Consistency analysis of triangular fuzzy reciprocal preference relations

In order to simulate the uncertainty associated with impression or vagueness, a decision maker may give her/his judgments by means of triangular fuzzy reciprocal preference relations in the process of decision making. The study of their consistency becomes a very important aspect to avoid a misleading solution. Based on the reciprocity property, this paper proposes a new definition of consistent triangular fuzzy reciprocal preference relations. The new definition is different from that reduced by consistent fuzzy reciprocal preference relations proposed by Buckley (1985). The properties of consistent triangular fuzzy reciprocal preference relations in the light of the new definition are studied in detail. In addition, the shortcomings of the proof procedure of the proposition given by Wang and Chen (2008) are pointed out. And the proposition is reproved by using the new definition of consistent triangular fuzzy reciprocal preference relations. Finally, using the (n − 1) restricted comparison ratios, a method for obtaining consistent triangular fuzzy reciprocal preference relations is proposed, and an algorithm is shown to make a consistent decision ranking. Numerical results are further calculated to illustrate the new definition and the obtained algorithm.     

Note on “Some models for deriving the priority weights from interval fuzzy preference relations”

Deriving accurate interval weights from interval fuzzy preference relations is key to successfully solving decision making problems. Xu and Chen (2008) proposed a number of linear programming models to derive interval weights, but the definitions for the additive consistent interval fuzzy preference relation and the linear programming model still need to be improved. In this paper, a numerical example is given to show how these definitions and models can be improved to increase accuracy. A new additive consistency definition for interval fuzzy preference relations is proposed and novel linear programming models are established to demonstrate the generation of interval weights from an interval fuzzy preference relation.     

Some models for deriving the priority weights from interval fuzzy preference relations

Interval fuzzy preference relation is a useful tool to express decision maker’s uncertain preference information. How to derive the priority weights from an interval fuzzy preference relation is an interesting and important issue in decision making with interval fuzzy preference relation(s). In this paper, some new concepts such as additive consistent interval fuzzy preference relation, multiplicative consistent interval fuzzy preference relation, etc., are defined. Some simple and practical linear programming models for deriving the priority weights from various interval fuzzy preference relations are established, and two numerical examples are provided to illustrate the developed models.     

Derivation of intuitionistic fuzzy weights based on intuitionistic fuzzy preference relations

This paper proposes linear goal programming models for deriving intuitionistic fuzzy weights from intuitionistic fuzzy preference relations. Novel definitions are put forward to define additive consistency and weak transitivity for intuitionistic fuzzy preference relations, followed by a study of their corresponding properties. For any given normalized intuitionistic fuzzy weight vector, a transformation formula is furnished to convert the weights into a consistent intuitionistic fuzzy preference relation. For any intuitionistic fuzzy preference relation, a linear goal programming model is developed to obtain its intuitionistic fuzzy weights by minimizing its deviation from the converted consistent intuitionistic fuzzy preference relation. This approach is then extended to group decision-making situations. Three numerical examples are provided to illustrate the validity and applicability of the proposed models.     

偏好信息为模糊互反判断矩阵的模糊多属性决策法

研究只有部分权重信息且决策者对方案的偏好信息以模糊互反判断矩阵形式给出的模糊多属性决策问题。提出了一种基于目标规划模型的模糊多属性决策方法。该法首先基于模糊互反判断矩阵，利用转换函数将决策信息一致化，建立了一个目标规划模型．通过求解该模型确定属性的权重，然后运用加性加权法求出各方案的模糊综合属性值，并利用已有的三角模糊数排序公式求得决策方案的排序。文章最后把该法应用于解决风险投资领域中的项目评估问题。     

A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making

In decision making problems, there may be the cases where the decision makers express their judgements by using preference relations with incomplete information. Then one of the key issues is how to estimate the missing preference values. In this paper, we introduce an incomplete interval multiplicative preference relation and give the definitions of consistent and acceptable incomplete ones, respectively. Based on the consistency property of interval multiplicative preference relations, a goal programming model is proposed to complement the acceptable incomplete one. A new algorithm of obtaining the priority vector from incomplete interval multiplicative preference relations is given. The goal programming model is further applied to group decision-making (GDM) where the experts evaluate their preferences as acceptable incomplete interval multiplicative preference relations. An interval weighted geometric averaging (IWGA) operator is proposed to aggregate individual preference relations into a social one. Furthermore, the social interval multiplicative preference relation owns acceptable consistency when every individual one is acceptably consistent. Two numerical examples are carried out to show the efficiency of the proposed goal programming model and the algorithms.     

