Fuzzy aggregation operators in decision making with Dempster-Shafer belief structure

We study the decision-making problem with Dempster-Shafer theory of evidence. We analyze how to deal with this model when the available information is uncertain and it can be represented with fuzzy numbers. We use different types of aggregation operators that aggregate fuzzy numbers such as the fuzzy weighted average (FWA), the fuzzy ordered weighted averaging (FOWA) operator and the fuzzy hybrid averaging (FHA) operator. As a result, we get the belief structure fuzzy weighted average (BS-FWA), the belief structure fuzzy ordered weighted averaging (BS-FOWA) operator and the belief structure fuzzy hybrid averaging (BS-FHA) operator. We further generalize this new approach by using generalized and quasi-arithmetic means. We also develop an illustrative example regarding the selection of investments where we can see the different results obtained by using different types of fuzzy aggregation operators.     

Evaluation method based on fuzzy relations between Dempster–Shafer belief structure

Many relations in the real world can be described by mathematical language. Fuzzy set theory can transform human language into mathematical language and use membership degree function to describe relations between events. Dempster–Shafer evidence theory provides basic probability assignment (BPA), which can describe the occurrence rate of attributes in basic events. Based on the known membership degree function and BPA distribution, a new evaluation method is proposed in this paper to analyze decision making. Given the relations among relevant events, which are expressed by BPA distribution and membership degree function, the relations among basic events and top event can be obtained. The Dempster's combination rule and pignistic probability transformation are used to transform BPA distribution into probability distribution. The belief measure is applied to deal with these fuzzy relations. Some numerical examples are given in this paper to illustrate the proposed evaluation methodology.     

A class of fuzzy measures generated from a Dempster–Shafer belief structure

Here the Dempster–Shafer belief structure is viewed as providing partial information about the underlying fuzzy measure associated with a uncertain variable. In this perspective there exists many possible fuzzy measures that can be associated with a Dempster–Shafer belief structure. Typically only two of these measures have been made explicit, those being the measure of belief and plausibility. Here we introduce a whole class of fuzzy measures that can be associated with a Dempster–Shafer belief structure. As an aid to choosing between these myriad of possibilities we discuss the entropy of a fuzzy measure. ©1999 John Wiley & Sons, Inc.     

An improved soft likelihood function for Dempster–Shafer belief structures

Information fusion is an important research direction. In the field of information fusion, there are many methods for evidence combination. Recently, Yager proposed a method of soft likelihood function to combine probabilistic evidence effectively. Considering that basic probability assignment (BPA) can deal with uncertainty information more effectively, in this paper, we extend Yager's soft likelihood function to combine BPA. First, according to the BPA evaluations of evidence sources, belief function and plausibility function on each alternative are calculated. Then, interval numbers are constructed by the obtained belief function and plausibility function to indicate the belief interval on each alternative. Next, the descending sorting of interval numbers is aggregated by the ordered weighted averaging operator. Finally, by sorting the result of the aggregation, the ordering of alternatives is obtained. A numerical example and an example of application in Iris data set classification illustrate the effectiveness of the improved method.     

TOPSIS with belief structure for group belief multiple criteria decision making

The technique for order performance by similarity to ideal solution(TOPSIS)is one of the major techniques in dealing with multiple criteria decision making(MCDM)problems, and the belief structure(BS)model has been used successfully for uncertain MCDM with incompleteness, impreciseness or ignorance. In this paper, the TOPSIS method with BS model is proposed to solve group belief MCDM problems. Firstly, the group belief MCDM problem is structured as a belief decision matrix in which the judgments of each decision maker are described as BS models, and then the evidential reasoning approach is used for aggregating the multiple decision makers' judgments. Subsequently, the positive and negative ideal belief solutions are defined with the principle of TOPSIS. To measure the separation from ideal solutions, the concept and algorithm of belief distance measure are defined, which can be used for comparing the difference between BS models. Finally, the relative closeness and ranking index are calculated for ranking the alternatives. A numerical example is given to illustrate the proposed method.     

