A Multi-Objective Dynamic Mission-Scheduling Algorithm Considering Perturbations for Earth Observation Satellites

Abstract

The number of real-time dynamic satellite observation missions has been rapidly increasing recently, while little attention has been paid to the dynamic mission-scheduling problem. It is crucial to reduce perturbations to the initial scheduling plan for the dynamic mission-scheduling as the perturbations have a significant impact on the stability of the Earth observation satellites (EOSs). In this paper, we focus on the EOS dynamic mission-scheduling problem, where the observation profit and perturbation are considered simultaneously. A multi-objective dynamic mission-scheduling mathematical model is first formulated. Then, we propose a multi-objective dynamic mission-scheduling algorithm (MODMSA) based on the improved Strength Pareto Evolutionary Algorithm (SPEA2). In the MODMSA, a novel two-stage individual representation, a minimum perturbation random initialization, multi-point crossover, and greedy mutation are designed to expand the search scope and improve the search efficiency. In addition, a profit-oriented local search algorithm is introduced into the SPEA2 to improve the convergence speed. Furthermore, an adaptive perturbation control strategy is adopted to improve the diversity of non−dominated solutions. Extensive experiments are conducted to evaluate the performance of the MODMSA. The simulation results show that the MODMSA outperforms other comparison algorithms in terms of solution quality and diversity, which demonstrates that the MODMSA is promising for practical EOS systems.

1. Introduction

Earth observation satellites (EOSs) are adopted to take photographs of Earth’s surface through their optical and infrared cameras, which are widely used in weather forecasting, environmental protection, and disaster monitoring. Although a large number of EOSs have been launched in recent years, the current EOSs are still struggle to meet the explosively increased observation demands [1,2,3]. Therefore, the EOS mission-scheduling, which allocates scarce EOS resources to observation missions, plays a significant role in improving the resource utilization efficiency of the EOS systems.

Over the past few decades, extensive research has been conducted on EOS mission-scheduling problems. To obtain an optimal observation plan, some exact algorithms such as the branch and bound algorithm have been proposed for the satellite observation-mission-scheduling problem [4,5,6,7]. As the EOS mission-scheduling problem is a complex non-deterministic polynomial time hard (NP-Hard) problem [8], these exact optimization algorithms struggle to find the optimal solution in a tolerable time in the scheduling scenarios with large-scale missions. Hence, researchers have turned to heuristic algorithms and machine learning algorithms to design the EOS mission-scheduling algorithms. The genetic algorithm (GA) [9,10,11], simulated annealing (SA) [12,13,14], and ant colony optimization (ACO) [15,16,17] have been extensively used to solve the EOS mission-scheduling problem. The local search-based mission-scheduling algorithms such as the simulated annealing algorithm and neighborhood search algorithm [18,19,20] are also popular in solving the EOS mission-scheduling problem. In addition, some learning-based EOS mission-scheduling algorithms based on deep neural networks [21,22,23] and deep reinforcement learning [24,25,26] have been proposed recently.

The existing EOS mission-scheduling algorithms mainly focus on static observation missions, which are submitted before the beginning of the scheduling period. Recently, with the expansion of observation applications such as dynamic observation requests from mobile terminals and dynamic target monitoring [27,28,29], the demand for real-time Earth observation information is urgent and enormous. These real-time observation demands, which arrive dynamically during the scheduling horizon and require being performed within a short deadline, are usually defined as dynamic observation missions [30]. Compared with static missions, the profit of dynamic observation missions is higher and dynamic missions have an urgent deadline constraint for observation [30,31,32]. In recent years, the scheduling of dynamic observation missions, referred to as dynamic scheduling, has become a hot research topic [33,34,35,36,37]. Wu et al. proposed a hybrid ACO method combined with iteration local search (ACO-ILS) for the satellite mission-scheduling problem with emergency and common missions [33]. For the distributed imaging satellite constellation, Li et al. proposed a satellite dynamic scheduling algorithm based on deep reinforcement learning [34]. In addition, a transformer-based fast scheduling algorithm [35], a hybrid genetic tabu search algorithm [36], and a two-phase mission-planning method [37] are proposed for the dynamic mission-scheduling problem.

In dynamic scheduling, the perturbation is an important indicator used to measure the difference between the scheduling plan generated by dynamic scheduling and the initial scheduling plan. Although the perturbation directly affects the stability of the EOS system, most of the existing studies on the dynamic observation-mission-scheduling problem ignore perturbations. To the best of our knowledge, only Wang et al. [38] and Yang et al. [39] have investigated the dynamic observation-mission-scheduling problem considering the perturbation. Wang et al. [38] proposed a dynamic scheduling algorithm with a mission merging strategy with the aims of the minimum perturbation, maximum satisfaction ratio, and minimum energy consumption. Yang et al. [39] established a multi-objective scheduling model for the observation-mission-scheduling problem in uncertain environments and proposed a hybrid local replanning strategy. It is worth noting that these two multi-objective dynamic scheduling algorithms [38,39] search for the optimal solution by simple heuristic operators and can only find one or a few optimal non-dominated solutions. However, in practical observational applications, multiple optimal alternative observation plans are necessary and crucial. The multiple optimal alternative solutions facilitate the decision-maker of the EOS system in selecting the most appropriate observation plan based on the current status of the EOS and the observation environment. Therefore, it is urgent and meaningful to design an effective multi-objective scheduling algorithm, which can optimize the profit and perturbation simultaneously.

