Transit Vehicle Scheduling: Problem Description

Transit Vehicle Scheduling: Problem Description Outline Problem Characteristics Service Planning Hierarchy (revisited) Vehicle Scheduling /24/03.224J/ESD.204J

Problem Characteristics Consolidated Operations (vs. Direct Operations) Passengers (vs freight) being moved Urban vs. intercity (short vs. long trip lengths) Relatively high service frequency (several trips per hour vs one trip per day) High temporal variation in demand within day Feasible speeds vary by time of day if vehicles affected by traffic congestion Operations/service plan is stable over a period of months Different type of competition May be a public agency or a private company Crew costs are significant fraction of total costs Routes have many nodes /24/03.224J/ESD.204J 2

Temporal Variation in Vehicle Requirements and Vehicle Blocks Vehicles in service 900 600 300 0 6:00 2:00 5:00 2:00 Time of day /24/03.224J/ESD.204J 3

Service Planning Hierarchy Input Function Output Demand characteristics Infrastructure Resources Policies (e.g. coverage) Demand characteristics Resources Policies (e.g. headways and pass loads) Route travel times Demand characteristics Resources Policies (e.g. reliability) Route travel times Resources Policies (e.g. reliability) Work rules and pay provisions Resources Policies Network and Route Design Frequency Setting Timetable Development Vehicle Scheduling Crew Scheduling Set of routes Service Frequency by route, day, and time period Departure/Arrival times for individual trips on each route (per stop) Revenue and Non-revenue Activities by Vehicle Crew duties /24/03.224J/ESD.204J 4

Service Planning Hierarchy Network Design Frequency Setting Timetable Development Vehicle Scheduling Crew Scheduling Service and Lost Infrequent Considerations Incremental Analysis Decisions both significant Methods Dominate Frequent Decisions Cost Considerations Dominate Computer Dominates Optimization-Based Methods /24/03.224J/ESD.204J 5

Input: Vehicle Scheduling Problem The timetable: a set of vehicle revenue trips to be operated, each characterized by: - starting point and time - ending point and time Possible layover/recovery arcs between the end of a trip and the start of a (later) trip at the same location Possible deadhead arcs connecting: - depot to trip starting points - trip ending points to depot - trip ending points to trips starting at a different point /24/03.224J/ESD.204J 6

Observations: there are many feasible but unattractive deadhead and layover arcs, generate only plausible non-revenue arcs layover time affects service reliability, set minimum layover (recovery) time Objective: define vehicle blocks (sequences of revenue and non-revenue activities for each vehicle) covering all trips so as to: - - Observation: Vehicle Scheduling Problem minimize fleet size (i.e. minimize #crews) minimize non-revenue time (i.e. minimize extra crew time) these are proxies for cost, but a large portion of cost will depend on crew duties which are unknown at this stage of solution. /24/03.224J/ESD.204J 7

Vehicle Scheduling Problem (continued) Constraints: Minimum vehicle block length Maximum vehicle block length Variations: Each vehicle restricted to a single line vs. interlining permitted Single depot vs multi-depot Vehicle fleet size constrained at depot level Routes (trips) assigned to specific depot Multiple vehicle types /24/03.224J/ESD.204J 8

Example: Single Route AB A B (Central City) Results of earlier planning and scheduling analysis: AM Peak Period Base Period (6-9 AM) (after 9 AM) Headways 20 min 30 min Scheduled trip time 40 min 35 min (A B or B A) Minimum layover time 0 min 0 min Dominant direction of travel in AM is A B /24/03.224J/ESD.204J 9

Timetable and Vehicle Block Development Depart A 6:00 6:20 0:00 0:30 :00 Arrive B 9:20 9:35 0:05 0:25 :05 :35 /24/03.224J/ESD.204J 0

Timetable and Vehicle Block Development Depart A Arrive B Depart B Arrive A 6:00 6:50 7:30 6:20 7:0 7:30 8:0 8:0 9:0 9:5 9:50 9:20 9:35 9:45 0:20 0:05 0:5 0:50 0:00 0:25 0:45 :20 0:30 :05 :5 :50 :00 :35 :45 2:20 /24/03.224J/ESD.204J

Timetable and Vehicle Block Development Veh # Depart A Arrive B Depart B Arrive A x-->6:00 6:50 7:30---> 6:20 7:0 7:30 8:0 8:0 9:0 9:5 9:50 9:20 9:35 9:45 0:20 0:05 0:5 0:50 0:00 0:25 0:45 :20 0:30 :05 :5 :50 :00 :35 :45 2:20 x = from depot /24/03.224J/ESD.204J 2

