Summary
A new column enumeration algorithm for solving the Set-Partitioning Problem is presented. It is not based on the staircase form of the coefficient matrix. Rather, it uses a preordering of the variables with respect to their cost of covering one row that is a supposition of a new strong lower bound concept. The enumeration process itself is organized similar to a general branch-and-bound concept. The performance of the algorithm is evaluated on the basis of a systematic comparison with different variants of the wellknown algorithms by Pierce and Garfinkel-Nemhauser. The computational experiences indicate that the new algorithm is superior for problems with moderatly dense coefficient matrices.
Zusammenfassung
In diesem Artikel wird ein neuer spaltenenumerierender Algorithmus zur Lösung des Set-Partitioning-Problems beschrieben, der nicht die übliche treppenförmige Anordnung der Koeffizientenmatrix benutzt. Das Verfahren geht vielmehr von einer Sortierung der Variablen nach aufsteigenden Kosten zur Bedeckung einer Zeile aus. Unter der Voraussetzung einer solchen Ordnung läßt sich dann eine scharfe untere Schranke berechnen. Die Enumeration selber wird ähnlich wie in einem generellen Branch-and-Bound-Verfahren vollzogen. Die Güte dieses Algorithmus wird auf der Grundlage eines systematischen Vergleichs mit verschiedenen Varianten der bekannten Verfahren von Pierce und Garfinkel-Nemhauser überprüft. Die Ergebnisse zeigen, daß der neue Algorithmus für Probleme mit mittlerer Dichte der Koeffizientenmatrix überlegen ist.
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Albers, S. Implicit enumeration algorithms for the Set-Partitioning Problem. OR Spektrum 2, 23–32 (1980). https://doi.org/10.1007/BF01720155
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DOI: https://doi.org/10.1007/BF01720155