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Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size
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Webinar: Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and fleet size

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2014-08-01 webinar by Omar Jorge Ibarra Rojas

2014-08-01 webinar by Omar Jorge Ibarra Rojas

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  • 1. webinar of the ALC BRT - COE july 2014 Integrating timetabling and vehicle scheduling to analyze the trade-off between transfers and the fleet size Omar Jorge Ibarra Rojas
  • 2. Outline 2 / • Context • Transit network characteristics • Timetabling problem • Vehicle scheduling problem • Integrated approach • Conclusions and future research 40
  • 3. Transit network planning 3 / context Frequency setting Timetabling Vehicle scheduling Crew assignment tactical decisions operational decisions 40
  • 4. Transit network planning 3 / context Frequency setting Timetabling Vehicle scheduling Crew assignment tactical decisions operational decisions level of service 40
  • 5. Transit network planning 3 / context Frequency setting Timetabling Vehicle scheduling Crew assignment tactical decisions operational decisions costs $$$ level of service 40
  • 6. How to solve the planning problem? 4 / context Frequency setting Timetabling Vehicle scheduling Crew assignment solution feedback solution feedback solution feedback 40
  • 7. Drawbacks of sequential approaches 5 / context • Suboptimal solutions, even for subproblems. • Restrictive for the last subproblems solved due to solution of previous subproblems. • Defining feedback. 40
  • 8. Drawbacks of sequential approaches 5 / context • Suboptimal solutions, even for subproblems. • Restrictive for the last subproblems solved due to solution of previous subproblems. • Defining feedback. Alternative: Integrate subproblems to jointly determine their decisions 40
  • 9. Motivation 6 / Our goal: help to decision makers of transport system management by integrating subproblems of the planning problem through operations research techniques context Frequency setting Integrated Timetabling and Vehicle scheduling Crew assignment 40
  • 10. Integrated approach 7 / context • Advantage: possible to find optimal solution for each subproblem considering the degrees of freedom of the integrated subproblems. • Handicaps: Exploring a large solution space and to defining a proper objective function. 40
  • 11. Transit network characteristics 8 /40
  • 12. Passengers demand 9 / Transit Network • Each day can be divided into different planning periods such as morning peak- hour, morning non peak-hour, afternoon peak hour, and so on. • Constant passenger demand in each period => regular service is desired. • The number of passengers transferring from one line to another is proportional to the bus load of the feeding line. • Frequency setting previously solved => the number of trips is given for each line and planning period (no capacity issues). • Small delays (up to 10% of the even headway) do not affect the passengers demand. 40
  • 13. Bus lines 10/ • There are planning periods with mid/low frequencies where well-timed transfers are needed. • Passengers may transfer from a line A to a line B and not necessarily vice versa. • Buses can not be held at stops. • Lines start and end at the same point. • Accurate estimation of the travel times from depot to each transfer node, for all lines and periods. Transit Network 40
  • 14. 11/ Timetabling problem 40
  • 15. 12/ Timetabling problem Problem definition Determine the departure times for all trips that maximizes the number of passengers benefit from well-timed passenger transfers. 40
  • 16. 13/ Timetabling problem Input Set of lines Set of planning periods for each Frequency of line i for period s Stops where passengers transfer from i to j Number of passengers that need to transfer from line i to line j at stop b in planning period s considering a regular service. S I fi s Bij [as, bs] s 2 S paxijb s 40
  • 17. 14/ Timetabling problem Input as = 8 : 00 bs = 8 : 40 a) Even headway hi s = bs as fi s 8 : 05 8 : 15 8 : 25 8 : 35 as = 8 : 00 bs = 8 : 40 b) Almost even headway times. Flexibility parameter [ ] i s = 1 min [ ][ ] [ ] Di 2 = [8 : 14, 8 : 16] i s 40
