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Conditional probability based multi-objective cooperative task assignment for heterogeneous UAVs

August 2023
Engineering Applications of Artificial Intelligence 123(1):106404

August 2023
123(1):106404

DOI:10.1016/j.engappai.2023.106404

Authors:

Lei Wang

Dalian University of Technology

Xinyong Yu

Dalian University of Technology

Xi-chao Su

Yantai University

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Conditional probability based multi-objective cooperative task

assignment for heterogeneous UAVs

Xiaohua Gaoa, Lei Wanga, Xinyong Yua, Xichao Sub, Yu Dingc,d,e , Chen Luc,d,e, Haijun

Pengf, Xinwei Wangf,∗

aSchool of Mathematical Science, Dalian University of Technology, Dalian, Liaoning 116024, China.

bDepartment of Airborne Vehicle Engineering, Naval Aeronautical and Astronautical University, Yantai

264001, China.

cScience and Technology on Reliability and Environmental Engineering Laboratory, Beijing 100191, China.

dInstitute of Reliability Engineering, Beihang University, Beijing 100191, China.

eSchool of Reliability and Systems Engineering, Beihang University, Beijing 100191, China.

fDepartment of Engineering Mechanics, State Key Laboratory of Structural Analysis for Industrial

Equipment, Dalian University of Technology, Dalian, Liaoning 116024, China.

Abstract

In actual air combat, there is an inevitable risk that an unmanned aerial vehicle (UAV) will be

destroyed. However, this risk is rarely considered in the mission planning phase. In this paper,

we focus on cooperative mission assignment for heterogeneous UAVs. We develop a multi-objective

optimization model to ﬁnd a balance between mission gains and UAV losses. The objective function

is expressed using conditional probability theory by introducing the probabilities of mission success

and UAV loss. Munitions loading capacity, time constraints, and priority constraints are modeled

as constraints. To solve this combinatorial problem, an improved multi-objective genetic algorithm,

which incorporates a natural chromosome encoding format and specially designed genetic operators,

is developed. An eﬃcient unlocking method is constructed to address the unavoidable dead-lock

phenomenon meanwhile maintaining the population randomness. Numerical simulations for diﬀerent

problem sizes and ammunition stocks are performed, and the proposed algorithm is compared with

the Multi-objective Particle Swarm Optimization and the Multi-objective Grey Wolf Optimization,

respectively, using diﬀerent unlocking approaches. The simulation and comparison results demonstrate

the practical value and eﬀectiveness of the developed model and the proposed algorithm.

Keywords: unmanned aerial vehicle; cooperative task assignment; multi-objective

optimization; genetic algorithm.

1. Introduction

1.1. Background and motivations

UAVs have the advantage of signiﬁcantly reducing cost and risk to human lives, and

they are ideal for performing dangerous and tedious tasks (Bethke et al.,2008). Nowadays,

UAVs are widely used in various ﬁelds, such as disaster area inspection (Song et al.,2022),

power lines inspection (Korki et al.,2019), aerial surveillance (Coombes et al.,2017), targets

∗Corresponding author E-mail: wangxinwei@dlut.edu.cn

surveillance (Gu et al.,2018). To improve the task completion eﬃciency, compared with a

single UAV, multiple UAV systems or UAV swarms are generally used due to their superiority

in redundancy, compatibility and scalability (Zhen et al.,2018;Tahir et al.,2019). There are

many decision problems in various ﬁelds in the real world, and it is meaningful to investigate

models and methods with risk and uncertainty environments (Deng et al.,2022a,2022b;Wang

et al.,2022;Wang et al.,2022,2023). For a multiple UAV system, path planning (Chen et al.,

2021a,2021b) and task assignment (Zhao et al.,2021) are two important research directions,

and the key to improve the eﬃciency of task execution is to obtain a task assignment scheme

based on task decisions to optimize a certain objective or multiple objectives, provided that

the constraints are satisﬁed.

