Multi-criteria decision-making method based on interval-valued intuitionistic fuzzy sets

Out of several generalizations of fuzzy set theory for various objectives, the notions introduced by Atanassov, 1983, Atanassov and Gargov, 1989 in defining intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets are interesting and very useful in modeling real life problems. Ranking of interval-valued intuitionistic fuzzy sets plays a vital role in decision-making, data analysis, artificial intelligence and socioeconomic system and it was studied in Xu, 2007c, Xu and Chen, 2007a, Ye, 2009. In this paper a new method for ranking interval-valued intuitionistic fuzzy sets has been introduced and studied. The method is illustrated by numerical examples and compared with other methods. And then a new method for handling multi-criteria fuzzy decision-making problems based on interval-valued intuitionistic fuzzy sets is presented in which criterion values for alternatives are interval-valued intuitionistic fuzzy sets. The method proposed here can provide a useful way to efficiently help the decision-maker to make his decision. An illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

Multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets

In this paper, we present a new method for multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets, where interval-valued intuitionistic fuzzy values are used to represent evaluating values of the decision-maker with respect to alternatives. First, we propose a new method for ranking interval-valued intuitionistic fuzzy values. Based on the proposed fuzzy ranking method of interval-valued intuitionistic fuzzy values, we propose a new method for multicriteria fuzzy decision making. The proposed multicriteria fuzzy decision making method outperforms Ye’s method (2009) due to the fact that the proposed method can overcome the drawback of Ye’s method (2009), where the drawback of Ye’s method is that it can not distinguish the ranking order between alternatives in some situations. The proposed method provides us with a useful way for dealing with multicriteria fuzzy decision making problems based on interval-valued intuitionistic fuzzy sets.     

Robot selection by using generalized interval-valued fuzzy numbers with TOPSIS

The aim of this paper is to propose a method to aggregate the opinion of several decision makers on different criteria, regarding a set of alternatives, where the judgment of the decision makers are represented by generalized interval-valued trapezoidal fuzzy numbers. A generalized interval valued trapezoidal fuzzy number based technique for order preference by similarity to ideal solution is proposed that can reflect subjective judgment and objective information in real life. The weights of criteria and performance rating values of criteria are linguistic variables expressed as generalized interval-valued trapezoidal fuzzy numbers. Finally, an illustrative example is provided to elaborate the proposed method for the selection of a suitable robot according to our requirements.     

Fuzzy multicriteria decision making method based on the improved accuracy function for interval-valued intuitionistic fuzzy sets

A new accuracy function for the theory of interval-valued intuitionistic fuzzy set, which overcomes some difficulties arising in the existing methods for determining rank of interval-valued intuitionistic fuzzy numbers, is proposed by taking into account the hesitancy degree of interval-valued intuitionistic fuzzy sets. By comparing it with several proposed accuracy functions, the necessity and efficiency of our accuracy function are provided by giving related examples. A fuzzy multicriteria decision making method is established to select the best alternative in multicriteria decision making process which is taken as interval-valued intuitionistic fuzzy set of criterion values for alternatives. While aggregating the interval-valued intuitionistic fuzzy information corresponding to each alternative, we utilize the interval-valued intuitionistic fuzzy weighted aggregation operators. Then the accuracy degree of the aggregated interval-valued intuitionistic fuzzy information is computed via the new proposed accuracy function. Thus, we can rank all the alternatives according to the accuracy function and choose the optimal one(s). Finally, an illustrative example is given to demonstrate the practicality and effectiveness of the proposed approach.     

A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making

The ranking of interval-valued intuitionistic fuzzy sets (IVIFSs) is very important for the interval-valued intuitionistic fuzzy decision making. From the probability viewpoint, the possibility degree of comparison between two interval-valued intuitionistic fuzzy numbers (IVIFNs) is defined by using the notion of 2-dimensional random vector, and a new method is then developed to rank IVIFNs. Hereby the ordered weighted average operator and hybrid weighted average operator for IVIFNs are defined based on the Karnik–Mendel algorithms and employed to solve multi-attribute group decision making problems with IVIFNs. The individual overall attribute values of alternatives are obtained by using the weighted average operator for IVIFNs. By using the hybrid weighted average operator for IVIFNs, we can obtain the collective overall attribute values of alternatives, which are used to rank the alternatives. A numerical example is examined to illustrate the effectiveness and flexibility of the proposed method in this paper.     

