more attractve travel mode by mprovng the transt servce qualty and reducng the operaton cost. In the past decades, several research efforts have examned the publc transt network optmzaton problem and many optmzaton approaches have been proposed. Dubos et al. [7] desgned transt network by dentfyng the roads needed for bus routes and choosng the set of bus routes. Then, frequences of the desgned routes were computed through a model amng to mnmze user watng tme. Hasselstrom [8] proposed a mathematcal programmng approach for transt network desgn by choosng the routes and determnng frequences concurrently. Ceder and Wlson [9] presentedanewapproachandanalgorthmtodesgn bus routes based on both passenger and operator nterests. Baa and Mahmassan [10] argued that a bus network could be generated by optmzng the route and the frequency, smultaneously. Van Nes et al. [11] presented a transt route desgn method, n whch route or frequency optmzaton was based on an economc crteron. Pattnak et al. [12]presented a genetc algorthm GA-based optmzaton method to desgn transt network. The obectve of ther optmzaton model was to mnmze the total cost of user and operator. Agrawal and Mathew [13] presented an optmzatonmodelfortranst network amng to mnmze the total system cost whch s thesumoftheoperatngcostandthegeneralzedtravelcost. Bell et al. [14]developedaheurstcbasedonGAtodesgn transt network to mnmze the average travel tme and management cost. Zhao [15] proposed a model for large-scale transt network amng to mnmze transfers and optmze the route/network drectness. Yang et al. [16] proposed a mathematcal model for transt network desgn amng to maxmze drect traveler densty that meant the number of drect travelers carred by per unt length of a route.

2 2 Mathematcal Problems n Engneerng However, almost all the studes focused on the benefts of travelers and operators, meanng that the tradtonal way of optmzng transt network s to take the maxmum benefts of travelers and operators as the optmzaton goal. Consderng the development trend of low-carbon transport, ths paper attempts to combne the concept of lowcarbon transport wth the tradtonal way of transt network optmzaton problem. To solve ths problem, a new methodology s proposed n whch the envronment effect ssue s explctly consdered n the process of network optmzaton for the frst tme. The proposed approach s presented usng a blevel optmzaton formulaton. The outlne of the paper s as follows: the next secton descrbes the detaled optmzaton obectves and gves the representaton of the branch obectve functons. In the model descrpton secton, a blevel optmzaton model s presented amng to mnmze the overall generalzed cost of provdng transt servces. After that a soluton algorthm s adopted for the blevel programmng approach based on the dfferental evoluton (DE) algorthm. Numercal example secton outlnes the applcaton of the proposed method to an example network. In the last secton, the results are dscussed and the maor fndngs of ths research are summarzed. 2. Transt Network Optmzaton Problem Statement 2.1. The Basc Outlne of Optmzaton Problem. The purpose of the low-carbon-based blevel model s to determne a route network layout that mnmzes the overall generalzed cost of provdng transt servces, ncludng the traveler cost, the operator cost, and the envronmental cost. More specfcally, the model wll mplement the obectves as follows: (1) mnmzng traveler cost: that s to reduce the total travel tme and ncrease the rate of drect passengers by mprovng the densty and servce coverage of transt network, (2) mnmzng operator cost: that s to reduce the runnng cost by mprovng the operaton effcency and ncrease the profts of publc transt enterprse by mprovng the transport capacty and the load factor, (3) mnmzng envronmental cost: that s to reduce the pollutant emsson and energy consumpton by mprovng the operaton effcency and capacty of network Descrpton of the Branch Obectve Functon. In ths secton, accordng to the three obectves mentoned above, some ndcators ncludng travel tme, drect passengers, load factor, energy consumpton, pollutant emsson, and traffc effcency are defned as the key branch obectve functons of the optmzaton model to evaluate the mpacts of the optmzed transt network on the travelers, operators, and envronment, respectvely Representaton of Traveler