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.     

A rough set approach to intuitionistic fuzzy soft set based decision making

The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. Since its appearance, there has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making. The intuitionistic fuzzy soft set is a combination of an intuitionistic fuzzy set and a soft set. The rough set theory is a powerful tool for dealing with uncertainty, granuality and incompleteness of knowledge in information systems. Using rough set theory, this paper proposes a novel approach to intuitionistic fuzzy soft set based decision making problems. Firstly, by employing an intuitionistic fuzzy relation and a threshold value pair, we define a new rough set model and examine some fundamental properties of this rough set model. Then the concepts of approximate precision and rough degree are given and some basic properties are discussed. Furthermore, we investigate the relationship between intuitionistic fuzzy soft sets and intuitionistic fuzzy relations and present a rough set approach to intuitionistic fuzzy soft set based decision making. Finally, an illustrative example is employed to show the validity of this rough set approach in intuitionistic fuzzy soft set based decision making problems.     

基于异或软集合的不完备信息系统约减

软集合理论是处理不确定问题的一种新兴数学理论.作为软集合重要的应用领域之一,软集合的参数约减的研究大都基于完备信息系统.介绍了异或软集合的改进运算以及异或软集合决策系统,提出了一种基于异或软集合的不完备信息系统约减方法,同时与拓扑方法相比较.结果表明,算法得出的约减软集合是拓扑法求出的约减软集合的一个子集,即该算法在对数据刻画方面较为细致和全面.     

Group decision making methods based on intuitionistic fuzzy soft matrices

Soft set theory was originally proposed by Molodtsov as a general mathematical tool for dealing with uncertainty in 1999. Recently, researches of decision making based on soft sets have got some progress, but few people consider multi-experts situation. As such, this paper discusses multi-experts group decision making problems. Firstly, we give a concept of intuitionistic fuzzy soft matrix (IFSM) and prove some relevant properties of IFSM. Then, an adjustable approach is presented by means of median level soft set and p-quantile level soft set for dealing with decision making problems based on IFSM. Thirdly, we study aggregation methods of IFSM, give two kinds of aggregation operators and methods that how to determine experts’ weights under different situation with programming models, four corresponding algorithms have been proposed, too. Finally, a practical example has been demonstrated the reasonability and efficiency of these new algorithms.     

广义多值直觉模糊软集及其在多属性决策中的应用

软集理论作为一种处理不确定问题的数学新工具弥补了传统模糊理论在参数工具上的不足。本文将广义模糊软集与多值直觉模糊集相结合,提出了广义多值直觉模糊软集的概念,给出了广义多值直觉模糊软集的交、并、补、“且”、“或”运算的概念,并基于这些概念研究其若干相关性质。此外,还将广义多值直觉模糊软集应用到多属性决策问题中去,给出相应的算法。并进行实例分析说明该决策方法的可行性和有效性。     

A fuzzy multi-criteria decision-making model based on lower and upper boundaries

Many researches generalized classical multi-criteria decision-making (MCDM) methods under uncertain environment for fuzzy multi-criteria decision-making (FMCDM), such that Chen, Liang, Raj and Kumar, and Wang et al. extended TOPSIS into FMCDM methods. However, there are some problems in their methods. For instance, intersection of fuzzy numbers is a null set, calculation is a complex work for approximate triangular fuzzy numbers or trapezoidal fuzzy numbers, or criteria values of ideal/anti-ideal solution may not exist in feasible alternatives. In addition, steps of the methods are too many to realize or apply for decision-makers. To improve above shortcomings, we propose a FMCDM method based on lower and upper boundaries. Practically, the proposed method is not totally generalized TOPSIS under fuzzy environment. Furthermore, the method is aggregated into a FMCDM model to simplify FMCDM computation. Since the FMCDM model can be clearly expressed by a simple formulation, it is easy to utilize the model to solve FMCDM problems.     

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.     

The trapezoidal fuzzy soft set and its application in MCDM

Soft set theory is a newly emerging mathematical tool to deal with uncertain problems. Since the trapezoidal fuzzy number, as a vital concept of fuzzy set, can express linguistic assessments by transforming them into numerical variables objectively, this paper aims to extend classical soft sets to trapezoidal fuzzy soft sets based on trapezoidal fuzzy numbers. Then, some operations on a trapezoidal fuzzy soft set are defined, such as complement operation, “AND” operation, and “OR” operation. Finally, a Multiple Criterion Decision-Making (MCDM) problem under a fuzzy environment is analyzed by trapezoidal fuzzy soft sets with the demonstration of a numerical example. This paper also uses traditional fuzzy soft sets to deal with the MCDM problem. The result shows that the method provided by this paper outperforms the traditional one.     

