An Algorithm for Constructing a Hereditarily Minimax Network with Predefined Vector of Node Degrees

Authors: Selin P.S., Tsurkov V.I., Gurchenkov A.A. Published: 14.02.2017
Published in issue: #1(70)/2017  
DOI: 10.18698/1812-3368-2017-1-43-58

Category: Mathematics | Chapter: Computational Mathematics  
Keywords: network optimization, transport type problems, minimax, minimax network, hereditarily minimax network, uniform network, transportation polyhedrons, network polyhedrons, predefined node degrees, fixed node degrees

In contrast to the classical transportation problem, where supply and demand points are known, and it is required to minimize the transportation cost, we consider a minimax criterion. In particular, a matrix with the minimal largest element is sought in the class of nonnegative matrices with given sums of row and column elements. In this case, the concept of the minimax criterion can be interpreted as follows. Suppose that the shipment time from a supply point to a demand point is proportional to the amount to be shipped. Then, the minimax is the minimal time required to transport the total amount. It is a common situation that the decision maker does not know the tariff coefficients. In other situations, they do not have any meaning, and neither do nonlinear tariff objective functions. In such cases, the minimax interpretation leads to an effective solution. For the classes of undirected networks with predefined vector of node degrees (transport and network polyhedrons) by using a characteristic functions the analytical formulas of calculating the minimax values expressed in terms of the vector coordinates and a nonnegative parameter are obtained. The minimax values determine the necessary and sufficient conditions under which the truncated polyhedrons are not empty sets. Finally, we obtained an algorithm for constructing a hereditarily-minimax network in network polyhedrons.


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