## 3 Problem Definition and Formulation
### Problem Formulation and Objective Function  
This section formulates the resource allocation and task scheduling problem as a multi-objective optimization problem in satellite networks. The goal is to maximize the total resource allocation of tasks across satellite nodes to enhance the network throughput, ensuring efficient data transmission within the network; simultaneously, by optimizing resource allocation and path selection, minimize task transmission latency to ensure timely completion of tasks; and reduce computational complexity by controlling the extent of resource adjustments, avoiding excessive computational burdens, and thereby improving system responsiveness, stability, and real-time performance. The following explains each part of the objective function and the meaning of the variables. Network throughput represents the total amount of data successfully transmitted within the network, which depends on the resources allocated to each task at each satellite node. Let \(t_{ij}\) denote the resources allocated to task \(j\) at satellite node \(i\). The network throughput is expressed as \(\sum_{i=1}^{n}\sum_{j=1}^{m}t_{ij}\), where \(n\) is the number of satellite nodes, and \(m\) is the number of tasks. By maximizing this sum, the optimization goal is to increase the total data transmission, thereby improving the network throughput. Computational complexity measures the computational load during the resource allocation process. Frequent adjustments in resource allocation lead to a higher computational complexity. This is quantified by the difference between the original resource allocation \(t_{ij}\) and the adjusted allocation \(t^{\prime}_{ij}\), expressed as \(\sum_{i=1}^{n}\sum_{j=1}^{m}|t_{ij}-t^{\prime}_{ij}|\). The total transmission latency \(D\) for all tasks should be minimized, where \(d_{j}\) represents the transmission latency for task \(j\). The total latency can be formulated as \(\sum_{j=1}^{m}d_{j}\). The optimization problem aims to maximize network throughput, while minimizing both computational complexity and transmission latency. This is achieved by optimizing resource allocation and task scheduling to enhance the overall transmission capacity, reduce the computational load, and ensure timely task completion. The objective function is formulated as  
\[\max_{\mathbf{x}}\ f=\alpha \cdot\sum_{i=1}^{n}\sum_{j=1}^{m}t_{ij}-\gamma\cdot\sum_{i=1}^{n} \sum_{j=1}^{m}|t_{ij}-t^{\prime}_{ij}|-\beta\cdot\sum_{j=1}^{m}d_{j}\] (10) s.t. \[\sum_{j=1}^{m}t_{ij}\leq R_{i},\quad\forall i=1,2,\ldots,n\] \[\sum_{i=1}^{n}t_{ij}=D_{j},\quad\forall j=1,2,\ldots,m\] \[d_{j}\leq D_{\text{max}},\quad\forall j=1,2,\ldots,m\] \[|t_{ij}-t^{\prime}_{ij}|\leq C_{\text{max}},\quad\forall i=1,2, \ldots,n,\forall j=1,2,\ldots,m\] \[P_{j}\in\mathcal{P},\quad\forall j=1,2,\ldots,m\] \[\text{Path(s_{j},t_{j})}\in\mathcal{G},\quad\forall j=1,2,\ldots,m\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters Name** & **Description** \\ \hline \(\mathcal{P}\) & Set of available paths. \\ \(\mathcal{G}\) & Network graph, representing the connectivity between the source and destination nodes. \\ \(\alpha\) & Weighting factor for throughput in the objective function. \\ \(\beta\) & Weighting factor for latency in the objective function. \\ \(\gamma\) & Weighting factor for computational complexity in the objective function. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._\[t_{ij}\leq\sum_{k=1}^{K_{j}}x_{jk}\cdot R_{i}\cdot\mathbb{I}(i\in P_{jk}),\quad \forall i\in\{1,...,n\},\forall j\in\{1,...,m\}\]  
where \(\alpha\), \(\beta\), and \(\gamma\) are weighting factors balancing the importance of throughput, latency, and computational complexity, respectively. \(\mathbf{X}\) represents all combinations of resource allocation, task scheduling, and path selection that satisfy the constraints.