Maintenance Routing

Gábor Maróti

CWI, Amsterdam

and

NS Reizigers, Utrecht

G.Maroti@cwi.nl

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Maintenance Routing

Gábor Maróti

Leo Kroon

Astrid Roelofs

CWI, Amsterdam

NS Reizigers, Utrecht

Erasmus University, Rotterdam

NS Reizigers, Utrecht

Free University, Amsterdam

NS Reizigers, Utrecht

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Maintenance Routing Problem formulation

successive shortest paths

Computational results

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

What do the planners now do?

A graph representation

Models

multicommodity flow network flow and node potential

Future

Maintenance Routing

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Problem formulation

successive shortest paths

Computational results

What do the planners now do?

A graph representation

Models

multicommodity flow network flow and node potential

Future

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

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Problem formulation

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

(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

Problem formulation

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Input: duties: sequences of tasks on each day

list of urgent units

deadlines

the actual operational plan

Output: new operational plan, such that

the urgent units can reach the maintenance station

“the cost is minimal”

Problem formulation

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

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

delays

shortage of crew

shortage of rolling stock

necessary changes

in the plan

cancelled trains

Maintenance Routing

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Problem formulation

successive shortest paths

Computational results

What do the planners now do?

A graph representation

Models

multicommodity flow network flow and node potential

Future

What do they now do?

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

1. Assign a most urgent unit to a first available maintenance job

If no solution, change a bit the deadlines (1 day).

2. Try to route it there

3. Call the local sunting crew: “Is the route feasible?”

4. Iterate this process

What do they now do?

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

1

3

2

Deadlinesfor urgent units

Days

Nights

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

What do they now do?

Urgentunit

Assigned maintenance job

Night change

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

What do they now do?

Urgentunit

Assigned maintenance job

Night change

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

What do they now do?

Urgentunit

Assigned maintenance job

Daily change

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

What do they now do?

Urgentunit

Assigned maintenance job

Daily change: maybe possible

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

What do they now do?

Urgentunit

Assigned maintenance job

Daily change: maybe possible

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

What do they now do?

Urgentunit

Assigned maintenance job

Daily change

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

What do they now do?

Urgentunit

Assigned maintenance job

Daily change (and a night change)

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

What do they now do?

Urgentunit

Assigned maintenance job

Empty train movement

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

What do they now do?

Urgentunit

Assigned maintenance job

Empty train movement

(taking care of the balance)

Maintenance Routing

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Problem formulation

successive shortest paths

Computational results

What do the planners now do?

A graph representation

Models

multicommodity flow network flow and node potential

Future

A graph representation

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Nodes: arrival and departure events

Arcs: operational plan + extra possibilities

A graph representation

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

“Grey box”: permitted or forbidden arcs

A perfect matching is required

Night arcs:

A graph representation

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Night arcs:

Assumption: a small number of changes can be carried out

A graph representation

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Day arcs:

Simple daily change possibility

A graph representation

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Day arcs:

If we allow only one change for each train unit

it is enough to insert all these arcs

A graph representation

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Day arcs:

In case we allow also more complex changes

the graph becomes more complicated.

A graph representation

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Day arcs:

However, we did not implement multiple changes because

they did not give any extra possibility (in the test data)

A graph representation

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Empty train arcs:

extra arcs between the boxes: all or some of them

(a small number is enough)

Maintenance Routing

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Problem formulation

successive shortest paths

Computational results

What do the planners now do?

A graph representation

Models

multicommodity flow network flow and node potential

Future

Models

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

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.

Models

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

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) 0 otherwise

Minimize the total sum of arc costs.

Idea: “the closer to the original plan the better”

Models

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Example:

StationUtrecht

expensive

cheap expensive

Models

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

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

Models

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

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

0 50km

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Maintenance Routing

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Problem formulation

successive shortest paths

Computational results

What do the planners now do?

A graph representation

Models

multicommodity flow network flow and node potential

Future

Successive shortest paths

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

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

Algorithm

Successive shortest paths

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Easy, simple, very fast

Takes no care of matching conditions (day, night)

Ad hoc ideas are necessary

Maintenance Routing

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Problem formulation

successive shortest paths

Computational results

What do the planners now do?

A graph representation

Models

multicommodity flow network flow and node potential

Future

Solution = Perfect matching in each box

1

3

2

s.t. the deadline conditions are fullfilled

Multicommodity flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

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

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Multicommodity flow

A 1-flow for each urgent unit

1

3

2

Multicommodity flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

2

Possible terminal nodes

Deadline

Multicommodity flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Variables:

matching variables m flows x1, x2, x3, …

Constraints:

matching constraints

conservation rule for each flow

starting and terminal constraints for the flows

xi m(e)

Multicommodity flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Multicommodity flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Objective function:

minimize (c (a) m(a) : a Night or Day Arcs)

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.

Multicommodity flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Maintenance Routing

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Problem formulation

successive shortest paths

Computational results

What do the planners now do?

A graph representation

Models

multicommodity flow network flow and node potential

Future

Having fixed a matching m,

s

t

set two new nodes s and t,set all arc capacities 1.

Does there exist an s-t network flow of value 3?

Network flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

x : Arcs[0 ; 1]

conservation rule for every nodess, t

the flow value is 3 (# of urgent units)

x(e)m(e) for Night Arcs

Network flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Network flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Given a digraph G and a function C : ArcsR

is a node potential (for the longest path)

if

C(uv)(u)(v)

u v

for every arc uv.

Network flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Having fixed a matchig m:

longest path = the only path

is an upper bound on the distance from the maintenance nodes (with appropriate initial values) (C 1)

0

00

BigBigBig

Big

Network flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

1

32

Instead of the deadlines:

1

53

distances.

d(u) := 2 deadline(u)1

Network flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

The important inequalities:

(u) (v) Big m(uv)) for Day and Night Arcs

(u) d(u) for urgent unit starting nodes

LB(v) (v)UB(v) for each node

The bounds LB and UB from the graph structure

Then Big := UB(v) LB(u) 1

Network flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

matching variables m flow variables x potential variables

integral

may be chosen float

Variables:

Network flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Constraints

matching variables m flow variables x potential variables

Variables:

matching constraints flow constraints potential constraints

Network flow

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Objective function:

minimize (c (a) m(a) : a Night or Day Arcs)

Maintenance Routing

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Problem formulation

successive shortest paths

Computational results

What do the planners now do?

A graph representation

Models

multicommodity flow network flow and node potential

Future

Computational results

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

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

MF NFNP

3 units 17 sec 10 sec

5 units Nr. 1. 4 – 10 sec 10 sec

5 units Nr. 2 22 sec 12 sec

Computational results

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Only night connections (5 nights):

NFNP

3 nights 10 sec

4 nights 15 sec

5 nights 15 - 500 sec

Computational results

All possibilities:

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Maintenance Routing

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

Problem formulation

successive shortest paths

Computational results

What do the planners now do?

A graph representation

Models

multicommodity flow network flow and node potential

Future

Future

Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001

A lot cooperation with planners and shunting crew in

modelling the night shunting possibilities (costs)

determining the practical relevance of the solutions

finding the set of day connections

New criteria for the rolling stock scheduling

Thank you.