Group decision making based on intuitionistic multiplicative aggregation operators

Preference relations are the most common techniques to express decision maker’s preference information over alternatives or criteria. To consistent with the law of diminishing marginal utility, we use the asymmetrical scale instead of the symmetrical one to express the information in intuitionistic fuzzy preference relations, and introduce a new kind of preference relation called the intuitionistic multiplicative preference relation, which contains two parts of information describing the intensity degrees that an alternative is or not priority to another. Some basic operations are introduced, based on which, an aggregation principle is proposed to aggregate the intuitionistic multiplicative preference information, the desirable properties and special cases are further discussed. Choquet Integral and power average are also applied to the aggregation principle to produce the aggregation operators to reflect the correlations of the intuitionistic multiplicative preference information. Finally, a method is given to deal with the group decision making based on intuitionistic multiplicative preference relations.     

Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations

The aim of this paper is to present a logarithmic least squares method (LLSM) to priority for group decision making with incomplete fuzzy preference relations. We give a reasonable definition of multiplicative consistent for incomplete fuzzy preference relation. We develop the acceptable fuzzy consistency ratio (FCR for short), which is simple and similar to Saaty’s consistency ratio CR for multiplicative fuzzy preference relations. We also extend the LLSM method to the case of individual preference relation with complete information. Finally, some examples are illustrated to show that our method is simple, efficient, and can be performed on computer easily.     

A chi-square method for obtaining a priority vector from multiplicative and fuzzy preference relations

Decision makers (DMs)’ preferences on decision alternatives are often characterized by multiplicative or fuzzy preference relations. This paper proposes a chi-square method (CSM) for obtaining a priority vector from multiplicative and fuzzy preference relations. The proposed CSM can be used to obtain a priority vector from either a multiplicative preference relation (i.e. a pairwise comparison matrix) or a fuzzy preference relation or a group of multiplicative preference relations or a group of fuzzy preference relations or their mixtures. Theorems and algorithm about the CSM are developed. Three numerical examples are examined to illustrate the applications of the CSM and its advantages.     

A group decision making approach based on aggregating interval data into interval-valued intuitionistic fuzzy information

Group decision making is one of the most important problems in decision making sciences. The aim of this article is to aggregate the interval data into the interval-valued intuitionistic fuzzy information for multiple attribute group decision making. In this model, the decision information is provided by decision maker, which is characterized by interval data. Based on the idea of mean and variance in statistics, we first define the concepts of satisfactory and dissatisfactory intervals of attribute vector against each alternative. Using these concepts, we develop an approach to aggregate the attribute vector into interval-valued intuitionistic fuzzy number under group decision making environment. A practical example is provided to illustrate the proposed method. To show the validity of the reported method, comparisons with other methods are also made.     

A least deviation method for priority derivation in group decision making with incomplete reciprocal preference relations

In this paper, based on the transfer relationship between reciprocal preference relation and multiplicative preference relation, we proposed a least deviation method (LDM) to obtain a priority vector for group decision making (GDM) problems where decision-makers' (DMs') assessments on alternatives are furnished as incomplete reciprocal preference relations with missing values. Relevant theorems are investigated and a convergent iterative algorithm about LDM is developed. Using three numerical examples, the LDM is compared with the other prioritization methods based on two performance evaluation criteria: maximum deviation and maximum absolute deviation. Statistical comparative study, complexity of computation of different algorithms, and comparative analyses are provided to show its advantages over existing approaches.     

A goal programming approach to group decision making based on multiplicative preference relations and fuzzy preference relations

This paper analyses the mechanisms through which binding finance constraints can induce debt-constrained firms to improve technical efficiency to guarantee positive profits. This hypothesis is tested on a sample of firms belonging to the Italian manufacturing. Technical efficiency scores are computed by estimating parametric production frontiers using the one stage approach as in Battese and Coelli [Battese, G., Coelli, T., 1995. A model for technical efficiency effects in a stochastic frontier production function for panel data. Empirical Economics 20, 325–332]. The results support the hypothesis that a restriction in the availability of financial resources can affect positively efficiency.     