The motivation for this paper consists of two main aspects: i) to construct a multi-

objective optimization model capable of measuring the risk of UAV destruction and various

resource constraints for the cooperative multi-task assignment problem (MOCTAP); and ii)

to propose an evolutionary algorithm with eﬃcient unlocking capability to avoid deadlock

situations. Although many models and algorithms for cooperative task assignment problems

(CTAP) have been developed, there is still a need to further improve the accuracy of modeling

and the applicability of multi-UAV MOCTAP in order to adapt to the increasingly complex

battleﬁeld environment. On the one hand, in order to reduce the complexity of the models and

the computational diﬃculty of the algorithms, existing studies have formulated the problem

with appropriate assumptions at the mission, UAV, and munitions levels, such as the UAV

must be able to complete the mission and survive, the munitions must be suﬃcient, and so

on. However, these assumptions lead to an idealized model that is hardly representative of

CTAP in real battleﬁeld environments. On the other hand, for infeasible individuals that

violate priority constraints (i.e., deadlock phenomenon), current research usually adopts a

stochastic-based unlocking approach, which has the disadvantage of consuming signiﬁcant

runtime and is computationally ineﬃcient, especially when dealing with large-scale problems.

Improving the practicality of the MOCTAP model as well as the eﬃciency and applicability

of the algorithm for multiple UAV systems is the main work of this paper.

1.2. Related work

1.2.1. Model and algorithm for CTAP

In recent years, CTAP for UAVs has attracted a lot of attention, where the main focus

is on optimization model formulation and algorithm design (Zhang et al.,2018;Issac et al.,

2020). CTAP is essentially a combinatorial optimization problem, and the classical basic mod-

els include the traveling salesman problem (TSP) (Dantzig et al.,1954), the vehicle routing

problem (VRP) (Laporte,1992), and the generalized assignment problem (GAP) (Cattrysse

and Van Wassenhove,1992). As the size of the CTAP increases, the established optimiza-

tion model comes with more design variables and constraints. Under such circumstances,

deterministic algorithms(e.g., Branch and Bound algorithm (Rasmussen and Shima,2006),

dynamic programming (Alighanbari et al.,2005) usually do not perform well in terms of com-

putational eﬃciency, or even ﬁnd reasonable solutions. Compared to deterministic algorithms,

heuristics show great potential in terms of eﬃciency and robustness for complex CTAPs, and

common approaches include particle swarm optimization (PSO) and its variants (Sujit et al.,

2008;Zeng et al.,2022), ant colony optimization (ACO) and its variants (Fei et al.,2008),

and genetic algorithm (GA) and its variants (Darrah et al.,2006), etc.

1.2.2. Single-objective optimization for CTAP with multiple UAVs

Early investigations on CTAP for multiple UAVs mainly focus on homogeneous UAV

systems. For the multi-UAV reconnaissance task assignment problem, Wang et al. (2018)

proposed a model and an opposition-based genetic algorithm involving double-chromosomes

encoding and multiple mutation operators. Velhal and Sundaram (2020) formulated the re-

stricted airspace protection as a multi-UAV spatio-temporal task assignment problem and

proposed a modiﬁed consensus-based bundled auction method. Considering the limitation of

the total ﬂight distance of UAVs, Wang et al. (2020) established a model with the total weight-

ed cost of target value and distance cost as the objective, and presented an improved genetic

algorithm based on the beetle antennae search algorithm. As modern aerial warfare becomes

complex in both the spatial and temporal dimension, a group of heterogeneous UAVs fre-

quently constitute a ﬂeet to complete a series missions. Research under such a heterogeneous

framework becomes the mainstream in the last two decades. For example, Ye et al. (2021)

proposed a modiﬁed consensus-based bundle algorithm with task coupling constraints for C-

TAP. And in (Ye et al.,2020), they further developed an adaptive GA for the suppression of

enemy air defense (SEAD) mission. Afghah et al. (2018) established a leader-follower coalition

formation model for CTAP in heterogeneous UAV networks, and proposed a reputation-based

mechanism to monitor the cooperative behavior of UAVs. Taking the eﬀective task execution

scope and the task priority into consideration, Fan et al. (2021) proposed a comprehensive

solution framework that integrates obstacle-avoidance distance estimation, crossover-based

updating and search intensity adaptive mechanism. Ye et al. (2020) developed a improved

genetic algorithm that includes multi-type-gene chromosome encoding strategy for CTAP.

Zhen et al. (2020) developed an intelligent cooperative mission planning scheme based on a

hybrid artiﬁcial potential ﬁeld and ant colony optimization method.

1.2.3. Multi-objective optimization for CTAP with multiple UAVs

It should be noted that researches mentioned above are developed under the framework of

single-objective optimization. In practical aerial warfare, commanders always expect to max-

imize the beneﬁt with minimum loss. The multi-objective optimization framework, which has

been used in many diﬀerent domains, provides a feasible technique to take these two conﬂict-

ing objectives into account simultaneously (Li et al.,2019;Zeng et al.,2021;Li et al.,2022).