It is evident that the initial observation plan is the scheduling plan with the minimum perturbation and lowest profits as there are no dynamic tasks scheduled. In addition, it is essential to take both the initial observation plan and dynamic tasks into consideration in order to achieve an observation scheme with higher profit in the mission observation scheduling. Consequently, significant perturbations are unavoidable for the scheduling plans with high profits. Therefore, in the dynamic mission scheduling considering the profit and perturbation (DMP-PP), maximizing profit and minimizing perturbation are strongly opposed to each other. However, the existing multi-objective evolutionary algorithms (MOEAs) perform poorly in addressing the DMP-PP, particularly in terms of solution diversity. The main reasons are that evolutionary operators, such as crossover and mutation, will introduce significant perturbations, and these perturbations will accumulate during the iteration process. Consequently, the non-dominated solutions obtained by existing MOEAs are mostly distributed in solution spaces with high perturbations. To address this issue, we develop a multi-objective dynamic mission-scheduling algorithm (MODMSA) based on the SPEA2. The core idea of the MODMSA is to enhance the exploration of solution space with low perturbations by searching the solution space in ascending order according to the perturbations. Specifically, the MODMSA gradually expands the maximum allowable perturbation with the iterations and retains the non-dominated solutions within the limited perturbation in the archive. In the MODMSA, a novel two-stage individual representation and a minimum perturbation random initialization are designed to reduce perturbation generated during the population initialization and decoding. The multi-point crossover and greedy mutation are improved based on the features and domain knowledge of the DMP-PP. In addition, a profit-oriented local search algorithm is introduced into the SPEA2 to further improve the quality of the solutions. Most importantly, an adaptive perturbation control strategy is presented to gradually increase the maximum allowable perturbation with the iterations. The main contributions and innovations of this paper are as follows:

• A mathematical model for the multi-objective dynamic scheduling problem is formulated to maximize the observation profit and minimize the perturbation.
• A multi-objective dynamic mission-scheduling algorithm (MODMSA) is developed for the DMP-PP. To improve the diversity and quality of non-dominated solutions, a two-stage individual representation, a minimum perturbation random initialization, multi-point crossover, and greedy mutation are designed based on the characteristic of the DMP-PP. A profit-oriented local search algorithm is proposed to improved the profits of the solutions, and an adaptive perturbation control strategy is developed to enhance the exploitation in the solution space.
• The MODMSA is evaluated by extensive simulation experiments. The results demonstrate that the MODMSA outperforms other comparison algorithms especially in the diversity of non-dominated solutions.

The remainder of this paper is oriented as follows: In Section 2, the EOS dynamic mission-scheduling problem is described and the multi-objective mission scheduling mathematical model is formulated. In Section 3, the MODMSA is proposed for the DMP-PP. The simulation experiments and discussion are introduced in Section 4 and Section 5 respectively. Finally, the conclusions are described in Section 6.

2. Problem Description

The observation mission scheduling assigns an EOS and a corresponding observation time window (OTW) to each observation mission. In the EOS system, an initial observation plan for static observation missions is firstly generated at the beginning of the scheduling horizon. Meanwhile, a large number of dynamic observation requests for real-time observation information are randomly submitted to the EOS system during the scheduling horizon. To fully utilize the resources of the EOS system to serve these dynamic missions, this paper investigates the EOS dynamic mission-scheduling problem. An example of dynamic observation mission scheduling is illustrated in Figure 1. The dynamic observation plan, represented in red, is generated for the dynamic observation missions based on the initial observation plan, which is shown in blue. We can easily find that the EOS system can execute more missions by dynamic scheduling. In addition, the profit and stability are vital for the EOS system, and the perturbation significantly affects stability of the EOS system. Thus, the observation profit and perturbation are optimized simultaneously in this paper. This section first introduces variables and assumptions for the multi-objective dynamic scheduling problem (DMP-PP) and then formulates the multi-objective dynamic mission-scheduling mathematical model.

2.1. Variables

An observation demand is regarded as an observation mission, and the set of missions is defined as $T=\left\{t_1,\cdots ,t_N\right\}$, where N is the sum of the number of static and dynamic missions. Each mission $t_i=\left\{po{s}_i,p_i,d_i,arr{T}_i,ET{W}_i\right\}\in T$ is composed of five tuples:

• $po{s}_i$: the location information of the mission including the latitude, longitude, and altitude.
• $p_i$: the profit level, as defined by the Earth observation system, serves to reflect the importance of the mission. The profit level of the dynamic mission is usually higher than that of the static mission.
• $d_i$: the required observation duration, which determines the size of the observed data.
• $arr{T}_i$: the arrival time of the mission. The arrival time of the dynamic mission is randomly distributed over the scheduling horizon, while the arrival time of the static mission is considered the start time of the scheduling horizon.
• $ET{W}_i=\left\{eo{t}_i,lo{t}_i\right\}$: the effective observation time window (ETW). $es{t}_i$ and $le{t}_i$ are the earliest effective observation time and the latest effective observation time of the mission. The observation for each mission must be completed within the ETW. Compared to the static missions, the ETW length of the dynamic missions is usually shorter because the dynamic missions have an urgent observation deadline.

The set of EOSs is denoted as $S=\left\{s_1,\cdots ,s_M\right\}$, where $s_i$ is the i-th EOS and M is the number of EOSs. The predefined scheduling horizon is divided into several orbits based on satellite orbit parameters, and the set of orbits for EOS $s_i$ is represented as $O_i=\left\{o_{i1},\cdots ,o_{i{m}_i}\right\}$, where $o_{ij}$ is the j-th orbit for EOS $s_i$ and $m_i$ is the number of orbits for EOS $s_i$.

Once the mission set T and EOS set S are determined, all visible time windows (VTWs) between missions and EOSs can be calculated. We denote the set of VTWs as $W=\left\{W_i,\cdots ,W_N\right\}$, and $W_i=\left\{w_{i1},\cdots ,w_{i{n}_i}\right\}$ is the set of VTWs of the mission $t_i$, where $w_{ij}$ is the j-th VTW of mission $t_i$ and $n_i$ is the number of VTWs of mission $t_i$. Each VTW can be defined as $w_{ij}=\left\{s_{ij},e_{ij},SI{D}_{ij},OI{D}_{ij},{\gamma }_{ij},{\pi }_{ij},{\psi }_{ji}\right\}$:

• $s{t}_{ij}$ and $e{t}_{ij}$: the start time and end time of VTW $w_{ij}$.
• $SI{D}_{ij}$ and $OI{D}_{ij}$: the identifiers of the EOS and orbit for VTW $w_{ij}$
• ${\gamma }_{ij}$, ${\pi }_{ij}$, and ${\psi }_{ji}$: the look angles for the roll, pitch, and yaw in VTW $w_{ij}$

