## 4 Multi-Objective Scheduling Optimization Model
### Multi-Objective Model Solution
#### 4.2.2 Weight Calculation  
Rough sets can be used to process incomplete and inaccurate data by learning, induction, and excavation and can be transformed into relatively clear and concise data systems to support decision analyses. The detailed theory about the rough set theory is detailed in the literature [26]. When the rough set theory is used to calculate the weight, the weight problem is transformed into a multi-attributed evaluation problem. Correspondingly, the knowledge dependency and attribute importance evaluation methods were constructed for the weight calculations. The detailed process is as follows:  
(1) Construction of the relationship data model  
It is assumed that the weight coefficient of \(f_{i}\) is \(1/\mathrm{I}\). Then, the value of the integrated single objective \(\widetilde{F}\) can be calculated. \(\widetilde{F}\) is regarded as a decision attribute, and \(D=\left\{\widetilde{F}\right\}\) is regarded as a decision attribute set. \(u_{k}\) is the optimal value of the comprehensive objective, which is an information point of the study object \(F\). Obviously, \(u_{j}=(f_{1j},f_{2j},\ldots,f_{mj};\widetilde{F}_{j})\). \(U=\left\{u_{1},u_{2},\ldots,u_{j}\right\}\) is the main domain, namely, the sample set. Therefore, the attributes of \(u_{k}\) are \(f_{i}(u_{j})=v_{ij}\) and \(F_{i}(u_{j})=\widetilde{F}_{j}\). It is assumed that \(R_{V}\) and \(R_{\mathrm{D}}\) are knowledge bases.  
(2) Calculation of the dependence degree of \(R_{V}\) on \(R_{\mathrm{D}}\), that is,  
\[r_{R_{V}}(R_{\mathrm{D}})=\frac{\sum\rho[R_{V}([\widetilde{F}]_{R_{\mathrm{D} }})]}{\rho(U)} \tag{36}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Functions** & \(F_{1}\) & \(F_{2}\) & \(\ldots\) & \(F_{1}\) \\ \hline \(F_{1}\) & \(F_{11}^{*}\) & \(F_{12}\) & \(\ldots\) & \(F_{\mathrm{II}}\) \\ \(F_{2}\) & \(F_{21}\) & \(F_{22}^{*}\) & & \(F_{\mathrm{2I}}\) \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \(F_{I}\) & \(F_{\mathrm{II}}\) & \(F_{\mathrm{I2}}\) & \(\ldots\) & \(F_{\mathrm{II}}^{*}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The payoff table of the proposed multi-objective model.  
where \(\rho(\cdot)\) represents the set base, and \(p_{v}(D)\) is the knowledge of all individuals using the classification \(U/C\) in the universe \(U\), which can correctly classify the equivalence classes of object sets in the \(U/D\).  
(3) Calculation of the dependence degree of \(R_{V-|v_{i}|}\) on \(R_{\rm D}\), that is,  
\[r_{R_{V-|v_{i}|}}(R_{\rm D})=\frac{\sum\rho[R_{V-|v_{i}|}([\bar{\bar{F}}]_{R_{ \rm D}})]}{\rho(U)} \tag{37}\]  
where \(p_{V-|v_{i}|}(D)\) is the knowledge of all individuals using the classification \(U/C\) in the universe \(U\) after removing the indicator \(V_{i}\).  
(4) Calculation of the weight value of objective \(i\), namely,  
\[\sigma_{\rm D}(D)=r_{R_{V}}(D)-r_{R_{V-|v_{i}|}}(D) \tag{38}\]  
\[\lambda_{i}=\sigma_{\rm D}(v_{i})/\sum_{i=1}^{I}\sigma_{\rm D}(v_{i}) \tag{39}\]  
where \(\sigma_{\rm D}(D)\) calculates the importance of objective function \(I\). \(\lambda_{i}\) is the weight of objective \(i\).