Nonlinear optimization models for multiple attribute group decision making with intuitionistic fuzzy information

Intuitionistic fuzzy numbers are very useful for experts to depict in depth their fuzzy preference information over objects. In this work, we investigate multiple attribute group decision‐making problems in which the attribute values provided by experts are expressed in intuitionistic fuzzy numbers, each of which is composed of a membership degree, a nonmembership degree and a hesitancy degree, and the weight information about both the experts and the attributes is to be determined. We first make different types of attribute values uniform so as to facilitate interattribute comparisons and employ the simple additive weighting method to fuse all the individual opinions into the group one. We then develop two nonlinear optimization models, one minimizing the divergence between each individual opinion and the group one, and the other minimizing the divergence among the individual opinions, from which two exact formulae can be obtained to derive the weights of experts. Similarly, from the viewpoint of maximizing group consensus, we establish a nonlinear optimization model based on all the individual intuitionistic fuzzy decision matrices to determine the weights of attributes. The simple additive weighting method is used to aggregate all the intuitionistic fuzzy attribute values corresponding to each alternative, and then the score function and the accuracy function are employed to rank and select the given alternatives. Moreover, we extend all the above results to interval intuitionistic fuzzy situations, and finally apply the developed models to an air‐condition system selection problem. © 2010 Wiley Periodicals, Inc.     

A new fuzzy MCDM approach based on centroid of fuzzy numbers

This paper proposes a new fuzzy MCDM (FMCDM) approach based on centroid of fuzzy numbers for ranking of alternatives. The FMCDM approach allows decision makers (DMs) to evaluate alternatives using linguistic terms such as very high, high, slightly high, medium, slightly low, low, very low or none rather than precise numerical values, allows them to express their opinions independently, and also provides an algorithm to aggregate the assessments of alternatives. Three numerical examples are investigated using the FMCDM approach to illustrate its applications. It is shown that the FMCDM approach offers a flexible, practical and effective way of group decision making.     

An integrated approach to machine selection problem using fuzzy SMART-fuzzy weighted axiomatic design

In respond to new market requirements and competitive positioning of manufacturing companies selecting optimal machines that are consistent with manufacturing goals is of crucial importance. As it involves multiple conflicting criteria and inherent ambiguity and vagueness, election of a suitable machine can be regarded as a fuzzy multi-criteria decision making problem. In this study, for the first time in the literature, an integrated approach consisting of fuzzy simple multiattribute rating technique (SMART) approach and fuzzy weighted axiomatic design (FWAD) approach is proposed to determining the optimal continuous fluid bed tea dryer for a privately owned tea plant operating in Turkey. The weights of the evaluation criteria are calculated via fuzzy SMART and then FWAD is utilized to rank competing machine alternatives in terms of their overall performance. In the FWAD application phase, five experts have determined functional requirements (FRs) and have rated alternatives. Therefore, individual fuzzy opinions were required to be aggregated in order to set up a group consensus. A group decision analysis, referred to as the least squares distance method is used to aggregating the ratings of FRs and alternatives. It is concluded that the proposed hybrid methodology is a robust decision support tool for ranking machine alternatives under fuzzy environment and furthermore, it can be exploited for other fuzzy decision making problems, as well.     

Ranking of alternatives with ordered weighted averaging operators

Multiattribute decision making is an important part of the decision process for both individual and group problems. We incorporate the fuzzy set theory and the basic nature of subjectivity due to ambiguity to achieve a flexible decision approach suitable for uncertain and fuzzy environments. Let us consider the analytic hierarchy process (AHP) in which the labels are structured as fuzzy numbers. To obtain the scoring that corresponds to the best alternative or the ranking of the alternatives, we need to use a total order for the fuzzy numbers involved in the problem. In this article, we consider a definition of such a total order, which is based on two subjective aspects: the degree of optimism/pessimism reflected with the ordered weighted averaging (OWA) operators. A numerical example is given to illustrate the approach. © 2004 Wiley Periodicals, Inc.     