The transition time between two adjacent observation missions is necessary for EOSs to adjust the observation angle, and the transition time between mission $t_i$ and $t_{i^{\prime }}$ is calculated by:

$$\begin{array}{c}\hfill {\mathrm{z}}_{i{i}^{\prime }}=\left\{\begin{array}{c}\begin{array}{cc}{a}_1+{\theta }_{i{i}^{\prime }}/v_1& {\theta }_{i{i}^{\prime }}\le 15\end{array}\hfill \\ \begin{array}{cc}{a}_2+{\theta }_{i{i}^{\prime }}/v_2& 15<{\theta }_{i{i}^{\prime }}\le 40\end{array}\hfill \\ \begin{array}{cc}{a}_3+{\theta }_{i{i}^{\prime }}/v_3& 40<{\theta }_{i{i}^{\prime }}\le 90\end{array}\hfill \\ \begin{array}{cc}{a}_4+{\theta }_{i{i}^{\prime }}/v_4& {\theta }_{i{i}^{\prime }}>90\end{array}\hfill \end{array}\right.\end{array}$$

(1)

where ${\theta }_{{\mathit{ii}}^{\prime }}$ is the total of the transition angle between the observation for two adjacent missions $t_i$ and $t_{i^{\prime }}$, $v_1,v_2,v_3,v_4$ are the angular velocities of attitude rotation, and $a_1,a_2,a_3,a_4$ are constant terms corresponding to different angular velocities indicating the rotation capacity of the EOS’s camera.

The perturbation $\delta$ is defined to measure the distance between a new observation plan and the initial observation plan. Similar to [38], the distances are summarized into the following three types of variance:

• 1. Variance of the observation time within the same VTW.
• 2. Variance of the observation time among different VTWs.
• 3. Rejection.

Thus, the perturbations to the initial observation plans can be calculated by:

$$\begin{array}{c}\hfill \delta =\sum _{u=1}^{N_s}\sum _{v=1}^3{w}_v{\sigma }_u^v\end{array}$$

(2)

where $N_s$ is the number of static observation missions, $w_{\nu }$, $\nu =1,2,3$ represents the perturbation degree of type $\nu$ variance, ${\sigma }_u^v$ is the binary variable, and ${\sigma }_u^v=1$ indicates the type $\nu$ variance happening in mission $t_u$.

Finally, the decision variables are defined as follows:

• $x_{ij}$: a binary decision variable. $x_{ij}$ is equal to 1 if and only if the mission $t_i$ is observed in VTW $w_{ij}$.
• $s{t}_i$: the observation start time of mission $t_i$.
• ${\rho }_{i{i}^{\prime }}^{jk}$: a binary decision variable for the observation order. ${\rho }_{i{i}^{\prime }}^{jk}=1$ represents that mission $t_{i^{\prime }}$ is observed immediately after task $t_i$ on orbit $o_{jk}$.

2.2. Assumptions

According to previous studies and engineering experience, several reasonable assumptions and simplifications are made to simplify the EOS scheduling problem:

• Only the spot targets are considered in this paper, and the polygon targets can be viewed as multiple independent spot targets. Each spot target only needs to be observed with one pass.
• The dynamic missions are assumed to arrive in batch style, which indicates that dynamic scheduling is only triggered when a certain number of dynamic tasks are submitted, rather than being triggered by one or a few tasks.
• All Earth observation satellites have enough memory and power on each orbit.
• Relay satellites and ground stations are assumed to be sufficient resource to download all observation data. Therefore, the download scheduling for the observation data is not considered.

2.3. Mathematical Model

Here, we establish a multi-objective mathematical model for the dynamic observation-mission-scheduling problem. Two dummy missions without profit, i.e, a dummy start mission s and a dummy end mission e, are introduced for the ease of modeling. The multi-objective mathematical model is described as follows:

$$\begin{array}{cc}\hfill & max{f}_1=\sum _{i=1}^N\sum _{j=1}^{n_i}{x}_{ij}·p_i\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & min{f}_2=\sum _{u=1}^{N_s}\sum _{v=1}^3{w}_v{\sigma }_u^v\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & \sum _{j=1}^{n_i}{x}_{ij}\le 1,\forall t_i\in T\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & eo{t}_i\le s{t}_i,s{t}_i+d_i\le lo{t}_i,\forall i\in T\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & s_{ij}\le s{t}_i,s{t}_i+d_i\le e_{ij},\forall t_i\in T,\mathrm{if}{x}_{ij}=1,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & s{t}_i+d_i+z_{i{i}^{\prime }}-s{t}_{i^{\prime }}\le V\left(1-{\rho }_{i{i}^{\prime }}^{jk}\right),\forall t_i,t_{i^{\prime }}\in T\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & \sum _{i^{\prime }\in T\cup \left\{s\right\}}{\rho }_{i^{\prime }i}^{jk}⩽1,\forall t_i\in T,s_j\in S,o_{jk}\in O_j\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & \sum _{i^{\prime }\in T\cup \left\{e\right\}}{\rho }_{i{i}^{\prime }}^{jk}⩽1,\forall t_i\in T,s_j\in S,o_{jk}\in O_j\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \sum _{i^{\prime }\in T\cup \left\{s\right\}}{\rho }_{i^{\prime }i}^{jk}=\sum _{i^{\prime }\in T\cup \left\{e\right\}}{\rho }_{i{i}^{\prime }}^{jk},\forall t_i\in T,s_j\in S,o_{jk}\in O_j\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & \sum _{i\in T}{\rho }_{si}^{jk}=\sum _{i\in T}{\rho }_{ie}^{jk}=1,\forall s_j\in S,o_{jk}\in O_j\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & x_{ij}\in \left\{0,1\right\},\forall t_i\in T,j\le n_i\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & {\rho }_{i{i}^{\prime }}^{jk}\in \left\{0,1\right\},\forall t_i,t_{i^{\prime }}\in T,s_j\in S,O_{ik}\in O_j\hfill \end{array}$$

