Summary
This paper treats five constrained shortest-route problems: 1) determining the shortest route when it is constrained to pass through a given set of specified nodes; 2) determining the shortest route when it is constrained to pass through a given set of specified nodes and the specified nodes are to be visited in a fixed order; 3) finding an optimal route for the travelling-salesman problem; 4) determining the shortest route throughK sets of specified nodes when at least one node of every set of specified nodes is to occur on the shortest route; and 5) finding the shortest route through the sets of specified nodes when at least one node of every set of specified nodes is to occur on the shortest route and the sets of specified nodes are to be visited in a fixed order. The functional equation technique of dynamic programming is employed to solve problems 1), 3), and 4), while problems 2) and 5) are solved through simpler algorithms. The methods are illustrated by examples.
Zusammenfassung
Es werden fünf Kürzeste-Wege-Probleme mit besonderen Bedingungen behandelt: 1) Bestimmung des kürzesten Weges unter der Bedingung, daß die in einer gegebenen Menge spezifizierten Knoten passiert werden müssen, 2) Bestimmung des kürzesten Weges, wobei die in einer gegebenen Menge spezifizierten Knoten in einer bestimmten Reihenfolge passiert werden müssen, 3) Ermittlung der optimalen Rundreise im Travelling-Salesman Problem, 4) Bestimmung des kürzesten Weges durchK Mengen von spezifizierten Knoten, wobei aus jeder Menge wenigstens ein Knoten passiert werden muß, und 5) Bestimmung des kürzesten Weges durch Mengen von spezifizierten Knoten, wobei aus jeder Menge wenigstens ein Knoten passiert und die Mengen in einer vorgegebenen Reihenfolge aufgesucht werden müssen. Zur Lösung der Probleme 1), 3) und 4) wird die Technik der dynamischen Optimierung angewandt, während die Probleme 2) und 5) mit einfacheren Algorithmen behandelt werden. Die Methoden werden an Beispielen erläutert.
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Bajaj, C.P. Some constrained shortest-route problems. Unternehmensforschung Operations Research 15, 287–301 (1971). https://doi.org/10.1007/BF01939836
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DOI: https://doi.org/10.1007/BF01939836