General Share-A-Ride Problem

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1 General Share-A-Rde Problem Sesya Sr Purwant Department of Industral Engneerng, Insttut Tenolog Bandung, Bandung, Indonesa Department of Industral Management, Natonal Tawan Unversty of Scence and Technology, Tape, Tawan Emal : sesya.purwant@gmal.com A. A. N. Perwra Red Department of Industral Management Natonal Tawan Unversty of Scence and Technology, Tape, Tawan Emal : wra.red@mal.ntust.edu.tw Vncent F. Yu Department of Industral Management Natonal Tawan Unversty of Scence and Technology, Tape, Tawan Emal: vncent@mal.ntust.edu.tw Abstract. Ineffcences n transportaton have resulted n economc and envronmental problems. Hgh levels of traffc congeston ncrease waste of resources and polluton. To address such ssue, the Share-A-Rde Problem (SARP) has been developed. In SARP, a tax could serve passenger and parcel requests at the same tme. However, SARP has certan lmtatons, such as assumng that a tax should not serve more than one passenger at the same tme and should observe the maxmum number of parcel requests that could be nserted between the pcup and drop-off / delvery ponts of a passenger. Ths study ntroduces a generalzed model for SARP, called General Share-A-Rde Problem (G-SARP), whch allows the tax to serve more than one passenger request at the same tme and has no restrcton n terms of the number of parcel requests that could be served between the pcup and drop-off/delvery ponts of a passenger. The objectve of G-SARP s to maxmze total proft obtaned from servng people and parcel requests. We present a mathematcal model for G-SARP and perform a numercal study usng CPLEX to solve small G-SARP nstances. Keywords: General share-a-rde problem, Rde sharng, Dal-a-rde problem 1. INTRODUCTION Traffc congeston and ar polluton are common problems n the urban area. The man cause of these problems s the ncreasng number of vehcles on the road. Rde-sharng, whch ams at mnmzng the number of vacant seats n vehcles, thereby reducng the number of requred vehcles, has emerged as an essental practce that could reduce traffc and fuel costs as well as support green ntatves that am at mprovng urban lfe and moblty condtons. The traffc and polluton problem affects transportaton of people as well as goods or pacages. In a practcal stuaton, onlne shoppng has become popular, and same-day delvery has become a common feature of onlne shoppng (B. L, Krunshnsy, D., Woensel, T.V., Rejers, H.A., 2016). Road congeston s a serous obstacle for achevng ths feature. New cty logstc approaches are needed to ensure effcent urban moblty for transportng people and goods. Recently, study ntegrated people and freght transportaton usng a model called Share-a-Rde Problem (SARP) (L, 2014). The SARP model was mplemented on conventonal tax operatons, and assumed that the same tax would be unable to respond to a two-passenger request at the same tme. Therefore, the SARP model taes the approach of combnng one passenger trp wth a goods delvery request, but ensures that the passenger has the hghest prorty. Sharng a tax has recently become popular n several countres. In Europe, a moble applcaton can assst a passenger n fndng a tax for sharng nearest to hs/her locaton. Tang advantage of ths opportunty, ths study

2 ntroduces a new approach for mprovng moblty condton by combnng sharng tax and goods transportaton servce. The proposed servce s called General Share-a-Rde Problem (G-SARP). From the modelng pont of vew, ths problem s a generalzed verson of SARP and a varant of Dal-a-Rde Problem (DARP). The am of ths model s to plan a set of maxmum proft vehcle routes that could accommodate all passenger and pacage requests. The ey contrbuton of ths paper s that the generalzed model of SARP could maxmze profts for tax companes because ths model relaxes constrants that relate to customer prorty and sharng rdes among passengers. Ths paper organzed as follows. Secton 2 presents an overvew of the related lterature on share-a-rde and dal-arde problem. Secton 3 descrbes the model formulaton of General Share-a-Rde Problem. Expermental results and comparson wth other approaches are presented n Secton 4. Fnally, conclusons and future research are dentfed n Secton LITERATURE REVIEW For several years, the rde-sharng problem has been wdely studed. Most ssues are derved from the Dal-a-Rde Problem, whch s the extended verson of the Pcup and Delvery Problem (PDP) (Braeers, 2014; Coredeau, 2003; Marovc, 2015; Parragh, 2010). The purpose of ths model s to desgn routes capable of accommodatng all requests whle mnmzng total