Consistency test and weight generation for additive interval fuzzy preference relations

Some simple yet pragmatic methods of consistency test are developed to check whether an interval fuzzy preference relation is consistent. Based on the definition of additive consistent fuzzy preference relations proposed by Tanino (Fuzzy Sets Syst 12:117–131, 1984), a study is carried out to examine the correspondence between the element and weight vector of a fuzzy preference relation. Then, a revised approach is proposed to obtain priority weights from a fuzzy preference relation. A revised definition is put forward for additive consistent interval fuzzy preference relations. Subsequently, linear programming models are established to generate interval priority weights for additive interval fuzzy preference relations. A practical procedure is proposed to solve group decision problems with additive interval fuzzy preference relations. Theoretic analysis and numerical examples demonstrate that the proposed methods are more accurate than those in Xu and Chen (Eur J Oper Res 184:266–280, 2008b).     

A new method of obtaining the priority weights from an interval fuzzy preference relation

In analyzing a multiple criteria decision-making problem, the decision maker may express her/his opinions as an interval fuzzy or multiplicative preference relation. Then it is an interesting and important issue to investigate the consistency of the preference relations and obtain the reliable priority weights. In this paper, a new consistent interval fuzzy preference relation is defined, and the corresponding properties are derived. The transformation formulae between interval fuzzy and multiplicative preference relations are further given, which show that two preference relations, consistent interval fuzzy and multiplicative preference relations, can be transformed into each other. Based on the transformation formula, the definition of acceptably consistent interval fuzzy preference relation is given. Furthermore a new algorithm for obtaining the priority weights from consistent or inconsistent interval fuzzy preference relations is presented. Finally, three numerical examples are carried out to compare the results using the proposed method with those using other existing procedures. The numerical results show that the given procedure is feasible, effective and not requisite to solve any mathematical programing.     

A note on group decision-making procedure based on incomplete reciprocal relations

In a very recent paper by Xu and Chen (Soft Comput 12:515–521, 2008), a novel procedure for group decision making with incomplete reciprocal relations was developed. In this note, we examine the function between the fuzzy preference relation and its corresponding priority vector developed by Xu and Chen with a numerical example and show that the function does not hold in general cases. Then, we deduce an exact function between the additive transitivity fuzzy preference relation and its corresponding priority vector. Based on this, we develop a procedure for the decision making with an incomplete reciprocal relation and also develop a procedure for the group decision making with incomplete reciprocal relations. In order to compare the performances of our method with Xu and Chen’s method in fitting the reciprocal relation, we introduce some criteria. Theoretical analysis and numerical examples have shown that the function deduced by us is more reasonable and effective than Xu and Chen’s.     

A decision support model for group decision making with intuitionistic fuzzy linguistic preferences relations

As a new preference structure, the intuitionistic fuzzy linguistic preference relation (IFLPR) was introduced to efficiently cope with situations in which the membership degree and non-membership degree are represented as linguistic terms. For group decision making (GDM) problems with IFLPRs, two significant and challenging issues are individual consistency and group consensus before deriving the reliable priority weights of alternatives. In this paper, a novel decision support model is investigated to simultaneously deal with the individual consistency and group consensus for GDM with IFLPRs. First, the concepts of multiplicative consistency and weak transitivity for IFLPRs are introduced and followed by a discussion of their desirable properties. Then, a transformation approach is developed to convert the normalized intuitionistic fuzzy priority weights into multiplicative consistent IFLPR. Based on the distance of IFLPRs, the consistency index, individual consensus degree and group consensus degree for IFLPRs are further defined. In addition, two convergent automatic iterative algorithms are proposed in the investigated decision support model. The first algorithm is utilized to convert an unacceptable multiplicative consistent IFLPR to an acceptable one. The second algorithm can assist the group decision makers to achieve a predefined consensus level. The main characteristic of the investigated decision support model is that it guarantees each IFLPR is still acceptable multiplicative consistent when the predefined consensus level is achieved. Finally, several numerical examples are provided, and comparative analyses with existing approaches are performed to demonstrate the effectiveness and practicality of the investigated model.

     

Goal programming approaches to obtain the priority vectors from the intuitionistic fuzzy preference relations

The priority method on the intuitionistic fuzzy preference relation (IFPR) is proposed. In order to avoid the operational difficulty in dealing with the intuitionistic sets, the equivalent interval matrices of the IFPR are introduced. Based on the multiplicative consistent definition of the fuzzy interval preference relation (FIPR), the goal programming models for deriving the priority vector of the IFPR have been put forward by analyzing the relation between the IFNPR and the IFPR. This goal programming method is generalized to the case of group decision making with the weight information defined by each DM. Two numerical examples are provided to illustrate the application of the proposed models.     

