Algorithm for improving additive consistency of linguistic preference relations with an integer optimization model

Linguistic preference relation (LPR) composed by linguistic terms can well express decision makers’ (DMs’) qualitative preference opinion by comparing alternatives with each other. The investigation of its consistency becomes an important issue to guarantee the rationality of the decision making solutions. Therefore, it is significant to investigate the consistency measure and the consistency improving approach for LPRs. In this paper we present a new method for group decision making (GDM) with LPRs. First, an additive consistency index is introduced on the basis of the information of the original LPR to check whether a LPR is acceptably additive consistency. For unacceptably additively consistent LPR, an integer optimization model is further developed to obtain the acceptably additively consistent LPR. Moreover, the optimization model can guarantee the integrity of the information of the LPR with acceptably additive consistency. Then, with respect to GDM with LPRs, an entropy weight method is proposed to determine the weights of DMs. Finally, the proposed methods are implemented in two numerical examples including a GDM problem. Meanwhile, the comparative analysis with existing methods are discussed in detail to demonstrate the validity of the proposed 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.     

Group decision making with incomplete fuzzy preference relations based on the additive consistency and the order consistency

In this paper, we present a new method for group decision making with incomplete fuzzy preference relations based on the additive consistency and the order consistency. We estimate unknown preference values based on the additive consistency and then construct the consistency matrix which satisfies the additive consistency and the order consistency simultaneously for aggregation. The existing group decision making methods may not satisfy the order consistency for aggregation in some situations. The proposed method can overcome the drawback of the existing methods. It provides us with a useful way for group decision making with incomplete fuzzy preference relations based on the additive consistency and the order consistency.     

Approaches to decision making with linguistic preference relations based on additive consistency

The linguistic preference relation (LPR) is introduced to efficiently deal with situations in which the decision makers (DMs) provide their preference information by using linguistic labels over paired comparisons of alternatives. However, the lack of consistency in decision making with LPRs can lead to inconsistent conclusions. In this paper, two new decision making methods are developed to improve the additive consistency of LPRs until they are acceptable, and eventually obtain the reliable decision making results. First, the new concepts of order consistency and additive consistency of LPRs are introduced, and followed by a discussion of the characterization about additive consistent LPRs. Then, a consistency index is defined to measure whether an LPR is of acceptable additive consistency. For an unacceptable additive consistent LPR, two automatic iterative algorithms are further proposed to help DMs improve additive consistency level until it is acceptable. In addition, the proposed algorithms can derive the priority weight vector from LPRs and obtain the ranking of the alternatives. Finally, the proposed methods are applied to an emergency operating center (EOC) selection problem. The comparative analysis demonstrates the applicability and effectiveness of the proposed methods.     

加型Pythagorean模糊偏好关系的多属性决策方法

考虑Pythagorean模糊偏好关系的多属性决策问题,提出了加性Pythagorean模糊偏好关系的多属性决策方法。基于加性一致性Pythagorean模糊偏好关系提出一种新的Pythagorean模糊权重确定模型。给出了可接受加性一致性Pythagorean模糊偏好关系的定义,并针对不满足可接受加性一致性的Pythagorean模糊偏好关系,提出一种加性一致性调整算法。给出基于Pythagorean模糊偏好关系加性一致性的多属性决策方法,并通过实例分析提出的新方法的可行性和合理性。     

模糊偏好关系一致性改进算法及其决策应用

模糊偏好关系是处理决策问题的一种有效工具。针对模糊偏好关系,研究了加性一致性模糊偏好关系的若干判定条件,构造了满足加性一致性的特征模糊偏好关系,并提出一致性指数、满意加性一致性等概念。在此基础上,构建了不满足加性一致性模糊偏好关系的改进算法,论证了算法的收敛性,该算法使得改进后的模糊偏好关系具有满意一致性条件,进而使得决策者获得合理可靠的决策结果。最后建立了基于模糊偏好关系加性一致性的决策模型。实例分析说明提出的模糊偏好关系决策模型是可行和有效的。     

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。     

基于犹豫语言判断矩阵的数据产品选择研究

犹豫语言判断矩阵作为一种新的判断矩阵,能够有效地处理决策信息为语言变量且决策者态度犹豫不行的决策问题。针对犹豫模糊语言信息环境下的数据产品选择问题,构建了一种基于犹豫语言判断矩阵的数据产品选择方法。该方法引入了犹豫语言判断矩阵的一些相关概念,包括加性一致性、一致性指数、可接受一致性;研究了犹豫语言判断矩阵一致性判定方法和特征矩阵的构造方法,并设计了一种收敛性算法用以改进犹豫语言判断矩阵的一致性;建立了基于犹豫语言判断矩阵的决策模型,并通过数据产品的选择实例说明提出的决策方法是合理和有效的。     

Two flexibility degrees-driven consensus model in group decision making with intuitionistic fuzzy preference relations

