An integrated model-based interactive approach to FMAGDM with incomplete preference information

Incomplete fuzzy preference relations, incomplete multiplicative preference relations, and incomplete linguistic preference relations are very useful to express decision makers’ incomplete preferences over attributes or alternatives in the process of decision making under fuzzy environments. The aim of this paper is to investigate fuzzy multiple attribute group decision making problems where the attribute values are represented in intuitionistic fuzzy numbers and the information on attribute weights is provided by decision makers by means of one or some of the different preference structures, including weak ranking, strict ranking, difference ranking, multiple ranking, interval numbers, incomplete fuzzy preference relations, incomplete multiplicative preference relations, and incomplete linguistic preference relations. We transform all individual intuitionistic fuzzy decision matrices into the interval decision matrices and construct their expected decision matrices, and then aggregate all these expected decision matrices into a collective one. We establish an integrated model by unifying the collective decision matrix and all the given different structures of incomplete weight preference information, and develop an integrated model-based approach to interacting with the decision makers so as to adjust all the inconsistent incomplete fuzzy preference relations, inconsistent incomplete linguistic preference relations and inconsistent incomplete multiplicative preference relations into the ones with acceptable consistency. The developed approach can derive the attribute weights and the ranking of the alternatives directly from the integrated model, and thus it has the following prominent characteristics: (1) it does not need to construct the complete fuzzy preference relations, complete linguistic preference relations and complete multiplicative preference relations from the incomplete fuzzy preference relations, incomplete linguistic preference relations and incomplete multiplicative preference relations, respectively; (2) it does not need to unify the different structures of incomplete preferences, and thus can simplify the calculation and avoid distorting the given preference information; and (3) it can sufficiently reflect and adjust the subjective desirability of decision makers in the process of interaction. A practical example is also provided to illustrate the developed approach.     

On Method for Uncertain Multiple Attribute Decision Making Problems with Uncertain Multiplicative Preference Information on Alternatives

The uncertain multiple attribute decision making (UMADM) problems are investigated, in which the information about attribute weights is known partly and the attribute values take the form of interval numbers, and the decision maker (DM) has uncertain multiplicative preference information on alternatives. We make the decision information uniform by using a transformation formula, and then establish an objective-programming model. The attribute weights can be determined by solving the developed model. The concept of interval positive ideal point of alternatives (IPIPA) is introduced, and an approach based on IPIPA and projection to ranking alternatives is proposed. The method can avoid comparing and ranking interval numbers, and can reflect both the objective information and the DMs subjective preferences.     

A lexicographical compromise method for multiple criteria group decision problems with imprecise information

This paper addresses multiple criteria group decision making problems where each group member offers imprecise information on his/her preferences about the criteria. In particular we study the inclusion of this partial information in the decision problem when the individuals’ preferences do not provide a vector of common criteria weights and a compromise preference vector of weights has to be determined as part of the decision process in order to evaluate a finite set of alternatives. We present a method where the compromise is defined by the lexicographical minimization of the maximum disagreement between the value assigned to the alternatives by the group members and the evaluation induced by the compromise weights.     

部分权重信息下对方案有互补偏好的多属性决策法

研究了只有部分权重信息且对方案的偏好信息以模糊互补判断矩阵形式给出的多属性决策问题.首先,基于模糊互补判断矩阵的主观偏好信息,利用转换函数将客观决策信息一致化,建立一个目标规划模型,通过求解该模型得到属性权重,从而利用加性加权法获得各方案的综合属性值,并以此对方案进行排序或择优.提出了一种基于目标规划的多属性决策方法.该方法具有操作简便和易于上机实现的特点.最后,通过实例说明模型及方法的可行性和有效性.     

Decision-aid for discrete multiple criteria decision making problems with imprecise data

We describe ways of aiding decision making with a discrete set of alternatives. In many decision situations, it is not possible to obtain explicit preference information from the decision makers. Instead, useful decision-aid can be provided to the decision makers by describing what kind of weighting of the criteria result in certain choices of the alternatives. The suggested treatment is based on the basic ideas of the ELECTRE III method. The modelling of the preferences by pseudo-criteria is especially helpful in case the data, that is, the criterion values are imprecise. Unlike ELECTRE III, no ranking of the alternatives is produced. Based on a minimum-procedure in the exploitation of the outranking relations, we provide information about the weights of the criteria that make a certain alternative the best. We also present an interactive searching procedure in the weighting space. The auxiliary optimization problems to be solved are nondifferentiable. Cases with both single and multiple decision makers are considered.     