(14)

The objective function $f_1$ aims to maximize the total profits of the scheduled missions and the objective function $f_2$ aims to minimize the perturbation to the initial observation plan. Constraint (5) denotes that each observation mission can only be observed at most once, and Constraint (6) limits the observation time for each mission to its effective time window. Constraint (7) indicates that each mission must be observed in the VTW. Constraint (8) is the transition time constraint, which indicates that there must be a sufficient time interval for the EOS to observe any two adjacent observation missions, where V is a sufficiently large positive number. Constraint (9) and Constraint (10) denote that there is at most one predecessor mission and one successor mission for each real observation mission. Constraint (11) is the observation mission flow balance limit, which indicates that the number of predecessor missions is equal to the number of successor mission for each mission. Constraint (12) guarantees that the observation on each orbit start from the dummy start mission s and ends at the dummy end mission e. Constraints (13–14) define the domains of the decision variables.

3. Multi-Objective Dynamic Mission Scheduling Algorithm (MODMSA)

In this section, the multi-objective dynamic mission-scheduling algorithm (MODMSA) is introduced in detail. The general framework of the MODMSA is described first, and the improved evolutionary operators in the SPEA2 are then proposed. Afterwards, a profit-oriented local search algorithm is illustrated, and an adaptive perturbation control strategy is presented.

3.1. Framework

The framework of the MODMSA is shown in Figure 2. In the MODMSA, a minimum perturbation random initialization is first adopted to generate the population with least perturbation while maintaining the diversity of the population. Then, evolutionary operators in the SPEA2, such as fitness assignment, environment selection, multi-point crossover, and greedy mutation, are performed to search for better scheduling solutions, and the profit of the current population is further improved by the profit-oriented local search algorithm at the end of each iteration. The minimum perturbation random initialization, multi-point crossover, and greedy mutation are improved based on the characteristic of the DMP-PP, while other operators such as environment selection are the same as the SPEA2 [40]. Afterwards, an adaptive perturbation control strategy is performed to limit the maximum allowable perturbation, and the archive sets are updated based on the obtained non-dominated solutions at the current iteration. The above procedures repeat until the termination condition is met and output the non-dominated solutions in the archive.

3.2. Two-Stage Individual Representation

Individual representation directly affects the crossover and mutation in evolutionary algorithms, which has a significant impact on the search efficiency and performance of evolutionary algorithms. In the MODMSA, each individual is represented in a two-stage manner, which are the VTW allocation stage and the single-orbit missionvscheduling stage. Specifically, each individual is represented by a VTW chromosome V and a single-orbit observation plan $\Pi$.

(1) Encoding: As shown in Figure 3, the encoding of the individual is carried out in two stages. In the first stage, each individual is encoded as a VTW chromosome $V=\left\{v_1,\cdots ,v_N\right\}$, and the gene $v_i$ in the VTW chromosome V indicates that VTW $w_{i{v}_i}$ is allocated to mission $t_i$ for observation. Thus, we can obtain the VTW allocation result $W_M$ based on the VTW chromosome V. In the second stage, each individual is further represented as the single-orbit observation plan set $\Pi =\left\{{\Pi }_{11},\cdots ,{\Pi }_{M{m}_M}\right\}$ based on the VTW allocation result $W_M$. ${\Pi }_{ij}=\left\{{\Pi }_{ij}^O,{\Pi }_{ij}^U\right\}$ is the observation plan on the orbit $o_{ij}$, which consists of a single-orbit observation sequence ${\Pi }_{ij}^O$ and an unscheduled mission set ${\Pi }_{ij}^U$. The single-orbit observation sequence ${\Pi }_{ij}^O$ includes all scheduled missions and observation times on the orbit $o_{ij}$, and the unscheduled mission set ${\Pi }_{ij}^U$ consists of missions assigned to orbit $o_{ij}$ but not executed.

(2) Decoding: The decoding of each individual is also performed in two satges. Firstly, each mission is assigned a VTW based on VTW chromosomes V and generate the VTW allocation result $W_M$. In the single-orbit mission-scheduling stage, if the mission is allocated to the same VTW as the previous iteration and is observed at the previous iteration, the mission will be added to the single-orbit observation sequence and is allocated the same observation time window as the previous iterations. Otherwise, the missions are added to the corresponding unscheduled mission set. Afterwards, we attempt insert the missions in the unscheduled mission set into the current observation sequence in ascending order of the mission arrival time. The earliest feasible observation time window (OTW) is selected when inserting the unscheduled mission.

3.3. Minimum Perturbation Random Initialization

Random initialization is effective and popular in multi-objective optimization algorithms, whereas the random initialization will introduce an intolerable perturbation in the MODMSA. To generate an initial population with less perturbations and good diversity, a minimum perturbation random initialization is proposed in this section. In the minimum perturbation random initialization, we first allocate the scheduled missions in the initial observation plan ${\Pi }_{ini}$ the same VTW as in the initial observation plan and randomly allocate a VTW to the unscheduled missions and the dynamic missions. Then, the VTW chromosomes can be determined according to the VTW allocation result $W_M$. Secondly, all missions are allocated to the unscheduled mission set of corresponding orbits based on the VTW chromosomes. Thirdly, the scheduled missions in the initial observation plan ${\Pi }_{ini}$ are first allocated the same OTW as that in ${\Pi }_{ini}$. After all the scheduled missions are allocated, other missions in the unscheduled mission set are attempted to be inserted directly into the current observation sequence on the orbit in descending order of the profits. Therefore, the minimum perturbation random initialization effectively avoids perturbation and maintains population diversity.