两类模糊数判断矩阵的转换关系及一致性研究

研究了三角模糊数互反和互补判断矩阵的相互转换和一致性问题.提出了三角模糊数互反判断矩阵完全一致性的定义以及三角模糊数互补判断矩阵加性一致性和乘性一致性的定义,给出了两类模糊数判断矩阵相互转化的公式,论证了转换公式对判断矩阵一致性的保持关系.最后,基于一致性模糊数判断矩阵元素和排序权值的关系,建立了两个方案排序的非线性规划模型.     

Interval weight generation approaches for reciprocal relations

In this paper, we propose methods to derive interval weight vectors from reciprocal relations for reflecting the inconsistency when decision makers provide preferences over alternatives (or criteria). Several goal programming models are established to minimize the inconsistency based on multiplicative and additive consistency, respectively. Especially, if we obtain a crisp weight vector from a reciprocal relation, then it is consistent. Then, we extend the proposed methods to incomplete reciprocal relations and interval reciprocal relations and develop the corresponding models to derive interval weight vectors. Several examples are also given to compare the developed methods with the existing ones.     

犹豫直觉模糊偏好关系积性一致性及其在群决策中的应用

模糊偏好关系在群决策中得到了广泛研究,针对犹豫直觉模糊集既能反映决策者偏好和非偏好的信息,又能描述其犹豫心理的特点,提出了犹豫直觉模糊偏好关系及其积性一致性的定义。为了修复不一致的犹豫直觉模糊偏好关系,先构建积性一致性指标,然后提出两种修复方法。最后,将犹豫直觉模糊偏好关系应用到群决策中,通过实例和比较说明了两种修复方法的有效性和合理性。     

Iterative algorithms for improving consistency of intuitionistic preference relations

Consistency of preference relations is an important research topic in decision making with preference information. The existing research about consistency mainly focuses on multiplicative preference relations, fuzzy preference relations and linguistic preference relations. Intuitionistic preference relations, each of their elements is composed of a membership degree, a non-membership degree and a hesitation degree, can better reflect the very imprecision of preferences of decision makers. There has been little research on consistency of intuitionistic preference relations up to now, and thus, it is necessary to pay attention to this issue. In this paper, we first propose an approach to constructing the consistent (or approximate consistent) intuitionistic preference relation from any intuitionistic preference relation. Then we develop a convergent iterative algorithm to improve the consistency of an intuitionistic preference relation. Moreover, we investigate the consistency of intuitionistic preference relations in group decision making situations, and show that if all individual intuitionistic preference relations are consistent, then the collective intuitionistic preference relation is also consistent. Moreover, we develop a convergent iterative algorithm to improve the consistency of all individual intuitionistic preference relations. The practicability and effectiveness of the developed algorithms is verified through two examples.     

An integrated model-based interactive approach to FMAGDM with incomplete preference information

