Most industrial products degrade with usage. To capture degradation patterns and schedule condition-based maintenance actions, it is necessary to model the degradation process and predict the system remaining useful life (RUL). In practice, the operating environments and loading conditions are dynamic and stochastic, which introduces extra randomness in the degradation process. To account for the dynamic environmental impacts, this article proposes a nonlinear Wiener process model with a random time-varying covariate. We model the dynamic covariate with an Ornstein–Uhlenbeck process, and link it to the time-varying degradation rate by an exponential form covariate-effect function. Therefore, the proposed Wiener process has a time-varying degradation rate impacted by the dynamic environment. We propose to estimate the model parameters by maximum likelihood estimation based on given observations of the degradation and covariate. We also propose a simulation-based procedure to obtain the RUL for the proposed model. The model is verified by Monte Carlo simulations. We apply the proposed model to a dust-cleaning fan degradation dataset and a lithium-ion battery degradation dataset, and the results show that the proposed model outperforms the existing degradation models in fitting the real data and RUL prediction.

The rescue behavior and information dissemination behavior of entrepreneurs affect the number of infected enterprises and thus the credit risk contagion. We construct an epidemic model considering entrepreneurial behavior. A double -layer network consisting of entrepreneur social network and enterprise network is built based on the small world network. Through theoretical analysis of the threshold and simulation of the model, the impact of entrepreneurs’ rescue behavior and information dissemination behavior on credit risk contagion is studied. The results are as follows. Both the rescue and the information dissemination behavior of entrepreneurs are conducive to restraining the contagion of credit risk, while the rescue behavior has a better influence on controlling the contagion. In addition, the tendency of rescue objectives will affect the credit risk contagion. Moreover, improving the social rate among entrepreneurs is conducive to curbing the credit risk contagion. Last but not least, reducing the similarity between entrepreneur social network and enterprise network is of great significance to control the credit risk contagion.

This paper proposes an integrated preventive maintenance and economic production quantity model. A condition-based maintenance policy is described by a random coefficient regression model, based on which the monitored condition is divided into two parts: the actual condition and random error. Products are produced in batches and the system is monitored at the end of each batch. If the observed system condition either reaches or exceeds the critical level, the system should be renewed by preventive maintenance. However, if the actual system condition reaches the failure level during the production process, the system fails and should be renewed immediately. Based on these two renewal situations, we construct a model of expected cost per unit time using the renewal reward theory. The critical level and production lot size are decision variables, which can be obtained by minimizing the cost model. We also develop a simulation process to obtain the optimal results in another way and validate our proposed cost model. Finally, a real case study is given to demonstrate the model and the simulation process.