n. bianchessi ([email protected]) g. righini ([email protected])

A Mathematical Programming Algorithm for

Planning and Scheduling an Earth Observing SAR Constellation

IWPSS 2006, Baltimore

N. Bianchessi ([email protected])G. Righini ([email protected])

Dipartimento di Tecnologie dell’Informazione Università degli Studi di Milano

IWPSS 2006 - October 22-25, 2006, Baltimore, USA 2

Earth Observation Satellites (EOSs)

CustomersImage Programming and Processing Center

Earth observation requests

Processed images

Ground stations

Satellites constellation

Daily plan

Images observed

Planning and Scheduling Problem


Outline

Literature Overview Problem

Description Formulation

Lagrangean Relaxation Almost Feasible Solutions Computational Results Conclusions Future Developments


Literature Overview

S.A. Harrison, M.E. Price, Task scheduling for satellite based imagery, in Proc. Eighteenth Workshop of the UK Planning and Scheduling Special Interest Group, University of Salford, Uk, 1999, pp 64-78.

A. Globus, J. Crawford, J. Lohn and A. Pryor, Scheduling Earth observing satellites with evolutionary algorithms, in Proceedings of the International Conference on Space Mission Challenges for Information Technology, Pasadena, California, 2003.

J. Frank, A. Jonsson, R. Morris, D.E. Smith, Planning and scheduling for fleets of Earth observing satellites, NASA Ames Research Center, 2001.

N. Bianchessi, V. Piuri, G. Righini, M. Roveri, G. Laneve, A. Zigrino, An optimization approach to the planning of Earth observing satellites, in Proceedings of the Fourth International Workshop on Planning and Scheduling for Space (ESA - ESOC). Darmstadt, Germany, 2004.

S. DeFlorio, T. Neff, T. Zehetbauer, Optimal operations planning for sar satellite constellations in low Earth orbit, in Proceedings of the Sixth International Symposium on Reducing the Costs of Spacecraft Ground Systems and Operations (ESA - ESOC). Darmstadt, Germany, 2005.

A. Globus, J. Crawford, J. Lohn, A. Pryor, A comparison of techniques for scheduling Earth observing satellites, in Proceedings of the Sixteenth Innovative Applications of Artificial Intelligence Conference (IAAA-04), American Association for Artificial Intelligence, San Jose, California, 2004.

N. Bianchessi, Planning and Scheduling Problems for Earth Observation Satellites: Models and Algorithms, Ph.D. Dissertation, Dipartimento di Tecnologie dell’Informazione, Università degli Studi di Milano, 2006.


Problem Description

These figures are taken from (Lemaître et al., 2002)

COSMO-SkyMed (ASI): 4 satellites equipped with SAR (Synthetic Aperture Radar) instruments performing about 15 orbits per day.

Requests acquisitions: images (swaths); Data Take Opportunities (DTOs) of given duration; transition time.

Satellite setup: two orientations: right-looking and left-looking. SAR instrument: look-angle, operating mode.

Memory device of given capacity. Satellite-station connections for transmission: Down Link

Opportunities (DLOs). Constraints on energy consumption (operational profiles).


Problem Description Operational Profile Constraints

There are NARROWFIELD (NF), WIDEFIELD (WF) acquisitions. Nominal profiles constraints:

they apply to every time window 1 orbit large; the time spent to acquire WF images cannot be greater than , and the

number of NF acquired images cannot be greater than ; Peak profiles:

for a time interval 1 orbit large (peak orbit), the overall acquisition time cannot exceed 4 (NF acquisitions are converted through a factor s.t. = ).

time

+

2( + )


Operational profiles

2τ /δ 4τ /δ σ= τ /δN. of NF images

Tim

e in

WF

m

od

e

4τ

2τ

τ

Peak orbits


Activity-time graph

O 1 2 3 4 5 d

(b1 + t15) a5: the satellite can take DTO 5 after DTO 1

time

1

23 5

4

Task-on-arc formulation (see Desaulniers et al. 1998). An acyclic di-graph Gk=(Vk, Ak) for each satellite k K. Vertex = state (i.e. time and set-up). Profit and resource consumption for each arc.

