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Equalities in transportation problems and characterizations of optimal solutions

✍ Scribed by Kenneth O. Kortanek; Maretsugu Yamasaki

Publisher
John Wiley and Sons
Year
1980
Tongue
English
Weight
304 KB
Volume
27
Category
Article
ISSN
0894-069X

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✦ Synopsis

Abstract

This paper considers the classical finite linear transportation Problem (I) and two relaxations, (II) and (III), of it based on papers by Kantorovich and Rubinstein, and Kretschmer. Pseudo‐metric type conditions on the cost matrix are given under which Problems (I) and (II) have common optimal value, and a proper subset of these conditions is sufficient for Problems (II) and (III) to have common optimal value. The relationships between the three problems provide a proof of Kantorovich's original characterization of optimal solutions to the standard transportation problem having as many origins as destinations. The result are extended to problems having cost matrices which are nonnegative row‐column equivalent.

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Fenchel duality and smoothness of soluti
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A linear state and control constrained problem arising in optimal routing in communication networks is investigated by Fenchel duality methods. The problem reduces to a dual program having a particularly simple solution.