Cost Functon. The mpact on travelers refers to whether the optmzed network can provde more convenent servce for travelers by reducng the total travel tme and transfer tme. Therefore, the travel tme and rate of drect passengers are used as the obectve functons to represent the traveler cost. Travel tme: f 1 =, N q t, N q, (1) t =λ 1 t 1 +λ 2 t 2 +λ 3 t 3 +λ 4 t 4. (2) Rate of drect passengers: f 2 =, N V, N q. (3) Representaton of Operator Cost Functon. The mpact on operators refers to whether the optmzed network can ncrease the profts of enterprse and reduce the runnng cost of vehcles to mnmze the operator cost. Therefore, the load factorandrevenuerateareusedtomeasuretheoperatorcost. Load factor: f 3 = k R, N q,kl k k R, N Q,k l k. (4) Revenue rate: f 4 = C 1, N p q =M+. (5) C 2 k R c r q k l k +C s +C V Representaton of Envronmental Cost Functon. The mpact on envronment refers to whether the optmzed network can reduce the emsson and energy consumpton by mprovng the effcency of network. Therefore, the pollutant emsson, energy consumpton as well as network effcency are used to measure the generalzed envronmental cost. Pollutant emsson rate: f 5 = h H k R b B l kq b k σb h (Vb k ) (6) h H k R b B l k q b k σb hs (Vb k ). Energy consumpton: Network effcency: f 6 = γl k q b k τ b (V b k ). (7) k R b B f 7 = k R, N q,kδ,k k R l k. (8) 3. Blevel Model Formulaton 3.1. Upper-Level Formulaton. The upper level model s artculated n accordance wth the concept of low-carbon transport, the am of whch s the low emssons and low energy consumpton caused by publc transt. Therefore, the upper-level formulaton s proposed n order to mnmze

3 Mathematcal Problems n Engneerng 3 the envronmental cost. The obectve functon of upper-level model takes nto account the emsson, the energy consumpton, and the operaton effcency, whch can represent the envronmental cost. Accordng to formulatons (6), (7), and (8), the obectve functon can be expressed as mn S (x) =w 5 α 5 f 5 +w 6 α 6 f 6 w 7 α 7 f 7, (9) where w 5, w 6,andw 7 are the weght coeffcents and α 5, α 6, and α 7 are transformaton coeffcents to convert the unts of each term n the obectve functon (determned by AHP method). In the upper-level model, the purpose of the obectve functon s to acheve the envronment optma by mnmzng the emsson and energy consumpton and maxmzng the operaton effcency, whch can meet the demand of lowcarbon transport Lower-Level Formulaton. The lower-level model s formulated to optmze the transt network by mnmzng both the traveler cost and operator cost. Therefore, the obectve functon of lower-level model s composed of the travel tme and the drect passengers, whch represent the traveler cost, as well as other evaluaton ndctors to represent the operator cost, ncludng the load factor and revenue rate. Accordng to formulatons (1), (3), (4), and (5), the obectve functon of lower-level model s expressed as mn Z (x) =w 1 α 1 f 1 w 2 α 2 f 2 w 3 α 3 f 3 w 4 α 4 f 4, (10) where w 1, w 2, w 3,andw 4 are the weght coeffcents and α 1, α 2, α 3,andα 4 are the transformaton coeffcents. Inthelower-levelmodel,thepurposeoftheobectve functon s to acheve the optma for both travelers and operators by mnmzng the travel tme and maxmzng the rate of drect passengers, load factor, and revenue rate. 4. Soluton Algorthm 4.1. Dfferental Evoluton Algorthm. The dfferental evoluton (DE) algorthm was frst proposed by Storn and Prce [17]. As a stochastc and parallel searched algorthm, the DE algorthm has been demonstrated to be an effectve and robust method for global optmzaton. The DE algorthm s a populaton-based algorthm, whch combnes smple arthmetc operators wth the classcal events of crossover, mutaton, and selecton to evolve from randomly generated ntal populaton to fnal ndvdual soluton [18]. In detal, the mutaton and crossover operators areusedtogeneratethetralvectors,andselectonsthen used to determne whether the new generated vectors can survve the next generaton. Because t has smple structure and local searchng property and requres few control parameters, fast convergence, the DE algorthm s regarded as one of the best evolutonary algorthms and wdely used to solve optmzaton problems. Accordng to some studes [19 21], DE algorthm can obtan a better soluton and has the better performance than other populaton-based evolutonary algorthms