In this paper, a methodbased on Choquetintegral is proposed to solve the intervalintuitionisticfuzzy multiple attributegroupdecisionmaking problems. Firstly, some concepts about intervalintuitionisticfuzzy measure are deﬁned, through the strict mathematical reasoning to prove the measure we proposed satisfying the axiomatic system of fuzzy measure. Then, on the basis of fuzzy measure and game theory, we propose two models to determine fuzzy measure based on intervalintuitionisticfuzzy entropy and weight information matrix. By calculating the Shapely value to determine expert weights, we establish a linear programming model based on relative entropy to determine the fuzzy measure of attribute weights to reﬂect the interactive characteristics among the criteria, using the Choquetintegral to aggregate the decision-making information. Finally, we give the process of decisionmaking in details.
In many decision areas, such as multi-period investment and personnel dynamic examination, the decision information is usually collected at different periods. Thus, it is necessary to develop some dynamic decisionmaking models to deal with these multi-period and multi-attributedecisionmaking prob- lems (also known as Dynamic Multi-Attribute, Decision Mak- ing (DMADM) problems [ 12 ]). Recently, research on DMADM problems has received some attention [ 12–15 ]. Xu [ 12 ] devel- oped a multi-period and multi-attributedecisionmaking model based on a simple additive weighting method. Lin et al. [ 13 ] pro- posed a dynamic multi-attributedecisionmaking model, where the attribute values are firstly aggregated into an overall evalu- ation value at each period, then all evaluation values are aggre- gated into an overall score of all alternatives. Xu and Yager [ 14 ] investigated a dynamic multi-attributedecisionmaking prob- lem where the decision information takes the form of the inter- val uncertain information. Wei [ 15 ] developed two aggregation operators to solve a dynamic multi-attributedecisionmaking problem where the decision information also takes the form of the interval uncertain information. All existing research focuses on DMADM problems where the decision information takes the form of a real number or interval uncertain information. Nev- ertheless, in many practical cases, the available decision infor- mation is usually difficult to judge precisely; instead, they can be easily characterized by some fuzzy linguistic terms, such as ‘‘good’’, ‘‘poor’’ and so on. In addition, triangular intuitionis- tic fuzzy numbers in the IntuitionisticFuzzy sets (IFs) can not only deal with vagueness information, but also express more
Supplier selection can be considered as a complicated multi criteria decision-making problem. In this paper the problem of supplier selection is studied in the presence of conflicting evaluations and insufficient information about the criteria and different attitudes of decision makers towards the risk. Most of fuzzy approaches used in multi- criteria groupdecisionmaking (MCGDM) are non-intuitionistic, which significantly restricts their application areas. Because of considering belongingness and non- belongingness of the issue in a same time, intuitionisticfuzzy sets can better encounter with a real supplier selection problem. Also to deal with different attitudes of decision makers toward the risk, the proposed approach in this paper employs a new decision function to participate this factor in decision process. In order to integrate fuzzy information, interval-valued intuitionisticfuzzy ordered weighted aggregation (IIFOWA) is applied to aggregate the obtained preferences. The influence of unfair arguments in final results can be reduced by assigning low weights to the “optimistic” or “pessimistic” discretions. Ranking process is based on the two indices, weighted score function and weighted accuracy function. To demonstrate the efficiency of the proposed approach, it is implemented to supplier selection in a project-based company.
Guiwu Wei and Gang Lan proposed a modified grey relational analysis (GRA) method and used the traditional GRA method for calculating steps for solving interval-valued intuitionisticfuzzy multiple attributedecision-making problems with known weight information. The degree of grey relation between every alternative and positive ideal solution and negative ideal solution are calculated. Then, according to the concept of the GRA, a relative relational degree is defined to determine the ranking order of all alternatives by calculating the degree of grey relation to both the positive-ideal solution (PIS) and negative-ideal solution (NIS) simultaneously. Wei Yang and Yongfeng Pang  also worked on GRA method with IVIFS to solve MCDM problems.