Atencia et al. (2019) proposed a novel weighted random generator for the formation of new

individuals, which was used to reduce the convergence rate of the multi-objective evolution-

ary algorithm (MOEA) solver. Chen et al. (2019) studied the multi-objective optimization

for the reconnaissance task assignment problem and developed a modiﬁed Multi-Objective

Symbiotic Organisms Search algorithm. Wang et al. (2020) considered a high-dimensional

multi-objective optimization problem containing four objectives, and described an improved

multi-objective quantum-behaved particle swarm optimization algorithm. Cheng et al. (2020)

constructed a heuristic algorithm based on the task assignment, non-dominated sort steps and

diﬀerential evolution-aided for the multi-objective optimization problem.

However, ammunition shortage scenarios, risk of UAV destruction, and mission incomple-

tion are rarely considered by studies of multi-UAV CTAP. The constructed algorithms are also

not directly applicable to solve multi-UAV CTAP for various ammunition scenarios. More-

over, the ineﬃciency of the current unlocking methods when the problem size is large and the

chromosome dimension is high will directly aﬀect the optimization process of the algorithms.

Therefore, it is imperative to develop a risk-based model and design a multi-objective opti-

mization algorithm for CTAP suitable for the multiple munitions case, which also needs to

be able to unlock chromosomes eﬃciently in large-scale problems.

1.3. Contributions

In this paper, the MOCTAP is investigated with heterogeneous UAVs (i.e., surveillance

UAV, combat UAV and munition UAV) and multiple task types (i.e., classiﬁcation task,

attack task and veriﬁcation task). The task assignment model may become idealized if the

task assignment ignores the actual risks, i.e., the risk of the UAV being destroyed, the risk

of the task not being completed. Considering them would result in a more practical task

assignment strategy than ignoring the risks in the model. The most important innovations in

this paper are the construction of a multi-objective optimization model for cooperative multi-

UAV task assignment based on various risks and the design of a multi-objective optimization

algorithm that includes a fast-unlocking method. The details are as follows.

First, to address the risk of UAV destruction and mission non-completion, a risk-based

optimization model framework for MOCTAP based on conditional probability theory is de-

veloped by comprehensively considering various risks. Maximizing the value of the attack

target and minimizing the value of the UAV being destroyed are taken as objectives. Ammu-

nition, time, and coordination constraints are all considered as constraints in the optimization

problem.

Second, to solve this problem, we propose an improved genetic algorithm (MOCTAP-

GA) based on the fast non-dominated sorting approach and an elite retention strategy. A

chromosome encoding format that only considers the assignment task is used, which reduces

the size of each chromosome. Compared with the encoding method that includes all tasks, this

encoding method can improve the computational eﬃciency of the algorithm. The specially

designed genetic operator can handle both ammunition suﬃcient and ammunition insuﬃcient

cases. This solves the shortcomings of a single crossover operator and a single mutation

operator.

Finally, a Fast Positioning Unlocking Method (FPUM) is proposed to handle deadlock

chromosomes. This method combines two stages of judging and unlocking chromosomes and

is able to unlock chromosomes exactly at the locking point to achieve an eﬃcient unlocking

process and maintain the randomness of the population. The high unlocking eﬃciency of

FPUM improves the eﬃciency of the constructed optimization algorithm.

1.4. Organization

The remainder of the paper is organized as follows. Section 2 describes the CTAP for

heterogeneous UAVs. Section 3 gives the multi-objective optimization model of CTAP. Section

4 provides the multi-objective optimization strategy and the unlocking method. The numerical

examples are given in Section 5. Finally, Section 6 concludes the paper.

2. Description of the UAV cooperative task assignment problem

For convenience, let In:= {1,2,· · · , n},n∈N+.

2.1. Enemy targets

Suppose that there are NTtargets on the battleﬁeld, and we denote the target set as

T={T1, T2,· · · , TNT},

where Tj(j∈INT) represents the j-th targets. Each target includes three types of indispens-

able tasks: classiﬁcation task, attack task, and veriﬁcation task, which are abbreviated as C,

A and V, respectively. The set of task type is denoted as

Stypes ={C, A, V }.