  • 18. 15/ Timetabling problem Input b line i line j timeib p timejb q ⇥ MinWijb pq , MaxWijb pq ⇤ 40
  • 19. 16/ Timetabling problem Decisions • : Departure time for each trip p of line i • :Auxiliary variable to identify if separation time between trip q of line j and trip p of line i at node b are within • : Number of passengers transferring from trip p of line i to line j at node b considering the departure time Xi p Y ijb pq ⇥ MinWijb pq , MaxWijb pq ⇤ PAXijb p PAXijb p := paxijb s 1 + Xi p Xi p 1 hi s hi s ! 40
  • 20. 17/ Timetabling problem Mathematical formulation max FTT(X) = X i2I X j2J(i) X b2Bij fi X p=1 PSijb p Xi p 2 Di p (Xj q + tjb q ) (Xi p + tib p ) 2 ⇥ MinWijb pq , MaxWijb pq ⇤ ! Y ijb pq = 1 PSijb p = PAXijb p fj X q=1 Y ijb pq (1) (2) (3) 40
  • 21. 18/ Vehicle scheduling problem 40
  • 22. 19/ Problem definition Determine the trip-vehicle assignment to minimize the fleet size Vehicle scheduling problem Vehicle Scheduling I: Fixed Schedules It is better to doubt what is true than accept what isn’t No. of vehicles Scheduler Gantt chart Time 7 40
  • 23. Input 20/ Vehicle scheduling problem ri p • A timetable. • F: Set of fleets where each fleet f cover a set of lines L(f) • :Turnaround time for trip p of line i 40
  • 24. Decisions 21/ Vehicle scheduling problem o o’i(1) i(2) i(fi ) j(1) j(fj ). . . . . . 40
  • 25. Decisions 21/ Vehicle scheduling problem o o’i(1) i(2) i(fi ) j(1) j(fj ). . . . . . V ijf pq = ⇢ 1 if a vehicle of fleet f makes trip j(q) after finishing trip i(p), 0 otherwise, 40
  • 26. Mathematical formulation 22/ Vehicle scheduling problem X j2I(f) fj X q=1 V ijf pq = X j2I(f) fj X q=1 V jif qp = 1 8f, p, i (4) min FV S(V ) = X f2F X i2I(f) fi X p=1 V if op 40
  • 27. 23/ Integrated Approach 40
  • 28. Common approaches 24/ Integrated Approach Sequential or min w1FT T (X) + w2FV S(V ) X 2 X V 2 V Guihaire and Hao (2010) Fleurent et al. (2009) Guihaire and Hao (2008) van den Heuvel et al. (2008) Liu and Shen (2007) 40
  • 29. Objectives conflict nature 25/ Integrated Approach Users costs Agency costs 100 60 70 100 Which is the best solution? 40
  • 30. Pareto front 26/ Integrated Approach Analyze the trade-off between criteria by finding efficient solutions Feasible solution space Non-convex Pareto curve FT T FV S F✏ T T (x) ✏ Efficient solutions Dominated solutions 40
  • 31. Common approach drawbacks 27/ Integrated Approach • It misses solution points on the non-convex part of the Pareto surface. • Even distribution of weights does not translate to uniform distribution of the solution points. • The distribution of solution points is highly dependent on the relative scaling of the objective. • Misinterpretation of the theoretical and practical meaning of the weights can make the process of intuitively selecting non-arbitrary weights an inefficient chore. 40
  • 32. Our integrated formulation 28/ Integrated Approach Timetabling constraints Vehicle scheduling constraints (1)-(3) Xj q Xi p + ri p M 1 V ijf pq (5)8 f, i, j, p, q (4) [max FT T (X), min FV S(V )] Text + epsilon constraint 40
  • 33. Solution approach: epsilon-constraint 29/ Integrated Approach Feasible solution space Non-convex Pareto curve FT T FV S F✏ T T (x) ✏ Algorithm 1 : ✏-constraint for TT-VS Input: TT-VS instance Output: ListPareto: Pareto optimal points 1: ListPareto = ; 2: Find V S⇤ = {min FVS(V ) : (1)-(5)} 3: Find TT⇤ = {max FTT(X) : (1)-(5)} 4: Find P⇤ 1 = {max FTT(X) : (1)-(5), FVS(V )  V S⇤ } 5: Find P⇤ 2 = {min FVS(V ) : (1)-(5), FTT(X) TT⇤ } 6: ListPareto = ListPareto [ {(TT⇤ , P⇤ 2 ) , (P⇤ 1 , V S⇤ )} 7: Let ✏ = P⇤ 2 1 8: while ✏ > V S⇤ do 9: Find P⇤ ✏ = {max FTT(X) : (1)-(5), FVS(V )  ✏} 10: Update ListPareto considering (P⇤ ✏ , ✏) 11: ✏ = ✏ 1 12: end while 4. Experimental Study94 findextremepointsfillParetofront 40