基于直觉模糊集和证据理论的群决策方法

针对属性值和权重均为直觉模糊数的多属性决策问题,提出一种基于直觉模糊集和证据理论的群决策方法.首先,对专家给出的每个方案的属性值和属性权重进行证据合成,在此基础上合成每个方案的所有属性值;然后,基于直觉模糊集相似度确定专家的相对权重,修正方案证据,并合成所有专家证据,得到方案的信任区间,根据信任区间的大小对方案进行排序;最后,通过数值案例验证了所提出方法的有效性和合理性.     

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.     

A new approach for multiple attribute group decision making with interval-valued intuitionistic fuzzy information

This paper proposes a novel method for multiple attribute group decision making (MAGDM) with interval-valued intuitionistic fuzzy information. The interval-valued intuitionistic fuzzy numbers of each expert preference matrix are first mapped into two dimensions. Thus, the values of each membership degree and non-membership degree are considered as points in the two-dimensional representation. Moreover, the distance between the points represents the variance among the different experts preferences. The preference points of the same character are considered as a point set. We employ the plant growth simulation algorithm (PGSA) to calculate the optimal rally points of every point set, the sum of whose Euclidean distances to other given points is minimal, and these optimal rally points reflect the preferences of the entire expert group. These points are used to establish an expert preference aggregation matrix. Suitable points from the matrix are chosen to constitute an ideal point matrix, a projection method is employed to calculate the sum of its Euclidean distance to the expert preference aggregation matrix, and the score of each alternative is evaluated. Finally, the overall ranking of alternatives is obtained. In addition, this study develops a process to evaluate the pros and cons of different aggregation methods. Two typical examples are presented to illustrate the feasibility and effectiveness of the proposed approach.     

A new linguistic MCDM method based on multiple-criterion data fusion

Multiple-criteria decision-making (MCDM) is concerned with the ranking of decision alternatives based on preference judgements made on decision alternatives over a number of criteria. First, taking advantage of data fusion technology to comprehensively consider each criterion data is a reasonable idea to solve the MCDM problem. Second, in order to efficiently handle uncertain information in the process of decision making, some well developed mathematical tools, such as fuzzy sets theory and Dempster Shafer theory of evidence, are used to deal with MCDM. Based on the two main reasons above, a new fuzzy evidential MCDM method under uncertain environments is proposed. The rating of the criteria and the importance weight of the criteria are given by experts’ judgments, represented by triangular fuzzy numbers. Then, the weights are transformed into discounting coefficients and the ratings are transformed into basic probability assignments. The final results can be obtained through the Dempster rule of combination in a simple and straight way. A numerical example to select plant location is used to illustrate the efficiency of the proposed method.     

一种基于灰色关联度的多属性群决策方法

基于灰色关联度的思想提出了一种群决策方法。该方法根据各专家给出的决策矩阵,利用灰色关联度的思想确定专家权重,得出群决策矩阵,基于偏差最大化的思想计算属性权重,计算各方案的综合关联度,据此确定方案的排序。算例表明了利用此方法得出的方案综合关联度之间相对差距较大,提高了决策的科学性和有效性。指出了徐选华等人提出的算例中的错误,予以纠正。     

大群体应急决策中考虑属性关联的偏好信息融合方法

针对大群体应急决策中属性之间相互关联的问题,提出一种新的偏好信息融合方法.首先,利用最优离散拟合模型测度专家的风险偏好,提出一种考虑专家风险偏好的犹豫模糊元补充方法;其次,运用TF-IDF算法获得相互关联的事件属性集;再次,结合传统的主成分分析法与误差理论,提出基于犹豫模糊语言的主成分分析模型,获得几个互不关联的主属性及其权重,进而进行信息集结和方案择优;最后,通过广西洪涝灾害事件验证所提方法的可行性和有效性.     