__________________________________________________________________________________________ Abstract: Out of several higher order fuzzy sets , the concept of an intuitionisticfuzzy set (IFS)  introduced by Atanassov has been found to be highly useful to deal with vagueness and imprecision. IFS theory has been extensively applied to areas like Artificial Intelligence, networking, Soft decisionmaking, Programming logic, operational research etc. One the promising role of IFS has been emerged in Decisionmaking problems specially groupdecisionmaking and multi-attributedecisionmaking. In some real-life situations, decision makers may not be able to accurately express their view for the problem as they may not possess a precise or sufficient level of knowledge of the problem or the decision makers are unable to discriminate explicitly the degree to which one alternative are better than others in such cases, the decision maker may provide their preferences for alternatives to a certain degree, but it is possible that they are not so sure about it . Thus, it is very suitable to express the decision maker preference values with the use of fuzzy/intuitionisticfuzzy values rather than exact numerical values or linguistic variables . To satisfy the need of decisionmaking problem with imprecision and uncertainty many researchers have been concentrated on IFS theory. In this paper we reviewed the development of different approaches for solving decisionmaking problem using IFS theory and a brief introduction on the role of Interval Valued IF sets (IVIFS)  in multiattribute decisionmaking.
With the development of economy and society, the competition between enter- prises is no longer the unilateral competition between price and quality, but the competition between supply chains. The supplier is at the source of the supply chain and plays a very core role in the whole supply chain. Choosing the right supplier is a good foundation for enterprise development. The evaluation and selection of suppliers is not the individual behavior of purchasers, but actually a complex multi-attributegroupdecision-making problem. Generally, there are not only quantitative indicators in the index system, such as product price and market share, but also qualitative indicators, such as product research and de- velopment capacity, after-sales maintenance level, etc., and the relationship be- How to cite this paper: Song, D.Y. and
In Section 2, some basic concepts are briey re- viewed, including Interval-Valued IntuitionisticFuzzy Sets (IVIFSs), fuzzy measures, the Choquetintegral, two Choquetintegral operators, and the generalized Shapley function. Meanwhile, it analyzes the lim- itations of the previous operations on IVIFSs. In Section 3, the IG-SIVIFCS operator is dened, and several important cases are investigated. Furthermore, some desirable properties are studied. In Section 4, several distance measure-based models for determining the optimal fuzzy measure on the attribute set are established, and an approach to interval-valued intu- itionistic fuzzymulti-attributedecision-making with incomplete weighting information is developed that considers the interactions. In Section 5, an illustrative example is provided to show the concrete application
Multi-criteria decision-making (MCDM) methods are very useful in the real-world decision-making problems. We are usually confronted with the decision-making process in an uncertain environment, and the fuzzy set theory is an effi cient tool to handle this uncertainty. Interval type-2 fuzzy sets are one of the extensions of the fuzzy sets which are very fl exible to model an uncertain environment. This study is related to MCDM problems within the context of interval type-2 fuzzy sets (IT2FSs). The evaluation based on distance from average solution (EDAS) method is a new and effi cient MCDM method, and assessment of alternatives in this method is based on the distance of them from average solution with respect to all criteria. In the EDAS method, each alternative has positive and negative distances which are used to determine the appraisal score of it. In this research, we present an extended EDAS method, which is named EDAS-IT2FSs, for dealing with multi- criteria groupdecision-making problems with interval type-2 fuzzy sets. Basic concepts of interval type-2 fuzzy sets and the arithmetic operations of trapezoidal IT2FSs are used to develop the extended EDAS method. A numerical example of multi-criteria subcontractor evaluation problem is used to illustrate the process of using the extended EDAS method. The example involves eight subcontractors that need to be evaluated with respect to seven criteria. A comparison and a sensitivity analysis based on different sets of criteria weights are also performed to show the validity of the proposed method. The results of these analyses show the effi ciency and stability of the extended EDAS method.
some decision problems. Then some aggregation operators based on these were proposed by Xu [5-6] and some methods for MADM with IFS were proposed in [7-8]. Furthermore, Atanassov and Gargov [4,9] extended the membership function and non-membership function to interval numbers and proposed interval-value intuitionisticfuzzy set (IVIFS). But IFS and IVIFS can only deal with incomplete information, but not uncertain and inconsistent information.
