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.     

A method to aggregate crisp values into interval-valued intuitionistic fuzzy information for group decision making

Owing to the increasing complexity in modern society, aggregating group's knowledge and experiences to make an appropriate decision is an important research topic. The aim of this article is to present a soft computing model for multiple attribute group decision-making problems. This model aggregates all individual decisions on an attribute into an interval-valued intuitionistic fuzzy number (IVIFN), in which each individual decision as an attribute value is expressed in crisp value. Furthermore, we obtain a collective decision matrix, in which the attribute values are expressed by the aggregated IVIFNs. Then make a decision under intuitionistic fuzzy environment. To illustrate the decision process of the developed approach, we give an example for supplier selection and a sensitivity analysis with different attribute weights. Finally, we show a comparison with another group decision-making method from relevant literature.     

基于TOPSIS 的区间直觉模糊数排序法

基于传统的逼近理想解排序法(TOPSIS) 思想, 运用区间直觉模糊数的欧氏距离, 给出区间直觉模糊数相对于最大区间直觉模糊数的贴近度公式, 并给出区间直觉模糊数贴近度所具有的优良性质, 这些性质表明贴近度作为排序指标是合理的. 通过与文献中有关区间直觉模糊数排序法的对比分析, 表明基于贴近度的排序方法具有更高的区分能力. 运用新的排序指标提出一种区间直觉模糊多属性决策方法, 并通过实例表明了所提出方法的有效性．

     

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.     

A method of determining attribute weights in evidential reasoning approach based on incompatibility among attributes

Decisions on attribute weights are important problems in multiple attribute decision making. Many methods have been proposed to create attribute weights which are used to aggregate attributes in a simple additive weighting way. In this paper, a method of deriving attribute weights from incompatibility among attributes and possible constraints on the weights is developed based on the evidential reasoning approach in which attribute aggregation is nonlinear rather than linear. The incompatibility is a flexible combination of deviation incompatibility and decision incompatibility with a relaxation coefficient. The deviation incompatibility measures differences between assessments of alternatives on each attribute and the decision incompatibility quantifies differences between assessments of alternatives on one attribute and the aggregated assessments of the alternatives. For a specific alternative, two pairs of optimization problems with a constraint on the difference between potential weights and the combination of deviation incompatibility and decision incompatibility are designed to generate the favorable intervals of attribute weights and those of utilities of assessment grades. A problem of car performance assessment is investigated to demonstrate the applicability of the proposed method. The method is validated by comparison with other methods of producing attribute weights using the problem.     

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.     

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.     

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.     

Generalized score functions on interval-valued intuitionistic fuzzy sets with preference parameters for different types of decision makers and their application

In this paper, the interval-valued intuitionistic fuzzy sets (IVIFSs) are studied from the viewpoint of the decision makers’ preference. Firstly, two series of principles are proposed to guide the ranking of interval-valued intuitionistic fuzzy numbers (IVIFNs), and two kinds of illustrative generalized score functions on IVIFSs are proposed according to the newly proposed principles. Secondly, two kinds of generalized score functions on IVIFSs are proposed based on decision-makers’ preference. The two generalized score functions are both of two parameters, which represent the decision makers’ attitudinal characters on the classical score values and the classical accuracy values on IVIFNs, respectively. Thirdly, two kinds of generalized score functions on IVIFSs, which are suitable for ranking IVIFNs when there is no information about the importance weights of the classical score values and accuracy values on IVIFNs, are proposed based on integral. Fourthly, three kinds of multi-criteria decision-making (MCDM) methods in interval-valued intuitionistic fuzzy setting are proposed. Finally, an example shows that when a novel generalized score function on IVIFSs is proposed, its suitable application environments should also be pointed out.     

A stochastic multicriteria acceptability analysis–evidential reasoning method for uncertain multiattribute decision‐making problems

Evidential reasoning (ER) is an effective approach for assessing alternatives with uncertain attribute values in the context of decision making. For the ER approach to be able to handle variations in the weights of uncertain attributes in an appropriate manner, this paper proposes a method to solve problems of uncertain multiattribute decision making that involve both uncertain attribute values and uncertain attribute weights, which this method does by combining the ER approach and stochastic multicriteria acceptability analysis‐2 (SMAA‐2). First, the uncertainty in attribute values is described by using a belief decision matrix as in the ER approach. The analytical ER algorithm is then used to create the utility function in the SMAA‐2 model, and that function is used to calculate the probability of different sorting positions of the decision units under weight‐related restrictions. Finally, the results of ranking are obtained by combining the sorting weights. An example is provided to verify the effectiveness of the proposed 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.     

主客观权重集成及扩展VIKOR的多属性群决策方法

针对评价信息、属性权重均为不同粒度语言短语的多属性群决策问题,提出一种基于主客观权重集成及扩展多准则协调优化解（VIKOR）的多属性群决策方法。由基本语言评价集实现对多粒度语言评价矩阵的一致化,基于同一粒度的语言决策矩阵计算群体对属性的评价偏差,基于群体评价意见的一致性原则得到属性客观权重,通过二元语义加权算术平均（T-WAA）算子得到属性主观权重,从而集成主、客观权重求得属性综合权重。集结转化后的单个评价矩阵得到群体评价矩阵及其导出矩阵,由扩展VIKOR方法,根据群效用值、个体遗憾值及综合评价值分别对方案进行排序,获得折衷方案。算例分析表明该方法的有效性与可行性。     

基于IVIF-AHP与改进CRITIC法的配电网规划方案综合评估

针对难以量化配电网规划方案的权重问题,提出IVIF-AHP法与改进CRITIC法结合的综合评估配电网规划方案的方法.首先,从配电网供电可靠性、经济性、供电质量及安全性四个方面建立综合评估指标体系;其次,通过三角模糊数确定决策者自身属性,引入评估结果相似度与决策者犹豫度共同决定加权因子矩阵,根据评估矩阵聚合的IVIFN计...     