A method for group decision making with multi-granularity linguistic assessment information

This paper proposes a method to solve the group decision making (GDM) problems with multi-granularity linguistic assessment information. In the method, the multi-granularity linguistic information provided by experts is firstly expressed in the form of fuzzy numbers. In order to make the collective opinion close to each expert’s opinion, a linear goal programming model is constructed to integrate the fuzzy assessment information and to directly compute the collective ranking values of alternatives without the need of information transformation. Then, a fuzzy preference relation on the pairwise comparisons of the collective ranking values of alternatives is constructed using the dominance possibility degree of the comparison between the fuzzy numbers. By applying a non-dominance choice degree to this fuzzy preference relation, the ranking of alternatives is determined and the most desirable alternative(s) is selected. An example is used to illustrate the applicability of the proposed method and its advantages.     

基于模糊目标和模糊约束的满意控制

研究在预测控制框架下进行模糊决策问题,提出一种基于模糊目标和模糊约束的预测控制方法。其目标函数以决策者的控制要求和最终控制的满意度来表示,比传统的加权方差具有更多的自由度;与基于二次型性能指标的预测相比,该方法可使系统设计更加灵活。仿真结果表明了该方法的有效性。     

模糊动态环境下复杂系统的满意优化控制

提出一种在满意控制框架下进行模糊决策的方法,将控制目标和系统约束模糊化,形 成多目标的优化问题,通过模糊规划方法求解,与基于二次型性能指标的预测控制相比,该方法 可使得系统设计更灵活.     

Supplier selection in a fuzzy group setting: A method using grey related analysis and Dempster–Shafer theory

Supplier selection in a fuzzy group setting is a very important strategic decision involving decisions balancing a number of conflicting criteria and opinions from different experts. This paper uses grey related analysis and Dempster–Shafer theory to deal with this fuzzy group decision making problem. First, in the individual aggregation, grey related analysis is employed as a means to reflect uncertainty in multi-attribute models through interval numbers. Second, in the group aggregation, the Dempster–Shafer (D–S) rule of combination is used to aggregate individual preferences into a collective preference, by which the candidate alternatives are ranked and the best alternative(s) are obtained. The proposed approach uses both quantitative and qualitative data for international supplier selection. It provides alternative tools to evaluate and improve supplier selection decisions in an uncertain global market.     

A consensus‐ and harmony‐based feedback mechanism for multiple attribute group decision making with correlated intuitionistic fuzzy sets

In this study, an interactive consensus model is proposed for correlated multiple attribute group decision making (MAGDM) problems with intuitionistic triangular fuzzy numbers (ITFNs). The harmony degree (HD) is investigated to determine the degree of maintaining experts' original information while the consensus level is defined as the proximity degree (PD) between an expert and other experts on three levels: evaluation elements of alternatives, alternatives, and decision matrices. Combining HD and PD, a three‐dimensional feedback mechanism is proposed to identify discordant experts, alternatives, and corresponding preference values that contribute less to consensus, and provides advice to reach a higher consensus level. Additionally, visual representation of experts' consensus position within the group is provided. Furthermore, a graphical simulation of future consensus and harmony status, if the recommended values were to be implemented, is also provided. Therefore, our proposed feedback mechanism guarantees that it increases the consensus level of the set of experts while maintaining, as much as possible, experts' original information. Then, the PD‐induced intuitionistic triangular fuzzy correlated averaging (PD‐IITFCA) operator is investigated to aggregate the interactive individual opinions between experts. Finally, the intuitionistic triangular fuzzy correlated averaging (ITFCA) operator is developed to aggregate the evaluation elements of alternatives under correlative attributes. Based on the score and accurate functions of ITFNs, an order relation is proposed to obtain the final solution of alternatives.     