However, apart from that Technique for Or- der Preference with respect to the Similarity to the Ideal Solution (TOPSIS), developed by Hwang and Yoon , there is a well-known Multi-Criteria Decision-Making (MCDM) method. The aim of this method is to choose the best alternative whose distance from its positive ideal solution is the shortest. After their existence, numerous attempts are made by the researchers to apply the TOPSIS method under the fuzzy and IFS environment. For instance, Szmidt and Kacprzyk  dened the concept of distance measure between the IFSs. Hung and Yang  presented the similarity measures between the two dierent IFSs based on Hausdor distance. Boran et al.  applied the TOPSIS method to solve the problem of human resource personnel selection. Dugenci  presented a distance measure for IVIF set and their application to MCDM with incomplete weight information. Garg  presented a generalized improved score function for IVIFSs and their TOPSIS-basedmethod for solving the DM problems. Mohammadi et al.  presented a gray relational analysis and TOPSIS approach to solving the DM problems. Garg et al.  presented a generalized entropy measure of order and degree under the IFS environment and applied it to solve the DM problems. Biswas and Kumar  presented an integrated TOPSIS approach for solving the DM prob- lems with IVIFS environment. Vommi  presented a TOPSIS method using statistical distances to solve DM problems. Singh and Garg  developed the distance measures between the type-2 IFS. Li  presented a nonlinear programming methodology-based TOPSIS method for solving Multi-AttributeDecisionMaking (MADM) problems under IVIFS environment. Garg and Arora  extended the Li  approach to the interval-valued intuitionisticfuzzy soft set environ- ment. Lu and Ye  developed logarithm similarity measures to solve the problems under interval-valued fuzzy set environment. Garg and Kumar  presented new similarity measures for IFSs based on the connec- tion number of the set pair analysis. Askarifar et al.  presented an approach to studying the framework of Iran's seashores using TOPSIS and best-worst MCDM methods. In [33,34], the authors developed a group DM method under IVIF environment by integrating extended TOPSIS and linear programming methods. Kumar and Garg [35,36] presented the TOPSIS ap- proach for solving DM problems by using connection number of the set pair analysis theory.
In this paper, we explored some generalized Einstein aggregation operators based on IVIFNs and applied them to the multi-attributegroupdecisionmaking problems where attribute values are the IVIFNs. Firstly, Interval-Valued IntuitionisticFuzzy General- ized Einstein Weighted Averaging (IVIFGEWA) oper- ator, Interval-Valued IntuitionisticFuzzy Generalized Einstein Ordered Weighted Averaging (IVIFGEOWA) operator, and Interval-Valued IntuitionisticFuzzy Gen- eralized Einstein Hybrid Weighted Averaging (IV- IFGEHWA) operator were proposed. Some of their general properties such as idempotency, commutativ- ity, monotonicity, and boundedness, were studied, and some special cases of them were analyzed. Further- more, a method to multi-criteria groupdecisionmakingbased on these operators was developed, and the operational processes were illustrated in detail. Finally, an illustrative example was given to show the decision steps of the proposed method and to demonstrate their eectiveness. In further research, it is necessary and signicant to give the applications of these operators to the other domains such as pattern recognition, fuzzy cluster analysis, uncertain programming, etc.
