Deriving Weights from Interval Multiplicative Preference Relations in Group Decision Making

In this article, we investigate group decision making problems with interval multiplicative preference relations (including complete interval multiplicative preference relations and incomplete interval multiplicative preference relations). On the basis of the number of judgments and the consistency degree of each interval multiplicative preference relation, we first give a combined weighting method to derive the weights of decision makers. Then, we establish two linear programming models to derive the weight intervals of alternatives from all individual consistent interval multiplicative preference relations and utilize the continuous ordered weighted averaging operator or the continuous ordered weighted geometric operator to aggregate all the values in each weight interval. In addition, we establish a more general model to check the consistency of all individual interval multiplicative preference relations. In the cases where the optimal objective value of the model is not zero, we can get the optimal weights of alternatives directly, and then utilize these optimal weights and the optimal deviation values derived from the model to construct consistent interval multiplicative preference relations. Furthermore, we discuss some special cases of the established models and illustrate our models with a practical example.     

Consistency in MCGDM Problems with Intuitionistic Fuzzy Preference Relations Based on an Exponential Score Function

This paper investigates an approach for multiple criteria decision making problems with intuitionistic fuzzy preference relations (IFPRs). An exponential score function for intuitionistic fuzzy numbers is developed. Its novelty is that it can convert an IFPR into a multiplicative preference relation. Then, the exponential score preference relation is introduced and its associated optimal transfer matrix is proposed. The optimal transfer matrix can guarantee the consistency via the additive transitivity property. Furthermore, an optimal model for aggregating group IFPRs to reach consistency is proposed. Finally, a group decision making approach and a multi-criteria decision making approach with IFPRs are presented.     

Compatibility Analysis of Intuitionistic Fuzzy Preference Relations in Group Decision Making

Compatibility analysis is an efficient and important tool used to measure the consensus of opinions within a given group of individuals. In this paper, we give a compatibility measure between intuitionistic preference values and a compatibility measure between intuitionistic preference relations, respectively, and study their properties. It is shown that each individual intuitionistic preference relation and the collective intuitionistic preference relation is perfectly compatible if and only if all the individual intuitionistic preference relations are perfectly compatible. Based on the compatibility measures, a consensus reaching procedure in group decision making with intuitionistic preference relations is developed, and a method for comparing intuitionistic fuzzy values is pointed out, by which the considered objects are ranked and selected. In addition, we extend the developed measures, procedure and method to accommodate group decision making situations with interval-valued intuitionistic preference relations. Numerical analysis on our results through an illustrative example is also carried out.     

On Consensus of Group Decision Making with Interval Utility Values and Interval Preference Orderings

Uncertainty is a common phenomenon in our real world. Interval utility values and interval preference orderings are two of the simplest and most convenient tools to describe uncertain preferences in decision making. In this paper, we investigate consensus problems in group decision making with interval utility values and interval preference orderings. We first establish their transformation relations, and give a formula for calculating the association coefficients of individual uncertain preferences and group ones. We then develop a consensus procedure for group decision making with interval utility values and interval preference orderings, which takes interval utility values as the uniform preference representation. This procedure can be reduced to a series of processes for dealing with some special group decision making situations, such as: group decision making with utility values and preference orderings, group decision making with interval utility values, group decision making with interval preference orderings, etc. Finally, we illustrate the applications of the developed procedures with two practical examples.     

Logarithm Compatibility of Interval Multiplicative Preference Relations with an Application to Determining the Optimal Weights of Experts in the Group Decision Making

We propose the new compatibility of interval multiplicative preference relations (IMPRs) in the group decision making (GDM) and apply it to determine the weights of experts. Firstly, we introduce the operation of interval numbers and define the new conception of logarithm compatibility degree of two interval multiplicative preference relations. Then, we prove the properties of logarithm compatibility of IMPR. It is pointed that if IMPR provided by every expert and its characteristic matrix are of acceptable compatibility, then the synthetic preference relation and the synthetic characteristic matrix are also of acceptable compatibility. Furthermore, we construct a mathematical programming model to determine the optimal weights of experts by minimizing the square logarithm compatibility in the GDM with IMPR and discuss the solution to the model. Finally, a numerical example is illustrated to show that the model is feasible.     