Targets with high values generally result in signiﬁcant risks, whereas targets with low values

typically result in low risks. In this paper, we make the assumption that the attack numbers on

low-value and high-value targets, respectively, are 1 and 2, in order to increase the successful

strike rate. Let V alueTj(j∈INT) represents the value of Tj∈T.

Hence, the task set of Tj∈T(j∈INT) is denoted as follows.

MTj=({C, A1, V },if Tjis a low-value target,

{C, A1, A2, V },otherwise,

where A1and A2represent the ﬁrst and the second (if necessary) attack on Tj. A target

is successfully executed only if all its tasks have been completed. Suppose that there are L

(L≤NT) high-value targets in T, then the total number of tasks Nall is 3NT+L.

As shown in Fig. 1, classiﬁcation task, attact task(s) and veriﬁcation task for each target

are executed successively. Let timeC

Tj,timeA1

Tj,timeA2

Tjand timeV

Tjrepresent the execution

C A1V

CA1A2V

(a) Low-value targets (b) High-value targets

Fig. 1. Order of execution of tasks

time instants of C,A1,A2and Vof Tj, respectively; and let 4tC

Tj,4tA1

Tj,4tA2

Tjand 4tV

represent the time durations required to perform C,A1,A2and Vof Tj, respectively.

Table 1. The capabilities of heterogeneous UAVs

Types of UAVs Symbol Types of tasks that can be performed

Surveillance UAV s C, V

Combat UAV c C, A, V

Munition UAV m A

2.2. Heterogeneous UAVs

Suppose that the UAV ﬂeet consists of NUUAVs. The set of UAVs is deﬁned as

U={U1, U2,· · · , UNU},

where Uirepresents the i-th UAV. Surveillance, combat and munition UAVs are included in

the UAV ﬂeet. For each type of UAV, its abbreviation and abilities are listed in Table 1.

The task set of Ui(i∈INU) is denoted as MUi,

MUi={taskUi

1, taskUi

2,· · · , taskUi

NUi},

where NUiis the number of tasks for Uiand taskUi

l(l∈INUi) represent the l-th task performed

by Ui.

The value of Ui(i∈INT) is denoted as V alueUi. Diﬀerent UAVs have diﬀerent probabil-

ities of success and survival when performing tasks to diﬀerent targets. Let SV k

ij (i∈INU,

j∈INT,k∈I3) represents the probability that Tjis destroyed by Uiwith task type k,

and k= 1,2,3 represent the task type of C,Aand V, respectively. Let KI k

ij denotes the

probability of Uibeing destroyed when Uiperforms the task type kfor Tj. If Uiis of type

s, then it cannot complete the attack task; if Uiis type of m, then it can only complete the

attack task. Consequently, we can make the following natural assumption. If the type of Uiis

s, then SV 2

ij =KI2

ij = 0, and while the type of Uiis m, then SV 1

ij =SV 3

ij =KI1

ij =KI3

ij = 0,

j∈INT.

3. Establishment of the multi-objective optimization model

3.1. Connected digraph

To represent the order in which the UAVs perform their tasks, a connected graph (Jia et

al.,2018) is introduced to reﬂect the execution of the UAV from the current task to the next

task. Let the set of vertices be deﬁned as V={T1, T2,· · · , TNT}. And the set of edges be

deﬁned as E={(Vm, Vn)|Vm∈V, Vn∈V, m 6=n}. Then the connected digraph can be

given as G= (V, E).

3.2. Objective functions

A task assignment plan with small losses and large proﬁts is expected. Therefore, the

maximization of expectation of the proﬁts and the minimization of expectation of the costs

are used as indicators to measure the task assignment plan. Since the UAV can continue to

perform the next task only if it is not destroyed, conditional probabilities are used in the

modeling. Only the following two conditions are satisﬁed, taskUi

j(i∈INU,j∈INUi) can be

executed.

1) The tasks before taskUi

jin the target set corresponding to taskUi

jhave been completed;

2) The taskUi

1,taskUi

2,· · · ,taskUi

j−1in task set MUihave been completed.

Two expectations are considered in the construction of the objective function, namely the

expectation of the value of the completed targets and the expectation of the value of the

destroyed UAVs. The speciﬁc formulations are as follows.

