Introducing Multi-Criteria Path Planning

with Terrain Visibility Constraints: The Optimal Searcher Path Problem with Visibility

Michael Morin

Department of Computer Science and Software Engineering

Université Laval, Québec, QC, Canada

XV ELAVIO Summer School

August 2-6, 2010

Project funded by MITACS and DRDC-Valcartier Extended version with references

Michael Morin

Presentation Outline • Introduction

• Literature review

• Methodology – The Optimal Searcher Path problem with Visibility

– A multi-criteria extension

– Solving single criterion OSPV problems

– Solving multi-criteria OSPV problems

• Results (overview)

• Conclusion

• Discussion

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Michael Morin

Introduction

• The goal

– To plan an optimal (or efficient) search plan(s) for

a constrained search unit (searcher) in order to

locate a lost search object (moving or not)

• The motivations

– Search operations (e.g., search and rescue)

– Surveillance operations (e.g., security)

– Unmanned vehicles, robotized search, etc.

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Michael Morin

Literature review • Search theory: the theory of optimal search

– Optimize the allocation of the search effort

– Search and Screening (Koopman, 1946)

• The Optimal Searcher Path (OSP) problem (Trummel et al., 1989)

– Find a lost search objet in a discrete environment under accessibility constraints

– A path planning problem under uncertainty

• Usual OSP solving techniques (Benkoski et al., 1991)

– Branch and bound, dynamic programming

4 See (Benkoski et al., 1991) for more information on the search theory litterature.

Michael Morin

Methodology

The OSPV problem • Real life practical environments are continuous

– They contain accessibility and visibility obstacles

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• Discrete space – A set R of N discrete regions

• Discrete time – A set I of T time steps

• Discrete search effort allocations – Q discrete effort units

Michael Morin

Methodology – The OSPV problem

The environment • A fictive indoor environment

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Michael Morin

Methodology – The OSPV problem

The OSPV problem data

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Q = 1, T = 4 and the object is stationary

t

t Search unit’s

position at time t

Searched region

at time t

1.0, ( ) 0: , : {0,..., }: ( , , )

0.0, ( ) 0t

r V s et I s r R e Q pod s r e

r V s e

Michael Morin

Methodology – The OSPV problem

The objective

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What is the optimal 4 steps plan P with Q = 1 considering a stationary object?

An optimal search plan

maximizes the COS criterion defined has

t

t Search unit’s

position at time t

Searched region

at time t

1 2 1 2[ , ,..., ] [ , ,..., ] ( , : {0,..., })T T t tP y y y e e e y R e R Q

( ) ( ).tt I r RCOS P pos r

Michael Morin

Methodology – The OSPV problem

A search plan example

9

What is the optimal 4 steps plan P with Q = 1 considering a stationary object?

1

1

t

t Search unit’s

position at time t

Searched region

at time t

= .1 + …

1 2 3 4

1 1 2 2

( ) ( ) ( ) ( ) ( )

( ) ( , ,1) ( ) ( , ,1) ...

COS P pos F pos E pos J pos G

poc F pod J F poc E pod E E

Michael Morin

Methodology – The OSPV problem

A search plan example

10

What is the optimal 4 steps plan P with Q = 1 considering a stationary object?

1

1

t

t Search unit’s

position at time t

Searched region

at time t

= .1 + .05 + …

1 2 3 4

1 1 2 2

( ) ( ) ( ) ( ) ( )

( ) ( , ,1) ( ) ( , ,1) ...

COS P pos F pos E pos J pos G

poc F pod J F poc E pod E E

2 2

Michael Morin

Methodology – The OSPV problem

A search plan example

11

What is the optimal 4 steps plan P with Q = 1 considering a stationary object?

1

1

t

t Search unit’s

position at time t

Searched region

at time t

= .1 + .05 + .1 + …

1 2 3 4

1 1 2 2

( ) ( ) ( ) ( ) ( )

( ) ( , ,1) ( ) ( , ,1) ...

COS P pos F pos E pos J pos G

poc F pod J F poc E pod E E

2 2

3

3

Michael Morin

Methodology – The OSPV problem

A search plan example

12

What is the optimal 4 steps plan P with Q = 1 considering a stationary object?

1

1

t

t Search unit’s

position at time t

Searched region

at time t

= .1 + .05 + .1 + .2

1 2 3 4

1 1 2 2

( ) ( ) ( ) ( ) ( )

( ) ( , ,1) ( ) ( , ,1) ...

COS P pos F pos E pos J pos G

poc F pod J F poc E pod E E

2 2

3

3

4

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Michael Morin

Methodology – The OSPV problem

A search plan example

13

What is the optimal 4 steps plan P with Q = 1 considering a stationary object?

1

1

t

t Search unit’s

position at time t

Searched region

at time t

= .1 + .05 + .1 + .2 = .45

1 2 3 4

1 1 2 2

( ) ( ) ( ) ( ) ( )

( ) ( , ,1) ( ) ( , ,1) ...

COS P pos F pos E pos J pos G

poc F pod J F poc E pod E E

2 2

3

3

4

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Michael Morin

Methodology – A multi-criteria extension

The search unit’s safety

• Minimize the cumulative probability of hazard (CH)

– In function of the terrain, of the topology, of the events, of

the search unit’s capacity, etc.

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( ) Pr( hazard in ) 1 (1 ( ))t tt It ICH P y poh y

Michael Morin

Methodology – A multi-criteria extension

The search plan’s complexity

• A complex search plan is more error prone.