Generalized interval-valued fuzzy variable precision rough sets determined by fuzzy logical operators

The fuzzy rough set model and interval-valued fuzzy rough set model have been introduced to handle databases with real values and interval values, respectively. Variable precision rough set was advanced by Ziarko to overcome the shortcomings of misclassification and/or perturbation in Pawlak rough sets. By combining fuzzy rough set and variable precision rough set, a variety of fuzzy variable precision rough sets were studied, which cannot only handle numerical data, but are also less sensitive to misclassification. However, fuzzy variable precision rough sets cannot effectively handle interval-valued data-sets. Research into interval-valued fuzzy rough sets for interval-valued fuzzy data-sets has commenced; however, variable precision problems have not been considered in interval-valued fuzzy rough sets and generalized interval-valued fuzzy rough sets based on fuzzy logical operators nor have interval-valued fuzzy sets been considered in variable precision rough sets and fuzzy variable precision rough sets. These current models are incapable of wide application, especially on misclassification and/or perturbation and on interval-valued fuzzy data-sets. In this paper, these models are generalized to a more integrative approach that not only considers interval-valued fuzzy sets, but also variable precision. First, we review generalized interval-valued fuzzy rough sets based on two fuzzy logical operators: interval-valued fuzzy triangular norms and interval-valued fuzzy residual implicators. Second, we propose generalized interval-valued fuzzy variable precision rough sets based on the above two fuzzy logical operators. Finally, we confirm that some existing models, including rough sets, fuzzy variable precision rough sets, interval-valued fuzzy rough sets, generalized fuzzy rough sets and generalized interval-valued fuzzy variable precision rough sets based on fuzzy logical operators, are special cases of the proposed models.     

An interval-valued intuitionistic fuzzy permutation method with likelihood-based preference functions and its application to multiple criteria decision analysis

This paper presents an interval-valued intuitionistic fuzzy permutation method with likelihood-based preference functions for managing multiple criteria decision analysis based on interval-valued intuitionistic fuzzy sets. First, certain likelihood-based preference functions are proposed using the likelihoods of interval-valued intuitionistic fuzzy preference relationships. Next, selected practical indices of concordance/discordance are established to evaluate all possible permutations of the alternatives. The optimal priority order of the alternatives is determined by comparing all comprehensive concordance/discordance values based on score functions. Furthermore, this paper considers various preference types and develops another interval-valued intuitionistic fuzzy permutation method using programming models to address multiple criteria decision-making problems with incomplete preference information. The feasibility and applicability of the proposed methods are illustrated in the problem of selecting a suitable bridge construction method. Moreover, certain comparative analyses are conducted to verify the advantages of the proposed methods compared with those of other decision-making methods. Finally, the practical effectiveness of the proposed methods is validated with a risk assessment problem in new product development.     

Interval-valued intuitionistic fuzzy mathematical programming method for hybrid multi-criteria group decision making with interval-valued intuitionistic fuzzy truth degrees

