In this paper we introduce concepts of pseudo-triangular entropy as a supplement measure of Uncertainty in the Uncertain portfolio optimization. We first prove that logarithm entropy and triangular entropy for Uncertain variables sometimes may fail to measure the Uncertainty of an Uncertain variable. Then, we propose a definition of pseudo-triangular entropy as a supplement measure to characterize the Uncertainty of Uncertain variables and we derive its mathematical properties. We also give a formula to calculate the pseudo-triangular entropy of Uncertain variables via inverse Uncertainty distribution. Moreover, we use the pseudo-triangular entropy to characterize portfolio risk and establish some Uncertain portfolio optimization models based on different types of entropy. A genetic algorithm (GA) is implemented in MATLAB software to solve the corresponding problem. Numerical results show that pseudo-triangular entropy as a quantifier of portfolio risk outperforms logarithm entropy and triangular entropy in the Uncertain portfolio optimization.

Uncertainty theory is a mathematical methodology for dealing with non-determinate phenomena in nature. As we all know, Uncertain pro-cess and Uncertain integral are important contents of Uncertainty theory, so it is necessary to have deep research. This paper presents the definition and discusses some properties of strong comonotonic Uncertain process. Besides, some useful formulas of Uncertain integral such as nonnegativity, monotonicity, intermediate results are studied.

Recently, a new paradigm for Uncertain information has been proposed that can effectively handle various types of Uncertainty in decision-making problems. This approach utilizes a certainty degree, which is represented by a real number indicating the level of certainty associated with input values. However, just like intuitionistic fuzzy information can handle more problems that cannot be well modeled by fuzzy information, the certainty degree in basic Uncertain information can also be intuitionistic fuzzy granule, which allows it to handle more Uncertainty involved decision making situations. In this paper, we introduce the concept of intuitionistic fuzzy type basic Uncertain information and explain its parameters. We also define a weighted arithmetic mean for aggregating this type of information and discuss different approaches for allocating induced weights based on trust preferred preference from four perspectives: (i) preference for higher certainty degrees,(ii) aversion to higher levels of Uncertainty,(iii) preference for greater differences in certainty degrees,and (iv) preference for intuitionistic fuzzy certainties. Additionally, we explore trichotomic rules-based decision making using intuitionistic fuzzy type basic Uncertain information. Finally, we present an objective-subjective evaluation numerical example utilizing these methods.

Catastrophe bonds are among the essential instruments in providing a financial hedge for insurance companies and their policyholders. Catastrophic events are rare, and the shortage of data turns using probability theory indefensible. On the other hand, Uncertainty theory is a reliable alternative to deal with these kinds of indeterminacies. We model the problem of pricing catastrophe bonds as an Uncertain optimization problem where the maximization of the cedent insurance company’s profit is constrained to the Uncertain measure of ruin defined for the investors. Consequently, one could provide a tradeoff between being profitable for the ceding company and having reasonable protection for the investors. A solution to the optimization problem will be considered as the spread over the LIBOR, leading to a complete determination of the bond price. The results suggest the practicality of the model, especially the application of Uncertainty theory in pricing catastrophe bonds. Finally, the Uncertain ruin index is calculated for a real-world problem, and the results are compared with those obtained by probability theory.

Age replacement policy is concerned with finding an optional time to minimize the cost, at which time the unit is replaced even if it does not fail.So far, age replacement policy involving random age has been proposed. This paper will assume the age of the unit is an Uncertain variable, and find the optimal time to replace the unit.