Aggregating decision information into Atanassov's intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making

The aim of this paper is to propose a new aggregation method to solve heterogeneous MAGDM problem which involves real numbers, interval numbers, triangular fuzzy numbers (TFNs), trapezoidal fuzzy numbers (TrFNs), linguistic values and Atanassov's intuitionistic fuzzy numbers (AIFNs). Firstly, motivated by the relative closeness of technique for order preference by similarity to ideal solution (TOPSIS), we propose a new general method for aggregating crisp values, TFNs, TrFNs and linguistic values into AIFNs. Thus all the group decision matrices for each alternative which involves heterogeneous information are transformed into an Atanassov's intuitionistic fuzzy decision matrix which only contains AIFNs. To determine the attribute weights, a multiple objective Atanassov's intuitionistic fuzzy programming model is constructed and solved by converting it into a linear program. Subsequently, comparison analyses demonstrate that the proposed aggregated technology can overcome the drawbacks of existing methods. An example about cloud computing service evaluation is given to verify the practicality and effectiveness of the proposed method.     

A Novel Approach for Linguistic Group Decision Making Based on Generalized Interval‐Valued Intuitionistic Fuzzy Linguistic Induced Hybrid Operator and TOPSIS

In the multiple attribute linguistic group decision making analysis with interval‐valued intuitionistic fuzzy linguistic information, seeking highly efficient aggregation method and order relation play a crucial role. In this paper, we redefine an interval‐valued intuitionistic fuzzy linguistic variable that considers principal component and propose generalized interval‐valued intuitionistic fuzzy linguistic induced hybrid aggregation (GIVIFLIHA) operator with entropic order‐inducing variable and interval‐valued intuitionistic fuzzy linguistic technique for order preference by similarity to an ideal solution (TOPSIS) order relation based on interval‐valued intuitionistic fuzzy linguistic distance measure. Then, some primary properties of the GIVIFLIHA operator are discussed, and a linguistic group decision‐making approach based on GIVIFLIHA operator and interval‐valued intuitionistic fuzzy linguistic TOPSIS order relation is proposed. Finally, a numerical example concerning the investment strategy is given to illustrate the validity and applicability of the proposed method, and then the method is compared with the existing method to further illustrate its flexibility.     

对方案有偏好的直觉模糊多属性决策方法

对属性权重信息不完全、属性值和决策者对方案的偏好信息均以直觉模糊数表示的多属性决策问题提出一种决策方法。首先根据决策者对方案的偏好信息建立多目标规划模型,求出属性权重,接着利用觉模糊加权算术平均算子求出方案的综合属性值,由直觉模糊数的得分函数和精确函数确定方案的排序,最后通过实例证明了该方法的实用性和有效性。     

Linguistic hesitant intuitionistic fuzzy decision-making method based on VIKOR

Linguistic hesitant intuitionistic fuzzy set, which allows an element having several linguistic evaluation values and each linguistic argument having several intuitionistic fuzzy memberships, is a power tool to model uncertain information existing in multiple attribute decision-making problems. In this paper, we propose new methods by using TOPSIS and VIKOR for multiple attribute decision-making problems, in which evaluation values are in the form of linguistic hesitant intuitionistic fuzzy elements. Different situations of attribute weight information are considered. If attribute weights are partly known, a linear programming model is set up based on the idea that reasonable weights should make the relative closeness of each alternative evaluation value to the linguistic hesitant intuitionistic fuzzy positive ideal solution as large as possible. If attribute weights are unknown completely, an optimization model is set up based on the maximum deviation method. A numerical example is presented to illustrate feasibility and practical advantages of the proposed method. We compare the alternatives’ rankings derived from the linguistic hesitant intuitionistic fuzzy TOPSIS method with those derived from the hesitant fuzzy linguistic TOPSIS and the hesitant intuitionistic fuzzy TOPSIS approach to further illustrate their advantages.

