Execution of arithmetic operations involving the second-order fuzzy numbers

In real operating conditions of complex systems, random changes in their possible states occur in the course of their operation. The traditional approach to describing such systems uses Markov models. However, the real non-deterministic mechanism that controls the duration of the system's stay in each of its possible states predetermines the insufficient adequacy of the models obtained in this case. This circumstance makes it expedient to consider models that are more general than Markov ones. In addition, when choosing such models, one should take into account the fundamental often manifested feature of the statistical material actually used in the processing of an array of observations, their small sample. All this, taken together, makes it relevant to study the possibility of developing less demanding, tolerant models of the behavior of complex systems. A method for the analysis of systems described under conditions of initial data uncertainty by semi-Markov models is proposed. The main approaches to the description of this uncertainty are considered: probabilistic, fuzzy, and bi-fuzzy. A procedure has been developed for determining the membership functions of fuzzy numbers based on the results of real data processing. Next, the following tasks are solved sequentially. First, the vector of stationary state probabilities of the Markov chain embedded in the semi-Markov process is found. Then, a set of expected values for the duration of the system's stay in each state before leaving it is determined, after which the required probability distribution of the system states is calculated. The proposed method has been developed to solve the problem in the case when the parameters of the membership functions of fuzzy initial data cannot be clearly estimated under conditions of a small sample

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“…The average values of fuzzy numbers a 0 , b 0 , c 0 , a 1 , b 1 , c 1 are determined by the relations [16]:…”

Section: Mathematicsmentioning

confidence: 99%

Analysis of semi-Markov systems with fuzzy initial data

et al. 2022

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“…, ,..., , which minimizes the integrated criteria, which is to be built as follows. By using an analytical expression for the objective function, as well as the membership function of the problem's fuzzy parameters, based on the rules of operations over fuzzy numbers [15,16], we find the membership function of the fuzzy value of the objective function of the problem μ f X A , .…”

Section: A Formal Description Of the Computational Scheme Of Problmentioning

confidence: 99%

“…In this situation, the canonical transport problem is transformed to the following: it is required to find a trans-portation plan X x ij = ( ) , minimizing (15) and satisfying constraints (12) to (14). The resulting problem is solved by the standard method of potentials.…”

Section: A Formal Description Of the Computational Scheme Of Problmentioning

confidence: 99%

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Universal method for solving optimization problems under the conditions of uncertainty in the initial data

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Sukhomlyn

et al. 2021

EEJET

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This paper proposes a method to solve a mathematical programming problem under the conditions of uncertainty in the original data. The structural basis of the proposed method for solving optimization problems under the conditions of uncertainty is the function of criterion value distribution, which depends on the type of uncertainty and the values of the problem’s uncertain variables. In the case where independent variables are random values, this function then is the conventional theoretical-probabilistic density of the distribution of the random criterion value; if the variables are fuzzy numbers, it is then a membership function of the fuzzy criterion value. The proposed method, for the case where uncertainty is described in the terms of a fuzzy set theory, is implemented using the following two-step procedure. In the first stage, using the membership functions of the fuzzy values of criterion parameters, the values for these parameters are set to be equal to the modal, which are fitted in the analytical expression for the objective function. The resulting deterministic problem is solved. The second stage implies solving the problem by minimizing the comprehensive criterion, which is built as follows. By using an analytical expression for the objective function, as well as the membership function of the problem’s fuzzy parameters, applying the rules for operations over fuzzy numbers, one finds a membership function of the criterion’s fuzzy value. Next, one calculates a measure of the compactness of the resulting membership function of the fuzzy value of the problem’s objective function whose numerical value defines the first component of the integrated criterion. The second component is the rate of deviation of the desired solution to the problem from the previously received modal one. Absolutely similarly designed is the computational procedure for the case where uncertainty is described in the terms of a probability theory. Thus, the proposed method for solving optimization problems is universal in relation to the nature of the uncertainty in the original data. An important advantage of the proposed method is the ability to use it when solving any problem of mathematical programming under the conditions of fuzzily assigned original data, regardless of its nature, structure, and type

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Execution of arithmetic operations involving the second-order fuzzy numbers

Cited by 2 publications

References 15 publications

Analysis of semi-Markov systems with fuzzy initial data

Analysis of semi-Markov systems with fuzzy initial data

Universal method for solving optimization problems under the conditions of uncertainty in the initial data

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