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With the increasing deployment of network technologies in industrial control systems (ICSs), cybersecurity has become a challenge in ICSs. Cybersecurity risk assessment (CRA) plays an important role in cybersecurity protection of ICSs. However, the weights of risk indices are constants in traditional CRA methods, and they do not fully consider the requirements of risk identification. In this paper, we define a novel order- $\alpha $ divergence measure for interval-valued intuitionistic fuzzy numbers (IVIFNs) and further develop a novel CRA approach for ICSs based on the proposed divergence measure under an interval-valued intuitionistic fuzzy environment to contribute to the research gap. First, an order- $\alpha $ divergence measure for IVIFNs is defined considering flexibility and robustness of divergence measures with the parameter. Next, a variable weight-based CRA approach for ICSs is developed. In this approach, IVIFNs are adopted to describe evaluation values of risk indices. The weights of risk indices are variable weight vectors and they are determined by the relative divergence closeness. Integration approaches of each node and each attack path in attack-defense trees (ADTs) are proposed based on the operations of IVIFNs, and risk scores of each attack path are calculated by using the score function. Finally, we apply the proposed method to the CRA of a civil aviation fuel supply automatic control system and verify its effectiveness and advantages by comparing it with other methods. This method can dynamically adjust the weights of risk indices considering the relationship between each risk index and the highest risk, and therefore, it can more effectively recognize the highest risk of ICSs than the traditional CRA method. In addition, it can also match the risk attitude of decision-makers by adjusting the parameter $\alpha $ .

In this paper, we expand the Muirhead mean (MM) operator and dual Muirhead mean (DMM) operator with interval-valued intuitionistic fuzzy numbers (IVIFNs) to propose the interval -valued intuitionistic fuzzy Muirhead mean (IVIFMM) operator, interval-valued intuitionistic fuzzy weighted Muirhead mean (IVIFWMM) operator, interval-valued intuitionistic fuzzy dual Muirhead mean (IVIFDMM) operator and interval-valued intuitionistic fuzzy weighted dual Muirhead mean (IVIFWDMM) operator. Then the MADM methods are proposed with these operators. In the end, we utilize an applicable example for green supplier selection in green supply chain management to prove the proposed methods. [ABSTRACT FROM AUTHOR]

Recently, the VIKOR has been used to solve multiple attribute decision making (MADM) problems. The interval-valued intuitionistic fuzzy sets (IVIFSs) are useful tools to depict uncertainty of the MADM problems. In this paper, we will extend VIKOR method to the MADM with the interval-valued intuitionistic fuzzy numbers (IVIFNs). Firstly, the definition, comparison and distance of IVIFNs are briefly introduced. Then, the extended classical VIKOR method is proposed to deal with MADM problems with the IVIFNs, and its significant characteristic is that it can fully consider the decision makers' bounded rationality which is a real action in decision making. Finally, a numerical example for financing risk assessment of rural tourism projects is proposed and a comparative analysis is also given. [ABSTRACT FROM AUTHOR]

Recently, the TODIM has been used to solve multiple attribute decision making (MADM) problems. The interval-valued intuitionistic fuzzy sets (IVIFSs) are useful tools to depict uncertainty of the MADM. In this paper, we will extend TODIM method to the MADM with the interval-valued intuitionistic fuzzy numbers (IVIFNs). Firstly, the definition, comparison and distance of IVIFNs are introduced, and the steps of the classical TODIM method for MADM problems are presented. Then, the extended classical TODIM method is proposed to deal with MADM problems with the IVIFNs, and its significant characteristic is that it can fully consider the decision makers' bounded rationality which is a real action in decision making. Finally, a numerical example for performance appraisal on social-integration-based rural reconstruction is proposed and a comparative analysis is also given. [ABSTRACT FROM AUTHOR]

In this paper, we expand the Hamy mean (HM) operator and dual Hamy mean (DHM) operator with IVIFNs to propose the interval -valued intuitionistic fuzzy Hamy mean (IVIFHM) operator, interval-valued intuitionistic fuzzy weighted Hamy mean (IVIFWHM) operator, interval-valued intuitionistic fuzzy dual Hamy mean (IVIFDHM) operator and interval-valued intuitionistic fuzzy weighted dual Hamy mean (IVIFWDHM) operator. Then the MADM models are designed with IVIFWHM and IVIFWDHM operators. Finally, we gave an example for competiveness evaluation of tourist destination to show the proposed models. [ABSTRACT FROM AUTHOR]

In this paper, we expand the Hamy mean (HM) operator and Dombi operations with interval-valued intuitionistic fuzzy numbers (IVIFNs) to propose the interval-valued intuitionistic fuzzy Dombi Hamy mean (IVIFDHM) operator, interval-valued intuitionistic fuzzy weighted Dombi Hamy mean (IVIFWDHM) operator, interval-valued intuitionistic fuzzy dual Dombi Hamy mean (IVIFDDHM) operator, and interval-valued intuitionistic fuzzy weighted dual Dombi Hamy mean (IVIFWDDHM) operator. Then the MADM models are designed with IVIFWDHM and IVIFWDDHM operators. Finally, we gave an example for evaluating the elderly tourism service quality in tourism destination to show the proposed models.

