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The flow maximization problem being a leading problem in network optimization is widely studied with many applications. In this work, we find the maximum flow pulled out from the source and sent to the sink using the uncertainty theory. We develop a flow function to get maximum flow for the network having uncertain arc capacities and sufficient storage capacities at the intermediate vertices. The centiles-method is used to find the maximum flow value and an algorithm is proposed to get non-conservative maximum flow. The result is also illustrated by an example. The flow values are compared graphically to see the dominance of the non-conservative maximum flow over the conservative maximum flow.

Uncertainty theory, as a mathematical tool to rationally handle degrees of belief of human beings, has been widely applied in science and engineering. As a basic component of uncertainty theory, uncertainty distribution is a carrier to describe the characterization of an uncertain variable. Moreover, inverse uncertainty distribution provides an easy way to calculate the functions of uncertain variables as well as their expected value. This paper mainly provides the formulas to calculate the expected value, variance, moments and entropy of lognormal uncertain variable. Besides, some operational laws of lognormal uncertain variables are proved.

Uncertain statistics is a set of mathematical techniques for collecting, analyzing and interpreting data by uncertainty theory. There are mainly three modeling methods in uncertain statistics: uncertain time series analysis, uncertain regression analysis, and uncertain differential equation. This paper applies these tools to modeling China’s per capita disposable income, and employs uncertain hypothesis test to determine whether the estimated uncertain statistical models fit China’s per capita disposable income. In addition, this paper shows that it is necessary to use uncertain statistics instead of probability statistics to model China’s per capita disposable income by investigating the corresponding residuals.

In this paper, we introduce a novel methodology for calibrating European option pricing within the context of uncertain financial markets. Our approach leverages an artificial neural network, where each input neuron corresponds to the option price function derived from the uncertain stock model. We investigate our method against traditional calibration techniques, including those based on uncertain differential equations and the Black–Scholes model. Numerical experiments demonstrate that the proposed neural network-based strategy significantly enhances the accuracy and performance of option price calibration, yielding improved results for both in-sample and out-of-sample datasets.

With the increasingly severe global climate change issue, carbon emission allowances trading has become one of the important means for the international community to reduce greenhouse gas emissions. However, this process also exposes the high costs and risks faced by emission control enterprises. To cope with this risk, carbon options, as an innovative tool in the financial derivatives market, allow holders to buy and sell carbon emission rights at a predetermined price at a specific time in the future. By purchasing carbon options, companies can lock in future carbon emission costs, thereby avoiding the risk of price fluctuations, which is of great significance for solving the high cost and high risk problems faced by emission control companies in emission reduction activities. The primary task is to price carbon options reasonably. Most existing methods are based on probability theory, and sometimes fail due to the lack of fulfillment of the necessary prerequisites for probability theory. This paper studies the pricing problem of carbon options based on exponential uncertain differential equations within the framework of uncertainty theory and derives calculation formulas. Furthermore, real data analysis based on China’s carbon trading market is conducted to illustrate our methodology in detail.

It is natural to seek a measure for assessing the influence of individual component within a system. In this paper, we define the influence index of uncertain systems and introduce several key properties of this index. Furthermore, we investigate the influence indexes of specific types of uncertain reliability systems, including uncertain series, parallel, parallel-series, and series-parallel configurations. Some formulas are also derived to calculate the influence index of a component in uncertain Boolean systems.

Nonnegative variables exist widely in the renewal processes such as queueing process, insurance process, and production process and play an important role in the practical application of related fields. As a powerful tool for modeling nonnegative variables, this paper gives a form of truncated normal uncertainty distribution, and gives the expressions or calculation formulas of its inverse uncertainty distribution, expected value, variance, and moments. In order to better apply the truncated normal uncertainty distribution to practice, this paper also studies the parameter estimation of the truncated normal uncertainty distribution based on the method of moments, the method of maximum likelihood and the method of least squares, and derives the rejection region of the corresponding hypothesis testing. Finally, this paper also illustrates the above results with two real data examples.

The interest rate is a key factor in the financial market. Various widely used interest rate derivatives, such as zero-coupon bond, interest rate ceiling, interest rate floor and currency exchange rate, are based on the interest rate. This paper assumes that the interest rate follows the uncertain mean-reverting differential equation and introduces some formulas for pricing zero-coupon bond, interest rate ceiling and interest rate floor within this model. Additionally, this paper provides numerical examples to show how these pricing formulas can be applied.

Heat equation is a partial differential equation describing the temperature change of an object with time. In the traditional heat equation, the strength of heat source is assumed to be certain. However, in practical application, the heat source is usually influenced by noise. To describe the noise, some researchers tried to employ a tool called Winner process. Unfortunately, it is unreasonable to apply Winner process in probability theory to modeling noise in heat equation because the change rate of temperature will tend to infinity. Thus, we employ Liu process in uncertainty theory to characterize the noise. By modeling the noise via Liu process, the one-dimensional uncertain heat equation was constructed. Since the real world is a three-dimensional space, the paper extends the one-dimensional uncertain heat equation to a three-dimensional uncertain heat equation. Later, the solution of the three-dimensional uncertain heat equation and the inverse uncertainty distribution of the solution are given. At last, a paradox of stochastic heat equation is introduced.

