## 3 Solving Method
### Two-Phase Hybrid Optimization  
According to the MDP in this study, the decision process of the MAEOSP is decomposed into a task sequence decision and a resource allocation decision. We propose a two-phase hybrid optimization that includes an encoder-decoder network for task sequence decision and a heuristic algorithm for resource allocation decision. Since DRL has demonstrated a strong ability in sequential decision-making, we propose an encoder-decoder network to make decision in the task sequence. Figure 3 shows the architecture of the proposed encoder-decoder network. The encoder architecture comprises three parallel neural network layers, each specialized for encoding static elements, dynamic elements, and scheduling sequences. The decoder is structured as a stacked series of fully connected layers integrated with activation functions and incorporates the softmax-based normalization to probabilistically constrain the output space. Given their distinct characteristics, static and dynamic elements undergo separate feature encoding through individual fully connected layers. To provide scheduling sequences to the encoder-decoder network, the GRU receives historical static elements from previously scheduled tasks. \(X=transpose([x_{1},\ x_{2},\cdots,x_{M}])\) is the matrix of the static elements, \(D_{t}=transpose([d_{1}^{t},\ d_{2}^{t},\cdots,d_{M}^{t}])\) is the matrix of the dynamic elements, \(a_{t-1}\) is the task action selected in the last time step, \(t-1\), and \(x(a_{t-1})\) is the \(a_{t-1}\)-th element in \(X\). According to Equations (10) and (11), the dimension of \(X\) is \((2,M)\), the dimension of \(D_{t}\) is \((5,M)\), and the dimension of \(x(a_{t-1})\) is \((2,1)\). \(X\), \(D_{t}\), and \(x(a_{t-1})\) are transposed before encoding, and after encoding, their dimensions are \((M,hd)\), \((M,\ hd)\), and \((1,\ hd)\), respectively. \(hd\) is the hidden dimensions of the fully connected layers and the GRU. We use a repeat operation to expand the dimension of encoded \(x(a_{t-1})\) from \((1,\ hd)\) to \((M,\ hd)\). Then, the encoded vectors are concatenated column-wise. In order not to lose information, \(X\) and \(D_{t}\) are encoded by fully connected networks. To save sequence information, \(x(a_{t-1})\) is encoded by a one-dimensional convolutional network and GRU. The encoded inputs are concatenated as illustrated in Figure 3, after three linear layers and two activation layers, the value of each task action is obtained. The values are mapped to \((0,1)\) by applying the Softmax function, and then the probability of each task action is obtained, based on which the task action selection is determined via sampling. All neural network components, including fully connected layers, convolutional layers, GRUs, activation functions, and the Softmax layer are implemented using PyTorch 2.5.1. The specific implementation commands and corresponding parameters for each network layer are detailed in Table A1 in the Appendix A.  
Figure 3: The proposed encoder-decoder network architecture. \(Mask_{t}\) is a mask matrix used to mask unfeasible task actions. \(Mask_{t}=\{m_{i}|i=1,\ldots,M\}\), \(m_{i}=1\) if task \(i\) is feasible at time step \(t\) and has not been scheduled, and \(m_{i}=0\) otherwise. The probability of an unfeasible task is modified to zero by the mask matrix. Task \(i\) is feasible if least one satellite has feasible time windows for task \(i\) and the remaining capacity of the satellite is no less than the storage consumed by task \(i\). The task action is determined by the proposed encoder-decoder network, and then the resource action and beginning time decisions for task execution are needed to update the state. When allocating a satellite to observe the determined task, we consider decreasing the impact of resource occupation by the determined task on the feasibility of other unscheduled tasks. We propose a heuristic rule denoted as MRC that indicates a feasible satellite with maximum residual capacity is allocated to the determined task. The MRC rule is illustrated in Algorithm 1.