3.4. Crossover and Mutation

In evolutionary algorithms, crossover and mutation operations are used to generate new individuals in search of the optimal solution. Perturbations are inevitable in the crossover and mutation, and the degree of the perturbations is determined by the crossover probability $P_C$ and mutation probability $P_M$. As introduced in Section 3.2, the individual is encoded as the VTW chromosome and single-orbit mission scheduling. Correspondingly, the crossover and mutation occur on the VTW chromosome and affect the mission scheduling on the corresponding orbit. The detailed procedures of the crossover and mutation are elaborated as follows:

(1) Multi-point crossover: In this paper, multi-point crossover is adopted. Firstly, randomly select $P_C·N_{pop}$ individuals from the population to form the crossover pool, where $N_{pop}$ is the number of the individuals. Secondly, randomly select two individuals from the crossover pool as parents and, remove them from crossover pool. Then, $P_C·N$ genes are selected randomly to exchange between the parents. Moreover, the missions corresponding to the exchanged genes are removed from the single-orbit observation sequence and are reallocated to the unscheduled mission set of the new orbits according to the new genes.

(2) Greedy mutation: In the greedy mutation, to improve search speed and efficiency, all the genes corresponding to the unscheduled missions will greedily mutate while the genes corresponding to the scheduled missions mutate with a mutation probability of $P_M$. A new gene is randomly generated for each mutated gene. In addition, each mission corresponding to the mutated gene is removed from the single-orbit observation plan of the previous orbit and is added to the unscheduled mission set of the new orbit based on the mutated gene.

3.5. Profit-Oriented Local Search Algorithm

After decoding the individual into the multiple single-orbit observation plan, a profit-oriented local search algorithm is adopted to improve the profit of each single-orbit observation plan. The detailed procedures of the profit-oriented local search algorithm are summarized in Algorithm 1. In lines 3–4, for each orbit, the maximum observation profit $P_{max}$, the best single-orbit observation plan ${\Pi }_{ij}^B$, and the current single-orbit observation plan ${\Pi }_{ij}^C$ are initialized. Then, destroy operation and repair operation are executed iteratively to search for the solution with the highest profit. In each iteration, the indicator removal and random removal are performed first (line 6–7), and then, the insertion operation is executed only when the repair probability $P_R$ exceeds its insertion execution probability (line 8–16). Finally, $P_{max}$ and ${\Pi }_{ij}^B$ are greedily updated according to their profits (line 17–21).

Algorithm 1 Profit-oriented local search algorithm.

Input: Single-orbit observation plan set $\Pi$, maximum iteration K

Output: Improved single-orbit observation plan set ${\Pi }^{\prime }$

1: for $i=1$ to N do 2:    for $j=1$ to $m_i$ do 3:      Calculate the profit of single-orbit observation plan ${\Pi }_{ij}^O$ as the maximum observation profit $P_{max}$ 4:      Initialize the initial single-orbit observation plan ${\Pi }_{ij}^O$ as the best single-orbit observation plan ${\Pi }_{ij}^B$ and the current single-orbit observation plan ${\Pi }_{ij}^C$ 5:      for $k=1$ to K do 6:         Perform indicator removal on ${\Pi }_{ij}^C$ 7:         Perform random removal on ${\Pi }_{ij}^C$ 8:         if $P_R\ge P_{I1}$ then 9:           Perform direct insertion on ${\Pi }_{ij}^C$ 10:        else if PR ≥ PI2 then 11:           Perform direct insertion and shifting insertion on ${\Pi }_{ij}^C$ 12:        else if PR ≥ PI3 then 13:           Perform direct insertion, shifting insertion, and backtracking insertion on ${\Pi }_{ij}^C$ 14:        else if PR ≥ PI4 then 15:           Perform direct insertion, shifting insertion, backtracking insertion, and deletion insertion on ${\Pi }_{ij}^C$ 16:         end if 17:         Calculate the profit $P^C$ of the current single-orbit observation plan ${\Pi }_{ij}^C$ 18:         if $P^C\ge P_{max}$ then 19:           $P_{max}=P^C$, ${\Pi }_{ij}^O={\Pi }_{ij}^C$ 20:         else 21:           ${\Pi }_{ij}^C={\Pi }_{ij}^O$ 22:         end if 23:      end for 24:    end for 25: end for

3.5.1. Destroy Operation

In the destroy operation, several scheduled missions are removed to facilitate the insertion of high-profit missions. Two removal operators are defined in the destroy operation, and each scheduled mission is removed with probability $P_D$ in two removal operators:

(1) Random removal: The random removal removes scheduled missions randomly from the single-orbit observation sequence ${\Pi }_{ij}^O$ and adds the removed missions into the unscheduled mission set ${\Pi }_{ij}^U$, where $|N_{ij}|$ is the number of scheduled missions in the current observation plan ${\Pi }_{ij}^O$.

(2) Indicator removal: In the indicator removal, we define the mission fitness value to guide the removal of the scheduled missions. The mission fitness value $t{f}_k$ is calculated by:

$$\begin{array}{c}\hfill t{f}_k={\displaystyle \frac{p_k}{d_k·L_{w_k}·c{f}_k}}\end{array}$$

(15)

where $p_k$ and $d_k$ are the profit and duration of the scheduled mission $t_i$, $L_{w_k}$ is the length of the VTW for mission $t_k$ on the current orbit, and $c{f}_k$ is the VTW conflict degree between the scheduled mission $t_k$ and all missions in the unscheduled missions set ${\Pi }_{ij}^U$. The VTW conflict degree $c{f}_k$ is calculated by:

$$\begin{array}{c}\hfill c{f}_k=\sum _l^{N_u^{ij}}{\displaystyle \frac{p_l·{\Phi }_{kl}}{L_{w_l}}}\end{array}$$

(16)

where $N_u^{ij}$ is the number of missions in the unscheduled mission set ${\Pi }_{ij}^U$, $p_k$ and $d_k$ are the profit and duration of the unscheduled mission $t_l$, and ${\Phi }_{kl}$ is the length of the overlap time between the VTW of mission $t_k$ and the VTW of mission $t_l$. The mission with high profit and low duration should be prioritized for observation as a high profit can be achieved with less observation time cost. The longer length of the VTW indicates that the mission has more opportunities to be observed again after it is removed, and the scheduled mission with a larger conflict time degree is more likely to hinder the insertion of other unscheduled missions. Thus, missions with a small mission fitness value are prioritized for removal in the indicator removal.