when appled to solve dverse combnatoral optmzaton problems wth contnuous varables. So ths paper attempts to use DE algorthm to solve the blevel optmzaton model Applcaton of DE Algorthm to Blevel Model. The blevel model even wth lnear obectve functons and constrants at both levels s an NP-hard problem and dffcult to solve. Moreover, there are many varables n the model proposed n ths paper, and the soluton doman and obectve functon vary wth the change of feature vectors; the tradtonal determnstc methods cannot guarantee the global optmum. So the DE algorthm, due to ts global search capablty ndependent of gradent nformaton, s appled to solve ths blevel optmzaton problem. The detaled DE algorthm can be descrbed as follows. () Parameters Intalzaton. The man parameters of DE algorthm are populaton sze N, length of the chromosome D, the mutaton factor F, thecrossoverratecr,andthe maxmum generatons number G. The mutaton factor F s selected n [0, 2];thecrossoverrateCRsselectedn[0, 1]. () Populaton Intalzaton. The ntal populaton s randomly generated wthn the boundary usng the followng formulaton: x 0 =xmn + rand (x max where = 1,2,...,N, = 1,2,...,D, x mn x mn ), (11) and x max are the mnmum and maxmum lmts of th dmenson, respectvely, and rand denotes a unform random number between [0, 1]. () Mutaton. The mutaton operaton creates a new vector by addng the weghted dfference of two random vectors to a thrd vector. For each vector x G n generaton G, themutant vector V G+1 s created accordng to the followng equaton: V G+1 =x G r 1 +F(x G r 2 x G r 3 ), (12) where F s a mutaton factor used to control the amplfcaton of the dfferental varaton; G s the current generaton number; and r 1, r 2,andr 3 are three dstnct random numbers and none of them concdes wth the current target number (r 1 =r 2 =r 3 =). (v) Crossover. Crossover operaton can ncrease the dversty of the populaton. The tral vector u G+1 s generated by mxng the mutated vectors V G+1 wth the target vectors x G accordng to the followng rules: = { f rand () CR, = rand n (t), { x G otherwse, { u G+1 V G+1 (13) where rand () [0, 1] s a randomly generated number wth unform dstrbuted; represents the th dmenson; and rand n(t) [1,2,...,D] s a randomly selected nteger to ensure that the tral vector gets at least one parameter from mutated vector.

4 4 Mathematcal Problems n Engneerng Start Intalze all the parameters and randomly generate the ntal populaton wth G=0 Calculate the ftness of the ntal populaton Update the populaton Endng condtons are satsfed or not? No =1 Yes Save the optmal value G=G+1 Mutaton operaton Crossover operaton Calculate the temporary populaton and complete the selecton accordng to the ftness Calculate the value of obectve functon and end Yes =NP? No =+1 Fgure 1: The flowchart of the DE-based soluton approach. (v) Selecton. Selecton operaton retans the better offsprng n the next generaton. The generated offsprng u G+1 replaces the parent x G, only f the ftness of the offsprng f(ug+1 ) s better than that of the parent f(x G ): x G+1 ={ ug+1 x G f f(u G+1 ) f(x G ), otherwse. (14) (v)thedetermnatonofweghtcoeffcent.theweght coeffcent of obectve functon s determned by the entropyweght method, whch s descrbed as H u = ( ln n) 1 n p ue ln p ue, e=1 w u =1 H u m u=1 (1 H u), (15) where w u s the weght coeffcent of the uth ndcator f u of obectve functon and m u=1 w u =1; H u s the entropy values; and p ue =r ue / n e=1 r ue, r ue [0, 1]. (v) The Calculaton of Obect Functon. Thevalueofthe obectve functon can be calculated as f (x) = m u=1 w u f u. (16) Fgure 2: The urban transt network of Bao. In ths blevel model, the optmzaton problem of upper level model s defned as mn f(x) and can be solved wth the algorthm mentoned prevously, whch s also applcable for the lower level subproblem. The flowchart of the DE-based soluton approach s llustrated n Fgure Numercal Example In ths secton, the proposed model and method are appled toarealtranstnetworknbaocty,chna.fgure 2 shows the layout of the network. There are 38 bus routes and 418 bus stops, whch extends km, and 865 buses carryng mllon passengers a year. The other detaled data used n ths example, such as the passenger stop OD matrx, the densty of transt network, and the nonlnear coeffcent of bus route, can be obtaned from our former research and found n [22].