As an important component of group decision making, the hybrid multi-criteria group decision making (MCGDM) is very complex and interesting in real applications. The purpose of this paper is to develop a novel interval-valued intuitionistic fuzzy (IVIF) mathematical programming method for hybrid MCGDM considering alternative comparisons with hesitancy degrees. The subjective preference relations between alternatives given by each decision maker (DM) are formulated as an IVIF set (IVIFS). The IVIFSs, intuitionistic fuzzy sets (IFSs), trapezoidal fuzzy numbers (TrFNs), linguistic variables, intervals and real numbers are used to represent the multiple types of criteria values. The information of criteria weights is incomplete. The IVIFS-type consistency and inconsistency indices are defined through considering the fuzzy positive and negative ideal solutions simultaneously. To determine the criteria weights, we construct a novel bi-objective IVIF mathematical programming of minimizing the inconsistency index and meanwhile maximizing the consistency index, which is solved by the technically developed linear goal programming approach. The individual ranking order of alternatives furnished by each DM is subsequently obtained according to the comprehensive relative closeness degrees of alternatives to the fuzzy positive ideal solution. The collective ranking order of alternatives is derived through establishing a new multi-objective assignment model. A real example of critical infrastructure evaluation is provided to demonstrate the applicability and effectiveness of this method.     

一种新的区间直觉模糊集决策方法: 区间证据组合的角度

基于证据理论,提出一种新的区间直觉模糊集决策模型.首先采用区间直觉模糊集表示属性值,将区间直觉模糊数转换为区间BPA;然后利用基于区间数的组合规则进行融合;最后将融合后的区间BPA转换为经典BPA用于决策,可直接方便地实现多属性数据的融合.该模型的优点在于:简单直观,能更有效地反映原始信息的不确定度;通用性好,可以推广到其他区间直觉模糊集的应用领域.算例结果表明了所提出模型的有效性.     

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.     

Application of correlation coefficient to interval-valued intuitionistic fuzzy multiple attribute decision-making with incomplete weight information

With respect to multiple attribute decision-making problems with interval-valued intuitionistic fuzzy information, some operational laws of interval-valued intuitionistic fuzzy numbers, correlation and correlation coefficient of interval-valued intuitionistic fuzzy sets are introduced. An optimization model based on the negative ideal solution and max-min operator, by which the attribute weights can be determined, is established. We utilize the interval-valued intuitionistic fuzzy weighted averaging operator proposed by Xu (Control Decis 22(2):215–219, 2007) to aggregate the interval-valued intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the correlation coefficient. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

Imprecise information in Bayes classifier

The paper considers the problem of classification error in pattern recognition. This model of classification is primarily based on the Bayes rule and secondarily on the notion of intuitionistic or interval-valued fuzzy sets. A probability of misclassifications is derived for a classifier under the assumption that the features are class-conditionally statistically independent, and we have intuitionistic or interval-valued fuzzy information on object features instead of exact information. A probability of the intuitionistic or interval-valued fuzzy event is represented by the real number. Additionally, the received results are compared with the bound on the probability of error based on information energy. Numerical example concludes the work.     

Soft computing based on new interval-valued fuzzy modified multi-criteria decision-making method

In this paper, a new interval-valued fuzzy modified TOPSIS (IVFM-TOPSIS) method is proposed that can reflect both subjective judgment and objective information in real life situations. This proposed method is based on concepts of the positive ideal and negative ideal solutions for solving multi-criteria decision-making (MCDM) problems in a fuzzy environment. The performance rating values and weights of criteria are linguistic variables expressed as triangular interval-valued fuzzy numbers. Furthermore, we appraise the performance of alternatives against both subjective and objective criteria with multi-judges for decision-making problems. Finally, for the purpose of proving the validity of the proposed method a numerical example is presented for a robot selection problem.     

Extension of the ELECTRE method based on interval-valued fuzzy sets

Decision-making is the process of finding the best option among the feasible alternatives. In classical multiple criteria decision-making (MCDM) methods, the ratings and the weights of the criteria are known precisely. However, if decision makers cannot reach an agreement on the method of defining linguistic variables based on the fuzzy sets, the interval-valued fuzzy set theory can provide a more accurate modeling. In this paper, the interval-valued fuzzy ELECTRE method is presented aiming at solving MCDM problems in which the weights of criteria are unequal, using interval-valued fuzzy set concepts. For the purpose of proving the validity of the proposed model, we present a numerical example and build a practical maintenance strategy selection problem.     

