Consistency improvement under a personalized individual semantics context in distributed linguistic group decision making

Consistency measurement is a significant issue in linguistic decision making when preferences are expressed via linguistic preference relations. However, the extant literatures on linguistic consistency generally overlook the fact that even the same word can have diverse meanings for different people, which indicates that people usually possess personalized individual semantics (PISs) over words. Furthermore, with the complexity of the practical decision-making problem increases, decision makers become more likely to be uncertain and hesitant to make their preferences due to the lack of knowledge, therefore, their linguistic preferences may be represented through distributed linguistic representations. However, there are few consistency improving studies on distributed linguistic representations. Therefore, in this study we devise a novel consistency improving approach for distribution linguistic preference relations under a PISs context. Furthermore, the usability and effectiveness of the PISs based consistency improvement method are verified through the detail numerical analysis and comparative study.     

Deriving priority weights from intuitionistic fuzzy multiplicative preference relations

Intuitionistic fuzzy multiplicative preference relations (IFMPRs), as an extension of multiplicative preference relations, can denote the decision-makers’ (DMs’) preferred and nonpreferred degrees simultaneously. Just as any other type of preference relations, consistency is crucial to guarantee the rational ranking orders. Thus, this paper introduces a new consistent concept for IFMPRs that is a natural extension of crisp case and overcomes the issues in the previous concepts of consistency. To judge the consistency of IFMPRs, several programming models are constructed, and an approach to deriving completely consistent IFMPRs is presented. Considering incomplete case, consistency-based models are built to determine missing values that can address incomplete IFMPRs with the ignored objects, namely, all information for them is unknown. After that, group decision-making with IFMPRs is studied. To measure the agreement degree between the DMs’ individual IFMPRs, a new consensus index is defined, and an interactive algorithm to improve the consensus is offered. Based on the consistency and consensus analysis, a new method to group decision-making with IFMPRs is developed. Finally, case studies are offered to show the application of the new procedure and to compare it with previous methods.     

Consistency and consensus improvement models driven by a personalized normalization method with probabilistic linguistic preference relations

Probabilistic linguistic preference relation (PLPR) provides an effective and flexible tool with which preference degrees of decision-makers can be captured when they vacillatingly express linguistic preference values among several linguistic terms. Individual consistency and group consensus are two important research topics of PLPRs in group decision making (GDM). Considering the problems associated with these two topics, this study proposes a novel GDM framework with consistency-driven and consensus-driven optimization models based on a personalized normalization method for managing complete and incomplete PLPRs. First, existing limitations of the traditional normalization method for probabilistic linguistic term sets (PLTSs) managing ignorance information are specifically discussed. Given the potential valuable information hidden in PLTSs, a personalized normalization method is newly proposed through a two-stage decision-making process with a comprehensive fusion mechanism. Then, based on the proposed normalization method for PLTSs, consistency-driven optimization models that aim to minimize the overall adjustment amount of a PLPR are constructed to improve consistency. Moreover, the developed models are extended to improve consistency and estimate the missing values of an incomplete PLPR. Subsequently, a consensus-driven optimization model that aims to maximize group consensus by adjusting experts’ weights is constructed to support the consensus-reaching process. Finally, an illustrative example, followed by some comparative analyses is presented to demonstrate the application and advantages of the proposed approach.     

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.     

On compatibility of uncertain multiplicative linguistic preference relations based on the linguistic COWGA

The aim of this work is to develop a new compatibility for the uncertain multiplicative linguistic preference relations and utilize it to determine the optimal weights of experts in the group decision making (GDM). First, the compatibility degree and compatibility index for the two multiplicative linguistic preference relations are proposed. Then, based on the linguistic continuous ordered weighted geometric averaging (LCOWGA) operator, some concepts of the compatibility degree and compatibility index for the two uncertain multiplicative linguistic preference relations are presented. We prove the property that the synthetic uncertain linguistic preference relation is of acceptable compatibility under the condition that the uncertain multiplicative linguistic preference relations given by experts are all of acceptable compatibility with the ideal uncertain multiplicative linguistic preference relation, which provides a theoretic basis for the application of the uncertain multiplicative linguistic preference relations in GDM. Next, an optimal model is constructed to determine the weights of experts based on the criterion of minimizing the compatibility index in GDM. Moreover, an approach to GDM with uncertain multiplicative linguistic preference relations is developed, and finally, an application of the approach to supplier selection problem with uncertain multiplicative linguistic preference relations is pointed out.     

