2.1 Time relationship

The time instants $t_{i,p}^{initial}$ and $t_{i,p}^{end}$ marking the initial and end of the $p$ th sub-mission for the $i$ th servicing satellite ( $1 \le p \le {P_i}$ ) are determined by

(1) \begin{align} \begin{array}{*{20}{l}}{t_{i,p}^{initial}} { = \left\{ {\begin{array}{*{20}{l}}{t_i^{parking}} {\left( {p = 1} \right)}\\[3pt] {t_{i,p - 1}^{end}} {(1 \lt p \le {P_i})}\end{array}} \right.}\\[3pt] {t_{i,p}^{end}} { = t_{i,p}^{initial} + t_{i,p}^{lambert} + t_{i,p,k}^{service}}\end{array} \end{align}

The time instant, $t_i^{end}$ , for servicing satellite $i$ to return to the space station after accomplishing all on-orbit services can be formulated as

(2) \begin{align} t_i^{end} = t_{i,{P_i}}^{end} + t_{i,back}^{lambert}. \end{align}

2.2 Orbital element vector of a target after time ${\boldsymbol{\Delta }}t$

The dynamics of an object in space can be delineated through the orbital element vector $x = \left[ {a,e,{\Omega},i,\omega, \theta } \right]$ , where $a$ represents the semimajor axis, $e$ signifies the eccentricity, ${\Omega}$ represents the right ascension of the ascending node, $i$ stands for the inclination, $\omega $ denotes the argument of perigee, and $\theta $ represents the true anomaly. When targets revolve in a specific Keplerian orbit, it is primarily the true anomaly that undergoes variations. Given a target state $x\left( {{t_0}} \right)$ at ${t_0}$ and a time interval ${\Delta}t$ , we can calculate $\boldsymbol{{x}}\!\left( {{t_0} + {\Delta}t} \right)$ as:

(3) \begin{align} \boldsymbol{{x}} \!\left( {{t_0} + {\Delta}t} \right) = {\mathcal G}\!\left( {\boldsymbol{{x}}\left( {{t_0}} \right),{\Delta}t} \right) \end{align}

For a detailed derivation, see the Appendix 6.

2.3 Lambert-based manoeuver

Each servicing satellite needs to perform orbital transfers during the mission to rendezvous with the service target. In this study, servicing satellites undergo orbital transfer procedures based on Lambert’s theorem. Two manoeuvers are taken into account for each orbital transfer. Provided with the initial position vector ${\boldsymbol{{r}}_1}$ , final position vector ${\boldsymbol{{r}}_2}$ , and the transfer time ${\Delta}t$ of the transfer orbit, the necessary velocity changes for the two manoeuvers can be computed as:

(4) \begin{align} {\boldsymbol{{v}}_1} = {{\mathcal{L}}_1}\!\left( {{\boldsymbol{{r}}_1},{\boldsymbol{{r}}_2},{\Delta}t} \right),{\boldsymbol{{v}}_2} = {{\mathcal{L}}_2}\!\left( {{\boldsymbol{{r}}_1},{\boldsymbol{{r}}_2},{\Delta}t} \right) \end{align}

For a detailed derivation, see the Appendix.

In addition, if the orbital element $x$ of an object at a specific point along an orbit are known, it allows for the determination of the object’s position and velocity vectors $\boldsymbol{{r}}$ and $\boldsymbol{{v}}$ at that point, and vice versa. We represent this functional relationship as:

(5) \begin{align} \begin{array}{*{20}{l}}{\boldsymbol{{x}} = {{\mathcal T}_x}( {\boldsymbol{{r,\,v}}} ),\boldsymbol{{r}} = {{\mathcal T}_r}( \boldsymbol{{x}}),\boldsymbol{{v}} = {{\mathcal T}_v}( \boldsymbol{{x}}).}\end{array} \end{align}

