# Fuel-Efficient on-Orbit Service Vehicle Allocation Based on an Improved Discrete Particle Swarm Optimization Algorithm

Authors: Wu Jian; Liu Qingguo; Liu Xinxue; Li Yaxiong
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Given the limited fuel capacity of an on-orbit service vehicle (OSV), proper OSV allocation to satellites during each service mission is critical for economic fuel consumption. This allocation problem can be formulated as an optimization problem with many continuous and discrete design variables of wide domains. This problem can be effectively handled through the proposed approach that combines the tabu search with the discrete particle swarm optimization algorithm (DPSO-TS). First of all, Pontryagin’s minimum principle and genetic algorithm (GA) are exploited to find the most fuel-efficient transfer trajectory. This fuel efficiency maximization can then serve as the performance index of the OSV allocation optimization model problem. In particular, the maximization of the minimum residual fuel over individual OSVs is proposed as a performance index for OSV allocation optimization. The optimization problem is numerically solved through the proposed DPSO-TS algorithm. Finally, the simulation results demonstrate that the DPSO-TS algorithm has a higher accuracy compared to the DPSO, the DPSO-PDM and the DPSO-CSA algorithms in the premise that these four algorithms have the basically same computational time. The DPSO-TS algorithm can effectively solve the OSV allocation optimization problem.


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## Body

## 1. Introduction

As humans explore space deeper, more and more diverse satellites are deployed into space. On-orbit service of satellites is a type of space operations performed by astronauts, robots, or both to extend the lives of satellites and improve their abilities to perform various tasks [1]. On-orbit service vehicles (OSV) are particularly useful as an effective tool to extend lives of satellites. These vehicles perform several tasks including on-orbit assembly, maintenance, and logistic support [2]. Many important problems of the OSV are investigated in the existing literature. The study of transfer trajectory is the basic work of allocation of OSVs. Direct methods [3–5], indirect methods [6, 7], and hybrid methods [8, 9] are used to study the optimization of transfer trajectories of different types of space vehicles. The hybrid methods, one of which is adopted in this paper to optimize the transfer trajectories, combine the advantages of the direct and indirect methods.In general, multiple satellites need to be maintained in one service mission. The allocation of OSVs to satellites is an important problem that has the characteristics of a general resource allocation problem. However, the OSV allocation problem is distinguished by two aspects: (a) there are many optimization variables with a wide search scope for each, and (b) the optimization variables are of both the continuous and discrete types. Allocation optimization problem is a complex combinatorial optimization problem and known to be an NP hard, and discrete particle swarm optimization (DPSO) is a commonly used algorithm to solve allocation optimization problems. The traditional DPSO algorithm [10–12] is easy to be trapped in the local optimum, and hence some improved DPSO algorithms are studied. Inertia weight of DPSO is used to balance the global and local search capacity, and then the way of designing the adjustment strategy of inertia weight to solve the problem of falling into local optimum is widely studied. Li et al. [13] propose DPSO-PDM algorithm by the hybridizing adjustment strategy based on new particle diversity and adaptive mutation strategy, which effectively solves the problem of complex network community detection. Xin et al. [14] propose an improved DPSO algorithm by designing the linearly decreasing inertia weight, which effectively solves the problem of scheduling MIMO radar tasks. Nagra et al. [15] propose an improved a hybrid self-inertia weight adaptive PSO algorithm. The methods mentioned above improve the performance of DPSO algorithm by adjusting the inertia weight. There are some methods combining DPSO with other algorithms to improve the performance of DPSO algorithm. Wang et al. [16] propose SA-DPSO algorithm which combines the DPSO algorithm and simulated annealing algorithm. Vairam et al. [17] propose CHIDPSO algorithm by combining DPSO algorithm and cyber swarm algorithm, which improves the local search capacity of DPSO algorithm. Vasudevan and Sinha [18] proposed a hybrid optimization algorithm by combining GA and PSO. Shayeghi et al. [19] improve the mutation of DPSO based on similarity algorithm for optimization of transmission lines loading. Qi [20] proposes an improved DPSO algorithm by combing the iterated local search. Yang and Xu [21] propose DPSO-MFO algorithm, which is able to provide excellent performances as a nature-inspired heuristic algorithm. The methods mentioned above improve the performance of DPSO by enhancing the exploration and exploitation abilities, but the balance of exploration and exploitation is not controlled. Too much exploitation makes the algorithm converge quickly often to a local optimum, and too much exploration slows down the convergence of the procedure although it increases the probability of finding the global or near-optimum solution of the optimization problem.The tabu search (TS) algorithm is a hill-climbing algorithm, which is simple and easy to implement [22–24]. In order to keep the balance of exploration and exploitation abilities, we introduce TS algorithm into DPSO algorithm to improve the performance of DPSO algorithm. The proposed method keeps the balance of exploration and exploitation abilities by controlling the inertia weight and the length of the tabu table. The background of this paper is that multiple satellites need to be maintained in one service mission. As fuel is extremely important for OSVs, the fuel-efficient problem becomes the focus of allocation optimization problem. This paper is organized as follows. First, we establish in Section 2 an optimization model for the minimum duration transfer trajectory by using a hybrid method. Then, we propose in Section 3 an algorithm that combines DPSO algorithm with TS algorithm. This DPSO-TS algorithm is used for solving the OSV allocation optimization problem. Finally, in Section 4, we carry on several simulations to demonstrate the superiority of the DPSO-TS algorithm over DPSO algorithm and some, and the results demonstrated have higher convergence accuracy than DPSO algorithm and two relevant representative algorithms including DPSO-PDM algorithm and CHIDPSO algorithm.In order to make the paper readable, the nomenclature of variables is shown in Table1 before the optimization models are established.Table 1
Nomenclature of variables.
Variable
Explanation
r
Position vector
v
Velocity vector
F
Thrust
m
OSV mass
I
sp
Specific impulse
α
Thrust direction
t
f
Terminal time of transfer trajectory
λ
r
Adjoint variables of position vector
λ
v
Adjoint variables of velocity vector
λ
m
Adjoint variable of the OSV mass
α
∗
Optimal thrust direction
F
max
Maximum of thrust
t
off
1
Switching-off time of the 1st thrust segment
t
on
2
Switching-on time of the 2nd thrust segment
t
off
2
Switching-off time of the 2nd thrust segment
β
Design parameters of transfer trajectory
P
T
Population size of GA
P
c
Crossover probability of GA
P
m
Mutation probability of GA
N
t
Maximum iterations of GA
N
Count of OSVs
M
Count of satellites
O
Denotation of OSV
S
Denotation of satellites
X
Allocation matrix
t
m
Maneuvering time
m
remain
k
Remaining fuel mass ofOk after maintaining
M
B
Population size of DPSO
K
Number of satellites needed to be maintained
T
D
Maximum iterations of DPSO
ω
Inertia weight
c
1
Cognitive coefficient
c
2
Social coefficient
h
Iteration counter
δ
Time step
t
max
Mission duration
P
i
h
Individual fitness extreme
P
g
h
Global fitness extreme
l
T
Length of the tabu table
M
T
Size of the candidate set
T
T
Maximum iterations of TS
Openlist
Candidate set
Tabulist
Tabu table


## 2. Transfer Trajectory Optimization Based on a Hybrid Method

In this section, we provide the differential equations of motion of an OSV in an Earth-centred inertial (ECI) coordinate system. Then, an optimization model of the OSV transfer trajectory is established.


### 2.1. OSV Equations of Motion

The differential equations of motion [25] are(1)r˙=v,v˙=−μr3r+Fmα,m˙=−Fg0Isp,where r is the position vector, v is the velocity vector, F is the thrust, m is the OSV mass, Isp is the specific impulse of the engine, and α=αx,αy,αzT is the unit vector of in the thrust direction.