Incomplete fuzzy preference relations, incomplete multiplicative preference relations, and incomplete linguistic preference relations are very useful to express decision makers’ incomplete preferences over attributes or alternatives in the process of decision making under fuzzy environments. The aim of this paper is to investigate fuzzy multiple attribute group decision making problems where the attribute values are represented in intuitionistic fuzzy numbers and the information on attribute weights is provided by decision makers by means of one or some of the different preference structures, including weak ranking, strict ranking, difference ranking, multiple ranking, interval numbers, incomplete fuzzy preference relations, incomplete multiplicative preference relations, and incomplete linguistic preference relations. We transform all individual intuitionistic fuzzy decision matrices into the interval decision matrices and construct their expected decision matrices, and then aggregate all these expected decision matrices into a collective one. We establish an integrated model by unifying the collective decision matrix and all the given different structures of incomplete weight preference information, and develop an integrated model-based approach to interacting with the decision makers so as to adjust all the inconsistent incomplete fuzzy preference relations, inconsistent incomplete linguistic preference relations and inconsistent incomplete multiplicative preference relations into the ones with acceptable consistency. The developed approach can derive the attribute weights and the ranking of the alternatives directly from the integrated model, and thus it has the following prominent characteristics: (1) it does not need to construct the complete fuzzy preference relations, complete linguistic preference relations and complete multiplicative preference relations from the incomplete fuzzy preference relations, incomplete linguistic preference relations and incomplete multiplicative preference relations, respectively; (2) it does not need to unify the different structures of incomplete preferences, and thus can simplify the calculation and avoid distorting the given preference information; and (3) it can sufficiently reflect and adjust the subjective desirability of decision makers in the process of interaction. A practical example is also provided to illustrate the developed approach.     

A consistency and consensus-based method to group decision making with interval linguistic preference relations

Preference relations are a powerful tool to address decision-making problems. In some situations, because of the complexity of decision-making problems and the inherent uncertainty, the decision makers cannot express their preferences by using numerical values. Interval linguistic preference relations, which are more reliable and informative for the decision-makers’ preferences, are a good choice to cope with this issue. Just as with the other types of preference relations, the consistency and consensus analysis is very importance to ensure the reasonable ranking order by using interval linguistic preference relations. Considering this situation, this paper introduces a consistency concept for interval linguistic preference relations. To measure the consistency of interval linguistic preference relations, a consistency measure is defined. Then, a consistency-based programming model is built, by which the consistent linguistic preference relations with respect to each object can be obtained. To cope with the inconsistency case, two models for deriving the adjusted consistent linguistic preference relations are constructed. Then, a consistency-based programming model to estimate the missing values is built. After that, we present a group consensus index and present some of its desirable properties. Furthermore, a group consensus-based model to determine the weights of the decision makers with respect to each object is established. Finally, an approach to group decision making with interval linguistic preference relations is developed, which is based on the consistency and consensus analysis. Meanwhile, the associated numerical examples are offered to illustrate the application of the procedure.     

Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations

The aim of this paper is to present a logarithmic least squares method (LLSM) to priority for group decision making with incomplete fuzzy preference relations. We give a reasonable definition of multiplicative consistent for incomplete fuzzy preference relation. We develop the acceptable fuzzy consistency ratio (FCR for short), which is simple and similar to Saaty’s consistency ratio CR for multiplicative fuzzy preference relations. We also extend the LLSM method to the case of individual preference relation with complete information. Finally, some examples are illustrated to show that our method is simple, efficient, and can be performed on computer easily.     

三角模糊数互补判断矩阵排序的目标规划法

针对决策者以三角模糊数互补判断矩阵形式给出的多目标决策问题.给出三角模糊数加性一致性互补判断矩阵的判定定理.利用该定理基于最小偏差建立一个目标规划模型而解得三角模糊数互补判断矩阵的权重向量,从而使用三角模糊数排序公式对方案排序,提出了基于目标规划的三角模糊数互补判断矩阵排序法.最后,将模型与方法应用于项目投资决策中.     

三角模糊偏好下冲突型多属性群决策方法研究

针对三角模糊偏好下冲突型群决策问题,本文提出一种新的决策方法。在冲突消解阶段,用三角模糊数表示决策专家偏好,定义两三角模糊数型偏好矢量间的相似度,通过计算专家对各个方案的偏好矢量与各方案的群偏好矢量间的相似度,以此为基础定义专家的冲突测度。给出阈值和协商机制调控专家的冲突测度,直到所有的专家的冲突测度都小于给定阈值,进入决策阶段。在决策阶段,利用三角模糊数的期望函数确定属性权重,计算各个方案群偏好矢量与理想方案偏好矢量之间的加权相似度,由加权相似度大小排列决策,选出最优方案。最后给出案例应用,利用Matlab画出各方案的冲突测度图,数值结果表明本文方法的可行性及有效性。     