Suppose that the $p$ th sub-mission of the $i$ th servicing satellite requires a manoeuver to rendezvous with the $j$ th target. The initiation time of this sub-mission is denoted as $t_{i,p}^{initial}$ , and the duration of the Lambert-based manoeuver is $t_{i,p}^{lambert}$ . The orbital elements for the $i$ th servicing satellite and the $j$ th target at $t_{i,p}^{initial}$ are known and designated as ${\boldsymbol{{x}}_i}\left( {t_{i,p}^{initia{\rm{l}}}} \right)$ and ${\boldsymbol{{x}}_j}\left( {t_{i,p}^{initial}} \right)$ . Combining the previously derived information, the change in velocity vector $\delta {v_{i,p}}$ required for $i$ th servicing satellite to manoeuver during the $p$ th sub-mission is

(6) \begin{align} {v_{i,p}} =\,& \parallel {{\mathcal{L}}_2}\left( {{{\mathcal T}_r}\left( {{\boldsymbol{{x}}_i}\left( {t_{i,p}^{initia{\rm{l}}}} \right)} \right),{{\mathcal T}_r}\left( {{\mathcal G}\left( {{\boldsymbol{{x}}_j}\left( {t_{i,p}^{initial}} \right),t_{i,p}^{lambert}} \right)} \right),{\Delta}t} \right) \nonumber \\[4pt] & - {{\mathcal T}_v}\left( {{\mathcal G}\left( {{\boldsymbol{{x}}_j}\left( {t_{i,p}^{initial}} \right),t_{i,p}^{lambert}} \right)} \right)\parallel + \parallel {{\mathcal T}_v}\left( {{\boldsymbol{{x}}_i}\left( {t_{i,p}^{initial}} \right)} \right) \nonumber\\[3pt] & - {{\mathcal{L}}_1}\left( {{{\mathcal T}_r}\left( {{\boldsymbol{{x}}_i}\left( {t_{i,p}^{initial}} \right)} \right),{{\mathcal T}_r}\left( {{\mathcal G}\left( {{\boldsymbol{{x}}_j}\left( {t_{i,p}^{initial}} \right),t_{i,p}^{lambe{\rm{r}}t}} \right)} \right),{\Delta}t} \right)\parallel \end{align}

Given that the initial mass of the $i$ th servicing spacecraft is ${m_{i,0}}$ and its mass after completing the $p$ th sub-mission is ${m_{i,p}}$ , the mass of fuel consumed by the $i$ th servicing spacecraft in the $p$ th sub-mission is

(7) \begin{align} \delta {m_{i,p}} = {m_{i,p - 1}} - {m_{i,p - 1}}\exp\!\left( {\frac{{ - \delta {v_{i,p}}}}{{{g_0}{I_{sp}}}}} \right) \end{align}

where ${g_0}$ is the sea-level standard acceleration of gravity, and ${I_{sp}}$ is the specific impulse of thrusters.

3.1 Design variables

As previously mentioned, the completed on-orbit service operation comprises $m$ independent service sub-missions, each corresponding to a target sequence ${{\textbf{S}}_i}$ , where $i$ represents the serial number of the sub-mission and also the serial number of the corresponding servicing satellite. Each sequence ${{\textbf{S}}_i}$ encompasses two pieces of information, comprising the transfer sequence

\begin{align*} {\textbf{S}}_i^{trans} = \left[ {s_{i,1}^{trans},s_{i,2}^{trans}, \cdots, s_{i,p}^{trans}, \cdots, s_{i,{P_i}}^{trans}} \right] \end{align*}

and the time sequence

\begin{align*} {\textbf{S}}_i^{time} = \left[ {t_i^{parking},t_{i,1}^{lambert}, \cdots, t_{i,{P_i}}^{lambert},t_{i,1,k}^{service}, \cdots, t_{i,{P_i},k}^{service},t_{i,back}^{lambert}} \right]. \end{align*}

where $s_{i,p}^{trans}$ signifies the serial number of the target for servicing satellite $i$ in the $p$ th sub-mission. The explanations for all other variables have been provided in the preceding section. Consequently, the optimisation variables for the multi-objective on-orbit service problem can be expressed as:

(8) \begin{align} {\textbf{S}} = \mathop \sum \limits_{i = 1}^m {{\textbf{S}}_i} = \mathop \sum \limits_{i = 1}^m \left\{ {{\textbf{S}}_i^{trans},{\textbf{S}}_i^{time}} \right\}. \end{align}