Measuring consistency of linguistic preference relations: a 2-tuple linguistic approach

By using the 2-tuple fuzzy linguistic representation model and by proposing the concept of the individual consistency evaluations, this paper develops a 2-tuple linguistic index to measure the consistency degree of linguistic preference relations. Comparing with the existing numerical consistency indexes, the proposed linguistic index not only measures the consistency degree of linguistic preference relations via a linguistic way, but also can reflect the individual difference on consistency degree of linguistic preference relations. Further, this paper proposes an algorithm to improving consistency degree in linguistic preference relations, and the proof of the convergence of this algorithm is also given. The results in this paper provide a theoretic basis for the application of consistent linguistic preference relations in decision making problems.     

On compatibility of uncertain additive linguistic preference relations based on the linguistic COWA operator

We develop a new compatibility for the uncertain additive linguistic preference relations and study its properties which are very suitable to deal with group decision making (GDM) problems involving uncertain additive linguistic preference relations. Based on the linguistic continuous ordered weighted averaging (LCOWA) operator, we present some concepts of the compatibility degree and compatibility index for the two uncertain additive linguistic preference relations. Then, we study some desirable properties including the property that the synthetic uncertain linguistic preference relation is of acceptable compatibility under the condition that uncertain additive linguistic preference relations given by experts are all of acceptable compatibility with the ideal uncertain linguistic preference relation, which provides a theoretic basis for the application of the uncertain additive linguistic preference relations in GDM. In order to determine the weights of experts, we construct an optimal model based on the criterion of minimizing the compatibility index in GDM. Finally, we propose a new approach based on the compatibility index and the expected additive linguistic preference relation to GDM and develop an application of the optimal weights approach compared with the equal weights approach where we analyze a GDM regarding the evaluation of schools in a university.     

Solving multi-criteria decision making with incomplete linguistic preference relations

When there are n criteria or alternatives in a decision matrix, a pairwise comparison methodology of analytic hierarchy process (AHP) with the time of n(n ? 1)/2 is frequently used to select, evaluate or rank the neighboring alternatives. But while the number of criteria or comparison level increase, the efficiency and consistency of a decision matrix decrease. To solve such problems, this study therefore uses horizontal, vertical and oblique pairwise comparisons algorithm to construct multi-criteria decision making with incomplete linguistic preference relations model (InLinPreRa). The use of pairwise comparisons will not produce the inconsistency, even allows every decision maker to choose an explicit criterion or alternative for index unrestrictedly. When there are n criteria, only n ? 1 pairwise comparisons need to be carried out, then one can rest on incomplete linguistic preference relations to obtain the priority value of alternative for the decision maker’s reference. The decision making assessment model that constructed by this study can be extensively applied to every field of decision science and serves as the reference basis for the future research.     

Group decision making with incomplete fuzzy linguistic preference relations

The aim of this paper is to propose a procedure to estimate missing preference values when dealing with incomplete fuzzy linguistic preference relations assessed using a two‐tuple fuzzy linguistic approach. This procedure attempts to estimate the missing information in an individual incomplete fuzzy linguistic preference relation using only the preference values provided by the respective expert. It is guided by the additive consistency property to maintain experts' consistency levels. Additionally, we present a selection process of alternatives in group decision making with incomplete fuzzy linguistic preference relations and analyze the use of our estimation procedure in the decision process. © 2008 Wiley Periodicals, Inc.     

