Fuzzy set based models and methods of multicriteria group decision making

In the paper, the term consensus scheme is utilized to denote a dynamic and iterative process where the experts involved discuss a multicriteria decision problem. This discussion process is conducted by a human or artificial moderator, with the purpose of minimizing the discrepancy between the individual opinions.During the process of decision making, each expert involved must provide preference information. The information format and the circumstances where it must be given play a critical role in the decision process. This paper analyses a generic consensus scheme, which considers many different preference input formats, several possible interventions of the moderator, as well as admitting several stop conditions for interrupting the discussion process. In addition, a new consensus scheme is proposed with the intention of eliminating some difficulties met when the traditional consensus schemes are utilized in real applications. It preserves the experts’ integrity through the intervention of an external person, to supervise and mediate the conflicting situations. The human moderator is supposed to interfere in the discussion process by adjusting some parameters of the mathematical model or by inviting an expert to update his opinion. The usefulness of this consensus scheme is demonstrated by its use to solve a multicriteria group decision problem, generated applying the Balanced Scorecard methodology for enterprise strategy planning. In the illustrating problem, the experts are allowed to give their preferences in different input formats. But the information provided is made uniform on the basis of fuzzy preference relations through the use of adequate transformation functions, before being analyzed. The advantage of using fuzzy set theory for solving multiperson multicriteria decision problems lies in the fact that it can provide the flexibility needed to adequately deal with the uncertain factors intrinsic to such problems.     

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.     

Consensus reaching via a GDSS with fuzzy majority and clustering of preference profiles

An interactive DSS for consensus reaching is presented. Experts provide their testimonies as fuzzy preference relations. The consensus reaching process is supervised by a moderator (super-expert). A degree of consensus, based on the concept of a fuzzy majority given as a linguistic quantifier is employed. Algorithms of cluster analysis are used to find groups of experts having similar preferences.     

The OWA-based consensus operator under linguistic representation models using position indexes

When using linguistic approaches to solve decision problems, we need linguistic representation models. The symbolic model, the 2-tuple fuzzy linguistic representation model and the continuous linguistic model are three existing linguistic representation models based on position indexes. Together with these three linguistic models, the corresponding ordered weighted averaging operators, such as the linguistic ordered weighted averaging operator, the 2-tuple ordered weighted averaging operator and the extended ordered weighted averaging operator, have been developed, respectively. In this paper, we analyze the internal relationship among these operators, and propose a consensus operator under the continuous linguistic model (or the 2-tuple fuzzy linguistic representation model). The proposed consensus operator is based on the use of the ordered weighted averaging operator and the deviation measures. Some desired properties of the consensus operator are also presented. In particular, the consensus operator provides an alternative consensus model for group decision making. This consensus model preserves the original preference information given by the decision makers as much as possible, and supports consensus process automatically, without moderator.     

Multicriteria decision support using rules that represent rough-graded preference relations

The approach described in this paper aims to support multicriteria choice and ranking of actions when the input preference information acquired from the decision maker is a graded comprehensive pairwise comparison (or ranking) of reference actions. It is based on decision-rule preference model induced from a rough approximation of the graded comprehensive preference relation among the reference actions. The set of decision rules applied to a new set of actions provides a graded fuzzy preference relation, which can be exploited by weighted-fuzzy net flow score or lexicographic-fuzzy net flow score procedure to obtain a final recommendation in terms of the best choice or of the ranking.     

Multicriteria sorting using a valued indifference relation under a preference disaggregation paradigm

Most multicriteria decision methods need the definition of a significant amount of preferential information from a decision agent. The preference disaggregation analysis paradigm infers the model’s parameter values from holistic judgments provided by a decision agent. Here, a new method for inferring the parameters of a fuzzy outranking model for multicriteria sorting is proposed. This approach allows us to use most of the preferential information contained in a reference set. The central idea is to characterize the quality of the model by measuring discrepancies and concordances amongst (i) the preference relations derived from the outranking model, and (ii) the preferential information contained in the reference set. The model’s parameters are inferred from a multiobjective optimization problem, according to some additional preferential information from a decision agent. Once the model has been fitted, sorting decisions about new objects are performed by using a fuzzy indifference relation. This proposal performs very well in some examples.     

