Delayed Renewal Process with Uncertain Random Inter-Arrival Times

An uncertain random variable is a tool used to research indeterminacy quantities involving randomness and uncertainty. The concepts of an ’uncertain random process’ and an ’uncertain random renewal process’ have been proposed in order to model the evolution of an uncertain random phenomena. This paper designs a new uncertain random process, called the uncertain random delayed renewal process. It is a special type of uncertain random renewal process, in which the first arrival interval is different from the subsequent arrival interval. We discuss the chance distribution of the uncertain random delayed renewal process. Furthermore, an uncertain random delay renewal theorem is derived, and the chance distribution limit of long-term expected renewal rate of the uncertain random delay renewal system is proved. Then its average uncertain random delay renewal rate is obtained, and it is proved that it is convergent in the chance distribution. Finally, we provide several examples to illustrate the consistency with the existing conclusions.

Cross Entropy Chance Distribution Model of Uncertain Random Shortest Path Problem

The shortest path problem (SPP) is one of the most typical and basic optimization problems in network theory for decades, and it covers a series of practical application problems, such as urban planning, logistics transportation, engineering and power grid strain analysis, etc. The circumstance where the weight of arcs in a network contains both randomness and uncertainty is considered, and the case of the weights of arcs with uncertain random variables is focused on in this paper. Here, we introduced a new model of the SPP which is based on the new definition of uncertain random variables cross entropy, and the newly established model can be used to find the path with the closest chance distribution to the ideal SP. The efficiency of this model is also evaluated in the final part.

The Scalar Mean Chance and Expected Value of Regular Bifuzzy Variables

As a natural extension of the fuzzy variable, a bifuzzy variable is defined as a mapping from a credibility space to the collection of fuzzy variables, which is an appropriate tool to model the two-fold fuzzy phenomena. In order to enrich its theoretical foundation, this paper explores some important measures for regular bifuzzy variables, the most commonly used type of bifuzzy variables. Firstly, we introduce the regular bifuzzy variables’ mean chance measure and some properties, including self-duality and its calculation formulas. Furthermore, we also investigate the mean chance distribution for strictly monotone functions of regular bifuzzy variables based on the proposed operational law. Finally, we present the expected value operator as well as equivalent analytical formulas of the expected value of regular bifuzzy variables and their strictly monotone functions.

A simulation algorithm with uncertain random variables

In many situations, uncertainty and randomness concurrently occur in a system. Thus this paper presents a new concept for uncertain random variable. Also, a simulation algorithm based on uncertain random variables is presented to approximate the chance distribution using pessimistic value and optimistic value. An example is also given to illustrate how to use the presented simulation algorithm.

Entropy of Uncertain Random Variables wi h Application to Minimum Spanning Tree Problem

Entropy is a measure of the uncertainty associated with a variable whose value cannot be exactly predicted. Based on the notion of chance measure, a concept of uncertain random entropy is introduced and used to provide a quantitative measurement of the uncertainty associated with uncertain random variables and its properties are studied in this paper. Relative entropy is a measure of the difference between two distribution functions. In order to deal with the divergence of uncertain random variables via chance distributions, this paper proposes also the relative entropy for uncertain random variables, as well as it investigates some mathematical properties of this concept. As an application, a model is presented to formulate a minimum spanning tree problem with uncertain random edge weights involving a relative entropy chance distribution. Finally, a numerical example of an uncertain random network is put forward to illustrate the effectiveness of the proposed model.