Deriving topological relations between regions from direction relations

The integration of topological and direction relations plays an important role in many applications, like spatial databases and pictorial retrieval systems. The method for deriving composition of binary topological relations cannot always yield unique or interesting results. Therefore, to integrate efficiently topological and direction relations, some new mechanisms are required to derive topological relations from direction cases when the above situations occur. This paper presents the computation methods for deriving topological relations from direction relations. The methods fall into two categories: the derivation of topological relations from one direction relation and two direction relations. Our methods can provide topological information when topological relations are unavailable, or more precise results are expected. Thus they are helpful in the integration of the calculi for topological and direction relations.     

Multiplicative consistency-based decision support system for incomplete linguistic preference relations

The experts may have difficulty in expressing all their preferences over alternatives or criteria, and produce the incomplete linguistic preference relation. Consistency plays an important role in estimating unknown values from an incomplete linguistic preference relation. Many methods have been developed to obtain a complete linguistic preference relation based on additive consistency, but some unreasonable values may be produced in the estimation process. To overcome this issue, we propose a new characterisation about multiplicative consistency of the linguistic preference relation, present an algorithm to estimate missing values from an incomplete linguistic preference relation, and establish a decision support system for aiding the experts to complete their linguistic preference relations in a more consistent way. Some examples are also given to illustrate the proposed methods.     

Group decision making based on multiple types of linguistic preference relations

In this paper, we investigate group decision making problems with multiple types of linguistic preference relations. The paper has two parts with similar structures. In the first part, we transform the uncertain additive linguistic preference relations into the expected additive linguistic preference relations, and present a procedure for group decision making based on multiple types of additive linguistic preference relations. By using the deviation measures between additive linguistic preference relations, we give some straightforward formulas to determine the weights of decision makers, and propose a method to reach consensus among the individual preferences and the group’s opinion. In the second part, we extend the above results to group decision making based on multiple types of multiplicative linguistic preference relations, and finally, a practical example is given to illustrate the application of the results.     

Consensus reaching models of linguistic preference relations based on distance functions

In group decision making (GDM) using linguistic preference relations to obtain the maximum degree of agreement, it is desirable to develop a consensus process prior to the selection process. This paper proposes two consensus models with linguistic information to support the GDM consensus reaching process. Two different distance functions between linguistic preference relations are introduced to measure both individual consistency and group consensus. Based on these measures, the consensus reaching models are developed. The two models presented have the same concept that the expert whose preference is farthest from the group preference needs to update their opinion according to the group preference relation. In addition, the convergence of the models is proved. After achieving the predefined consensus level, each expert’s consistency indexes are still acceptable under the condition that the initial preference relations are of satisfactory consistency. Finally, an example is given to show the effectiveness of the models and to verify the theoretical results.     

Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP

The lack of consistency in decision making can lead to inconsistent conclusions. In fuzzy analytic hierarchy process (fuzzy AHP) method, it is difficult to ensure a consistent pairwise comparison. Furthermore, establishing a pairwise comparison matrix requires judgments for a level with n criteria (alternatives). The number of comparisons increases as the number of criteria increases. Therefore, the decision makers judgments will most likely be inconsistent. To alleviate inconsistencies, this study applies fuzzy linguistic preference relations (Fuzzy LinPreRa) to construct a pairwise comparison matrix with additive reciprocal property and consistency. In this study, the fuzzy AHP method is reviewed, and then the Fuzzy LinPreRa method is proposed. Finally, the presented method is applied to the example addressed by Kahraman et al. [C. Kahraman, D. Ruan, I. Do?an, Fuzzy group decision making for facility location selection, Information Sciences 157 (2003) 135-153]. This study reveals that the proposed method yields consistent decision rankings from only n − 1 pairwise comparisons, which is the same result as in Kahraman et al. research. The presented fuzzy linguistic preference relations method is an easy and practical way to provide a mechanism for improving consistency in fuzzy AHP method.     

Group decision making with 2-tuple intuitionistic fuzzy linguistic preference relations

The aim of this paper is to propose a new type of preference relation, the intuitionistic fuzzy linguistic preference relation (IFLPR). Taking as base the 2-tuple fuzzy linguistic representation model, we introduce the definition of the IFLPR, and its transitivity properties. We present an approach to group decision making based on IFLPRs and incomplete-IFLPRs, respectively. The score function and accuracy function are applied to the ranking and selection of alternatives. Finally, we give an example of IFLPRs in group decision making, and a comparative of the exploitation of the IFLPR with the exploitation of the traditional fuzzy linguistic preference relations.     