3.2 Constraint conditions

The transfer sequence ${\textbf{S}}_i^{trans}$ is insufficient for detailing the constraints on transfer paths in the TRCTSP model. Consequently, we employ the binary variable $a_{q,w}^i \in \left\{ {0,1} \right\}$ to elucidate the transfer path of the $i$ th servicing satellite and provide the associated constraints. In this context, $q,w \in \left\{ {0,1,2, \ldots, n} \right\}$ , where $\left\{ 0 \right\}$ denotes the space station, and $\left\{ {1,2, \ldots, n} \right\}$ denote the serial numbers of the targets. $a_{q,w}^i = 1$ if the $i$ th servicing satellite possesses a direct manoeuvering path between target $q$ and target $w$ ; and otherwise, $a_{q,w}^i = 0$ . As an illustrative example to elucidate this concept, let’s consider the transfer sequence of the 1st servicing satellite ${\textbf{S}}_1^{trans} = \left[ {2,1,5} \right]$ . In this case, we have: $a_{0,2}^1 = 1$ , $a_{2,1}^1 = 1$ , $a_{1,5}^1 = 1$ , $a_{5,0}^1 = 1$ , and all other relevant binary variables are 0. For the sake of simplicity in our description, we denote $\left\{ {1,2, \cdots, n} \right\}$ as $\langle n\rangle $ . The multi-objective on-orbit service problem is governed by the following constraint conditions:

1. (a) All servicing satellites are required to embark from and return to the space station.

(9) \begin{align} \mathop \sum \limits_{w = 1}^n a_{0,w}^i = 1,\forall i \in \langle m\rangle, \forall w \in \langle n\rangle \\[-25pt] \nonumber \end{align}

(10) \begin{align} \mathop \sum \limits_{q = 1}^n a_{q,0}^i = 1,\forall i \in \langle m\rangle, \forall q \in \langle n\rangle \end{align}

Allow the service capacity set of the satellite and the service requirement set of the target to be denoted as $c_i^{satellite}$ and $c_q^{target}$ , respectively, with $c_i^{satellite},c_q^{target} \in \left\{ {1,2,3,4} \right\}$ . In accordance with the problem scenario outlined in Section 2, each target has a solitary requirement, while each satellite can possess either one or two service capabilities. If $c_q^{target} \in c_i^{satellite}$ , it signifies that servicing satellite $i$ possesses the capability to service target $q$ . Subsequently, we can derive the serial number set of targets accessible by the $i$ th satellite as ${V_i} = \left\{ {q:c_q^{target} \in c_i^{satellite},q \in 1,2, \cdots, n} \right\}$ .

1. (b) Servicing satellite $i$ cannot access targets for which it lacks the capability to provide service:

(11) \begin{align} \mathop \sum \limits_{q = 1}^n \mathop \sum \limits_{w = 1}^n a_{q,w}^i = 0,\forall i \in \langle m\rangle, \forall q \in \langle n\rangle \backslash {V_i},\forall w \in \langle n\rangle \\[-25pt] \nonumber \end{align}

(12) \begin{align} \mathop \sum \limits_{q = 1}^n \mathop \sum \limits_{w = 1}^n a_{q,w}^i = 0,\forall i \in m,\forall q \in n,\forall w \in n\backslash {V_i}. \end{align}

1. (c) Each target must be accessed once and only once:

(13) \begin{align} \mathop \sum \limits_{q = 1}^n \mathop \sum \limits_{i = 1}^m a_{q,w}^i = 1,\forall i \in \langle m\rangle, \forall q \in \langle n\rangle, \forall w \in \langle n\rangle, q \ne w \\[-25pt] \nonumber \end{align}

(14) \begin{align} \mathop \sum \limits_{ = 1}^n \mathop \sum \limits_{i = 1}^m a_{q,w}^i = 1,\forall i \in \langle m\rangle, \forall q \in \langle n\rangle, \forall w \in \langle n\rangle, q \ne w \\[-25pt] \nonumber \end{align}