(i) Maximizing the expectation of the value of the destroyed targets:

max J1:=

j=1

i=1 n

NUi

l=1 h(1 −KIiai

l)SViai

liIFioV alueTj,(1)

(ii) Minimizing the expectation of the value of the destroyed UAVs:

min J2:=

i=1 hV alueUi−

NUi

l=1

(1 −KIiai

l)V alueUii,(2)

where ai

l(l∈INUi,i∈INU) represents the target number of the l-th task in MUi, and

Xi,k

(Vai

l−1

,Vj)∈ {0,1}is introduced to represent whether Uiﬂies from Tai

l−1to Tjto perform

task type kfor Tjor not. Thus, ai

l:=

j=1

k=1

jXi,k

(Vai

l−1

,Vj), l ∈INUi, i ∈INU.

For convenience, we let ai

0:= 0. And Viai

l,KIiai

lare given as follows,

SViai

l:= 





SV k

iai

, if Xi,k

(Vai

l−1

,Vai

)= 1, k ∈I3, ai

l=j,

1, else,

and

KIiai

l:= 





KIk

iai

, if Xi,k

(Vai

l−1

,Vai

)= 1, k ∈I3,

0, else,

and

IFi:= 









NUi

l=1

1/(1 −KIiai

l), if ai

l6=j, ∀l∈INUi,

NUi

m=˜

1/(1 −KIiai

m),˜

l= max{l|ai

l=j, l ∈INUi}, else.

(iii) Minimizing the the expectation of the value of the undestroyed targets:

min J3:=

j=1

V alueTj−J1.(3)

Let f= (f1, f2)>, where f1=J3and f2=J2. Then the objective function is to minimize

the vector function f.

3.3. Constraints

The following constraints are taken into account in CTAP based on the characteristics of

CTAP and the analysis mentioned above.

(1) Ammunition constraints

The number of the task type Aassigned to UAVs cannot exceed the ammunition capacity

of the UAVs. And the number for type Cand Vperformed by a UAV is not limited by the

UAV itself.

NUi

l=1

jl=1

Xi,2

(Vai

l−1

,Vjl)≤AMi,∀i∈INU,(4)

where AMirepresents the ammunition capacity of Ui. If the type of a UAV is s, then its

ammunition capacity is recorded as 0.

(2) Coordination constraints

In order to shorten the dwell time of the same UAV at a location and reduce the risk

of UAV being discovered and destroyed, we assume that the number of attacks on the same

target by the same UAV does not exceed one, that is,

NUi

l=1

Xi,2

(Vai

l−1

,Vjl)≤1,∀jl∈INT.(5)

Therefore, attack tasks for high-value target will be assigned to diﬀerent UAVs.

(3) Task timing constraints

The tasks for each target need to be executed in order. For tasks of the same target, task

Acan only be executed after task Cis completed, and task Vcan only be executed after task

Ais completed. Therefore, the task assignment needs to satisfy the following constraints,

timeC

Tj+4tC

Tj< timeA

Tj, timeA

Tj+4tA

Tj< timeV

Tj,∀j∈INT,(6)

where 4tA

Tj= max{4tA1

Tj,4tA2

Tj}when Tjis a high-value target, otherwise, 4tA

Tj=4tA1

Tj.

(4) UAV execution timing constraints

For a certain UAV, it needs to follow the task sequence in its task set. For example,

taskUi

j∈MUi(j∈INUi,i∈INU) can only be executed after the previous task taskUi

j−1∈MUi

has been completed. Therefore, the following constraints need to be considered,

timej

Ui< timej+1

Ui,∀j∈INUi−1,∀i∈INU,(7)

where timej

Uirepresents the execution time instant of taskUi

j∈MUi.

3.4. Multi-objective optimization model

Based on the above description, the MOCTAP is to ﬁnd task assignment plans to minimize

the objective function funder the certain constraints (4)-(7).

(MOCTAP) min f= (f1, f2)>

s.t. (4)−(7).(8)

4. Multi-objective optimization algorithm of MOCTAP

Among MOEAs, the non-dominated sorting genetic algorithm II (NSGA-II) is widely used

due to its advantages of the fast non-dominated sorting approach and the elite-preserving

strategy (Deb et al.,2002). In this section, we modify the chromosome encoding format

and genetic operators and propose an eﬃcient unlocking method to deal with the deadlock

chromosomes.