基于方案偏好和部分权重信息的模糊多属性决策方法

研究了只有部分权重信息且决策者对方案的偏好信息以三角模糊数互反判断矩阵形式给出的模糊多属性决策问题.首先为得到属性权重,给出一种结合主观模糊偏好信息和客观决策信息的极小化极大偏差模型;然后,运用加性加权法求出各方案的模糊综合属性值,并利用已有的三角模糊数排序公式求得决策方案的排序;最后,通过算例说明了该方法的可行性和有效性.     

Fuzzy extension of TOPSIS model for group decision making under multiple criteria

Multi criteria group decision making methods are broadly used in the real-world decision circumstances for homogeneous groups. Significant and vital decisions are usually made by the heterogeneous groups of managers or experts in organizations. In many situations, experts may decide on the basis of imprecise information coming from a variety of sources about alternatives. In fact, some criteria are completely quantifiable, some partially quantifiable, and others completely subjective. This paper proposes a fuzzy extension of TOPSIS method for heterogeneous group decision making models under fuzzy environment. It converts the decision makers’ fuzzy decision matrices into an aggregated decision matrix to determine the most preferable choice among all possible alternatives. The results of a numerical example of proposed method have been highly consistent based on the Spearman’s rank correlation coefficient and showed good agreement with other methods.     

Interval programming method for hesitant fuzzy multi-attribute group decision making with incomplete preference over alternatives

The aim of this study is to employ the main structure of LINMAP (LINear programming technique for Multidimensional Analysis of Preference) to propose an interval programming method for solving multi-attribute group decision making (MAGDM) problems in which the ratings of alternatives are taken as hesitant fuzzy elements (HFEs) and all pair-wise comparison judgments over alternatives are represented by interval numbers. The contribution of this study is fivefold: (1) we define the new consistency and inconsistency indices; (2) we construct an interval programming model to determine the hesitant fuzzy positive ideal solution and the optimal weights of attributes, and at the same time present a decision algorithm; (3) we discuss several special cases of the proposed model in detail; (4) we show that compared with intuitionistic fuzzy LINMAP method (Li et al., 2010), the proposed approach reveals more useful information including the interval preference information, and does not need to transform HFEs into intuitionistic fuzzy numbers but directly deals with MAGDM problems and thus obtains better final decision results; and (5) we demonstrate the applicability and implementation process of the proposed approach by using an energy project selection example.     

An automatic approach to reaching consensus in multiple attribute group decision making

In this work, we consider the problem of consensus of multiple attribute group decision making, and develop an automatic approach to reaching consensus among group opinions. In the process of group decision making, each expert provides his/her preferences over the alternatives with respect to each attribute, and constructs an individual decision matrix. The developed approach first aggregates these individual decision matrices into a group decision matrix by using the additive weighted aggregation (AWA) operator, and then establishes a convergent iterative algorithm to gain a consentaneous group decision matrix. Then based on the consentaneous group decision matrix, the approach utilizes the AWA operator to derive the overall attribute values of alternatives, by which the most desirable alternative can be found out. Finally, we detailedly expound the implementation process of the approach with a practical example.     

Group decision making model and approach based on interval preference orderings

In group decision making under uncertainty, interval preference orderings as a type of simple uncertain preference structure, can be easily and conveniently used to express the experts’ evaluations over the considered alternatives. In this paper, we investigate group decision making problems with interval preference orderings on alternatives. We start by fusing all individual interval preference orderings given by the experts into the collective interval preference orderings through the uncertain additive weighted averaging operator. Then we establish a nonlinear programming model by minimizing the divergences between the individual uncertain preferences and the group’s opinions, from which we derive an exact formula to determine the experts’ relative importance weights. After that, we calculate the distances of the collective interval preference orderings to the positive and negative ideal solutions, respectively, based on which we use a TOPSIS based approach to rank and select the alternatives. All these results are also reduced to solve group decision making problems where the experts’ evaluations over the alternatives are expressed in exact preference orderings. A numerical analysis of our model and approach is finally carried out using two illustrative examples.