Compared with the Euclidean distance or the Hamming distance, projection measurement is a com- prehensive consideration. It has been widely used in many decisionmaking problems. For example, using projection measure instead of distance measure, Yue and Jia  introduced a GDM model with hybrid intuitionisticfuzzy information. Xu and Hu  established two projection models for GDM problems. Wei  proposed an MADM methodbased on the projection technique, in which the attribute values were characterized by IFNs. Zheng et al.  developed an application of improved grey projection method. Yue  proposed a GDM methodbased on the projection measurement. Xu and Liu  described a GDM approach based on a projection method in uncertain fuzzy environment. Xu and Da  modeled an uncertain MADM method. Yue  suggested a GDM methodbased on a projection method. Yue and Jia  proposed a direct projection-based GDM methodology with crisp values and interval data.
Abstract. In today’s competitive global markets, selection of a potential supplier plays an important role to cut production costs as well as material costs of the company. This leads to successful survival and sustainability in a competitive marketplace. Therefore, evaluation and selection of an appropriate supplier has become an important part of supply chain management. The nature of the supplier selection process is a complex multi-attributegroupdecisionmaking (MAGDM) problem which deals with both quantitative and qualitative factors may be conflicting in nature as well as contain incomplete and uncertain information. In order to solve such a kind of MAGDM problems, the development of an effective supplier selection model is evidently desirable. In this paper, an application of the VIKOR method combined with fuzzy logic has been used to solve supplier selection problems with confliting and non-commensurable (different units) criteria, assuming that compromising is acceptable for conflict resolution. The decision maker wants a solution, which must be closest to the ideal, and the alternatives are evaluated according to all established criteria. Linguistic values are used to assess the ratings and weights for the conflicting factors. These linguistic ratings can be expressed in triangular fuzzy numbers. Then, a hierarchy MAGDM model based on fuzzy sets theory and the VIKOR method has been proposed to deal with the supplier selection problems in the supply chain system. A case study has been illustrated as an application of the proposed model.
Multiple attributedecisionmaking (MADM) is an important part of modern decision science. It has been extensively applied to various areas, such as society, economics, management, military and engineering technology. For example, the investment decision- making, project evaluation, the economic evaluation, the personnel evaluation etc. Since the object things are fuzzy, uncertainty and human thinking is ambiguous, the majority of the multi-attributedecision-making is uncertain and ambiguous, which is called the fuzzy multiple attributedecision-making (FMADM). Since Bellmanhe and Zadeh  initially proposed the basic model of fuzzydecisionmakingbased on the theory of fuzzy mathematics, FMADM has been receiving more and more attentions. Many achievements have been made on FMADM problems [2-5,7-21].
Abstract. The trustworthy service selection is a typical Multi-AttributeGroupDecision- Making (MAGDM) problem. The aim of this paper is to develop a novel method for MAGDM with Triangular IntuitionisticFuzzy Numbers (TIFNs) and apply it to the trustworthy service selection problem. Firstly, we dene the mean-index, variance-index, and standard deviation of TIFN. Moreover, a new distance measure of TIFNs is proposed, and the corresponding proofs are given. Based on these concepts of mean-index and standard deviation, a ranking method for TIFNs is developed considering the risk preference of Decision Maker (DM). Further, according to the crisp relative closeness coecient matrix with respect to the normalized TIFNs decision matrix, we use entropy measure to obtain attribute weights. The DMs' weights are calculated by the similarity between the individual and the average decisions. Then, a decision procedure is described to solve the MAGDM under triangular intuitionisticfuzzy environment. Finally, a real trustworthy service selection example is analyzed to verify the practicality and eectiveness of the developed method.
