fast reoptimization algorithms(1,3) 08:15 (5,1) 2 a tour-variable model contains a variable for each...

Fast Reoptimization Algorithmsfor Multi-Elevator Systems

Jörg Rambau

Konrad-Zuse-Zentrum für Informationstechnik Berlin

. Problem Description: Elevator Group Control

. Efficiency of Reoptimization: Modeling, Algorithm, Result

. Effectiveness of Reoptimization: Simulation Results

. Conclusion

Joint work withPhilipp Friese

Based on joint work withBenjamin Hiller, Sven O. Krumke, Luis M. Torres(ADAC team)

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PROBLEM DESCRIPTION

THE PRACTITIONER’ S V IEW

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PROBLEM DESCRIPTION


Real System:

Task:

Online aspect:

Realtime aspect:

Currently:

Problem:

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PROBLEM DESCRIPTION


Real System:pallet transportation Herlitz PBS AG,

Falkensee near Berlin

Task:

Online aspect:

Realtime aspect:

Currently:

Problem:

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Task: requests→ elevators, scheduling

minimize flow times/empty moves

Online aspect:

Realtime aspect:

Currently:

Problem:

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Online aspect:future requests unknown

Realtime aspect:

Currently:

Problem:

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Realtime aspect:response time≤ 1s

Currently:

Problem:

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Currently: choice between:

FIFO oderNEARESTNEIGHBOR

Problem:

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Problem: FIFO:

inefficient

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Problem: FIFO:

inefficient

NEARESTNEIGHBOR:

forgotten pallets

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PROBLEM DESCRIPTION

REOPTIMIZATION: USE OPTIMAL SOLUTION FOR SNAPSHOTPROBLEM

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PROBLEM DESCRIPTION


Reoptimization:

Maintain dispatch to control server:

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Reoptimization:


new event

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Reoptimization:


new event↓snapshot of available data

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Reoptimization:


new event↓snapshot of available data↓

offline-optimization w.r.t. aux.-problem

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PROBLEM DESCRIPTION


Reoptimization:


new event↓snapshot of available data↓

offline-optimization w.r.t. aux.-problem↓dispatch

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HOW GOOD IS EXACT REOPTIMIZATION?

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QUESTIONS OF STUDY:

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PROBLEM DESCRIPTION


QUESTIONS OF STUDY:

IS REOPTIMIZATION EFFICIENT?

(IS REOPTIMIZATION REALTIME -COMPLIANT?)

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PROBLEM DESCRIPTION


QUESTIONS OF STUDY:

IS REOPTIMIZATION EFFICIENT?

(IS REOPTIMIZATION REALTIME -COMPLIANT?)

IS REOPTIMIZATION EFFECTIVE?

(REOPTIMIZATION OF OBJECTIVE =⇒ GOOD LONG-TERM VALUE OF OBJECTIVE?)

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EFFICIENCY OF REOPTIMIZATION: MODELING, ALGORITHM, RESULT

A TOUR-BASED INTEGERL INEAR PROGRAM

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The transport graph for one elevator.

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Some requests.

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08:03

07:45

08:15 Their time stamps.

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08:03

07:45

08:15 The position of the elevator.

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08:03

08:15

07:45

(4,6)

Each request can be seen as a node.

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08:15

07:45

(4,6)

08:03

(1,3)


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07:45

(4,6)

08:03

(1,3)

08:15

(5,1)


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07:45

(4,6)

08:03

(1,3)

08:15

(5,1)

2

The elevator is a special node.

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07:45

(4,6)

08:03

(1,3)

08:15

(5,1)

2

The connecting moves between requests . . .

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07:45

(4,6)

08:03

(1,3)

08:15

(5,1)

2

. . . are arcs . . .

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07:45

(4,6)

08:03

(1,3)

08:15

(5,1)

2

. . . whose weights are empty moves.

[Ascheuer 1996]

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1

3

122

0

1

5

07:45

(4,6)

08:03

(1,3)

08:15

(5,1)

2

A feasible dispatch

is atour through all nodes

starting at the elevator’s node

with precedence conditionson each floor.

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07:45

(4,6)

08:03

(1,3)

08:15

(5,1)

2

We do not show the arcs anymore

since they are implicitely given.

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5

07:45

(4,6)

08:03

(1,3)

08:15

(5,1)

2

If there is another elevator . . .

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5

07:45

(4,6)

08:03

(1,3)

08:15

(5,1)

2

. . . then a feasible solution is a

partitioning

of requests into tours

with precedence conditions

on each floor.

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5

x(R)=1

x(S)=1

07:45

(4,6)

08:03

(1,3)

08:15

(5,1)

2

A tour-variable model contains a variable for

each feasible tour of a server.

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5

x(R)=1

x(S)=1

07:45

(4,6)

08:03

(1,3)

08:15

(5,1)

2



Problem:

astronomic number of variables

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5

x(R)=1

x(S)=1

07:45

(4,6)

08:03

(1,3)

08:15

(5,1)

2



Problem:


Solution:

dynamic column generation

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5

x(R)=1

x(S)=1

07:45

(4,6)

08:03

(1,3)

08:15

(5,1)

2



Problem:


Solution:

dynamic column generation

Precedence Conditions:

Good for us!

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DYNAMIC COLUMN GENERATION

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Maintain a feasiblereduced LP(RLP) with fewer variables. Then

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1. Solve (RLP)

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1. Solve (RLP)

2. Extractdual values

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1. Solve (RLP)


3. Variables (columns) withnegative reduced costs?

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1. Solve (RLP)



No: Return current solutionandstop.