     

An interval-valued intuitionistic fuzzy multiattribute group decision making framework with incomplete preference over alternatives

This article proposes a framework to handle multiattribute group decision making problems with incomplete pairwise comparison preference over decision alternatives where qualitative and quantitative attribute values are furnished as linguistic variables and crisp numbers, respectively. Attribute assessments are then converted to interval-valued intuitionistic fuzzy numbers (IVIFNs) to characterize fuzziness and uncertainty in the evaluation process. Group consistency and inconsistency indices are introduced for incomplete pairwise comparison preference relations on alternatives provided by the decision-makers (DMs). By minimizing the group inconsistency index under certain constraints, an auxiliary linear programming model is developed to obtain unified attribute weights and an interval-valued intuitionistic fuzzy positive ideal solution (IVIFPIS). Attribute weights are subsequently employed to calculate distances between alternatives and the IVIFPIS for ranking alternatives. An illustrative example is provided to demonstrate the applicability and effectiveness of this method.     

基于前景和量子理论的模糊多属性决策方法

针对属性权重不完全确定且属性偏好值为区间直觉模糊数的多属性决策问题,提出一种基于前景理论和量子进化算法的模糊多属性决策方法。该方法根据前景理论及模糊数距离公式,定义区间直觉模糊数的前景价值函数,同时将决策者对方案的风险偏好纳入决策行为中,以此来构建方案综合前景值最大化的非线性规划模型。通过引入量子进化算法,求解模型得出最优权重向量。最终根据方案前景值确定出方案的排序。该方法适用于模糊决策环境,能满足决策者不提供确定属性权重的要求,并充分考虑决策者风险心理因素对决策行为的影响,具有广泛的应用价值。数值算例说明了该方法的有效性和可行性。     

Nonlinear optimization models for multiple attribute group decision making with intuitionistic fuzzy information

Intuitionistic fuzzy numbers are very useful for experts to depict in depth their fuzzy preference information over objects. In this work, we investigate multiple attribute group decision‐making problems in which the attribute values provided by experts are expressed in intuitionistic fuzzy numbers, each of which is composed of a membership degree, a nonmembership degree and a hesitancy degree, and the weight information about both the experts and the attributes is to be determined. We first make different types of attribute values uniform so as to facilitate interattribute comparisons and employ the simple additive weighting method to fuse all the individual opinions into the group one. We then develop two nonlinear optimization models, one minimizing the divergence between each individual opinion and the group one, and the other minimizing the divergence among the individual opinions, from which two exact formulae can be obtained to derive the weights of experts. Similarly, from the viewpoint of maximizing group consensus, we establish a nonlinear optimization model based on all the individual intuitionistic fuzzy decision matrices to determine the weights of attributes. The simple additive weighting method is used to aggregate all the intuitionistic fuzzy attribute values corresponding to each alternative, and then the score function and the accuracy function are employed to rank and select the given alternatives. Moreover, we extend all the above results to interval intuitionistic fuzzy situations, and finally apply the developed models to an air‐condition system selection problem. © 2010 Wiley Periodicals, Inc.     

Prioritized intuitionistic fuzzy aggregation operators

In some multi-attribute decision making problems, distorted conclusions will be generated due to the lack of considering various relationships among the attributes of decision making. In this paper, we investigate the prioritization relationship of attributes in multi-attribute decision making with intuitionistic fuzzy information (i.e., partial or all decision information, like attribute values and weights, etc., is represented by intuitionistic fuzzy values (IFVs)). Firstly, we develop a new method for comparing two IFVs, based on which the basic intuitionistic fuzzy operations satisfy monotonicities. In addition, we devise a method to derive the weights with intuitionistic fuzzy forms, which can indicate the importance degrees of the corresponding attributes. Then we develop a prioritized intuitionistic fuzzy aggregation operator, which is motivated by the idea of the prioritized aggregation operators [R.R. Yager, Prioritized aggregation operators, International Journal of Approximate Reasoning 48 (2008) 263–274]. Furthermore, we propose an intuitionistic fuzzy basic unit monotonic (IF-BUM) function to transform the derived intuitionistic fuzzy weights into the normalized weights belonging to the unit interval. Finally, we develop a prioritized intuitionistic fuzzy ordered weighted averaging operator on the basis of the IF-BUM function and the transformed weights.     

