The multi-fuzzy soft set and its application in decision making

Molodtsov’s soft set theory was originally proposed as a general mathematical tool for dealing with uncertainty. By combining the multi-fuzzy set and soft set models, the purpose of this paper is to introduce the concept of multi-fuzzy soft sets. Some operations on a multi-fuzzy soft set are defined, such as complement operation, “AND” and “OR” operations, Union and Intersection operations. Then, the DeMorgan’s laws are proved. Finally, by means of level soft set, an algorithm is presented, and a decision problem is analyzed using multi-fuzzy soft set.     

Soft set theory based on its extension

In this paper, we point out a defect that the definitions of soft subset and soft intersection in Maji’s paper are partial definitions, and then we introduce extension of soft set to redefine and simplify operations between soft sets. At last, we obtain some algebraic properties of new soft operations.     

A note on the paper “The trapezoidal fuzzy soft set and its application in MCDM”

We propose a new concept which is a generalization of fuzzy soft subset and fuzzy soft equal. Using such notions, we will be able to consider the distributive law of fuzzy soft sets. Using the distributive law of fuzzy soft sets, we point out that the distributive law of trapezoidal fuzzy soft sets as proposed by Xiao et al. (2012) is not true. The correction will further improve further extensions of the results of Xiao et al. (2012). We will also establish the generalized distributive law of trapezoidal fuzzy soft sets along with illustrative examples.     

A new evaluation method based on D–S generalized fuzzy soft sets and its application in medical diagnosis problem

Soft set theory, initiated by Molodtsov, is a general mathematical tool for dealing with uncertain problems. In this paper, we first point out that the similarity measure in a previous paper by Majumdar and Samanta [P. Majumdar, S.K. Samanta, Generalized fuzzy soft sets, Comput. Math. Appl. 59 (2010) 1425–1432] is limited by two counterexamples. To deal with the problems of subjective evaluation and uncertain knowledge, this paper proposes the concept of D–S generalized fuzzy soft sets by combining Dempster–Shafer theory of evidence and generalized fuzzy soft sets. We study some of its operations and basic properties, and the relationship between generalized fuzzy soft sets and D–S generalized fuzzy soft sets are introduced. Then we propose the concept of the similarity between two D–S generalized fuzzy soft sets. At last, we present a new method of evaluation based on D–S generalized fuzzy soft sets and apply it into a medical diagnosis problem.     

Soft ordered semigroups

Molodtsov introduced 1999 the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to ordered semigroups. The notions of (trivial, whole) soft ordered semigroup, soft ordered subsemigroup, soft left (right) ideal, and left (right) idealistic soft ordered semigroup are introduced, and various related properties are investigated (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A soft version of the Knaster–Tarski fixed point theorem with applications

In this paper, first, we define a partial order on a soft set (F, A) and introduce some related concepts. Then, using the concept of a soft mapping introduced by Babitha and Sunil (Comput Math Appl 60(7):1840–1849, 2010), a soft version of Knaster–Tarski fixed point theorem is obtained. Some examples are presented to support the concepts introduced and the results proved herein. As an application of our result, we show that the soft Knaster–Tarski fixed point theorem ensures the existence of a soft common fixed point for a commuting family of order-preserving soft mappings.     

An adjustable approach to fuzzy soft set based decision making

Molodtsov’s soft set theory was originally proposed as a general mathematical tool for dealing with uncertainty. Recently, decision making based on (fuzzy) soft sets has found paramount importance. This paper aims to give deeper insights into decision making based on fuzzy soft sets. We discuss the validity of the Roy-Maji method and show its true limitations. We point out that the choice value designed for the crisp case is no longer fit to solve decision making problems involving fuzzy soft sets. By means of level soft sets, we present an adjustable approach to fuzzy soft set based decision making and give some illustrative examples. Moreover, the weighted fuzzy soft set is introduced and its application to decision making is also investigated.     