TOPSIS with fuzzy belief structure for group belief multiple criteria decision making

TOPSIS is one of the major techniques in dealing with multiple criteria decision making (MCDM) problems, and Belief Structure (BS) model and Fuzzy BS model have been used successfully for uncertain MCDM with incompleteness, impreciseness or ignorance. In this paper, the TOPSIS method with Fuzzy BS model is proposed to solve Group Belief MCDM problems. Firstly, the Group Belief MCDM problem is structured as a fuzzy belief decision matrix in which the judgments of each decision maker are described as Fuzzy BS models, and then the Evidential Reasoning approach is used for aggregating the multiple decision makers’ judgments. Subsequently, the positive and negative ideal belief solutions are defined with the principle of TOPSIS. In order to measure the separation from the ideal belief solutions, the concept and algorithm of Belief Distance Measure are introduced to compare the difference between Fuzzy BS models. Using the Belief Distance Measure, the relative closeness and ranking index can be calculated for ranking the alternatives. A numerical example is finally given to illustrate the proposed method.     

Discriminative aggregation operators for multi criteria decision making

A general aggregation formalism for multi criteria decision making (MCDM) applications is presented. Using this formalism, we derive the existing aggregation operators, and also develop some new ones. The proposed general formalism is further extended to develop discriminative class of aggregation operators for aiding MCDM. The proposed discriminative aggregation operators are based on the consideration of the variability in the various evaluations of a criterion. Four families of discriminative aggregation operators are developed using the extended formalism. These operators and applied in a managerial real world case-study.     

Handling of Dependence in Dempster–Shafer Theory

Dempster's rule of combination can only be used when the bodies of evidence are assumed to be independent. However, such an assumption is often unrealistic. This paper proposes a systematic approach to handle dependence in evidence theory. It includes both the representation of dependence among information sources and the aggregation of the dependent evidence. For the representation of the dependence, the proposed methodology is able to capture both inner dependence (i.e., dependence among features of a system) and outer dependence (i.e., dependence among the evidence sources during the information propagating and evaluating process). We suggest dealing with the inner dependence by applying the analytic network process model, and modeling the outer dependence based on the intersection situations of the identified influencing factors. Then for the combination of dependent evidence, the strategy is to use discounting aggregation where the discounting coefficients are related to the degree of both outer and inner dependence. The discounting operator helps reduce the duplicate calculations in the fusion of dependent evidence and relax the assumption of independence when using Dempster's rule. A case study of transportation project evaluation is used to illustrate the proposed methodology.     

Hesitant fuzzy Hamacher aggregation operators for multicriteria decision making

As a fuzzy set extension, the hesitant set is effectively used to model situations where it is allowable to determine several possible membership degrees of an element to a set due to the ambiguity between different values. We first introduce some new operational rules of hesitant fuzzy sets based on the Hamacher t-norm and t-conorm, in which a family of hesitant fuzzy Hamacher operators is proposed for aggregating hesitant fuzzy information. Some basic properties of these proposed operators are given, and the relationships between them are shown in detail. We further discuss the interrelations between the proposed aggregation operators and the existing hesitant fuzzy aggregation operators. Applying the proposed hesitant fuzzy operators, we develop a new technique for hesitant fuzzy multicriteria decision making problems. Finally, the effectiveness of the proposed technique is illustrated by mean of a practical example.     

Linguistic group decision making with induced aggregation operators and probabilistic information

A new approach for linguistic group decision making by using probabilistic information and induced aggregation operators is presented. It is based on the induced linguistic probabilistic ordered weighted average (ILPOWA). It is an aggregation operator that uses probabilities and OWA operators in the same formulation considering the degree of importance that each concept has in the formulation. It uses complex attitudinal characters that can be assessed by using order inducing variables. Furthermore, it deals with an uncertain environment where the information cannot be studied in a numerical scale but it is possible to use linguistic variables. Several extensions to this approach are presented by using moving averages and Bonferroni means. The applicability of this approach is also studied with a focus on multi-criteria group decision making by using multi-person aggregation operators in order to deal with the opinion of several experts in the analysis. An illustrative example regarding importation strategies in the administration of a country is developed.     