A new preference scale mcdm method based on interval-valued intuitionistic fuzzy sets and the analytic hierarchy process

The intuitionistic fuzzy analytic hierarchy process (IF-AHP), in which intuitionistic fuzzy numbers are utilized in defining decision makers’ linguistic judgment, has been used to solve various multi-criteria decision-making problems. Previous theories have suggested that interval-valued intuitionistic fuzzy numbers (IVIFN) with hesitation degree can act as alternative fuzzy numbers that can handle vagueness and uncertainty. This paper proposes a new preference scale in the framework of the interval-valued intuitionistic fuzzy analytic hierarchy process (IVIF-AHP). The comparison matrix judgment is expressed in IVIFN with degree of hesitation. The proposed new preference scale concurrently considers the membership function, the non-membership function and the degree of hesitation of IVIFN. To define the weight entropy of the aggregated matrix of IVIFN, a modified interval-valued intuitionistic fuzzy weighted averaging is proposed, by considering the interval number of the hesitation degree. Three multi-criteria decision-making problems are used to test the proposed method. A comparison of the results is also presented to check the feasibility of the proposed method. It is shown that the ranking order of the proposed method is slightly different from that of the other two methods because of the inclusion of the hesitation degree in defining the preference scale.     

A novel risk attitudinal ranking method for intuitionistic fuzzy values and application to MADM

The ranking of intuitionistic fuzzy sets (IFSs) is very important for the intuitionistic fuzzy decision making. The aim of this paper is to propose a new risk attitudinal ranking method of IFSs and apply to multi-attribute decision making (MADM) with incomplete weight information. Motivated by technique for order preference by similarity to ideal solution (TOPSIS), we utilize the closeness degree to characterize the amount of information according to the geometrical representation of an IFS. The area of triangle is calculated to measure the reliability of information. It is proved that the closeness degree and the triangle area just form an interval. Thereby, a new lexicographical method is proposed based on the intervals for ranking the intuitionistic fuzzy values (IFVs). Furthermore, considered the risk attitude of decision maker sufficiently, a novel risk attitudinal ranking measure is developed to rank the IFVs on the basis of the continuous ordered weighted average (C-OWA) operator and this interval. Through maximizing the closeness degrees of alternatives, we construct a multi-objective fractional programming model which is transformed into a linear program. Thus, the attribute weights are derived objectively by solving this linear program. Then, a new method is put forward for MADM with IFVs and incomplete weight information. Finally, an example analysis of a teacher selection is given to verify the effectiveness and practicability of the proposed method.     

A complex multi-attribute large-group decision making method based on the interval-valued intuitionistic fuzzy principal component analysis model

In the complex multi-attribute large-group decision making (CMALGDM) problems where attribute values are interval-valued intuitionistic fuzzy numbers (IVIFNs), the number of decision attributes is often large and their correlation degrees are high, which increase the difficulty of decision making and thus influence the accuracy of the result. To solve this problem, this paper proposes the interval-valued intuitionistic fuzzy principal component analysis (IVIF-PCA) model. This model represents major information of original attributes, effectively reduces dimensions of attribute space, and synthesizes original attributes into several relatively independent principal components (PCs). The basic thought of this model is as follows: first, we use thoughts of ‘equivalency’ and ‘order invariance’ to transform IVIFN samples into interval number samples; subsequently, we use the ‘error theory’ to replace interval numbers with their middle points, and combine the middle points with the traditional PCA to obtain the PC scores of interval number samples; finally, we adopt the thought of ‘equivalency’ to obtain the PC scores of IVIFN samples. Moreover, based on the IVIF-PCA model, we give a decision making method for the CMALGDM problem. The feasibility and validity of the decision making method is investigated through a numerical example.     