(15) \begin{align} \mathop \sum \limits_{q = 1}^n a_{qw}^i = \mathop \sum \limits_{l = 1}^n a_{wl}^i,q \ne w \ne l,\forall w,q,l \in \langle n\rangle, \forall i \in \langle m\rangle \end{align}

1. (d) The incorporation of subloops in the transfer path of servicing satellite manoeuvers is strictly forbidden:

(16) \begin{align} u_q^i - u_w^i + n \times a_{q,w}^i \le n - 1,\forall q,w \in n,q \ne w,\forall i \in m, \end{align}

where $u_q^i$ represents the count of targets serviced by satellite $i$ from the station to target $q$ .

1. (e) Considering the restrictions on the fuel capacity of each servicing satellite, the total fuel consumption during manoeuvers will not surpass a maximum value $\delta {m_{i,max}}$ :

(17) \begin{align} \delta {m_{i,back}} + \mathop \sum \limits_{p = 1}^{{P_i}} \delta {m_{i,p}} \le \delta {m_{i,max}}, \end{align}

1. (f) The parking time, manoeuver time, and on-orbit service time for each servicing spacecraft are restricted:

(18) \begin{align} \begin{array}{c@{\quad}c}t_{min}^{final} \le {t_{max}} \le & t_{max}^{final}\\[3pt] t_{i,min}^{lambert} \le t_{i,p}^{lambert} \le & t_{i,max}^{lambert}\\[3pt] t_{i,min}^{lambert} \le t_{i,back}^{lambert} \le & t_{i,max}^{lambert}\\[3pt] t_{i,min}^{parking} \le t_i^{parking} \le & t_{i,max}^{parking}\\[3pt] t_{i,p,k,min}^{service} \le t_{i,p,k}^{service} \le & t_{i,p,k,max}^{service} \end{array} \end{align}

where $t_{min}^{final}$ , $t_{max}^{final}$ , $t_{i,min}^{lambert}$ , $t_{i,max}^{lambert}$ , $t_{i,min}^{r{\rm{k}}ing}$ , $t_{i,max}^{parking}$ , $t_{i,p,k,min}^{service}$ , and $t_{i,p,k,max}^{service}$ are known constants. If $t_{i,p,k}^{service}$ exceeds $t_{i,p,k,min}^{service}$ , it indicates that the $i$ th satellite will park in the target orbit upon accomplishing its $i$ th service target sub-mission, with a parking duration of $t_{i,p,k}^{service} - t_{i,p,k,min}^{service}$ .

3.3 The cost function

This multi-objective optimisation function is normalised prior to summation. The total cost ${\textbf{C}}$ of the campaign is defined as:

(19) \begin{align} {\textbf{C}} = \kappa {{{t_{max}} - t_{min}^{final}} \over {t_{max}^{final} - t_{min}^{final}}} + \left( {1 - \kappa } \right){{\sum\limits_{i = 1}^m {\left( {\delta {m_{i,back}} + \sum\limits_{p = 1}^{{P_i}} \delta {m_{i,p}}} \right)} } \over {m\delta {m_{i,max}}}} + \varepsilon {n_{violate}} \end{align}

where ${t_{max}} = max\left( {t_{1,{P_1}}^{end},t_{2,{P_2}}^{end}, \cdots, t_{i,{P_i}}^{end}, \cdots, t_{m,{P_m}}^{end}} \right)$ represents the duration required to accomplish the entire task, $\kappa $ denotes the weight parameter, $\varepsilon $ is the penalty coefficient, and ${n_{violate}}$ represents the number of targets in the planned path that violate the reachability variances restriction. The penalty coefficient $\varepsilon $ should take a value much larger than the normalised multi-objective function to ensure that the relevant constraints are achieved. $\delta {m_{min}}$ is the minimum mass consumption of the task estimated by advance computation. Our optimisation objective encompasses two facets: minimissing mission completion time and reducing fuel consumption for servicing satellite manoeuvers. Unfortunately, these objectives are inherently in conflict; shorter mission completion times typically result in higher fuel consumption, necessitating a trade-off between the two. The weighting parameter $\kappa $ captures this trade-off, where a higher value of $\kappa $ indicates a greater emphasis on swift mission completion at the cost of increased fuel consumption. We will discuss the effect of the value of the weighting parameter $\kappa $ on the experimental results in the subsequent section.

