## 2 Problem Formulation  
### Problem Formulation with the Extended E-CARGO Model  
The factors related to EMRS in cloud manufacturing are complex and various. It is difficult to describe the characteristics of various elements and the relationships between them at the same time when using conventional modeling methods. In this paper, the E-CARGO model [37] is used to formally model the factors to be considered in the EMRS problem, which is abstracted as a role coordination problem in order to provide model support for better resource scheduling. Therefore, a formal description of EMRS in cloud manufacturing based on the E-CARGO model is proposed in this paper. When equipment fails, the maintenance demand is transmitted to the cloud manufacturing platform, which must then urgently allocate resources to complete equipment maintenance and ensure that the equipment is put back into production. When multiple equipment maintenance tasks arrive at the cloud manufacturing platform at the same time, how to distribute maintenance resources is an important issue that the cloud manufacturing platform needs to resolve. According to the research needs of this paper, the basic definition of the E-CARGO model is simplified and improved. Some elements that do not directly affect the research content of this paper are not described here, while related elements are expanded according to the research background of this paper. **Definition 1**. : _Roles are used to represent equipment maintenance tasks; these are defined as \(R\). Assuming that a batch of faulty equipment generates \(m\) maintenance tasks, then \(R~{}=~{}\{r_{1},r_{2}\ldots,r_{m}\}\)._  
**Definition 2**. : _Agents are used to represent maintenance resources, denoted by \(H\). For a maintenance task, \(n\) resource groups can be configured to complete the task, and the maintenance resource set is \(H~{}=~{}\{h_{1},h_{2}\ldots,h_{n}\}\)._  
**Definition 3**. : _The allocation matrix is denoted by \(T\), which represents the matching relationships between resource groups and maintenance tasks. The matrix values are represented by \(t_{ij}\). For an equipment maintenance task with \(m\) maintenance tasks and \(n\) resource groups, \(T\) should be a matrix with \(n\) rows and \(m\) columns. The elements of the matrix take a value of 0 or 1, where \(t_{ij}\) = 0 indicates that resource group \(h_{i}\) is not assigned to maintenance task \(r_{j}\), while \(t_{ij}\) = 1 indicates that resource group \(h_{i}\) is assigned to the maintenance task \(r_{j}\). The distribution matrix \(T\) can be represented as follows:_  
\[T~{}=~{} \begin{bmatrix}t_{11}&\cdots&t_{1m}\\ \vdots&\ddots&\vdots\\ t_{n1}&\cdots&t_{nm}\end{bmatrix} .\]  
**Definition 4**. : _The evaluation matrix \(Q\) represents the benefit of completing a maintenance task for each resource group. For \(m\) maintenance tasks and \(n\) resource groups, \(Q\) should be a matrix with \(n\) rows and \(m\) columns, which values are expressed as \(Q_{ij}\). According to the actual condition of the equipment maintenance, all benefit values are non-negative, and a value of 0 indicates that the resource group cannot be used for the corresponding maintenance task. The larger the value, the better the benefit of the task calling for the corresponding resources. The evaluation matrix \(Q\) can be expressed as follows:_  
\[Q~{}=~{} \begin{bmatrix}q_{11}&\cdots&q_{1m}\\ \vdots&\ddots&\vdots\\ q_{n1}&\cdots&q_{nm}\end{bmatrix} .\]  
For a set of maintenance tasks, there may be different evaluation matrices, according to different types of maintenance benefits. Users are mainly concerned about the completion time and price of maintenance, while the cloud manufacturing platform is mainly concerned about the cost and completion time. Therefore, we expand the evaluation matrix into time, cost, and price evaluation matrices. When there are multiple evaluation matrices, in order to ensure that the dimensionality of each matrix is consistent, the value of each column is normalized.