Uncertain Power Average Operators for Aggregating Interval Fuzzy Preference Relations

In this paper, we investigate group decision making problems based on interval fuzzy preference relations. We define an uncertain power weighted average (UPWA) operator and an uncertain power ordered weighted average (UPOWA) operator, on the basis of the power average operator of Yager (IEEE Trans Syst Man Cybern A 31:724–731, 1988) and the uncertain geometric mean. In the situations where the weights of experts are known, we develop a method based on the UPWA operator for group decision making with interval fuzzy preference relations; and in the situations where the weights of experts are unknown, we develop a method based on the UPOWA operator for group decision making with interval fuzzy preference relations.     

A Note on Linguistic Hybrid Arithmetic Averaging Operator in Multiple Attribute Group Decision Making with Linguistic Information

In this paper, we propose a linguistic hybrid arithmetic averaging (LHAA) operator, which is based on linguistic weighted arithmetic averaging (LWAA) operator and extended ordered weighted averaging (EOWA) operator, and study some desirable properties of the LHAA operator. The LHAA operator can not only reflect the importance degrees of both the given argument and its ordered position, but also relieve the influence of unfair arguments on the decision results by weighting these arguments with small values. Based on the LWAA and LHAA operators, we develop a practical approach to multiple attribute group decision making under linguistic environment. The approach first aggregates the individual linguistic preference values into a collective linguistic preference value for each alternative by using the LWAA and LHAA operators (it is worth pointing out that the aggregation process does not produce any loss of linguistic information), and then orders the collective linguistic preference values to obtain the best alternative(s). Finally, an illustrative example is also given to verify the approach and to demonstrate its feasibility and practicality.     

An Interactive Approach to Multiple Attribute Group Decision Making with Multigranular Uncertain Linguistic Information

The multiple attribute group decision making (MAGDM) problems having multiple sources of uncertain linguistic information assessed in different linguistic label sets are investigated. The existing linguistic labels in a linguistic label set are uniformly and symmetrically distributed, but in many real-life situations, the unbalanced linguistic information appears due to the nature of the linguistic variables used in the problems (Herrera and Herrera-Viedma, Proceedings of 4th international workshop on preferences and decisions, Trento, Italy, 2003). In this paper, we first define some unbalanced linguistic label sets, and then develop some transformation functions to unify the given multigranular linguistic labels in a unique linguistic label set without loss of information. Moreover, we utilize the uncertain linguistic weighted averaging operator to aggregate all individual uncertain linguistic decision matrices into a collective one, and define two similarity measures, one for measuring the similarity degree between each pair of uncertain linguistic variables, and the other for checking the consensus degrees among the individual uncertain linguistic decision matrices and the collective uncertain linguistic decision matrix. Finally, we develop an interactive approach to MAGDM with multigranular uncertain linguistic information and illustrate the developed approach with an application example.     

Prioritization and Aggregation of Intuitionistic Preference Relations: A Multiplicative-Transitivity-Based Transformation from Intuitionistic Judgment Data to Priority Weights

This article proposes a goal programming framework for deriving intuitionistic fuzzy weights from intuitionistic preference relations (IPRs). A new multiplicative transitivity is put forward to define consistent IPRs. By analyzing the relationship between intuitionistic fuzzy weights and multiplicative consistency, a transformation formula is introduced to convert normalized intuitionistic fuzzy weights into multiplicative consistent IPRs. By minimizing the absolute deviation between the original judgment and the converted multiplicative consistent IPR, two linear goal programming models are developed to obtain intuitionistic fuzzy weights from IPRs for both individual and group decisions. In the context of multicriteria decision making with a hierarchical structure, a linear program is established to obtain a unified criterion weight vector, which is then used to aggregate local intuitionistic fuzzy weights into global priority weights for final alternative ranking. Two numerical examples are furnished to show the validity and applicability of the proposed models.     

A Fuzzy Multi-Criteria Decision Making Model for IS Student Group Project Assessment

To accommodate the criterion-referenced student group project assessment approach, this paper proposes a fuzzy group Multi-Criteria Decision Making (MCDM) model. The proposed model can be used to solve student group project assessment problem in particular and generic MCDM problems in general, where the criteria used are often different and can be scored with multiple preference formats. The proposed fuzzy group MCDM model supports seven different preference formats, including preference ordering, utility vector, linguistic term vector, selected subset, fuzzy selected subset, fuzzy preference relation, and normal preference relation. It has been firstly used in the context of Information Systems (IS) student group project assessment.     