4.1 Positional substitution mutation strategy

During the update process of firefly individuals, we introduce PSMS. Let the solution of firefly individual $i$ after the $k$ th iteration be denoted as $s_i^k$ . We randomly select two elements within $s_i^k$ and exchange their positions, resulting in $\bar s_i^k$ . Subsequently, the values of the consumption functions, denoted as ${\textbf{C}}\left( {s_i^k} \right)$ and ${\textbf{C}}\left( {\bar s_i^k} \right)$ , are calculated. If ${\textbf{C}}\left( {\bar s_i^k} \right) \lt {\textbf{C}}\left( {s_i^k} \right)$ , we replace the solution $s_i^k$ with $\bar s_i^k$ . This positional substitution operation is repeated ${N_s}$ times.

On one hand, given that this operation occurs subsequent to individual position updates and is nested within the population update process, the value of ${N_s}$ must not be overly large. Excessive values would augment the computational intricacy of the algorithm, consequently diminishing its convergence speed. On the other hand, if the frequency of these operations is too scant, the alterations in individual positions will be negligible, rendering the escape from local optima unattainable and impairing the precision of the optimisation search. Therefore, it is recommended that ${N_s}$ be confined within the range $2 \le {N_s} \le 5$ .

4.2 Differential evolution strategy

To maximise the utilisation of population-wide optimal information and enhance the performance of the algorithm, DES is incorporated into the population updating process. After $k$ iterations, the best solution $s_{best}^k$ within the firefly population is recorded. Subsequently, two solutions of fireflies, $s_i^k$ and $s_j^k$ , are randomly selected from the population to compute $\bar s_{best}^k$ using the following formula:

(25) \begin{align} \bar s_{best}^k = s_{best}^k + \eta \left( {s_i^k - s_j^k} \right) \end{align}

In this equation, $\eta $ is a variable following a normal distribution. Subsequently, the values of the consumption functions, denoted as ${\textbf{C}}\left( {s_{best}^k} \right)$ and ${\textbf{C}}\left( {\bar s_{best}^k} \right)$ , are calculated. If ${\textbf{C}}\left( {\bar s_{best}^k} \right) \lt {\textbf{C}}\left( {s_{best}^k} \right)$ , we replace the solution $s_{best}^k$ with $\bar s_{best}^k$ . The differential evolution operation is repeated ${N_d}$ times.

As there is no circular nesting in this operation, ${N_d}$ can be chosen to be a relatively larger value, enabling the utilisation of historical optimal information to guide the search process and enhance the algorithm’s convergence accuracy. Taking cues from the differential evolution algorithm, the value of ${N_d}$ can be set equal to the population size, denoted as ${M_P}$ .

The pseudo-code of EFA is depicted in Algorithm 1. DES is outlined in Algorithm 2, and PSMS is detailed in Algorithm 3. Here, the function ${\textbf{RandomNaturalNumber}}\left( {{M_P}} \right)$ denotes the generation of random natural numbers that are less than or equal to ${M_P}$ .

Algorithm 1:

The enhanced firefly algorithm

Algorithm 2:

Function of ${\textbf{DifferentialEvolution}}\left( {\left\{ {s_1^k, \cdots, s_N^k} \right\}} \right)$

Algorithm 3:

Function of ${\textbf{SubstitutionMutation}}\left( {s_i^k} \right)$

5.1 Experimental settings

Each servicing satellite has a total mass of 1,000 kg and carries 500 kg of chemical fuel, with the thruster parameter ${g_0}{I_{sp}}$ set at 3,000 m/s [Reference Zhang, Parks, Luo and Tang9]. The initial solutions are generated randomly, and the parameter settings used during the simulation are detailed in Table 1. The experimental setup includes an Intel(R) Core(TM) i7-13700 CPU.

Table 1.

