operations research models for railroad routing and scheduling problems xuesong zhou university of...
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Operations Research Models for Railroad Routing and Scheduling Problems
Xuesong Zhou
University of Utah
Outline
Background on railroad planning and operations Traffic OD demand estimation problem Railroad blocking problem and line planning problem Train timetabling and dispatching problems Research directions
Railroad Network Capacity Line capacity
Single or double-track -> meet-pass plans Signal control type -> minimal headways Locomotive power -> speed, acceleration/deceleration time loss Train schedules -> overall throughput
Node capacity (yards, terminals / sidings) Track configuration (# of tracks, track lengths, switcher engines) Locomotive power-> car processing time Yard make-up plans, terminal operating plans
-> overall throughput
Railroad Planning and Operations
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OD Demand ->Routes-> Blocks-> Trains Origin destination (OD) traffic demand
Commodities, commercial unit traffic groups, car equipment requirement Route plan
Single route or multiple routes for each OD pair Blocking plan
Block origin, block destination, commodities, final destinations Standard, local, unit
Make-up plan Block-to-train assignment
Train schedule Arrival and departure times, station dwell times Block assignment and order Local/express, freight/passenger
Trip plan Distance, transit time Block sequence and intermediate classification yards
How to Estimate Freight OD Demand Matrix?
Input-output models (Leontieff, 1936)
Spatial interaction models Gravity models Direct demand models ( functions of socio-economic characteristics)
Trip-based spatial interaction models Trip generation, trip distribution, trip assignment
Commodity-based spatial interaction models Commodity generation, commodity distribution, mode split, trip
assignment
Origin destination demand estimation based on link counts
Traffic OD Demand Estimation Problem Given
Network Network zone structure Socio-economic data and interview samples Historical demand information Link counts cl
Find Freight/ Passenger OD matrix di,j
where i,j = zone index l = link index pl,i,j= link flow proportion, i.e. fraction of traffic demand flow from OD pair( i, j) contributing to the flow on link l
= measurement noise
Measurement Equations cl=Σi,j,pl, i,j× di,j+
jid ,ˆ
Freight OD Demand Estimation Models Linear programming model with multiple vehicle classes
List and Turnquist, 1994 Entropy model with given link flow proportion matrix
Jornsten and Wallace,1993 Linear programming model with simple route choice model
Sherali et al.1994 Bi-level model for multimode multi-product freight transportation
Crainic et al. 2001 Upper level:
Lower level: Pl,i,j = System optimum traffic assignment (di,j) Holguin-Veras, et al. 2004
Lower level: market equilibrium
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Railroad Blocking Problem Given
Network G=(N,A) Freight shipments with different origins/destinations
Find Blocking plan that can route all shipments over the network
Which blocks to build at each yard Sequences of blocks to deliver each shipment
Constraints Node capacity (yard processing volume) Maximal # of outbound arcs (blocks) for each node
# of classification tracks at a yard Objective Functions
Minimize total travel distance Minimize intermediate handlings cost Minimize delay cost
Network representation Classification yards: nodes->arcs Candidate blocks: express arcs that bypass yards
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Network Design FormulationLinear cost, multi-commodity capacitated network design problem Minimize Minimize (i,j)A fijyij + kK (i,j) cij xk
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subject to: Flow conservation constraint
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Yard capacity constraintkK xk
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(i,j)O(i) yij ≤ ≤ vi for all i Nwhere
k = commodity/product index (origin, destination, traffic class)yij = 0-1 design variable for opening block (i,j) or notxk
ij= amount of flow of commodity k using block (i,j)fij= fixed cost of opening block (i,j)
cij= transportation cost of moving per unit of commodity on arc (i,j)dk
i=demand of commodity k at node i uij=capacity of link (i,j)
Blocking Plan Optimization Models (I) Bodin, Golden and Schuster (1980)
Arc-based multi-commodity flow formulation No freight priorities
Assad (1983) Dynamic programming formulation for a rail line
Crainic, Ferland and Bousseau (1984) Consider blocking, train routing, and block-to-train assignment Nonlinear objective with queuing-based delay estimate Column generation to introduce attractive block and train