A bijective enumeration of tiered trees

Tiered trees were introduced by Dugan–Glennon–Gunnells–Steingrímsson as a generalization of intransitive trees that were introduced by Postnikov, the latter of which have exactly two tiers. Tiered trees arise naturally in counting the absolutely indecomposable representations of certain quivers, and enumerating torus orbits on certain homogeneous varieties over finite fields. By employing generating function arguments and geometric results, Dugan et al. derived an elegant formula concerning the enumeration of tiered trees, which is a generalization of Postnikov’s formula for intransitive trees. In this paper, we provide a bijective proof of this formula by establishing a bijection between tiered trees and certain rooted labeled trees. As an application, our bijection also enables us to derive a refinement of the enumeration of tiered trees with respect to level of the node 1.     

模糊软映射及其应用

在模糊软集合的基础上,提出了模糊软关系和模糊软映射的定义,并研究了它们的性质,最后给出了模糊软映射在医疗诊断中的应用实例.     

随机值Ⅱ-型模糊集及其不确定性知识表示

针对人工智能中的不确定性问题,在研究了Ⅱ-型模糊集理论的基础上,提出了一种随机值Ⅱ-型模糊集概念,给出了正态随机值Ⅱ-型模糊集数学模型及其生成算法,并对正态随机值Ⅱ-型模糊集的参数及形态特征进行解析.实例证明随机值Ⅱ-型模糊集可以用来处理社会和自然科学中诸多不确定性问题.     

A novel approach to interval-valued intuitionistic fuzzy soft set based decision making

The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. The interval-valued intuitionistic fuzzy soft set is a combination of an interval-valued intuitionistic fuzzy set and a soft set. The aim of this paper is to investigate the decision making based on interval-valued intuitionistic fuzzy soft sets. By means of level soft sets, we develop an adjustable approach to interval-valued intuitionistic fuzzy soft sets based decision making and some numerical examples are provided to illustrate the developed approach. Furthermore, we also define the concept of the weighted interval-valued intuitionistic fuzzy soft set and apply it to decision making.     

Numerical solutions to large-scale differential Lyapunov matrix equations

In this paper, we introduce an extension of multiple set split variational inequality problem (Censor et al. Numer. Algor. 59, 301–323 2012) to multiple set split equilibrium problem (MSSEP) and propose two new parallel extragradient algorithms for solving MSSEP when the equilibrium bifunctions are Lipschitz-type continuous and pseudo-monotone with respect to their solution sets. By using extragradient method combining with cutting techniques, we obtain algorithms for these problems without using any product space. Under certain conditions on parameters, the iteration sequences generated by the proposed algorithms are proved to be weakly and strongly convergent to a solution of MSSEP. An application to multiple set split variational inequality problems and a numerical example and preliminary computational results are also provided.     

Comment on “A fuzzy soft set theoretic approach to decision making problems”

The algorithm for identification of an object in a previous paper of A.R. Roy et al. [A.R. Roy, P.K. Maji, A fuzzy soft set theoretic approach to decision making problems, J. Comput. Appl. Math. 203(2007) 412–418] is incorrect. Using the algorithm the right choice cannot be obtained in general. The problem is illustrated by a counter-example.     

A method for solving fuzzy goal programming problems based on MINMAX approach

Narasimhan incorporated fuzzy set theory within goal programming formulation in 1980. Since then numerous research has been carried out in this field. One of the well-known models for solving fuzzy goal programming problems was proposed by Hannan in 1981. In this paper the conventional MINMAX approach in goal programming is applied to solve fuzzy goal programming problems. It is proved that the proposed model is an extension to Hannan model that deals with unbalanced triangular linear membership functions. In addition, it is shown that the new model is equivalent to a model proposed in 1991 by Yang et al. Moreover, a weighted model of the new approach is introduced and is compared with Kim and Whang’s model presented in 1998. A numerical example is given to demonstrate the validity and strengths of the new models.     