With respect to multiple attribute group decision-making problems in which attribute values take the form of the interval-valued intuitionistic fuzzy numbers, the group decision-making methods based on some Hamacher aggregation operators, which extended the algebraic aggregation operators and Einstein aggregation operators, are developed. First, an interval-valued intuitionistic fuzzy Hamacher weighted averaging operator, interval-valued intuitionistic fuzzy Hamacher-ordered weighted averaging operator, interval-valued intuitionistic fuzzy Hamacher hybrid weighted averaging operator, interval-valued intuitionistic fuzzy Hamacher geometric weighted averaging operator, interval-valued intuitionistic fuzzy Hamacher geometric-ordered weighted averaging operator, and interval-valued intuitionistic fuzzy Hamacher geometric hybrid weighted averaging operator are proposed. Some desirable properties of these operators, such as commutativity, idempotency, monotonicity, and boundedness, are studied, and some special cases in these operators are analyzed. Furthermore, two methods to multicriteria decision group making based on these operators are developed. Finally, an illustrative example is given to verify the proposed methods and to demonstrate their practicality and effectiveness. [ABSTRACT FROM AUTHOR]

Abstract: This article proposes a framework to handle multiattribute group decision making problems with incomplete pairwise comparison preference over decision alternatives where qualitative and quantitative attribute values are furnished as linguistic variables and crisp numbers, respectively. Attribute assessments are then converted to interval-valued intuitionistic fuzzy numbers (IVIFNs) to characterize fuzziness and uncertainty in the evaluation process. Group consistency and inconsistency indices are introduced for incomplete pairwise comparison preference relations on alternatives provided by the decision-makers (DMs). By minimizing the group inconsistency index under certain constraints, an auxiliary linear programming model is developed to obtain unified attribute weights and an interval-valued intuitionistic fuzzy positive ideal solution (IVIFPIS). Attribute weights are subsequently employed to calculate distances between alternatives and the IVIFPIS for ranking alternatives. An illustrative example is provided to demonstrate the applicability and effectiveness of this method. [Copyright &y& Elsevier]

Abstract: This article proposes an approach to handle multi-attribute decision making (MADM) problems under the interval-valued intuitionistic fuzzy environment, in which both assessments of alternatives on attributes (hereafter, referred to as attribute values) and attribute weights are provided as interval-valued intuitionistic fuzzy numbers (IVIFNs). The notion of relative closeness is extended to interval values to accommodate IVIFN decision data, and fractional programming models are developed based on the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method to determine a relative closeness interval where attribute weights are independently determined for each alternative. By employing a series of optimization models, a quadratic program is established for obtaining a unified attribute weight vector, whereby the individual IVIFN attribute values are aggregated into relative closeness intervals to the ideal solution for final ranking. An illustrative supplier selection problem is employed to demonstrate how to apply the proposed procedure. [Copyright &y& Elsevier]

With China's sustained economic development and constant increase in national income, Chinese nationals' tourism consumption rate increases. As a major Chinese economic development engine, the domestic tourism industry has entered a transition period operation pattern featured by diversified products. Among them, as a new hot spot of the tourism industry in China, ecological tourism has enjoyed rapid development, with great potential. Thus, the ecological value evaluation of forest ecological tourism demonstration areas is very important to the domestic tourism industry. In this paper, we propose some Dombi Heronian mean operators with interval-valued intuitionistic fuzzy numbers (IVIFNs). Then, two MADM (multiple attribute decision making) methods are proposed based on IVIFWDHM (interval-valued intuitionistic fuzzy weighted Dombi Heronian mean) and IVIFWDGHM (interval-valued intuitionistic weighted Dombi geometric Heronian mean) operators. Finally, we gave an experimental case for evaluating the ecological value of forest ecological tourism demonstration area to show the proposed decision methods., Competing Interests: The authors declare no conflict of interest.

In this paper, we expand the Hamy mean (HM) operator and Dombi operations with interval-valued intuitionistic fuzzy numbers (IVIFNs) to propose the interval-valued intuitionistic fuzzy Dombi Hamy mean (IVIFDHM) operator, interval-valued intuitionistic fuzzy weighted Dombi Hamy mean (IVIFWDHM) operator, interval-valued intuitionistic fuzzy dual Dombi Hamy mean (IVIFDDHM) operator, and interval-valued intuitionistic fuzzy weighted dual Dombi Hamy mean (IVIFWDDHM) operator. Then the MADM models are designed with IVIFWDHM and IVIFWDDHM operators. Finally, we gave an example for evaluating the elderly tourism service quality in tourism destination to show the proposed models. [ABSTRACT FROM AUTHOR]