## 2 Problem Definition
### Problem Modeling
#### 2.2.5 Constraints  
The constraints corresponding to the objective function are defined in Equation (4) through (8):  
\[n_{i}^{min}\leq \sum_{j\in I}\sum_{k\in V_{ij}}x_{ijk}\leq n_{i}^{max}\quad\text{ for }i\in I \tag{4}\] \[\sum_{i\in I}\sum_{j\in I}\sum_{k\in V_{ij}}\tau_{ijk}^{o}x_{ijk} \leq\sum_{j\in I}d_{j}\] (5) \[t_{jl}^{s}\leq\tau_{ijk}^{s}\quad\text{and}\quad\tau_{ijk}^{e} \leq t_{jl}^{e}\quad\text{for }i\in I,\quad j\in J,\quad k\in V_{ij},\quad l\in T_{j}\] (6) \[\tau_{ijk}^{s}+\tau_{ijk}^{o}\leq t_{jl}^{e}\quad\text{and}\quad t_{jl}^{e}=t _{jl+1}^{s}\quad\text{for }i\in I,\quad j\in J,\quad k\in V_{ij},\quad l\in T_{j}\] (7) \[\tau_{ijk}^{e}+\tau_{j}^{g}\leq\tau_{ijk}^{s}\quad\text{for }i\
eq i^{\prime},\quad i,\,i^{\prime}\in I,\quad j\in J,\quad k\in V_{ij}, \quad l\in T_{j} \tag{8}\]  
Equation (4) represents the constraint regarding the minimum and maximum number of times that a target is observed. This constraint is modeled within a range to  
Figure 4: Example of VTW count (case: single satellite and 1000 targets during 7 days).  
accommodate multiple imaging of the same target as per user requirements, and it is not set as a constant to allow for future model scalability. Equation (5) pertains to the maximum observation time available for a satellite during one pass over the mission area. Various limitations of the satellite, such as power, thermal balance, and memory capacity, identified from the satellite development stage, are integrated into a single variable termed \"duty time \((d_{j})\)\". Therefore, this constraint ensures that the actual imaging mission time of the satellite does not exceed this duty time. Equation (6) is a constraint necessary for the application of the MDP optimization algorithm used in this study. It relates to each time interval in the sub-problem formation, indicating that the start time of the time interval should be set before the imaging starts, and the end time of the interval after the imaging ends. Equation (7), also related to the MDP algorithm, expresses the constraint that the imaging mission must be completed within each respective time interval. These time intervals serve as a fundamental criterion for segmenting the total targets within the mission area into several grouped segments. This segmentation considers the VTW, observation time, and gap time, ensuring that the end of the \(l\)th interval aligns seamlessly with the start of the \((l+1)\)th interval. Such alignment is critical for the effective implementation of dynamic programming, which tackles the larger problem by sequentially addressing these interconnected sub-problems, each defined by its distinct segment. Equation (8) addresses the requirement for a guaranteed gap time between consecutive imaging targets. This is a critical condition, especially for multi-imaging missions using active phased array SAR sensors, and is essential in determining the next target post-imaging of the current one. Figure 5 illustrates the constraints mentioned above, including the VTW, observation duration time, gap time, and time intervals. It visualizes various conditions required during consecutive imaging missions. In Figure 5, satellite \(j\) selects and images the \(k\)th VTW of target \(i\), \(i+2\), and \(i+4\), ensuring an appropriate gap time between the targets, as depicted schematically.