A Practical Model for Minimizing Waiting Time in a Transit Network

Transcription

1 A Practical Model for Minimizing Waiting Time in a Tranit Network Leila Dianat, MASc, Department of Civil Engineering, Sharif Univerity of Technology, Tehran, Iran Youef Shafahi, Ph.D. Aociate Profeor, Department of Civil Engineering, Sharif Univerity of Technology, Tehran, Iran Introduction Nowaday urban planner are attempting to improve attraction of public tranportation. The mot important reaon that impede paenger from chooing public tranportation a their trip mode i waiting time in tranit ytem. According to thi fact need of decreaing ytem waiting time and tranfer coordination i obviou. Tranit waiting time ha two major component: boarding waiting time in the firt tation of the trip and tranfer waiting time in the middle tation. Tranfer waiting time caue more inconvenience for uer a it occur in the middle of the trip. The firt component depend motly on the headway of the line and will be reduced by decreaing the headway while the econd part varie with everal parameter and i more difficult to deal with. The primary purpoe of thi reearch i to create a model to decreae the tranfer waiting time of a ytem with preet headway but to 1

2 make the model more flexible, we let headway vary in a mall range. We put boarding waiting time in the model to minimize both waiting time imultaneouly. In ection 2, we reviewed ome previou reearche on thi ubject. Section 3 i the repreentation of the model. In ection 4 we decribed the olution method and ection 5 combine unreal and real life example to how efficiency of the model. Finally concluion are explained in ection 6. Literature review Recent tudie in tranfer optimization have been reviewed and a ummary i explained. Ceder et al. (2000) preented a MIP model to maximize the number of imultaneou arrival of bue from different line at tranfer tation. In thi model headway are not preet but they hould vary between a maximum and minimum value. The deciion variable i the departure time of i-th vehicle in line k (x ik ) and the difference between x (i+1)k and x ik hould be in the range [Hmax k, Hmin k ]. The travel time of each vehicle i aumed to be determinitic and predictable. Small network may be olved directly with thi model but to olve large cale one heuritic approache are ued. Quak (2003) changed the objective function of Ceder model to minimize paenger waiting time and olved hi model by ome change in Ceder heuritic model. Ting (1997) preented two model for tranfer coordination. He aimed to minimize cot of ytem by optimizing headway and dwell time. In hi firt model travel time are conidered to be determinitic and analytical method are ued to olve the model while in the econd model travel time are probabilitic and heuritic method are applied to olve it. Fleurent et al (2004) introduced three concept of tranfer waiting time: minimum, ideal and maximum tranfer waiting time. They made a compoite quality index for ynchronization and entered it in 2

3 the cot function a well a other cot. Finally they olved the model uing a Lagrangean relaxation and heuritic mechanim. Cevallo and Zhao (2006) attempted to hift the exiting timetable. Their model ued riderhip data at all tranfer tation and conidered randomne of bu arrival at tation. They olved a network with 80 line and 255 tranfer tation baed on genetic algorithm. The reult howed 12.1% reduction in the tranfer waiting time. Chung and Shalaby (2008) preented an optimization model to modify the exiting timetable of the tranit network and exerted extra dwell time in tranfer tation. They aumed that bue arrival time at tranfer tation followed a log-normal ditribution. Thi model wa olved uing genetic algorithm approach. Shafahi and Khani (2010) propoed two IP model to minimize paenger tranfer waiting time in the network. The variable of the firt model i the tart time of the firt vehicle of each line from the firt tation. In the econd model the top time of each vehicle at tranfer tation i alo conidered. Shafahi and Khani olved their model by CPLEX package for mall network and Genetic Algorithm for large network. Mollanejad (2010) propoed a MIP model to minimize the total waiting time of paenger in tranfer node of the tranit network. The variable of thi model i the headway of the line which i aumed to be uneven. He olved hi model uing CPLEX package for mall network and imulated annealing algorithm for large network. Thi reearch attempt to deal with waiting time and preent a mathematical formulation and an efficient olution procedure for tranit network of every ize. Waiting time optimization model The propoed model for minimizing waiting time in a public tranportation ytem are two mixed integer non-linear program (MINLP). Objective function of both model i um of tranfer and 3

