Summary
The placement of telecommunication satellites in the geostationary orbit (GSO) gives rise to NP-hard optimization problems usually approached with iterative neighborhood (possibly tabu) search schemes. A typical iteration thereof consists in fixing an order for the satellites and determining their actual locations by linear programming. In such procedures it is crucial to efficiently solve the very large number of arising special linear programs. We describe those linear programs, characterize their duals as special network flow problems with one side constraint and then present an efficient network simplex method to solve them. Since these problems can be highly degenerate, we generalize Cunningham's concept of strongly feasible bases to our case and present a procedure based thereupon which prevents cycling. Computational experience with our algorithms substantiates our efficiency claims.
Zusammenfassung
Die Placierung von Telekommunikationssatelliten im geostationären Orbit (GSO) führt auf NP-schwierige Optimierungsprobleme, die z. B. mit Nachbarschafts-Suchverfahren, insbes. Tabuverfahren angegangen werden können. Ein typischer Schritt darin besteht in der Festlegung einer Reihenfolge der Satelliten und anschließender Placierung mit linearer Programmierung. Bei solchen Verfahren ist die effiziente Lösung der großen Anzahl auftretender spezieller linearer Programme ausschlaggebend. Wir beschreiben diese linearen Programme, charakterisieren deren Dualprobleme als Netzwerkflußaufgaben mit einer Nebenbedingung und geben dafür ein effizientes Simplex-Verfahren an. Da diese Aufgaben meist stark degeneriert sind, erweitern wir Cunninghams Begriff einer stark zulässigen Basis auf unseren Fall und geben ein darauf basierendes, Zyklen vermeidendes Lösungsverfahren an. Numerische Experimente belegen unsere Behauptungen zur Effizienz der Verfahren.
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Spälti, S.B., Liebling, T.M. Modeling the satellite placement problem as a network flow problem with one side constraint. OR Spektrum 13, 1–14 (1991). https://doi.org/10.1007/BF01719766
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DOI: https://doi.org/10.1007/BF01719766