Parameter settings

In scenario 1, the space station, all servicing satellites, and all targets are operating in geostationary Earth orbit (GEO). Drawing inspiration from the GEO orbital element settings in the literature [Reference Han, Guo, Li, Zhi and Lv17, Reference Yan, Guo, Li and Song24], the corresponding orbital elements are randomly generated within the following ranges: $a \in \left[ {42,164,{\rm{\;}}42,166} \right]{\rm{km}}$ , $e \in \left[ {0,0.0004} \right]$ , ${\Omega} \in {\left[ {0,360} \right]^ \circ }$ , $i \in {\left[ {0,0.05} \right]^ \circ }$ , $\omega \in {\left[ {0,360} \right]^ \circ }$ , $\theta \in {\left[ {0,360} \right]^ \circ }$ . Experiments will be conducted under scenario 1 with three cases: 3 satellites serving 8 targets, 4 satellites serving 15 targets, and 6 satellites serving 20 targets. Table 2 outlines the service capabilities of the servicing satellites in these three cases, where ID represents the number of the servicing satellite. The initial orbital parameters for the three cases in scenario 1, along with the specific on-orbit service requirements, are presented in Tables 7, 10 and 11 in the Appendix, where ID denotes the number of each target, and ID 0 represents the space station.

Table 2.

Servicing capabilities of servicing satellites

In scenario 2, the space station, all servicing satellites, all targets are operating on randomly generated orbits. The corresponding orbital elements are randomly generated in the following range: $a \in \left[ {6,800,{\rm{\;}}42,000} \right]\;{\rm{km}}$ , $e \in \left[ {0,0.2} \right]$ , ${\Omega} \in {\left[ {0,360} \right]^ \circ }$ , $i \in {\left[ {0,180} \right]^ \circ }$ , $\omega \in {\left[ {0,360} \right]^ \circ }$ , $\theta \in {\left[ {0,360} \right]^ \circ }$ . We will conduct experiments under scenario 2 with 3 satellites serving 8 targets, 4 satellites serving 15 targets, and 6 satellites serving 20 targets, for a total of three cases. Table 2 outlines the service capabilities of the servicing satellites in three cases under scenario 2, where ID is the number of the serving satellite. The initial orbital parameters for the three cases under scenario 2 and the specific on-orbit service requirements are presented in Tables 8, 12 and 13 in the Appendix, where ID is the number of each target and ID 0 represents the space station.

In scenario 3, the space station, all servicing satellites, all targets are operating on low Earth orbits. The corresponding orbital elements are randomly generated in the following range: $a \in \left[ {7,380,{\rm{\;}}7,400} \right]{\rm{km}}$ , $e \in \left[ {0,0.0004} \right]$ , ${\Omega} \in {\left[ {99,101} \right]^ \circ }$ , $i \in {\left[ {81,82} \right]^ \circ }$ , $\omega \in {\left[ {0,0} \right]^ \circ }$ , $\theta \in {\left[ {0,360} \right]^ \circ }$ [Reference Jing, Xiaoqian and Lihu14]. We will conduct experiments under scenario 3 with 3 satellites serving 8 targets, 4 satellites serving 15 targets, and 6 satellites serving 20 targets, for a total of three cases. Table 2 outlines the service capabilities of the servicing satellites in three cases under scenario 3, where ID is the number of the serving satellite. The initial orbital parameters for the three cases under scenario 3 and the specific on-orbit service requirements are presented in Tables 9, 14 and 15 in the Appendix, where ID is the number of each target and ID 0 represents the space station.

5.2 Optimisation results

Tables 16, 17 and 18 give the optimisation results for the three cases of scenario 1, including transfer sequence, time of arrival, time of departure, velocity change and total fuel mass consumption.

Here, we will use the information of servicing satellite 1 from the optimisation results under case 1, scenario 1, as an example to elucidate the significance of the optimisation outcome.

Transfer sequence 0-7-4-1-0 indicates that servicing satellite 1 departs from the space station, sequentially provides on-orbit services to targets 7, 4 and 1, and ultimately returns to the space station.