4 non-tranfer waiting time in the ytem. Deciion variable in the firt model are headway and departure time of the firt vehicle of each line from the firt tation. In the econd model which i formulated by expanding the firt model, an extra top time i conidered for line in tranfer tation, o there would be more ucceful tranfer. Model aumption Some aumption are made to make the model impler. The main one are a below: The tranit network and fleet ize are given. Headway of each line i uniform during the planning duration and varie in a mall determined range o that fleet ize doe not change. Travel time in each ection i given and conidered to be contant. Travel demand and paenger choen route are given and are independent of ytem characteritic. A primary top time i conidered for vehicle in each tranfer tation, o paenger can get off and on the vehicle. In the econd model thi value i the output of the model. Paenger tranfer time between line in each tranfer tation i a given contant value. Thi time i the minimum poible time that paenger can tranfer between vehicle. In each tranfer tation, tranferring paenger elect the firt vehicle of the target line for tranferring in order to reduce their tranfer waiting time. Paenger average boarding waiting time for each line i aumed to be half of the headway of that line. Variable and input parameter The input parameter of the model are: R: et of all line in the tranit network; i, j and k are line indice, 4

5 S: et of all tranfer tation in the tranit network; i the tranfer tation index, S j : et of all tranfer tation in tranit network in line j, h0 i : Current headway of line i (in minute), t i : Travel time of vehicle from tarting point of line i to tranfer tation, dt i : Stopping time of vehicle in line i at tranfer tation, tp ij : Number of paenger tranferring from line i to line j at tranfer tation during the planning duration, p i : Total number of paenger in line i not tranferring at tranfer tation during the planning duration (taying aboard), tt ij : required tranfer time for paenger tranferring from line i to line j at tranfer tation. Thi value i the time needed for a paenger to walk from the vehicle in line i to the vehicle in line j. p i : Total paenger of line i including tranfer and non-tranfer paenger during the planning duration. Deciion variable in the firt propoed model are: x k = Departure time of the firt vehicle of line i from it firt tation, h k = Adjuted headway of line k; Other variable are: WT ij = minimum waiting time of paenger tranferring from line i to line j in tranfer tation, AWt ij = average waiting time of paenger tranferring from line i to line j in tranfer tation, 5

6 g ij = greatet common divior of adjuted headway h i and h j, y1 ij, y2 ij = integer variable, w ij = binary variable, Z: objective function; um of both tranfer and non-tranfer waiting time in all origin and tranfer tation, Problem formulation Conider paenger of line i in their origin, we aumed an average boarding waiting time equal to h i /2 for them. So the total boarding waiting time of paenger of line i during planning duration would be equal to p i h i /2 and the total boarding waiting time in the network would be i p i h i /2. Now let u conider paenger tranferring from line i to line j, their tranfer waiting time would be equal to (Shafahi and Khani, 2010): WT ij = x j + t j + dt j + y2 ij h j x i + t i + tt ij + y1 ij h i, i, jϵr, ϵs (1) In thi equation (x i + t i + tt ij ) i the arrival time of the paenger of the firt vehicle of line i and (x j + t j + dt j ) i the departure time of the firt vehicle of line j. If the tranfer paenger from line i reach the firt vehicle of line j the difference between thee two entence would be their tranfer waiting time; however, we hould conider two cae here: firt, the paenger of the firt vehicle of line i may mi their tranfer to the firt vehicle of line j; under thi condition, they hould wait for the next vehicle of line j. To calculate their waiting time under thi circumtance, we added y2 ij h j to the equation. Second, thi tranfer waiting time may exceed h i. A we are calculating minimum tranfer waiting time, we hould conider paenger of all of the vehicle from line i that have arrived during thi period, thu, we added y1 ij h i to the model. 6

7 y1 ij and y2 ij are calculated to gain the minimum waiting time. If headway of line i and j were not equal, the minimum tranfer waiting time from line i to line j would not be equal for all the vehicle of line i, o an average waiting time i calculated a below (Shafahi and Khani,2010): AWT ij = WT ij + (h j i, jϵr, ϵs (2) g ij )/2, Tranfer waiting time for all the paenger changing their line from i to j in the tranfer tation would be AWT ij tp ij during planning duration. By umming thi value for all poible ij- in tranfer tation we can calculate total tranfer waiting time in tranfer paenger and by umming thi value for all tranfer tation in the network we will calculate the total tranfer waiting time in the network. Now we can propoe the firt model: Minimize z = tp ij AWT ij + p ih i 2 i j WT ij = x j + t j + dt j + y2 ij g ij x i + t i + tt ij + y1 ij g ij, i (3) i, jϵr, ϵs (4) WT ij 0, i, jϵr, ϵs (5) WT ij < g ij, i, jϵr, ϵs (6) AWT ij = WT ij + (h j g ij /2), i, jϵr, ϵs (7) AWT ij h j, i, jϵr, ϵs (8) x k < h k, kϵr x k 0, kϵr minh k h k max h k, kϵr y1 ij : integer, i, jϵr, ϵs : integer, i, jϵr, ϵs y2 ij (9) (10) (11) (12) (13) 7