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1. Solve (RLP)




Yes: Add all/some/one such variable(s) to (RLP) andrepeat

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1. Solve (RLP)





I F USED “C OTS”:

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1. Solve (RLP)





I F USED “C OTS”:

SLOW CONVERGENCEIN THE BEGINNING =⇒ NOT REAL-TIME COMPLIANT

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DYNAMIC PRICING CONTROL (ADAC-PROJECT)

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Tours can be searched in aprefix tree.

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The root corresponds to the

“don’t-move-tour”.

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The next level corresponds to

all tours visiting one request.

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The next level corresponds to

all tours visiting two requests.

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And so forth . . .

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. . . until all request sequences

have been enumerated.

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Task:

find (all) paths in prefix tree

that have negative/minimum reduced

cost.

Problem:

in the beginning,

input data (dual prices) far off.

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Ingredients of

Dynamic Pricing Control:

Dynamic Size Of Search Space

Dynamic Variation Of Search Space

Dynamic Threshold For Reduced Cost

Many Iterations Of (RLP)-Runs

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Restriction ofdegreeanddepth:

degree1, depth1.

If columns found, reoptimize (RLP).

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degree1, depth2.


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degree1, depth3.


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degree1, depth4.


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degree2, depth1.

We greedily select the

2 best next requests.


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degree2, depth2.

By changing the2-best-next criterion

we obtain variations in search space.


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degree2, depth3.

By changing the2-best-next criterion

we obtain variations in search space.


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degree2, depth4.

Usually, we stop increasing depth

when no new columns are found.


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degree2, depth4.

Usually, we stop increasing depth

when no new columns are found.


Crucial:

Extremely fast Reoptimization of

(RLP)

CPLEX 8.x

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DUMMY CONTRACTOR

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DUMMY CONTRACTOR

Problem: Too many requests/server=⇒ long tours in optimum=⇒ DPC insufficient

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DUMMY CONTRACTOR


Rescue:Dummy contractor: service after fixed delay

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DUMMY CONTRACTOR



Effect: Cut-off of tail requests in dispatch

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DUMMY CONTRACTOR




Benefit: Tour lengths in optimum predictable

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DUMMY CONTRACTOR




Benefit: Tour lengths in optimum predictable

RESULT:

EFFICIENT FOR ALL MODELS WITH EFFECTIVE TIME WINDOWS FOR REQUESTS.

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FEATURES OFIMPLEMENTATION

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. Configurable runtime bound

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. Parametrized objective function consisting of terms for

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. time window violations for requests

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. time window violations for servers

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. server movements

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. server movements

. service

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. server movements

. service

. Various reoptimization strategies includingREPLAN andIGNORE

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. server movements

. service


. Start heuristics (BESTINSERTION, 2OPT)

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. server movements

. service


. Start heuristics (BESTINSERTION, 2OPT)

. Embedded in simulation environment

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RESULT

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RESULT

SOUND BITES :

OPTIMAL SOLUTIONS WITH PROOF IN 100MS FOR UP TO 7 REQUESTSPER SERVER

REASONABLE SOLUTION AFTER 5MS

(EFFECTIVE TIME WINDOWS NECESSARY)

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EFFECTIVENESSOF REOPTIMIZATION: SIMULATION RESULTS

SIMULATION ENVIRONMENT

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SIMULATION ENVIRONMENT

AMSEL Simulation Library

[Ascheuer]

Elevator Simulation

[Rambau]

Herlitz System

[Hauptmeier, Müller, Rambau]

Multi-Capacity Elevators

[Rambau, Weider]

Multiple Elevators

[Friese, Rambau]

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INVESTIGATED ONLINE-ALGORITHMS

Whenever a new requests occurs, dispatch it so as to . . .

GREEDYDRIVE greedily minimize empty moves (no reopimization)

REOPTDRIVE minimize empty moves plusε-lateness (complete reoptimization)

XLOADDRIVE dto. with squared lateness and Dummy Contractor (complete reoptimization)

GREEDYLATE greedily minimize lateness (no reoptimization)

REOPTLATE minimize lateness plusε-empty-moves (complete reoptimization)

XLOADLATE dto. with squared lateneess and Dummy Contractor (complete reoptimization)

REOPTMIX minimize lateness plus empty-moves (complete reoptimization)

XLOADMIX dto. with squared lateness and Dummy Contractor (complete reoptimization)

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SIMULATION RESULTS (VARIOUS LOADS, 10 SAMPLES À 24H EACH)

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SOUND BITES :

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SOUND BITES :

GREEDILY M INIMIZING EMPTY MOVES IS UNACCEPTABLE

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SOUND BITES :


GREEDILY M INIMIZING LATENESSIS GOOD

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SOUND BITES :



REOPTIMIZATION DELIVERS BEST CORR. LONG-TERM OBJECTIVE

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SOUND BITES :



REOPTIMIZATION DELIVERS BEST CORR. LONG-TERM OBJECTIVE

M INIMIZING SQUARED LATENESSDELIVERS QOS-STABILIZATION

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CONCLUSION

EXPERIMENTAL EVIDENCE :

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CONCLUSION


REOPTIMIZATION IS EFFICIENT AND EFFECTIVE

FOR ONLINE GROUPCONTROL OF TRANSPORTELEVATORS

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CONCLUSION




STILL M ISSING:

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CONCLUSION




STILL M ISSING:

COMPUTABLE MATHEMATICAL PERFORMANCEGUARANTEES

=⇒ RESEARCHIN PROGRESS(DFG CENTER C6)

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CONCLUSION




STILL M ISSING:

COMPUTABLE MATHEMATICAL PERFORMANCEGUARANTEES

=⇒ RESEARCHIN PROGRESS(DFG CENTER C6)

THANK YOU FOR YOUR ATTENTION !

fast reoptimization algorithms(1,3) 08:15 (5,1) 2 a tour-variable model contains a variable for each...

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