Multi-objective Optimization of Earth Observing Satellite Missions

MULTI-OBJECTIVE OPTIMIZATION OF EARTH OBSERVING SATELLITE

MISSIONSBy

Panwadee Tangpattanakul

Thesis supervisorsPierre Lopez

Nicolas Jozefowiez

26th September 2013

AGILE EARTH OBSERVING SATELLITE (AGILE EOS)

2

MissionObtain photographs of the Earth surface to satisfy users

requirements

Satellite direction

Captured photographCandidate photographs

Earth surface

WORK OVERVIEW

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Observation scheduling problem of agile EOS&

Multi-objective optimization

Biased random key genetic algorithm

(BRKGA)

Indicator-based multi-objective

local search(IBMOLS)

OUTLINE Problem statement Multi-objective optimization Biased random key genetic algorithm Indicator-based multi-objective local search BRKGA vs. IBMOLS Conclusions & perspectives

4

PROBLEM STATEMENT

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PROPERTIES OF AGILE EOSEx. PLEIADES CNES (French Center for Space Studies)

• One fixed camera on-board• The whole satellite can move in 3 degrees of freedom

• Problem data’s description was proposed (Verfaillie et al., 2002)

• Several methods were used to solve the problem, e.g. greedy algorithm, dynamic programming, constraint programming, local search (Lemaître et al., 2002)

• ROADEF 2003 challenge• Simulated annealing (Kuipers, 2003)• Tabu search (Cordeau & Laporte, 2005)

Literature of PLEIADES observation scheduling problem

7

MULTI-USER OBSERVATION SCHEDULING PROBLEM FOR AGILE EOS

8

User 1 User 2 User n

Select

Schedule

&

Ground station

Requests

Acquisitions

Satellitecapacitylimitation

Profit

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Bataille et al., 1999Optimize 2 objectives: fairness and efficiency Use 3 strategies:

Give priority to fairness Give priority to efficiencyConsider 2 objectives, but search for 1 solution

Gabrel & Vanderpooten, 2002Optimize 3 objectives: maximize the number of shots,

maximize the total profit, and minimize the satellite use

Select a satisfactory efficient path in a graph without circuit

Literature

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Bianchessi et al., 2007 Multiple satellites, multiple orbits, and multiple users 3 phases

Select users depending on priority Select requests from the subset of users Allocate the remaining capacities of the satellites

between all users A single objective:

Weighted sum of the normalized utilities of the users Tabu search

Literature


11

The obtained sequence has to optimize 2 objectives:

Sequence (SA1, SA2,…, SAn)

Maximize the total profit

Minimize the maximum profit difference between users ensure fairness of resource sharing

𝑓 1=∑𝑖=1

𝑛

𝑝𝑖

𝑓 2= max(𝑖 , 𝑗 )∈ ⟦1 ,𝑛𝑢⟧ 2

¿ 𝑖≠ 𝑗

|𝑢𝑖−𝑢 𝑗|

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Constraints Time windows

No overlapping acquisitions


Acquisition

Possible starting time

Duration time

time

time

Acquisition1

Acquisition2

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Constraints Sufficient transition times


Earth surface

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Constraints Two acquisitions may be exclusive

Two acquisition may be linked


time

Acqusition2E

Acqusition1E

time

Acqusition2L

Acqusition1L


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Request from

Time

User 2

User 1

Acq3-1L

Acq4Acq3-2L

Acq2-2E

Acq1 Acq2-1E

P3 = 20P4 = 10

P1 = 4

P2-1 = 5

P2-2 = 5

Solution 1: (Acq3-1L & Acq3-2L & Acq4)Total profit = 30, Max profit difference = 30Solution 2: (Acq1 & Acq2-1E & Acq4)Total profit = 19, Max profit difference = 1Solution 3: (Acq3-1L & Acq3-2L & Acq2-1E)Total profit = 25, Max profit difference = 15

Max profit difference

Total profit

MULTI-OBJECTIVE OPTIMIZATION

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MULTI-OBJECTIVE PROBLEM

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with: n ≥ 2: number of objectives F = (f1, f2,…,fn): vector of functions to

optimize Ω: set of feasible solutions y = F(Ω): objective space

The considered problem needs to maximize f1 (x), minimize f2

(x)