### 2.2. Transfer Trajectory Optimization

Assuming that the maneuvering time of each OSV is known or has been already determined, an optimization model for the transfer trajectory between each pair of an OSV and a satellite is established based on a hybrid method, which exploits Pontryagin’s minimum principle [26, 27] and genetic algorithm (GA) [28]. The more fuel one OSV has retained, the longer it can stay in the orbit and the more satellites it can serve. Therefore, the mass of the remaining fuel can be taken as the optimization index to be maximized, namely,(2)maxJ=mtf,where tf is the terminal time of each transfer trajectory and m is a function of the change in the OSV fuel mass.Based on the OSV equations of motion in formula (1), the Hamilton function can be derived as(3)H=λrv+λv−μr3r+Fma−λmFg0Isp,where λr and λv are the adjoint variables of the state variables and λm is the adjoint variable of the OSV mass m.According to Pontryagin’s minimum principle, the optimal thrust directionα∗ is obtained by sovling α=1 and ∂H/∂α=0. Then, the optimal thrust direction is(4)a∗=−λvλv.The partial derivatives of the Hamiltonian with respect to the adjoint variables are thus obtained. Consequently, the adjoint variables can be shown to satisfy the following differential equations:(5)λ˙r=μr3λv−3μ⋅λvr5r,λ˙v=−λr,λ˙m=−Fm2λv.Initial boundary conditions are(6)rt0=r0,vt0=v0,mt0=m0,and the terminal boundary constraints are(7)rtf=rf,vtf=vf.The optimal control under finite-thrust conditions is the switching mode of the engine strategy (bang-bang control). In this paper, the adopted switching mode is the on-off-on mode of the engine working sequence, and the switching moments are taken as the optimization variables. In the switching-on segment, the thrust is set toFmax. In the switching-off segment, the thrust is set to zero.For the GA-based hybrid method, the orbital optimization problem is converted into an optimization problem with parameters, including the initial values of the adjoint variables, the switching-off timetoff1 of the first thrust segment, the switching-on time ton2 of the second thrust segment, and the switching-off time toff2 of the second thrust segment. For maximizing the mass of the remaining OSV fuel under the given constraints, seventeen parameters β=λ1t0,λ2t0,toff1,ton2,toff2 are taken as optimization variables. The solution of this optimization problem using genetic algorithm is carried out as follows:Step 1: the design parametersβ=λ1t0,λ2t0,toff1,ton2,toff2 are encoded. Thirty chromosomes are generated randomly to form the initial population. The population size PT, the crossover probability Pc, the mutation probability Pm, and the maximum number of iterations Nt are set.Step 2: for each chromosome, the equations are solved to get the optimal thrust direction under the conditions (6) and (7). Then, the equations of motion (1) are solved to obtain the transfer trajectory. The fitness values of all chromosomes in the current generation are computed based on the fitness function in formula (2).Step 3: for the next generation, chromosomes are selected using the roulette-wheel selection method. Some genes on two different chromosomes reciprocally cross according to the crossover probability and mutate according to the mutation probability. The execution of the selection, crossover, and mutation operations produces the next generation population.Step 4: steps 2 and 3 are repeated until the GA fitness function values go belowε=10−6 or the maximum number of iterations N is reached.The hybrid method results in the OSV transfer trajectory with the maximum mass of the remaining fuel at a given maneuvering moment.


## 2.1. OSV Equations of Motion

The differential equations of motion [25] are(1)r˙=v,v˙=−μr3r+Fmα,m˙=−Fg0Isp,where r is the position vector, v is the velocity vector, F is the thrust, m is the OSV mass, Isp is the specific impulse of the engine, and α=αx,αy,αzT is the unit vector of in the thrust direction.


## 2.2. Transfer Trajectory Optimization

Assuming that the maneuvering time of each OSV is known or has been already determined, an optimization model for the transfer trajectory between each pair of an OSV and a satellite is established based on a hybrid method, which exploits Pontryagin’s minimum principle [26, 27] and genetic algorithm (GA) [28]. The more fuel one OSV has retained, the longer it can stay in the orbit and the more satellites it can serve. Therefore, the mass of the remaining fuel can be taken as the optimization index to be maximized, namely,(2)maxJ=mtf,where tf is the terminal time of each transfer trajectory and m is a function of the change in the OSV fuel mass.Based on the OSV equations of motion in formula (1), the Hamilton function can be derived as(3)H=λrv+λv−μr3r+Fma−λmFg0Isp,where λr and λv are the adjoint variables of the state variables and λm is the adjoint variable of the OSV mass m.According to Pontryagin’s minimum principle, the optimal thrust directionα∗ is obtained by sovling α=1 and ∂H/∂α=0. Then, the optimal thrust direction is(4)a∗=−λvλv.The partial derivatives of the Hamiltonian with respect to the adjoint variables are thus obtained. Consequently, the adjoint variables can be shown to satisfy the following differential equations:(5)λ˙r=μr3λv−3μ⋅λvr5r,λ˙v=−λr,λ˙m=−Fm2λv.Initial boundary conditions are(6)rt0=r0,vt0=v0,mt0=m0,and the terminal boundary constraints are(7)rtf=rf,vtf=vf.The optimal control under finite-thrust conditions is the switching mode of the engine strategy (bang-bang control). In this paper, the adopted switching mode is the on-off-on mode of the engine working sequence, and the switching moments are taken as the optimization variables. In the switching-on segment, the thrust is set toFmax. In the switching-off segment, the thrust is set to zero.For the GA-based hybrid method, the orbital optimization problem is converted into an optimization problem with parameters, including the initial values of the adjoint variables, the switching-off timetoff1 of the first thrust segment, the switching-on time ton2 of the second thrust segment, and the switching-off time toff2 of the second thrust segment. For maximizing the mass of the remaining OSV fuel under the given constraints, seventeen parameters β=λ1t0,λ2t0,toff1,ton2,toff2 are taken as optimization variables. The solution of this optimization problem using genetic algorithm is carried out as follows:Step 1: the design parametersβ=λ1t0,λ2t0,toff1,ton2,toff2 are encoded. Thirty chromosomes are generated randomly to form the initial population. The population size PT, the crossover probability Pc, the mutation probability Pm, and the maximum number of iterations Nt are set.Step 2: for each chromosome, the equations are solved to get the optimal thrust direction under the conditions (6) and (7). Then, the equations of motion (1) are solved to obtain the transfer trajectory. The fitness values of all chromosomes in the current generation are computed based on the fitness function in formula (2).Step 3: for the next generation, chromosomes are selected using the roulette-wheel selection method. Some genes on two different chromosomes reciprocally cross according to the crossover probability and mutate according to the mutation probability. The execution of the selection, crossover, and mutation operations produces the next generation population.Step 4: steps 2 and 3 are repeated until the GA fitness function values go belowε=10−6 or the maximum number of iterations N is reached.The hybrid method results in the OSV transfer trajectory with the maximum mass of the remaining fuel at a given maneuvering moment.


## 3. OSV Allocation Optimization Based on the DPSO-TS Algorithm

The OSV allocation optimization problem can be solved through the DPSO algorithm. The inertia weightω is an important parameter of the DPSO algorithm. The larger the inertia weight ω is, the stronger the global search capability is. TS algorithm is a hill-climbing algorithm with good local search performance, and it improves the efficiency of DPSO algorithm by exploiting the neighborhood of the results obtained by DPSO algorithm. In order to keep the balance between exploration and exploitation in DPSO-TS algorithm, we set the inertia weight ω to a larger value to get strong exploration ability. We propose to use TS algorithm in each iteration of DPSO algorithm to improve local search capacity on conditions that the exploitation ability is limited by setting the appropriate length of the tabu table.


### 3.1. Mathematical Optimization Model

Assume there areN OSVs denoted by O=O1,O2,…,ON, and M satellites denoted by S=S1,S2,…,SM. Each OSV serves at most a satellite in one mission, and each satellite can be served at most once. The allocation matrix X can be written as(8)xijtm=1,OiservingSjwith manoeuvringtimetm,0,Oinot servingSjwith manoeuvring timetm,where i=1,2,…,N, j=1,2,…,N and tm is maneuvering time. X includes the allocation plan of OSVs and their maneuevring times. The mathematical model of OSV allocation problem can be written as(9)maxfX=maxminmremain1X,mremain1X,…,mremainsX,s≤Ns.t.∑j=1Mxijtm≤1,∀i=1,2,…,N0<∑i=1Nxijtm≤1,∀j=1,2,…,Mmremaink≥mkeep,∀k=1,2,…,N,where mremaink is the remaining fuel mass of the kth OSV after maintaining, and mkeep is the mass of the fuel used to keep the orbit. The inner min operator minmremain1X,mremain1X,…,mremainsX returns the minimum mass of the remaining fuel among individual OSVs used to maintain satellites according to X. The outer max operator maxminmremain1X,mremain1X,…,mremainsX searches for the allocation plan X that maximizes the minimum mass value. The optimization index reflects the overall service capability of OSVs.


### 3.2. Optimization Problem Solution Based on the DPSO-TS Algorithm

The OSV allocation optimization problem can be solved using the proposed DPSO-TS algorithm. Details of the DPSO-TS algorithm are given for two parts: the implementation steps of DPSO-TS algorithm and DPSO algorithm and some steps and strategies of DPSO-TS algorithm.


#### 3.2.1. The Implementation Steps of DPSO-TS Algorithm

The implementation steps of DPSO-TS algorithm areStep 1: set the DPSO-TS parameters including the population sizeMB of the particle swarm, the OSV count N, the number of satellites K need to be maintained, the maximum number of iterations TD, the inertia weight ω, the cognitive coefficient c1, the social coefficient c2, and the iteration counter h = 1. Select a certain time step δ to discretize the mission duration tmax.Step 2: initialize the OSV particle swarm population. Set parameter values for each particle including an OSV serial number and a randomly generated maneuvering moment. For computational efficiency, maneuvering moments and OSV numbers of OSVs are separately used for the DPSO update by formula (12). The fitness function values fPsi of all particles are calculated, and the individual fitness extreme Pih and the global fitness extreme Pgh are recorded.Step 3: the particle positions are updated by formula (12). The fitness function values fPsi of all particles are calculated. The individual fitness extreme Pih and the global fitness extreme Pgh are updated.Step 4: the parameters of TS algorithm are set including the lengthlT of the tabu table, the size MT of the candidate set, and the maximum number of iterations TT.Step 5: set the initial solutionPT=Pgh and the optimal solution PTbest=Pgh of the TS algorithm.Step 6: according to the centrality and diversity search strategies, generate the neighborhood of the current solution, and selectNT particles from the neighborhood as the candidate set OpenlistY.Step 7: according to the fitness function valuefYi, judge whether the amnesty criterion is met. If so, the corresponding candidate solution Yi will be placed in the tabu table TabulistY and the solution in the tabu table will be released. Otherwise, keep the tabu table intact.Step 8: check whether the TS terminal conditions are satisfied. If so, outputPgh=PTbest. Otherwise, go to Step 6.Step 9: increase the iteration number,h=h+1. Check whether the DPSO termination conditions are satisfied. If so, output Pgh. Otherwise, go to Step 3.The flow chart of DPSO-TS algorithm is shown in Figure1.Figure 1
Flow chart of DPSO-TS.