A fuzzy goal programming method with imprecise goal hierarchy

Two most widely used approaches to treating goals of different importance in goal programming (GP) are: (1) weighted GP, where importance of goals is modelled using weights, and (2) preemptive priority GP, where a goal hierarchy is specified implying infinite trade-offs among goals placed in different levels of importance. These approaches may be too restrictive in modelling of real life decision making problems. In this paper, a novel fuzzy goal programming method is proposed, where the hierarchical levels of the goals are imprecisely defined. The imprecise importance relations among the goals are modelled using fuzzy relations. An additive achievement function is defined, which takes into consideration both achievement degrees of the goals and degrees of satisfaction of the fuzzy importance relations. Examples are given to illustrate the proposed method.     

模糊判断矩阵一致性逼近及排序方法

根据一致性模糊判断矩阵定义,提出了一种求取一致性判断矩阵及方案排序的新方法,该方法是通过建立一个线性目标规划模型来得到排序向量,并相应地得到逼近于决策偏好的一致性判断矩阵,最后给出了一个算例。     

部分权重信息下对方案有互补偏好的多属性决策法

研究了只有部分权重信息且对方案的偏好信息以模糊互补判断矩阵形式给出的多属性决策问题.首先,基于模糊互补判断矩阵的主观偏好信息,利用转换函数将客观决策信息一致化,建立一个目标规划模型,通过求解该模型得到属性权重,从而利用加性加权法获得各方案的综合属性值,并以此对方案进行排序或择优.提出了一种基于目标规划的多属性决策方法.该方法具有操作简便和易于上机实现的特点.最后,通过实例说明模型及方法的可行性和有效性.     

Stackelberg solutions for fuzzy random two-level linear programming through probability maximization with possibility

This paper considers Stackelberg solutions for decision making problems in hierarchical organizations under fuzzy random environments. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced into the formulated fuzzy random two-level linear programming problems. On the basis of the possibility and necessity measures that each objective function fulfills the corresponding fuzzy goal, together with the introduction of probability maximization criterion in stochastic programming, we propose new two-level fuzzy random decision making models which maximize the probabilities that the degrees of possibility and necessity are greater than or equal to certain values. Through the proposed models, it is shown that the original two-level linear programming problems with fuzzy random variables can be transformed into deterministic two-level linear fractional programming problems. For the transformed problems, extended concepts of Stackelberg solutions are defined and computational methods are also presented. A numerical example is provided to illustrate the proposed methods.     

Extension of the LINMAP for multiattribute decision making under Atanassov’s intuitionistic fuzzy environment

The aim of this article is further extending the linear programming techniques for multidimensional analysis of preference (LINMAP) to develop a new methodology for solving multiattribute decision making (MADM) problems under Atanassov’s intuitionistic fuzzy (IF) environments. The LINMAP only can deal with MADM problems in crisp environments. However, fuzziness is inherent in decision data and decision making processes. In this methodology, Atanassov’s IF sets are used to describe fuzziness in decision information and decision making processes by means of an Atanassov’s IF decision matrix. A Euclidean distance is proposed to measure the difference between Atanassov’s IF sets. Consistency and inconsistency indices are defined on the basis of preferences between alternatives given by the decision maker. Each alternative is assessed on the basis of its distance to an Atanassov’s IF positive ideal solution (IFPIS) which is unknown a prior. The Atanassov’s IFPIS and the weights of attributes are then estimated using a new linear programming model based upon the consistency and inconsistency indices defined. Finally, the distance of each alternative to the Atanassov’s IFPIS can be calculated to determine the ranking order of all alternatives. A numerical example is examined to demonstrate the implementation process of this methodology. Also it has been proved that the methodology proposed in this article can deal with MADM problems under not only Atanassov’s IF environments but also both fuzzy and crisp environments.