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Many concepts of problem solving theory are better understood in an abstract algebraic framework which also applies in automata theory. Because many systems of practical interest fall outside the scope of linear theory, it is desirable to enlarge as much as possible the class of systems for which a complete structure theory is available. The fuzzy system approach is presented as a basis for the design of systems far superior in artificial intelligence to those we can conceive today. The concepts of controllability, observability and minimality are developed, and conditions for the realization of an input‐output map by such a system are given. Several problems, all directly or indirectly related to fuzzification, arise in considering this broader class of systems.

In this paper representation theorems are given for L‐sets. In a particular case, by choosing a suitable L, fuzzy sets and flou sets are obtained and the connection of these concepts with the continuous logic and n‐valued logics is shown. Representation theorems of the same type are given for L‐topological subspaces and L‐algebraic substructures. The possibility of generalizing these results is taken into consideration.

By fuzzy optimization we here mean optimization in a fuzzy environment, i.e., optimization with fuzzy constraints. Such a problem can be reduced to a family of ordinary optimization problems by using the representation theorem which states that a fuzzy set is a family of ordinary sets. Since it is difficult to work with a family of sets, in this paper a fuzzy set is approximated by an ordinary set. The Chebyshev norm is introduced into the set of all fuzzy sets, and a set is said to approximate a fuzzy set if the norm of a difference of its characteristic functions is smaller than a given number.

We investigated a fuzzy programming problem in which the constraints are a fuzzy subset over the alternatives and the objective is in the form of a linear ordering. A fuzzy subset is developed which reflects the objects ranking information in terms of grades of membership of the constraints. These two fuzzy subsets then are combined via intersection operation to form a fuzzy decision function.

Provides a personal view of the development of the theories and applications of fuzzy systems which were first introduced in the 1960s. Details the interrelationships between the pioneering proponents of fuzzy theory. Concentrates in part I on the historical beginnings of the field and in part II continues to provide personal insights into contemporary studies.

The use of deterministic or stochastic models to describe an observable phenomenon is considered to present enormous difficulty when evaluating the subjective parameters. In those cases where randomness cannot be established, or when there are not sufficient data available, it is believed that fuzzy theory is the appropriate approach to the problem. Utilizes the classical stochastic Boltzman model to compare the stochastic and fuzzy techniques. Gives a comparison of a specific case study: the life expectancy of individuals whose causa mortis is affected by poverty.

Provides a personal view of the development of the theories and applications of fuzzy systems which were first introduced in the 1960s. Details the interrelationships between the pioneering proponents of fuzzy theory. Concentrates in part I on the historical beginnings of the field and in part II continues to provide personal insights into contemporary studies.

This paper proposes a fuzzy random multi-objective portfolio model with different entropy measures and designs a hybrid algorithm to solve the proposed model.

Because random uncertainty and fuzzy uncertainty are often combined in a real-world setting, the security returns are considered as fuzzy random numbers. In the model, the authors also consider the effects of different entropy measures, including Yager's entropy, Shannon's entropy and min-max entropy. During the process of solving the model, the authors use a ranking method to convert the expected return into a crisp number. To find the optimal solution efficiently, a fuzzy programming technique based on artificial bee colony (ABC) algorithm is also proposed.

(1) The return of optimal portfolio increases while the level of investor risk aversion increases. (2) The difference of the investment weights of the optimal portfolio obtained with Yager's entropy are much smaller than that of the min–max entropy. (3) The performance of the ABC algorithm on solving the proposed model is superior than other intelligent algorithms such as the genetic algorithm, differential evolution and particle swarm optimization.

To the best of the authors' knowledge, no effect has been made to consider a fuzzy random portfolio model with different entropy measures. Thus, the novelty of the research is constructing a fuzzy random multi-objective portfolio model with different entropy measures and designing a hybrid fuzzy programming-ABC algorithm to solve the proposed model.

This paper is a continuation of our paper10,11 and formulates a fuzzy team decision problem of type 2. The concept of fuzzy sets of type 2 is introduced to formulate the team decision processes which contain fuzzy‐fuzzy states, fuzzy‐fuzzy information functions, fuzzy‐fuzzy information signals, fuzzy‐fuzzy decision functions and fuzzy‐fuzzy actions. After some definitions of fuzzy‐fuzzy relations and fuzzy‐fuzzy mappings, a model of fuzzy team decision of type 2 is proposed.

The purpose of this paper is to study free L‐fuzzy left R‐module, using the language of categories and functors for the general description of L‐fuzzy left R‐modules generated by L‐fuzzy set. In the language of categories and functors, an L‐fuzzy left R‐modules generated by L‐fuzzy set is called a free object in the category of L‐fuzzy left R‐modules determined by L‐fuzzy set.

Category theory is used to study the existent quality, unique quality and material structure of L‐fuzzy left R‐modules generated by L‐fuzzy set.

The paper gives the uniqueness, structure and existence theorems of free object in the category of L‐fuzzy left R‐modules determined by L‐fuzzy set, and the authors prove that the fuzzy free functor is left adjoint to the fuzzy underlying functor.

Some property of free L‐fuzzy left R‐modules will need to be further researched.

The paper defines a new class of L‐fuzzy left R‐modules, i.e. free L‐fuzzy left R‐modules, research and explore free L‐fuzzy left R‐modules in theory.