cost, under a set of constrants, such as vehcle capacty, route duraton, and maxmum rdng tme for each customer. A common DARP applcaton s for door-todoor transportaton servce for elderly and the dsabled (Laporte, 2007). The dfferences between DARP and other VRP varances s a trade-off between transportaton cost and user convenence that are consdered when desgnng a soluton gven that the models allow more than one passenger to use the same vehcle at the same tme. Coredeau (2003) proposed a DARP model wth capactated vehcle and tme wndows. The maxmum total duraton for each vehcle was added as a constrant. Small nstances were solved usng the Tabu Search heurstc. Usng other heurstc methods for solvng the DARP problem, Parragh (2010) suggested a compettve Varable Neghborhood Search-based heurstc usng three classes of the neghborhood. Ths study provded 16 new best solutons f compared wth the benchmar study accordng to Coredeau (2003). The other verson of DARP s Shared-Tax Problem. Ths model has several advantages that could mnmze vacant seats n a tax and reduce tax operatonal costs. Consequently, tax fares wll be lower for passengers and traffc congeston reduced. The shared-tax problem sees to determne the optmal assgnment of passengers to taxs as well as the optmal route for each tax. Each tax passenger request has to defne a pcup and a drop off locaton, and the earlest acceptable pcup tme and latest acceptable drop off tme (Hosn, 2014). The applcaton of rde sharng s used by travelers. A moble applcaton was developed to facltate travelers wth smlar tnerares and tme schedules on short notce. Ths problem s dentfed as dynamc-rde-sharng. The system may provde sgnfcant socetal and envronmental benefts by reducng the number of cars used for personal travel and mprovng the utlzaton of avalable seat capacty (Agatz, 2012). The mpact of drver and rder flexblty on the dynamc rde-sharng problem could ncrease the performance of a rde-sharng system (Stglc, 2016). Recently, an extended model of DARP, called SARP, was proposed by L et al. SARP combnes passenger and pacage request at the same tax. Gven that the system uses a tax as a transportaton mode, ths model cannot allow more than one passenger to rde the same tax at the same tme. Moreover, the passenger has hgher prorty over pacage requests. Passenger and pacage requrements have several dfferences. For nstance, the pcup and delvery tmes for passengers are more crtcal than for pacages. Hence, not tang parcel requests and obtanng dfferent costs and benefts from pacage and passenger servces are obvous. The objectve of ths study s to maxmze total proft obtaned from servng passenger and pacage requests. The small nstances used n ths study were solved by GUROBI (L, 2014). The other research solves ths problem by usng Adaptve Large Neghborhood Search heurstc. Ths study compared the solutons wth DARP proposed by Laporte et al., and showed that ther method could delver better solutons n terms of tme and qualty (B. L, Krunshnsy, D., Woensel, T.V., Rejers, H.A., 2016). Recently, L et al. developed a new varant of share-a-rde problem that consdered stochastc travel tme and stochastc delvery locatons. The objectve of ths research s to maxmze expected proft of servng passengers and parcels usng heterogeneous vehcles. The result of ths research suggested that stochastc nformaton s valuable n real lfe and could dramatcally mprove the performance of tax sharng system, compared to determnstc solutons (B. L, Krunshnsy, D., Woensel, T.V., Rejers, H.A., 2016) 3. MODEL FORMULATION The General Share-a-Rde Problem (G-SARP) s a generalzed model from SARP proposed by L. Ths model allows more than one passenger rdng a tax at the same tme, le Tax Sharng Problem or Dal-a-Rde Problem. Moreover, no restrcton s mposed on the maxmum number of requests that could be nserted nto the passenger servce as long as these trps do not exceed capacty constrants. The am of ths problem s to maxmze total profts that the tax company wll obtan after servng all requests whle consderng penalty costs ncurred from exceedng the drect travel tme for