Optimal consensus models for group decision making under linguistic preference relations

This paper proposes an optimal consensus model to derive weights for linguistic preference relations (LPRs). Two indexes, an individual‐to‐group consensus index (ICI) and a collective consensus index (CCI), are introduced. An iterative algorithm is presented to describe the consensus reaching process. By changing the weights and modifying a pair of individuals' comparison judgments—which have largest deviation value to the group judgments—the consensus reaching process can terminate, while both ICI and CCI are controlled with predefined thresholds. The algorithm aims to preserve the decision makers’ original information as much as possible. The model and algorithm are then extended to handle the uncertain additive LPRs. Finally, two examples are given to show the effectiveness of the proposed methods.     

An automatic iterative decision-making method for intuitionistic fuzzy linguistic preference relations

As a new preference structure, the intuitionistic fuzzy linguistic preference relation (IFLPR) was recently introduced to efficiently deal with situations in which the membership and non-membership are represented as linguistic terms. In this paper, we study the issues of additive consistency and the derivation of the intuitionistic fuzzy weight vector of an IFLPR. First, the new concepts of order consistency, additive consistency and weak transitivity for IFLPRs are introduced, and followed by a discussion of the characterisation about additive consistent IFLPRs. Then, a parameterised transformation approach is investigated to convert the normalised intuitionistic fuzzy weight vector into additive consistent IFLPRs. After that, a linear optimisation model is established to derive the normalised intuitionistic fuzzy weights for IFLPRs, and a consistency index is defined to measure the deviation degree between an IFLPR and its additive consistent IFLPR. Furthermore, we develop an automatic iterative decision-making method to improve the IFLPRs with unacceptable additive consistency until the adjusted IFLPRs are acceptable additive consistent, and it helps the decision-maker to obtain the reasonable and reliable decision-making results. Finally, an illustrative example is provided to demonstrate the validity and applicability of the proposed method.     

求解VRPTW问题的不确定性目标偏好蚁群算法*

通过分析多目标的、有时间窗的车辆路径问题,对各个目标进行多属性不确定性语言评判,结合相关专家的综合意见以及决策者自身对专家意见的偏好,将决策者对目标属性的离散意见转换为对各目标的综合意见;通过定义一种综合排序指标来确定决策者对各目标的偏好权重,依据目标权重和各目标函数的规范化处理值,构建评价有时间窗的车辆路径问题的多目标偏好的综合适应度函数,将多目标问题转换为单目标问题,进而采用最大—最小蚂蚁系统算法对该问题进行求解;最后通过一个算例来说明该算法的有效性。     

基于模糊语言偏好关系一致性改进的决策方法

针对决策者提供的偏好信息为语言标量的决策问题,首先引入了语言偏好关系的有序一致性和加性一致性的定义,研究了语言偏好关系加性一致性的判定方法,构建了满足加性一致性的诱导语言偏好关系,提出一致性指数和满意一致性的概念;然后建立了基于语言偏好关系一致性改进的决策算法,并证明了算法的收敛性,同时通过该算法改进后的语言偏好关系满足满意一致性条件。最后通过数据库系统的选择实例说明提出的决策算法是合理的和有效的。     

一种不确定语言偏好排序的多目标粒子群算法

为了将彼此冲突的多目标问题转换为单目标问题,并充分考虑专家决策过程的不确定性和思维的模糊性,将不确定语言信息转换为不确定语言变量,利用不确定语言变量运算法则进行计算,通过可能度的定义来建立可能度互补判断矩阵,采用多指标不确定性排序法确定决策者权重,从而将专家对各目标的离散意见转换为综合意见,进而确定各目标权重。通过对各目标值进行规范化处理,综合各目标权重得到决策者不确定性偏好排序的目标综合适应度函数,将多目标问题转换为单目标问题,进而采用粒子群算法对该问题进行求解。最后通过一个算例来说明该算法的实用性和有效性。