Narrow Results

To strengthen the ideal theory in BCI-algebras, the more general concept $(\in ,\in \vee ({\kappa }^{\ast },q_{\kappa }))$-fuzzy $a$-ideals in BCI-algebras is proposed. It is shown that $(\in ,\in \vee q)$-fuzzy $a$-ideals are $(\in ,\in \vee ({\kappa }^{\ast },q_{\kappa }))$-fuzzy $a$-ideals; however, the converse is not valid. Following that, the concept of $(\in \vee ({\kappa }^{\ast },q_{\kappa }),\in \vee ({\kappa }^{\ast },q_{\kappa }))$-fuzzy $a$-ideals is introduced. We demonstrate that $(\in \vee ({\kappa }^{\ast },q_{\kappa }),\in \vee ({\kappa }^{\ast },q_{\kappa }))$-fuzzy $a$-ideals are $(\in ,\in \vee ({\kappa }^{\ast },q_{\kappa }))$-fuzzy $a$-ideals. The converse is not true, and an example is given to support it. An equivalent condition for $(\in ,\in \vee ({\kappa }^{\ast },q_{\kappa }))$-fuzzy $a$-ideals is provided. We prove that the $(\in ,\in \vee ({\kappa }^{\ast },q_{\kappa }))$-fuzzy $a$-ideals are $(\in ,\in \vee ({\kappa }^{\ast },q_{\kappa }))$-fuzzy $p$-ideals and $(\in ,\in \vee ({\kappa }^{\ast },q_{\kappa }))$-fuzzy $q$-ideals. Furthermore, $(\in ,\in \vee ({\kappa }^{\ast },q_{\kappa }))$-fuzzy $a$-ideals are characterized in terms $a$-ideals.

A new technique for nonlinear sensitivity analysis of geophysical models for small size ensembles of model outputs has been developed. Such an analysis utilizes the following metrics: (a) Sobol–Saltelli sensitivity indices and cumulative distribution functions if perturbations of model parameters are random, and (b) a Hartley-like measure if perturbations of model parameters are nonrandom and parametrized through fuzzy sets. The indices and the Hartley-like measure allow for ranging model parameters along their significance to the model output. Our calculations demonstrate that accurate estimates of the sensitivity indices are possible even if an ensemble of random perturbations contains considerably less than 100 members. Some calculations were successfully provided for random ensembles with 20–30 members only but, in general 50–100 member ensembles are required to get robust and significant estimations of model sensitivity. The fuzzy set concept allows for robust estimations for small size nonrandom ensembles of model outputs (50–100 members) and accounts for additional a priori information on model sensitivity coming from different sources. The Lorenz 63 model (a few degrees of freedom) and the ocean component (POP) of the Community Climate System Model (CCSM3) (several thousand degrees of freedom) are used to illustrate the sensitivity analysis based on this approach.

This paper presents three main ideas. They are the Metatheorem, the lattice embedding for sets, and the lattice embedding for algebras.

The Metatheorem allows you to convert existing theorems about classical subsets into corresponding theorem about fuzzy subsets. The concept of a fuzzyfiable operation on a powerset is defined. The main result states that any implication or identity which can be stated using fuzzyfiable operations is true about fuzzy subsets if and only if it is true about classical subsets.

The lattice embedding theorem for sets shows that for any set X, there is a set Y such that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of Y. In fact it is further proved that if X is infinite, then we can choose Y = X and get the surprising result that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of X itself. The idea is illustrated with an example explicitly showing how the lattice of fuzzy subsets of the closed unit interval 𝕀 = [0,1] embeds into the lattice of classical subsets of 𝕀.

The lattice embedding theorem for algebras shows that under certain circumstances the lattice of fuzzy subalgebras of an algebra A embeds into the lattice of classical subalgebras of a closely related algebra A′. The following sample use of this embeding theorem is given. It is a well known fact that the lattice of normal subgroups of a group is a modular lattice. The embeding theorem is used here to conclude that lattice of fuzzy normal subgroups of a group is a modular lattice too.

The majority rule is frequently presented as a cornerstone of any democratic society, guiding many group decision-making processes where final decision requires the agreement of more than half the people involved. But sometimes, some key decisions require a higher level of agreement. In such cases, an added value would be to reach some consensus about the decision-making problem. Decision making under consensus drives to decisions which are better accepted and appreciated. But it also implies a greater complexity and time consuming process to reach a final decision, and it may even lead to a deadlock or unsuccessful results, whenever the searched agreement is not achieved. Meanwhile, these problems arise because the requirements to achieve the consensus are too strong, and different processes have softened their requirements. In particular, soft consensus is one of the most widespread consensus reaching processes that uses fuzzy logic to soften the consensus requirements. However, several problems still persist despite the softening of the requirements.

In this paper, we are going to make a brief revision of the different concepts about consensus and about different consensus reaching processes, both in the crisp and fuzzy environment. We shall then analyze how to overcome their lacks, indicating the challenges facing these processes in order to obtain successful results in those group decision problems in which they are required to make a decision under consensus.

Fuzzy set was introduced in 1965 by Prof. A. Lotfi Zadeh to deal with uncertainty. Fuzzy set is an important tool for solving real life problems. Similarly, hesitant fuzzy set is the extended tool of fuzzy set which plays an important role to deal with uncertainty, imprecision and vagueness more clearly and accurately. In this paper, hesitant fuzzy base rule system is proposed which is the extension of fuzzy base rule system. A new dimension of hesitant fuzzy set i.e. hesitant fuzzy membership line (HFML) is defined and the HFML is classified into different classes (Good, Fair, Poor) according to Quality Index Parameter (QIP) which is calculated by the expert only with the best of their knowledge. Also this paper consists of two newly defined operations AND and OR operations on hesitant fuzzy sets. The effective criteria like Ecology, Hostility, Cost, Water Quality and Air Quality are considered so that decision makers make appropriate site selection of the power plant in more rational and easy evaluation method. Using all the newly proposed methods in this paper, the policy makers are able to select the best power plant most easily and effectively without any big calculation. Finally, the power plant sites are ranked according to the highest value given by a score function.

An interval-valued hesitant fuzzy set (IVHFS) is a best tool to address uncertainty and hesitation of a production planning problem (PPP) which appears in engineering, agriculture, and industrial sectors. Often, a PPP is formulated as a multiobjective linear programming problem (MOLPP) and therefore, it is very necessary to develop a suitable and realistic method to deal MOLPP with uncertainty and hesitation. In this paper, we define a set of possible interval-valued hesitant fuzzy degrees for all objectives, and using this, a MOLPP is converted into a interval-valued hesitant fuzzy linear programming (IVHFLPP). Further, we introduce a new optimization technique based on a new operation of IVHFS, and later it is implemented in a computational method to search a Pareto optimal solution of the considered problem. Further, a PPP is solved by using the proposed method and the result shows the superiority of the proposed computational method over the existing methods.

This work proposes an heuristic approach based on evolutionary computation, whose goal is to find a set of k-spanning trees with lowest costs in graphs that contain uncertainties in their parameters. In order to avoid the high complex ordinary resolution, this work presents a genetic approach to explore the space of solutions looking for satisfactory results, without the necessity of comparing all possible solutions.