Similarity in soft set theory

We introduce and study the concept of similarity between soft sets, which is an extension of the equality for soft set theory. We also introduce the concepts of conjunction parameter (α∧β) and disjunction parameter (α∨β) of ordered pair parameter 〈α,β〉 for soft set theory, and we investigate modified operations of soft set theory in terms of ordered parameters.     

Intuitionistic Fuzzy Soft K-Algebras

Intuitionistic fuzzy sets and soft sets are two different soft computing models for representing vagueness and uncertainty. We apply these soft computing models in combination to study vagueness and uncertainty in K-algebras. We first introduce the notion of ${(\in, \in\vee q)}$ -intuitionistic fuzzy K-algebras and discuss some of their properties. Then we introduce intuitionistic fuzzy soft K-algebras and investigate some of their properties. Finally, we introduce ${(\in, \in \vee q)}$ -intuitionistic fuzzy soft K-algebras and present some of their related properties.     

The characterization of Γ-modules in terms of fuzzy soft Γ-submodules

In this paper, we focus on combining the theories of fuzzy soft sets with Γ-modules, and establishing a new framework for fuzzy soft Γ-submodules. The main contributions of the paper are 3-fold. First, we present the concepts of (R, S)-bi-Γ-submodules, quasi-Γ-submodules and regular Γ-modules. Meanwhile, some illustrative examples are given to show the rationality of the definitions introduced in this paper. Second, several new kinds of generalized fuzzy soft Γ-submodules are proposed, and related properties and mutual relationships are also investigated. Third, we discover some intrinsic connections between the generalized fuzzy soft Γ-submodules presented in this paper and crisp Γ-submodules, and describe the relationships between regular Γ-modules and the generalized fuzzy soft Γ-submodules presented in this paper.     

Some formal relationships among soft sets,fuzzy sets,and their extensions

We prove that every hesitant fuzzy set on a set E can be considered either a soft set over the universe $[0,1]$ or a soft set over the universe E. Concerning converse relationships, for denumerable universes we prove that any soft set can be considered even a fuzzy set. Relatedly, we demonstrate that every hesitant fuzzy soft set can be identified with a soft set, thus a formal coincidence of both notions is brought to light. Coupled with known relationships, our results prove that interval type-2 fuzzy sets and interval-valued fuzzy sets can be considered as soft sets over the universe $[0,1]$. Altogether we contribute to a more complete understanding of the relationships among various theories that capture vagueness and imprecision.     

Recent Developments in Critical Management Science

For much of its history management science had a quantitative and technical emphasis. More recently, there has been a move towards more subjective approaches such as ‘soft OR’ and ‘soft systems’. Currently, there is interest in ‘critical management science’ drawing on critical theory, particularly the work of Habermas. This paper reviews developments in critical management science, in particular critiques of traditional and ‘soft’ management science; Jackson and Keys' system of systems methodology; critical management science methodologies and the problem of power in bringing about change; and the postmodernist critique.     

Multimethodology in series and parallel: strategic planning using hard and soft OR

This paper examines two distinct ways in which hard and soft operational research (OR) methodologies can be combined, in series and in parallel. Multimethodology in series is acknowledged as the simpler and more common approach. Multimethodology in parallel is identified as having the potential to provide significant benefits to projects in political, changing, or ‘wicked’ contexts that multimethodology in series cannot. Observations regarding these approaches to multimethodology are examined in light of an information systems strategic planning project in the Australian public sector. Two distinct methodologies were combined in the project: soft systems methodology and project management. These methodologies are based on the soft and hard paradigms, respectively. However, findings in this paper have the potential to be transferred to combinations of other hard and soft OR methodologies.     