8 The objective function i the um of both tranfer and non-tranfer waiting time in the network. Contraint 4 calculate the minimum waiting time. Contraint 5 and 6 how the upper and lower bound of the minimum waiting time, repectively. Contraint 7 i the definition of average waiting time. Contraint 8 i the upper bound of average waiting time. Contraint 9 guarantee the practicality of the model. Contraint 11 how the range in which headway can change to gain the minimum waiting time without cauing change in the frequency. To reduce the olving time an upper bound i conidered for y1 ij and a below (Shafahi and Khani, 2010): y2 ij y1 ij. up = t j t i + h ih j /g g ij ij y2 ij. up = t j t i + h ih j /g g ij ij (14) (15) We can alo include contraint 16 and 17. A a reult, one of the y1 ij or y2 ij would be zero and the other one would have a non-zero value. y1 ij M(W ij ), i, jϵr, ϵs (16) M(1 W ij ), i, jϵr, ϵs (17) y2 ij A an expanion of the firt model we propoed a econd model. In the firt model we conidered a contant top time for vehicle in the tranfer tation, in the econd model we added the extra top time (edt j ) to the deciion variable of the model. Therefore, paenger who mi their tranfer with a mall gap would have more ucceful tranfer; However, thi extra waiting time hould not exceed an upper bound, becaue the aboard paenger travel time would increae and thi decreae their tendency toward uing public tranportation; Moreover, next vehicle would arrive and thi extra top time would be uele. By entering the new variable to the model ome change are made. Firt, the definition of minimum tranfer waiting time will change a below: WT ij = x j + t j + dt j + edt j + y2 ij g ij x i + t i + tt ij + y1 ij g ij, (18) 8

9 i, jϵr, ϵs In which t j i the travel time from the firt tation to the tranfer tation and i calculated a below: t j = t n j + edt j, nϵsb j jϵr, ϵs (19) where Sb j i the et of all tranfer tation in the tranit network placed in line i before tation. Upper bound that are conidered for extra waiting time in each tranfer tation and alo total extra waiting time in the network are hown in contraint 20, 21 repectively. edt j max_edt j, jϵr, ϵs (20) edt j ϵs j max_tedt j, jϵr, ϵs (21) Finally we hould add the extra travel time of the non-tranfer aboard paenger when the vehicle top for a longer time in tranfer tation. Thi time i equal to p j edt j, therefore, the total extra travel time in all tranfer tation i j p j edt j. The propoed model i repreented here: Minimize z = tp ij AWT ij i j + p ih i 2 i + p j edt j j (22) Contraint Contraint 5-13 Contraint Genetic Algorithm Approach Complexity of the model i motly caued by calculating integer variable y1 ij, y2 ij and alo g ij. Conidering run time and lack of memory, model cannot be olved by common olver package. On the ground of thi fact, we needed a heuritic approach to olve the model. A genetic algorithm i created to olve even very large network uch a urban metropolitan. Thi approach i briefly decribed here. 9

10 The deciion variable of the firt model are departure time of the firt vehicle of each line from the firt tation (x k ) and headway of each line (h k ), o chromoome include thee two variable a their gene in the GA. Fitne function i the total waiting time in the network which i the ame a the firt model objective function. x k are defined a double variable which can change in the range [0, h k ] and h k are defined a integer variable that vary in the range [hmin k, hmax k ]. The firt population i created randomly. To create the population of next generation we ued linear croover and mutation operator. Through the linear croover value of the ame gene of two chromoome are replaced, o the value of gene are till in their range. A croover operation may reult in local optimum, we hould exert mutation to earch the feaible region completely. Through mutation random value are added to the gene which are choen randomly. Thi may caue gene value to exceed their bound. To prevent thi we divided gene value to their range and replaced them with the remainder of thi diviion to aure that the value are in the feaible region. The other proce in the genetic algorithm i the election of the chromoome for the next generation. The common election procee are Roulette wheel and Elititim. We ued both of the procedure in thi reearch. The final tep i to determine termination criteria of the algorithm. The three criteria that we ued in thi reearch are: The bet olution doe not change after a given number of iteration. The difference between the bet and wort olution in a population i le than a given value, i.e., 1%. A maximum number of iteration i reached. To run the algorithm we need to determine value of algorithm parameter. One of thee parameter i the ratio of croovermutation, we gained that by running the algorithm with different ratio for contant number of iteration and invetigate the convergence in each of the run and finally we choe the bet rate. The other parameter wa number of chromoome in each generation. We found the bet population ize following the ame methodology a wa decribed for the croover-mutation ratio. Optimal number of iteration i alo determined conidering run time and convergence. 10