基于区间模糊数的多阶段冲突型大群体应急决策方法

针对阶段权重未知且偏好信息表示为区间模糊数的多阶段大群体应急决策问题,提出一种新的群决策方法。首先给出了区间模糊数相似度公式,利用该公式对各阶段的专家偏好信息进行聚类;然后构建相对熵优化模型对聚集权重和阶段权重进行求解,得到整个决策过程的综合群体偏好,根据综合群体偏好对备选方案进行排序,确定最佳方案;最后通过算例对该方法的有效性和可行性进行验证。     

A flexible approach to ranking with an application to MBA Programs

We develop a model for flexibly ranking multi-dimensional alternatives/units into preference classes via Mixed Integer Programming. We consider a linear aggregation model, but allow the criterion weights to vary within pre-specified ranges. This allows the individual alternatives/units to play to their strengths. We illustrate the use of the model by considering the Financial Times Global MBA Program rankings and discuss the implications. We argue that in many applications neither the data nor the weights or the aggregation model itself is precise enough to warrant a complete ranking, providing an argument for sorting or what we call flexible ranking.     

Fuzzy relations for the analysis of traders’ preferences in an information market game

In this paper we focus on preference and decision data gathered during a computer-supported information market game in which 35 students participated during seven consecutive trading sessions. The participants’ individual preferences on the market shares are collected to calculate a collective preference ranking using the Borda social choice method. Comparing this preference ranking to the shares’ actual market ranking resulting from the participants’ trading, we find a statistically significant difference between both rankings. As the preferences established by market behavior cannot be adequately explained through a social choice rule, we propose an alternative explanation based on the herd behavior phenomenon where traders imitate the most successful trader in the market. Using a decision analysis technique based on fuzzy relations, we study the participants’ rankings of the best share in the market during 7 weeks and compare the most successful trader to the other traders. The results from our analysis show that a substantial number of traders is indeed following the market leader.     

Extension of the LINMAP for multiattribute decision making under Atanassov’s intuitionistic fuzzy environment

The aim of this article is further extending the linear programming techniques for multidimensional analysis of preference (LINMAP) to develop a new methodology for solving multiattribute decision making (MADM) problems under Atanassov’s intuitionistic fuzzy (IF) environments. The LINMAP only can deal with MADM problems in crisp environments. However, fuzziness is inherent in decision data and decision making processes. In this methodology, Atanassov’s IF sets are used to describe fuzziness in decision information and decision making processes by means of an Atanassov’s IF decision matrix. A Euclidean distance is proposed to measure the difference between Atanassov’s IF sets. Consistency and inconsistency indices are defined on the basis of preferences between alternatives given by the decision maker. Each alternative is assessed on the basis of its distance to an Atanassov’s IF positive ideal solution (IFPIS) which is unknown a prior. The Atanassov’s IFPIS and the weights of attributes are then estimated using a new linear programming model based upon the consistency and inconsistency indices defined. Finally, the distance of each alternative to the Atanassov’s IFPIS can be calculated to determine the ranking order of all alternatives. A numerical example is examined to demonstrate the implementation process of this methodology. Also it has been proved that the methodology proposed in this article can deal with MADM problems under not only Atanassov’s IF environments but also both fuzzy and crisp environments.     

Aggregation of ordinal data using ordered weighted averaging operator weights

In multi-criteria decision-making problems, ordinal data themselves provide a convenient instrument for articulating preferences but they impose some difficulty on the aggregation process since ambiguity prevails in the preference structure inherent in the ordinal data. One of the key concerns in the aggregation of ordinal data is to differentiate among the rank positions by reflecting decision-maker??s preferences. Since individual attitude is fairly different, it is presumable that each ranking position has different importance. In other words, the quantification schemes among the rank positions could vary depending on the individual preference structure. We find that, among others, the ordered weighted averaging (OWA) operator can help to take this concept into effect on several reasons. First, the OWA operator provides a means to take into account a discriminating factor by introducing the measure of attitudinal character. Second, it can produce appropriate ranking weights corresponding to each rank position by solving a mathematical program subject to the constraint of attitudinal character. To better understand the attitudinal character playing a role as a discriminating factor, we develop centered ranking weights from ordinal weak relations among the ranking positions and then investigate their properties to relate them with the OWA operator weights having the maximum entropy. Finally, we present a method for generating the OWA operator weights via rank-based weighting functions.     

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。     

Eliciting and mapping qualitative preferences to numeric rankings in group decision making

Group work is becoming the norm in organizations. From strategy planning committees to quality management teams, organizational members are collaborating on problem solving. One area of team support that is often desired is the scoring and ranking of decision alternatives on qualitative/subjective domains, and the aggregation of individual preferences into group preferences. In this paper we present a new conceptual approach to qualitative preference elicitation and aggregation. This approach is based on well established decision analysis techniques. It significantly advances the state of the art of group decision making by addressing four common limitations: (1) the inability to deal with vagueness of human decision makers in articulating preferences; (2) difficulties in mapping qualitative evaluation to numeric estimates; (3) problems in aggregating individual preferences into meaningful group preference; and (4) the lack of simple user friendly techniques for dealing with a large number of decision alternatives. Our approach is easy to implement in stand alone personal computers and groupware. We illustrate this with a real-world problem.     

