Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht [email protected] Models...

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Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht [email protected] Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30 - 10 - 2001
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Transcript of Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht [email protected] Models...

Page 1: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

Maintenance Routing

Gábor Maróti

CWI, Amsterdam

and

NS Reizigers, Utrecht

[email protected]

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

Page 2: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 3: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 4: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 5: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 6: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 7: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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”

Page 8: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 9: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 10: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 11: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

What do they now do?

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

1

3

2

Deadlinesfor urgent units

Days

Nights

Page 12: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

What do they now do?

Urgentunit

Assigned maintenance job

Night change

Page 13: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

What do they now do?

Urgentunit

Assigned maintenance job

Night change

Page 14: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

What do they now do?

Urgentunit

Assigned maintenance job

Daily change

Page 15: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

What do they now do?

Urgentunit

Assigned maintenance job

Daily change: maybe possible

Page 16: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

What do they now do?

Urgentunit

Assigned maintenance job

Daily change: maybe possible

Page 17: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

What do they now do?

Urgentunit

Assigned maintenance job

Daily change

Page 18: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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)

Page 19: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

What do they now do?

Urgentunit

Assigned maintenance job

Empty train movement

Page 20: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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)

Page 21: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 22: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

A graph representation

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

Nodes: arrival and departure events

Arcs: operational plan + extra possibilities

Page 23: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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:

Page 24: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 25: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

A graph representation

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

Day arcs:

Simple daily change possibility

Page 26: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 27: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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.

Page 28: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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)

Page 29: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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)

Page 30: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 31: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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.

Page 32: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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”

Page 33: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

Models

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

Example:

StationUtrecht

expensive

cheap expensive

Page 34: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 35: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 36: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

0 50km

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

Page 37: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 38: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 39: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 40: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 41: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 42: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 43: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 44: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

A 1-flow for each urgent unit

1

3

2

Multicommodity flow

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

Page 45: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

2

Possible terminal nodes

Deadline

Multicommodity flow

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

Page 46: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 47: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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)

Page 48: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 49: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 50: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 51: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 52: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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.

Page 53: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 54: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 55: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 56: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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:

Page 57: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 58: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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)

Page 59: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 60: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 61: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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):

Page 62: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 63: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 64: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

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

Page 65: Maintenance Routing Gábor Maróti CWI, Amsterdam and NS Reizigers, Utrecht G.Maroti@cwi.nl Models for Maintenance Routing 2nd AMORE Seminar, Partas, 30.

Thank you.