Maintenance Routing

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Maintenance Routing

• Gábor Maróti

CWI, Amsterdam and NS Reizigers, Utrecht
G.Maroti_at_cwi.nl
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Maintenance Routing
Leo Kroon
Erasmus University, Rotterdam NS Reizigers,
Utrecht
Astrid Roelofs
Free University, Amsterdam NS Reizigers, Utrecht

• Gábor Maróti

CWI, Amsterdam NS Reizigers, Utrecht
3
Maintenance Routing

• Problem formulation

• What do the planners now do?

• A graph representation

• Models

• successive shortest paths

• multicommodity flow

• network flow and node potential

• Computational results

• Future

4
Maintenance Routing

• Problem formulation

?

• What do the planners now do?

• A graph representation

• Models

• successive shortest paths

• multicommodity flow

• network flow and node potential

• Computational results

• Future

5
Problem formulation
Train units
After reaching a kilometer limit, they have to be
checked.
In practice the most urgent units go for
maintenance.
The operational plan must be changed.
Bottleneck shunting
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Problem formulation
(Very) naive idea solve the shunting problem at
each station
Natural decomposition solve the problem
separately
for the rolling stock types
(and try to estimate the shunting difficulty)
Solution new rolling stock schedule in the
planning horizon
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Problem formulation
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Problem formulation

• delays
• shortage of crew
• shortage of rolling stock

The planning horizon is short (e.g. 3 days).
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Maintenance Routing

• Problem formulation

?

• What do the planners now do?

• A graph representation

• Models

• successive shortest paths

• multicommodity flow

• network flow and node potential

• Computational results

• Future

10
What do they now do?
1. Assign a most urgent unit to a first available
maintenance job
2. Try to route it there
3. Call the local sunting crew Is the route
feasible?
4. Iterate this process
If no solution, change a bit the deadlines (1
day).
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What do they now do?
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What do they now do?
Assigned maintenance job
Night change
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What do they now do?
Assigned maintenance job
Night change
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What do they now do?
Assigned maintenance job
Daily change
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What do they now do?
Assigned maintenance job
Daily change maybe possible
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What do they now do?
Assigned maintenance job
Daily change maybe possible
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What do they now do?
Assigned maintenance job
Daily change
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What do they now do?
Assigned maintenance job
Daily change (and a night change)
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What do they now do?
Assigned maintenance job
Empty train movement
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What do they now do?
Assigned maintenance job
Empty train movement
(taking care of the balance)
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Maintenance Routing

• Problem formulation

• What do the planners now do?

?

• A graph representation

• Models

• successive shortest paths

• multicommodity flow

• network flow and node potential

• Computational results

• Future

22
A graph representation
Nodes arrival and departure events
Arcs operational plan extra possibilities
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A graph representation
Night arcs
Grey box permitted or forbidden arcs
A perfect matching is required
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A graph representation
Night arcs
Assumption a small number of changes can be
carried out
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A graph representation
Day arcs
Simple daily change possibility
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A graph representation
Day arcs
If we allow only one change for each train unit¼
¼it is enough to insert all these arcs
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A graph representation
Day arcs
In case we allow also more complex changes¼
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A graph representation
Day arcs
However, we did not implement multiple changes
because
they did not give any extra possibility (in the
test data)
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A graph representation
Empty train arcs
extra arcs between the boxes all or some of them
(a small number is enough)
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Maintenance Routing

• Problem formulation

• What do the planners now do?

• A graph representation

?

• Models

• successive shortest paths

• multicommodity flow

• network flow and node potential

• Computational results

• Future

31
Models
Solution a new operational plan, i.e.

• perfect matching on the Night Arcs
• perfect matching on the Day Arcs

such that each urgent unit gets to the
maintenance facility.
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Models
Quality of a solution the extra shunting cost
Linear cost function cost on the arcs
c (a) 0 if a is in the original plan
c (a) ³ 0 otherwise
Minimize the total sum of arc costs.
Idea the closer to the original plan the better
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Models
Example
Station Utrecht
cheap
expensive
expensive
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Models
Night arcs cheap, not too expensive or almost
impossible
Day arcs typically more expensive (more risky)
Empty train arcs very expensive

• carriage kilometer
• crew schedule

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Models
Test data rolling stock type Sprinter

• 52 units (duties)
• 1 maintenance job on each workday
• 1 maintenance station
• 10 terminal stations
• 2 further possible (daily) shunting stations

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Maintenance Routing

• Problem formulation

• What do the planners now do?