The aim of this paper is to present a novel method for solving the minimum cost flow problem on networks with uncertain-random capacities and costs. The objective function of this problem is an uncertain random variable and the constraints of the problem do not make a deterministic feasible set. Under the framework of uncertain random programming, a corresponding $\alpha$-minimum cost flow model with a prespecified confidence level $\alpha$, is formulated and its main properties are analyzed. It is proven that there exists an equivalence relationship between this model and the classical deterministic minimum cost flow model. Then an algorithm is proposed to find the maximum amount of $\alpha$ such that for it, the feasible set of $\alpha$-minimum cost flow model is nonempty. Finally, a numerical example is presented to illustrate the efficiency of our proposed method.

Uncertain fractional differential equation driven by Liu process plays a significant role in depicting the memory effects of uncertain dynamical systems. This paper mainly investigates the stability problems for the Caputo type of uncertain fractional differential equations with the order $0<p\le 1$. The concept of stability in measure of solutions to uncertain fractional differential equation is proposed based on uncertainty theory. Several sufficient conditions for ensuring the stability of the solutions are derived, respectively, in which the systems are divided into two cases with order $\frac{1}{2}<p\le 1$ and $0<p\le \frac{1}{2}$. Some illustrative examples are performed to display the effectiveness of the proposed results.

In this paper, the finite-time stability in mean for the uncertain fractional order linear time-invariant discrete systems is investigated. First, the uncertain fractional order difference equations with the nabla operators are introduced. Then, some conditions of finite-time stability in mean for the systems driven by the nabla uncertain fractional order difference equations with the fractional order $0<\nu <1$ are obtained by the property of Riemann–Liouville-type nabla difference and the generalized Gronwall inequality. Furthermore, based on these conditions, the state feedback controllers are designed. Finally, some examples are presented to illustrate the effectiveness of the results.

In this paper, we discuss some properties in uncertainty theory when uncertain measure is continuous. Firstly, the judgement conditions of continuous uncertain measure are proposed. Secondly, basic properties of uncertainty distribution and critical values of uncertain variable are proved. Finally, the convergence theorems for expected value are discussed.

Liu's inference is a process of deriving consequences from uncertain knowledge or evidence via the tool of conditional uncertainty. Using membership functions, this paper derives some expressions of Liu's inference rule for uncertain systems. This paper also discusses Liu's inference rule with multiple antecedents and with multiple if-then rules.

In practical applications of graph theory, non-deterministic factors are frequently encountered. This paper employs uncertainty theory to deal with non-deterministic factors in problems of graph connectivity. The concepts of uncertain graph and connectedness index of uncertain graph are proposed in this paper. It presents two algorithms to calculate connectedness index of an uncertain graph.

The traditional single period inventory problem assumes that the market demand is a random variable. However, as an empirical or subjective estimation, market demand is better to be regarded as an uncertain variable. This paper is concerning with single period inventory problem under two main assumptions that (i) the market demand is an uncertain variable and (ii) a setup cost and an initial stock exist. Under the framework of uncertainty theory, the optimal inventory policy for uncertain single period inventory problem with an initial stock and a setup cost is derived, which is of (s,S) type. Also, some expansions are obtained.

In this paper, several useful inequalities for uncertain variables are proved. A Borel-Cantelli lemma for uncertain measures is obtained and some convergence theorems for continuous uncertain measures are derived. Finally, these theorems are applied to compute the uncertainty distribution of Liu integral. We prove that the uncertain integral of a deterministic function with respect to a Liu process has a normal uncertainty distribution.

In many systems, randomness and uncertainty are present simultaneously. Uncertain random variables provide a tool to deal with these inexact phenomena. This paper proposes a concept of risk index to quantify the risk of an uncertain random system. In addition, a risk index theorem is proved in order to calculate the risk index, and is applied to series systems, parallel systems and standby systems. Finally, a concept of expected loss is suggested for an uncertain random system.

For uncertain variable sequences, conditions of convergences such as Cauchy convergence in measure, convergence almost surely and convergence uniformly almost surely are given. Consequently, the relationships among convergences of uncertain variable sequences are shown. These results have not been proposed in literature so far.

This paper investigates the uncertain minimum spanning tree (UMST) problem where the edge weights are assumed to be uncertain variables. In order to propose effective solving methods for the UMST problem, path optimality conditions as well as some equivalent definitions for two commonly used types of UMST, namely, uncertain expected minimum spanning tree (expected UMST) and uncertain α-minimum spanning tree (α-UMST), are discussed. It is shown that both the expected UMST problem and the α-UMST problem can be transformed into an equivalent classical minimum spanning tree problem on a corresponding deterministic graph, which leads to effective algorithms with low computational complexity. Furthermore, the notion of uncertain most minimum spanning tree (most UMST) is initiated for an uncertain graph, and then the equivalent relationship between the α-UMST and the most UMST is proved. Numerical examples are presented as well for illustration.