#### 3.2.2. Some Steps and Strategies of DPSO-TS Algorithm

(1) Coding of Particles. Particles of the OSV allocation problem are encoded in a decimal system where each particle has a length of 2×N. Each particle is encoded as shown in Figure 2, where O1,…,ON are N OSVs, and t1,…,tN are the corresponding maneuvering moments. The value of Oi indicates the number of satellites served by the ith OSV. This value belongs to the set 0,1,2,…,M, where 0 means no satellite is served. We set the duration of the mission to 0,tmax. If Oi=0, set ti=−1.Figure 2
Coding of a particle in the OSV allocation problem.(2) Calculation of the Fitness Function. Objective function (9) is taken as the fitness function for the particles(10)maxfPsi=maxminmremain1Psi,mremain1Psi,…,mremainsPsi,s≤N,where Psi is the ith particle.(3) Particle Position and Velocity Update. The position of a particle is determined by the particle velocity, the individual fitness extreme, and the global fitness extreme [29]. The particle position and velocity in a PSO algorithm are updated as (11)vih+1=ω⋅vih+c1⋅r1⋅Pih−xih+c2⋅r2⋅Pgh−xih,xih+1=xih+vih+1,where Pih is the individual fitness extreme of the ith particle in the hth iteration; Pgh is the global fitness extreme in the hth iteration, xih is the fitness function value of the ith particle in the hth iteration; xih+1 is the fitness function value of the ith particle in the (h + 1)th iteration; vih is the particle velocity value of the ith particle in the hth iteration; vih+1 is the particle velocity value of the ith particle in the (h + 1)th iteration; ω is the inertia weight which reflects the extent to which a particle keeps the current velocity; c1 is the cognitive coefficient, which measures how far a particle gets close to the individual fitness extreme; c2 is the social coefficient, which measures how far a particle gets close to the global fitness extreme; and both r1 and r2 are random numbers in the interval [0, 1].On the basis of the PSO, we propose the following DPSO update rules for the particle position and velocity:(12)Psih+1=c2⊗F3c1⊗F2ω⊗F1Psih,Pih,Pgh,where Psih and Psih+1 are the fitness function values of the ith particle in the hth and (h + 1)th iterations, respectively. Formula (12) includes three formulas which are written as formulas (13)–(15).(13)Ψih=ω⊗F1Psih=F1Psih,rand<ω,Psih,rand≥ω,(14)Φih=c1⊗F2Ψih,Pih=F2Ψih,Pih,rand<c1,Ψih,rand≥c1,(15)Psih+1=c2⊗F3Φih,Pgh=F3Φih,Pgh,rand<c2,Φih,rand≥c2.In formula (13), Ψih is the value of the ith particle in the hth iteration and is obtained via the product ω⊗F1Psih. The function rand generates a random number from a uniform distribution on [0, 1]. If rand<ω, the function F1Psih is computed leading to the random generation of two integers a and b, from the set 0,1,2,…,M, and hence the interchange of the values of the ith particle at positions a and b. If rand≥ω, the output is Psih which means that the coding values of the ith particle are kept unchanged.In formula (14), Φih is the value of the ith particle in the hth iteration, and is obtained via the product c1⊗F2Ψih,Pih, where Pih is the individual fitness extreme position in the hth iteration. If rand<c1, the function F2Ψih,Pih is computed leading to the random generation of two integers a and b, from the set 0,1,2,…,M, and hence the interchange of the values of Ψih and Pih at positions a and b. If rand≥c1, the output is Ψih, which means that the coding values of Ψih are kept unchanged.In formula (15), Pgh is the global fitness extreme position in the hth iteration. If rand<c2, function F3Φih,Pgh is computed leading to the random generation of two integers a and b, from the set 0,1,2,…,M, and hence the interchange of the values of Φih and Pgh at positions a and b. If rand≥c2, the output is Φih which means that the coding values on Φih are kept unchanged.(4) Some Strategies of TS Algorithm. In this paper, the initial solution of the TS algorithm is the global extremum. Pg in each iteration of the DPSO algorithm. The fitness function is written as in formula (10). TS algorithm includes some strategies: if the solution is superior to the current optimal solution, the optimal solution should be amnesty; tabu tables which are first-in first-out (FIFO) queue are used to prevent repeated search for visited solutions and to avoid falling into local loops; the centralized search strategy refers to strengthening search in the neighborhood of good solutions. A diversity search strategy looks for solutions in unknown or highly unexplored regions.


## 3.1. Mathematical Optimization Model

Assume there areN OSVs denoted by O=O1,O2,…,ON, and M satellites denoted by S=S1,S2,…,SM. Each OSV serves at most a satellite in one mission, and each satellite can be served at most once. The allocation matrix X can be written as(8)xijtm=1,OiservingSjwith manoeuvringtimetm,0,Oinot servingSjwith manoeuvring timetm,where i=1,2,…,N, j=1,2,…,N and tm is maneuvering time. X includes the allocation plan of OSVs and their maneuevring times. The mathematical model of OSV allocation problem can be written as(9)maxfX=maxminmremain1X,mremain1X,…,mremainsX,s≤Ns.t.∑j=1Mxijtm≤1,∀i=1,2,…,N0<∑i=1Nxijtm≤1,∀j=1,2,…,Mmremaink≥mkeep,∀k=1,2,…,N,where mremaink is the remaining fuel mass of the kth OSV after maintaining, and mkeep is the mass of the fuel used to keep the orbit. The inner min operator minmremain1X,mremain1X,…,mremainsX returns the minimum mass of the remaining fuel among individual OSVs used to maintain satellites according to X. The outer max operator maxminmremain1X,mremain1X,…,mremainsX searches for the allocation plan X that maximizes the minimum mass value. The optimization index reflects the overall service capability of OSVs.


## 3.2. Optimization Problem Solution Based on the DPSO-TS Algorithm

The OSV allocation optimization problem can be solved using the proposed DPSO-TS algorithm. Details of the DPSO-TS algorithm are given for two parts: the implementation steps of DPSO-TS algorithm and DPSO algorithm and some steps and strategies of DPSO-TS algorithm.


### 3.2.1. The Implementation Steps of DPSO-TS Algorithm

The implementation steps of DPSO-TS algorithm areStep 1: set the DPSO-TS parameters including the population sizeMB of the particle swarm, the OSV count N, the number of satellites K need to be maintained, the maximum number of iterations TD, the inertia weight ω, the cognitive coefficient c1, the social coefficient c2, and the iteration counter h = 1. Select a certain time step δ to discretize the mission duration tmax.Step 2: initialize the OSV particle swarm population. Set parameter values for each particle including an OSV serial number and a randomly generated maneuvering moment. For computational efficiency, maneuvering moments and OSV numbers of OSVs are separately used for the DPSO update by formula (12). The fitness function values fPsi of all particles are calculated, and the individual fitness extreme Pih and the global fitness extreme Pgh are recorded.Step 3: the particle positions are updated by formula (12). The fitness function values fPsi of all particles are calculated. The individual fitness extreme Pih and the global fitness extreme Pgh are updated.Step 4: the parameters of TS algorithm are set including the lengthlT of the tabu table, the size MT of the candidate set, and the maximum number of iterations TT.Step 5: set the initial solutionPT=Pgh and the optimal solution PTbest=Pgh of the TS algorithm.Step 6: according to the centrality and diversity search strategies, generate the neighborhood of the current solution, and selectNT particles from the neighborhood as the candidate set OpenlistY.Step 7: according to the fitness function valuefYi, judge whether the amnesty criterion is met. If so, the corresponding candidate solution Yi will be placed in the tabu table TabulistY and the solution in the tabu table will be released. Otherwise, keep the tabu table intact.Step 8: check whether the TS terminal conditions are satisfied. If so, outputPgh=PTbest. Otherwise, go to Step 6.Step 9: increase the iteration number,h=h+1. Check whether the DPSO termination conditions are satisfied. If so, output Pgh. Otherwise, go to Step 3.The flow chart of DPSO-TS algorithm is shown in Figure1.Figure 1
Flow chart of DPSO-TS.