Some new Pythagorean fuzzy Choquet–Frank aggregation operators for multi‐attribute decision making

Pythagorean fuzzy set (PFS) whose main feature is that the square sum of the membership degree and the non‐membership degree is equal to or less than one, is a powerful tool to express fuzziness and uncertainty. The aim of this paper is to investigate aggregation operators of Pythagorean fuzzy numbers (PFNs) based on Frank t‐conorm and t‐norm. We first extend the Frank t‐conorm and t‐norm to Pythagorean fuzzy environments and develop several new operational laws of PFNs, based on which we propose two new Pythagorean fuzzy aggregation operators, such as Pythagorean fuzzy Choquet–Frank averaging operator (PFCFA) and Pythagorean fuzzy Choquet–Frank geometric operator (PFCFG). Moreover, some desirable properties and special cases of the operators are also investigated and discussed. Then, a novel approach to multi‐attribute decision making (MADM) in Pythagorean fuzzy context is proposed based on these operators. Finally, a practical example is provided to illustrate the validity of the proposed method. The result shows effectiveness and flexible of the new method. A comparative analysis is also presented.     

Adaptive linguistic weighted aggregation operators for multi-criteria decision making

In this paper, we propose new aggregation operators for multi-criteria decision making under linguistic settings. The proposed operators are based on two sets of criteria weights. Besides the primary conventional criteria weights, we introduce a method to deduce secondary criteria weights from the criteria evaluations, which reflect the role of the different criteria in discriminating among the alternatives. The properties of the proposed operators are investigated. An approach for the application of the said operators in a group multi-criteria decision making problem is presented. Following the same, the proposed operators are applied in a case study on supplier selection. The empirical validation of the proposed operators is performed on a set of 12 real datasets.Note: All usages of he, him, his in the paper, also refer to she, and her.     

毕达哥拉斯模糊交叉影响集成算子及其决策应用

定义毕达哥拉斯模糊数的交叉影响加法、数乘、乘法及幂运算,提出毕达哥拉斯模糊交叉影响加权平均算子(PFIWA)、毕达哥拉斯模糊交叉影响有序加权平均算子(PFIOWA)、毕达哥拉斯模糊交叉影响加权几何算子(PFIWG)及毕达哥拉斯模糊交叉影响有序加权几何算子(PFIOWG),推导出它们的数学表达式,并研究其性质.提出基于毕达哥拉斯模糊交叉影响集成算子的决策方法,并通过决策实例验证所提出方法的稳定性和有效性.     

Pythagorean fuzzy power aggregation operators in multiple attribute decision making

In this paper, we utilize power aggregation operators to develop some Pythagorean fuzzy power aggregation operators: Pythagorean fuzzy power average operator, Pythagorean fuzzy power geometric operator, Pythagorean fuzzy power weighted average operator, Pythagorean fuzzy power weighted geometric operator, Pythagorean fuzzy power ordered weighted average operator, Pythagorean fuzzy power ordered weighted geometric operator, Pythagorean fuzzy power hybrid average operator, and Pythagorean fuzzy power hybrid geometric operator. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the Pythagorean fuzzy multiple attribute decision‐making problems. Finally, a practical example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

广义毕达哥拉斯模糊集成算子及其决策应用

研究毕达哥拉斯模糊决策环境下的集成算子及其决策应用.给出拟加权几何集成算子和拟有序加权几何算子的概念, 并分析 它们的性质.将有序加权平均算子、有序加权几何算子、拟有序加权平均算子和拟有序加权几何算子推广到毕达哥拉斯 模糊决策环境, 定义毕达哥拉斯模糊有序加权平均算子、广义毕达哥拉斯模糊有序加权平均算子、毕达哥拉斯模糊有序加权几何算子、广义毕达哥拉斯模糊有序加权几何 算子、拟毕达哥拉斯模糊有序加权平均算子和拟毕达哥拉斯模糊有序加权几何算子.提出基于广义毕达哥拉斯模糊集成算子的决策方法, 并通过实例验证其可行性.     

Decision Making with Ordinal Payoffs Under Dempster–Shafer Type Uncertainty

Our focus is on decision making in uncertain environments. We first introduce the Dempster–Shafer framework to model the uncertainty associated with possible outcomes. We then describe an approach for decision making when our uncertainty is captured using the Dempster–Shafer model and where the payoffs are numeric values. An important part of this approach is the role of the decision attitude as well as the aggregation of the possible payoffs. We then look at the situation where the payoffs, rather than being numbers, are values drawn from an ordinal scale. This requires us to provide appropriate operations for combining payoffs drawn from an ordinal scale.     