A solution x dominates a solution y iff f1 (x) and f2 (x)

or f1 (x) and f2 (x)

PARETO DOMINANCE & HYPERVOLUME

18A

C

E

BD

f1(x)

f2(x) Reference point

A

C

E

f1 (x)

f2 (x)

IMPLEMENTATION Conduct on realistic instances 4-user modified ROADEF 2003 challenge instances (Subset A) The instance sizes are between 4 to 1,068

acquisitions Implement via C++ language 10 runs/instance are tested Reference point of the hypervolume

The worst values of both objectives19

BIASED RANDOM KEY GENETIC ALGORITHM

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Biased random key genetic algorithm Gonçalves et al. (2002)

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Random key (Bean, 1994)Chromosomes are represented as a vector of

randomly generated real numbers in the interval [0,1].

EncodingDecision variable

Decoding

Chromosome

Solution

ENCODING

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Acq1 Acq2-1E

Acq2-2E

Acq3-1L

Acq3-2L

Acq4

0.6984 0.9939 0.6485 0.2509 0.7593 0.4236Random key chromosome

Request from

Time

User 2

User 1

Acq3-1L

Acq4Acq3-2L

Acq2-2E

Acq1 Acq2-1E

Biased random key genetic algorithm Gonçalves et al. (2002)

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Random key (Bean, 1994)Chromosomes are represented as a vector of

randomly generated real numbers in the interval [0,1].

EncodingDecision variable

Decoding

Chromosome

Solution

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Random keychromosome

Ordered list ofacquisitions

Multi-user observation scheduling problem

Sequence ofselected acquisitions

Priority computation Assign the acquisition, which satisfies all constraints

DECODING

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• Basic decoding (D1)• The priority is equal to its gene value

Priorityj = genej

• The priority to assign each acquisition in the sequenceAcq2-1E, Acq3-2L, Acq1, Acq2-2E, Acq4, Acq3-1L

Acq1 Acq2-1E

Acq2-2E

Acq3-1L

Acq3-2L

Acq4

0.6984 0.9939 0.6485 0.2509 0.7593 0.4236Random key chromosome

Example

PRIORITY COMPUTATION

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• Decoding of gene value and ideal priority combination (D2)• The priority is

Priorityj = ideal priority * f(genej)

• Concept of ideal priority• The acquisition, which has the earliest possible starting time,

should be selected firstly and be scheduled in the beginning of the solution sequence


𝐼𝑑𝑒𝑎𝑙𝑝𝑟𝑖𝑜𝑟𝑖𝑡𝑦 𝑗=𝑇𝑚𝑎𝑥𝐿−𝑇𝑚𝑖𝑛 𝑗

𝑇𝑚𝑎𝑥𝐿

• The ideal priority values of Acq3-1L = Acq3-2L > Acq1 > Acq2-1E > Acq2-2E > Acq4 27

Request from

Time

User 2

User 1

Acq3-1L

Acq4Acq3-2L

Acq2-2E

Acq1 Acq2-1E

Example


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Random keychromosome

Multi-user observation scheduling problem

Sequence ofselected acquisitions

Priority computation Assign the acquisition, which satisfies all constraints

DECODING

Ordered list ofacquisitions

BRKGA GENERATION

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POPULATION

Generation i

ELITE

CROSSOVEROFFSPRING

MUTANT

Generation i+1

ELITE

NON-ELITE

X

ADAPTATION TO MULTI-OBJECTIVE Q: How to select the elite set? A: Borrow selection methods from efficient MOEAs, e.g. NSGA-II, SMS-EMOA, IBEA

Q: Could chromosome be associated to several nondominated solutions? A: Use a multiple decoding

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ELITE SET SELECTIONS

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Ref: Deb et al. (2002)

Fast nondominated sorting and crowding distance assignment

f2 (x)

f1 (x)

Rank1Rank2Rank3


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Ref: Deb et al. (2002)

Fast nondominated sorting and crowding distance assignment

Rank 1 Nondominated solutions

f1 (x)

f2 (x)

i-1i i+1

solutions in Rank 1


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Ref: Beume et al. (2007)

S-metric selection evolutionary multiobjective optimization algorithm (SMS-EMOA)

Rank 1 Nondominated solutions

solutions in Rank 1

f1 (x)

f2 (x)