### 3.2.2. Some Steps and Strategies of DPSO-TS Algorithm

(1) Coding of Particles. Particles of the OSV allocation problem are encoded in a decimal system where each particle has a length of 2×N. Each particle is encoded as shown in Figure 2, where O1,…,ON are N OSVs, and t1,…,tN are the corresponding maneuvering moments. The value of Oi indicates the number of satellites served by the ith OSV. This value belongs to the set 0,1,2,…,M, where 0 means no satellite is served. We set the duration of the mission to 0,tmax. If Oi=0, set ti=−1.Figure 2
Coding of a particle in the OSV allocation problem.(2) Calculation of the Fitness Function. Objective function (9) is taken as the fitness function for the particles(10)maxfPsi=maxminmremain1Psi,mremain1Psi,…,mremainsPsi,s≤N,where Psi is the ith particle.(3) Particle Position and Velocity Update. The position of a particle is determined by the particle velocity, the individual fitness extreme, and the global fitness extreme [29]. The particle position and velocity in a PSO algorithm are updated as (11)vih+1=ω⋅vih+c1⋅r1⋅Pih−xih+c2⋅r2⋅Pgh−xih,xih+1=xih+vih+1,where Pih is the individual fitness extreme of the ith particle in the hth iteration; Pgh is the global fitness extreme in the hth iteration, xih is the fitness function value of the ith particle in the hth iteration; xih+1 is the fitness function value of the ith particle in the (h + 1)th iteration; vih is the particle velocity value of the ith particle in the hth iteration; vih+1 is the particle velocity value of the ith particle in the (h + 1)th iteration; ω is the inertia weight which reflects the extent to which a particle keeps the current velocity; c1 is the cognitive coefficient, which measures how far a particle gets close to the individual fitness extreme; c2 is the social coefficient, which measures how far a particle gets close to the global fitness extreme; and both r1 and r2 are random numbers in the interval [0, 1].On the basis of the PSO, we propose the following DPSO update rules for the particle position and velocity:(12)Psih+1=c2⊗F3c1⊗F2ω⊗F1Psih,Pih,Pgh,where Psih and Psih+1 are the fitness function values of the ith particle in the hth and (h + 1)th iterations, respectively. Formula (12) includes three formulas which are written as formulas (13)–(15).(13)Ψih=ω⊗F1Psih=F1Psih,rand<ω,Psih,rand≥ω,(14)Φih=c1⊗F2Ψih,Pih=F2Ψih,Pih,rand<c1,Ψih,rand≥c1,(15)Psih+1=c2⊗F3Φih,Pgh=F3Φih,Pgh,rand<c2,Φih,rand≥c2.In formula (13), Ψih is the value of the ith particle in the hth iteration and is obtained via the product ω⊗F1Psih. The function rand generates a random number from a uniform distribution on [0, 1]. If rand<ω, the function F1Psih is computed leading to the random generation of two integers a and b, from the set 0,1,2,…,M, and hence the interchange of the values of the ith particle at positions a and b. If rand≥ω, the output is Psih which means that the coding values of the ith particle are kept unchanged.In formula (14), Φih is the value of the ith particle in the hth iteration, and is obtained via the product c1⊗F2Ψih,Pih, where Pih is the individual fitness extreme position in the hth iteration. If rand<c1, the function F2Ψih,Pih is computed leading to the random generation of two integers a and b, from the set 0,1,2,…,M, and hence the interchange of the values of Ψih and Pih at positions a and b. If rand≥c1, the output is Ψih, which means that the coding values of Ψih are kept unchanged.In formula (15), Pgh is the global fitness extreme position in the hth iteration. If rand<c2, function F3Φih,Pgh is computed leading to the random generation of two integers a and b, from the set 0,1,2,…,M, and hence the interchange of the values of Φih and Pgh at positions a and b. If rand≥c2, the output is Φih which means that the coding values on Φih are kept unchanged.(4) Some Strategies of TS Algorithm. In this paper, the initial solution of the TS algorithm is the global extremum. Pg in each iteration of the DPSO algorithm. The fitness function is written as in formula (10). TS algorithm includes some strategies: if the solution is superior to the current optimal solution, the optimal solution should be amnesty; tabu tables which are first-in first-out (FIFO) queue are used to prevent repeated search for visited solutions and to avoid falling into local loops; the centralized search strategy refers to strengthening search in the neighborhood of good solutions. A diversity search strategy looks for solutions in unknown or highly unexplored regions.


## 3.2.1. The Implementation Steps of DPSO-TS Algorithm

The implementation steps of DPSO-TS algorithm areStep 1: set the DPSO-TS parameters including the population sizeMB of the particle swarm, the OSV count N, the number of satellites K need to be maintained, the maximum number of iterations TD, the inertia weight ω, the cognitive coefficient c1, the social coefficient c2, and the iteration counter h = 1. Select a certain time step δ to discretize the mission duration tmax.Step 2: initialize the OSV particle swarm population. Set parameter values for each particle including an OSV serial number and a randomly generated maneuvering moment. For computational efficiency, maneuvering moments and OSV numbers of OSVs are separately used for the DPSO update by formula (12). The fitness function values fPsi of all particles are calculated, and the individual fitness extreme Pih and the global fitness extreme Pgh are recorded.Step 3: the particle positions are updated by formula (12). The fitness function values fPsi of all particles are calculated. The individual fitness extreme Pih and the global fitness extreme Pgh are updated.Step 4: the parameters of TS algorithm are set including the lengthlT of the tabu table, the size MT of the candidate set, and the maximum number of iterations TT.Step 5: set the initial solutionPT=Pgh and the optimal solution PTbest=Pgh of the TS algorithm.Step 6: according to the centrality and diversity search strategies, generate the neighborhood of the current solution, and selectNT particles from the neighborhood as the candidate set OpenlistY.Step 7: according to the fitness function valuefYi, judge whether the amnesty criterion is met. If so, the corresponding candidate solution Yi will be placed in the tabu table TabulistY and the solution in the tabu table will be released. Otherwise, keep the tabu table intact.Step 8: check whether the TS terminal conditions are satisfied. If so, outputPgh=PTbest. Otherwise, go to Step 6.Step 9: increase the iteration number,h=h+1. Check whether the DPSO termination conditions are satisfied. If so, output Pgh. Otherwise, go to Step 3.The flow chart of DPSO-TS algorithm is shown in Figure1.Figure 1
Flow chart of DPSO-TS.


## 3.2.2. Some Steps and Strategies of DPSO-TS Algorithm

(1) Coding of Particles. Particles of the OSV allocation problem are encoded in a decimal system where each particle has a length of 2×N. Each particle is encoded as shown in Figure 2, where O1,…,ON are N OSVs, and t1,…,tN are the corresponding maneuvering moments. The value of Oi indicates the number of satellites served by the ith OSV. This value belongs to the set 0,1,2,…,M, where 0 means no satellite is served. We set the duration of the mission to 0,tmax. If Oi=0, set ti=−1.Figure 2
Coding of a particle in the OSV allocation problem.(2) Calculation of the Fitness Function. Objective function (9) is taken as the fitness function for the particles(10)maxfPsi=maxminmremain1Psi,mremain1Psi,…,mremainsPsi,s≤N,where Psi is the ith particle.(3) Particle Position and Velocity Update. The position of a particle is determined by the particle velocity, the individual fitness extreme, and the global fitness extreme [29]. The particle position and velocity in a PSO algorithm are updated as (11)vih+1=ω⋅vih+c1⋅r1⋅Pih−xih+c2⋅r2⋅Pgh−xih,xih+1=xih+vih+1,where Pih is the individual fitness extreme of the ith particle in the hth iteration; Pgh is the global fitness extreme in the hth iteration, xih is the fitness function value of the ith particle in the hth iteration; xih+1 is the fitness function value of the ith particle in the (h + 1)th iteration; vih is the particle velocity value of the ith particle in the hth iteration; vih+1 is the particle velocity value of the ith particle in the (h + 1)th iteration; ω is the inertia weight which reflects the extent to which a particle keeps the current velocity; c1 is the cognitive coefficient, which measures how far a particle gets close to the individual fitness extreme; c2 is the social coefficient, which measures how far a particle gets close to the global fitness extreme; and both r1 and r2 are random numbers in the interval [0, 1].On the basis of the PSO, we propose the following DPSO update rules for the particle position and velocity:(12)Psih+1=c2⊗F3c1⊗F2ω⊗F1Psih,Pih,Pgh,where Psih and Psih+1 are the fitness function values of the ith particle in the hth and (h + 1)th iterations, respectively. Formula (12) includes three formulas which are written as formulas (13)–(15).(13)Ψih=ω⊗F1Psih=F1Psih,rand<ω,Psih,rand≥ω,(14)Φih=c1⊗F2Ψih,Pih=F2Ψih,Pih,rand<c1,Ψih,rand≥c1,(15)Psih+1=c2⊗F3Φih,Pgh=F3Φih,Pgh,rand<c2,Φih,rand≥c2.In formula (13), Ψih is the value of the ith particle in the hth iteration and is obtained via the product ω⊗F1Psih. The function rand generates a random number from a uniform distribution on [0, 1]. If rand<ω, the function F1Psih is computed leading to the random generation of two integers a and b, from the set 0,1,2,…,M, and hence the interchange of the values of the ith particle at positions a and b. If rand≥ω, the output is Psih which means that the coding values of the ith particle are kept unchanged.In formula (14), Φih is the value of the ith particle in the hth iteration, and is obtained via the product c1⊗F2Ψih,Pih, where Pih is the individual fitness extreme position in the hth iteration. If rand<c1, the function F2Ψih,Pih is computed leading to the random generation of two integers a and b, from the set 0,1,2,…,M, and hence the interchange of the values of Ψih and Pih at positions a and b. If rand≥c1, the output is Ψih, which means that the coding values of Ψih are kept unchanged.In formula (15), Pgh is the global fitness extreme position in the hth iteration. If rand<c2, function F3Φih,Pgh is computed leading to the random generation of two integers a and b, from the set 0,1,2,…,M, and hence the interchange of the values of Φih and Pgh at positions a and b. If rand≥c2, the output is Φih which means that the coding values on Φih are kept unchanged.(4) Some Strategies of TS Algorithm. In this paper, the initial solution of the TS algorithm is the global extremum. Pg in each iteration of the DPSO algorithm. The fitness function is written as in formula (10). TS algorithm includes some strategies: if the solution is superior to the current optimal solution, the optimal solution should be amnesty; tabu tables which are first-in first-out (FIFO) queue are used to prevent repeated search for visited solutions and to avoid falling into local loops; the centralized search strategy refers to strengthening search in the neighborhood of good solutions. A diversity search strategy looks for solutions in unknown or highly unexplored regions.