In the fuzzy set theory  there were no scopes to think about the hesitation in the membership degree, which arise in various real life situations. To overcome these situations Atanassov  introduced theory of intuitionisticfuzzy set in 1986 as a generalization of fuzzy set.Most of the problems in engineering, medical science, economics, environments etc have various uncertainties. Molodtsov initiated the concept of soft set theory as a mathematical tool for dealing with uncertainties. Research works on soft set theory are progressing rapidly. Maji et al. defined several operations on soft set theory. Combining soft sets with fuzzy sets and intuitionisticfuzzy sets, Feng et al. and Maji et al.[9,10] defined fuzzy soft sets and intuitionisticfuzzy soft sets which are rich potentials for solving decisionmaking problems.Matrices play an important role in the broad area of science and engineering. The classical matrix theory cannot solve the problems involving various types of uncertainties. In  Yang et al, initiated a matrix representation of a fuzzy soft set and applied it in certain decisionmaking problems. The concept of fuzzy soft matrix theory was studied by Borah et al. in . In , Chetia et al. and in  Rajarajeswari et al. defined intuitionisticfuzzy soft matrix.Again it is well known that the matrices are important tools to model/study different mathematical problems specially in linear algebra. Due to huge applications of imprecise data in the above mentioned areas, hence are motivated to study the different matrices containing these data. Soft set is also one of the interesting and popular subject, where different types of decisionmaking problem can be solved. So attempt has been made to study the decisionmaking problem by using intuitionisticfuzzy soft aggregation operator. Das and Kar  proposed an algorithmic approach for groupdecisionmakingbased on IF soft set. The authors  have used cardinality of IF soft set as a novel concept for assigning confident weight to the set of experts. Cagman and Enginogh[3 ,4] pioneered the concept of soft matrix to represent a soft set. Mao et al. presented the concept of intuitionisticfuzzy soft matrix(IFSM) and applied it in groupdecisionmaking problem.
Deschrijver and Kerre  have shown that IFSs are equivalent to IVFSs (also called vague sets ) and both can be regarded as L-fuzzy sets in the sense of Goguen . In reality, it may not be easy to identify exact values for the membership and non-membership degrees of an element to a set. In this case, a range of values may be a more appropriate measurement to accommodate the vagueness. As such, Atanassov and Gargov  introduce the notion of IVIFS:
Abstract. An Interval-Valued Trapezoidal IntuitionisticFuzzy Number (IVTrIFN) is a special case of an IntuitionisticFuzzy Set (IFS), which is dened on a real number set. From a geometric viewpoint, the expectation and expectant score of an IVTrIFN are dened using the notion of a barycenter, and a new method is developed to rank IVTrIFNs. Hereby, some generalized aggregation operators of IVTrIFNs are dened, including the generalized ordered weighted averaging operator and the generalized hybrid weighted averaging operator, which are employed to solve multi-attributegroupdecisionmaking problems. Using the weighted average operator of IVTrIFNs, the attribute values of alternatives are integrated into the individual comprehensive ratings, which are further aggregated into the collective one by the generalized hybrid weighted averaging operator of IVTrIFNs. The ranking orders of alternatives are then generated according to the expectation and expectant score of the collective comprehensive ratings of alternatives. A numerical example is examined to demonstrate the applicability and implementation process of the decisionmethod proposed in this paper.
Abstract – In this paper, we use the Choquetintegral to propose the normal distribution interval number Choquet ordered averaging operator. The operator not only considers the importance of the elements, but also can reflect the correlations among the elements. It is worth pointing out that most of the existing normal distribution interval numbers averaging operators are special cases of our operator. Finally an illustrative example is given to use the operator in the range of uncertain multi-attributedecision-making. The results show that the method proposed in this paper is feasible.
Evaluating the performance of the available Cloud services with respect to multiple, usually conflicting criteria in a specific situation is always challenging due to (a) the availability of multiple Cloud services; (b) the multi-dimensional nature of the decisionmaking problem; (c) the involvement of multiple decision makers; and (d) the presence of subjectivity and imprecision involved in the decisionmaking process [ 34 ]. To overcome these concerns, this paper presents a fuzzymulti-criteria decisionmakingmethodbased on the fusion of several concepts including (a) the TOPSIS method; (b) the Choquetintegral operator; and (c) intuitionisticfuzzy numbers.