A hierarchical family of integrals based on a fixed copula is introduced and discussed. The extremal members of this family correspond to the inner and outer extension of integrals of basic functions, the copula under consideration being the corresponding multiplication. The limits of the members of the family are just copula-based universal integrals as recently introduced in Klement et al. (IEEE Trans Fuzzy Syst 18:178---187, 2010). For the product copula, the family of integrals considered here contains the Choquet and the Shilkret integral, and it belongs to the class of decomposition integrals proposed in Even and Lehrer (Econ Theory, 2013) as well as to the class of superdecomposition integrals introduced in Mesiar et al. (Superdecomposition integral, 2013). For the upper Frechet-Hoeffding bound, the corresponding hierarchical family contains only two elements: all but the greatest element coincide with the Sugeno integral.

This paper discusses a principal---agent problem with multi-dimensional incomplete information between a principal and an agent. Firstly, how to describe the incomplete information in such agency problem is a challenging issue. This paper characterizes the incomplete information by uncertain variable, because it has been an appropriate tool to depict subjective assessment and model human uncertainty. Secondly, the relevant literature often used expected-utility-maximization to measure the two participators' goals. However, Ellsberg paradox indicates that expected utility criterion is not always appropriate to be regarded as decision rule. For this reason, this paper presents another decision rule based on confidence level. Instead of expected-utility-maximization, the principal's aim is to maximize his potential income under the acceptable confidence level, and the agent's aim depends on whether he has private information about his efforts. According to the agent's different decision rules, three classes of uncertain agency (UA) models and their respective optimal contracts are presented. Finally, a portfolio selection problem is studied to demonstrate the modeling idea and the viability of the proposed UA models.

Group decision making is the process to explore the best choice among the screened alternatives under predefined criteria with corresponding weights from assessment of a group of decision makers. The Fuzzy TOPSIS taking an evaluated fuzzy decision matrix as input is a popular tool to analyze the ideal alternative. This research, however, finds that the classical fuzzy TOPSIS produces a misleading result due to some inappropriate definitions, and proposes the rectified fuzzy TOPSIS addressing two technical problems. As the decision accuracy also depends on the evaluation quality of the fuzzy decision matrix comprising rating scores and weights, this research applies compound linguistic ordinal scale as the fuzzy rating scale for expert judgments, and cognitive pairwise comparison for determining the fuzzy weights. The numerical case of a robot selection problem demonstrates the hybrid approach leading to the much reliable result for decision making, comparing with the conventional fuzzy Analytic Hierarchy Process and TOPSIS.

This paper presents an uncertain stock model with periodic dividends based on uncertainty theory. Some option pricing formulae related to the proposed model are investigated and several numerical examples are discussed to illustrate the related formula.

Uncertain calculus is a branch of mathematics that deals with differentiation and integration of function of uncertain processes. As a fundamental concept, uncertain integral has been defined with respect to canonical process. However, emergencies such as economic crisis and war occur occasionally, which may cause the uncertain process a sudden change. So far, uncertain renewal process has been employed to model these jumps. This paper will present a new uncertain integral with respect to renewal process. Besides, this paper will propose a type of uncertain differential equation driven by both canonical process and renewal process.

Uncertain set is a set-valued function on an uncertainty space, and attempts to model "unsharp concepts" that are essentially sets but their boundaries are not sharply described. This paper will propose a concept of membership function and define the independence of uncertain sets. This paper will also present an operational law of uncertain sets via membership functions or inverse membership functions. Finally, the linearity of expected value operator is verified.

This paper develops a robust method to describe fuzzy returns by employing parametric possibility distributions. The parametric possibility distributions are obtained by equivalent value (EV) reduction methods. For common type-2 triangular and trapezoidal fuzzy variables, their reduced fuzzy variables are studied in the current development. The parametric possibility distributions of reduced fuzzy variables are first derived, then the second moment formulas for the reduced fuzzy variables are established. Taking the second moment as a new risk measure, the reward-risk and risk-reward models are developed to optimize fuzzy portfolio selection problems. The mathematical properties of the proposed optimization models are analyzed, including the analytical representations for the second moments of linear combinations of reduced fuzzy variables as well as the convexity of second moments with respect to decision vectors. On the basis of the analytical representations for the second moments, the reward-risk and risk-reward models can be turned into their equivalent parametric quadratic convex programming problems, which can be solved by conventional solution methods or general-purpose software. Finally, some numerical experiments are performed to demonstrate the new modeling ideas and the efficiency of solution method.

Fuzzy analytic hierarchy process (FAHP) has increasingly been applied in many areas. Extent analysis method is the popular tool for prioritization in FAHP, although significant technical errors are identified in this study. With addressing the errors, this research proposes membership maximization prioritization methods (MMPMs) using different membership functions as the novel solutions. As a lack of research about effectiveness measurement on the crisp/fuzzy prioritization methods, this study proposes membership fitness index to evaluate the effectiveness of the prioritization methods. Comparisons with the other popular fuzzy/crisp prioritization methods including modified fuzzy preference programming, Direct least squares, and Eigen value are conducted and analyses indicate that MMPMs lead to much more reliable result in view of membership fitness index. A numerical example demonstrates the usability of MMPMs for FAHP, and thus MMPMs can effectively be applied to various decision analysis applications.