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Soft set theory, originally proposed by Molodtsov, has become an effective mathematical tool to deal with uncertainty. A type-2 fuzzy set, which is characterized by a fuzzy membership function, can provide us with more degrees of freedom to represent the uncertainty and the vagueness of the real world. Interval type-2 fuzzy sets are the most widely used type-2 fuzzy sets. In this paper, we first introduce the concept of trapezoidal interval type-2 fuzzy numbers and present some arithmetic operations between them. As a special case of interval type-2 fuzzy sets, trapezoidal interval type-2 fuzzy numbers can express linguistic assessments by transforming them into numerical variables objectively. Then, by combining trapezoidal interval type-2 fuzzy sets with soft sets, we propose the notion of trapezoidal interval type-2 fuzzy soft sets. Furthermore, some operations on trapezoidal interval type-2 fuzzy soft sets are defined and their properties are investigated. Finally, by using trapezoidal interval type-2 fuzzy soft sets, we propose a novel approach to multi attribute group decision making under interval type-2 fuzzy environment. A numerical example is given to illustrate the feasibility and effectiveness of the proposed method. 相似文献
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Zhang and Zhang (2013) proposed the arithmetic operations of trapezoidal interval type-2 fuzzy numbers having different left and right heights and hence the arithmetic operations of trapezoidal interval type-2 fuzzy soft sets having different left and right heights. In this paper, it is pointed out that the complement operation of a trapezoidal interval type-2 fuzzy number, proposed by Zhang and Zhang, is not valid and hence, the complement operation of trapezoidal interval type-2 fuzzy soft set as well as all the results, proposed by Zhang and Zhang in which complement operation is used, are not valid. The results, proposed by Zhang and Zhang, are valid only for such trapezoidal interval type-2 fuzzy numbers and trapezoidal interval type-2 fuzzy soft sets in which left and right heights are equal. 相似文献
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In a recently published paper “A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets””, Khalil and Hassan pointed out that assertions (3) and (4) of Theorem 3.2 in our previous paper “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets” are not true [2]. Furthermore, they introduced the notions of a generalized trapezoidal interval type-2 fuzzy soft subset and a generalized trapezoidal interval type-2 fuzzy soft equal and used these two notions to correct the flaw in assertions (3) and (4) of Theorem 3.2 in our previous paper. In this paper, we show by a counterexample that Khalil and Hassan's correction is incorrect and provide the modified versions of assertions (3) and (4) of Theorem 3.2, along with a strict proof. In addition, Khalil and Hassan pointed out by a counterexample that assertions (3)–(6) of Theorem 3.5 in our paper are not true and proposed the corrections of those assertions. In this paper, we show that Khalil and Hassan's counterexample and corrections are incorrect and provide a new example to verify the inaccuracies of assertions (3) and (5) of Theorem 3.5 in our paper. Moreover, we offer the modified versions of assertions (3) and (5) of Theorem 3.5 and prove them. Finally, Khalil and Hassan's statement that assertions (4) and (6) of Theorem 3.5 in our previous paper are not true is proven to be incorrect, i.e. assertions (4) and (6) of Theorem 3.5 in our previous paper are true. 相似文献
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Behnam Vahdani M. Zandieh R. Tavakkoli-Moghaddam 《Applied Mathematical Modelling》2011,35(3):1396-1412
In this paper, we consider the problem of alternative-fuel buses selection using two novel fuzzy multiple criteria decision-making (MCDM) methods. Several types of fuels are considered as fuel modes, i.e., electricity, fuel cell (hydrogen), and methanol. For the purpose of determining appropriate fuel buses, many factors including quantitative and qualitative ones such as efficiency, price, and capability must be taken into account. In the first proposed method, linguistic variables are used to assess the ratings and weights for factors. These linguistic ratings can be expressed in trapezoidal or triangular fuzzy numbers. Then, a hierarchical MCDM method based on fuzzy-sets theory is proposed to deal with the fuel buses selection problem. To simultaneously determine the ranking order of all alternatives based on the concept of the TOPSIS by calculating the distances to the both fuzzy positive-ideal solution (FPIS) and fuzzy negative-ideal solution (FNIS), a closeness coefficient is defined. In the second presented method we extend preference selection index (PSI) method for fuzzy environment. In this method performance ratings of criteria evaluate by linguistic variable which can be expressed in trapezoidal fuzzy numbers. Finally, an example is shown to highlight the procedure of the proposed methods and compare the results of these methods with each other. 相似文献
