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.     

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.     

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.     

Minimizing Group Discordance Optimization Model for Deriving Expert Weights

This paper focuses on the problem of how to determine expert weights in multiple attribute group decision making. We first aggregate all the individual decision matrices into the collective decision matrix by means of the weighted arithmetic averaging operator, and then from the angle of minimizing group discordance, we establish a general nonlinear optimization model based on deviation function, and employ a genetic algorithm to solve our model so as to find the optimal expert weights. Moreover, we extend our model to uncertain multiple attribute group decision making, where the attribute values are interval numbers, and finally, apply our model to the plan evaluation of new model of cars of an investment company.     

Multi-criteria Group Decision-Making Method Based on Intuitionistic Interval Fuzzy Information

For problems in multi-criteria group decision-making (MCGDM), this paper defines intuitionistic interval numbers, and the operational laws and comparison method of it. Some intuitionistic interval information aggregation operators are proposed, such as intuitionistic interval weighted arithmetic averaging operator, intuitionistic interval weighted geometric averaging operator, intuitionistic interval ordered weighted averaging operator, intuitionistic interval heavy averaging operator and intuitionistic interval aggregating operator. Then, based on intuitionistic interval fuzzy information, a method is developed to handle the problems in MCGDM. In this method, by applying the knowledge level of the experts to the decision making problem, the model of maximizing comprehensive membership coefficient is constructed to determine the weights of decision makers. By calculating the distances to the ideal and negative ideal solutions, the comprehensive attribute values and the rank of the alternatives can be obtained. Finally, an example is provided to demonstrate the feasibility and effectiveness of the proposed method.     

Some Intuitionistic Fuzzy Weighted Distance Measures and Their Application to Group Decision Making

Xu and Chen (J Syst Sci Syst Eng 17:432–445, 2008) introduced a new decision-making technique called the ordered weighted distance (OWD) measure, having been proved useful for the treatment of situation where the available information is represented in exact numerical values. In this paper, we consider the situations with intuitionistic fuzzy and interval-valued intuitionistic information, and develop some intuitionistic fuzzy weighted distance measures such as intuitionistic fuzzy ordered weighted distance (IFOWD) measure, interval-valued intuitionistic fuzzy ordered weighted distance (IVIFOWD) measure, intuitionistic fuzzy hybrid weighted distance (IFHWD) measure and interval-valued intuitionistic fuzzy hybrid weighted distance (IVIFHWD) measure. These developed weighted distance measures are very suitable to deal with the situation where the input data are represented in intuitionistic fuzzy numbers or interval-lvalued intuitionistic fuzzy numbers. Then we present a consensus reaching process for group decision making with intuitionistic fuzzy preference information based on the developed distance measures. Finally, a practical application of he developed approach to the problem of evaluating university faculty for tenure and promotion is given.     

A Deviation-Based Approach to Intuitionistic Fuzzy Multiple Attribute Group Decision Making

The aim of this article is to investigate the approach to multiple attribute group decision making (MAGDM) with intuitionistic fuzzy information. We first introduce a deviation measure between two intuitionistic fuzzy numbers, and then utilize the intuitionistic fuzzy hybrid aggregation operator to aggregate all individual intuitionistic fuzzy decision matrices into a collective intuitionistic fuzzy decision matrix. Based on the deviation measure, we develop an optimization model by which a straightforward formula for deriving attribute weights can be obtained. Furthermore, based on the intuitionistic fuzzy weighted averaging operator and information theory, we utilize the score function and accuracy function to give an approach to ranking the given alternatives and then selecting the most desirable one(s). In addition, we extend the above results to MAGDM with interval-valued intuitionistic fuzzy information.     