• A graph representation

• Models

• successive shortest paths

?

• multicommodity flow

• network flow and node potential

• Computational results

• Future

39
Successive shortest paths
Algorithm
1. Match the urgent train units to the
maintenance jobs
2. For each urgent unit
determine a shortest path in the graph
delete this path from the graph
take the next urgent unit
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Successive shortest paths
Easy, simple, very fast
Takes no care of matching conditions (day, night)
Ad hoc ideas are necessary
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Maintenance Routing

• Problem formulation

• What do the planners now do?

• A graph representation

• Models

• successive shortest paths

?

• multicommodity flow

• network flow and node potential

• Computational results

• Future

42
Multicommodity flow
1
3
2
Solution Perfect matching in each box
s.t. the deadline conditions are fullfilled
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Multicommodity flow
Matching variables m on the Night Arcs and Day
Arcs (0-1 valued).
Still needed
linear inequalities expressing that
1. each urgent unit reaches the maintenance
facility
2. in the time limit
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Multicommodity flow
1
3
2
A 1-flow for each urgent unit
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Multicommodity flow
2
Deadline
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Multicommodity flow
Variables

• matching variables m
• flows x1, x2, x3,

Constraints

• matching constraints
• conservation rule for each flow
• starting and terminal constraints for the flows
• å xi m(e)

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Multicommodity flow
Objective function
S
(c (a) m(a) a Î Night or Day Arcs)
minimize
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Multicommodity flow
If m is integral, the values x may be chosen
float (read-valued).
If x and m are integral on the Day Arcs,
the other variables may be chosen
float.
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Maintenance Routing

• Problem formulation

• What do the planners now do?

• A graph representation

• Models

• successive shortest paths

• multicommodity flow

?

• network flow and node potential

• Computational results

• Future

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Network flow
Having fixed a matching m,
set all arc capacities 1.
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Network flow

• x Arcs 0 1
• conservation rule for every nodes ¹ s, t
• the flow value is 3 ( of urgent units)
• x(e) m(e) for Night Arcs

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Network flow
Given a digraph G and a function C Arcs R,
p is a node potential (for the longest path)
if
C(uv) p(u) - p(v)
for every arc uv.
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Network flow
Having fixed a matchig m

• longest path the only path

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Network flow
1
3
2
Instead of the deadlines
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Network flow
The important inequalities
p(u) d(u) for urgent
unit starting nodes
p(u) - p(v) ³ 1 - Big (1 - m(uv)) for Day and
Night Arcs
LB(v) p(v) UB(v) for each
node
The bounds LB and UB from the graph structure
Then Big UB(v) - LB(u) 1
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Network flow
Variables

• matching variables m
• flow variables x
• potential variables p

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Network flow
Variables

• matching variables m
• flow variables x
• potential variables p

Constraints

• matching constraints
• flow constraints
• potential constraints

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Network flow
Objective function
S
(c (a) m(a) a Î Night or Day Arcs)
minimize
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Maintenance Routing

• Problem formulation

• What do the planners now do?

• A graph representation

• Models

• successive shortest paths

• multicommodity flow

• network flow and node potential

• Computational results

?

• Future

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Computational results

• Test data rolling stock type Sprinter
• 3 - 5 days planning horizon
• 3 - 5 urgent units
• IBM PC, Pentium III 900 MHz, 256 MB RAM
• Software ILOG OPL Studio 3.0, CPLEX 7.0

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Computational results
Only night connections (5 nights)
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Computational results
All possibilities
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Maintenance Routing

• Problem formulation

• What do the planners now do?

• A graph representation

• Models

• successive shortest paths

• multicommodity flow

• network flow and node potential

• Computational results

?

• Future

64
Future
A lot cooperation with planners and shunting crew
in

• modelling the night shunting possibilities
  (costs)

• finding the set of day connections

• determining the practical relevance of the
  solutions

New criteria for the rolling stock scheduling
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Thank you.