A group of experts are commonly invited to find an optimal solution to a complex decision making problem. When the bipolarity of decision information should be considered in group decision making (GDM), intuitionistic fuzzy values (IFVs) have the capability to model such opinions of decision makers (DMs). This paper develops a consensus model in GDM under intuitionistic fuzzy environments with flexibility. First, it is assumed that the initial opinions of DMs are expressed as intuitionistic fuzzy preference relations (IFPRs). A novel additive consistency index is constructed to measure the deviation degree of IFPRs from fuzzy preference relations (FPRs) with additive consistency, where the non-determinacy degree of IFPRs is incorporated. The thresholds of the proposed index corresponding to IFPRs with acceptable additive consistency are discussed and computed. Second, the consensus level of DMs is defined using the similarity degree between two IFVs. An optimization problem is established by maximizing the fitness function, which is constructed by linearly combining the proposed additive consistency index and consensus level. Two flexibility degrees are offered to each DM such that the initial opinions with the bipolarity can be adjusted correspondingly. Third, individual IFPRs in GDM are optimized using the particle swarm optimization (PSO) algorithm. Numerical examples are carried out to illustrate the proposed consensus model by comparing with the existing ones. The obtained results reveal that the proposed additive consistency index can reflect the inherent property of IFPRs. Different with the previous studies, two original flexibility degrees are proposed to characterize the multi-granularity of decision information in GDM.     

Linguistic intuitionistic fuzzy preference relations and their application to multi-criteria decision making

Linguistic intuitionistic fuzzy sets can be regarded as a qualitative form of intuitionistic fuzzy sets. This type of fuzzy sets uses a linguistic membership degree and a linguistic non-membership degree to represent the qualitative preferred and non-preferred judgments of decision makers. Preference relation is a useful and efficient tool for decision making that only requires the decision makers to compare two objects at one time. Taking the advantages of linguistic intuitionistic fuzzy sets and preference relations, this paper introduces linguistic intuitionistic fuzzy preference relations (LIFPRs) and studies their application to decision making. To ensure the ranking of objects reasonably, an additive consistency concept is introduced, and several of its desirable properties are discussed. To cope with inconsistent and incomplete LIFPRs, programming model-based methods to derive additively consistent LIFPRs and determine missing values are constructed, respectively. Subsequently, an approach to multi-criteria decision making with LIFPRs is offered, and the application of the new approach is illustrated by using a decision-making problem about evaluating mobile phones.     

Consistency-driven data quality management of networked sensor systems

With more and more real deployments of wireless sensor network applications, we envision that their success is nonetheless determined by whether the sensor networks can provide a high quality stream of data over a long period. In this paper, we propose a consistency-driven data quality management framework called Orchis that integrates the quality of data into an energy efficient sensor system design. Orchis consists of four components, data consistency models, adaptive data sampling and process protocols, consistency-driven cross-layer protocols and flexible APIs to manage the data quality, to support the goals of high data quality and energy efficiency. We first formally define a consistency model, which not only includes temporal consistency and numerical consistency, but also considers the application-specific requirements of data and data dynamics in the sensing field. Next, we propose an adaptive lazy energy efficient data collection protocol, which adapts the data sampling rate to the data dynamics in the sensing field and keeps lazy when the data consistency is maintained. Finally, we conduct a comprehensive evaluation to the proposed protocol based on both a TOSSIM-based simulation and a real prototype implementation using MICA2 motes. The results from both simulation and prototype show that our protocol reduces the number of delivered messages, improves the quality of collected data, and in turn extends the lifetime of the whole network. Our analysis also implies that a tradeoff should be carefully set between data consistency requirements and energy saving based on the specific requirements of different applications.     

Multi-criteria group decision making with incomplete hesitant fuzzy preference relations

In order to simulate the hesitancy and uncertainty associated with impression or vagueness, a decision maker may give her/his judgments by means of hesitant fuzzy preference relations in the process of decision making. The study of their consistency becomes a very important aspect to avoid a misleading solution. This paper defines the concept of additive consistent hesitant fuzzy preference relations. The characterizations of additive consistent hesitant fuzzy preference relations are studied in detail. Owing to the limitations of the experts’ professional knowledge and experience, the provided preferences in a hesitant fuzzy preference relation are usually incomplete. Consequently, this paper introduces the concepts of incomplete hesitant fuzzy preference relation, acceptable incomplete hesitant fuzzy preference relation, and additive consistent incomplete hesitant fuzzy preference relation. Then, two estimation procedures are developed to estimate the missing information in an expert's incomplete hesitant fuzzy preference relation. The first procedure is used to construct an additive consistent hesitant fuzzy preference relation from the lowest possible number, (n − 1), of pairwise comparisons. The second one is designed for the estimation of missing elements of the acceptable incomplete hesitant fuzzy preference relations with more known judgments. Moreover, an algorithm is given to solve the multi-criteria group decision making problem with incomplete hesitant fuzzy preference relations. Finally, a numerical example is provided to illustrate the solution processes of the developed algorithm and to verify its effectiveness and practicality.     

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.     