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重大灾害下应急物资储备决策是阻止突发灾难蔓延的有效手段之一。针对救灾信息具备不确定性与复杂性特点,构建基于区间二型梯形模糊集的应急物资储备动态协同决策模型,并给出应急物资储备策略。利用区间二型模糊集理论的决策方案并结合比例分析法(COPRAS),构建常态环境下应急物资供应商选择的群决策模型,解决不相容群决策属性之间的冲突问题;进而,充分考虑“救灾阶段性动态时间因素”对储备决策的影响,构建动态救灾环境下应急物资储备结构模糊优化模型,实现常态决策与非常态应急决策之间的动态协同;最后,以2012年云南丽江“6.14”突发特大山洪灾害为实例进行数值分析,验证该动态协同决策模型的合理性与可行性,能有效解决动态救灾环境下应急物资储备结构优化问题。 相似文献
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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. 相似文献
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Zhi Xiao Xianglei Yang Qing Niu Yuanxiang Dong Ke Gong Sisi Xia Ying Pang 《Applied Mathematical Modelling》2012
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. 相似文献
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Fuzzy sets have been used successfully in order to deal with imprecise data, linguistic terms or not well-defined concepts. Recently, considerable effort has been made in the direction of combining the neural network approach with fuzzy sets. In this paper a fuzzy feed-forward neural network, able to process trapezoidal fuzzy sets, has been investigated. Normalized trapezoidal fuzzy sets have been considered. The fuzzy generalized delta rule with different back-propagation algorithms is discussed. The more interesting and characteristic property of the proposed architecture is the ability of each node to process fuzzy sets or linguistic terms, preserving the simplicity of the back-propagation algorithm. Consequently, the resulting architecture is able to cope with problems in which the input parameters and the desired targets are described by linguistic terms. This methodology has the further interesting characteristic of being able to operate at the linguistic level rather than at the numerical level, that is it can work at a higher data abstraction level. An example in computerized electrocardiography will be illustrated in order to test the proposed approach. 相似文献
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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. 相似文献
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We propose an approach to solve a nonlinear multi-objective problem subject to fuzzy relation inequalities with max-Archimedean-t-norm composition by a genetic algorithm. The additive generator of Archimedean t-norms is utilized to reform the existent genetic algorithm to solve the constrained nonlinear multi-objective optimization problems. We consider thoroughly the feasible set of the fuzzy relation inequality systems in three possible cases, namely “≤”, “≥” and the combination of them. In general, their feasible sets are nonconvex which are completely determined by one vector as their maximum solution and a finite number of minimal solutions. The maximum and minimal solutions are formulated by using the additive generator. Additionally, we present some conditions for each case under which the problem can be reduced. Finally, each reduced problem is solved by the genetic algorithm and the efficiency of the proposed method is shown by some numerical examples. 相似文献
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Supplier selection problem, considered as a multi-criteria decision-making (MCDM) problem, is one of the most important issues for firms. Lots of literatures about it have been emitted since 1960s. However, research on supplier selection under operational risks is limited. What’s more, the criteria used by most of them are independent, which usually does not correspond with the real world. Although the analytic network process (ANP) has been proposed to deal with the problems above, several problems make the method impractical. This study first integrates the fuzzy cognitive map (FCM) and fuzzy soft set model for solving the supplier selection problem. This method not only considers the dependent and feedback effect among criteria, but also considers the uncertainties on decision making process. Finally, a case study of supplier selection considering risk factors is given to demonstrate the proposed method’s effectiveness. 相似文献