A mathematical model to study the soil arching effect in stone column-supported embankment resting on soft foundation soil

Soil arching is a common phenomena in pile or columnar (vibroconcrete columns, soil–cement columns by mixing or grouting, stone columns) supported geosynthetic-reinforced or unreinforced embankments resting on soft soil. Due to soil arching, stress acting on soft soil or geosynthetic reinforcement decreases and stress on piles or columns increases. In this paper, using mechanical elements (such as spring, dashpot), a generalized mathematical model has been developed to study the soil arching effect in stone column-supported geosynthetic-reinforced and unreinforced embankments resting on soft soil. Pasternak model concept has been used to model the embankment soil. The soft soil has been idealized by spring-dashpot system to include the time-dependent behavior. The stone columns and geosynthetic reinforcement are idealized by stiffer nonlinear springs and rough elastic membrane, respectively. The consolidation effect of soft soil due to inclusions of stone columns has also been included in the model to study its effect on soil arching. Plane strain condition has been considered in the analysis. A finite difference scheme has been used to solve the governing differential equations and results are presented in non-dimensional form. It has been observed that the height of embankment, degree of consolidation of soft soil, stiffness of the stone column material, spacing between the stone columns, use of geosynthetic reinforcement and properties of soft and embankment soils (such as ultimate bearing capacity of soft soil, shear modulus and ultimate shearing resistance of embankment soil) significantly influence the degree of soil arching.     

Soft Ideals and Arithmetic Mean Ideals

This article investigates the soft-interior (se) and the soft-cover (sc) of operator ideals. These operations, and especially the first one, have been widely used before, but making their role explicit and analyzing their interplay with the arithmetic mean operations is essential for the study in [10] of the multiplicity of traces. Many classical ideals are ‘soft’, i.e., coincide with their soft interior or with their soft cover, and many ideal constructions yield soft ideals. Arithmetic mean (am) operations were proven to be intrinsic to the theory of operator ideals in [6, 7] and arithmetic mean operations at infinity (am-∞) were studied in [10]. Here we focus on the commutation relations between these operations and soft operations. In the process we characterize the am-interior and the am-∞ interior of an ideal. Both authors were partially supported by grants of the Charles Phelps Taft Research Center; the second named author was partially supported by NSF Grants DMS 95-03062 and DMS 97-06911.     

Micro-mechanical modeling of anisotropic inelastic response of soft tissues under supra-physiological loading

Under supra-physiological loading, soft tissues exhibit many inelastic phenomena, such as stress softening, hysteresis and permanent set [1]. Knowledge of the mechanical response of soft tissues under such a large range of deformation is vital for optimizations of vascular medical devices or improvement of injury prevention techniques. In this work, a micro-mechanical model is proposed for soft collagenous tissues. Besides the anisotropy the model can also describe the aforementioned inelastic effects under extremal loadings. To this end, the dispersion of collagen fibers in the soft tissues is captured by a probability distribution of fiber orientations around preferred directions. The deformation induced damage inside the tissue is assumed to take place between collagen fibrils and is included within a statistical mechanical framework. Finally, the accuracy of the model is assessed by comparison with experimental data available in the literature. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

BCI代数的软关联理想和软正定关联理想

给出BCI代数的软关联理想和软正定关联理想的概念,讨论软理想、软关联理想和软正定关联理想三者之间的关系,研究了两个软关联理想(软正定关联理想)的扩展交、限制交、限制并和限制差分的性质。     

From Optimizing to Learning: A Development of Systems Thinking for the 1990s

In any subject concerned with rational intervention in human affairs, theory must lead to practice; but practice is the source of theory: neither theory nor practice is prime. We can examine this ‘groundless’ relation by asking what intellectual framework F is applied in what methodology M to what area of application A? If we do this for O.R., systems analysis, systems engineering etc., we see that F and M have changed dramatically between the 1950s and the 1980s, yielding the ‘hard’ and ‘soft’ traditions of systems thinking. The ‘hard’ tradition, based on goal seeking, is examined in the work of Simon and contrasted with the ‘soft’ tradition, based on learning, as exemplified in the work of Vickers and the development of soft systems methodology. The two are complementary, but the relation between them is that the ‘hard’ is a special case of ‘soft’ systems thinking. This analysis makes sense of the recent history of management science and helps to prepare us for the 1990s.