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Ref: Zitzler et al. (2004)

Indicator-based evolutionary algorithm based on the hypervolume concept (IBEA)

f1 (x) f1 (x)

f2 (x) f2 (x)

IHD(B,A) > 0

IHD(A,B) > 0

A

B

A

BIHD(A,B) = - IHD(B,A) > 0

Binary tournament on all individuals in P Compute the fitness

ADAPTATION TO MULTI-OBJECTIVE Q: How to select the elite set? A: Borrow selection methods from efficient MOEAs, e.g. NSGA-II, SMS-EMOA, IBEA

Q: Could chromosome be associated to several nondominated solutions? A: Use a multiple decoding

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MULTIPLE DECODING

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• Hybrid decoding (HD)Chromosome

Basic decoding(D1)

Decoding of gene value and ideal priority combination

(D2)

Solution 1 Solution 2

?

HYBRID DECODING

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• Elite set management – Method 1 (M1)

D1Population

Elite setPreferred chromosome

sD2chromosome

solution 1 solution 2

Dominance relation

Dominant solution

HYBRID DECODING

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D1Population

Elite setPreferred chromosome

sD2chromosome

solution 1 solution 2

Select randomly

Selected solution

HYBRID DECODING

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D1

Population

Elite setPreferred chromosom

es

D2

chromosome

solution 1

solution 2

HYBRID DECODING

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D1

Population

D2

chromosome

solution 1

solution 2

Elite setPreferred chr.

Preferred chr.

IMPLEMENTATION

In each iteration, Nondominated solutions are stored in an archive

set A If at least one solution from P can dominate some

solutions in the archive set A Update the archive set A 41

Parameter ValuePopulation size (p) p = n,

where n is the length of the chromosome

Elite set size (pe) 0.15p ≤ pe Mutant set size (pm) pm = 0.30pProb. elite inherent (ρe)

ρe = 0.6

Stopping criteria Number of iterations since the last archive

improvement Value = 50

Computation time limitation Depending on the instances size

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IMPLEMENTATION

COMPARISON OF TWO DECODINGS (D1, D2) AND THREE ELITE SET SELECTIONS (S1, S2, AND S3)

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For all instances• All methods obtain

similar results• Each method has the

advantage in different instances

S1, S2, and S3

Large-size instances• D1 obtains better median value• D2 can reduce the range

COMPARISON OF TWO DECODINGS (D1, D2) AND THREE ELITE SET SELECTIONS (S1, S2, AND S3)

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Medium-size instances• D2 obtains better results

• Median value• Std. deviation• Computation time

D1, D2

HYBRID DECODING – COMPARISON OF ELITE SET MANAGEMENT

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M1 M2 M3Hypervolume Median O O O

Standard deviation O O O

Computation timeO

Time outLarge instances

(S3)

Time outSmall instances

(S1, S2)

Since M1 spends less computation time for all elite set selection methods, its results will be used to compare with the results from the two single decoding

BRKGA – COMPARISON OF TWO SINGLE DECODING AND HYBRID DECODING

46

• HD obtains results close to the best one, when comparing the two single decoding

• HD can preserve the advantages of each single decoding• D1 vs. HD, HD can reduce the range• D2 vs. HD, HD can avoid to entrap in local optima

INDICATOR-BASED MULTI-OBJECTIVE LOCAL SEARCH

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INDICATOR-BASED MULTI-OBJECTIVE LOCAL SEARCH (BASSEUR AND BURKE, 2007)

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Basic Local search&

Binary indicator from IBEA

Each iteration

Update the approximate Pareto front

Initial population generation

Fitness computation

Local search step

STRATEGIES

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Initial population generationFirst iteration Random generation Using data of problem instancesOther iterations Random generation Perturbation

Neighborhood structure and dynamic stopping value Insert, remove, and replace & Stop value = 10 Insert, remove & Stop value = 10 Insert, remove & Stop value = 50

STRATEGIES

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Feasibility checking Method 1 Method 2

Stopping criterion Dynamic stopping Fixed computation time Fixed number of visited neighbors


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Each iteration



Fitness computation

Local search step

INITIAL POPULATION GENERATION

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First iteration - Random generation For all candidate acquisitions are considered depending on a random order