基于累积共识贡献的不确定语言共识决策方法

针对不确定加型语言偏好信息下的群决策问题,提出一种基于累积共识贡献的自适应式语言共识决策方法。首先,将不确定加型语言偏好转化为不确定二元语义偏好,定义个体一致度与个体共识偏度,并利用它们构建确定专家初始权重的优化模型;然后,利用不确定二元语义的可能度构造集结模糊评价矩阵以及方案的集结群体偏好,提出专家累积共识贡献测度和群体共识测度,通过对拥有较少合作的专家权重进行惩罚让群体自适应地达成共识,无需强迫专家修改其观点,提出一种群体共识决策方法对方案排序择优。最后,通过一个算例说明方法的可行性和有效性。     

Linguistic fuzzy consensus model for collaborative development of fuzzy cognitive maps: a case study in software development risks

Fuzzy cognitive maps (FCMs) have been widely used in several domains for information processing, such as, data fusion, decision making. Although several methods to automatically learn FCMs are recognized from the scientific literature, the most used approach to build an FCM relies on a collaborative task involving single person or, more suitably, group of experts. Collaborative development increases reliability and robustness of the resulting FCM, but rises some problems in terms of group decision making to aggregate different perspectives of the problem representation. This paper proposes to support collaborative development of FCMs introducing knowledge engineering process that relies on Linguistic Fuzzy Consensus Model. In the proposed approach, each expert builds the own version of the FCM. When all different versions are available, a Group Decision Making process is activated in order to reach the consensus on conflictual modeling opinions. The result is a unique final version of the FCM that is not a simple aggregation of the versions provided by the experts but is the result of a well-suited mathematical model. In addition, this work adopts consensus model with incomplete preference relations scheme to address knowledge harmonization issues. Finally, advantages and the limitations of the proposed framework are argued.     

Multicriteria analysis in decision making under information uncertainty

This paper presents results of research related to multicriteria decision making under information uncertainty. The Bellman–Zadeh approach to decision making in a fuzzy environment is utilized for analyzing multicriteria optimization models (X,M models) under deterministic information. Its application conforms to the principle of guaranteed result and provides constructive lines in obtaining harmonious solutions on the basis of analyzing associated maxmin problems. This circumstance permits one to generalize the classic approach to considering the uncertainty of quantitative information (based on constructing and analyzing payoff matrices reflecting effects which can be obtained for different combinations of solution alternatives and the so-called states of nature) in monocriteria decision making to multicriteria problems. Considering that the uncertainty of information can produce considerable decision uncertainty regions, the resolving capacity of this generalization does not always permit one to obtain unique solutions. Taking this into account, a proposed general scheme of multicriteria decision making under information uncertainty also includes the construction and analysis of the so-called X,R models (which contain fuzzy preference relations as criteria of optimality) as a means for the subsequent contraction of the decision uncertainty regions. The paper results are of a universal character and are illustrated by a simple example.     

基于模糊聚类的不同形式偏好信息的集结方法研究

针对具有5种不同形式偏好信息的群决策问题给出了一种分析方法.当专家给出的偏好信息是模糊互补判断矩阵、区间值、正互反矩阵、序关系值以及效用值时,首先把不同形式的偏好信息转化为模糊互补判断矩阵,然后,再根据模糊互补判断矩阵得出每个专家的方案排序值,据此对专家进行模糊聚类,根据聚类结果确定专家的权重,进而进行信息合成和方案选优,并用算例进行了验证.     

区间直觉模糊判断矩阵群决策中的逆判问题

研究了区间直觉模糊判断矩阵的群决策问题.定义了两种区间直觉模糊集相似度公式,给出两种与决策群体意见一致性程度最高的理想区间直觉模糊判断矩阵构造优化方法.利用矩阵对不同专家判断矩阵中相同位置元素的一致性进行分析,并对不同专家的判断信息进行整体相似程度分析,最后通过算例说明了该方法的有效性和实用性.     

Fuzzy ordered distance measures

Xu and Chen (J Syst Sci Syst Eng 17:432–445, 2008) introduced a type of ordered weighted distance measures called ordered weighted distance (OWD) measures whose fundamental aspect is the reordering step, which can be used in many actual fields, including group decision making, medical diagnosis, data mining, and pattern recognition, etc. The OWD measures are very suitable to deal with situations where the input data are expressed in exact numerical values. In this paper, we consider situations with linguistic, interval or fuzzy preference information, and develop some fuzzy ordered distance measures, such as linguistic ordered weighted distance measure, uncertain ordered weighted distance measure, linguistic hybrid weighted distance measure, and uncertain hybrid weighted distance measure, etc. After that, based on hybrid weighted distance measures, we establish a consensus reaching process of group decision making with linguistic, interval, triangular or trapezoidal fuzzy preference information.     

