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.

A Novel Probability of Detection Assessment Considering Model Uncertainty for Lamb Wave Detection

Uncertainty in Non-Destructive Evaluation (NDE) arises from many sources, e.g., manufacturing variability, environmental noise, and inadequate measurement devices. The reliability of the NDE measurements is typically quantified by the probability of detection (POD). With the advent and technical developments of the simulation method and computer science, efforts have been devoted to generating and estimating the POD curve for Lamb wave damage detection. However, few studies have been reported on the POD evaluation considering model selection uncertainty. This paper presents a novel POD assessment method incorporating model selection uncertainty for Lamb wave damage detection. By treating the flaw quantification model as a discrete uncertain variable, a hierarchical probabilistic model for Lamb wave POD is formulated in the Bayesian framework. Uncertainties from the model choice, model parameters, and other variables can be explicitly incorporated using the proposed method. The Bayes factor is used to evaluate the performance of models. The posterior distributions of model parameters and the model fusion results are calculated through the Bayesian update using the reversible jump Markov chain Monte Carlo method. A fatigue problem with naturally developed cracks is used to demonstrate the proposed method.

Incentive Contract Design for Supply Chain Enterprise’s Pollution Abatement with Carbon Tax

This paper applies mechanism design to the supply chain enterprise’s pollution abatement problem with carbon tax. To maximize the government’s expected utility, an uncertain contract model is presented in the framework of principal-agent theory, where the government’s assessment of the supply chain enterprise’s carbon emission level is described as an uncertain variable. Afterwards, the equivalent model is provided to obtain the optimal contract for the uncertain pollution abatement problem. The results demonstrate that the supply chain enterprise’s optimal output decreases with the carbon emission level. Furthermore, the government’s optimal transfer payment decreases with the carbon emission level if the carbon tax is low. In contrast, if the carbon tax is high, the optimal transfer payment increases with the carbon emission level. In addition, an increase in the carbon emission level decreases the optimal utilities of both the government and the supply chain enterprise and also leads to the supply chain enterprise’s incremental marginal utility. Finally, we provide a numerical example, which illustrates the effectiveness and practicability of the proposed model.

Finite time stability and sliding mode control for uncertain variable fractional order nonlinear systems

This paper deals with the finite time stability and control for a class of uncertain variable fractional order nonlinear systems. The variable fractional Lyapunov direct method is developed to provide the basis for the stability proof of the system considered. The sliding mode control method is applied for robust control of uncertain variable fractional order systems; furthermore, the chattering phenomenon is avoided. And the finite time stability of the systems under control law is proved based on the proposed stability criterion. Finally, numerical simulations are proposed and the efficiency of the controller is verified.

A new measure of indeterminacy for uncertain variables with application to portfolio selection

Entropy is a measure for characterizing indeterminacy of a random variable or an uncertain variable with respect to probability theory and uncertainty theory, respectively. In order to characterize indeterminacy of uncertain variables, the concept of exponential entropy for uncertain variables is proposed. For computing the exponential entropy for uncertain variables, a formula is derived via inverse uncertainty distribution. As an application of exponential entropy, portfolio selection problems for uncertain returns are optimized via exponential entropy-mean models. For better understanding, several examples are provided.

Almost convergence of complex uncertain double sequences

Convergence of real sequences, as well as complex sequences are studied by B. Liu and X. Chen respectively in uncertain environment. In this treatise, we extend the study of almost convergence by introducing double sequences of complex uncertain variable. Almost convergence with respect to almost surely, mean, measure, distribution and uniformly almost surely are presented and interrelationships among them are studied and depicted in the form of a diagram. We also define almost Cauchy sequence in the same format and establish some results. Conventionally we have, every convergent sequence is a Cauchy sequence and the converse case is not true in general. But taking complex uncertain variable in a double sequence, we find that a complex uncertain double sequence is a almost Cauchy sequence if and only if it is almost convergent. Some suitable examples and counter examples are properly placed to make the paper self sufficient.

Maximal covering location problem of smart recycling infrastructure for recyclable waste in an uncertain environment

To incentivise citizens to return recyclable waste (RW) through formal recycling channels, the Chinese government has introduced smart recycling facilities for RW. To maximise the satisfaction of citizens’ demand for returning RW, the proper arrangement of recyclable facilities is a problem worth studying in cases in which there is a lack of data on the citizens’ behaviour. Thus, to help government decision-makers rationally arrange smart recycling facilities, this paper discusses how to properly allocate the recycling infrastructure to cover the maximum demand with a limited number of smart recycling facilities in an uncertain environment. According to the uncertainty theory, the service cost is taken as an uncertain variable. Then, an uncertain maximal covering location problem model and an extended uncertain maximal covering location problem model are constructed, and solution procedures of the two models are provided. The effectiveness of the proposed mathematical formulations and solution procedures were validated through two application cases.

Digital twin-based production scheduling system for heavy truck frame shop

Existing scheduling systems cannot respond quickly and efficiently to the appearance of uncertain variable interference factors in the scheduling process. The efficiency of the enterprise is hence seriously affected by these factors. The concept of the digital twin provides new research directions for scheduling systems. In order to improve the overall performance of the production scheduling system in a frame shop, virtual-real fusion technology utilizing the digital twin is first introduced to integrate the information and logistics flows in the scheduling process as part of the manufacturing execution system-based production scheduling mechanism. The balance optimization of the mixed-flow production line is then carried out, and the mixed-flow production line balance and sequencing models are established. A scheduling system for the frame shop combining various system functions is finally designed based on the sequencing model and production status. The scheduling system comprehensively considers the two uncertain factors of materials and orders to generate scheduling information through an intelligent scheduling mode while realizing real-time synergy between the information and logistics flows. The system has achieved excellent implementation results in the frame shop and provides a reference for the application of digital twins in production workshops.