基于相对熵的区间Pythagorean模糊多属性AQM决策方法及其应用

针对决策信息为区间Pythagorean模糊数,属性权重不完全确定的多属性决策问题,提出了一种基于相对熵的AQM决策方法。首先,提出区间Pythagorean模糊数的相对熵,计算了各方案与区间Pythagorean模糊正理想方案和负理想方案间的相对熵,据此构建了基于方案相对满意度最大的非线性规划属性权重确定模型;其次,针对每个属性,利用新的区间Pythagorean模糊数得分函数计算方案的0-1优先关系矩阵,依据AQM方法对所有0-1优先关系矩阵进行融合得到合成0-1优先关系矩阵,并确定了方案的综合度,由此获得方案的排序。最后,以软件开发项目的选取为实例说明了该方法的可行性和有效性。     

基于二元联系数距离的不确定语言多属性决策模型

研究了属性权重范围已知,方案主观偏好值为语言变量,决策信息为不确定语言决策矩阵的多属性决策问题.在给出不确定语言变量转换为二元联系数的公式以及二元联系数距离公式的基础上,将方案主观偏好语言评价值转换为二元联系数,将不确定语言决策矩阵转换为二元联系数决策矩阵,从而得到方案的二元联系数综合属性值,通过最小化方案的二元联系数综合属性值和主观偏好值之间距离,建立多目标优化模型,并将其转换为一个单目标规划模型计算出属性权重.然后,通过对方案的二元联系数综合属性值进行不确定性分析,得到各方案的排序总数,利用排序总数对方案进行排序择优.应用实例表明该决策方法可行有效.     

Performance evaluation: An integrated method using data envelopment analysis and fuzzy preference relations

In a multi-attribute decision-making (MADM) context, the decision maker needs to provide his preferences over a set of decision alternatives and constructs a preference relation and then use the derived priority vector of the preference to rank various alternatives. This paper proposes an integrated approach to rate decision alternatives using data envelopment analysis and preference relations. This proposed approach includes three stages. First, pairwise efficiency scores are computed using two DEA models: the CCR model and the proposed cross-evaluation DEA model. Second, the pairwise efficiency scores are then utilized to construct the fuzzy preference relation and the consistent fuzzy preference relation. Third, by use of the row wise summation technique, we yield a priority vector, which is used for ranking decision-making units (DMUs). For the case of a single output and a single input, the preference relation can be directly obtained from the original sample data. The proposed approach is validated by two numerical examples.     

IS/IT项目选择决策的灰色逼近理想解排序法

IS/IT项目选择决策是一个多属性决策问题.针对传统逼近理想解排序法(TOPSIS)在确定属性权重系数上的缺陷,并考虑到在实际IS/IT项目选择决策过程中部分决策信息的不足,提出了基于灰色TOPSIS改进算法.算法运用区间灰数表达指标权重和指标评价值,定义备择项目与正、负理想解的灰色关联度,依此计算各备则项目的贴近度并实现最终排序.仿真实例验证了该方法的合理和有效性.     

基于分段策略评分的冲突主体偏好排序方法及其应用

冲突中各利益主体的偏好信息对冲突局势的演变和纠纷调解具有重要影响。现有的冲突偏好排序方法主要基于决策者对冲突局势或状态、策略权重和声明信息的主观判断和理解,缺乏科学的数据来源支撑。为准确获取冲突主体的偏好信息,本文提出了一种基于调查法的分段策略冲突偏好排序方法。首先,根据决策者类别将冲突策略集合进行分段,并通过问卷、调研等方法获取每个冲突主体对所有分段策略的重要度评分信息。在此基础上,计算决策者对各个冲突状态的综合偏好评分,进而得到状态偏好的排序结果。最后以医患纠纷为例,对比分析了传统策略权重法和分段策略评分法的偏好排序和稳定性分析结果,进一步验证了所提方法的有效性。     

A fuzzy ranking method with range reduction techniques

For ranking alternatives based on pairwise comparisons, current analytic hierarchy process (AHP) methods are difficult to use to generate useful information to assist decision makers in specifying their preferences. This study proposes a novel method incorporating fuzzy preferences and range reduction techniques. Modified from the concept of data envelopment analysis (DEA), the proposed approach is not only capable of treating incomplete preference matrices but also provides reasonable ranges to help decision makers to rank decision alternatives confidently.     

On the maximum number of feasible ranking sequences in multi-criteria decision making problems

In many decision problems the focus is on ranking a set of m alternatives in terms of a number, say n, of decision criteria. Given are the performance values of the alternatives for each one of the criteria and the weights of importance of the criteria. This paper demonstrates that if one assumes that the criteria weights are changeable, then the number of all possible rankings may be significantly less than the upper limit of m!. As a matter of fact, this paper demonstrates that the number of possible rankings is a function of the number of alternatives and the number of criteria. These findings are important from a sensitivity analysis point of view or when a group decision making environment is considered.