Van Dyke (1986, 1999) Improve existing blocking plan Augmented shortest path
Keaton (1989,1992) Consider blocking, train routing and block-to-train assignment Dual adjustment procedure based on Lagrangian relaxation
Blocking Plan Optimization Models (II) Huntley (1995)
Meta-heuristic solution algorithms
Newton (1996), Newton et al. (1998), Barnhart et al. (2000) Path-based multi-commodity flow formulation Branch-and-price, branch-and-cut Dual-based Lagrangian relaxation with sub-gradient optimization
Ahuja, et al. (2004) Very Large-Scale Neighborhood (VLSN) search Start with feasible solutions Improve/re-optimize solutions for each node (yard) Multiple passes over the nodes
Line Planning
Given Network G=(N,A) Traffic demand with different origins/destinations
Find Routes, frequencies, and stop schedules
Constraints Train capacity
Objective Functions Minimize total travel time Minimize total number of train units
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Frequency Service Network Design Formulation Minimize Minimize sS fsys + kK sS c(xk
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subject to: Flow conservation constraint
sS σ(k,s)xk = dk for all k K Service capacity constraint
kK xks ≤ ≤ usys for all s S
where k = index for commodity (origin, destination, traffic class)s = index for service that defines train origin, destination, route, intermediary terminals with stopsys = frequency of service s (integer variable)xk
s= amount of flow of commodity k using service sfs= fixed cost of operating one train for service sc (xk
s, ys)= transportation cost/delay of moving commodity k by using service sσ(k,s) = commodity-service indicator, 1: commodity k can use service s, 0 otherwise dk=demand of commodity kus= capacity of one train for service s
Frequency Service Network Design Models
Crainic and Rousseau (1996) Network design formulation Decomposition solution approach Column generation to find attractive itineraries
Bussieck and Zimmermann (1997), Bussieck (1998) Line planning problem
Passenger rail network with periodic schedules Maximize direct travelers Mixed integer programming formulation Branch-and-cut solution method to use utilize polyhedral structure
Goossens, Hoesel and Kroon (2004) Branch-and-cut with valid inequalities
Dynamic Service Network Design Time-space network
Physical network represented at each period Service arcs: represent each possible departure of each service Waiting/holding (inventory) arcs: connect two nodes representingthe same terminals in consecutive periods
Decision variables 0-1 variables: whether or not the service leaves at the specified time Continuous path flow through the service network
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Dynamic Service Network Design Models Morlok and Peterson (1970)
Integrate blocking, train formulation, and train scheduling Large mixed integer programming formulation
Haghani (1989) Integrate train make-up, routing and empty car distribution Empty and loaded car flows, engine flows
Gorman (1998) Consider operating plan that minimizes operating costs, meets
customer’s service requirements Generate candidate train schedules, and then evaluate
economic, service, and operational performance
Train Timetabling and Dispatching Problem Given
Line track configuration Minimum segment and station headways # of trains and their arrival times at origin stations
Find Timetable: Arrival and departure times of each train at each station
Objectives (Planning) Minimize the transit times and overall operational costs,
performance and reliability (Dispatching) Minimize the deviation between actual schedules and
planned schedule
Resource Constrained Project Scheduling Formulation Project i = train i with a sequence of K activities Activity (i, k) = train i traveling section k Renewable resources k: entering times and leaving times of
station k Mode m = stop-stop, stop-pass, pass-stop, pass-pass Processing time pt(i,k,m) = traveling time with acceleration and
deceleration time losses Precedence constraint: minimum headway constraints for two
consecutive activities at the same station
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Integer Programming Formulation Minimize
Departure time constraint
Dwell time constraint
Travel time on sections
Minimum headway constraint (N× (N-1) ×K× 4 constraints)
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Branch-and-Bound Solution Algorithm (I) Step 1: (Initialization)
Create a new node, in which contains the first task of all trains. Set the departure time for this train and insert this node into active node list (L).
Step 2: (Node Selection) Select an active node from L according to a given node selection rule.
Step 3: (Stopping criterion)If all of active nodes in L have been visited, then terminate.
