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

Maintenance RoutingGábor Maróti

CWI, Amsterdamand

NS Reizigers, Utrecht

[email protected]@[email protected]@cwi.nl

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

Maintenance Routing

Gábor Maróti

Leo Kroon

Astrid Roelofs

CWI, AmsterdamNS Reizigers, Utrecht

Erasmus University, RotterdamNS Reizigers, Utrecht

Free University, AmsterdamNS Reizigers, Utrecht


Maintenance Routing♦ 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

Maintenance Routing


! ♦ Problem formulation





♦ Models


♦ 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


Problem formulation


(Very) naive idea: solve the shunting problem at each station

Natural decomposition: solve the problem separatelyfor the rolling stock types

(and try to estimate the shunting difficulty)

Solution: new rolling stock schedule in the planning horizon

Problem formulation


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


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


!

♦ Problem formulation



♦ What do the planners now do?♦ A graph representation

♦ Models


♦ Future

What do they now do?


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



1

3

2

Deadlinesfor urgent units

Days

Nights



Urgentunit

Assigned maintenance job

Night change



Urgentunit


Night change



Urgentunit


Daily change



Urgentunit


Daily change: maybe possible



Urgentunit


Daily change: maybe possible



Urgentunit


Daily change



Urgentunit


Daily change (and a night change)



Urgentunit


Empty train movement



Urgentunit


Empty train movement

(taking care of the balance)

Maintenance Routing


!





♦ A graph representation♦ Models


♦ Future

A graph representation


Nodes: arrival and departure events

Arcs: operational plan + extra possibilities



“Grey box”: permitted or forbidden arcs

A perfect matching is required

Night arcs:



Night arcs:

Assumption: a small number of changes can be carried out



Day arcs:

Simple daily change possibility



Day arcs:

If we allow only one change for each train unit…

…it is enough to insert all these arcs



Day arcs:

In case we allow also more complex changes…

…the graph becomes more complicated.



Day arcs:

However, we did not implement multiple changes because

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



Empty train arcs:

extra arcs between the boxes: all or some of them

(a small number is enough)

Maintenance Routing


!






♦ Models


♦ Future

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.

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) ≥ 0 otherwise

Minimize the total sum of arc costs.

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

Models


Example:

StationUtrecht

expensive

cheap expensive

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

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

0 50km



Maintenance Routing


!






♦ Models


♦ Future

Successive shortest paths


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


Easy, simple, very fast

Takes no care of matching conditions (day, night)

Ad hoc ideas are necessary

Maintenance Routing


!






♦ Models


♦ Future

Solution = Perfect matching in each box

1

3

2

s.t. the deadline conditions are fullfilled

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


Multicommodity flow

A 1-flow for each urgent unit

1

3

2

Multicommodity flow


2

Possible terminal nodes

Deadline

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♦ ∑ xi ≤ m(e)

Multicommodity flow


Multicommodity flow


Objective function:

minimize Σ (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


Maintenance Routing


!






♦ Models


♦ 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


♦ x : Arcs → [0 ; 1]

♦ conservation rule for every nodes ≠ s, t

♦ the flow value is 3 (# of urgent units)

♦ x(e) ≤ m(e) for Night Arcs

Network flow


Network flow


Given a digraph G and a function C : Arcs → R,

π is a node potential (for the longest path)

if

C(uv) ≤ π(u) − π(v)u v

for every arc uv.

Network flow


Having fixed a matchig m:

♦ longest path = the only path

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

0

00

BigBigBig

Big

Network flow


1

32

Instead of the deadlines:

1

53

distances.

d(u) := 2 deadline(u) − 1

Network flow


The important inequalities:

π(u) − π(v) ≥ 1 −Big⋅ (1 −m(uv)) for Day and Night Arcs

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

LB(v) ≤ π(v) ≤ UB(v) for each node

The bounds LB and UB from the graph structure

Then Big := UB(v) − LB(u) + 1

Network flow


♦ matching variables m♦ flow variables x♦ potential variables π

integral

may be chosen float

Variables:

Network flow


Constraints

♦ matching variables m♦ flow variables x♦ potential variables π

Variables:

♦ matching constraints♦ flow constraints♦ potential constraints

Network flow


Objective function:

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

Maintenance Routing


!






♦ Models


♦ Future

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

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


Only night connections (5 nights):

NFNP

3 nights 10 sec

4 nights 15 sec

5 nights 15 - 500 sec

Computational resultsAll possibilities:


Maintenance Routing


!






♦ Models


♦ Future

Future


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.

Documents

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