Uncertainty Theory Based Partitioning for Cyber-Physical Systems with Uncertain Reliability Analysis

Reasonable partitioning is a critical issue for cyber-physical system (CPS) design. Traditional CPS partitioning methods run in a determined context and depend on the parameter pre-estimations, but they ignore the uncertainty of parameters and hardly consider reliability. The state-of-the-art work proposed an uncertainty theory based CPS partitioning method, which includes parameter uncertainty and reliability analysis, but it only considers linear uncertainty distributions for variables and ignores the uncertainty of reliability. In this paper, we propose an uncertainty theory based CPS partitioning method with uncertain reliability analysis. We convert the uncertain objective and constraint into determined forms; such conversion methods can be applied to all forms of uncertain variables, not just for linear. By applying uncertain reliability analysis in the uncertainty model, we for the first time include the uncertainty of reliability into the CPS partitioning, where the reliability enhancement algorithm is proposed. We study the performance of the reliability obtained through uncertain reliability analysis, and experimental results show that the system reliability with uncertainty does not change significantly with the growth of task module numbers.

On Multi-Objective Minimum Spanning Tree Problem under Uncertain Paradigm

Minimum spanning tree problem (MSTP) has allured many researchers and practitioners due to its varied range of applications in real world scenarios. Modelling these applications involves the incorporation of indeterminate phenomena based on their subjective estimations. Such phenomena can be represented rationally using uncertainty theory. Being a more realistic variant of MSTP, in this article, based on the principles of the uncertainty theory, we have studied a multi-objective minimum spanning tree problem (MMSTP) with indeterminate problem parameters. Subsequently, two uncertain programming models of the proposed uncertain multi-objective minimum spanning tree problem (UMMSTP) are developed and their corresponding crisp equivalence models are investigated, and eventually solved using a classical multi-objective solution technique, the epsilon-constraint method. Additionally, two multi-objective evolutionary algorithms (MOEAs), non-dominated sorting genetic algorithm II (NSGAII) and duplicate elimination non-dominated sorting evolutionary algorithm (DENSEA) are also employed as solution methodologies. With the help of the proposed UMMSTP models, the practical problem of optimizing the distribution of petroleum products was solved, consisting in the search for symmetry (balance) between the transportation cost and the transportation time. Thereafter, the performance of the MOEAs is analyzed on five randomly developed instances of the proposed problem.

Some Theorems for Inverse Uncertainty Distribution of Uncertain Processes

In real life, indeterminacy and determinacy are symmetric, while indeterminacy is absolute. We are devoted to studying indeterminacy through uncertainty theory. Within the framework of uncertainty theory, uncertain processes are used to model the evolution of uncertain phenomena. The uncertainty distribution and inverse uncertainty distribution of uncertain processes are important tools to describe uncertain processes. An independent increment process is a special uncertain process with independent increments. An important conjecture about inverse uncertainty distribution of an independent increment process has not been solved yet. In this paper, the conjecture is proven, and therefore, a theorem is obtained. Based on this theorem, some other theorems for inverse uncertainty distribution of the monotone function of independent increment processes are investigated. Meanwhile, some examples are given to illustrate the results.

A New Formula for Calculating Uncertainty Distribution of Function of Uncertain Variables

As a mathematical tool to rationally handle degrees of belief in human beings, uncertainty theory has been widely applied in the research and development of various domains, including science and engineering. As a fundamental part of uncertainty theory, uncertainty distribution is the key approach in the characterization of an uncertain variable. This paper shows a new formula to calculate the uncertainty distribution of strictly monotone function of uncertain variables, which breaks the habitual thinking that only the former formula can be used. In particular, the new formula is symmetrical to the former formula, which shows that when it is too intricate to deal with a problem using the former formula, the problem can be observed from another perspective by using the new formula. New ideas may be obtained from the combination of uncertainty theory and symmetry.

Risk-Neutral Pricing Method of Options Based on Uncertainty Theory

In order to rationally deal with the belief degree, Liu proposed uncertainty theory and refined into a branch of mathematics based on normality, self-duality, sub-additivity and product axioms. Subsequently, Liu defined the uncertainty process to describe the evolution of uncertainty phenomena over time. This paper proposes a risk-neutral option pricing method under the assumption that the stock price is driven by Liu process, which is a special kind of uncertain process with a stationary independent increment. Based on uncertainty theory, the stock price’s distribution and inverse distribution function under the risk-neutral measure are first derived. Then these two proposed functions are applied to price the European and American options, and verify the parity relationship of European call and put options.

Stability Analysis for Linear Uncertain Switched Systems in Infinite-time Domain

In this work, an uncertain switched system expressed as a series of uncertain differential equations is considered in depth. Stability issues have been widely investigated on switched systems while few results related to stability analysis for uncertain switched systems can be found. Due to such fact, three different stabilities, including stability in measure, almost sure stability and stability in mean, are comprehensively studied for linear uncertain switched systems in infinite-time domain. Internal property of the systems is able to be illustrated from different perspectives with the help of above stability analysis. By employing uncertainty theory and the feature of switched systems, corresponding judgement theorems of these stabilities are proposed and verified. An example with respect to stability in measure is provided to display the validness of the results derived.

Structural Credit Risk Model with Jumps Based on Uncertainty Theory

Traditional finance studies of credit risk structured models are based on the assumption that the price of the underlying asset obeys a stochastic differential equation. However, according to behavioral finance, the price of the underlying asset is not entirely stochastic, and the credibility of financial investors also plays a very important role in asset prices. In this paper we introduce uncertainty theory to describe these credibility of investors and propose a new credit risk structured model with jumps based on the assumption that the underlying asset is described by an uncertain differential equation with jumps. The company default belief degree formula, zero coupon bond value and stock value formula are formulated. Company bond credit spread and credit default swap (CDS) pricing are studied as applications of the proposed model in uncertain markets.