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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.

A novel approach of weighted support vector regression (WSVR) technique with applied chance theory was proposed to build a robust forecasting model, called the chance weighted support vector regression (chWSVR) model. In order to forecast the particulate matter air pollutant of diameter less than 10 micrometers (PM₁₀) one hour advance in the Greater Cairo Metropolitan Area (GCMA) in Egypt. The chance theory has advanced concepts pertinent to treat cases where both randomness and fuzziness play simultaneous roles at one time. The basic idea based on the proposed chWSVR model is assigning the chance weight value of the target variable, based on the chance theory, to its corresponding dataset point to become minimized in the objective function making that point more significant during the training process. Measuring data were collected and reprocessed from four monitoring stations located in GCMA and relative to the springs during the period from 2007 to 2010. The results of such model compared to similar ones built by other machine learning techniques, Random Forest and Bootstrap aggregating techniques. In all stations, comparing such models revealed that the proposed chWSVR model findings were promising in the forecasting of PM₁₀ hourly concentration.

Some complex systems may suffer from failure processes arising from soft failures and hard failures. The existing researches have shown that the reliability of a dynamic system is not constant under uncertain random environments. First, two types of uncertain random optimization models are proposed where reliability index is quantified by chance measure based on whether soft and hard failures are independent or not. It is considered that internal degradation is driven by left Caputo fractional linear difference equation, while shocks are defined as discrete i.i.d. random variables. The shocks may generate additional uncertain degradation shifts when considering the competing dependent failure processes. Then, two proposed optimization reliability problems may be transformed into their equivalent deterministic forms on the basis of $\alpha$-path, and improved gradient descent method is applied to obtain optimal solutions. Finally, the numerical example of a micro-engine indicates that the optimization models are beneficial to the reliability of engineering systems.

The particulate matter air pollutant of diameter less than 10 micrometers (PM${}_{10}$), a category of pollutants including solid and liquid particles, can be a health hazard for several reasons: it can harm lung tissues and throat, aggravate asthma and increase respiratory illness. Accurate prediction models of PM${}_{10}$ concentrations are essential for proper management, control, and making public warning strategies. Therefore, machine learning techniques have the capability to develop methods or tools that can be used to discover unseen patterns from given data to solve a particular task or problem. The chance theory has advanced concepts pertinent to treat cases where both randomness and fuzziness play simultaneous roles at one time. The main objective is to study the modification of a single machine learning algorithm — support vector machine (SVM) — applying the chance weight of the target variable, based on the chance theory, to the corresponding dataset point to be superior to the ensemble machine learning algorithms. The results of this study are outperforming of the SVM algorithms when modifying and combining with the right theory/technique, especially the chance theory over other modern ensemble learning algorithms.

Partial entropy is a device to measure uncertainty of an uncertain random variable. In order to characterize indeterminacy of uncertain random variables, the concept of partial exponential entropy for uncertain random variables is presented. By invoking inverse uncertainty distribution, we derive a formula for computing partial exponential entropy for uncertain random variables. As an application of partial exponential entropy, portfolio selection problems for uncertain random returns are optimized via partial entropy-mean models. For better understanding, several examples are provided.