  • 34. Test instances 30/ Integrated Approach Instances T1 T2 T3 T4 T5 T6 |I| 10 50 10 50 10 50 |B| 1 5 1 5 1 5 100 i s hi s 2 [7.5,12.5] [7.5,12.5] [11.25,18.75] [11.25,18.75] [15,25] [15,25] Table 1: Instance types and parameter values. 4.2. Analysis of Results328 Our ✏-constraint algorithm described by Algorithm 1 was implemented on a Macbook air329 1.3 GHz Intel Core i5 processor with 4 GB 1600 MHz of RAM. We used the integer linear330 programming solver CPLEX 12.6. Table 2 shows the computational time in seconds (Time)331 Instances based on a transit network in Mexico (Ibarra-Rojas et al., 2014) 40
  • 35. Numerical results 31/ Integrated Approachinstances. Note that our ✏-constraint algorithm is capable of finding the Pareto optimal333 solutions for all instances of our case study. T1 T2 T3 T4 T5 T6 time PF time PF time PF time PF time PF time PF 1 26.25 1 569.66 2 15.66 1 586.85 1 123.96 2 73093.8 5* 2 42.06 1 246.51 1 31.06 1 237.18 1 375.062 2 21313.9 2 3 28.52 1 384.69 2 28.71 1 1030.64 2 152 2 25493.9 3 4 49.65 1 381.59 1 88.93 2 508.43 2 304.99 3* 45338.4 3 5 319.30 2* 265.82 1 266.81 2* 990.71 2 139.33 1 25408.7 3 6 34.04 1 3957.69 3 36.74 1 1175.55 2 7484.51 2 33401.3 3 7 44.84 2 305.49 2 49.02 2 2307.14 3 186.30 1 25420 3 8 42.49 1 1120.39 1 41.63 1 571.80 2 19035.7 2 23678.8 4 9 226.29 1 1851.18 3* 161.96 1 3848.58 5* 4080.64 2 45109 3 10 14.60 1 1093.22 3 16.59 1 843.19 2 192.68 1 39750.3 4 Table 2: Computational results using our ✏-constraint algorithm for instances T1–T6. 40
  • 36. Some Pareto fronts 32/ Integrated Approach 1010 369 370 371 372 5300 5675 6050 6425 6800 Pareto front of T2_9 Numberofbuses Passenger Transfers 40
  • 37. Some Pareto fronts 33/ Integrated ApproachPassenger Transfers 1430 362 363 364 365 366 367 368 7140 7435 7730 8025 8320 Pareto front of T4_9 Numberofbuses Passenger Transfers 40
  • 38. Some Pareto fronts 34/ Integrated Approach 2780 358 360 361 363 10400 10463 10525 10588 10650 Pareto front of T6_1 Numberofbuses Passenger Transfers 40
  • 39. Using one more vehicle yields . . . 35/ Integrated Approach 0 2 5 7 10 12 14 17 19 22 24 [0,50] [51,100] [101,150] [151,200] [201,250] [300, 350] [600,700] [1100,1200] Passengers benefited by using one more vehicle 40
  • 40. 36/ Conclusions 40
  • 41. Conclusions 37/ • It is possible to identify instances where the conflict of objectives is present. • It is possible to measure the “cost” of a vehicle in terms of well-timed passenger transfers. • Computational times are acceptable since the input (lines and frequency) are modified in long periods, e.g., once every six months. Conclusions 40
  • 42. Future research 38/ • Heterogeneous fleets. • Multiple-depots. • Other criteria such as total waiting time for larger flexibility parameters and deadhead costs for vehicles. Conclusions 40
  • 43. 39/ References Ibarra-Rojas, O., Giesen, R., Ríos-Solis,Y.A. An integrated approach for timetabling and vehicle scheduling problems to analyze the trade-off between level of service and operating costs of transit networks. under revision in Transportation Research B. Ibarra-Rojas, O., López-Irarragorri, F., Rios-Solis,Y.A., (2014). Multiperiod synchronization bus timetabling.Transportation Science (in press). Ibarra-Rojas, O., Rios-Solis,Y.A., (2012). Synchronization of bus timetabling.Transportation Research B: Methodological 46, 599-614. Guihaire, V., Hao, J.K., (2010). Transit network timetabling and vehicle assignment for regulating authorities. Computers and Industrial Engineering 59, 16-23. Fleurent, C., Lessard, R., (2009). Integrated Timetabling andVehicle Scheduling in Practice.Technical Report. GIRO Inc. Montreal, Canada. van den Heuvel, A., van den Akker, J., van Kooten, M., (2008). Integrating timetabling and vehicle scheduling in public bus transportation. Technical Report UUCS-2008-003. Department of Information and Computing Sciences, Utrecht University, Utrecht,The Netherlands. Guihaire, V., Hao, J.K., (2008). Transit network re-timetabling and vehicle scheduling, in: Le Thi, H.A., Bouvry, P., Pham Dinh, T. (Eds.), Modelling, Computation and Optimization in Information Systems and Management Sciences. Springer Berlin Heidelberg. volume 14 of Communications in Computer and Information Science, pp. 135-144. Liu, Z., Shen, J., (2007). Regional bus operation bi-level programming model integrating timetabling and vehicle scheduling. Systems Engineering-Theory & Practice 27, 135-141. 40
  • 44. 40/ FIN 40

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