11 To olve the econd model we alo ued the ame genetic algorithm with ome change. We hould add extra top time of vehicle in tranfer tation to the variable. A all the line do not pa all the tranfer tation, we only conidered thoe i and for which p i i defined. We replaced thi variable in the algorithm with y i. y i i defined a integer variable which can change in a pecified range. Therefore, the vector formed the gene of the chromoome include (x i, h i, y i ). The fitne function i the ame a objective function of the econd model. The other procedure are the ame a the firt model. A Real Cae Study: City Of Mahhad Mahhad real life network i ued to evaluate the efficiency of model and genetic algorithm approach. Mahhad network i a large network with 139 two way (278 one way) bu line and 3618 bu top of which 841 one are tranfer tation. Figure 1 how Mahhad network. To etimate the value of the model parameter we performed tranit aignment uing Optimal Strategy in TranCAD oftware. Then, we applied genetic algorithm to olve the model of thi network. Headway of thee line vary between 2 and 165. We conidered a ±10% change in the headway value in order not to change the fleet ize. Alo the upper bound of the edt i wa defined a below: edt i. up = min 4, h i 4 (23) We etimated GA parameter a decribed before. We ran the GA for croover mutation ratio of 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8 with 2000 iteration, the optimal croover mutation ratio wa 0.5 conidering the GA convergence, then we ran GA with thi optimal croover mutation ratio with population ize of 10, 20, 30 and the optimal population ize wa found 20 regarding run time. We alo conidered 5000 iteration for termination. 11

12 Figure 1- Mahhad City bu network Finally 3 cenario were olved uing GA. The reult are hown in table 1. Table 1 Mahhad tranit network cheduling reult cenario Objective function value Objective function improvement No planning % Firt model % Second model % By the current condition of the network, waiting time in the whole network during the planning duration i minute. By exerting the new condition that i the reult of the firt model the waiting time would increae to minute, thi mean 12.5% improvement in the ytem which i equal to minute aving time for paenger. Finally by olving the econd model the objective 12

13 function value would reduce to minute. The improvement i 13.16% or minute. Comparing two propoed model, the econd model i 0.8% more efficient than the firt one; however, we hould notice that thee reult are not the optimal olution. The trend of the firt and econd model objective function improvement veru iteration i hown in figure 2 and 3 repectively. Finally we can conclude that both model are efficient in reducing waiting time of the paenger in the tranit network and GA i an appropriate approach to olve the large cale network a it doe not have limitation for the number of variable and contraint. fitne function number of iteration Figure 2- Trend of firt model objective function improvement by number of iteration for Mahhad city tranit network uing the genetic algorithm 13

14 fitne function number of iteration Figure 3- Trend of econd model objective function improvement by number of iteration for Mahhad city tranit network uing the genetic algorithm Concluion In thi paper we repreented two model to minimize tranfer and non-tranfer waiting time in a tranit network imultaneouly. According to the high number of variable uing a heuritic approach to olve the model wa neceary and we verified the reult with the reult of Shafahi and Khani (2010) olution. Finally, we created a genetic algorithm and applied it for a large real life tranit network. We made the following concluion: By a mall change in headway, there would be a reaonable reduction in the network waiting time; it i noticeable that we aumed uniform headway for each tranit line. Improvement of the econd model in comparing with the firt model how that extra top time in the tranfer tation caue more ucceful tranfer and reduce waiting time in the ytem. More ignificant origin and tranfer tation can be weighted in the model baed on their numbr of paenger. Both of the propoed model reduce the waiting time ignificantly. 14

15 Finally for eae of the modeling in thi reearch it i aumed that demand of each line i fixed (independent of it characteritic), a an extenion of thi reearch, the influence of the changed parameter on demand can be tudied. Alo tochatic travel time can be conidered intead of contant one. Other heuritic approache or a combination of them can be ued to olve the model. Moreover, in thi model half of the headway of each line i conidered a it average boarding waiting time, it can be replaced with better value. Bibliography Ceder A., Golany B. and Tal O., 2000, Creating bu timetable with maximal ynchronization, Tranportation Reearch - Part A 35, Cevallo, F.,Zhao, F., 2006, Minimizing Tranfer Time in a Public Tranit Network with a Genetic Algorithm. Tranportation Reearch Record, No. 1971, Wahington D.C., 2006,pp Fleurent, C., Leerd, R., and Segiun, L., Tranit Timetable Synchronization : Evaluation and Optimization, 9th International Conference on Computer Aided Scheduling in Public Tranport, San Diego California, Augut, Mollanejad, M., 2010, Creating Bu Timetable with Maximum Synchronization, MSC. Thei, Sharif Univerity of Technology, Iran. Quak, C.B., Bu line planning. Shafahi,Y., Khani, A., 2010, A practical model for tranfer optimization in a tranit network: Model formulation and olution, Tranportation Reearch, Part A, NO.44, pp Ting Ching-Jung.,1997, Tranfer coordination in tranit network, Ph.D diertation, Univerity of Maryland at College Park, MD. 15