## 4. Simulations and Analyses of Results

### 4.1. Simulations

The parking orbit elements of satellites (at 0:00 on July 1, 2099) in need of service are shown in Table2. The parking orbit elements of the OSVs (at 0:00 on July 1, 2099) are shown in Table 3. The maintaining time for a satellite is an hour ,and the on-orbit service mission should be achieved in 24 hours (From 0:00 on July 1, 2099 to 0:00 on July 2, 2099), which means tm+ttra (ttra is the duration of OSV and satellite from maneuvering time to rendezvous time) is in 0,23×3600s. The mass of each OSV is 1000 kg, each OSV carries 600 kg fuel the engine thrust is 490 N, and the gas jet velocity of 3000 m/s. mkeep is 100 kg. The population size PT is 20, the crossover probability Pc is 0.7, the mutation probability Pm is 0.1, and the maximum number of iterations Nt is 30. The population size MB is 20, maximum number of iterations TD is 30, the inertia weight ω is 0.8, the cognitive coefficient c1 is 0.6, and the social coefficient c2 is 0.6. The length lT of the tabu table is 2, the size MT of candidate is 5, and the maximum number of iterations TT is 8. Time step δ is 1 s, and I=tm/δ. The simulation performs ten independent experiments. In order to reduce computational time, each trajectory is computed by one independent blade of blade servers.Table 2
Parking orbit elements of satellites.
Satellite
A
km
e
I
°
α
°
Ω
°
f
°
S
1
9306
0.238550
28.321
224.138
307.927
135.862
S
2
7059
0.000225
98.084
76.815
185.081
283.332
S
3
9237
0.000165
52.006
171.386
350.005
188.701
S
4
7828
0.001219
101.618
87.562
293.441
282.681
S
5
7707
0.054088
99.031
56.562
347.938
308.882
S
6
7554
0.002882
90.045
245.844
85.215
113.968
S
7
8428
0.201645
82.982
262.555
29.901
44.341
S
8
9501
0.213200
28.226
34.607
152.421
325.392
S
9
9401
0.016556
64.822
302.192
337.387
56.304
S
10
9939
0.321639
56.917
231.629
218.498
128.371
S
11
10687
0.355917
57.022
347.224
317.457
122.776
S
12
6903
0.000549
96.390
110.033
325.324
203.070Table 3
Parking orbit elements of OSVs.
OSV
A
km
e
I
°
α
°
Ω
°
f
°
O
1
7751
0.000120
32.223
56.232
76.223
255.520
O
2
8400
0.000225
78.002
102.332
253.010
122.243
O
3
8526
0.004120
76.924
154.257
73.020
243.000
O
4
9201
0.370056
55.223
180.922
223.001
95.620
O
5
7002
0.004021
87.235
102.556
330.765
88.598
O
6
8002
0.529030
145.667
156.007
200.321
270.321
O
7
9923
0.052301
120.023
254.124
122.232
55.220
O
8
7125
0.003924
42.084
51.815
255.081
30.200
O
9
8838
0.000249
98.178
69.604
155.872
150.009
O
10
9123
0.013793
52.674
109.398
23.270
200.020
O
11
7685
0.004693
37.477
52.707
158.017
60.000
O
12
9215
0.489300
55.371
128.871
305.501
80.000
O
13
7453
0.004218
23.618
77.562
293.441
50.101
O
14
9675
0.054089
99.031
56.562
187.938
160.032
O
15
6874
0.000555
97.420
104.012
316.284
204.781
O
16
7534
0.003002
91.145
239.735
84.325
113.000We design comparison experiments with DPSO algorithm, DPSO-PDM algorithm, and CHIDPSO algorithm. The parameters of DPSO algorithm are the same as those of DPSO-TS algorithm. The parameters of DPSO algorithm, part of DPSO-PDM algorithm, are the same as those of DPSO-TS algorithm, and the other parameters are the same as those in the article written by Li et al. [13]. The parameters of the DPSO algorithm part of CHIDPSO algorithm are the same as those of DPSO-TS algorithm, and the other parameters are the same as those in the article written by Vairam et al. [17].