乘型一致性毕达哥拉斯模糊偏好关系

毕达哥拉斯模糊偏好关系(PFPR)是直觉模糊偏好关系的推广,也是毕达哥拉斯模糊集的重要研究领域.相对于其他模糊偏好关系而言,毕达哥拉斯模糊偏好关系在表达决策者的模糊偏好时更加灵活有力.在乘型一致性区间模糊偏好关系和乘型一致性直觉模糊偏好关系研究成果的启发下,定义毕达哥拉斯模糊偏好关系的乘型一致性,并提出利用毕达哥拉斯模糊权重向量构造乘型一致性毕达哥拉斯模糊偏好关系的公式.以给定的毕达哥拉斯模糊偏好关系与构造的乘型一致性毕达哥拉斯模糊偏好关系的偏差最小为目标函数建立并求解优化模型,从而获取毕达哥拉斯模糊偏好关系的标准化权重向量,为方案排序提供一种可行的方法.计算实例分析表明,所提出方法是可行有效的.     

Some models for generating and ranking multiplicative weights

The ranking of multiplicative interval and fuzzy weights is often necessary in multiplicative analytic hierarchy process. The existing ranking method is found flawed and needs to be revised. Firstly, this paper presents a correct formula for ranking multiplicative interval weights, and offers the relevant properties and lemmas to support them. Secondly, since different rank orders of interval weights are derived by the two-stage logarithmic goal programming (TLGP) method under different α-cuts, an approximation and adjustment (AAM) method is developed to generate multiplicative triangular fuzzy weights. In order to compare two multiplicative triangular fuzzy weights, the geometric mean centroid of multiplicative triangular fuzzy weight is proposed. Thus, a practical algorithm for decision making is introduced based on the above model and formulas. Finally, two numerical examples are provided to illustrate the practicality and validity of the proposed method.     

A new procedure for hesitant multiplicative preference relations

Hesitant information is powerful and flexible to denote decision maker's judgments. Hesitant multiplicative preference relations (HMPRs) own the advantages of preference relations and hesitant fuzzy sets that permit the decision makers (DMs) to compare objects by using several values. Just as other types of preference relations, how to derive the priority weight vector is a crucial step. According to the principle of the consistency concept for multiplicative preference relations, this paper first introduces a new consistency concept for HMPRs, which avoids the disadvantages of the previous ones. Using the new concept, models to judge the consistency of HMPRs are built. Then, a consistency probability-based method to derive the hesitant fuzzy priority weight vector from HMPRs is offered. Considering the incomplete case, consistency-based programming models to determine the missing values are constructed. To address group decision making with HMPRs, a distance measure is defined to determine the weights of the DMs, and a consensus index is proposed. Then, a consistency and consensus-based group decision-making algorithm is performed. Finally, two practical examples, an investment problem and a water conservancy problem are offered to illustrate the feasibility and efficiency of the new algorithm. Comparison analysis from the numerical and theoretical aspects verifies the potential application of the new procedure.     

Incomplete Fuzzy Preference Matrix and Its Application to Ranking of Alternatives

A fuzzy preference matrix is the result of pairwise comparison of a powerful method in multicriteria optimization. When comparing two elements, a decision maker assigns the value between 0 and 1 to any pair of alternatives representing the element of the fuzzy preference matrix. Here, we investigate relations between transitivity and consistency of fuzzy preference matrices and multiplicative preference ones. The obtained results are applied to situations where some elements of the fuzzy preference matrix are missing. We propose a new method for completing fuzzy matrix with missing elements called the extension of the fuzzy preference matrix. We investigate some important particular case of the fuzzy preference matrix with missing elements. Consequently, by the eigenvector of the transformed matrix we obtain the corresponding priority vector. Illustrative numerical examples are supplemented.     