群决策中基于判断相似度的专家聚类及群体意见集结方法

针对大规模群决策问题,提出了一种基于专家意见相似度的群体判断信息逐步集结规划的方法。首先利用备选方案序关系向量的灰色关联度和夹角余弦构造两两专家判断信息的组合相似度;其次以判断相似度为标准,采用一种广度邻居搜索算法对专家进行聚类;然后以判断偏差最小为目标,构造非线性的约束规划模型对每一类专家意见进行集结,从而获得类内专家的集结信息;最后从专家数量最多的类别开始,依次对每类专家集结后的判断信息进行再次集结,从而获得最终的评判结果。该方法将大规模的复杂群决 策转化为低复杂度的多阶段专家信息集结问题,并保证了群体结果的一致性。算例分析验证了方法的可行性和有效性。     

权重信息未知对方案有偏好的vague集多指标决策

就指标权重未知,且对方案有偏好的vague集多指标决策问题,提出了通过使决策者的主观偏好值与属性值的相离度最小来建立最优化模型,从而获得指标的权重.通过将vague值转化为模糊值来建立模糊值矩阵,由模糊值矩阵按各指标对应值的大小对方案进行排序,形成多个线性序,进而由线性序来构造模糊优先矩阵,对其进行截割,得到最优方案.最后通过一个实例说明此方法的具体决策过程.     

基于直觉模糊距离的群决策专家意见聚合分析

提出了一种基于直觉模糊距离来聚合专家个人意见为一个优化的群体意见一致度的新方法.首先,根据直觉模糊集的几何意义,定义了两个直觉模糊集之间的距离;然后,利用直觉模糊距离来聚合专家两两之间对备选方案意见的一致度,综合考虑每位专家的相对重要权重,得到专家群体对备选方案意见的综合一致度;最后通过一个具体实例来说明这种方法的具体应用及计算过程.     

基于TOPSIS的模糊群体多属性决策方法

针对模糊群体多属性决策问题,给出一种基于理想点法(TOPSIS)的多属性决策方法.方法先用三角模糊数的形式表示专家评价值的模糊性和不确定性,而后考虑了专家在不同评价属性中的重要程度和意见的相似度,并将专家意见进行集结得到专家群体关于方案集的模糊决策矩阵,最后定义了三角模糊数形式的正负理想方案,通过计算各方案与正负理想方案的距离以及各方案与理想点的相对接近度,最终确定最优方案.通过实例分析说明了该方法的可行性和有效性.     

基于不同粒度语言判断矩阵的群体一致性分析方法

针对群决策中基于不同粒度语言判断矩阵形式偏好信息的群体一致性问题,提出了一种分析方法。首先,给出有关不同粒度语言判断矩阵和二元语义等若干定义,通过转换函数将不同粒度语言判断矩阵一致化为由二元语义表示的判断矩阵;然后,通过定义专家与群偏好的偏差矩阵以及各专家的总体偏差指标,给出了专家群体一致性的判别方法及专家群体判断不一致的调整方法;最后,通过一个算例说明了该方法的有效性。     

基于前景理论的区域绿色经济发展水平多指标评价

针对绿色经济发展水平评价指标存在模糊性、异质性等问题,提出一种规范化信息,并运用前景理论进行多指标评价的方法。首先,构建区域绿色经济发展水平评价指标体系。其次,规范化异质信息,并计算待比较方案的前景函数。然后,通过定义前景值排序与专家决策偏好的一致性和不一致性程度,建立决策优化模型,计算各方案的前景值,进行方案排序和优选。最后,以三明市县域绿色经济发展水平评价为例,说明该方法的有效性、合理性和实用性。     

Modeling subjective evaluation for fuzzy group multicriteria decision making

This paper presents a new fuzzy multicriteria decision making (MCDM) approach for evaluating decision alternatives involving subjective judgements made by a group of decision makers. A pairwise comparison process is used to help individual decision makers make comparative judgements, and a linguistic rating method is used for making absolute judgements. A hierarchical weighting method is developed to assess the weights of a large number of evaluation criteria by pairwise comparisons. To reflect the inherent imprecision of subjective judgements, individual assessments are aggregated as a group assessment using triangular fuzzy numbers. To obtain a cardinal preference value for each decision alternative, a new fuzzy MCDM algorithm is developed by extending the concept of the degree of optimality to incorporate criteria weights in the distance measurement. An empirical study of aircraft selection is presented to illustrate the effectiveness of the approach.