Abstract
In this paper we define a non-deterministic dynamic neighborhood model. As a special case, a linear neighborhood model is considered. When a non-deterministic neighborhood model functions, it is possible to introduce a restriction on the number of active layers, which will allow the variation of the non-determinism of the model at each moment of time. We give the notion of the non-determinacy measure and prove that it has the properties of a probability measure. We formulate the problem of reachability with partially specified parameters, layer priorities, and the non-determinacy measure. An algorithm for solving the attainability problem for a neighborhood model with variable indeterminacy and layer priorities is presented. An example of its solution is shown, which shows that when the priorities are compared and the measure of non-determinism is used, the solution of the problem can be obtained more quickly than by a method that does not use priorities.
Highlights
A further direction in the development of the theory of neighborhood systems is fuzzy neighborhood models, which take into account the degree of fuzzy influence of neighborhood elements on each other [2]
In order to vary the non-determinacy of a neighborhood model, we introduce the notion of a non-determinacy measure
Where Wkx [t + 1] ∈ Rn×n, Wkx [t] ∈ Rn×n —matrices of the coefficients of the k-th layer by states at the moments t + 1 and t, respectively; Wkv [t] ∈ Rn×n —matrix of the coefficients of the k-th layer by inputs at the moment t; X[t + 1] ∈ Rn, X[t] ∈ Rn —state vector of the neighborhood system at the moments t + 1 and t, respectively; V[t] ∈ Rn —input vector at the moment t; D[t] ∈ Rm —random vector consisting of zeros and one unit in the position corresponding to the selected k-th layer, according to the equations of which the node states of the neighborhood model are recalculated at the moment t + 1
Summary
Examples of the application of the method in question are the linear and bilinear neighborhood models of the wastewater treatment plant, the neighborhood model of transport systems, the model of the aeration tank functioning, and the dynamic non-deterministic neighborhood model of the functioning of cement production given in References [1,2,5]. We give the definition of non-deterministic dynamic neighborhood models in which there is some element of randomness of the system’s functioning and which make it possible to model stochastic parallel processes inherent in a significant part of production systems. Let us consider the formulation, the algorithm, and an example of solving the problem of reaching a given state of a neighborhood model system with given layer priorities and a non-determinacy measure
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