Abstract
In this paper the author presents an algorithm of optimization for a special class of networks not having the Markov property. A definition of the class of networks under consideration and a formulation of the optimization problem are given. A conception of the algorithm is discussed and next the general and detailed flow diagrams of the algorithm are offered. The realization of the algorithm is illustrated with a simple example showing the process of execution of the tasks included in the algorithm. Some possibilities of applying the algorithm in allocation problems and nonlinear integer programming are presented.
The computer program in FORTRAN IV for the execution of the algorithm is enclosed.
Zusammenfassung
In dieser Arbeit wird ein Optimierungsalgorithmus für nichtmarkowsche Netze dargestellt. Es wird die Definition des betrachteten Netzes gegeben und ein Optimierungsproblem formuliert. Es werden das allgemeine Konzept und dann das Ablaufschema des Algorithmus dargestellt. Die Realisierung des Algorithmus wird an einem einfachen Beispiel erklärt. Es werden auch die Möglichkeiten der Anwendung des Verfahrens für Allokationsprobleme und für nichtlineare Programmierung erwähnt.
Das Programm ist in FORTRAN IV geschrieben. Es ermöglicht die Realisierung des Verfahrens auf einer EDV-Anlage.
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Abbreviations
- \(\mathop \wedge \limits_x\) :
-
universal quantifier “for eachx”
- \(\mathop \vee \limits_x\) :
-
existential quantifier “there is suchx”
- ¬:
-
negation
- ⋏:
-
conjunction
- u∈U :
-
u belongs toU
- \(\{ U_1 ,...U_w ,...U_{w_x } \}\) :
-
the set consisting of the elements\(U_1 ,...U_w ,...U_{w_x }\)
- Γ u :
-
the set of the graph nodesu a ∈U for whichu \(\vec R\) u a
- \(\Gamma _u^{ - 1}\) :
-
the set of the graph nodesu b ∈U for whichu b \(\vec R\) u
- ∪:
-
union of set
- ∩:
-
intersection of set
- ⊘:
-
the empty set
References
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Hu, T. C.: Revised Matrix Algorithms for Shortest Paths in a Network. I. SIAM15, 207 to 218 (1967).
Kaufman, A.: Graphs, dynamic programming and finite games. (English translation.) New York-London: 1967.
Osyczka, A.: Statical Optimization of the Machine Tool Gear Boxes. D. Sc. Thesis, Politechnika Krakowska, 1973.
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Osyczka, A. An algorithm of optimization for a special class of networks. Computing 16, 77–97 (1976). https://doi.org/10.1007/BF02241982
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DOI: https://doi.org/10.1007/BF02241982