Axioms, Volume 7, Issue 1 (March 2018) – 19 articles

Soft sets (SSs), neutrosophic sets (NSs), and rough sets (RSs) are different mathematical models for handling uncertainties, but they are mutually related. In this research paper, we introduce the notions of soft rough neutrosophic sets (SRNSs) and neutrosophic soft rough sets (NSRSs) as hybrid models for soft computing. We describe a mathematical approach to handle decision-making problems in view of NSRSs. We also present an efficient algorithm of our proposed hybrid model to solve decision-making problems. Full article

We show that if a $C^1$ generic diffeomorphism of a closed smooth two-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it is Anosov. Moreover, if a $C^1$ generic vector field of a closed smooth three-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it satisfies singular Axiom A without cycles. Full article

A general characterization result of projective aggregation functions is shown, the proof of which makes use of the celebrated Arrow’s theorem, thus providing a link between aggregation functions theory and social choice theory. The result can be viewed as a generalization of a theorem obtained by Kim (1990) for real-valued aggregation functions defined on the n-dimensional Euclidean space in the context of measurement theory. In addition, two applications of the core theorem of the article are shown. The first is a simple extension of the main result to the context of multi-valued aggregation functions. The second offers a new characterization of projective bijection aggregators, thus connecting aggregation operators theory with social choice. Full article

Ordered fuzzy numbers are defined by Kosiński. In this way, he was going to supplement a fuzzy number by orientation. A significant drawback of Kosiński’s theory is that there exist such ordered fuzzy numbers which, in fact, are not fuzzy numbers. For this reason, a fully formalized correct definition of ordered fuzzy numbers is proposed here. Also, the arithmetic proposed by Kosiński has a significant disadvantage. The space of ordered fuzzy numbers is not closed under Kosiński’s addition. On the other hand, many mathematical applications require the considered space be closed under used arithmetic operations. Therefore, the Kosinski’s theory is modified in this way that the space of ordered fuzzy numbers is closed under revised arithmetic operations. In addition, it is shown that the multiple revised sum of finite sequence of ordered fuzzy numbers depends on its summands ordering. Full article

Bringing toghether mathematical and philosophical ideas related to transcendental numbers, this paper begins with a survey on transcendence and transcendental numbers, it then presents several properties of the transcendental numbers e and $\pi$ , and then it gives the proof of a new inequality for transcendental numbers. Also, in relationship with these topics, we study solutions to the Yang-Baxter equation from hyperbolic functions and from logical implication. Full article

Neutrosophic sets (NSs) handle uncertain information while fuzzy sets (FSs) and intuitionistic fuzzy sets (IFs) fail to handle indeterminate information. Soft set theory, neutrosophic set theory, and rough set theory are different mathematical models for handling uncertainties and they are mutually related. The neutrosophic soft rough set (NSRS) model is a hybrid model by combining neutrosophic soft sets with rough sets. We apply neutrosophic soft rough sets to graphs. In this research paper, we introduce the idea of neutrosophic soft rough graphs (NSRGs) and describe different methods of their construction. We consider the application of NSRG in decision-making problems. In particular, we develop efficient algorithms to solve decision-making problems. Full article

In practical situations, we often have to handle programming problems involving indeterminate information. Building on the concepts of indeterminacy I and neutrosophic number (NN) (z = p + qI for p, q ∈ ℝ), this paper introduces some basic operations of NNs and concepts of NN nonlinear functions and inequalities. These functions and/or inequalities contain indeterminacy I and naturally lead to a formulation of NN nonlinear programming (NN-NP). These techniques include NN nonlinear optimization models for unconstrained and constrained problems and their general solution methods. Additionally, numerical examples are provided to show the effectiveness of the proposed NN-NP methods. It is obvious that the NN-NP problems usually yield NN optimal solutions, but not always. The possible optimal ranges of the decision variables and NN objective function are indicated when the indeterminacy I is considered for possible interval ranges in real situations. Full article

A neutrosophic number (a + bI) is a significant mathematical tool to deal with indeterminate and incomplete information which exists generally in real-world problems, where a and bI denote the determinate component and indeterminate component, respectively. We define score functions and accuracy functions for ranking neutrosophic numbers. We then define a cosine function to determine the unknown weight of the criteria. We define the neutrosophic number harmonic mean operators and prove their basic properties. Then, we develop two novel multi-criteria group decision-making (MCGDM) strategies using the proposed aggregation operators. We solve a numerical example to demonstrate the feasibility, applicability, and effectiveness of the two proposed strategies. Sensitivity analysis with the variation of “I” on neutrosophic numbers is performed to demonstrate how the preference ranking order of alternatives is sensitive to the change of “I”. The efficiency of the developed strategies is ascertained by comparing the results obtained from the proposed strategies with the results obtained from the existing strategies in the literature. Full article

