## 2 Problem Formulation
### Resource Allocation Model Based on Platform and User Satisfaction  
In fact, different users have different requirements for the completion time and price of different tasks. For some tasks, users are more concerned about completion time and offer higher prices for this reason. For other tasks, users have limited funds and relatively low requirements for the completion time. However, other users are expected to have lower prices due to limited funds. To address the above issues, this article introduces user expectations for the time and cost of completing maintenance tasks, as well as cloud manufacturing platforms' expectations for the time and cost of completing tasks. The cloud manufacturing platform can use the dynamic pricing model mentioned above to control the relationship between cost, price, and time for maintenance tasks. The cloud manufacturing platform can provide the relationship between the completion time and price of maintenance tasks for users to choose from. The cloud manufacturing platform indicates the uptime and price benchmarks in order to help users to form judgments. In multi-objective programming problems, there are often strong conflicts between multiple objectives, and the optimal solution regarding a given objective may have poor performance in other objectives. In order to take into account the benefits of multiple objectives, the scheme decision maker can give the expected value for the individual objective function in combination with the specific planning scenario, in order to effectively reduce conflict between objectives. At the same time, the scheme decision maker should determine the limit of the results for each objective function. Based on [PERSON]'s fuzzy theory [30], a fuzzy objective is a decision-making objective combining tolerance and expectation. In this paper, 0 means \"completely unacceptable\", 1 means \"most satisfactory\", and other satisfaction degrees are expressed by numbers in the range [0, 1]. The difference between the actual completion and expected results of maintenance tasks is used to indicate satisfaction. The trapezoidal membership function is adopted in this paper, as shown below:  
\[\mu\ =\ \left\{ \begin{array}{l}0,\ \sigma<a\\ \\ \frac{\sigma-a}{b-a},\ a\leq\sigma\leq b\\ \\ 1,\ \sigma>b\end{array} \right., \tag{12}\]  
where \(a\) and \(b\) denote the user's tolerance range for the maintenance effect. As a positive evaluation is correlated with user and cloud manufacturing platform expectations, a value higher than \(b\) is considered to indicate complete satisfaction, while a value lower than \(a\) is considered to indicate complete dissatisfaction. \(\mu\) represents the satisfaction value, and \(\sigma\) represents the actual maintenance effect value. The membership curve is shown in Figure 2.  
Figure 2: Example membership function curve for the user decision satisfaction degree. Then, the multi-objective EMRS problem can be transformed into a maximum satisfaction problem based on the theoretical decision satisfaction model. Platform services are expected to improve the time and cost-effectiveness of resource use, so we propose the concept of platform satisfaction. Equation (7) is extended as follows:  
\[\text{max}F_{p}\ =\ w_{t}\times\mu_{ft}+w_{c}\times\mu_{fc}, \tag{13}\]  
subject to  
\[\mu_{ft}\ =\ \left\{ \begin{array}{l}0,\ \sigma_{t}<a_{ft}\\ \\ \frac{\sigma_{t}-a_{ft}}{b_{ft}-a_{ft}},\ a_{ft}\leq\sigma_{t}\leq b_{ft}\\ \\ 1,\ \sigma_{t}>b_{ft}\end{array} \right., \tag{14}\]  
\[\mu_{fc}\ =\ \left\{ \begin{array}{l}0,\ \sigma_{c}<a_{fc}\\ \\ \frac{\sigma_{c}-a_{fc}}{b_{fc}-a_{fc}},\ a_{fc}\leq\sigma_{c}\leq b_{fc}\\ \\ 1,\ \sigma_{c}>b_{f_{c}}\end{array} \right., \tag{15}\]  
where \(F_{p}\) denotes platform satisfaction, \(a_{ft}\) and \(b_{ft}\) are the lowest and highest platform expectations of the time benefit evaluation, respectively, and \(a_{fc}\) and \(b_{fc}\) are the lowest and highest platform expectations of the cost benefit evaluation, respectively.