Abstract
In this paper, the functional equation technique of dynamic programming is applied to solve the problems of a) determining an optimal path from a given origin to a fixed destination when the path is subject to a given number of improvements, b) finding an optimal path from a given origin to an assigned destination by passing at least once through each node of a set of specified nodes when the path is subject to a given number of improvements, c) obtaining an optimal path from a given origin to a fixed destination by passing at least once through at least one node of each ofK sets of specified nodes when the path is subject to a given number of improvements.
Zusammenfassung
In dieser Arbeit wird die Funktionalgleichung des dynamischen Programmierens verwendet, um folgende drei Netzwerkprobleme zu lösen: a) Bestimmung eines optimalen Pfades von einem gegebenen Anfangs- zu einem gegebenen Endknoten, wenn längs einem Pfad eine gewisse Anzahl von Verbesserungen möglich sind. b) Wie a), wobei zusätzlich der Pfad mindestens einmal durch jeden Knoten einer spezifizierten Knotenmenge gehen soll. c) Wie a), wobei der Pfad durch mindestens einen Knoten in jeder vonK spezifizierten Knotenmenge gehen soll.
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References
Bajaj, C.P.: Some Constrained Shortest-Route Problems. Unternehmensforschung15, 1971, 287–301.
Bansal, S.P., andS. Kumar: Shortest Route Subject to Improvements Through Dynamic Programming. SCIMA1 (1), 1972, 95–104.
Bellman, R.E.: Dynamic Programming. Princeton, N.J., 1957.
—: On a Routing Problem. Quart. Appl. Maths.16, 1958, 87–90.
—: Dynamic Programming Treatment of the Travelling Salesman Problem. J. of ACM9, 1962, 61–63.
Bellman, R.E., andS. Dreyfus: Applied Dynamic Programming. Princeton, N.J., 1962.
Brucker, P.: A Decomposition Algorithm for Shortest Paths in a Network with many stronger connected components. ZOR18, 1974, 177–180.
Dantzig, G.B.: Discrete Variable Extremum Problems. OR5, 1957, 266–277.
Dreyfus, S.E.: An Appraisal of Some Shortest Path Algorithms. OR17 (3), 1969, 395–412.
Farbey, B.A., A.H. Land andJ.D. Murchland: The cascade Algorithm for finding all Shortest Distances in a Directed Graph. Management Science14 (1), 1967, 19–28.
Held, M., andR.M. Karp: A Dynamic Programming Approach to Sequencing Problems, J. Soc. Indust. Appl. Math.10, 1962, 196–210.
Ibaraki, T.: Algorithms for obtaining Shortest Paths Visiting Specified Nodes. SIAM Review15 (2), 1973, 309–317.
Kalaba, R.: On Some Communication Network Problems. Combinatorial Anal. Proc. Symp. Appl. Math.10, 1960, 261–280.
Mills, G.: A Decomposition Algorithm for the Shortest-Route problem. OR14, 1966, 279–291.
Peart, R.M., P.H. Randolph andT.E. Bartlett: The shortest Route Problem. OR8, 1960, 866–868.
Pollack, M., andW. Wiebenson: Solution of the Shortest Route Problem — A review. OR8, 1960, 224–230.
Pollatschek andAvi-Itzhak: An Efficient Shortest Route Algorithm. Angewandte Informatik11, 1974.
Saxena, J.P., andS. Kumar: The Routing Problem withK Specified Nodes, OR14, 1966, 908–913.
Wagner, R.A.: A Shortest Path Algorithms for Edge-Sparse Graphs, J. of ACM23 (1), 1976, 50–57.
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Bajaj, C.P., Jain, J.P. Some optimal path problems subject to improvements. Zeitschrift für Operations Research 22, 115–129 (1978). https://doi.org/10.1007/BF01917653
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DOI: https://doi.org/10.1007/BF01917653