IWPSS 2006 - October 22-25, 2006, Baltimore, USA

Resource Consumption Operational profiles are checked only in a discrete number Z

of orbits. Ti

z: time spent in acquiring WF images up to node i;

Siz: # of NF images taken up to node i;

Piz: additional time spent up to node i.

additional amount of timeavailable for acquisitionsin a peak orbit

8


Resource Consumption Memory:

with each arc (i,j) we associate a consumption mij;

Qi:: amount of memory consumed up to node i;

R.E.F.: Qj = max {0, Qi + mij}.

Memory is regenerated (mij < 0) on arcs corresponding to

downlink operations. Feasibility conditions:

node i, z {1,…Z}, Qi Q, Tiz T, Si

z S, Piz P.

8


Mathematical Programming Model

A SPPRC k K

9


For each given > 0, the Lagrangean relaxation LR() is:

where for eack kK, zk is the optimal value of the following SPPRC:

Lagrangean Relaxation

10


Dynamic Programming

d

i

vPSTQ iiii ,,,,

)1()1()1()1()1()1( ,,,, vPSTQL iiii

)2()2()2()2()2()2( ,,,, vPSTQL iiii

)2()1()2()2()2()2()1()1()1()1()2()1( ,,,,,, vvPSTQPSTQLL iiiiiiii

Dominance:

“Efficient” labels Pareto-optimal set of paths to extend

...

...

...

*,,,, vPSTQ dddd

Optimal path

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Almost Feasible Solutions

Definition: infeasible solutions for the real problem; a subset of DTOs to acquire that can be used as a

guide by a heuristic. Procedure:

solve the K subproblems sequentially preventing multiple acquisitions of an image (consider only nominal profiles constraints);

apply a post-processing algorithm to exploit peak orbits for each satellite.

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Computational ResultsInstances

Instances from Alenia Spazio:

The resource consumptions are not monotone because of downlink operations and fluctuations in the workload.

Computing time refers to a 1.6 GHz Intel Pentium IV PC.

14


0

500

1000

1500

2000

2500

3000

1 2 3 4 5

instances

du

al b

ou

nd

s Serie1

Serie2

Serie3

Serie4

LR(1)()

LR(2)()

LR(3)()

LR(4)()

Computational results

Upper bounds from different relaxations (time-out = 6 hours): LR(1)(): only memory; LR(2)(): only operational profiles with Z=1 (3 resources); LR(3)(): like LR(2)(), but each orbit can be peak (1 resource); LR(4)(): WF images are combined linearly with NF images in

evaluating nominal profiles, Z=1 (2 resources);

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Computational ResultsLower bounds

The time needed to compute the almost-feasible solutions for instances 1 and 2 was less than 2 minutes.

This approach represents a valid alternative for relatively small instances.

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Computational Results

Due to the size of the considered problem instances, only for the relaxation LR(3)() we have always been able to approximate the optimal value of the associated Lagrangean dual.

LR(3)() allows each orbit to be a peak orbit; thus, with respect to the real operational profile constraints, the admissible workload increases by a factor of about 2.

16

Primal-dual gap


Conclusions

We have addressed the planning and scheduling problem with: multiple satellites; multiple orbits; very large scale instances.

Upper bounds have been computed through a multi-commodity flow formulation and Lagrangean relaxation. For small instances, the upper bounds are a significant benchmark for the heuristic solution.

Lower bounds were computed by an existing constructive algorithm, guided by the structure of the solutions of the Lagrangean relaxation. Almost-feasible solutions can guide the heuristic algorithm to discover better solutions than those computed from scratch.

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Future Developments

Tighter bounds: alternative problem relaxations, exploiting the special

structure of the time-activity digraph; exact algorithms to solve them: bi-directional bounded

dynamic programming and decremental state space relaxation (Righini and Salani, Discrete Optimization, 2006).

Better heuristics: exploitation of the structure of almost-feasible

solutions; local search.

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n. bianchessi ([email protected]) g. righini ([email protected])

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