广义对偶犹豫模糊软集及其在决策中的应用

犹豫模糊软集作为一种处理不确定问题的工具受到了广泛关注。本文将广义模糊软集与对偶犹豫模糊集相结合,提出了广义对偶犹豫模糊软集的概念,给出了广义对偶犹豫模糊软集的交、并、补、“且”、“或”运算的概念,并基于这些概念研究其若干相关性质。此外,定义了广义对偶犹豫模糊软集的信息能量、相关性和相关系数的概念并讨论了它的性质。最后,给出了基于广义对偶犹豫模糊软集的多属性决策方法,并通过实例说明了该方法的的可行性和有效性。     

Computing border bases using mutant strategies

Border bases, a generalization of Gröbner bases, have actively been addressed during recent years due to their applicability to industrial problems. In cryptography and coding theory a useful application of border based is to solve zero-dimensional systems of polynomial equations over finite fields, which motivates us for developing optimizations of the algorithms that compute border bases. In 2006, Kehrein and Kreuzer formulated the Border Basis Algorithm (BBA), an algorithm which allows the computation of border bases that relate to a degree compatible term ordering. In 2007, J. Ding et al. introduced mutant strategies bases on finding special lower degree polynomials in the ideal. The mutant strategies aim to distinguish special lower degree polynomials (mutants) from the other polynomials and give them priority in the process of generating new polynomials in the ideal. In this paper we develop hybrid algorithms that use the ideas of J. Ding et al. involving the concept of mutants to optimize the Border Basis Algorithm for solving systems of polynomial equations over finite fields. In particular, we recall a version of the Border Basis Algorithm which is actually called the Improved Border Basis Algorithm and propose two hybrid algorithms, called MBBA and IMBBA. The new mutants variants provide us space efficiency as well as time efficiency. The efficiency of these newly developed hybrid algorithms is discussed using standard cryptographic examples.     

Bell-shaped Fuzzy Soft Sets and Their Application in Medical Diagnosis

A soft set theory is a new general mathematical tool to dealing with imprecision. The theory differs from traditional mathematical tools to dealing with uncertainties in which many mathematical theories failed. In this paper, we proposed the concept of Bell shaped fuzzy soft set and its application in medical diagnosis using arithmetic operations of discrete- Gaussian fuzzy number, triangular-Cauchy fuzzy number and Gaussian-Cauchy fuzzy number. As an integral part of this paper, we discuss the parametric form of Gaussian membership function is discussed with some important properties.     

Hybrid flow shop scheduling with sequence dependent family setup time and uncertain due dates

This paper studies the scheduling problem in hybrid flow shop (HFS) environment. The sequence dependent family setup time (SDFST) is concerned with minimization of makespan and total tardiness. Production environments in real world include innumerable cases of uncertainty and stochasticity of events and a suitable scheduling model should consider them. Hence, in this paper, due date is assumed to be uncertain and its data follow a normal distribution. Since the proposed problem is NP-hard, two metaheuristic algorithms are presented based on genetic algorithm, namely: Non-dominated Sorting Genetic Algorithm (NSGAII) and Multi Objective Genetic Algorithm (MOGA). The quantitative and qualitative results of these two algorithms have been compared in different dimensions with multi phase genetic algorithm (MPGA) used in literature review. Experimental results indicate that the NSGAII performs very well when compared against MOGA and MPGA in a considerably shorter time.     

An approximation-based approach for fuzzy multi-period production planning problem with credibility objective

This paper develops a fuzzy multi-period production planning and sourcing problem with credibility objective, in which a manufacturer has a number of plants or subcontractors. According to the credibility service levels set by customers in advance, the manufacturer has to satisfy different product demands. In the proposed production problem, production cost, inventory cost and product demands are uncertain and characterized by fuzzy variables. The problem is to determine when and how many products are manufactured so as to maximize the credibility of the fuzzy costs not exceeding a given allowable invested capital, and this credibility can be regarded as the investment risk criteria in fuzzy decision systems. In the case when the fuzzy parameters are mutually independent gamma distributions, we can turn the service level constraints into their equivalent deterministic forms. However, in this situation the exact analytical expression for the credibility objective is unavailable, thus conventional optimization algorithms cannot be used to solve our production planning problems. To overcome this obstacle, we adopt an approximation scheme to compute the credibility objective, and deal with the convergence about the computational method. Furthermore, we develop two heuristic solution methods. The first is a combination of the approximation method and a particle swarm optimization (PSO) algorithm, and the second is a hybrid algorithm by integrating the approximation method, a neural network (NN), and the PSO algorithm. Finally, we consider one 6-product source, 6-period production planning problem, and compare the effectiveness of two algorithms via numerical experiments.