A constructive algorithm for max–min paths problems on energy networks
Section snippets
Introduction and problem formulation
In this paper we consider the max–min paths problem on energy networks, which generalizes the classical combinatorial problems of the shortest and the longest paths in weighted directed graphs [2]. This max–min paths problem arose as an auxiliary problem within the study of a game theoretic version of the discrete control problem from [8], [9], [12], [13] and within the examination of optimal stationary strategies of players in cyclic games [3], [5], [12], [16].
This approach extends the results
An algorithm for solving the problem on acyclic networks
The formulated problem for acyclic networks has been studied in [8].
Let be a finite directed graph without directed cycles and a given sink vertex . The partition () of the vertex set of is given and the cost function on the edges is defined. We consider the dynamic -game on with a given starting position .
It is easy to observe that for fixed strategies of the players and the subgraph has the structure of a directed tree with sink
The main results for the problem on an arbitrary network
First of all we give an example which demonstrates that equality (1) may fail to hold. In Fig. 1 a certain network is given with starting position and final position , where the positions of the first player are represented by circles and the positions of the second player are represented by squares; values of the cost functions are given on the edges.
It is easy to observe thatThe following theorem gives conditions for the
An application of the algorithm for solving zero value cyclic games
In this section we show that the zero value ergodic cycle game can be regarded as a max–min paths problem and therefore the proposed algorithm can be used for determining the optimal strategies of the players in such special cyclic games.
At first we summarize the formulation of cyclic games and some necessary preliminary results.
Summary
The approach presented here extends the results of a cooperative treatment of the technology emission means (TEM) model which was introduced to model and simulate investment strategies within energy markets. The TEM model and its mathematical structure are related to those general control problems. In [7] suitable transformation principles are developed which lead to an algorithmic determination of Pareto optima. Furthermore it is obvious that the mathematical assumptions are not too
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