5 Mathematcal Problems n Engneerng 5 Table 1: Comparson of the optmzaton model results wth the exstng transt network. Indcators f 1 /mn f 2 /% f 3 /% f 4 /% f 5 /% f 6 /10 4 ton f 7 /% Exstng results Optmal results Rato enhancement 13.4% 7.2% 10.8% 4.6% 9.1% 4.4% 8.5% The soluton process s as follows. Step 1 (determnng the weght coeffcent). Accordng to formula (15), the weght coeffcents of the ndcators n the model can be calculated as w 1 = , w 2 = , w 3 = , w 4 = , w 5 = , w 6 = , w 7 = (17) Step 2 (determnng the transformaton coeffcent). Correspondngly, the transformaton coeffcents of these ndcators can be determned by usng the analytc herarchy process: α 1 = , α 2 = , α 3 = , α 4 = , α 5 = , α 6 = , α 7 = (18) Step 3 (parameters calbraton). The parameters used n the DE algorthm are defned: populaton sze N=40,mutaton factor F = 1.3, crossoverratecr = 0.8, andmaxmum generatons number G = 200. Step 4 (mplementng the DE algorthm). The DE algorthm procedure, whch s proposed for solvng the blevel model of transt network optmzaton, s coded by MATLAB 2009 and mplemented on a computer wth a 2.2 GHz CPU. Table 1 presents the optmal results of evaluaton ndcators, whch arecalculatedfromtheoptmzatonmodel.forcomparson, the exstng results of correspondng ndcators are also ncluded, whch are the real data obtaned from the traffc survey. Accordng to Table 1,thefollowngcanbeclearlyseen. () For the travelers, the average travel tme decreases by 5.2 mn and the rate of drect passengers ncreases by 7.2%, whch ndcates that the optmzed transt network becomes more convenent for travelers by mprovng densty and servce coverage of the network and ensures the maxmum benefts of travelers. () For the operators, there s an mprovement of 10.8% fortheaverageloadfactorofnetworkand4.6%for revenue rate of transt enterprse, ndcatng that the optmzed network enables mprovng the operator benefts by ncreasng transport capacty and operaton effcency. () For the envronment, the amounts of pollutant emsson and energy consumpton decrease by 9.1% and 4.4%, respectvely, whle the operaton effcency of network ncreases by 8.5%. It shows that the optmzed network acheves the goal of low emsson, low energy consumpton, and hgh effcency. 6. Conclusons The transt network optmzaton problem s an extremely complex problem wth mult obectves and constrants. Ths paper combned the concept of low-carbon transportaton nto the transt network optmzaton problem, whch means that the envronment effect should be consdered n the process of network optmzaton. A low-carbon-based blevel optmzaton model was proposed amng to mnmze the overall generalzed cost of provdng transt servces, ncludng the traveler cost, the operator cost, and the envronmental cost. Then the model was solved wth the DE algorthm and appled to a real network of Bao cty. The applcaton results showed that the optmzaton model can not only make the transt network more convenent and effcent by mprovng the drect passengers and servce coverage, but can also ensure the envronmental benefts n terms of lower energy consumpton, polluton, and emsson. Notaton f 1 : Average travel tme of travelers f 2 : Rateofdrectpassengers f 3 : Load factor of the network f 4 : Revenue rate of operators f 5 : Pollutantemssonsrate f 6 : Energy consumpton f 7 : Network effcency,: Index of node n the transt network k: Index of transt route b: Index of vehcle type h: Index of pollutant type, h = 1, 2, 3, 4 represents CO, CO 2,NO x,hc, respectvely N: Set of nodes n the transt network R: Setoftranstroutes B: Set of vehcle types H: Set of pollutant types q : Number of trps orgnatng from node and destned for node q,k : Numberoftrpsfrom to on route k Q,k : Vehcle seatng capacty from to on route k V : Number of drect trps orgnatng from node and destned for node