Timetable and Vehicle Block Development Veh # Depart A Arrive B Depart B Arrive A x-->6:00 6:50 7:30---> 2 x-->6:20 7:0 7:30 8:0 8:0 9:0 2 -> y 9:5 9:50 9:20 9:35 9:45 0:20 0:05 0:5 0:50 0:00 0:25 0:45 :20 0:30 :05 :5 :50 :00 :35 :45 2:20 x = from depot /24/03.224J/ESD.204J 6

Timetable and Vehicle Block Development Veh # Depart A Arrive B Depart B Arrive A x-->6:00 6:50 7:30---> 2 x-->6:20 7:0 3 x--> 7:30 8:0 8:0 9:0 2 -->y 3 9:5 9:50 9:20 9:35 9:45 0:20 0:05 0:5 0:50 3 0:00 0:25 0:45 :20 0:30 :05 :5 :50 :00 :35 :45 2:20 x = from depot /24/03.224J/ESD.204J 7

Timetable and Vehicle Block Development Veh # Depart A Arrive B Depart B Arrive A x-->6:00 6:50 7:30---> 2 x-->6:20 7:0 3 x--> 7:30 8:0 4 x--> 8:0 9:0 2 -->y 3 9:5 9:50 4 9:20-> y 9:35 9:45 0:20 0:05 0:5 0:50 3 0:00 0:25 0:45 :20 0:30 :05 :5 :50 :00 :35 :45 2:20 x = from depot /24/03.224J/ESD.204J 8

Timetable and Vehicle Block Development Veh # Depart A Arrive B Depart B Arrive A x-->6:00 6:50 7:30---> 2 x-->6:20 7:0 3 x--> 7:30 8:0 4 x--> 5 x--> 8:0 9:0 2 -->y 3 9:5 9:50 4 9:20 ->y 5 9:35 9:45 0:20 0:05 0:5 0:50 3 0:00 0:25 0:45 :20 5 0:30 :05 :5 :50 :00 :35 :45 2:20 x = from depot /24/03.224J/ESD.204J 9

Example: Vehicle Blocks Block : Depot - A (6:00) - B (6:50) - A () - B () - A () B (0:5 ) - A (:00) - B (:45)... Block 2: Depot - A (6:20) - B (7:0) - A () - B () - Depot Block 3: Depot - A () - B (7:30) - A () - B (9:5) - A (0:00) B (0:45)... Block 4: Depot - A () - B () - A () - Depot Block 5: Depot - A () - B (8:0) - A () - B (9:45) - A (0:30) B (:5) -... /24/03.224J/ESD.204J 20

Vehicle Scheduling Model Approaches Heuristic approaches:. Define compatible trips at same terminal k such that trips i and j are compatible iff : t s j - t e i > M k t s j - t e i < 2 D k where t s j = starting time for trip j t e i = ending time for trip i M k = minimum recovery/layover time at terminal D k = deadhead time from terminal k to depot k /24/03.224J/ESD.204J 2

Vehicle Scheduling Model Approaches 2. Apply Restricted First-in-First-out rules at each terminal (a) Order arrivals and departures at the terminal chronologically (b) Start with (next) earliest arrival at terminal; if none, go to step (e) (c) Link to earliest compatible trip departure; if none, return vehicle to depot and return to step (b) (d) Check vehicle block length against constraint: if constraining, return vehicle to depot and return to step (b); otherwise return to step (c) with new trip arrival time (e) Serve all remaining unlinked departures from depot /24/03.224J/ESD.204J 22

Time-Space Network Representation Route Route N A B A N B N Depot Depot revenue arc Time of Day /24/03.224J/ESD.204J 23

Time-Space Network Representation Route Route N A B A N B N Depot Depot revenue arc layover arc Time of Day /24/03.224J/ESD.204J 24

Time-Space Network Representation Route Route N A B A N B N Depot Depot revenue arc layover arc deadhead arc Time of Day /24/03.224J/ESD.204J 25

Time Space Network Representation Detail Time of Day (,,0) (0,,c ij ) (,,0) (0,,c. j ) (0,,c j.) (,,0) (0,N k,0) D E P O T Storage at Depot (0,N,C) D E P O T (l ij,u ij,c ij ) ( minimum flow, maximum flow, cost per unit of flow) correspond to revenue trips deadhead trip to or from the depot or between routes, or layovers between revenue trips on same route /24/03.224J/ESD.204J 26

Minimum Cost Network Flow Formulation Minim iz e (i, j ) A c ij x ij s.t. x ij x ji = 0 { j: ( i, j ) A } { j: ( j,i ) A} i N...( ) x ij lij, (i, j ) A...( 2 ) x ij u ij, (i, j ) A...( 3 ) x ij Z +, (i, j ) A...( 4 ) /24/03.224J/ESD.204J 27