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.     

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.     

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.     

A robust additive consistency-based method for decision making with triangular fuzzy reciprocal preference relations

To express uncertain information in decision making, triangular fuzzy reciprocal preference relations (TFRPRs) might be adopted by decision makers. Considering consistency of this type of preference relations, this paper defines a new additive consistency concept, which can be seen as an extension of that for reciprocal preference relations. Then, a simple method to calculate the triangular fuzzy priority weight vector is introduced. When TFRPRs are inconsistent, a linear goal programming framework to derive the completely additive consistent TFRPRs is provided. Meanwhile, an improved linear goal programming model is constructed to estimate the missing values in an incomplete TFRPR that can address the situation where ignored objects exist. After that, an algorithm for decision making with TFRPRs is presented. Finally, numerical examples and comparison analysis are offered.     

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.     

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.     

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。     

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.     

Eigenvector method,consistency test and inconsistency repairing for an incomplete fuzzy preference relation

In this paper, we extend the eigenvector method (EM) to priority for an incomplete fuzzy preference relation. We give a reasonable definition of multiplicative consistency for an incomplete fuzzy preference relation. We also give an approach to judge whether an incomplete fuzzy relation is acceptable or not. We develop the acceptable consistency ratio for an incomplete multiplicative fuzzy preference relation, which is simple and similar to Saaty’s consistency ratio (CR) for the multiplicative preference relation. If the incomplete fuzzy preference relation is not of acceptable consistency, we define a criterion to find the unusual and false element (UFE) in the preference relation, and present an algorithm to repair an inconsistent fuzzy preference relation until its consistency is satisfied with the consistency ratio. As a result, our improvement method cannot only satisfy the consistency requirement, but also preserve the initial preference information as much as possible. Finally, an example is illustrated to show that our method is simple, efficiency, and can be performed on computer easily.     

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.     

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.     

Multiperson decision-making based on multiplicative preference relations

A multiperson decision-making problem, where the information about the alternatives provided by the experts can be presented by means of different preference representation structures (preference orderings, utility functions and multiplicative preference relations) is studied. Assuming the multiplicative preference relation as the uniform element of the preference representation, a multiplicative decision model based on fuzzy majority is presented to choose the best alternatives. In this decision model, several transformation functions are obtained to relate preference orderings and utility functions with multiplicative preference relations. The decision model uses the ordered weighted geometric operator to aggregate information and two choice degrees to rank the alternatives, quantifier guided dominance degree and quantifier guided non-dominance degree. The consistency of the model is analysed to prove that it acts coherently.     

Group decision making based on intuitionistic multiplicative aggregation operators

Preference relations are the most common techniques to express decision maker’s preference information over alternatives or criteria. To consistent with the law of diminishing marginal utility, we use the asymmetrical scale instead of the symmetrical one to express the information in intuitionistic fuzzy preference relations, and introduce a new kind of preference relation called the intuitionistic multiplicative preference relation, which contains two parts of information describing the intensity degrees that an alternative is or not priority to another. Some basic operations are introduced, based on which, an aggregation principle is proposed to aggregate the intuitionistic multiplicative preference information, the desirable properties and special cases are further discussed. Choquet Integral and power average are also applied to the aggregation principle to produce the aggregation operators to reflect the correlations of the intuitionistic multiplicative preference information. Finally, a method is given to deal with the group decision making based on intuitionistic multiplicative preference relations.     

An intuitionistic fuzzy programming method for group decision making with interval-valued fuzzy preference relations

The paper develops a new intuitionistic fuzzy (IF) programming method to solve group decision making (GDM) problems with interval-valued fuzzy preference relations (IVFPRs). An IF programming problem is formulated to derive the priority weights of alternatives in the context of additive consistent IVFPR. In this problem, the additive consistent conditions are viewed as the IF constraints. Considering decision makers’ (DMs’) risk attitudes, three approaches, including the optimistic, pessimistic and neutral approaches, are proposed to solve the constructed IF programming problem. Subsequently, a new consensus index is defined to measure the similarity between DMs according to their individual IVFPRs. Thereby, DMs’ weights are objectively determined using the consensus index. Combining DMs’ weights with the IF program, a corresponding IF programming method is proposed for GDM with IVFPRs. An example of E-Commerce platform selection is analyzed to illustrate the feasibility and effectiveness of the proposed method. Finally, the IF programming method is further extended to the multiplicative consistent IVFPR.     