Time of arrival refers to the moment when servicing satellite 1 reaches the target corresponding to the transfer sequence mentioned above. Specifically, servicing satellite 1 is stationed at the initial moment of 0. The subsequent arrival times at targets 7, 4 and 1 are 47.97, 86.37 and 127.96 h, respectively. Finally, the satellite returns to the station at 166.16 h.

Time of departure denotes the moment when servicing satellite 1 departs from the target corresponding to the transfer sequence mentioned above. Specifically, satellite 1 left the space station at 22.47 h and departed from targets 7, 4 and 1 at 63.97, 106.66 and 143.96 h, respectively.

It is noteworthy that the time of arrival and time of departure can be used to calculate both the duration required for each orbital transfer and the time spent on each on-orbit service. For instance, the time spent servicing target 7 by satellite 1 is determined by subtracting the time of arrival at target 7 from the time of departure from target 7, yielding 16 h. The time required for the orbital transfer from the station to target 7 is calculated by subtracting the time of departure from the station from the time of arrival at target 7, resulting in 25 h.

Additionally, we provide the velocity change required for each orbital transfer based on Lambert’s theory. For example, the velocity change required to maneuver from the station to target 7 is 178.04 m/s. Finally, the table details the fuel consumption of each servicing satellite upon completing its mission.

Tables 19, 20 and 21 present the optimisation results for the three cases in scenario 2. Tables 22, 23 and 24 present the optimisation results for the three cases in scenario 3. These results are displayed in the same manner as those for scenario 1.

It should be noted that scenario 2 is purely a mathematical simulation and does not account for practical engineering applications. When parameterising the target initial orbits for scenario 2, we did not reference values from commonly used orbits. The purpose of this setup is twofold: to verify the validity and superiority of the proposed algorithm from another perspective, and to lay the theoretical groundwork for potential spatial applications. The results indicate that the velocity change required for orbital transfer is substantial under the scenario 2 settings. Conventional chemical fuels would be insufficient to meet such demands. Therefore, in scenario 2, we assume that the servicing satellite is equipped with high-power ion thrusters, offering a specific impulse of up to 70,000 m/s [Reference Koroteev, Lovtsov, Vyacheslav, Muravlev, Andrey and Shagayda25].

5.3 Superiority experimental results

In this subsection, we conduct a comparative analysis between GA [Reference Mirjalili3], LNS-GA [Reference Han, Guo, Li, Zhi and Lv17], PSO [Reference Kennedy and Eberhart5], T-PSO [Reference Daneshjou, Mohammadi-Dehabadi and Bakhtiari21] and FA [Reference Fister, Fister, Yang and Brest4] as benchmark algorithms against EFA. Each algorithm undergoes execution on 10 randomly generated instances to comprehensively assess their optimisation capabilities. The rules for generating orbital elements for each instance are the same as for scenario 1 and scenario 2. Detailed parameters for the compared GA, and PSO algorithms are provided in Table 3, while parameters for the compared LNS-GA and T-PSO algorithms are elucidated in Table 4.

Table 3.

The parameters of the compared GA, PSO

Table 4.

The parameters of the compared LNG-GA, T-PSO

Table 5 presents a comparison of the average cost achieved by various algorithms. In case 1, marked by lower problem complexity, the differences in solutions generated by the three algorithms are subtle. However, in case 3, characterised by higher problem complexity, the EFA proposed herein demonstrates a significantly superior solution quality compared to the other algorithms. Additionally, the average number of iterations failing to meet the constraints provides further insight into the algorithms’ ability to explore a broad solution space, as shown in Table 6. Notably, both PSO and TPSO fail to produce solutions that adhere to the constraints in case 3. The reasons for this outcome will be elaborated upon in the following subsection.

Table 5.

Average cost for different algorithms

Table 6.

Average number of the iterations unsatisfying the constraints for different algorithms

5.4 Hyperparametric sensitivity experiments

In this subsection, we will analyse the effects of the weight parameter $\kappa $ and the penalty coefficient $\varepsilon $ on the optimisation results, using case 1 of scenario 1 as an example.

