## 2 Problem Formulation
### Dynamic Pricing Model  
Based on practical considerations, the equipment repaired on the cloud manufacturing platform is generally a more general intelligent device. Because the maintenance given by the cloud manufacturing platform is relatively standardized maintenance, the EMRS problem considered in this paper is based on equipment maintenance problems that can allocate maintenance resources through a cloud manufacturing platform. After the maintenance equipment breaks down, the cloud manufacturing platform charges for each maintenance task submitted. In actual maintenance scenarios, different users have different requirements for the completion time and price of different tasks. Therefore, this article comprehensively considers the interests of both cloud manufacturing platforms and users and designs a dynamic pricing model. This model can provide a transformation relationship between maintenance cost, price, and maintenance time. Especially for some users that require high completion times, by utilizing high-quality resources to minimize task completion time, users can accept higher prices. On the other hand, for tasks with limited costs, cloud platforms can utilize relatively low-quality resources to reduce maintenance costs, but the maintenance time will be extended. In other words, by using a pricing model, the cost can be reduced to a level that users believe is affordable. Maintenance resources in the cloud manufacturing platform are divided into two types: physical resources and virtual resources. Physical resources include maintenance accessories and equipment, the cost of which is generally fixed. Virtual resources include personnel skills, information, and so on. In general, the higher the resource allocation, the better the maintenance technology, the shorter the maintenance completion time, and the higher the maintenance price. Therefore,we adopt the virtual resource benefit to adjust the maintenance price in this paper. The following definitions are made:  
**Definition 7**. : _The average time taken for a single resource to complete the jth maintenance task is defined as_  
\[avgt_{j}\ =\ \frac{1}{n}\sum_{i\ =\ 1}^{n}t_{ij}, \tag{5}\]  
_where \(n\) indicates the number of resource groups, \(i\) indicates the number of maintenance resources, and \(j\) indicates the number of tasks._  
It is obvious that there is a negative correlation between the price of virtual resources and completion time, and the price goes down as time goes by. It should be noted that the longer maintenance time here is due to the insufficient efficiency of the maintenance resources allocated, but the price will not fall without limit. Symmetric sigmoid functions exhibit these properties well. In this paper, a dynamic pricing model based on sigmoid function is designed. The price required by resource \(i\) to repair equipment \(j\) is as follows:  
\[p_{ij}\ =\ c_{ij}+\rho_{ij}\times avgt_{j}\times(1-\frac{1}{1+e^{-(t_{ij}- avgt_{j})}}), \tag{6}\]  
where \(c_{ij}\) is the cost required to repair the \(j\)th item of equipment for the \(i\)th resource, and \(\rho_{ij}\) is the added value of the virtual resource cost per unit of maintenance time. The sigmoid function curve is shown in Figure 1.  
This curve is a center symmetric curve that effectively simulates the relationship between user price expectations and service quality. Equation (6) can express well the constraints and rules of the pricing model. The longer the completion time of the allocation, the lower the price. However, as the completion time of the allocation increases, the magnitude of the price decrease decreases until it approaches a constant. This demonstrates the rationality of the pricing model and ensures the benefits of cloud manufacturing platforms. The shorter the time the user expects to complete the task, the higher the price. The parameters of the resource pricing model in this paper are determined based on the general market conditions.