3 passenger request. In ths problem, a set of N passenger requests and M pacage requests are gven, whch correspond to demand type C = {1, 2,.., C }. For ths partcular study, c = 1 denotes passenger and c = 2 denotes pacage. The formulaton of G- SARP s defned on a complete undrected graph G = (V, A), where V s a set of vertex parttoned nto {V p, V f, {0,2σ+1}}. Subsets V p and V f correspond to passenger and parcel requests, respectvely, whle 0 and 2σ+1 represent orgn and destnaton depots for the vehcles, respectvely. Subset V and V f,0 represent the orgn nodes for each passenger and parcel request, respectvely, and the orgn nodes always precede destnaton nodes. Each vertex s assocated wth servce tme duraton s 0 wth s0 = s+σ = 0. Each arc (,j) ϵ A s assocated wth travel dstance dj and travel tme tj. The detaled formulaton of G- SARP s shown as follows: Parameters: m n c q t j d s j e, l c Q T Decson Varables: x j u Number of pacage requests Number of passenger requests Number of vehcles Total number of all request ( =m+n) A load quantty of stop for demand type c Travel tme between stop and j Travel dstance between stop and j Servce duraton of a request at stop Tme wndows at stop Maxmum capacty of vehcle for demand type c, for K and c C Maxmal duraton of route vehcle Intal fare charged for delverng one passenger Intal fare charged for delverng one parcel Fare charged for delverng one passenger per m Fare charged for delverng one parcel per m Average cost per lometer for delverng request Dscount factor for exceedng the drect delvery tme of passengers Bnary decson varables equal to 1 f stop and j served respectvely by vehcle Arrval tme for vehcle at stop j c w The load of vehcle for demand type c upon leavng stop r p Tme spent by request n vehcle Rato between actual passengers rdng tme wth ts drect travel tme Objectve Functon: max( ( d ) x V jv K f,0 V jv K 2, 1, ( d ) x d x ( p 1) j 3 j j 4 V jv K V Constrants: f,0 xj 1, V V (2) jv K x0 1, K (3) V x,2 1 1, K (4) V x x K (5) V V 0 2 1, 0, V x x j V V K (6) j f,0 j, j,, V p f xj x j, V V, K (7) jv jv u u s t M (1 x ), j j j p K, V, j V V c c c w w q M (1 x ), p f K, V, j V V, c C j j j r u u, K, V V (10) f,0 u u T, K (11) e u l, K, V (12) f (1) (8) (9)

4 c c c c c q w Q Q q max 0, mn,, K, V, c C u u ( s t ) x,, j f,0 K, V V, j V p V (13) (14) 1, (15) r p V, (16) K ( t, s ) x 0,1,, j V, K (17) j Objectve functon (1) maxmzes the total proft obtaned for passenger and parcel delvery, and comprses passenger fare, parcel fare, cost of dstance traveled, and penalty cost for passenger extra rdng tme compared to drect delvery. Constrant (2) ensures that all requests are served exactly once by the same vehcle. Constrants (3), (4) and (5) guarantee that each vehcle starts and ends ts route at the depot. Constrants (6) and (14) ensure that orgn and destnaton nodes for the same request vsted by the same vehcle and orgn node wll be vsted before destnaton node. Every stop, except for the depot, must have one precedng and one succeedng stop, whch s defned n constrant (7). Constrants (8), (9), and (10) defne the starts of servce tmes, vehcle loads, and rdng tmes of passenger request, respectvely. Total servce tme for each vehcle must not exceed maxmum vehcle operatonal tme, whch s defned by constrant (11). The tme wndow constrants for the request are defned n (12). Constrant (13) defnes the load of vehcle after vst, and vertex must not exceed the maxmum vehcle capacty. Constrants (15) and (16) ensure that the rato between a passengers rdng tme and the correspondng drect travel tme s greater than or equal to 1 so that the last term n the objectve functon has a postve value. Constrant (17) shows the decson bnary varable. The G-SARP model gnores customer prorty constrants n the SARP model. Those constrants defne that customer rdng tme should not exceed a specfc amount of tme, and restrct the same customer rde n the same tax at the same tme. Therefore, n ths model, passenger rdng tme s lmted only to the orgn and destnaton pont tme wndows, and allows the tax to serve more than one passenger at the same tme. Table 1: Parameters used for GSARP and SARP model Parameters Values Parameters Values K {1, 2} α 3.5 m + n(m = n) {4, 6} β 2.33 e U(0:00 h, 12:00 h) γ l U(8:00 h, 14:00h) γ q 1 γ Q 5 γ COMPUTATIONAL STUDY The computatonal experment s performed on Intel Xeon 3.70 GHz CPU 40 GB computer, under Wndows 7 operatng system. AMPL was used for solvng small nstances. The tested nstances are explaned n the next secton. Ths secton compares the solutons obtaned from G-SARP and SARP by solvng the same nstances. 4.1 Instance Desgn for the G-SARP model Table 1 shows the G-SARP nstances generated by usng parameters from L et al. The passenger and pacage request randomly generated the followng unform dstrbutons U([0,17] [0,10]) m. Each request has pcup and delvery nodes. Tme wndows for all of the request are drew unformly at random from ntervals between 0:00-12:00 h for the earlest tme wndow and 8:00-14:00 h for the latest tme wndow. The dstance between the nodes was calculated usng Manhattan dstance.