6 6 Mathematcal Problems n Engneerng t : Total travel tme between nodes and t 1 : Average walkng tme t 2 : Average watng tme t 3 : Average transfer tme t 4 : Average n-vehcle travel tme λ (=1,2,3,4): Adusted coeffcent (can be determned by Delph method) l k : Lengthofroutek l k : Length from to measured along the route k δ k : Percentage of the number of trps from to dstrbuted on route k C 1 : Total revenue of publc transt enterprse C 2 : Thetotalcost M: Subsdes provded by the government p : Tcketprcefrom to c r : Per-klometer operatng cost of a bus C s : Acquston cost of transt vehcles C V : Mantenance cost of transt vehcles q k : Numberofoperatngbusesonroutek q b k : Number of transt vehcles of type b on route k V b k : Average speed of transt vehcles of type b on route k σ b h (Vb k ): Actual concentraton of pollutant h at the speed of V b k for vehcle b σ b hs (Vb k ): Standard concentraton of pollutant h at the speed of V b k for vehcle b τ b (V b k ): Energy consumpton factor at the speed of V b k for vehcle b on the route k γ: Converson coeffcent of energy. Acknowledgments The authors are grateful to the edtor and anonymous revewers for ther valuable suggestons whch mproved the paper. Ths work s partly supported by Scence and Technology Program of Beng, Chna (Grant no. Z ). References [1] L. Zhang, The research on low-carbon transport stuaton and countermeasure n Chna, Energy Conservaton Technology, vol.3,no.1,pp.79 83,2013. [2] M. Meng, C. F. Shao, and X. Zhang, Research of traffc network equlbrum model wth low carbon emssons constrants, Dsaster Advances,vol.5,no.4,pp ,2012. [3] M.Su,R.L,W.Lu,C.Chen,B.Chen,andZ.Yang, Evaluaton of a low-carbon cty: method and applcaton, Entropy,vol.15, no. 4, pp , [4] T. Zhang, Research on urban low-carbon transport development ndex, Technology Economcs, vol. 32, no. 3, pp , [5] S. M. Feng and H. R. Chen, Study of publc transt network optmzaton method, Harbn Insttute of Technology, vol.37,no.5,pp ,2005. [6] Y. Zhao and S. Zhong, Optmzaton for the urban transt routng problem based on the genetc algorthm, Computer Engneerng and Scence,vol.34,no.4,pp ,2013. [7] D. Dubos, G. Bel, and M. Llbre, A set of methods n transportaton network synthess and analyss, the Operatonal Research Socety,vol.30,no.9,pp ,1979. [8] D. Hasselstrom, Publc transportaton plannng-a mathematcal programmng approach [Ph.D. thess], Unversty of Goteborg, Goteborg, Sweden, [9] A. Ceder and N. H. M. Wlson, Bus network desgn, Transportaton Research Part B,vol.20,no.4,pp ,1986. [10] M. H. Baa and H. S. Mahmassan, Hybrd route generaton heurstc algorthm for the desgn of transt networks, Transportaton Research Part C,vol.3,no.1,pp.31 50,1995. [11] R. van Nes, R. Hamerslag, and B. H. Immerse, Desgn of publc transportaton networks, Transportaton Research Record, vol. 1202, pp , [12]S.B.Pattnak,S.Mohan,andV.M.Tom, Urbanbustranst route network desgn usng genetc algorthm, Transportaton Engneerng,vol.124,no.4,pp ,1998. [13]J.AgrawalandT.V.Mathew, Transtroutenetworkdesgn usng parallel genetc algorthm, Computng n Cvl Engneerng,vol.18,no.3,pp ,2004. [14] M. Bell, M. Carama, and P. Carotenuto, Genetc algorthms n bus network optmzaton, Transportaton Research C, vol. 10,no.1,pp.19 34,2002. [15] F. Zhao, Transt network optmzaton-mnmzng transfers and optmzng route drectness, Publc Transportaton,vol.7,no.1,pp.63 82,2004. [16] Z. Yang, B. Yu, and C. Cheng, A parallel ant colony algorthm for bus network optmzaton, Computer-Aded Cvl and Infrastructure Engneerng,vol.22,no.1,pp.44 55,2007. [17] R. Storn and K. Prce, Dfferental evoluton: a smple and effcent heurstc for global optmzaton over contnuous spaces, Global Optmzaton, vol. 11, no. 4, pp , [18] L. Wang, C.-X. Dun, W.-J. B, and Y.-R. Zeng, An effectve and effcent dfferental evoluton algorthm for the ntegrated stochastc ont replenshment and delvery model, Knowledge- Based Systems,vol.36,pp ,2012. [19] Q. Feng and D. Y. Zhou, Tme optmal path plannng based on dfferent evaluaton algorthm, Computer Engneerng and Applcaton,vol.31,no.12,pp.74 76,2005. [20] J. J. Wang, L. L, D. Nu, and Z. Tan, An annual load forecastng model based on support vector regresson wth dfferental evoluton algorthm, Appled Energy,vol.94,pp.65 70,2012. [21] S. A. Taher and S. A. Afsar, Optmal locaton and szng of UPQC n dstrbuton networks usng dfferental evoluton algorthm, Mathematcal Problems n Engneerng, vol.2012, Artcle ID , 20 pages, [22] H. P. Lu et al., The Comprehensve Transport Plannng for Bao Cty, Insttute of Transportaton Engneerng, Tsnghua unversty, Beng, Chna, 2011.

7 Advances n Operatons Research Advances n Decson Scences Appled Mathematcs Algebra Probablty and Statstcs The Scentfc World Journal Internatonal Dfferental Equatons Submt your manuscrpts at Internatonal Advances n Combnatorcs Mathematcal Physcs Complex Analyss Internatonal Mathematcs and Mathematcal Scences Mathematcal Problems n Engneerng Mathematcs Dscrete Mathematcs Dscrete Dynamcs n Nature and Socety Functon Spaces Abstract and Appled Analyss Internatonal Stochastc Analyss Optmzaton