Deriving the priority weights from hesitant multiplicative preference relations in group decision making

In this paper, we investigate the deviation of the priority weights from hesitant multiplicative preference relations (HMPRs) in group decision-making environments. As basic elements of HMPRs, hesitant multiplicative elements (HMEs) usually have different numbers of possible values. To correctly compute or compare HMEs, there are two principles to normalize them, i.e., the α-normalization and the β-normalization. Based on the α-normalization, we develop a new goal programming model to derive the priority weights from HMPRs in group decision-making environments. Based on the β-normalization, a consistent HMPR and an acceptably consistent HMPR are defined, and their desired properties are studied. A convex combination method is then developed to obtain interval weights from an acceptably consistent HMPR. This approach is further extended to group decision-making situations in which the experts evaluate their preferences as several HMPRs. Finally, some numerical examples are provided to illustrate the validity and applicability of the proposed models.     

Additive consistency-based priority-generating method of q-rung orthopair fuzzy preference relation

The q-rung orthopair fuzzy set, whose membership function and nonmembership function belong to the interval [0,1], is more powerful than both intuitionistic fuzzy set and Pythagorean fuzzy set in expressing imprecise information of decision-makers. The aim of this paper is to investigate a method to determine the priority weights from individual or group q-rung orthopair fuzzy preference relations (q-ROFPRs). To do so, firstly, a new definition of additively consistent q-ROFPR is presented based on the preference relation of alternatives given by decision-makers. Afterward, according to individual and group q-ROFPRs, two kinds of goal programming models are proposed, respectively, to generate the q-rung orthopair fuzzy priority weight vector of the given q-ROFPR(s). Finally, two numerical examples are given to illustrate the effectiveness and superiority of the method proposed in this paper.     

Ratio-based similarity analysis and consensus building for group decision making with interval reciprocal preference relations

Similarity analysis and preference information aggregation are two important issues for consensus building in group decision making with preference relations. Pairwise ratings in an interval reciprocal preference relation (IRPR) are usually regarded as interval-valued And-like representable cross ratios (i.e., interval-valued cross ratios for short) from the multiplicative perspective. In this paper, a ratio-based formula is introduced to measure similarity between a pair of interval-valued cross ratios, and its desirable properties are provided. We put forward ratio-based similarity measurements for IRPRs. An induced interval-valued cross ratio ordered weighted geometric (IIVCROWG) operator with interval additive reciprocity is developed to aggregate interval-valued cross ratio information, and some properties of the IIVCROWG operator are presented. The paper devises an importance degree induced IRPR ordered weighted geometric operator to fuse individual IRPRs into a group IRPR, and discusses the derivation of its associated weights. By employing ratio-based similarity measurements and IIVCROWG-based aggregation operators, a soft consensus model including a generation mechanism of feedback recommendation rules is further proposed to solve group decision making problems with IRPRs. Three numerical examples are examined to illustrate the applicability and effectiveness of the developed models.     

A consistency model for group decision making problems with interval multiplicative preference relations

The main aim of this paper is to present a consistency model for interval multiplicative preference relation (IMPR). To measure the consistency level for IMPR, a referenced consistent IMPR of a given IMPR is defined, which has the minimum logarithmic distance from the given IMPR. Based on the referenced consistent IMPR, the consistency level of an IMPR can be measured and an IMPR with unacceptable consistency can be adjusted by a proposed algorithm such that the revised IMPR is of acceptable consistency. A consistency model for group decision making (GDM) problems with IMPRs is proposed to obtain the collective IMPR with highest consistency level. Numerical examples are provided to illustrate the validity of the proposed approaches in decision making.     

The induced continuous ordered weighted geometric operators and their application in group decision making

In [IEEE Trans. Syst., Man, Cybernet.––Part B 29 (1999) 141], a more general class of OWA operators called the induced ordered weighted averaging (IOWA) operators is developed. Later, Yager and Xu [Fuzzy Sets and Syst, 157 (2006) 1393–1402.] introduced the continuous ordered weighted geometric operator(COWG), which is suitable for individual decision making problems taking the form of interval multiplicative preference relation. The aim of this paper is to develop some induced continuous ordered weighted geometric (ICOWG) operators. In particular, we present the reliability induced COWG (R-ICOWG) operator, which applies the ordering of the argument values based upon the reliability of the information sources; and the relative consensus degree induced COWG (RCD-ICOWG) operator, which applies the ordering of the argument values based upon the relative consensus degree of the information sources. Some desirable properties of the ICOWG operators are studied, and then, the ICOWG operators are applied to group decision making with interval multiplicative preference relations.     

Multiple-attribute group decision making with different formats of preference information on attributes.