5 4.2 Comparson Result of G-SARP and SARP Model To compare the benefts of the G-SARP wth the SARP, eght small nstances were run usng AMPL. The two models ran under the same condtons. For each nstance, a dfferent number of customer requests, number of vehcles, and dfferent length of tme wndow were used. Table 2 reveals the results for G-SARP and SARP. The G-SARP result exhbted larger values for all solutons than the SARP model. Because the objectve of ths model s maxmzng proft, the G-SARP model gves a better soluton than the SARP model. Although the G-SARP model gnores customer prorty constrants of the SARP model, customers stll do not spend too much tme n the tax. Hgher proft was obtaned by the tax based on the formulaton for calculatng the profts and by ncludng penalty cost for exceedng the drect travel tme. Four of the eght nstances have narrow tme wndow (nstance 14n, nstance 24n, nstance 16n, and nstance 26n). The result of those nstances have smaller soluton values and larger gap percentage than the nstance wth wder tme wndows. Narrow tme wndows maes the hgh qualty solutons for the customers but maes more dffcult to get the optmal solutons. However, for both condtons the G-SARP model gves better soluton than the SARP model. Table 2: Comparson results for all nstances No. Instances G-SARP SARP Gap (% ) Soluton Tme (s) Soluton Tme (s) 1 nstance % 2 nstance14n % 3 nstance % 4 nstance24n % 5 nstance % 6 nstance16n % 7 nstance % 8 nstance26n % 5. CONCLUSIONS AND FUTURE DIRECTION Ths study proposed the G-SARP model to address the problems on passenger and parcel delvery servces. The objectve of ths model s to maxmze total proft for transportaton companes. Results ndcated that G-SARP yelded better solutons than SARP. Hence, the publc transportaton system could be managed more effcently, whle ncreasng profts and reducng traffc congeston and ar polluton. Future research could apply the model and the proposed metaheurstc algorthm for solvng bgger nstances. ACKNOWLEDGMENTS Ths research was partally supported by the Mnstry of Scence and Technology of the Republc of Chna (Tawan) under grant MOST E MY3. Ths support s gratefully acnowledged. REFERENCES Agatz, N., Erera, A., Savelsbergh, M., Wang, X. (2012). Optmzaton for dynamc rde-sharng: A revew. European Journal of Operatonal Research, Braeers, K., Cars, A., Janssens, G.K. (2014). Exact and metaheurstc approach for a general heterogeneous dal-a-rde problem wth multple depot. Transportaton Research Part B: Methodologcal, Coredeau, J. F., Laporte, G. (2003). A tabu heurstc for the statc mult-vehcle dal-a-rde problem. Transportaton Research Part B: Methodologcal, Hosn, H., Sawaya, J., Artal, H. (2014). The shared-tax problem: Formulaton and soluton methods. Transportaton Research Part B: Methodologcal, Laporte, J. C. G. (2007). The dal-a-rde problem: models and algorthms. Annals of Operatons Research, 153(1), L, B., Krunshnsy, D., Woensel, T.V., Rejers, H.A. (2014). The share-a-rde problem: people and parcels sharng taxs. European Journal of Operatonal Research, L, B., Krunshnsy, D., Woensel, T.V., Rejers, H.A. (2016). An adaptve large neghborhood search heurstc for the share-a-rde problem. Computers & Operatons Research, 66, L, B., Krunshnsy, D., Woensel, T.V., Rejers, H.A. (2016). The share-a-rde problem wth stochastc travel tmes and stochastc delvery locatons. transportaton

6 Research Part C: Emergng Technologes, Marovc, N., Nar, R., Schonfeld, P., Hoos, E., Mohebb, M.. (2015). Optmzng dal-a-rde servces n Maryland: Benefts of computerzed routng and schedulng. transportaton Research Part C: Emergng Technologes, Parragh, S. N., Doerner, K.F., Hartl, R. F. (2010). Varable neghborhood search for dal-a-rde problem. Computers & Operatons Research, Stglc, M., Agatz, N., Savelsbergh, M., Gradsar, M. (2016). Mang dynamc rde-sharng wor: The mpact of drver and rder flexblty. Transporttaon Research Part E: Logstcs and Transportaton Revew,