Considered acquisition

Check all constraints

Satisfy

Assign to the

sequence

Consider the next acquisition

Yes

No

INITIAL POPULATION GENERATION

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Other iterations – Perturbation Original individual selection Select randomly Element removing Repeat

Select the removing position Remove the acquisition Check the constraint of linked

acquisitions Until nmodify ≤ ¾ noriginal

Approximate Pareto front

A B C D

Individualselection

Positionselection

e.g. 2A C D

B


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Each iteration



Fitness computation

Local search step

FITNESS COMPUTATION

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Use the indicator based on the hypervolume concept

Perform binary tournaments for all individuals in P

f1 (x) f1 (x)

f2 (x) f2 (x)

IHD(B,A) > 0

IHD(A,B) > 0

A

B

A

BIHD(A,B) = - IHD(B,A) > 0

FITNESS COMPUTATION

56

Use the indicator based on the hypervolume concept

Perform binary tournaments for all individuals in P

Fitness value is


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Each iteration



Fitness computation

Local search step

bin

For all individuals x in P

LOCAL SEARCH STEP

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xy*

Population P

Worst solution

4 types of move Insert an acquisition i Remove an acquisition i Insert linked acquisitions i and j Remove linked acquisitions i and j

NEIGHBORHOOD STRUCTURE

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Feasibility checking – Method 2

FEASIBILITY CHECKING FOR INSERTION

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sa1 sa2 sa3 sa4 sa5

Insertion position

time

sa1 sa2 sa3 sa4 sa5

time

Acq k

The acquisitions, which stay behind the insertion position, are moved to the back as late as possible

BRKGA VS. IBMOLS

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BRKGA VS. IBMOLS

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Medium instances Large instances

IBMOLS obtains better results than BRKGA for all instances • Median value• Standard deviation

BRKGA VS. IBMOLS (IMPROVEMENT)

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0 2 4 6 8 10 12 14 16 182.0E+172.2E+172.4E+172.6E+172.8E+173.0E+173.2E+173.4E+17

Instance 68_12_106

Time (s)

0 50 100 150 200 250 3002.0E+162.5E+163.0E+163.5E+164.0E+164.5E+16

Instance 77_40_147

Time (s)

0 50 100 150 200 2502.0E+16

2.2E+17

4.2E+17

6.2E+17

8.2E+17Instance 218_39_295

Time (s)0 1000 2000 3000 4000

2.0E+16

5.2E+17

1.0E+18

1.5E+18

2.0E+18Instance 336_55_483

Time (s)

0 1002003004005006007008002.0E+16

5.2E+17

1.0E+18

1.5E+18

2.0E+18Instance 375_63_534

Time (s)0 500 1000 1500 2000 2500

2.0E+161.2E+172.2E+173.2E+174.2E+175.2E+176.2E+17

Instance 150_87_342

Time (s)

Medium instances Large instances

For medium instances, IBMOLS spends more computation time• From the start, IBMOLS obtains better hypervolume valueFor large instances, IBMOLS spends less computation time

BRKGAIBMOLS

CONCLUSIONS AND PERSPECTIVES

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CONCLUSIONS The multi-user observation scheduling problem of

an agile EOS is solved in this work. The obtained sequences have to optimize two

objectives and also satisfy all constraints. BRKGA and IBMOLS are applied to solve the

problem. All parameters of each algorithm are tested with

realistic instances. The results obtained from BRKGA and IBMOLS are

compared. IBMOLS converges faster and gives better results65

PERSPECTIVES Short term further works

BRKGA Modify the hybrid decoding

Each single objective may be re-defined to mainly consider one objective

Modify the elite set selection Other indicators can be used (e.g. from SPEA2)

IBMOLS Include other strategies for the initial population

generation by using data of the problem instances The number of removed elements in the

perturbation can also be modified.66

PERSPECTIVES Long term further works

BRKGA Apply more advanced decoding methods

e.g. consider the decoder as a full multi-objective problem

IBMOLS Use other perturbation rules

e.g. insert some feasible acquisitions for replacing the removed elements

Use other neighborhood structuresScheduling problem Other objective functions can be included

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THANK YOU FOR YOUR ATTENTION

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Documents

Multi-objective Optimization of Earth Observing Satellite Missions