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Leena Suhl University of Paderborn, Germany

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Presentation on theme: "Leena Suhl University of Paderborn, Germany"— Presentation transcript:

1 Branching Strategies to Improve Regularity of Crew Schedules in Ex-Urban Public Transit
Leena Suhl University of Paderborn, Germany joint work with Ingmar Steinzen and Natalia Kliewer International Graduate School of Dynamic Intelligent Systems

2 Outline Introduction Ex-urban vehicle and crew scheduling problem
Problem definition Irregular timetables Solution Approach Column Generation with Lagrangian relaxation Distance measure modified Ryan/Foster branching rule Local Branching Computational results

3 lines / service network
Introduction lines / service network timetable of one line service trip: 21: :00 from Westerntor to Liethstaudamm

4 line+frequency planning
Introduction crew scheduling timetabling vehicle scheduling crew rostering line+frequency planning timetable/service trips vehicle blocks/tasks relief points crew duties labour regulations crew rosters

5 Multi-Depot Vehicle Scheduling Problem (MDVSP)
Given: set of service trips of a timetable Task: find an assignment of trips to vehicles such that Each trip is covered exactly once Each vehicle performs a feasible sequence of trips (vehicle block) Each sequence of trips starts and ends at the same depot (vehicle capital and operational) costs are minimized block 1 block 2 block 3 D1 D2

6 Crew Scheduling Problem (CSP)
Given: set of tasks From vehicle blocks and relief points (sequential CSP) From timetable and relief points (integrated CSP) Task: assign tasks to crew duties at minimum cost such that Each task is covered (exactly) once Each duty starts/ends at the same depot Each duty satifies (complex) governmental and in-house regulations block 1 block 2 D1 break

7 Crew Scheduling Problem (CSP)
duty piece of work 1 piece of work 2 break task 1 task 4 trip deadhead relief point piece of work-related duty-related constraints

8 Crew Scheduling Problem (CSP)
Minimize total crew costs Constraints Cover all tasks of vehicle schedule (sequential) Cover all tasks of timetable (independent) I set of all tasks K set of all feasible duties K(i) set of all duties covering task i set partitioning or set covering formulation possible

9 Ex-urban Vehicle and Crew Scheduling Problem (VCSP)
Given: set of service trips of a timetable and set of relief points Task: find a set of vehicle blocks and crew duties such that Vehicle and crew schedule are feasible Vehicle and crew schedule are mutually compatible Sum of vehicle and crew costs is minimized Only few relief points in ex-urban settings Assumption: All relief points in depot (typical for ex-urban settings)

10 Irregular Timetables Timetable consists of
regular (daily) trips irregular trips (e.g. to school or plants): about 1-5% of all trips similar situation: timetable modifications similar and regular crew schedules easier to manage in crew rostering phase less error-prone for drivers regular trips trips day A trips day B

11 instance: Monheim (423 trips)
Irregular Timetables Naive approach: plan all periods sequentially, but Modifications of timetable have a strong impact on regularity of vehicle and crew scheduling solutions instance: Monheim (423 trips) timetable Monday timetable Tuesday 2% of trips different crew schedule 93% of crew duties different vehicle schedule crew schedule 66% of vehicle blocks different 100% of crew duties different

12 Irregular Timetables No literature on irregular timetables in public transport Simple heuristics from practice Solve problem with all trips of periods Solve problem with regular and irregular trips of periods separately  fix (regular) duties  C: set of remaining (unfixed) tasks small problems many deadheads, high costs large problems low regularity trade-off

13 Outline Introduction Ex-urban vehicle and crew scheduling problem
Problem definition Irregular timetables Solution Approach Column Generation with Lagrangian relaxation Distance measure modified Ryan/Foster branching rule Local Branching Computational results

14 Solution approach crew scheduling vehicle scheduling Volume Algorithm
Column generation in combination with Lagrangean relaxation Compute dual multipliers by solving Lagrangean dual problem with current set of columns while duties ≠  and no termination criteria satisfied duties = initial column set Delete duties with high positive reduced costs duties = Generate new negative reduced cost columns Add duties to master Find integer solution crew scheduling Volume Algorithm Partial Pricing with Dynamic Programming Algorithm Construct feasible vehicle schedule (pieces of work correspond to service trips) vehicle scheduling

15 Network Models for a Decomposed Pricing Problem
Piece generation network pieces of work pieces of work connection-based duty generation network (Freling et al. 1997, 2003) aggregated time-space duty generation network (Steinzen et al. 2006) Time Space network size: O(#tasks4) network size: O(#tasks2)

16 Guided IP Branch-and-Bound search
Average number of different optima for ICSP Idea: guide IP solution method to „favorable“ solutions (concerning distance to reference solution) Follow-on branching Adaptive local branching Adaptive local branching with follow-on branching tolerance #trips #instances 0% 0,01% 80 10 1052 1115 100 9 723 945 160 1807 2046 test set from Huisman, abort search after 2500 optima set partitioning, independent crew scheduling, variable costs

17 Distance measure for crew duties
2 6 9 14 21 56 84 24 service trips si ti timetable A crew schedule G crew schedule H timetable B 1 2 3 4 5 duties Gi 1 2 3 4 5 duties Hi trip chain T1={2,6,9} irregular trip Reference solution

18 Follow-on Branching Ryan/Foster branching rule for fractional solution of a set partitioning problem and two rows r and s Create two subproblems Choose r and s with max f(r,s) Follow-on branching: allow only consecutive tasks (rows)

19 Follow-on branching to create regular crew schedules
Follow-on branching strategies DEF: Original FOR1: Sequences from reference schedule FOR2: Piece of work from reference schedule FOR3: Maximum length sequence from reference schedule Initialize set Sk of trip chains Ti with Sk={Ti: 0<f(Ti)<1} Sk=? Initialize Skmax={Ti:max(|Ti|)} and branch on Ti Skmax with max(f(Ti)) Branch on trip chain (r,s) with 0<f(r,s)<1 and max(f(r,s)) No Yes FOR2

20 Local Branching Strategic local search heuristic controls „tactical“ MIP solver Local branching cuts equal Hamming distance with L0={kK: xk’=1} Exact solution approach

21 Local Branching to create regular crew schedules
Use local branching to search subspaces that contain „regular“ solutions first Initial solution modify cost function ck’ = ck+dk with dk distance of duty to reference crew schedule  weight of distance Adapt neighbourhood size if necessary (time limit exceeded) Optional: use follow-on branching in subproblem

22 Outline Introduction Ex-urban vehicle and crew scheduling problem
Problem definition Irregular timetables Solution Approach Column Generation with Lagrangian relaxation Distance measure modified Ryan/Foster branching rule Local Branching Computational results

23 Computational Results
Tests with both real-world and artificial data Artificial data generated like Huisman (2004) with 320/400/640/800 trips (two instances each), relief points only in depots Real-world data with ~430 trips (German town with ~ inh.) Irregular trips: 5% (artificial), 2-3% (real-world) Reference crew schedule is known for all instances All tests on Intel Pentium IV 2.2GHz/2 GB RAM with CPLEX 9.1.3 Limited branch-and-bound time to 2 hours

24 Computational Results (Column Generation)
irr% - percentage of irregular trips cpu_ma – cpu time (sec) for the master problem cpu_pr – cpu time (sec) for the pricing subproblem

25 Computational Results (Regularity of Crew Schedules)
prd% - percentage of duties (completely) preserved from reference crew schedule prp% - percentage of trip sequences preserved from reference avcl% - percentage of average trip sequence length preserved from reference

26 Thank you very much for your attention
International Graduate School of Dynamic Intelligent Systems


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