Group decision making problems in a linguistic and dynamic context

The aim of this paper is to present a new model of decision support system for group decision making problems based on a linguistic approach and dynamic sets of alternatives. The model incorporates a mechanism that allows to manage dynamic decision situations in which some information about the problem is not constant in time. We assume that the set of alternatives can change during the decision making process. The model is presented in a mobile and dynamic context where the experts’ preferences can be incomplete. The linguistic approach is used to represent both the experts’ preferences about the alternatives and the agreement degrees to manage the change of some alternatives. A prototype of such mobile decision support system in which the experts use mobile devices to provide their linguistic preferences at anytime and anywhere has been implemented. In such a way, we provide a new linguistic group decision making framework that is mobile and dynamic.     

Qualitative hesitant fuzzy group decision making: An additively consistent probability and consensus-based perspective

Hesitant fuzzy linguistic preference relations (HFLPRs) can efficiently denote the hesitant qualitative judgments of decision makers. Consistency and consensus are two critical topics in group decision making (GDM) with preference relations. This paper uses the additively consistent concept for linguistic fuzzy preference relations (LFPRs) to give an additive consistency definition for HFLPRs. To judge the additive consistency of HFLPRs, 0-1 mixed programming models (0-1-MPMs) are constructed. Meanwhile, additive-consistency-based 0-1-MPMs to ascertain missing values in incomplete HFLPRs are established. Following the consistent probability of LFPRs, an algorithm to calculate the linguistic priority weighting vector is presented. In consideration of the consensus of GDM, a consistency-probability-distance-measure-based consensus index is defined, and an interactive improving consensus method is provided. Finally, a method for GDM with HFLPRs is offered that can address incomplete and inconsistent cases. Meanwhile, numerical examples are offered, and comparative analysis is made.     

A comparative study for consistency-based decision making with interval multiplicative preference relations

ABSTRACT

To ensure the reasonable application and perfect the theory of decision making with interval multiplicative preference relations (IMPRs), this paper continues to discuss decision making with IMPRs. After reviewing previous consistency concepts for IMPRs, we find that Krej?í’s consistency concept is more flexible and natural than others. However, it is insufficient to address IMPRs only using this concept. Considering this fact, this paper researches inconsistent and incomplete IMPRs that are usually encountered. First, programming models for addressing inconsistent and incomplete IMPRs are constructed. Then, this paper studies the consensus of individual IMPRs and defines a consensus index using the defined correlation coefficient. When the consensus requirement does not satisfy requirement, a programming model for improving consensus level is built, which can ensure the consistency. Subsequently, a procedure for group decision making with IMPRs is offered, and associated examples are provided to specifically show the application of main theoretical results.     

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.     

Large-scale group decision making with multiple stakeholders based on probabilistic linguistic preference relation

A large number of stakeholders take part in the process of decision making, namely, large-scale group decision making (LGDM) problems. Every stakeholder utilises a linguistic preference relation (LPR) to represent her/his preference information for alternatives. Then, a probabilistic LPR (PLPR) is established to represent the group preference. However, some stakeholders may only provide partial preference information about the alternatives. Thus, a PLPR with incomplete probabilities can be used to manage LGDM problems in complex environments. Based on the defined expected multiplicative consistency of PLPR, a probability computation model is established by mathematical programming to derive the missing probabilities of PLPR. In addition, an iterative algorithm to improve the consistency is proposed to obtain the PLPR with satisfactory consistency. Finally, a real-world investment decision-making problem with multiple stakeholders is solved to demonstrate the effectiveness of the proposed method.     

混合非结构MAGDM的决策导向一致性检验方法

为有效应对现有群决策一致性检验方法的系列弊端,针对群决策的决策导向多元、决策方案众多、决策属性异构、决策信息多样等特征,在引入票权概念解析群决策一致性判定复杂性、刻画非结构多属性群决策合意信息表征假设情景的基础上,通过对常规混合非结构多属性群决策（MAGDM）问题进行公理化描述,并依据从方案层面到属性层面的整体决策信息判定策略,给出决策导向层面的整体判断信息一致性检验方法、多轮次非一致性决策信息调整策略及信息集结方法。案例应用结果表明提出的方法有效、可行。     

Consensus reaching for group decision making with multi-granular unbalanced linguistic information: A bounded confidence and minimum adjustment-based approach

In group decision making problems, there exist the situations that decision makers may use unbalanced linguistic term sets that are not uniformly and symmetrically distributed to provide their linguistic assessments over alternatives. Moreover, due to the difference in knowledge and culture backgrounds, it is also possible that multi-granular linguistic term sets may also be used by decision makers. How to manage multi-granular unbalanced linguistic information in consensus-based group decision making has becoming an important topic in linguistic decision making. In this paper, we first revise Herrera’s unbalanced linguistic term sets and propose a simplified linguistic computational model to fuse multi-granular unbalanced linguistic terms. Afterwards, for multi-criteria group decision making problems with multi-granular unbalanced linguistic information, we develop two optimization models to generate adjustment advice for decision makers who have to change his/her opinions in consensus reaching process, which consider both the bounded confidence levels and minimum adjustment of decision makers’ linguistic assessments. Moreover, an algorithm is further proposed to help decision makers reach consensus in group decision making. Eventually, an application example for ERP system supplier selection and some simulation results are presented to illustrate and justify the consensus reaching algorithm.