基于区间直觉模糊数的双向投影决策模型

研究了权重不完全确定,评价信息为区间直觉模糊数的多属性决策问题。提出了方案与理想方案、临界方案形成的向量表达方式,建立了针对区间直觉模糊信息的向量投影测度方法;构建了基于Jaynes最大熵原理和方案公平竞争下的非线性规划属性权重确定模型;提出了基于理想方案与临界方案的贴近度测算公式,以此对方案进行排序。通过算例对比分析说明了该方法的有效性和可行性。     

Analyzing fuzzy risk based on similarity measures between interval-valued fuzzy numbers

In this paper, we present a new method for handling fuzzy risk analysis problems based on the proposed new similarity measure between interval-valued fuzzy numbers. First, we present a new similarity measure between interval-valued fuzzy numbers. It considers the degrees of closeness between interval-valued fuzzy numbers on the X-axis and the degrees of differences between the shapes of the interval-valued fuzzy numbers on the X-axis and the Y-axis, respectively. We also prove three properties of the proposed similarity measure. Then, we make an experiment to compare the experimental results of the proposed method with the existing similarity measures between interval-valued fuzzy numbers. The proposed method can overcome the drawbacks of the existing methods. Finally, based on the proposed similarity measure between interval-valued fuzzy numbers, we present a new fuzzy risk analysis algorithm for dealing with fuzzy risk analysis problems. Because the proposed method allows the evaluating values of sub-components to be represented by interval-valued fuzzy numbers, it is more flexible than Chen and Chen’s method (2003).     

Generalized Dice measures for multiple attribute decision making under intuitionistic and interval-valued intuitionistic fuzzy environments

This paper proposes Dice measures of intuitionistic fuzzy sets (IFSs) and interval-valued intuitionistic fuzzy sets (IVIFSs) and generalized Dice measures of IFSs and IVIFSs, and then indicates the relations of the generalized Dice measures, Dice measures, and projection measures (asymmetric measures) of IFSs and IVIFSs. Furthermore, we develop the generalized Dice measures-based multiple attribute decision-making methods with intuitionistic and interval-valued intuitionistic fuzzy information. Through the weighted generalized Dice measures between each alternative and the ideal solution (ideal alternative) according to some parameter value selected by decision makers’ preference, all the alternatives can be ranked and the best one can be chosen as well. Finally, an actual example about the selection of manufacturing schemes is provided to demonstrate the applications of the proposed decision-making methods under intuitionistic and interval-valued intuitionistic fuzzy environments, and then, a comparison analysis is conducted between the developed approach and other existing methods to verify the effectiveness and flexibility of the proposed method.

     

Multiattribute decision making based on interval-valued intuitionistic fuzzy values

In this paper, we present a new multiattribute decision making method based on the proposed interval-valued intuitionistic fuzzy weighted average operator and the proposed fuzzy ranking method for intuitionistic fuzzy values. First, we briefly review the concepts of interval-valued intuitionistic fuzzy sets and the Karnik–Mendel algorithms. Then, we propose the intuitionistic fuzzy weighted average operator and interval-valued intuitionistic fuzzy weighted average operator, based on the traditional weighted average method and the Karnik–Mendel algorithms. Then, we propose a fuzzy ranking method for intuitionistic fuzzy values based on likelihood-based comparison relations between intervals. Finally, we present a new multiattribute decision making method based on the proposed interval-valued intuitionistic fuzzy weighted average operator and the proposed fuzzy ranking method for intuitionistic fuzzy values. The proposed method provides us with a useful way for multiattribute decision making based on interval-valued intuitionistic fuzzy values.     

区间直觉模糊信息下的双向投影决策模型

研究权重完全未知、评价信息为区间直觉模糊数的多准则决策问题. 考虑犹豫度影响, 给出备选方案与正理想方案、负理想方案形成的向量表达方式, 提出一种针对区间直觉模糊信息的向量投影测度方法; 构建基于方案投影总偏差最小的非线性规划准则权重确定模型; 给出基于方案投影的相对贴近度测算公式, 并以此对方案进行排序. 最后通过算例对比分析表明了所提出方法的有效性和可行性.