Step 4: (Conflict set construction)Update the schedulable set in the selected node .
Scan all the tasks this set , and find a task t(i,j) with the minimal end time and all the tasks that invalidate the headway constraint.
Insert these tasks and task t(i,j) into the current conflict set.
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Branch-and-Bound Solution Algorithm (II) Step 5: (Active schedule generation)
For each task in the conflict set, create a new active node and insert it into L. Fix a task in the conflict set and right-shift the start time of the other conflict tasks according to the headway requirement with respect to the fixed task.
Step 6: (Lower bound)Compute the lower bound , If LB>Z, fathom this node, go to step 2. If conflict set is empty, this indicates all the tasks have been finished and the related schedule is feasible. Furthermore, if the current schedule is a feasible one with a smaller objective function value Z’ , then update Z=Z’.
Step 7: (Dominance rule)Compare the current schedule with the other schedules at visited nodes, if the dominance conditions are satisfied, then dominated node is disabled, its associated successors will not be completed. Go to Step 2.
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Conditions for node a dominating node b
(1) Same set of finished trains
(2) Z (a) < Z (b) for finished trains
(3) The starting time for each unfinished activity in node a is no later than the counterpart in node b for each feasible mode
Exact Solution Algorithms Szpigel (1973)
Job-shop scheduling formulation Branch-and-bound solution algorithm
Chen and Harker (1990); Jovanovic and Harker (1991) Considers reliability (on-time performance) of planed schedules under
random travel times Higgins et al. (1995); Higgins and Kozan (1998)
Inexact lower bound that estimates possible delay Multi-criteria scheduling framework that considers fuel costs for
locomotives, labor costs for crews Brannlund et al. (1998)
Lagrangian lower bound that relaxes the segment capacity Peeters and Kroon (2001); Kroon and Peeters (2003)
Cyclic railway timetabling Zhou (2005)
Exact lower bound for estimating possible delay Dominance rules
Heuristic Solution Algorithms Simple priority rule-based methods (used in simulation
systems) Minimal-delay criterion (MinDelay); Shortest departure time rule (SDT) First-in-first-out rule (FIFO); Shortest processing time rule (SPT)
Improved priority rule-based methods used in the train scheduling field Look-ahead method (Sahin, 1999)
Evaluation horizon only covers all the conflict trains at the current node Global priority rule (Higgins et al., 1995)
Find a feasible solution by applying a simple local rule successively Use the corresponding objective function value as the estimation value
Backtracking method (Adenso-Diaz el at., 1999) Use a local rule to find a feasible schedule Backtrack the search tree until k nodes have been examined
Beam search (Zhou, 2005) Beam search algorithm uses a certain evaluation rule to select the α-
best nodes to be computed at next level Look-ahead method Global priority rule
Meta-heuristics (Higgins, et al. 1997)
Train Routing Problem at Terminals
Given Track configuration ( track lengths, switcher engines ) Signal configuration Inflow/Outflow (arrival and departure times of trains)
Find Train paths through a terminal Choke points System performance of a rail facility under a variety of conditions
Train Routing through Terminals Switch Grouping
Train Paths Train type I: switch groups a, b, d Train type II: switch groups c, d, e Train type III: switch groups f, g, h
Carey and Lockwood (1995); Carey (1994) Mixed integer programming formulation Heuristic solution algorithm
Zwaneveld, Kroon, Hoesel (2001); Kroon, Romeijn, Zwaneveld (1997) Complexity issues Node packing model
North American Railroad Network
5 major US railroads after years of consolidations:CSX, UP, CR, NS, BNSF
Research Directions Robust schedule design
Executable vs. recoverable, from planning to real-time decision Improve freight railroad service reliability
Disruption management under real time information Service networks (blocking and line planning) Train dispatching Rail network and terminal capacity recovery plan Locomotive and crew recovery plan
Integrated pricing and demand management model Long term and short term pricing schemes and cost structures
Separation of track from traction in Europe Impact on traffic demand and operating plans (train schedule, fleet
sizing and repositioning) Shipper logistics modeling Demand estimation and prediction model