Interval utility values, interval fuzzy preference relations, and interval multiplicative preference relations are three common uncertain-preference formats used by decision-makers to provide their preference information in the process of decision making under fuzziness. This paper is devoted in investigating multiple-attribute group-decision-making problems where the attribute values are not precisely known but the value ranges can be obtained, and the decision-makers provide their preference information over attributes by three different uncertain-preference formats i.e., 1) interval utility values; 2) interval fuzzy preference relations; and 3) interval multiplicative preference relations. We first utilize some functions to normalize the uncertain decision matrix and then transform it into an expected decision matrix. We establish a goal-programming model to integrate the expected decision matrix and all three different uncertain-preference formats from which the attribute weights and the overall attribute values of alternatives can be obtained. Then, we use the derived overall attribute values to get the ranking of the given alternatives and to select the best one(s). The model not only can reflect both the subjective considerations of all decision-makers and the objective information but also can avoid losing and distorting the given objective and subjective decision information in the process of information integration. Furthermore, we establish some models to solve the multiple-attribute group-decision-making problems with three different preference formats: 1) utility values; 2) fuzzy preference relations; and 3) multiplicative preference relations. Finally, we illustrate the applicability and effectiveness of the developed models with two practical examples.     

Note on “The induced continuous ordered weighted geometric operators and their application in group decision making”

In this short communication, we will show that the condition of the theorem does not hold in general cases in a recent paper “The induced continuous ordered weighted geometric operators and their application in group decision making” [Computers & Industrial Engineering 56 (2009) 1545–1552] by Wu et al., and also illustrate an example to show that we cannot construct a consistent interval multiplicative preference relation according to the condition of theorem in general case. Furthermore, we present a more reasonable condition to satisfy Theorem 1 so that we can construct a consistent interval multiplicative preference relation.     

Group consensus algorithms based on preference relations

In many group decision-making situations, decision makers’ preferences for alternatives are expressed in preference relations (including fuzzy preference relations and multiplicative preference relations). An important step in the process of aggregating preference relations, is to determine the importance weight of each preference relation. In this paper, we develop a number of goal programming models and quadratic programming models based on the idea of maximizing group consensus. Our models can be used to derive the importance weights of fuzzy preference relations and multiplicative preference relations. We further develop iterative algorithms for reaching acceptable levels of consensus in group decision making based on fuzzy preference relations or multiplicative preference relations. Finally, we include an illustrative example.     

A note on “Incomplete interval fuzzy preference relations and their applications”

In a very recent paper by Xu et al. (2014), an interval-arithmetic-based equation is introduced to define additively consistent interval fuzzy preference relations (IFPRs), and some properties of consistent IFPRs are put forward and used to define additive consistency for incomplete IFPRs. This note shows that such additive consistency definitions are technically wrong, and corrects errors in the definitions and properties of additively consistent IFPRs.     

Group decision making based on incomplete multiplicative and fuzzy preference relations

The main aim of this paper is to investigate the group decision making on incomplete multiplicative and fuzzy preference relations without the requirement of satisfying reciprocity property. This paper introduces a new characterization of the multiplicative consistency condition, based on which a method to estimate unknown preference values in an incomplete multiplicative preference relation is proposed. Apart from the multiplicative consistency property among three known preference values, the method proposed also takes the multiplicative consistency property among more than three values into account. In addition, two models for group decision making with incomplete multiplicative preference relations and incomplete fuzzy preference relations are presented, respectively. Some properties of the collective preference relation are further discussed. Numerical examples are provided to make a discussion and comparison with other similar methods.     

Aggregating information and ranking alternatives in decision making with intuitionistic multiplicative preference relations

The intuitionistic multiplicative preference relation (IMPR), whose all elements are measured by an unsymmetrical scale (Saaty's 1–9 scale) instead of the symmetrical scale in the intuitionistic fuzzy preference relation (IFPR), is suitable for describing the asymmetric preference information. In decision making process, one of the most crucial issues is how to rank alternatives from the given preference relation constructed by the decision maker. In this paper, two approaches are proposed for deriving the ranking orders of the alternatives from two different angles. To do it, a transformation mechanism is developed to transform an IMPR to a corresponding IFPR, and then all alternatives depicted by the given IMPR can be ranked via solving a familiar IFPR. In addition, the generalized intuitionistic multiplicative ordered weighted averaging (GIMOWA) and the geometric (GIMOWG) operators are given by taking fully account of the different weights associated with the particular ordered positions and their desirable properties are also discussed. After that, through a practical example, the proposed approaches are compared with the previous work and a numerical analysis of the results is also given.