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The multiple criteria decision making (MCDM) methods VIKOR and TOPSIS are all based on an aggregating function representing “closeness to the ideal”, which originated in the compromise programming method. The VIKOR method of compromise ranking determines a compromise solution, providing a maximum “group utility” for the “majority” and a minimum of an “individual regret” for the “opponent”, which is an effective tool in multi-criteria decision making, particularly in a situation where the decision maker is not able, or does not know to express his/her preference at the beginning of system design. The TOPSIS method determines a solution with the shortest distance to the ideal solution and the greatest distance from the negative-ideal solution, but it does not consider the relative importance of these distances. And, the hesitant fuzzy set is a very useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision makers. In this paper, we develop the E-VIKOR method and TOPSIS method to solve the MCDM problems with hesitant fuzzy set information. Firstly, the hesitant fuzzy set information and corresponding concepts are described, and the basic essential of the VIKOR method is introduced. Then, the problem on multiple attribute decision marking is described, and the principles and steps of the proposed E-VIKOR method and TOPSIS method are presented. Finally, a numerical example illustrates an application of the E-VIKOR method, and the result by the TOPSIS method is compared. 相似文献
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《International Journal of Approximate Reasoning》2007,44(3):268-285
Multicriteria decision-making (MCDM) problems often involve a complex decision process in which multiple requirements and fuzzy conditions have to be taken into consideration simultaneously. The existing approaches for solving this problem in a fuzzy environment are complex. Combining the concepts of grey relation and pairwise comparison, a new fuzzy MCDM method is proposed. First, the fuzzy analytic hierarchy process (AHP) is used to construct fuzzy weights of all criteria. Then, linguistic terms characterized by L–R triangular fuzzy numbers are used to denote the evaluation values of all alternatives versus subjective and objective criteria. Finally, the aggregation fuzzy assessments of different alternatives are ranked to determine the best selection. Furthermore, this paper uses a numerical example of location selection to demonstrate the applicability of the proposed method. The study results show that this method is an effective means for tackling MCDM problems in a fuzzy environment. 相似文献
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Many nearest parametric approximation methods of fuzzy sets are proposed in the literature. It is clear that the specific approximations may lead to the loss of information about fuzziness. To overcome this problem, most of these methods rely on the minimization of the distance between the original fuzzy set and its approximation. But these approximations mostly are not flexible to the decision maker's choice. Hence, in this paper, we offer a parametric fuzzy approximation method based on the decision maker's strategy as an extension of trapezoidal approximation of a fuzzy number. This method comprises the selection of the form of the parametric membership function and its evaluation. 相似文献
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本文从匹配意愿的视角研究了基于直觉模糊集信息的双边匹配问题。首先给出了直觉模糊集和双边匹配的概念;接着描述了考虑匹配意愿的直觉模糊双边匹配问题。为求解该问题,先将直觉模糊集矩阵转化为得分矩阵。以每个主体得分最大为目标,在一对一双边匹配约束条件下,建立了双边匹配模型。依据得分矩阵,计算匹配意愿矩阵;依据匹配意愿矩阵,将双边匹配模型转化为单目标优化模型;通过求解该模型获得“最佳”双边匹配。最后,通过一个人岗匹配实例说明了所提双边匹配决策的可行性和有效性。 相似文献
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《European Journal of Operational Research》1999,114(1):141-152
Minimum Spanning Tree (MST) problem is of high importance in network optimization. The multi-criteria MST (mc-MST) is a more realistic representation of the practical problem in the real world, but it is difficult for the traditional network optimization technique to deal with. In this paper, a genetic algorithm (GA) approach is developed to deal with this problem. Without neglecting its network topology, the proposed method adopts the Prüfer number as the tree encoding and applies the Multiple Criteria Decision Making (MCDM) technique and nondominated sorting technique to make the GA search give out all Pareto optimal solutions either focused on the region near the ideal point or distributed all along the Pareto frontier. Compared with the enumeration method of Pareto optimal solution, the numerical analysis shows the efficiency and effectiveness of the GA approach on the mc-MST problem. 相似文献
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Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. There has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making. In this paper we generalize the adjustable approach to fuzzy soft sets based decision making. Concretely, we present an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets and give some illustrative examples. The properties of level soft sets are presented and discussed. Moreover, we also introduce the weighted intuitionistic fuzzy soft sets and investigate its application to decision making. 相似文献