Aggregation of Individual Preference Structures in Ahp-Group Decision Making

This paper presents a new procedure, to which we have given the name Aggregation of Individual Preference Structures (AIPS), whose objective is to deal with multiactor decision making when using Analytic Hierarchy Process (AHP) as the methodological support. This procedure incorporates ideas similar to Borda count methods and transfers to the case of preference structures the principle of aggregation employed in the two approaches traditionally followed in AHP-group decision making (aggregation of individual judgments and aggregation of individual priorities). The new aggregation method allows us to capture: (i) the richness of uncertainty inherent to human beings; (ii) the vision of each decision maker within the context of the problem; (iii) the interdependencies between the alternatives being compared and (iv) the intensities of the preferences that each decision maker gives to these interdependencies. From the preference structure distribution associated to each decision maker, this new approach (AIPS) provides the holistic importance of each alternative and ranking, as well as the most representative preference structure distribution for the group. The knowledge derived from these could be employed as an initial step in the search for consensus, which characterises the negotiation processes followed by the actors involved in the resolution of decisional problems. An earlier version of this paper was presented at the GDN2005 Conference held at Vienna. The work has been partially funded under Research Projects “Electronic Government. Internet-based Complex Decision Making: e-democracy and e-cognocracy” (Ref. PM2004-052) and “Internet-based Complex Decision Making. Decisional Tools for e-cognocracy” (Ref. TSI2005-02511).     

A Consensus Based Method for Multi-criteria Group Decision Making Under Uncertain Linguistic Setting

A two-stage (a consensus process and a selection process) approach is proposed to solve multi-criteria group decision making problems under an uncertain linguistic environment. Since achieving general consensus is a desirable goal in group decision making, the proposed method first develops a consensus reaching process in order to reach a satisfactory consensus. Based on the partial order of uncertain linguistic variables, the superiority index of one alternative over another for a given criterion and the overall superiority index of one alternative are defined. Then a procedure based on the superiority indices is described to select the best alternative(s). Given the decision makers’ desire for a consensus solution, a common framework based on the previous consensus model and the selection process is presented. Finally, a practical application is demonstrated to show the effectiveness of the proposed method.     

Defining a right problem in group decision and negotiation: Feeling and evolutionary generating procedures

It is not unusual that decision makers define and solve a wrong problem. Here we develop an operational procedure for defining a right or correct problem. A problem may be represented as—a problem representation defining a problem consists of—two evolving hierarchies of relations, as discussed in the article. Rightness in a problem representation requires rightness in these relations which are beliefs held by a decision maker. Operational definition and validation of rightness in these relations, that is, retaining them as correct, is by feeling and by specified evolutionary generating procedures for examining, changing (evolving), and retaining these relations. Based on rightness in these relations, we discuss right group problem definition and solution in the general case where information is not fully shared (nonshared) among individuals in the group. Thus, our work contributes to procedural rationality—how decisions should be or are made—in individual and group decision-making and associated group decision and negotiation support systems (GDNSS).     

An Acceptable Consistency-Based Framework for Group Decision Making with Intuitionistic Preference Relations

This article studies acceptable consistency of intuitionistic preference relations (IPRs) and examines how to aggregate individual IPRs into a collective judgment in a group decision making (GDM) context. A consistency index is first introduced to measure the consistency level, thereby defining acceptable consistency for IPRs. If a decision-maker is unwilling or unavailable to revise his/her judgment for an IPR with unacceptable consistency, an automated approach is developed to improve its consistency to an acceptable level. The acceptably consistent IPRs are subsequently aggregated into a group opinion by using an induced ordered weighted averaging operator. A procedure is then proposed to solve GDM problems with IPRs. An illustrative example is presented to demonstrate the effectiveness and applicability of the proposed approach.     