A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets

Soft set theory, originally proposed by Molodtsov, has become an effective mathematical tool to deal with uncertainty. A type-2 fuzzy set, which is characterized by a fuzzy membership function, can provide us with more degrees of freedom to represent the uncertainty and the vagueness of the real world. Interval type-2 fuzzy sets are the most widely used type-2 fuzzy sets. In this paper, we first introduce the concept of trapezoidal interval type-2 fuzzy numbers and present some arithmetic operations between them. As a special case of interval type-2 fuzzy sets, trapezoidal interval type-2 fuzzy numbers can express linguistic assessments by transforming them into numerical variables objectively. Then, by combining trapezoidal interval type-2 fuzzy sets with soft sets, we propose the notion of trapezoidal interval type-2 fuzzy soft sets. Furthermore, some operations on trapezoidal interval type-2 fuzzy soft sets are defined and their properties are investigated. Finally, by using trapezoidal interval type-2 fuzzy soft sets, we propose a novel approach to multi attribute group decision making under interval type-2 fuzzy environment. A numerical example is given to illustrate the feasibility and effectiveness of the proposed method.     

A new method for ranking fuzzy numbers and its application to group decision making

In this paper, a new method for comparing fuzzy numbers based on a fuzzy probabilistic preference relation is introduced. The ranking order of fuzzy numbers with the weighted confidence level is derived from the pairwise comparison matrix based on 0.5-transitivity of the fuzzy probabilistic preference relation. The main difference between the proposed method and existing ones is that the comparison result between two fuzzy numbers is expressed as a fuzzy set instead of a crisp one. As such, the ranking order of n fuzzy numbers provides more information on the uncertainty level of the comparison. Illustrated by comparative examples, the proposed method overcomes certain unreasonable (due to the violation of the inequality properties) and indiscriminative problems exhibited by some existing methods. More importantly, the proposed method is able to provide decision makers with the probability of making errors when a crisp ranking order is obtained. The proposed method is also able to provide a probability-based explanation for conflicts among the comparison results provided by some existing methods using a proper ranking order, which ensures that ties of alternatives can be broken.     

A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making

In decision making problems, there may be the cases where the decision makers express their judgements by using preference relations with incomplete information. Then one of the key issues is how to estimate the missing preference values. In this paper, we introduce an incomplete interval multiplicative preference relation and give the definitions of consistent and acceptable incomplete ones, respectively. Based on the consistency property of interval multiplicative preference relations, a goal programming model is proposed to complement the acceptable incomplete one. A new algorithm of obtaining the priority vector from incomplete interval multiplicative preference relations is given. The goal programming model is further applied to group decision-making (GDM) where the experts evaluate their preferences as acceptable incomplete interval multiplicative preference relations. An interval weighted geometric averaging (IWGA) operator is proposed to aggregate individual preference relations into a social one. Furthermore, the social interval multiplicative preference relation owns acceptable consistency when every individual one is acceptably consistent. Two numerical examples are carried out to show the efficiency of the proposed goal programming model and the algorithms.     

Approach to group decision making based on determining the weights of experts by using projection method

The aim of this paper is to present a new approach for determining weights of experts in the group decision making problems. Group decision making has become a very active research field over the last decade. Especially, the investigation to determine weights of experts for group decision making has attracted great interests from researchers recently and some approaches have been developed. In this paper, the weights of experts are determined in the group decision environment via projection method. First of all, the average decision of all individual decisions is defined as the ideal decision. After that, the weight of expert is determined by the projection of individual decision on the ideal decision. By using the weights of experts, all individual decisions are aggregate into a collective decision. Then an ideal solution of alternatives of the collective decision, expressed by a vector, is determined. Further, the preference order of alternatives are ranked in accordance with the projections of alternatives on the ideal solution. Comparisons with an extended TOPSIS method are also made. Finally, an example is provided to illustrate the developed approach.     

Aggregation of fuzzy preference relations to multicriteria decision making

Weighted aggregation of fuzzy preference relations on the set of alternatives by several criteria in decision-making problems is considered. Pairwise comparisons with respect to importance of the criteria are given in fuzzy preference relation as well. The aggregation procedure uses the composition between each two relations of the alternatives. The membership function of the newly constructed fuzzy preference relation includes t-norms and t-conorms to take into account the relation between the criteria importance. Properties of the composition and new relation, giving a possibility to make a consistent choice or to rank the alternatives, are proved. An illustrative numerical study and comparative examples are presented.     

Fuzzy preference orderings in group decision making

In this paper, some use of fuzzy preference orderings in group decision making is discussed. First, fuzzy preference orderings are defined as fuzzy binary relations satisfying reciprocity and max-min transitivity. Then, particularly in the case where individual preferences are represented by utility functions (utility values), group fuzzy preference orderings of which fuzziness is caused by differences or diversity of individual opinions are defined. Those orderings might be useful for proceeding the group decision making process smoothly, in the same manner as the extended contributive rule method.     

