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。     

Incomplete interval-valued intuitionistic fuzzy preference relations

The aim of this paper is to investigate decision making problems with interval-valued intuitionistic fuzzy preference information, in which the preferences provided by the decision maker over alternatives are incomplete or uncertain. We define some new preference relations, including additive consistent incomplete interval-valued intuitionistic fuzzy preference relation, multiplicative consistent incomplete interval-valued intuitionistic fuzzy preference relation and acceptable incomplete interval-valued intuitionistic fuzzy preference relation. Based on the arithmetic average and the geometric mean, respectively, we give two procedures for extending the acceptable incomplete interval-valued intuitionistic fuzzy preference relations to the complete interval-valued intuitionistic fuzzy preference relations. Then, by using the interval-valued intuitionistic fuzzy averaging operator or the interval-valued intuitionistic fuzzy geometric operator, an approach is given to decision making based on the incomplete interval-valued intuitionistic fuzzy preference relation, and the developed approach is applied to a practical problem. It is worth pointing out that if the interval-valued intuitionistic fuzzy preference relation is reduced to the real-valued intuitionistic fuzzy preference relation, then all the above results are also reduced to the counterparts, which can be applied to solve the decision making problems with incomplete intuitionistic fuzzy preference information.     

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.     

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.     

Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making

With respect to multiple attribute group decision making (MAGDM) problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of intuitionistic fuzzy numbers or interval-valued intuitionistic fuzzy numbers, some new group decision making analysis methods are developed. Firstly, some operational laws, score function and accuracy function of intuitionistic fuzzy numbers or interval-valued intuitionistic fuzzy numbers are introduced. Then two new aggregation operators: induced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) operator and induced interval-valued intuitionistic fuzzy ordered weighted geometric (I-IIFOWG) operator are proposed, and some desirable properties of the I-IFOWG and I-IIFOWG operators are studied, such as commutativity, idempotency and monotonicity. An I-IFOWG and IFWG (intuitionistic fuzzy weighted geometric) operators-based approach is developed to solve the MAGDM problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of intuitionistic fuzzy numbers. Further, we extend the developed models and procedures based on I-IIFOWG and IIFWG (interval-valued intuitionistic fuzzy weighted geometric) operators to solve the MAGDM problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of interval-valued intuitionistic fuzzy numbers. Finally, some illustrative examples are given to verify the developed approach and to demonstrate its practicality and effectiveness.     

Multiple attribute group decision-making methods with unknown weights in intuitionistic fuzzy setting and interval-valued intuitionistic fuzzy setting

To better solve the corresponding multiple attribute group decision-making problem with unknown weights, multiple attribute group decision-making methods with completely unknown weights of decision-makers and incompletely known weights of attributes are proposed in intuitionistic fuzzy setting and interval-valued intuitionistic fuzzy setting. In the group decision-making method, two weight models are proposed based on the score function to determine the weights of both experts and attributes from the intuitionistic fuzzy decision matrices and the interval-valued intuitionistic fuzzy decision matrices. Then, overall evaluation formulas of weighted scores for each alternative are introduced in the intuitionistic fuzzy setting and the interval-valued intuitionistic fuzzy setting to obtain the ranking order of alternatives and the most desirable one(s). Finally, two numerical examples demonstrate the applicability and benefit of the proposed methods.     

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.     

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.     

属性权重不确定条件下的区间直觉模糊多属性决策

在区间直觉模糊集(Interval-valued intuitionistic fuzzy set, IVIFS)的框架内,重点研究了属性权重在一定约束条件下和属性权重完全未知的 多属性群决策问题.首先利用区间直觉模糊集成算子获得方案在属性上的综合区间直觉模糊决策矩阵,进一步依据逼近理想解排序法(Technique for order preference by similarity to an ideal solution, TOPSIS) 的思想计算候选方案和理想方案的加权距离,最后确定方案排序.其中针对属性权重在一定约束条件下的决策问题,提出了基于 区间直觉模糊集精确度函数的线性规划方法,用以解决属性权重求解问题.针对属性权重完全未知的决策问题,首先定义了区间直觉 模糊熵,其次通过熵衡量每一属性所含的信息量来求解属性权重.实验结果验证了决策方法的有效性和可行性.     

A mathematical programming approach to multi-attribute decision making with interval-valued intuitionistic fuzzy assessment information

This article proposes an approach to handle multi-attribute decision making (MADM) problems under the interval-valued intuitionistic fuzzy environment, in which both assessments of alternatives on attributes (hereafter, referred to as attribute values) and attribute weights are provided as interval-valued intuitionistic fuzzy numbers (IVIFNs). The notion of relative closeness is extended to interval values to accommodate IVIFN decision data, and fractional programming models are developed based on the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method to determine a relative closeness interval where attribute weights are independently determined for each alternative. By employing a series of optimization models, a quadratic program is established for obtaining a unified attribute weight vector, whereby the individual IVIFN attribute values are aggregated into relative closeness intervals to the ideal solution for final ranking. An illustrative supplier selection problem is employed to demonstrate how to apply the proposed procedure.     

An approach to multiattribute decision making with interval-valued intuitionistic fuzzy assessments and incomplete weights

This article proposes an approach to multiattribute decision making with incomplete attribute weight information where individual assessments are provided as interval-valued intuitionistic fuzzy numbers (IVIFNs). By employing a series of optimization models, the proposed approach derives a linear program for determining attribute weights. The weights are subsequently used to synthesize individual IVIFN assessments into an aggregated IVIFN value for each alternative. In order to rank alternatives based on their aggregated IVIFN values, a novel method is developed for comparing two IVIFNs by introducing two new functions: the membership uncertainty index and the hesitation uncertainty index. An illustrative investment decision problem is employed to demonstrate how to apply the proposed procedure and comparative studies are conducted to show its overall consistency with existing approaches.     