3.5.2. Repair Operation

The repair operation is executed to insert the unscheduled missions into the current observation sequence, which consists of direct insertion, shifting insertion, backtracking insertion, and deletion insertion. Each insertion operation corresponds to an insertion execution probability, i.e., $P_{I1}$, $P_{I2}$, $P_{I3}$, and $P_{I4}$. The insertion operations are executed if the repair probability $P_R$ exceeds its insertion execution probability. The above four insertion operations attempt to insert the unscheduled missions in descending order of priority and allocate the earliest feasible observation time window (OTW) for each unscheduled mission:

(1) Direct insertion: Direct insertion attempts to insert the unscheduled mission directly into the current observation sequence if there is enough idle time on the orbit.

(2) Shifting insertion: For each unscheduled mission, the shifting insertion first moves the OTW of the scheduled mission before the insertion position forward and moves the OTW of the scheduled mission behind the insertion position backward to free up more idle time without violating the observation constraints. Then, the shifting insertion attempts to insert the unscheduled mission at the insertion position directly, and all insertion locations are traversed to try to insert the unscheduled mission.

(3) Backtracking insertion: For each mission waiting to be inserted, the backtracking insertion first selects the conflict mission set, which consists of scheduled missions whose VTW overlaps with the VTW of the current waiting mission. Then, one scheduled mission in the conflict mission set is temporarily removed, and the waiting missions and the removed missions are attempted to be observed in turn. The backtracking insertion is successful only when both the waiting mission and the removed mission can be scheduled simultaneously. An example of backtracking insertion is shown in Figure 4. When attempting to insert the waiting mission $t_i$, we first remove the mission $t_2$ from the current observation sequence and then try to perform the mission $t_i$ at the idle time. If the mission $t_i$ is scheduled successfully, the removed mission $t_2$ is then reinserted in another location in the observation sequence.

(4) Deletion insertion: In the deletion insertion, the mission with less profit than the unscheduled mission is first deleted, and we then try to insert the unscheduled mission directly into the current observation plans. We attempt to insert the current waiting mission by deleting scheduled missions in descending order of profit until successful.

3.6. Adaptive Perturbation Control Strategy

An adaptive perturbation control strategy is developed to enhance exploration in the multi-objective solution space. Since higher observation profits are often accompanied by larger perturbations, the adaptive perturbation control strategy gradually increases the maximum allowable perturbation to control the MODMSA to search for different multi-objective solution space regions. In the MODMSA, the perturbation mainly occurs during the multi-point crossover and greedy mutation and the profit-oriented local search algorithm. In addition, the maximize allowable perturbations are determined by the crossover probability $P_C$, mutation probability $P_M$, removal probability $P_D$, and repair probability $P_R$. Therefore, the adaptive perturbation control strategy expands the maximum allowable perturbation by linearly increasing these probabilities. The probabilities are updated as follows:

$$\begin{array}{cc}\hfill & P_C=min\left(P_C^m,{\displaystyle \frac{n·{ϵ}_C}{N_{max}}}\right)\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & P_M=min\left(P_M^m,{\displaystyle \frac{n·{ϵ}_M}{N_{max}}}\right)\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & P_D=min\left(P_D^m,{\displaystyle \frac{n·{ϵ}_D}{N_{max}}}\right)\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & P_R=min\left(1,{\displaystyle \frac{k·{ϵ}_R}{N_{max}}}\right)\hfill \end{array}$$

(20)

where n and $N_{max}$ denote the current and maximum number of iterations of the MODMSA, ${ϵ}_C$, ${ϵ}_M$, ${ϵ}_D$, and ${ϵ}_R$ are the coefficients, and $P_C^m$, $P_M^m$, and $P_D^m$ are the maximum crossover probability, maximum mutation probability, and maximum removal probability, respectively.

4. Results

4.1. Experimental Setting

All the algorithms were run on a laptop with an Intel (R) Core (TM) i7-7600U CPU at 2.80 GHz 10 with 8 GB of RAM. The orbit parameters of the EOSs in the experiment were determined by referring to the orbit parameters in the studies [1,19,41], and the EOS orbit parameters are shown in

Table 1. Parameter setting.

Satellite Parameters	Value
Semimajor axis (km)	6678
Inclination (°)	96.576
Right ascension of the ascending node (°)	0, 18, 36
	54, 72, 90
	108, 126
	144, 162
Eccentricity (°)	0.000627
Argument of perigee (°)	0
Mean anomaly (°)	20, 38, 56
	74, 92, 110
	128, 146
	164, 2
$a_1,a_2,a_3,a_4$	5, 10, 16, 22
$v_1,v_2,v_3,v_4$ (°/s)	1.5, 2, 2.5, 3
Parameters related to the MODMSA	Value
Pop size	100
$it{e}_{max}$	200
${ϵ}_C$, ${ϵ}_M$, ${ϵ}_{RD}$	2, 2, 1.5
${ϵ}_{ID}$, ${ϵ}_R$	1.5, 1.5
$P_{Mu}$	0.8
$P_{I1}$, $P_{I2}$, $P_{I3}$, $P_{I4}$	0, 0.25
	0.50, 0.75
$P_C^m$, $P_M^m$	0.6, 0.4
$P_D^m$, $P_D^m$	0.2, 0.2

. The scheduling horizon was 86,400 s. The duration of each observation mission followed a uniform distribution with the range of [1, 10]. The profit of the static and dynamic missions were random values within [1, 10] and [5, 15]. Static missions were submitted before the start time of the scheduling horizon, and static missions were allowed to be observed during all the scheduling horizon, while the dynamic missions arrived randomly during the scheduling horizon and the length of ETW is generated in the uniform distribution with the range [15,000 s, 35,000 s]. The observation missions were randomly generated in the area of latitude 3° N–53° N and longitude 74° E–133° E. Scheduling scenarios with low, medium, and heavy loads were investigated, where the numbers of static missions were 400, 600, and 800, respectively. For each load situation, four test cases with different numbers of dynamic missions were studied where the ratios of the number of dynamic missions to the number of static missions were 25%, 50%, 75%, and 100%, respectively. Therefore, there were 12 test cases in the experiments and the test scenarios are denoted by “Case$\varsigma$_$\psi$” where $\varsigma$ and $\psi$ are the numbers of the static missions and dynamic missions.