Intuitionistic Fuzzy Interval‐Valued Linguistic Entropic Combined Weighted Averaging Operator for Linguistic Group Decision Making

Entropy, a basic concept of measuring the amount of information and the degree of confusion, has been applied in many weighted averaging operators in the linguistic group decision making. In the paper, we construct an intuitionistic fuzzy linguistic entropy based on the intuitionistic fuzzy entropy and the intuitionistic fuzzy linguistic variable. Then, inspired by operations of concentration and dilation (De SK, Biswas R, and Roy AR, Fuzzy Sets Syst. 2000;114(3):477?484), we extend the intuitionistic fuzzy linguistic entropy to the intuitionistic fuzzy interval‐valued linguistic entropy. After that, the intuitionistic fuzzy interval‐valued linguistic entropic combined weighted averaging (IFIVLECWA) operator is proposed for multiple attribute linguistic group decision making. Finally, a numerical example about the selection of optimal alternative(s) is presented to illustrate the applicability and effectiveness of the proposed method.     

考虑时间序列的动态大群体应急决策方法

针对专家权重和属性权重未知、阶段权重未知且与时间序列有关的动态大群体应急决策问题,提出一种考虑时间序列的动态大群体应急决策方法.首先,提出一个考虑区间直觉模糊数犹豫度的距离公式,定义区间直觉模糊数贴近度,综合考虑贴近度和相似度,用模糊聚类法对大群体专家偏好信息进行聚类;其次,基于现有区间直觉模糊熵公式的不足,提出一个新的区间直觉模糊熵公式,基于此公式考虑专家之间知识水平的差异和各个阶段偏好信息不具遗传性等特点,计算得出专家在不同属性下的权重和属性在各阶段下的权重;再次,考虑时间序列对各阶段权重的影响,构建相对熵模型,对阶段权重进行合理确定,进而利用加权平均算子得到整个决策过程中各方案的综合决策偏好;然后,利用区间直觉模糊数的得分函数和精确函数对方案进行排序,选出最优方案;最后,通过与以往文献的方法对比分析验证所提出方法的有效性和优越性.     

用区间直觉模糊集方法对属性权重未知的群求解其多属性决策

本文首先提出群区间直觉模糊有序加权几何(groupinterval-valuedintuitionistic fuzzy orderedweighted geometric,GIVIFOWG)算子和群区间直觉模糊有序加权平均(group interval-valued intuitionistic fuzzy ordered weighted averaging,GIVIFOWA)算子.利用GIVIFOWG算子或GIVIFOWA算子聚集群的决策矩阵以获得方案在属性上的综合区间直觉模糊决策矩阵(collectiveinterval-valuedintuitionistic fuzzy decision-matrix,CIVIFDM).然后定义了一个考虑犹豫度的区间直觉模糊熵(interval-valuedintuitionistic fuzzyentropy,IVIFE);通过熵衡量每个属性所含的信息来求解属性权重.最后,提出基于可能度的接近理想解的区间排序法(interval technique for order preference by similarity to an ideal solution,ITOPSIS)和区间得分函数法.在ITOPSIS法中,依据区间距离公式计算候选方案和理想方案的属性加权区间距离,进而采用ITOPSIS准则对各方案进行排序;在区间得分函数法中,算出CIVIFDM中各方案的得分值以及精确值,然后利用区间得分准则对各方案进行排序.实验结果验证了决策方法的有效性和可行性.     

A ranking method for multiple attribute decision-making problems based on the possibility degrees of trapezoidal intuitionistic fuzzy numbers

To solve multiple attribute decision-making problems with attribute values or decision values characterized by trapezoidal intuitionistic fuzzy numbers (TIFNs), we define a trapezoidal intuitionistic fuzzy induced ordered weighted arithmetic averaging (TIFIOWA) operator, which is an extension of the induced ordered weighted arithmetic averaging operator. We derive and prove some related properties and conclusions of the TIFIOWA operator. To compare the TIFNs, we define possibility degrees of the TIFNs. Based on the possibility degrees of the TIFNs and the TIFIOWA operator, we construct a new method to determine the order of alternatives in multiple attribute decision making and to choose the best alternative. Finally, a numerical example shows that the developed method is feasible and effective.     