When considering perceptions, the observation scale and resolution are closely related properties. There is consensus on considering resolution as the density of the elementary pieces of information in a specified information space. On the other hand, with the concept of scale, several conceptions compete for a consistent meaning. Scale is typically regarded as a way to indicate the degree of detail in which an observation is performed. Surprisingly, there is not a unified definition of scale as a description’s property. This paper offers a precise definition of scale and a method to quantify it as a property associated with the interpretation of a description. To complete the parameters needed to describe the perception of a description, the concepts of scope and resolution are also revealed with an exact meaning. A model describing a recursive process of interpretation, based on evolving steps of scale, scope and resolution, is introduced. The model relies on the conception of observation scale and its association to the selection of symbols. Five experiments illustrate the application of these concepts, showing that resolution, scale and scope integrate the set of properties to define any point of view from which an observation is performed and interpreted. The results obtained for descriptions expressed in one and two dimensions, are the basis for a comparison of the perceivable symbolic information from different interpretations of the same descriptions. In conclusion, this study provides a framework for building models of our interpretation process and suggests ways to understand some mechanisms in the formation of information from initially meaningless symbols. Full article

In this paper, the solvability of nonlinear fractional partial differential equations (FPDEs) with mixed partial derivatives is considered. The invariant subspace method is generalized and is then used to derive exact solutions to the nonlinear FPDEs. Some examples are solved to illustrate the effectiveness and applicability of the method. Full article

In this paper, we examine the real options approach to capital budgeting decision making in the presence of managerial adverse incentives. We show that real options have the potential to be value enhancing or value destroying depending on the managerial incentives that may result from having objectives different from firm value maximization. We further examine the possibility of using a generic residual income based rule of managerial compensation to induce the proper investment incentives and we seek to determine the cost-of-capital that must be employed in such a rule. Using numerical examples it is demonstrated that in general, a range of incentive compatible costs-of-capital exists across all managerial investment horizons but not across all managerial hurdle rates. Full article

In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In this paper, we describe a natural fuzzy analog of the category of sets and functions, and we show that, in this category, fuzzy relations (a natural fuzzy analogue of functions) can be determined in category terms—of course, modulo 1-1 mapping of the corresponding universe of discourse and 1-1 re-scaling of fuzzy degrees. Full article

As a new extension of a cubic set, the notion of a cubic interval-valued intuitionistic fuzzy set is introduced, and its application in $BCK/BCI$ -algebra is considered. The notions of $\alpha$ -internal, $\beta$ -internal, $\alpha$ -external and $\beta$ -external cubic IVIF set are introduced, and the P-union, P-intersection, R-union and R-intersection of $\alpha$ -internal and $\alpha$ -external cubic IVIF sets are discussed. The concepts of cubic IVIF subalgebra and ideal in $BCK/BCI$ -algebra are introduced, and related properties are investigated. Relations between cubic IVIF subalgebra and cubic IVIF ideal are considered, and characterizations of cubic IVIF subalgebra and cubic IVIF ideal are discussed. Full article

A rough neutrosophic set model is a hybrid model which deals with vagueness by using the lower and upper approximation spaces. In this research paper, we apply the concept of rough neutrosophic sets to graphs. We introduce rough neutrosophic digraphs and describe methods of their construction. Moreover, we present the concept of self complementary rough neutrosophic digraphs. Finally, we consider an application of rough neutrosophic digraphs in decision-making. Full article

The need for organizations to evaluate their environmental practices has been recently increasing. This fact has led to the development of many approaches to appraise such practices. In this paper, a novel decision model to evaluate company’s environmental practices is proposed to improve traditional evaluation process in different facets. Firstly, different reviewers’ collectives related to the company’s activity are taken into account in the process to increase company internal efficiency and external legitimacy. Secondly, following the standard ISO 14031, two general categories of environmental performance indicators, management and operational, are considered. Thirdly, since the assumption of independence among environmental indicators is rarely verified in environmental context, an aggregation operator to bear in mind the relationship among such indicators in the evaluation results is proposed. Finally, this new model integrates quantitative and qualitative information with different scales using a multi-granular linguistic model that allows to adapt diverse evaluation scales according to appraisers’ knowledge. Full article

The notion of a neutrosophic positive implicative $\mathcal{N}$ -ideal in $BCK$ -algebras is introduced, and several properties are investigated. Relations between a neutrosophic $\mathcal{N}$ -ideal and a neutrosophic positive implicative $\mathcal{N}$ -ideal are discussed. Characterizations of a neutrosophic positive implicative $\mathcal{N}$ -ideal are considered. Conditions for a neutrosophic $\mathcal{N}$ -ideal to be a neutrosophic positive implicative $\mathcal{N}$ -ideal are provided. An extension property of a neutrosophic positive implicative $\mathcal{N}$ -ideal based on the negative indeterminacy membership function is discussed. Full article

The aim of this paper is to present a trade-off ranking method in a fuzzy multi-criteria decision-making environment. The triangular fuzzy numbers are used to represent the imprecise numerical quantities in the criteria values of each alternative and the weight of each criterion. A fuzzy trade-off ranking method is developed to rank alternatives in the fuzzy multi-criteria decision-making problem with conflicting criteria. The trade-off ranking method tackles this type of multi-criteria problems by giving the least compromise solution as the best option. The proposed method for the fuzzy decision-making problems is compared against two other fuzzy decision-making approaches, fuzzy Technique for Order Preference by Similarity to the Ideal Solution (TOPSIS) and fuzzy VlseKriterijuska Optimizacija I Komoromisno Resenje (VIKOR), used for ranking alternatives. Full article