Scaled prioritized aggregation operators and their applications to decision making

As an extension of the prioritized aggregation operators by Yager (Int J Approx Reason 48:263–274, 2008), this paper uses the priority labels to express the prioritized relationship between criteria and presents some scaled prioritized aggregation operators, including the scaled prioritized score operator and the scaled prioritized averaging operator. Moreover, we consider the priority under uncertain environment and develop the uncertain prioritized aggregation operators, including the uncertain prioritized scoring operator and the uncertain prioritized averaging operator. We investigate the properties of these operators and build the models to derive the weights by maximizing square deviations from a possible range to distinguish the candidate alternatives mostly. Furthermore, approaches to multi-attribute decision making based on the proposed operators are given, which have benefits over the TOPSIS method (Behzadian, Expert Syst Appl 39:13051–13069, 2012) and the methods based on the OWA operator (Zhou and Chen, Fuzzy Sets Syst 168:18–34, 2011) when prioritized relationship between criteria is considered. Finally, examples are illustrated to show the feasibility and validity of the new approaches to the application of decision making.     

A spectrum of compromise aggregation operators for multi-attribute decision making

In many decision making problems, a number of independent attributes or criteria are often used to individually rate an alternative from an agent's local perspective and then these individual ratings are combined to produce an overall assessment. Now, in cases where these individual ratings are not in complete agreement, the overall rating should be somewhere in between the extremes that have been suggested. However, there are many possibilities for the aggregated value. Given this, this paper systematically explores the space of possible compromise operators for such multi-attribute decision making problems. Specifically, we axiomatically identify the complete spectrum of such operators in terms of the properties they should satisfy, and show the main ones that are widely used—namely averaging operators, uninorms and nullnorms—represent only three of the nine types we identify. For each type, we then go onto analyse their properties and discuss how specific instances can actually be developed. Finally, to illustrate the richness of our framework, we show how a wide range of operators are needed to model the various attitudes that a user may have for aggregation in a given scenario (bidding in multi-attribute auctions).     

Generalized power aggregation operators and their applications in group decision making

In this paper, we investigate a generalized power average (GPA) operator and its weighted form, which are on the basis of the power average (PA) operator and the generalized mean, and develop a generalized power ordered weighted average (GPOWA) operator based on the power ordered weighted average (POWA) operator. Then, we extend these operators to uncertain environments and present an uncertain generalized power average (UGPA) operator and its weighted form, and an uncertain generalized power ordered weighted average (UGPOWA) operator to aggregate the input arguments taking the form of interval of numerical values. We also extend the GPA operator and the GPOWA operator to intuitionistic fuzzy environment, and obtain the generalized intuitionistic fuzzy power averaging (GIFPA) operator and the generalized intuitionistic fuzzy power ordered weighted averaging (GIFPOWA) operator. Moreover, some properties of these operators are studied. We also present new approaches on the basis of the proposed operators in an example of strategic decision making.     

Multicriteria decision making with 2‐dimension linguistic aggregation techniques

In this paper, we represent evaluation information by 2‐dimension linguistic labels so as to avoid biased results and achieve high accuracy in multicriteria decision making. We analyze the relationship between a 2‐dimension linguistic label and a common linguistic label, and then quantify a certain 2‐dimension linguistic label by using a generalized triangular fuzzy number (TFN). On the basis of the mapping function from 2‐dimension linguistic labels to the corresponding generalized TFNs and its inverse function, we develop a 2‐dimension linguistic weighted averaging (2DLWA) operator and a 2‐dimension linguistic ordered weighted averaging (2DLOWA) operator. Furthermore, we verify the feasibility of the 2DLWA and 2DLOWA operators by discussing their properties and investigating their applications to produce reliable decision results in multicriteria decision making under linguistic evaluations. Finally, an example of selecting the outstanding postgraduate dissertation(s) is used to illustrate the practicability and validity of these two 2‐dimension linguistic aggregation techniques. © 2012 Wiley Periodicals, Inc.