零模的直觉模糊不确定语言集成算子研究

基于零模与共轭零模算子,探讨了直觉模糊不确定语言变量运算法则,得到了基于零模与共轭零模的直觉模糊不确定语言加权几何算子,并给出了一种使用直觉不确定语言变量的集成算子的多属性群决策方法,最后通过Matlab软件分析了直觉模糊不确定语言加权几何算子的K值与语言术语下标间关系。为多属性群决策提供了有价值的参考,有效地解决了一类具有直觉模糊不确定语言评估信息的多属性群决策问题。     

基于直觉模糊推理的多属性群决策方法研究*

针对属性取值以直觉模糊数形式给出的多属性决策问题,提出了基于直觉模糊推理的多属性群决策方法。首先针对专家的评价信息构建直觉决策推理规则,然后根据规则之间的关系给出了决策推理模型,进而给出了基于直觉模糊推理的决策方法;最后通过购房实例验证了该方法的正确性和有效性。     

Some induced intuitionistic fuzzy aggregation operators applied to multi-attribute group decision making

In this paper, we define various induced intuitionistic fuzzy aggregation operators, including induced intuitionistic fuzzy ordered weighted averaging (OWA) operator, induced intuitionistic fuzzy hybrid averaging (I-IFHA) operator, induced interval-valued intuitionistic fuzzy OWA operator, and induced interval-valued intuitionistic fuzzy hybrid averaging (I-IIFHA) operator. We also establish various properties of these operators. And then, an approach based on I-IFHA operator and intuitionistic fuzzy weighted averaging (WA) operator is developed to solve multi-attribute group decision-making (MAGDM) problems. In such problems, attribute weights and the decision makers' (DMs') weights are real numbers and attribute values provided by the DMs are intuitionistic fuzzy numbers (IFNs), and an approach based on I-IIFHA operator and interval-valued intuitionistic fuzzy WA operator is developed to solve MAGDM problems where the attribute values provided by the DMs are interval-valued IFNs. Furthermore, induced intuitionistic fuzzy hybrid geometric operator and induced interval-valued intuitionistic fuzzy hybrid geometric operator are proposed. Finally, a numerical example is presented to illustrate the developed approaches.     

Grey relational analysis model for dynamic hybrid multiple attribute decision making

In this paper, the dynamic hybrid multiple attribute decision making problems, in which the decision information, provided by decision makers at different periods, is expressed in real numbers, interval numbers or linguistic labels (linguistic labels can be described by triangular fuzzy numbers), respectively, are investigated. The method first utilizes three different GRA (grey relational analysis (real-valued GRA, interval-valued GRA and fuzzy-valued GRA) to calculate the individual grey relational degree of each alternative to the positive and negative ideal alternatives based on the decision information expressed in real numbers, interval numbers and linguistic labels, respectively, provided by each decision maker at each period, and then adopt the concept of fuzzy membership grade and clustering to aggregate the grey relational degree of all the evaluated periods. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

Enhancing PROMETHEE method with intuitionistic fuzzy soft sets

The notion of intuitionistic fuzzy soft sets (IFSSs) provides an effective tool for solving multiple attribute decision making with intuitionistic fuzzy information. The most crucial issue in decision making based on IFSSs is how to derive the ranking of alternatives from the information quantified in terms of intuitionistic fuzzy values. In this study, we propose a new extension of the preference ranking organization method for enrichment evaluation (PROMETHEE), by taking advantage of IFSSs. In addition to presenting a myriad of new notions, such as intuitionistic fuzzy membership (or nonmembership) deviation matrices, intuitionistic fuzzy membership (or nonmembership) preference matrices, and aggregated intuitionistic fuzzy preference matrices, we put more emphasis on the construction of three distinct preference structures and related utility functions on the corresponding weakly ordered sets by considering the positive, negative, and net flows of the alternatives based on the aggregated intuitionistic fuzzy preference matrix. We present a new algorithm for solving multiple attribute decision-making problems with the extended PROMETHEE method based on IFSSs. Moreover, a benchmark problem concerning risk investment is investigated to give a comparative analysis and show the feasibility of our approach.