An approach to group decision making based on interval multiplicative and fuzzy preference relations by using projection

This paper presents a consensus model for group decision making with interval multiplicative and fuzzy preference relations based on two consensus criteria: (1) a consensus measure which indicates the agreement between experts’ preference relations and (2) a measure of proximity to find out how far the individual opinions are from the group opinion. These measures are calculated by using the relative projections of individual preference relations on the collective one, which are obtained by extending the relative projection of vectors. First, the weights of experts are determined by the relative projections of individual preference relations on the initial collective one. Then using the weights of experts, all individual preference relations are aggregated into a collective one. The consensus and proximity measures are calculated by using the relative projections of experts’ preference relations respectively. The consensus measure is used to guide the consensus process until the collective solution is achieved. The proximity measure is used to guide the discussion phase of consensus reaching process. In such a way, an iterative algorithm is designed to guide the experts in the consensus reaching process. Finally the expected value preference relations are defined to transform the interval collective preference relation to a crisp one and the weights of alternatives are obtained from the expected value preference relations. Two numerical examples are given to illustrate the models and approaches.     

Deriving priority weights from intuitionistic multiplicative preference relations under group decision-making settings

The intuitionistic multiplicative preference relation (IMPR), which takes into account both the ratio degree to which an alternative is preferred to another and the ratio degree to which an alternative is non-preferred to another, is a useful tool for decision makers to elicit their preference information using Saaty’s 1–9 scale. In this paper, we focus on group decision making with IMPRs. First, we analyze the flaws of the consistency definition of an IMPR in previous work and then propose a new definition to overcome the flaws. On this basis, a linear programming-based algorithm is developed to check and improve the consistency of an IMPR. Second, we discuss the relationships between an IMPR and a normalized intuitionistic multiplicative weight vector and develop two approaches to group decision making based on complete and incomplete IMPRs, respectively. Based on the proposed algorithm and approaches, a general framework for group decision making with IMPRs is proposed. Finally, some numerical examples are provided to demonstrate the proposed approaches. The results show that the proposed approaches can deal with group decision-making problems with IMPRs effectively.     

A least deviation method for priority derivation in group decision making with incomplete reciprocal preference relations

In this paper, based on the transfer relationship between reciprocal preference relation and multiplicative preference relation, we proposed a least deviation method (LDM) to obtain a priority vector for group decision making (GDM) problems where decision-makers' (DMs') assessments on alternatives are furnished as incomplete reciprocal preference relations with missing values. Relevant theorems are investigated and a convergent iterative algorithm about LDM is developed. Using three numerical examples, the LDM is compared with the other prioritization methods based on two performance evaluation criteria: maximum deviation and maximum absolute deviation. Statistical comparative study, complexity of computation of different algorithms, and comparative analyses are provided to show its advantages over existing approaches.     

基于最大满意度的区间混合判断矩阵排序方法

区间混合判断矩阵对表达决策者的判断信息很有帮助,然而区间混合判断矩阵尚未引起学者的足够重视.定义了一致性乘型区间混合判断矩阵和一致性加型区间混合判断矩阵,并基于最大满意度的思想,提出了求解区间混合判断矩阵的乘型和加型最大满意度模型和一致性指标,分析讨论了最大满意度模型的优点,最后通过算例验证了该模型的正确性和适用性,具有一定的理论和实际应用价值.     

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。     

A chi-square method for obtaining a priority vector from multiplicative and fuzzy preference relations

Decision makers (DMs)’ preferences on decision alternatives are often characterized by multiplicative or fuzzy preference relations. This paper proposes a chi-square method (CSM) for obtaining a priority vector from multiplicative and fuzzy preference relations. The proposed CSM can be used to obtain a priority vector from either a multiplicative preference relation (i.e. a pairwise comparison matrix) or a fuzzy preference relation or a group of multiplicative preference relations or a group of fuzzy preference relations or their mixtures. Theorems and algorithm about the CSM are developed. Three numerical examples are examined to illustrate the applications of the CSM and its advantages.