• Minimize the total path length (TPL)

– In function of the travelled distance

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1( ) ( , ) t tt I

TPL P dist y y

Michael Morin

Methodology – Solving a single criterion OSPV problem

Elitist Ant Search

16 See (Dorigo et al., 2005) for a survey on the Ant Colony Optimization techniques.

Michael Morin

Methodology – Solving a single criterion OSPV problem

Elitist Ant Search

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Michael Morin

Methodology – Solving a single criterion OSPV problem

Elitist Ant Search

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Michael Morin

Methodology – Solving a single criterion OSPV problem

Elitist Ant Search

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Michael Morin

Methodology – Solving a single criterion OSPV problem

Elitist Ant Search

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Michael Morin

Methodology – Solving a single criterion OSPV problem

Elitist Ant Search

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t

t Search unit’s

position at time t

Searched region

at time t

1

1 2

2

3

3

4

4

Michael Morin

Methodology – Solving a single criterion OSPV problem

Elitist Ant Search

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1

1

2

2

3

3

4

4

t

t Search unit’s

position at time t

Searched region

at time t

Michael Morin

Methodology – Solving a single criterion OSPV problem

Elitist Ant Search

23

1

1

2 2

3 3

4 4

t

t Search unit’s

position at time t

Searched region

at time t

Michael Morin

Methodology – Solving a single criterion OSPV problem

Elitist Ant Search

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1

1

2

2

3

3

4 4

t

t Search unit’s

position at time t

Searched region

at time t

Michael Morin

Methodology

Solving a multi-criteria OSPV problem

• Search plan’s efficiency.

– For a given set of criteria F, we will find a set of

efficient solutions instead of one solution.

• The algorithm approximates the Pareto

optimal set (or the Pareto front).

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, ' : ' ( : ( ) ( ')) ( : ( ) ( '))

* : ( ' : ' )

* ( ) : *)

P P P P f F f P f P f F f P f P

PSET P P P P

PFRONT F x x PSET

See (Talbi, 2009) for more information on metaheuristics for multi-objective problems solving.

Michael Morin

Methodology – Solving a multi-criteria OSPV problem

Pareto Ant Search

• A Pareto ACO adaptation (Doerner et al., 2008)

• The algorithm

– approximates the Pareto optimal set (stores non

dominated search plans)

– uses two pheromone tables (as in Elitist Ant Search)

• When a non dominated search plan is encountered

– PAS updates the pheromone values using a

normalized quality indicators (one per objective) and

stores it in the archive (removing the dominated ones)

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Michael Morin

Methodology – Solving a multi-criteria OSPV problem

Lexicographic Ant Search • As we did in the PAS algorithm

– LAS stores non dominated search plans

• However, a different update process guides the search

• LAS uses

– a permutation of criteria representing the current priority order of the criteria

– an archive AL of search plans

• LAS updates the pheromone values when a search plan is lexicographically optimal or equal with respect to AL

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Michael Morin

Methodology – Solving a multi-criteria OSPV problem

Lexicographic Ant Search

• Lexicographically?

– Let Perm = [COS, TPL, CH]; Let f = COS; Let P be a search plan;

a. If P is better on f than at least one plan in AL or if AL is empty then P is lexicographically optimal with respect to AL; Stop;

b. If P is equal on f to at least one plan in AL then P is lexicographically equal with respect to AL; If f is the last criterion in Perm then Stop; Let f be equal to the next criterion in P; Goto a;

c. P is not lexicographically optimal or equal with resp. to AL; Stop;

Note: P updates the trails according to f when it is lexicographically optimal or equal with respect to AL 28

Michael Morin

• CPLEX vs Elitist Ant Search

– Randomly generated grid environments

Results

The experiment

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Michael Morin

Results – Results obtained by Elitist AS and ILOG CPLEX on single criterion OSPV instances

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The best results obtained by CPLEX (5 configurations)

and the average results obtained by Elitist Ant Search (10 runs)

prdm = 0.001

C = 1000

Michael Morin

Results

Results for a multi-criteria OSPV

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The Pareto Front for Pareto AS The Pareto Front for Lexicographic AS

• The environment is a square grid where N = T = 16 and Q

= 5 (15 minutes on environment 2).

• The search object is moving and the detection is imperfect.

Michael Morin

Conclusion

• Optimal Searcher Path with Visibility

– extends the classical OSP problem to use inter-

region visibility.

– A multi-criteria extension

• Stochastic local search solving techniques

– Elitist Ant Search (single criterion)

– Pareto and Lexicographic Ant Search (multi-

criteria)

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Michael Morin

Discussion

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by Yann Arthus-Bertrand

Unknown author

See (Morin, 2012) for more details on the formalisms and algorithms presented in this talk..

Michael Morin

References • S. Benkoski, J.R. Weisinger, and M.G. Monticino, "A Survey of the Search

Theory Literature," Naval Research Logistics, vol. 38, pp. 469-494, 1991.

• M. Dorigo and C. Blum, "Ant Colony Optimization Theory: A Survey," Theoretical Computer Science, vol. 344, no. 2-3, pp. 243-278, 2005.

• K.F. Doerner, W.J. Gutjahr, R.F. Hartl, C. Strauss, and C. Stummer, "Nature-Inspired Metaheuristics for Multiobjective Activity Crashing," Omega, vol. 36, no. 6, pp. 1019-1037, 2008.

• B.O. Koopman, Search and screening, Operations Evaluation Group, Office of the Chief of Naval Operations, Navy Department, 1946.

• M. Morin, "Multi-Criteria Path Planning with Terrain Visibility Constraints: The Optimal Searcher Path Problem with Visibility," Université Laval, Québec, QC, Canada, Master's thesis 2010.

• E.-G. Talbi, Metaheuristics: From Design to Implementation.: John Wiley & Sons, 2009.

• K.E. Trummel and J.R. Weisinger, "The Complexity of the Optimal Searcher Path Problem," Operations Research, vol. 34, no. 2, pp. 324-327, 1986.

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