The multi-objective optimization algorithms for comparison in this paper were as follows: SPEA2 [40], NSGA-II [42], MOEA/D [43], I-MOMA, and D-MOMA [44]. SPEA2 [40], NSGA-II [42], and MOEA/D [43] are the classical indicator-based MOEA, domination-based MOEA, and decomposition-based MOEA, respectively. I-MOMA and D-MOMA [44] are state-of-the-art multi-objective scheduling algorithms for the EOS-mission-scheduling problem, which combine a multi-objective evolutionary optimization algorithm and a local search algorithm. I-MOMA and D-MOMA are designed based on the indicator-based multi-objective optimization algorithm and the domination-based multi-objective optimization algorithm, respectively.

After testing the performance of different parameters related to the MODMSA, we chose the parameters with the best performance as the parameters for the MODMSA. The parameters related to the MODMSA are listed in Table 1. The population size for all multi-objective scheduling algorithms was 100, and the number of the archive size in the SPEA2 was 50. The maximum number of iterations was 200, and the number of iterations in the local search algorithm was 10. The other parameters remained the same as the default ones in the SPEA2, NSGA-II, MOEA/D, I-MOMA, and D-MOMA.

4.2. Experimental Results and Analysis

The pareto fronts (PFs) of reserved non-dominated solutions for the test cases with low, medium, and high loads are shown in Figure 5, Figure 6 and Figure 7, respectively. The x-axis and y-axis represent the additive inverse of the profit and the perturbation. We can easily find that, in all test scenarios, the MODMSA outperforms other multi-objective comparison algorithms in terms of the diversity and the quality of non-dominated solutions. It is obvious that the pareto fronts (PFs) of the MODMSA are well distributed in both the x-axis and y-axis, whereas the non-dominated solutions obtained by other comparison algorithms are centrally distributed, which proves the effectiveness of our proposed multi-objective optimization algorithm in improving the diversity of non–dominated solutions. Especially in test scenarios with a heavy load, the non-dominated solutions obtained by other comparison algorithms are almost the same as each other. In addition, it is worth noting that only the MODMSA can obtain non–dominated solutions with different perturbations, and the non–dominated solution obtained by other comparison algorithms are concentrated in areas with high perturbation. This is because the evolutionary operators such as crossover and mutation introduce large perturbation and the perturbations are accumulated continuously in these multi-objective comparison algorithm. Therefore, the MODMSA is promising for the practical EOS dynamic mission-scheduling problem as the MODMSA can provide more high-quality candidate observation plans with various perturbations for the decision-makers of the EOS system.

The optimal values of each objective obtained by all multi-objective scheduling algorithms in different test cases are listed in

Table 2. The maximum profit rates of the non-dominated solutions obtained by all multi-objective scheduling algorithms in different test cases.

Cases	MODMSA	SPEA2	NSGA-II	MOEA/D	I-MOMA	D-MOMA
Case400_100	91.16	89.31	88.66	83.72	90.22	90.34
Case400_200	90.84	87.38	85.86	79.05	88.86	88.67
Case400_300	89.94	84.81	82.33	75.63	87.37	86.50
Case400_400	89.83	81.03	80.60	72.97	85.53	84.05
Case600_150	89.40	80.63	79.21	74.67	83.42	83.75
Case600_300	88.29	76.33	72.57	69.16	82.10	79.29
Case600_450	87.04	71.45	67.67	65.76	76.68	76.59
Case600_600	85.38	69.32	65.49	61.78	73.97	73.12
Case800_200	87.53	72.59	71.16	67.11	77.89	76.45
Case800_400	84.85	65.19	64.44	61.23	71.12	71.33
Case800_600	82.12	61.04	60.28	57.41	67.28	65.69
Case800_800	79.65	56.70	55.56	52.90	61.31	61.17

and

Table 3. The minimum perturbation of the non-dominated solutions obtained by all multi-objective scheduling algorithms in different test cases.

Cases	MODMSA	SPEA2	NSGA-II	MOEA/D	I-MOMA	D-MOMA
Case400_100	0	3863	3820	4443	3871	3836
Case400_200	0	3930	3947	4587	3983	3984
Case400_300	0	3901	3968	4511	3868	3953
Case400_400	0	3984	4001	4615	3901	4029
Case600_150	0	6265	6304	6611	6359	6267
Case600_300	0	5945	6456	6787	6064	6248
Case600_450	0	6214	6291	6697	6031	6145
Case600_600	0	6210	6514	6902	6180	6240
Case800_200	0	8039	8076	8360	7906	7848
Case800_400	0	7939	7979	8379	7867	7917
Case800_600	0	8051	8043	8384	7960	8007
Case800_800	0	8188	8292	8686	8225	8268

. The MODMSA can simultaneously obtain the maximum profit and the minimum perturbation in all the test cases. Regarding the profit rates, the gap in the maximum profit rates between the MODMSA and other multi-objective comparison algorithms increases significantly with the the number of missions increases. Compared to comparative algorithms, the MODMSA improved by at least 2.42%, 8.40%, and 14.09% on average in the test cases with low, medium, and high load, respectively. The superior performance of our proposed algorithm in observation profit is due to the fact that the proposed evolutionary operators and profit-oriented local search algorithm can fully utilize the heuristic features of the DMP-PP. Moreover, I-MOMA and D-MOMA are superior to the other three classical multi-objective comparison algorithms, which indicates the advantage of the memetic-based multi-objective scheduling algorithm in solving the complex multi-objective observation-mission-scheduling problem. In terms of the perturbation, the MODMSA can obtain the non-dominated solution without perturbation, whereas the minimum perturbations of other comparison algorithms are much higher than the MODMSA. This is because the perturbations generated in evolutionary operators, such as crossover and mutation, are unavoidable and accumulated, whereas the non–dominated solutions generated at each iteration of the MODMSA are preserved in the archive set and do not participate in subsequent iterations.