The use of ordered weighted averaging method for decision making under uncertainty

A decision making under uncertainty (DMUU) prevails at the outset and often evolves into a decision making under partial uncertainty as information on the states of nature, for example, a probability distribution, is advanced. Many methods have emerged for solving the DMUU problems, which includes the classical decision criteria and the domain criterion. Yager (1988) introduced a new approach, the so‐called ordered weighted averaging (OWA) as a viable method for solving the DMUU problems. The OWA weights to be used in the aggregation are generated under the degree of optimism provided by a decision maker and then combined with the reordered payoffs to produce aggregated payoffs for each strategy. The reordering process, one of the characterizing features of the OWA method, enables us to perform various types of aggregations including maximax, maximin, and Hurwicz‐α index in conjunction with the generated weights. The OWA method obviously extends the Hurwicz approach by taking into account the tradeoffs among the entire payoffs while the Hurwicz approach considers a tradeoff only between the two extremes, the maximum and the minimum payoffs. In this paper, we examine the features of the OWA method in light of Milnor's set of requirements for reasonable decision criteria, thus providing a solid methodological foundation for the DMUU. The OWA method can also be used to solve a group DMUU problem by exploiting individual decision results in the situation when the use of a fuzzy majority is advocated.     

三参数区间数下非线性可拓关联度决策方法

针对属性值为三参数区间数和权重未知的多属性决策问题,提出一种新的基于非线性可拓简单关联度的多属性决策方法和框架.该方法基于可拓简单关联函数提出非线性可拓三参数区间关联度的计算方法,通过定义区间映射变换算子,将不同类型指标的关联度计算均变换为效益型指标的单调递增关联度计算.该方法内含偏好态度系数的设置以反映决策者态度倾向带来的决策不确定性,而且能够体现人们进行决策评价时的非线性思维.最后通过算例分析表明所提出方法的合理性和稳定性.     

基于前景理论的犹豫模糊TOPSIS多属性决策方法

针对属性权重未知、属性值为犹豫模糊集的决策问题,提出一种前景理论和逼近理想解(TOPSIS)相结合的多属性决策方法.考虑到决策者对指标集的不同偏好,利用犹豫模糊熵的相关理论,提出一种基于犹豫模糊熵的熵权法确定属性权重.将决策者的风险心理因素引入犹豫模糊多属性决策中,定义了犹豫模糊数的前景价值函数,并以此将犹豫模糊决策矩阵转化为价值矩阵,计算出各方案的收益损失比值.最终应用TOPSIS的基本思路,确定备选方案的优劣排序,并通过算例分析验证了所提出方法的有效性.     

一种考虑区间直觉模糊集的多属性匹配决策方法

针对属性及属性权重均为区间直觉模糊数（IVIFN）的多属性匹配决策问题,提出一种匹配决策方法．首先根据区间直觉模糊数加权绝对值距离的定义,以逼近理想解法的思想,构建一方主体与另一方潜在对象最优匹配度的分式规划模型,并通过Charnes-Cooper变换,将原模型化为线性规划模型并求解模型得到双方的匹配度矩阵;然后,以匹配度最大为目标,建立一种双目标区间优化模型,通过线性加权转为单目标优化模型并求解得到匹配结果．最后,算例说明了所提方法的可行性和有效性．     

2TLCGPOWA算子及其在多属性群决策中的应用

基于集对分析理论中二元联系数的不确定性,将联系变量引入到区间二元语义环境中,定义了二元语义联系变量,给出了二元语义联系变量的运算法则,并提出了几种新的算术集结算子。针对决策矩阵元素为区间二元语义变量和属性权重完全未知的不确定多属性群决策问题,提出了一种基于2TLCGPOWA算子的不确定多属性群决策方法。最后通过对某大学教师的任职和晋升考核来说明该方法的可行性和有效性。     

基于前景M-V准则的正态三角模糊随机多属性决策方法

针对具有正态三角模糊随机变量且属性权重未知的多属性决策问题, 提出基于前景均值-方差(M-V) 准则的正态三角模糊随机多属性决策方法. 该方法首先构建正态三角模糊随机决策矩阵, 进而通过运算得到属性值的期望与方差, 并将其转化为M-V 决策矩阵; 然后, 通过定义前景效应构建前景M-V 决策矩阵, 利用改进灰色系统理论模型求解属性权重值, 获取综合前景M-V 决策矩阵; 最后, 定义前景序关系, 两两比较前景M-V 价值获取方案排序. 在此基础上, 通过案例验证了所提出方法的可行性及有效性.

     

犹豫模糊语言后悔理论和ELECTRE决策方法及应用

针对决策信息为犹豫模糊语言元素形式、属性权重完全未知的多属性决策问题,提出了一种基于后悔理论和ELECTRE III的多属性决策方法。利用灰色关联分析和极大熵原理确定属性权重。确定犹豫模糊语言信息的后悔-欣喜函数,凭借该函数确定方案对的后悔-欣喜和谐指数与不和谐指数,进而确定方案对的可信度指数。通过方案对的可信度指数确定各方案的净可信度,依此对方案进行排序。通过算例说明了所提方法的可行性和有效性。由于该方法同时考虑了决策者的心理行为和属性间的部分可补偿性,因此决策结果更加贴近现实且更为合理。对后悔规避系数[μ]的灵敏度分析表明了所提方法的稳定性,与其他两种方法的对比分析展示了所提方法的优势。