Multiple Attribute Group Decision-Making Methods with Completely Unknown Weights in Intuitionistic Fuzzy Setting and Interval-Valued Intuitionistic Fuzzy Setting

The existing multiple attribute group decision-making approaches based on intuitionistic fuzzy sets (IFSs) or interval-valued intuitionistic fuzzy sets (IVIFSs) are considered as the situation that the weights of experts are given beforehand and the attribute weights are known or unknown. To better describe the uncertain decision environment and solve the corresponding decision problem, multiple attribute group decision-making methods with completely unknown weights of both experts and attributes are proposed in intuitionistic fuzzy setting and interval-valued intuitionistic fuzzy setting. Entropy weight models can be used to determine the weights of both experts and attributes from intuitionistic fuzzy decision matrices or interval-valued intuitionistic fuzzy decision matrices, and then the evaluation formulas of weighted correlation coefficients between alternatives and the ideal alternative are introduced in intuitionistic fuzzy setting and interval-valued intuitionistic fuzzy setting. The alternatives can be ranked and the most desirable one(s) can be selected according to the values of the weighted correlation coefficients for IFSs or IVIFSs. Finally, two numerical examples demonstrate the effectiveness of the proposed methods: they are capable for handling the multiple attribute group decision-making problems with completely unknown weights.     

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.     

Continuous Ordered Weighted Distance Measure and Its Application to Multiple Attribute Group Decision Making

We develop a new distance measure called the continuous ordered weighted distance (COWD) measure by using the continuous ordered weighted averaging (COWA) operator in the interval distance. We study some properties and different families of the COWD measure. We further generalize the COWD measure. The prominent characteristics of the COWD measure are that it is not only a generalization of some widely used distance measures and the continuous generalized OWA operator, but also it can deal with interval deviations in aggregation on interval arguments by using a controlled parameter. The desirable characteristics make the COWD measure be suitable to wide range situations, such as decision making, engineering and economics. In the end, we develop the new approach to group decision making in investment selection.     

Interval-Valued Intuitionistic Fuzzy WASPAS Method: Application in Reservoir Flood Control Management Policy

Interval-valued intuitionistic fuzzy sets (IVIFSs) are very flexible tool to cope with the uncertainty arises in multi-criteria decision making (MCDM) problems. In recent times, MCDM problems with interval-valued intuitionistic fuzzy information have achieved more attention from researchers in different areas and consequently, several MCDM methods have been extended for IVIFSs. In this paper, a novel approach based on WASPAS method is developed under IVIFSs. The developed method is based on the operators of IVIFSs, some amendments in the classical WASPAS method and a new process for calculation of criteria and decision experts’ weights. In process for calculating weights, new procedures is propoesd to compute the decision experts’ weights and criteria weights based on interval-valued intuitionistic fuzzy information measures (entropy, divergence and similarity measures) to achieve more realistic weights. Innovative information measures are developed based on the exponential function for IVIFSs to determine the weights of the criteria and decision experts. Since the uncertainty is an unavoidable feature of MCDM problems, the developed method can be a constructive tool for decision-making in an uncertain environment. Further, an uncertain decision making problem of reservoir flood control management policy is implemented with interval-valued intuitionistic fuzzy information, which reveals the effectiveness and reliability of the proposed IVIF-WASPAS method. To validate the result, comparative analysis with existing methods and sensitivity analysis are presented under interval-valued intuitionistic fuzzy environment.     

Generalized Multiple Averaging Operators and their Applications to Group Decision Making

In this paper, we present a new operator called the generalized ordered weighted multiple averaging (GOWMA) operator based on a minimization problem with penalty function, which unifies the geometric mean and the generalized ordered weighted averaging operator with the generalized ordered weighted harmonic averaging operator in the same formulation. We study different properties and families of the GOWMA operator and develop a generalization of this operator called the generalized hybrid multiple averaging operator. To determine the GOWMA operator weights, we propose the generalized least squares method which does not follow a regular distribution. Finally, we give a numerical example of an investment selection to illustrate the application of the GOWMA operator to multiple attribute group decision making.     