### 4.2. Analyses of Results

Results including allocation plans, computational time, and the mass of the remaining fuel obtained by DPSO-TS algorithm, DPSO algorithm, DPSO-PDM algorithm, and CHIDPSO algorithm in ten independent experiments are shown in Tables4–7, separately. The maneuvering time of OSV selected from the integer from the duration of on-orbit service mission is 0,23×3600s. The maneuvering time I obtained by these four algorithms in different experiments are shown in Tables 4–7, and the best results can be obtained at these maneuvering times. It can be seen from Tables 4–7 that the average mass of the remaining fuel obtained by DPSO-TS algorithm, DPSO algorithm, DPSO-PDM algorithm, and CHIDPSO algorithm is 538.5 kg, 467.8 kg, 488.9 kg, and 508.8 kg; the average mass of the remaining fuel obtained by DPSO-TS algorithm is better than that of the other algorithms. Except the ninth experiment of DPSO-TS algorithm, the mass of the remaining fuel obtained by DPSO-TS algorithm in the other experiments is superior to that obtained by the other algorithms in any experiment; the average computational time obtained by DPSO-TS algorithm, DPSO algorithm, DPSO-PDM algorithm, and CHIDPSO algorithm is 1792.3 s, 1623.2 s, 1769.1 s, and 1797.3 s. The average computational time obtained by DPSO-TS algorithm differs from the shortest average computational time by 169.1 s, which has no effect on the service mission because there is a long preparation time before the mission is executed. The computational time of these four algorithms can be considered basically the same.Table 4
Results of allocation optimization based on DPSO-TS algorithm.
No.
X
S
1
S
2
S
3
S
4
S
5
S
6
S
7
S
8
S
9
S
10
S
11
S
12
t
c
s
f
X
1
OSV
O
4
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
15
1801.4
540.3
I
2301
6415
67
167
2251
449
781
1
299
5233
649
3339
2
OSV
O
9
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
4
O
7
O
12
O
15
1788.9
537.4
I
4609
6422
1776
167
2254
2304
222
0
298
5233
1233
3339
3
OSV
O
4
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
15
1801.4
541.7
I
5521
6413
67
169
2217
32
781
0
299
5233
24
144
4
OSV
O
4
O
15
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
8
1791.7
539.3
I
221
6415
69
167
2251
667
781
5
299
4446
640
301
5
OSV
O
4
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
15
1739.5
540.0
I
2304
6392
67
167
2251
379
780
0
299
5233
649
3332
6
OSV
O
9
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
4
O
7
O
12
O
15
1820.9
549.6
I
4001
6409
378
166
2259
1339
239
15
298
5409
1602
3337
7
OSV
O
7
O
8
O
10
O
11
O
16
O
13
O
3
O
9
O
14
O
4
O
12
O
15
1800.9
539.8
I
5521
6413
25
169
2217
32
699
7
271
5233
755
144
8
OSV
O
4
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
15
1802.8
538.2
I
5522
6413
65
168
2216
32
781
2
299
5233
23
142
9
OSV
O
4
O
15
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
5
1772.3
520.4
I
5522
6413
65
168
2216
32
781
2
299
5233
23
142
10
OSV
O
4
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
15
1803.6
538.2
I
5524
6202
12
168
2109
32
789
0
297
5233
22
144Table 5
Results of allocation optimization based on DPSO algorithm.
No.
X
S
1
S
2
S
3
S
4
S
5
S
6
S
7
S
8
S
9
S
10
S
11
S
12
t
c
s
f
X
1
OSV
O
7
O
11
O
6
O
13
O
16
O
8
O
3
O
12
O
9
O
10
O
14
O
15
1622.0
474.2
I
6600
350
471
337
2535
107
4730
4
335
7774
6629
4209
2
OSV
O
7
O
8
O
6
O
13
O
16
O
11
O
3
O
14
O
9
O
10
O
12
O
15
1607.5
467.3
I
6630
1323
470
360
2535
24
4735
4701
350
7772
258
4311
3
OSV
O
4
O
15
O
6
O
11
O
16
O
13
O
3
O
12
O
9
O
7
O
14
O
8
1610.3
472.8
I
5501
6009
498
200
2215
33
781
7
299
5405
24
251
4
OSV
O
7
O
11
O
6
O
13
O
16
O
10
O
3
O
12
O
9
O
14
O
2
O
8
1626.7
478.3
I
5902
355
400
2099
2435
12
4688
2289
300
5405
18
1455
5
OSV
O
4
O
15
O
6
O
11
O
16
O
13
O
3
O
12
O
9
O
7
O
14
O
8
1611.2
470.6
I
5523
6010
0
1
2420
42
1733
5
249
5422
22
250
6
OSV
O
7
O
8
O
10
O
11
O
16
O
13
O
3
O
9
O
14
O
4
O
12
O
15
1629.9
449.4
I
5322
427
27
469
2015
125
621
4456
378
4236
765
742
7
OSV
O
4
O
11
O
10
O
13
O
16
O
15
O
3
O
12
O
9
O
14
O
2
O
8
1635.5
470.2
I
4755
327
447
2179
2336
120
4587
67
340
5425
2234
1709
8
OSV
O
10
O
15
O
4
O
11
O
16
O
13
O
3
O
12
O
9
O
7
O
14
O
8
1638.8
459.3
I
5090
7426
647
7162
3315
2
4908
3
292
4263
378
140
9
OSV
O
4
O
5
O
10
O
11
O
16
O
13
O
6
O
14
O
9
O
7
O
12
O
8
1627.5
468.1
I
4523
6423
62
1778
4213
310
1741
12
2690
4233
276
242
10
OSV
O
4
O
5
O
10
O
11
O
16
O
13
O
3
O
12
O
9
O
7
O
14
O
15
1622.9
468.0
I
6734
628
17
1556
6421
422
276
13
449
5345
212
260Table 6
Results of allocation optimization based on DPSO-PDM algorithm.
No.
X
S
1
S
2
S
3
S
4
S
5
S
6
S
7
S
8
S
9
S
10
S
11
S
12
t
c
s
f
X
1
OSV
O
4
O
11
O
10
O
13
O
16
O
15
O
3
O
12
O
9
O
14
O
2
O
8
1750.5
475.6
I
4635
127
825
4133
2001
127
2371
60
236
4416
2055
169
2
OSV
O
7
O
8
O
10
O
11
O
16
O
13
O
3
O
9
O
14
O
4
O
12
O
15
1789.0
502.4
I
6754
466
376
169
2167
4125
2621
456
366
4233
65
142
3
OSV
O
4
O
5
O
10
O
11
O
16
O
13
O
3
O
12
O
9
O
7
O
14
O
15
1752.9
481.0
I
2520
412
79
168
2255
321
780
650
498
5437
6663
4004
4
OSV
O
4
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
15
1752.4
480.2
I
2217
6330
162
165
2241
1438
3891
15
245
5204
620
2341
5
OSV
O
4
O
15
O
6
O
11
O
16
O
13
O
3
O
12
O
9
O
7
O
14
O
8
1749.5
499.1
I
2522
160
494
211
1238
4300
6750
334
498
5001
6678
123
6
OSV
O
4
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
15
1735.6
490.3
I
5467
3442
678
6637
217
360
4661
15
256
5109
259
341
7
OSV
O
4
O
15
O
10
O
12
O
16
O
13
O
3
O
9
O
14
O
7
O
6
O
8
1785.9
500.4
I
5361
412
624
279
327
329
6695
176
10
5408
259
3134
8
OSV
O
4
O
15
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
5
1812.4
492.3
I
3546
2455
167
163
6721
188
428
0
378
5013
343
667
9
OSV
O
4
O
15
O
6
O
11
O
16
O
13
O
3
O
12
O
9
O
7
O
14
O
8
1789.3
495.6
I
5411
132
600
234
2709
6035
490
29
306
5218
4345
176
10
OSV
O
3
O
6
O
10
O
11
O
5
O
13
O
3
O
14
O
9
O
7
O
12
O
8
1773.4
472.4
I
452
622
152
7065
2036
42
189
236
429
4986
231
744Table 7
Results of allocation optimization based on CHIDPSO algorithm.
No.
X
S
1
S
2
S
3
S
4
S
5
S
6
S
7
S
8
S
9
S
10
S
11
S
12
t
c
s
f
X
1
OSV
O
1
O
8
O
10
O
11
O
16
O
2
O
3
O
14
O
6
O
7
O
12
O
5
1827.3
511.6
I
4372
2276
567
5901
253
207
14
15
236
421
3690
1306
2
OSV
O
4
O
15
O
10
O
12
O
16
O
13
O
3
O
9
O
14
O
7
O
6
O
8
1792.9
506.7
I
5009
398
521
1277
320
308
650
179
45
5268
250
2971
3
OSV
O
2
O
9
O
10
O
15
O
16
O
13
O
3
O
14
O
8
O
7
O
11
O
12
1779.3
520.3
I
4538
203
617
7001
3246
319
483
16
99
4205
876
1641
4
OSV
O
7
O
8
O
11
O
12
O
16
O
13
O
3
O
9
O
14
O
4
O
15
O
1
1833.2
489.3
I
352
667
2396
4690
392
127
242
4101
326
646
6095
7020
5
OSV
O
4
O
15
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
5
1804.5
520.9
I
2598
1423
261
460
309
242
428
3356
372
4410
279
461
6
OSV
O
2
O
8
O
10
O
11
O
15
O
6
O
3
O
13
O
4
O
7
O
12
O
14
1790.3
499.6
I
3871
100
1278
473
4133
690
431
299
143
428
4638
230
7
OSV
O
3
O
6
O
10
O
11
O
5
O
13
O
3
O
14
O
9
O
7
O
12
O
8
1802.0
510.6
I
4328
3568
142
6011
230
108
48
4376
25
5099
4238
541
8
OSV
O
4
O
6
O
10
O
11
O
5
O
13
O
3
O
14
O
9
O
7
O
12
O
8
1739.8
511.5
I
2407
3699
183
5280
431
324
380
2325
421
5203
6218
1541
9
OSV
O
7
O
11
O
6
O
13
O
16
O
10
O
3
O
12
O
9
O
14
O
2
O
8
1788.4
499.6
I
4598
1256
427
3899
2408
19
722
2289
123
6263
701
23
10
OSV
O
4
O
12
O
10
O
11
O
16
O
1
O
3
O
5
O
9
O
7
O
13
O
14
1815.2
518.3
I
3911
5235
67
432
2231
370
699
22
491
536
272
4900Figures3–7 show the best, average, and worst fitness value curves obtained by DPSO-TS algorithm, DPSO algorithm, DPSO-PDM algorithm, and CHIDPSO algorithm in the experiments with top 5 optimization results in turn. Figures 3–7 are used to show the changes of results obtained by these four algorithms. No. 6, 3, 1, 5, and 7 experiments of DPSO-TS algorithm are shown in Figures 3–7 in turn. No. 4, 1, 3, 5, and 7 experiments of DPSO algorithm are shown in Figures 3–7 in turn. No. 2, 7, 5, 9, and 8 experiments of DPSO-PDM algorithm are shown in Figures 3–7 in turn. No. 5, 3, 10, 1, and 8 experiments of CHIDPSO algorithm are shown in Figures 3–7 in turn. Figures 8–12 show the comparison of best fitness value curves obtained by DPSO-TS algorithm, DPSO algorithm, DPSO-PDM algorithm, and CHIDPSO algorithm in the experiments with top 5 optimization results in turn, from which we can see that the DPSO-TS algorithm’s fitness value of each iteration is superior to the other algorithms after the eleventh iteration. Figures 8–12 show that the TS algorithm has strong local search capacity.Figure 3
Best, average, and worst fitness value curves in each of the four algorithms’ experiment with the first optimization results. (a) DPSO-TS algorithm. (b) DPSO algorithm. (c) DPSO-PDM algorithm. (d) CHIDPSO algorithm.
(a)
(b)
(c)
(d)Figure 4
Best, average, and worst fitness value curves in each of the four algorithms’ experiment with the second optimization results. (a) DPSO-TS algorithm. (b) DPSO algorithm. (c) DPSO-PDM algorithm. (d) CHIDPSO algorithm.
(a)
(b)
(c)
(d)Figure 5
Best, average, and worst fitness value curves in each of the four algorithms’ experiment with the third optimization results. (a) DPSO-TS algorithm. (b) DPSO algorithm. (c) DPSO-PDM algorithm. (d) CHIDPSO algorithm.
(a)
(b)
(c)
(d)Figure 6
Best, average, and worst fitness value curves in each of the four algorithms’ experiment with the fourth optimization results. (a) DPSO-TS algorithm. (b) DPSO algorithm. (c) DPSO-PDM algorithm. (d) CHIDPSO algorithm.
(a)
(b)
(c)
(d)Figure 7
Best, average, and worst fitness value curves in each of the four algorithms’ experiment with the fifth optimization results.
(a)
(b)
(c)
(d)Figure 8
Comparison of the best fitness value curves in each of the four algorithms’ experiment with the first optimization results.Figure 9
Comparison of the best fitness value curves in each of the four algorithms’ experiment with the second optimization results.Figure 10
Comparison of the best value curves in each of the four algorithms’ experiment with the third optimization results.Figure 11
Comparison of the best fitness value curves in each of the four algorithms’ experiment with the fourth optimization results.Figure 12
Comparison of the best fitness value curves in each of the four algorithms’ experiment with the fifth optimization results.The simulation results demonstrate that the DPSO-TS algorithm has a higher accuracy compared to the DPSO, the DPSO-PDM, and the DPSO-CSA algorithms and can effectively solve the OSV allocation optimization problem.


