Branching Strategies to Improve Regularity of Crew Schedules

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

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 Page: 2 ATMOS 2007 – Nov. 16, 2007

Introduction lines / service network timetable of one line service trip: 21: 45 -- 22: 00 from Westerntor to Liethstaudamm Page: 3 ATMOS 2007 – Nov. 16, 2007

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

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 D 1 block 1 D 1 block 2 D 2 block 3 Page: 5 ATMOS 2007 – Nov. 16, 2007

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 D 1 block 1 break D 1 block 2 Page: 6 ATMOS 2007 – Nov. 16, 2007

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

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 Page: 8 set partitioning or set covering formulation possible ATMOS 2007 – Nov. 16, 2007

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 exurban settings) Page: 9 ATMOS 2007 – Nov. 16, 2007

Irregular Timetables • Timetable consists of – regular (daily) trips – irregular trips (e. g. to school or plants): about 1 -5% of all trips regular trips day A trips day B • similar situation: timetable modifications • similar and regular crew schedules – easier to manage in crew rostering phase – less error-prone for drivers Page: 10 ATMOS 2007 – Nov. 16, 2007

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

Irregular Timetables • No literature on irregular timetables in public transport • Simple heuristics from practice – Solve problem with all trips of periods fix (regular) duties C: set of remaining (unfixed) tasks – Solve problem with regular and irregular trips of periods separately small problems many deadheads, high costs Page: 12 trade-off large problems low regularity ATMOS 2007 – Nov. 16, 2007

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 Page: 13 ATMOS 2007 – Nov. 16, 2007

Solution approach Column generation in combination with Lagrangean relaxation crew scheduling duties = initial column set while duties ≠ and no termination criteria satisfied Add duties to master Compute dual multipliers by solving Lagrangean dual problem with current set of columns Volume Algorithm Delete duties with high positive reduced costs duties = Generate new negative reduced cost columns Partial Pricing with Dynamic Programming Algorithm Find integer solution Construct feasible vehicle schedule (pieces of work correspond to service trips) vehicle scheduling Page: 14 ATMOS 2007 – Nov. 16, 2007

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

Guided IP Branch-and-Bound search • Average number of different optima for ICSP tolerance #trips #instances 0% 0, 01% 80 10 1052 1115 100 9 723 945 160 9 1807 2046 test set from Huisman, abort search after 2500 optima set partitioning, independent crew scheduling, variable costs • Idea: guide IP solution method to „favorable“ solutions (concerning distance to reference solution) – Follow-on branching – Adaptive local branching with follow-on branching Page: 16 ATMOS 2007 – Nov. 16, 2007

Distance measure for crew duties timetable A crew schedule G duties Gi 1 2 3 4 5 service trips si crew schedule 2 6 9 14 21 56 trip chain T 1={2, 6, 9} … timetable B H service trips ti duties Hi 1 2 2 6 3 84 4 9 5 24 … 56 Reference solution irregular trip Page: 17 ATMOS 2007 – Nov. 16, 2007

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) Page: 18 ATMOS 2007 – Nov. 16, 2007

Follow-on branching to create regular crew schedules • Follow-on branching strategies – – DEF: Original FOR 1: Sequences from reference schedule FOR 2: Piece of work from reference schedule FOR 3: Maximum length sequence from reference schedule Initialize set S k of trip chains Ti with Sk={Ti: 0<f(Ti)<1} Yes ? Branch on trip chain (r, s) with 0<f(r, s)<1 and max(f(r, s)) No Initialize Skmax={Ti: max(|Ti|)} and branch on Ti Skmax with max(f(Ti)) Page: 19 FOR 2 ATMOS 2007 – Nov. 16, 2007

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

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 Page: 21 ATMOS 2007 – Nov. 16, 2007

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 Page: 22 ATMOS 2007 – Nov. 16, 2007

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 ~45. 000 inh. ) – Irregular trips: 5% (artificial), 2 -3% (real-world) • Reference crew schedule is known for all instances • All tests on Intel Pentium IV 2. 2 GHz/2 GB RAM with CPLEX 9. 1. 3 • Limited branch-and-bound time to 2 hours Page: 23 ATMOS 2007 – Nov. 16, 2007

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 Page: 24 ATMOS 2007 – Nov. 16, 2007

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 Page: 25 ATMOS 2007 – Nov. 16, 2007

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