Some models for deriving the priority weights from interval fuzzy preference relations

Interval fuzzy preference relation is a useful tool to express decision maker’s uncertain preference information. How to derive the priority weights from an interval fuzzy preference relation is an interesting and important issue in decision making with interval fuzzy preference relation(s). In this paper, some new concepts such as additive consistent interval fuzzy preference relation, multiplicative consistent interval fuzzy preference relation, etc., are defined. Some simple and practical linear programming models for deriving the priority weights from various interval fuzzy preference relations are established, and two numerical examples are provided to illustrate the developed models.     

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.     

Analytic hierarchy process-hesitant group decision making

In this paper, we consider that the judgments provided by the decision makers (DMs) cannot be aggregated and revised, then define them as hesitant judgments to describe the hesitancy experienced by the DMs in decision making. If there exist hesitant judgments in analytic hierarchy process-group decision making (AHP-GDM), then we call it AHP-hesitant group decision making (AHP-HGDM) as an extension of AHP-GDM. Based on hesitant multiplicative preference relations (HMPRs) to collect the hesitant judgments, we develop a hesitant multiplicative programming method (HMPM) as a new prioritization method to derive ratio-scale priorities from HMPRs. The HMPM is discussed in detail with examples to show its advantages and characteristics. The practicality and effectiveness of our methods are illustrated by an example of the water conservancy in China.     

Intuitionistic and interval-valued intutionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group

Szmidt and Kacprzyk (Lecture Notes in Artificial Intelligence 3070:388–393, 2004a) introduced a similarity measure, which takes into account not only a pure distance between intuitionistic fuzzy sets but also examines if the compared values are more similar or more dissimilar to each other. By analyzing this similarity measure, we find it somewhat inconvenient in some cases, and thus we develop a new similarity measure between intuitionistic fuzzy sets. Then we apply the developed similarity measure for consensus analysis in group decision making based on intuitionistic fuzzy preference relations, and finally further extend it to the interval-valued intuitionistic fuzzy set theory.     

The extended COWG operators and their application to multiple attributive group decision making problems with interval numbers

The aim of this paper is to develop two extended continuous ordered weighted geometric (COWG) operators, such as the weighted geometric averaging COWG (WG-COWG) and ordered weighted geometric averaging COWG (OWG-COWG) operators. We study some desirable properties of the WG-COWG and OWG-COWG operators, and present their application to multiple attributive group decision making (MAGDM) problems with interval numbers. Finally, an illustrative numerical example is used to verify the developed approaches.     

Aggregating crisp values into intuitionistic fuzzy number for group decision making

This paper presents a multiple attribute group decision making model based on aggregating crisp values into intuitionistic fuzzy numbers. First, each alternative is evaluated with respect to their attributes, whose values are provided by decision maker as crisp numbers. Second, to make a reasonable normalization of attribute values in the group decision making environment, a maximum grade and a minimum grade are added to the attribute values. These normalized attribute values are then aggregated (per attribute) into an induced intuitionistic fuzzy number. Each alternative is then evaluated according to the induced intuitionistic fuzzy number. To show the major technical advances in this paper, comparisons with other methods are also made. Finally, an experimental analysis for supplier selection is given to illustrate the reasonableness and efficiency of the introduced method.     

Approaches to collective decision making with fuzzy preference relations

Recent experimental studies show that the predictive accuracy of many of the solution concepts derived from the collective decision making theory leaves much to be desired. In a previous paper the author attempted to explain some of the inaccuracies in terms of the fuzzy indifference regions of the individuals participating in the voting game. This paper gives straightforward generalizations of the solutions concepts in terms of the fuzzy social or individual preference relations. It turns out that some of these new solution concepts cotain their nonfuzzy counterparts as subsets. Others, in turn, are subsets of their nonfuzzy counterparts. We also discuss a method of aggregating individual nonfuzzy preferences so as to get a fuzzy social preference relation and, furthermore, a nonfuzzy social choice set.     

Developing a straightforward approach for group decision making based on determining weights of decision makers

Group decision making is an active area of research within multiple attribute decision making. This paper assumes that all the decision makers (DMs) are not equally qualified to contribute equitably to the decision process. The aim of this paper is to develop an approach to determine weights of DMs, in which the decision information on alternatives with respect to attributes, provided by each DM, is represented in the form of interval data. We define the average of all individual decisions as the positive ideal decision (PID), and the maximum separation from PID as the negative ideal decision, which are characterized by a matrix, respectively. The weight of each DM is determined according to the Euclidean distances between the individual decision and ideal decisions. By using the obtained weights of DMs, all individual decisions are aggregated into a collective decision. Then the alternatives is ranked based on the collective decision. Meanwhile, this paper also gives a humanized decision method by using an optimistic coefficient, which is used in adjusting the relative importance between profit and risk. Finally, we give an example to illustrate the developed approach.