An approach for multiple attribute group decision making problems with interval-valued intuitionistic trapezoidal fuzzy numbers

This article proposes an approach to resolve multiple attribute group decision making (MAGDM) problems with interval-valued intuitionistic trapezoidal fuzzy numbers (IVITFNs). We first introduce the cut set of IVITFNs and investigate the attitudinal score and accuracy expected functions for IVITFNs. Their novelty is that they allow the comparison of IVITFNs by taking into accounting of the experts’ risk attitude. Based on these expected functions, a ranking method for IVITFNs is proposed and a ranking sensitivity analysis method with respect to the risk attitude is developed. To aggregate the information with IVITFNs, we study the desirable properties of the interval-valued intuitionistic trapezoidal fuzzy weighted geometric (IVITFWG) operator, the interval-valued intuitionistic trapezoidal fuzzy ordered weighted geometric (IVITFOWG) operator, and the interval-valued intuitionistic trapezoidal fuzzy hybrid geometric (IVITFHG) operator. It is worth noting that the aggregated value by using these operators is also an interval-valued intuitionistic trapezoidal fuzzy value. Then, based on these expected functions and aggregating operators, an approach is proposed to solve MAGDM problems in which the attribute values take the form of interval-valued intuitionistic fuzzy numbers and the expert weights take the form of real numbers. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

基于改进熵和新得分函数的区间直觉模糊多属性决策

针对决策信息为区间直觉模糊数且属性权重完全未知的多属性决策问题, 提出基于改进的区间直觉模糊熵和新得分函数的决策方法. 首先, 利用改进的区间直觉模糊熵确定属性权重; 然后, 利用区间直觉模糊加权算术平均算子集成信息, 得到各备选方案的综合属性值, 进而指出现有得分函数存在排序失效或排序不符合实际的不足, 同时给出一个新的得分函数, 并以此对方案进行排序; 最后, 通过实例表明了所提出方法的有效性.

     

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.     

MADM method based on prospect theory and evidential reasoning approach with unknown attribute weights under intuitionistic fuzzy environment

This paper proposes an intuitionistic fuzzy decision method based on prospect theory and the evidential reasoning approach, aiming at analyzing multi-attribute decision making problems in which the criteria values are intuitionistic fuzzy numbers and the information of attributes weights is unknown. Firstly, the measures of entropy and cross entropy are defined for intuitionistic fuzzy sets by taking into consideration the preference of decision maker towards hesitancy degree. Secondly, combined with bounded rationality, the prospect decision matrix is calculated in the light of prospect theory and intuitionistic fuzzy distance. Thirdly, the correlational analyses are conducted between the attribute weights and three indicators which are entropy, cross entropy and prospect value, and optimization models for identifying attribute weights are built under the circumstances that the weights are incomplete and unknown. Finally, in order to avoid the loss of decision making information, the evidential reasoning approach is applied to the calculation of comprehensive prospective values for all alternatives. Following the value calculation, the ranking and the optimal alternative are determined based on the comprehensive prospective values. Illustrating examples demonstrate that the proposed method is reasonable and feasible.     

基于分式规划的区间直觉梯形模糊数多属性决策方法

针对属性值为区间梯形直觉模糊且属性权重为区间数的多属性决策问题,提出一种基于分式规划的决策方法.定义了区间梯形直觉模糊数的Hamming距离和Euclidean距离,采用优劣解距离法构建了相对贴近度的非线性分式规划模型,并通过Charnes and Cooper变换转化为线性规划模型求解,得到各方案相对贴近度的区间数,进而提出了决策方法.数值算例分析验证了所提出方法的有效性.     

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。     

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.     

区间值模糊随机多准则决策方法

针对不完全信息的区间值模糊随机多准则决策问题,提出了两种求解方法。第一种方法利用离差最大化构建区间参数线性规划,通过区间数运算法则和定位规划求得最优准则权重向量、状态集结值区间决策矩阵与期望值区间决策矩阵,根据决策者风险偏好水平得到各方案的期望集结值从而确定排序。第二种方法将区间值模糊数决策矩阵转化为直觉模糊数决策矩阵,利用不完全的准则权重,通过规划模型求解,获取各方案在各自然状态下的加权记分函数值与加权精确函数值的区间,利用不完全的状态概率,得到各方案的记分函数期望值与精确函数期望值的区间,根据决策者风险偏好水平,求得各方案的记分函数与精确函数的期望集结值,进而确定方案的排序结果。算例分析验证了两种方法的有效性和可行性。     

Some induced correlated aggregating operators with intuitionistic fuzzy information and their application to multiple attribute group decision making

In this paper, some multiple attribute group decision making (MAGDM) problems in which both the attribute weights and the expert weights are usually correlative, attribute values take the form of intuitionistic fuzzy values or interval-valued intuitionistic fuzzy values, are investigated. Firstly, some operational law, score function and accuracy function of intuitionistic fuzzy values or interval-valued intuitionistic fuzzy values are introduced. Then two new aggregation operators: induced intuitionistic fuzzy correlated averaging (I-IFCA) operator and induced intuitionistic fuzzy correlated geometric (I-IFCG) operator are developed and some desirable properties of the I-IFCA and I-IFCG operators are studied, such as commutativity, idempotency and monotonicity. An I-IFCA and IFCA (intuitionistic fuzzy correlated averaging) operators-based approach is developed to solve the MAGDM problems in which both the attribute weights and the expert weights usually correlative, attribute values take the form of intuitionistic fuzzy values. Then, we extend the developed models and procedures to the interval-valued intuitionistic fuzzy environment. Finally, some illustrative examples are given to verify the developed approach and to demonstrate its practicality and effectiveness.