N. Bianchessi (bianchessi@dti.unimi.it) G. Righini (righini@dti.unimi.it)

1. A Mathematical Programming Algorithm for Planning and Scheduling an Earth Observing SAR ConstellationIWPSS 2006, Baltimore N. Bianchessi (bianchessi@dti.unimi.it) G. Righini (righini@dti.unimi.it) Dipartimento di Tecnologie dell’Informazione Università degli Studi di Milano

2. Planning and Scheduling Problem Earth Observation Satellites (EOSs) Daily plan Ground stations Satellites constellation Images observed Processed images Image Programming and Processing Center Customers Earth observation requests IWPSS 2006 - October 22-25, 2006, Baltimore, USA

3. Outline • Literature Overview • Problem • Description • Formulation • Lagrangean Relaxation • Almost Feasible Solutions • Computational Results • Conclusions • Future Developments IWPSS 2006 - October 22-25, 2006, Baltimore, USA

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

5. Problem Description • 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). These figures are taken from (Lemaître et al., 2002) IWPSS 2006 - October 22-25, 2006, Baltimore, USA

6. 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.  = ). Problem Description Operational Profile Constraints 2( + )  +  IWPSS 2006 - October 22-25, 2006, Baltimore, USA

7. Operational profiles 4τ Time in WF mode 2τ Peak orbits τ σ= τ /δ 2τ/δ 4τ /δ N. of NF images IWPSS 2006 - October 22-25, 2006, Baltimore, USA

8. 1 4 2 3 5 time d 5 2 4 O 1 3 Activity-time graph • 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. (b1 + t15)  a5: the satellite can take DTO 5 after DTO 1 IWPSS 2006 - October 22-25, 2006, Baltimore, USA

9. Resource Consumption • Operational profiles are checked only in a discrete number Z of orbits. • Tiz: 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 time available for acquisitions in a peak orbit IWPSS 2006 - October 22-25, 2006, Baltimore, USA 8

10. 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,Siz  S,Piz  P. Resource Consumption IWPSS 2006 - October 22-25, 2006, Baltimore, USA 8

11. Mathematical Programming Model  A SPPRC k K IWPSS 2006 - October 22-25, 2006, Baltimore, USA 9

12. Lagrangean Relaxation For each given  > 0, the Lagrangean relaxation LR() is: where for eack kK, zk is the optimal value of the following SPPRC: IWPSS 2006 - October 22-25, 2006, Baltimore, USA 10

13. Optimal path Dominance: Dynamic Programming d ... i ... ... “Efficient” labels  Pareto-optimal set of paths to extend IWPSS 2006 - October 22-25, 2006, Baltimore, USA 11

14. 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. IWPSS 2006 - October 22-25, 2006, Baltimore, USA 13

15. 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. IWPSS 2006 - October 22-25, 2006, Baltimore, USA 14

16. 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); IWPSS 2006 - October 22-25, 2006, Baltimore, USA 15

17. Lower bounds Computational Results • 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 smallinstances. IWPSS 2006 - October 22-25, 2006, Baltimore, USA 16

18. Primal-dual gap 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. IWPSS 2006 - October 22-25, 2006, Baltimore, USA 16

19. 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. IWPSS 2006 - October 22-25, 2006, Baltimore, USA 17

20. 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. IWPSS 2006 - October 22-25, 2006, Baltimore, USA 18