## 4.1. Simulations

The parking orbit elements of satellites (at 0:00 on July 1, 2099) in need of service are shown in Table2. The parking orbit elements of the OSVs (at 0:00 on July 1, 2099) are shown in Table 3. The maintaining time for a satellite is an hour ,and the on-orbit service mission should be achieved in 24 hours (From 0:00 on July 1, 2099 to 0:00 on July 2, 2099), which means tm+ttra (ttra is the duration of OSV and satellite from maneuvering time to rendezvous time) is in 0,23×3600s. The mass of each OSV is 1000 kg, each OSV carries 600 kg fuel the engine thrust is 490 N, and the gas jet velocity of 3000 m/s. mkeep is 100 kg. The population size PT is 20, the crossover probability Pc is 0.7, the mutation probability Pm is 0.1, and the maximum number of iterations Nt is 30. The population size MB is 20, maximum number of iterations TD is 30, the inertia weight ω is 0.8, the cognitive coefficient c1 is 0.6, and the social coefficient c2 is 0.6. The length lT of the tabu table is 2, the size MT of candidate is 5, and the maximum number of iterations TT is 8. Time step δ is 1 s, and I=tm/δ. The simulation performs ten independent experiments. In order to reduce computational time, each trajectory is computed by one independent blade of blade servers.Table 2
Parking orbit elements of satellites.
Satellite
A
km
e
I
°
α
°
Ω
°
f
°
S
1
9306
0.238550
28.321
224.138
307.927
135.862
S
2
7059
0.000225
98.084
76.815
185.081
283.332
S
3
9237
0.000165
52.006
171.386
350.005
188.701
S
4
7828
0.001219
101.618
87.562
293.441
282.681
S
5
7707
0.054088
99.031
56.562
347.938
308.882
S
6
7554
0.002882
90.045
245.844
85.215
113.968
S
7
8428
0.201645
82.982
262.555
29.901
44.341
S
8
9501
0.213200
28.226
34.607
152.421
325.392
S
9
9401
0.016556
64.822
302.192
337.387
56.304
S
10
9939
0.321639
56.917
231.629
218.498
128.371
S
11
10687
0.355917
57.022
347.224
317.457
122.776
S
12
6903
0.000549
96.390
110.033
325.324
203.070Table 3
Parking orbit elements of OSVs.
OSV
A
km
e
I
°
α
°
Ω
°
f
°
O
1
7751
0.000120
32.223
56.232
76.223
255.520
O
2
8400
0.000225
78.002
102.332
253.010
122.243
O
3
8526
0.004120
76.924
154.257
73.020
243.000
O
4
9201
0.370056
55.223
180.922
223.001
95.620
O
5
7002
0.004021
87.235
102.556
330.765
88.598
O
6
8002
0.529030
145.667
156.007
200.321
270.321
O
7
9923
0.052301
120.023
254.124
122.232
55.220
O
8
7125
0.003924
42.084
51.815
255.081
30.200
O
9
8838
0.000249
98.178
69.604
155.872
150.009
O
10
9123
0.013793
52.674
109.398
23.270
200.020
O
11
7685
0.004693
37.477
52.707
158.017
60.000
O
12
9215
0.489300
55.371
128.871
305.501
80.000
O
13
7453
0.004218
23.618
77.562
293.441
50.101
O
14
9675
0.054089
99.031
56.562
187.938
160.032
O
15
6874
0.000555
97.420
104.012
316.284
204.781
O
16
7534
0.003002
91.145
239.735
84.325
113.000We design comparison experiments with DPSO algorithm, DPSO-PDM algorithm, and CHIDPSO algorithm. The parameters of DPSO algorithm are the same as those of DPSO-TS algorithm. The parameters of DPSO algorithm, part of DPSO-PDM algorithm, are the same as those of DPSO-TS algorithm, and the other parameters are the same as those in the article written by Li et al. [13]. The parameters of the DPSO algorithm part of CHIDPSO algorithm are the same as those of DPSO-TS algorithm, and the other parameters are the same as those in the article written by Vairam et al. [17].