We conducted experiments by varying the weight parameter $\kappa $ from 0 to 0.5 in intervals of 0.05. To mitigate the impact of random factors, each experiment was repeated 10 times. The experimental results are depicted in Fig. 2, where the dashed curves represent the results of the 10 individual experiments, and the red curve represents the average value across these experiments. As observed, an increase in the weight parameter $\kappa $ indicates that the optimisation algorithm is more inclined to select routes that minimise time at the expense of higher fuel consumption.

Figure 2.

Optimisation results under different weight parameters $\kappa $ .

For a single orbital transfer, it is possible to achieve a more time-efficient and fuel-efficient re-orbiting based on Lambert’s theory. However, as Fig. 2 illustrates, increasing $\kappa $ does not necessarily reduce mission time while increasing fuel consumption. Nonetheless, from a broader perspective, by analysing the average data from multiple experiments, we can draw some conclusions. If the priority is to conserve fuel while accepting longer mission durations, $\kappa $ should be decreased. Conversely, if a shorter mission time is preferred, even at the cost of higher fuel consumption, $\kappa $ should be increased.

We conducted experiments by varying the penalty coefficient $\varepsilon $ from 0.2 to 2 in intervals of 0.2. To mitigate the effects of random factors, each experiment was repeated 10 times. The experimental results are displayed in Fig. 3, where the dashed curves represent the results of the 10 individual experiments, and the red curve represents the average value across these experiments. When $\varepsilon $ is set to 0.2 and 0.4, the algorithm fails to produce optimisation results that meet the constraints. As $\varepsilon $ gradually increases, the optimization algorithm becomes more inclined to generate results that satisfy the constraints. Once $\varepsilon $ reaches a certain threshold, the algorithm consistently produces results that almost always satisfy the constraints. Therefore, we recommend choosing a larger value for $\varepsilon $ . Given that the objective function is normalised, we have set $\varepsilon = 2$ in this study.

Figure 3.

Optimisation results under different penalty coefficients $\varepsilon $ .

5.5 Discussion of results

Based on the experimental outcomes presented in the preceding subsection, it is clear that in cases with fewer optimisation variables — indicating a reduced number of service satellites — and lower complexity, the differences among the algorithms are minimal. Conversely, in cases of increased complexity, the proposed EFA demonstrates notably superior performance. Below, we provide a detailed analysis of the results for each algorithm.

The particles in PSO possess both velocity and position attributes, granting PSO robust local search capabilities. Consequently, PSO and T-PSO perform well in scenario 1, which is characterised by low model complexity. However, PSO lacks the stochastic elements found in GA, such as mutation and crossover operations, as well as the stochastic term $\alpha {\varepsilon _i}$ in FA’s update mechanism. This absence limits PSO’s ability to explore large solution spaces thoroughly. As a result, PSO and T-PSO struggle to generate solutions that adhere to constraints in scenario 3, which is of high complexity. Notably, T-PSO, optimised solely with hyper-parameters, does not significantly enhance PSO’s exploration or local search capabilities, leading to only a modest improvement over PSO.

GA exhibits a strong capacity for exploring expansive solution spaces due to its use of crossover and mutation operations. However, GA is known for its limited local search capabilities and tendency towards premature convergence [Reference Liu, Guo and Weng26]. Although LNS-GA was introduced by [Reference Han, Guo, Li, Zhi and Lv17] to improve GA’s local search performance, this enhancement does not fully address GA’s limitations. Consequently, both GA and LNS-GA underperform relative to FA. Nevertheless, GA’s ability to explore vast solution spaces allows it to find solutions that conform to constraints even in scenario 3, characterised by the highest complexity — unlike PSO, where exploration efforts are insufficient.

In contrast to PSO and GA, FA achieves a better balance between extensive exploration and local search capabilities. The incorporation of PSMS enhances FA’s ability to explore large solution spaces, while DES improves its local search performance, effectively addressing the challenges posed by TRCTSP. Notably, in scenario 3, the EFA demonstrates markedly superior performance compared to other meta-heuristics. We believe that EFA will continue to outperform other algorithms in cases with elevated complexity.