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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

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ATMOS 2007 – Nov. 16, 2007 Page: 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

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ATMOS 2007 – Nov. 16, 2007 Page: 3 Introduction lines / service network timetable of one line service trip: 21: :00 from Westerntor to Liethstaudamm

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ATMOS 2007 – Nov. 16, 2007 Page: 4 Introduction crew scheduling timetabling vehicle scheduling crew rostering line+frequency planning timetable/service trips vehicle blocks/tasks crew duties crew rosters labour regulations relief points

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ATMOS 2007 – Nov. 16, 2007 Page: 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

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ATMOS 2007 – Nov. 16, 2007 Page: 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

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ATMOS 2007 – Nov. 16, 2007 Page: 7 Crew Scheduling Problem (CSP) break piece of work 1 piece of work 2 duty trip deadhead relief point task 1 task 4 piece of work-related duty-related constraints

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ATMOS 2007 – Nov. 16, 2007 Page: 8 Crew Scheduling Problem (CSP) Minimize total crew costs Constraints –Cover all tasks of vehicle schedule (sequential) –Cover all tasks of timetable (independent) Iset 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

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ATMOS 2007 – Nov. 16, 2007 Page: 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)

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ATMOS 2007 – Nov. 16, 2007 Page: 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

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ATMOS 2007 – Nov. 16, 2007 Page: 11 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 Mondaytimetable Tuesday 2% of trips different vehicle schedule crew schedule 66% of vehicle blocks different 100% of crew duties different crew schedule 93% of crew duties different

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ATMOS 2007 – Nov. 16, 2007 Page: 12 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 Irregular Timetables fix (regular) duties C: set of remaining (unfixed) tasks small problems many deadheads, high costs large problems low regularity trade-off

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ATMOS 2007 – Nov. 16, 2007 Page: 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

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ATMOS 2007 – Nov. 16, 2007 Page: 14 Solution approach Construct feasible vehicle schedule (pieces of work correspond to service trips) Volume Algorithm Partial Pricing with Dynamic Programming 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 vehicle scheduling

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ATMOS 2007 – Nov. 16, 2007 Page: 15 Network Models for a Decomposed Pricing Problem Piece generation network pieces of work connection-based duty generation network (Freling et al. 1997, 2003) network size: O(#tasks 4 ) pieces of work aggregated time-space duty generation network (Steinzen et al. 2006) Time Space network size: O(#tasks 2 )

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ATMOS 2007 – Nov. 16, 2007 Page: 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#instances0%0,01% test set from Huisman, abort search after 2500 optima set partitioning, independent crew scheduling, variable costs

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ATMOS 2007 – Nov. 16, 2007 Page: 17 Distance measure for crew duties trip chain T 1 ={2,6,9} crew schedule G 12345…12345… duties G i crew schedule H 12345…12345… duties H i timetable Atimetable B service trips sisi titi irregular trip Reference solution

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ATMOS 2007 – Nov. 16, 2007 Page: 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)

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ATMOS 2007 – Nov. 16, 2007 Page: 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 S k of trip chains T i with S k ={T i : 0

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ATMOS 2007 – Nov. 16, 2007 Page: 20 Local Branching Strategic local search heuristic controls „tactical“ MIP solver Local branching cuts equal Hamming distance with L 0 ={k K: x k ’=1} Exact solution approach

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ATMOS 2007 – Nov. 16, 2007 Page: 21 Local Branching to create regular crew schedules Use local branching to search subspaces that contain „regular“ solutions first Initial solution –modify cost function c k ’ = c k + d k with d k 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

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ATMOS 2007 – Nov. 16, 2007 Page: 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

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ATMOS 2007 – Nov. 16, 2007 Page: 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 Limited branch-and-bound time to 2 hours

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ATMOS 2007 – Nov. 16, 2007 Page: 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

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ATMOS 2007 – Nov. 16, 2007 Page: 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

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

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