## 4.2. Analyses of Results

Results including allocation plans, computational time, and the mass of the remaining fuel obtained by DPSO-TS algorithm, DPSO algorithm, DPSO-PDM algorithm, and CHIDPSO algorithm in ten independent experiments are shown in Tables4–7, separately. The maneuvering time of OSV selected from the integer from the duration of on-orbit service mission is 0,23×3600s. The maneuvering time I obtained by these four algorithms in different experiments are shown in Tables 4–7, and the best results can be obtained at these maneuvering times. It can be seen from Tables 4–7 that the average mass of the remaining fuel obtained by DPSO-TS algorithm, DPSO algorithm, DPSO-PDM algorithm, and CHIDPSO algorithm is 538.5 kg, 467.8 kg, 488.9 kg, and 508.8 kg; the average mass of the remaining fuel obtained by DPSO-TS algorithm is better than that of the other algorithms. Except the ninth experiment of DPSO-TS algorithm, the mass of the remaining fuel obtained by DPSO-TS algorithm in the other experiments is superior to that obtained by the other algorithms in any experiment; the average computational time obtained by DPSO-TS algorithm, DPSO algorithm, DPSO-PDM algorithm, and CHIDPSO algorithm is 1792.3 s, 1623.2 s, 1769.1 s, and 1797.3 s. The average computational time obtained by DPSO-TS algorithm differs from the shortest average computational time by 169.1 s, which has no effect on the service mission because there is a long preparation time before the mission is executed. The computational time of these four algorithms can be considered basically the same.Table 4
Results of allocation optimization based on DPSO-TS algorithm.
No.
X
S
1
S
2
S
3
S
4
S
5
S
6
S
7
S
8
S
9
S
10
S
11
S
12
t
c
s
f
X
1
OSV
O
4
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
15
1801.4
540.3
I
2301
6415
67
167
2251
449
781
1
299
5233
649
3339
2
OSV
O
9
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
4
O
7
O
12
O
15
1788.9
537.4
I
4609
6422
1776
167
2254
2304
222
0
298
5233
1233
3339
3
OSV
O
4
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
15
1801.4
541.7
I
5521
6413
67
169
2217
32
781
0
299
5233
24
144
4
OSV
O
4
O
15
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
8
1791.7
539.3
I
221
6415
69
167
2251
667
781
5
299
4446
640
301
5
OSV
O
4
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
15
1739.5
540.0
I
2304
6392
67
167
2251
379
780
0
299
5233
649
3332
6
OSV
O
9
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
4
O
7
O
12
O
15
1820.9
549.6
I
4001
6409
378
166
2259
1339
239
15
298
5409
1602
3337
7
OSV
O
7
O
8
O
10
O
11
O
16
O
13
O
3
O
9
O
14
O
4
O
12
O
15
1800.9
539.8
I
5521
6413
25
169
2217
32
699
7
271
5233
755
144
8
OSV
O
4
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
15
1802.8
538.2
I
5522
6413
65
168
2216
32
781
2
299
5233
23
142
9
OSV
O
4
O
15
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
5
1772.3
520.4
I
5522
6413
65
168
2216
32
781
2
299
5233
23
142
10
OSV
O
4
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
15
1803.6
538.2
I
5524
6202
12
168
2109
32
789
0
297
5233
22
144Table 5
Results of allocation optimization based on DPSO algorithm.
No.
X
S
1
S
2
S
3
S
4
S
5
S
6
S
7
S
8
S
9
S
10
S
11
S
12
t
c
s
f
X
1
OSV
O
7
O
11
O
6
O
13
O
16
O
8
O
3
O
12
O
9
O
10
O
14
O
15
1622.0
474.2
I
6600
350
471
337
2535
107
4730
4
335
7774
6629
4209
2
OSV
O
7
O
8
O
6
O
13
O
16
O
11
O
3
O
14
O
9
O
10
O
12
O
15
1607.5
467.3
I
6630
1323
470
360
2535
24
4735
4701
350
7772
258
4311
3
OSV
O
4
O
15
O
6
O
11
O
16
O
13
O
3
O
12
O
9
O
7
O
14
O
8
1610.3
472.8
I
5501
6009
498
200
2215
33
781
7
299
5405
24
251
4
OSV
O
7
O
11
O
6
O
13
O
16
O
10
O
3
O
12
O
9
O
14
O
2
O
8
1626.7
478.3
I
5902
355
400
2099
2435
12
4688
2289
300
5405
18
1455
5
OSV
O
4
O
15
O
6
O
11
O
16
O
13
O
3
O
12
O
9
O
7
O
14
O
8
1611.2
470.6
I
5523
6010
0
1
2420
42
1733
5
249
5422
22
250
6
OSV
O
7
O
8
O
10
O
11
O
16
O
13
O
3
O
9
O
14
O
4
O
12
O
15
1629.9
449.4
I
5322
427
27
469
2015
125
621
4456
378
4236
765
742
7
OSV
O
4
O
11
O
10
O
13
O
16
O
15
O
3
O
12
O
9
O
14
O
2
O
8
1635.5
470.2
I
4755
327
447
2179
2336
120
4587
67
340
5425
2234
1709
8
OSV
O
10
O
15
O
4
O
11
O
16
O
13
O
3
O
12
O
9
O
7
O
14
O
8
1638.8
459.3
I
5090
7426
647
7162
3315
2
4908
3
292
4263
378
140
9
OSV
O
4
O
5
O
10
O
11
O
16
O
13
O
6
O
14
O
9
O
7
O
12
O
8
1627.5
468.1
I
4523
6423
62
1778
4213
310
1741
12
2690
4233
276
242
10
OSV
O
4
O
5
O
10
O
11
O
16
O
13
O
3
O
12
O
9
O
7
O
14
O
15
1622.9
468.0
I
6734
628
17
1556
6421
422
276
13
449
5345
212
260Table 6
Results of allocation optimization based on DPSO-PDM algorithm.
No.
X
S
1
S
2
S
3
S
4
S
5
S
6
S
7
S
8
S
9
S
10
S
11
S
12
t
c
s
f
X
1
OSV
O
4
O
11
O
10
O
13
O
16
O
15
O
3
O
12
O
9
O
14
O
2
O
8
1750.5
475.6
I
4635
127
825
4133
2001
127
2371
60
236
4416
2055
169
2
OSV
O
7
O
8
O
10
O
11
O
16
O
13
O
3
O
9
O
14
O
4
O
12
O
15
1789.0
502.4
I
6754
466
376
169
2167
4125
2621
456
366
4233
65
142
3
OSV
O
4
O
5
O
10
O
11
O
16
O
13
O
3
O
12
O
9
O
7
O
14
O
15
1752.9
481.0
I
2520
412
79
168
2255
321
780
650
498
5437
6663
4004
4
OSV
O
4
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
15
1752.4
480.2
I
2217
6330
162
165
2241
1438
3891
15
245
5204
620
2341
5
OSV
O
4
O
15
O
6
O
11
O
16
O
13
O
3
O
12
O
9
O
7
O
14
O
8
1749.5
499.1
I
2522
160
494
211
1238
4300
6750
334
498
5001
6678
123
6
OSV
O
4
O
8
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
15
1735.6
490.3
I
5467
3442
678
6637
217
360
4661
15
256
5109
259
341
7
OSV
O
4
O
15
O
10
O
12
O
16
O
13
O
3
O
9
O
14
O
7
O
6
O
8
1785.9
500.4
I
5361
412
624
279
327
329
6695
176
10
5408
259
3134
8
OSV
O
4
O
15
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
5
1812.4
492.3
I
3546
2455
167
163
6721
188
428
0
378
5013
343
667
9
OSV
O
4
O
15
O
6
O
11
O
16
O
13
O
3
O
12
O
9
O
7
O
14
O
8
1789.3
495.6
I
5411
132
600
234
2709
6035
490
29
306
5218
4345
176
10
OSV
O
3
O
6
O
10
O
11
O
5
O
13
O
3
O
14
O
9
O
7
O
12
O
8
1773.4
472.4
I
452
622
152
7065
2036
42
189
236
429
4986
231
744Table 7
Results of allocation optimization based on CHIDPSO algorithm.
No.
X
S
1
S
2
S
3
S
4
S
5
S
6
S
7
S
8
S
9
S
10
S
11
S
12
t
c
s
f
X
1
OSV
O
1
O
8
O
10
O
11
O
16
O
2
O
3
O
14
O
6
O
7
O
12
O
5
1827.3
511.6
I
4372
2276
567
5901
253
207
14
15
236
421
3690
1306
2
OSV
O
4
O
15
O
10
O
12
O
16
O
13
O
3
O
9
O
14
O
7
O
6
O
8
1792.9
506.7
I
5009
398
521
1277
320
308
650
179
45
5268
250
2971
3
OSV
O
2
O
9
O
10
O
15
O
16
O
13
O
3
O
14
O
8
O
7
O
11
O
12
1779.3
520.3
I
4538
203
617
7001
3246
319
483
16
99
4205
876
1641
4
OSV
O
7
O
8
O
11
O
12
O
16
O
13
O
3
O
9
O
14
O
4
O
15
O
1
1833.2
489.3
I
352
667
2396
4690
392
127
242
4101
326
646
6095
7020
5
OSV
O
4
O
15
O
10
O
11
O
16
O
13
O
3
O
14
O
9
O
7
O
12
O
5
1804.5
520.9
I
2598
1423
261
460
309
242
428
3356
372
4410
279
461
6
OSV
O
2
O
8
O
10
O
11
O
15
O
6
O
3
O
13
O
4
O
7
O
12
O
14
1790.3
499.6
I
3871
100
1278
473
4133
690
431
299
143
428
4638
230
7
OSV
O
3
O
6
O
10
O
11
O
5
O
13
O
3
O
14
O
9
O
7
O
12
O
8
1802.0
510.6
I
4328
3568
142
6011
230
108
48
4376
25
5099
4238
541
8
OSV
O
4
O
6
O
10
O
11
O
5
O
13
O
3
O
14
O
9
O
7
O
12
O
8
1739.8
511.5
I
2407
3699
183
5280
431
324
380
2325
421
5203
6218
1541
9
OSV
O
7
O
11
O
6
O
13
O
16
O
10
O
3
O
12
O
9
O
14
O
2
O
8
1788.4
499.6
I
4598
1256
427
3899
2408
19
722
2289
123
6263
701
23
10
OSV
O
4
O
12
O
10
O
11
O
16
O
1
O
3
O
5
O
9
O
7
O
13
O
14
1815.2
518.3
I
3911
5235
67
432
2231
370
699
22
491
536
272
4900Figures3–7 show the best, average, and worst fitness value curves obtained by DPSO-TS algorithm, DPSO algorithm, DPSO-PDM algorithm, and CHIDPSO algorithm in the experiments with top 5 optimization results in turn. Figures 3–7 are used to show the changes of results obtained by these four algorithms. No. 6, 3, 1, 5, and 7 experiments of DPSO-TS algorithm are shown in Figures 3–7 in turn. No. 4, 1, 3, 5, and 7 experiments of DPSO algorithm are shown in Figures 3–7 in turn. No. 2, 7, 5, 9, and 8 experiments of DPSO-PDM algorithm are shown in Figures 3–7 in turn. No. 5, 3, 10, 1, and 8 experiments of CHIDPSO algorithm are shown in Figures 3–7 in turn. Figures 8–12 show the comparison of best fitness value curves obtained by DPSO-TS algorithm, DPSO algorithm, DPSO-PDM algorithm, and CHIDPSO algorithm in the experiments with top 5 optimization results in turn, from which we can see that the DPSO-TS algorithm’s fitness value of each iteration is superior to the other algorithms after the eleventh iteration. Figures 8–12 show that the TS algorithm has strong local search capacity.Figure 3
Best, average, and worst fitness value curves in each of the four algorithms’ experiment with the first optimization results. (a) DPSO-TS algorithm. (b) DPSO algorithm. (c) DPSO-PDM algorithm. (d) CHIDPSO algorithm.
(a)
(b)
(c)
(d)Figure 4
Best, average, and worst fitness value curves in each of the four algorithms’ experiment with the second optimization results. (a) DPSO-TS algorithm. (b) DPSO algorithm. (c) DPSO-PDM algorithm. (d) CHIDPSO algorithm.
(a)
(b)
(c)
(d)Figure 5
Best, average, and worst fitness value curves in each of the four algorithms’ experiment with the third optimization results. (a) DPSO-TS algorithm. (b) DPSO algorithm. (c) DPSO-PDM algorithm. (d) CHIDPSO algorithm.
(a)
(b)
(c)
(d)Figure 6
Best, average, and worst fitness value curves in each of the four algorithms’ experiment with the fourth optimization results. (a) DPSO-TS algorithm. (b) DPSO algorithm. (c) DPSO-PDM algorithm. (d) CHIDPSO algorithm.
(a)
(b)
(c)
(d)Figure 7
Best, average, and worst fitness value curves in each of the four algorithms’ experiment with the fifth optimization results.
(a)
(b)
(c)
(d)Figure 8
Comparison of the best fitness value curves in each of the four algorithms’ experiment with the first optimization results.Figure 9
Comparison of the best fitness value curves in each of the four algorithms’ experiment with the second optimization results.Figure 10
Comparison of the best value curves in each of the four algorithms’ experiment with the third optimization results.Figure 11
Comparison of the best fitness value curves in each of the four algorithms’ experiment with the fourth optimization results.Figure 12
Comparison of the best fitness value curves in each of the four algorithms’ experiment with the fifth optimization results.The simulation results demonstrate that the DPSO-TS algorithm has a higher accuracy compared to the DPSO, the DPSO-PDM, and the DPSO-CSA algorithms and can effectively solve the OSV allocation optimization problem.


## 5. Conclusions and Future Research

Fuel-efficient OSV allocation is a basic work to accomplish the on-orbit service mission in the background of deploying a large number of satellites in space in the future. We establish a transfer trajectory optimization model to obtain the mass of remaining fuel, which lays a foundation of allocation model. Then, we propose DPSO-TS algorithm to solve the allocation model, and the simulation results demonstrate that the proposed algorithm is more effective than DPSO algorithm, DPSO-PDM algorithm, and CHIDPSO algorithm. The DPSO-TS algorithm can be applied to other allocation optimization problems.Although the proposed method has high accuracy, it still has some limitations. Further accelerating the computational time and investigating more algorithms with strong local search capacity in future research would be better.



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*Source: 5683698-2020-04-15.xml*