V. Cacchiani, ATMOS 2007, Seville1 Solving a Real-World Train
Unit Assignment Problem V. Cacchiani, A. Caprara, P. Toth
University of Bologna (Italy) European Project ARRIVAL Slide 2 V.
Cacchiani, ATMOS 2007, Seville2 Problem description A Graph
representation ILP models (arc formulation & path formulation)
Strong inequalities for the Capacity Constraints Maintenance
Constraints An LP-based heuristic algorithm Experimental Results
Conclusions Outline Slide 3 V. Cacchiani, ATMOS 2007, Seville3
Train Unit Assignment Problem (TUAP) Input: timetabled train trips
train unit types Output: Minimum Cost A ssignment of the train
units (TUs) to the trips required number of passenger seats for
each trip (possibly combining TUs ) maximum number of TUs that can
be assigned to each trip (generally 2) maximum number of TUs
available sequencing constraints between trips maintenance
constraints Constraints: Slide 4 V. Cacchiani, ATMOS 2007, Seville4
SV 352 MI 389 MI 356 MD 396 SA 437BA 464 BA 540DAT 552 BA 720DAT
732 360 720 516 876 Trips TUs 516 2 360 2 Output: A B C D E A A B
CD E E Input: Slide 5 V. Cacchiani, ATMOS 2007, Seville5 Graph
Representation node trip arcs for TUs of type k iff a TU of type k
can be assigned to i and then to j within the same day A E D C B
(and dummy nodes 0 and n+1) n+1 0 dummy start node dummy end node
arc n # of trips p # of TU types Slide 6 V. Cacchiani, ATMOS 2007,
Seville6 Arc Formulation number of times that arc a is selected in
the solution max # of paths for TU of type k capacity constraints
min of the total cost of the paths max # of TU for trip j # of
times that arc a is selected in the solution flow constraints Slide
7 V. Cacchiani, ATMOS 2007, Seville7 Path Formulation number of
times that path P is selected in the solution subcollection of
paths in that visit trip j, max # of paths for TU of type k
capacity constraints min of the total cost of the paths max # of TU
for trip j # of times that path P is selected in the solution
collection of paths from 0 to n+1 in Slide 8 V. Cacchiani, ATMOS
2007, Seville8 Strong capacity constraints An example number of TUs
of type k assigned to a trip j weak strong Slide 9 V. Cacchiani,
ATMOS 2007, Seville9 Strong capacity constraints The described
capacity constraints are the following: Assuming For a trip j
define: such that The derived strong inequalities describe the
convex hull of the capacity constraints. Slide 10 V. Cacchiani,
ATMOS 2007, Seville10 ILP Formulation number of paths for a TU of
type k which contain a maintenance arc maintenance every days
maintenance constraints collection of maintenance paths from 0 to
n+1 in Slide 11 V. Cacchiani, ATMOS 2007, Seville11 An LP-based
Heuristic Algorithm 0. initialize the current LP as the described
model (strong capacity constraints, a subset of variables) 1. solve
the current LP (CPLEX) 2. constructive heuristic (based on the
optimal dual solution) 4. if there are dual constraints violated
add some of the corresponding primal variables to the current LP
else stop5. if the current LP is infeasible if value of the
incumbent solution fixing stop else goto 1. 3. refine the solution
found Slide 12 V. Cacchiani, ATMOS 2007, Seville12 Experimental
Results on the Case Study Methods implemented in C, computational
tests on a Pentium IV, 3.2 GHz, 1 Gb Ram, Cplex 9.0 instArc LP Arc
LP Time (sec) Path LPPath LP Time (sec) Strong Arc LP Strong Arc LP
Time (sec) Strong Path LP Strong Path LP Time (sec) A (528,8)
57624571366212016250 B (662,10) 414724241174532690753150 C (660,10)
402384140179532735053177 instActual Solution LP Maint LP Maint Time
(sec) Heur Time (sec) A (528,8) 7262130633544 B (662,10)
7656196595471 C (660,10) 7455295588875 Slide 13 V. Cacchiani, ATMOS
2007, Seville13 Conclusions ILP formulations based on a Graph
representation of the TUAP LP-relaxation with strong capacity
constraints LP-based Heuristic Signifcant improvement over the
practitioners solution. Current Research (cooperation with Erasmus
University Rotterdam) Build a robust solution based on the
possibility of exchanging TUs in case of delays