Generalized Hybrid Aggregation Operators Based on the 2-Dimension Uncertain Linguistic Information for Multiple Attribute Group Decision Making

With respect to the multiple attribute decision making problems in which the attribute values take the form of the 2-dimension uncertain linguistic information, a new method based on the generalized hybrid operators is proposed. Firstly, the definition, properties, expectations and ranking method of 2-dimension uncertain linguistic information are introduced, and the operational laws the 2-dimension uncertain linguistic information are defined. Then some aggregation operators, including 2-dimension uncertain linguistic generalized weighted average operator, 2-dimension uncertain linguistic generalized ordered weighted average operator, and 2-dimension uncertain linguistic generalized hybrid weighted average operator, are developed, and some properties and special cases of them are also discussed. Finally, these operators are applied to multi-criteria decision making and an illustrative example is given to verify the developed approach and to demonstrate its effectiveness.     

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.     

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.     

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.     

A Practical Procedure for Group Decision Making under Incomplete Multiplicative Linguistic Preference Relations

The aim of this paper is to investigate a group decision making problem with incomplete multiplicative linguistic preference relations. We first define the concept of an incomplete multiplicative linguistic preference relation, and then develop a simple algorithm to extend each incomplete multiplicative linguistic preference relation to a complete multiplicative linguistic preference relation. Finally, we develop a practical procedure for group decision making under incomplete multiplicative linguistic preference relations, and give a numerical example to illustrate the developed procedure.     

A family of IOWA operators with reliability measurement under interval-valued group decision-making environment

Aggregation operators play an essential role in the aggregation of various individual input arguments in group decision-making (GDM). In this paper, we have proposed a family of IOWA operators with reliability measurement to aggregate uncertain decision information represented by interval numbers in GDM problems. In particular, we introduce the reliability-induced uncertain OWA (R-IUOWA) operator and the clusters’ reliability-induced uncertain OWA (CR-IUOWA) operator. This type of operators uses the reliability measurement representing the opinion consensus of individuals as the associated order-inducing variable and considers the decision-makers’ preference in the calculation of the position weights. Thus, the aggregation results have a higher consensus level. The R-IUOWA and CR-IUOWA operators have three primary properties such as commutativity, idempotency and boundness. The generalized formulas and some special cases of the two operators are outlined. Finally, the proposed operators are applied to a GDM problem regarding the selection of an investment company. The validity of the two operators is illustrated by comparing the aggregation results with that of other operators.

     

Developing Group Ordered Weighted Averaging Operator Weights for Group Decision Support

A few single decision-making methods under uncertainty (SDMUU) are available in the literature. The reason for such scarcity seems to be mainly due to too insufficient information to induce a reasonable result for effective decision support. Moreover their final outcomes on the same SDMUU problem may be different depending on which method is applied. A group decision-making method under uncertainty (GDMUU) extends a SDMUU in a sense that a group of individuals, each expressing different opinions, work together to solve a relevant problem. As in a SDMUU, we find that just a few methods are available to solve a GDMUU problem. In the paper, the ordered weighted averaging (OWA) method, originally devised for use with the SDMUU problem, is considered to deal with a GDMUU problem where individuals of a group express their degree of optimism in terms of attitudinal characters. We first find the extreme points corresponding to the attitudinal character and then solve a quadratic mathematical program which minimizes a squared distance from each extreme point identified. Thus the resulting OWA operator weights for the group are located at the center of the weights-space constructed by attitudinal characters. This idea is further extended to deal with uncertain attitudinal character expressed in the form of interval in situations where it is difficult to reliably obtain a precise attitudinal character due to time pressure and a limited domain knowledge and so on.     

Conflicting Bifuzzy Multi-attribute Group Decision Making Model with Application to Flood Control Project

We propose a group decision making model based on conflicting bifuzzy sets (CBFS) where evaluation are bi-valued in accordance to the subjective assessment obtained from the experts for the positive and negative views. This paper discusses the weighting methods for particular attribute and subattribute with emphasis given to the unification of subjective and objective weights. The integration of CBFS in the model is naturally done by extending the fuzzy evaluation in parallel with the intuitionistic fuzzy. We introduce a new technique to compute the similarity measure, being the degree of agreement between the experts. We end up the paper by demonstrating the applicability of the proposed model to the empirical case of flood control project, one of the project selection problems.