The hypervolumes (HVs) of all multi-objective scheduling algorithms in different test cases are shown in

Table 4. The HVs of all multi-objective scheduling algorithms in different test cases.

Cases	MODMSA	SPEA2	NSGA-II	MOEA/D	I-MOMA	D-MOMA
Case400_100	0.7605	0.1634	0.1597	0.0304	0.1722	0.1794
Case400_200	0.7163	0.1722	0.1529	0.0299	0.1735	0.1709
Case400_300	0.6751	0.1573	0.1292	0.0322	0.1798	0.1570
Case400_400	0.6271	0.1317	0.1255	0.0261	0.1708	0.1387
Case600_150	0.6528	0.0540	0.0445	0.0120	0.0595	0.0695
Case600_300	0.5690	0.0796	0.0297	0.0066	0.0955	0.0667
Case600_450	0.5381	0.0590	0.0422	0.0145	0.0922	0.0811
Case600_600	0.4932	0.0686	0.0371	0.0120	0.0872	0.0759
Case800_200	0.5382	0.0261	0.0188	0.0042	0.0472	0.0452
Case800_400	0.4725	0.0311	0.0262	0.0087	0.0500	0.0448
Case800_600	0.4381	0.0265	0.0285	0.0090	0.0436	0.0382
Case800_800	0.4331	0.0312	0.0241	0.0076	0.0390	0.0356

. The HV is an indicator used to measure the performance of the multi-objective optimization algorithm, and a larger HV reflects better comprehensive performance. The objective function values of the non-dominated solution are normalized before calculating the HV. Comparing the HVs of non-dominated solutions obtained by all algorithms, it is easy to find that the MODMSA performs best in terms of comprehensive performance. Specifically, compared to other multi-objective algorithms, the MODMSA improved by 527% on average in terms of the hypervolume.

Furthermore, experiments were conducted to evaluate the roles of the proposed improvement procedures, i.e., the minimum perturbation random initialization, the adaptive perturbation control strategy, and the profit-oriented local search algorithm. All algorithms were compared in the test cases with medium load. The comparison algorithms are briefly described as follows:

• MODMSA-INI, which applies the random initialization proposed in [44] and reserves the other procedures of the MODMSA.
• MODMSA-PPCS, which is a variant of the MODMSA, but without the adaptive perturbation control strategy.
• MODMSA-LS, which is a variant of the MODMSA, but without the profit-oriented local search algorithm.

As shown in Figure 8, the Pareto fronts MODMSA-INI and the MODMSA-PPCS are predominantly distributed in the regions with high profit and high perturbation, which demonstrates that the proposed initialization strategy and the adaptive perturbation control strategy are crucial in preserving the solution diversity. The non–dominated solutions solved by the MODMSA-LS are distributed in the region with lower profit and fewer perturbation, which shows that the local search algorithm can effectively improve the profit of solutions. Regarding the maximum profit obtained, the MODMSA-INI and MODMSA-PPCS can obtain higher profit than the MODMSA as the search space of the MODMSA is limited at the beginning stage. In addition, the MODMSA-LS can find better non–dominated solutions than the MODMSA in the region of low profit and fewer perturbations, which is because the local search algorithm in the MODMSA introduces additional perturbation. However, only the MODMSA can obtain non–dominated solutions with superior performance in quality and diversity. Therefore, every improvement procedure is necessary and indispensable in the proposed MODMSA.

5. Discussion

5.1. Efficiency of MODMSA in Solving Dynamic Mission Scheduling Problems

For the multi-objective dynamic observation-mission-scheduling problem, the proposed MODMSA demonstrates excellent performance in the diversity and quality of the non-dominated solutions. The performance gap between the MODMSA and other comparison algorithms increases significantly as the number of missions grows. Especially for the diversity of the non-dominated solutionss, only the MODMSA can obtain various non-dominated solutions, while the non-dominated solutions obtained by other comparison algorithms are concentrated in highly perturbed solutions. However, it is easy to find that the computational complexity of the proposed MODMSA is higher than that of the compared algorithms as the profit-oriented local search algorithm causes additional computational costs. In addition, the additional computational cost is acceptable since the computational complexity of the profit-oriented local search algorithm is at a polynomial level. Therefore, our proposed MODMSA can provided multiple high-quality alternative observation plans with acceptable computational costs for the EOS system decision-maker, which shows broad application prospects in practical EOS dynamic mission planning.

5.2. Effectiveness of Improvement Procedures

The main improvement procedures proposed in the MODMSA are minimum perturbation random initialization, adaptive perturbation control strategy, and profit-oriented local search algorithm. Based on the experiment results, we found that each improvement procedure is vital for the MODMSA. The minimum perturbation random initialization and adaptive perturbation control strategy determine the performance of the MODMSA in diversity. The profit-oriented local search algorithm is able to improve the profits of the non-dominated solutions further. In addition, the framework of the MODMSA and the proposed improvement procedures provide ideas for other dynamic mission-scheduling problems.

6. Conclusions

In this paper, we focused on the EOS dynamic mission-scheduling problem to optimize the observation profit and perturbation simultaneously. A multi-objective dynamic mission-scheduling algorithm (MODMSA) based on the SPEA2 was developed. In the MODMSA, a two-stage individual representation, multi-point crossover, and greedy mutation are proposed based on the characteristics of the dynamic mission-scheduling problem. A profit-oriented local search algorithm is introduced to further improve the observation profit. In addition, an adaptive perturbation control strategy is presented to enhance the exploration in the multi-objective solution space. The simulation results show that the MODMSA improves 8.30% and 527% on average in terms of the observation profit and the HV compared the other algorithms, which indicates that the MODMSA can provide decision-makers of the EOS system with high-quality alternative observation plans in practical engineering applications. In our future research, the dynamic mission-scheduling problem with large-scale EOSs and missions will be studied.