模糊群决策方法在中间商选择中的应用研究

企业在选择中间商时往往面临着候选伙伴评价指标信息获取的困难,为了使决策科学合理,采用群决策方法集成各领域不同专家的信息,通过语言变量把专家的主观判断转化为对应模糊数,建立相应的数学模型,并计算出用模糊数表示的候选者综合评价值。     

AHP-Group Decision Making: A Bayesian Approach Based on Mixtures for Group Pattern Identification

This paper proposes a Bayesian estimation procedure to determine the priorities of the Analytic Hierarchy Process (AHP) in group decision making when there are a large number of actors and a prior consensus among them is not required. Using a hierarchical Bayesian approach based on mixtures to describe the prior distribution of the priorities in the multiplicative model traditionally used in the stochastic AHP, this methodology allows us to identify homogeneous groups of actors with different patterns of behaviour for the rankings of priorities. The proposed procedure consists of a two-step estimation algorithm: the first step carries out a global exploration of the model space by using birth and death processes, the second concerns a local exploration by means of Gibbs sampling. The methodology has been illustrated by the analysis of a case study adapted from a real experiment on e-democracy developed for the City Council of Zaragoza (Spain). Partially funded under the research project Electronic Government. Internet-based Complex Decision Making: e-democracy and e-cognocracy (Ref. PM2004-052) approved by the Regional Government of Aragon (Spain) as part of the multi-disciplinary projects programme.     

A Spreadsheet Module for Consistent Consensus Building in AHP-Group Decision Making

This paper presents a spreadsheet module developed with Microsoft Excel and Visual Basic to search for consensus when dealing with AHP in group decision making. On the basis of the theoretical results included in the paper Moreno-Jiménez et al. (2003), this spreadsheet module allows us to construct, in a local context (one criterion), the Consistent Consensus Matrix (CCM) corresponding to the core of consistency associated with the actors involved in the decision making process. By means of simulation techniques, this module provides, for each of the different values of the inconsistency thresholds considered, the preference structures derived from the interval incomplete matrix (CCM) that represents the groups core of consistency. This information will be used to establish consensus paths between the actors involved in the resolution process, one of the most important goals in distributed and participatory decision making.This paper has received partial financial support from the Research Project of the University of Zaragoza (2002): Herramientas decisionales para el constructivismo cognitivo (ref. 228-056). A previous version of this paper has been presented at the Group Decision and Negotiation session of the EURO-INFORMS Congress in Istanbul, 2003.     

A Relative Entropy Approach to Group Decision Making with Interval Reciprocal Relations Based on COWA Operator

A new approach has been presented based on relative entropy to rank all the alternatives in the group decision making with interval reciprocal relations. First we introduced the continuous ordered weighted averaging operator and used it to aggregate all individual interval reciprocal relations to derive the priority vector. Then we define the consensus indicator of the interval reciprocal relations to determine the weights of experts in the group decision making. Based on the conception of relative entropy, we construct an optimization model to minimize the difference between the group priority vector and all individual priority vectors. We also give the solution to the optimal model, in which we obtain the formula to rank the given alternatives in the group decision making for the collective reciprocal relations and select the most desirable one. Finally, a numerical example shows that the developed approach is feasible and the result is credible.     

A logistic regression-based pairwise comparison method to aggregate preferences

In a group decision making process, several individuals or a committee have the responsibility to choose the best alternative from a set. The problem addressed in this paper is how to aggregate personal preferences to arrive at an optimal group decision. New technologies allow individuals that may seldom or never meet to make group decisions. This paper proposes a methodology to obtain the group preference ordering in two steps. Firstly, each individual studies the problem isolated, and then, in a possibly virtual meeting, the group must agree on the preferences on some pairs of alternatives. Then, the group criterion is achieved by using a logistic regression model within the pairwise comparison framework proposed here. Properties of the procedure are studied and two illustrative examples are presented.     

A linguistic decision process in group decision making

Assuming a set of linguistic preferences representing the preferences of the individuals, a linguistic choice process is presented. This is developed using the concept of fuzzy majority for deriving a collective linguistic preference, and the concept of nondominated alternatives for deriving the selected alternatives in the linguistic choice process. The fuzzy majorities are equated with fuzzy linguistic quantifiers. The collective linguistic preference is derived by means of a linguistic ordered weighted averaging operator whose weights are defined using a fuzzy linguistic quantifier. In order to obtain the nondominated alternatives, we present a novel reformulation of Orlovski's nondominance degree under linguistic information.