4.1. Chromosome encoding

Considering the heterogeneous UAVs and various types of tasks in MOCTAP, the chro-

mosome needs to contain the assignment information of UAVs, targets and tasks. At the

same time, information about unassigned targets and UAVs is not expected to be included in

the chromosome to reduce the chromosome dimension. However, the chromosome encoding

methods in the traditional GA cannot achieve these requirements, such as binary encoding

method, decimal encoding method and real number encoding method, etc.

The encoding of chromosomes is achieved by constructing genes and combining all genes.

Each gene fragment consists of three parts: target number, task type, and UAV number. All

generated genes are synthesized into a complete chromosome, as shown in Fig. 2. The genes of

each chromosome are generated sequentially, and the UAVs perform the corresponding tasks

based on the order in which the genes are generated. The dimension of the chromosome is

not ﬁxed, it depends on the number of tasks assigned. The dimension, denoted by DIM, is

calculated as follows,

DIM =









3NT+L, if all tasks are assigned,

i=1

AMi−2h, else, (9)

where h(0 < h ≤L≤[b/2]) represents the number of high-value targets assigned, and b

represents the total number of ammunition.

Ti2

C/A/V

Uj2

Ti1

C/A/V

Uj1

Ti3

C/A/V

Uj3

C/A/V

Target number

Type of task

UAV number

Gene

Gene Gene Gene

Chromosome

Fig. 2. The chromosome encoding method

For calculation purposes, all genes of the chromosome need to be grouped and sorted

according to the UAV number. This process is illustrated with an example where tasks are

all assigned. Suppose there are 4 UAVs and 4 targets, and the speciﬁc information of UAVs

and targets is shown in Table 2and Table 3. A chromosome that satisﬁes constraints (4)-(7)

is shown in Fig. 3. Based on (9), the dimension of the chromosome is 14.

Table 2. The information of UAVs

UAVs Types Ammunition UAVs Types Ammunition

U1s0U3c2

U2c2U4m4

Table 3. The information of targets

Targets Task sets Targets Task sets

T1{C, A1, V }T3{C, A1, A2, V }

T2{C, A1, V }T4{C, A1, A2, V }

Gene1 Gene2 Gene3 Gene4 Gene5 Gene6 Gene7 Gene8 Gene9 Gene10 Gene11 Gene12 Gene13 Gene14

Fig. 3. A chromosome in the population

In Fig. 3, it is not quite readily apparent what speciﬁc tasks each UAV accomplished.

Therefore, based on the tasks accomplished by each UAV, the genes of the chromosome are

regrouped and reordered in Fig. 4. The task sets of U1,U2,U3and U4are as follows,

MU1={TV

3, T C

4, T C

1}, MU2={TA2

3, T C

2, T A1

2, T V

4},

MU3={TC

3, T V

2, T A1

4, T V

1}, MU4={TA1

3, T A2

4, T A1

1}.

In order to ensure randomness, the targets and UAVs are randomly selected in the chromosome

encoding. The speciﬁc process is shown in Algorithm 1.

4.2. Crossover operator

The crossover operator play an important role in GA for generating the oﬀspring popu-

lation. Some diﬃculties lead to the fact that the traditional crossover operators cannot be

applied to MOCTAP. For example, 1) the selection of the crossover point will be problematic

if the crossover parents have diﬀerent chromosome dimensions; 2) the same task of the same

target in the chromosomes of the crossover oﬀspring is performed multiple times, etc.

According to Eq. (9), the chromosome dimension is related to the number of ammunition

and the number of attack tasks, so the crossover operators are constructed according to the

relationship between these two factors. The probability of crossover is denoted by Pc. Let F1

and F2respectively denote the two parents of the crossover operation, and RMidenotes the

Subchromosome1 Subchromosome2 Subchromosome3 Subchromosome4

Chromosome

Fig. 4. New form of chromosome

Algorithm 1: Chromosome encoding process

Input:Am,T S,AU ,C V U(the ammunition of the UAVs, the set of discovered targets, the set of

UAVs of type cand m, and the set of UAVs of type sand c)

Output: Chromosome

1Chromosome ←initialize() {Initially empty archive};

2Randomly select a target from T S;

3while length[T S]6=0 and Am 6=0 do

4Randomly select a UAV from CV U, and let Gene ←[T arg et,C,UAV ];

5Chromosome ←Chromosome ∪Gene;

6Randomly select a UAV from AU , and let Gene ←[T arg et,A1,UAV ];

7Chromosome ←Chromosome ∪Gene;

8Am[UAV ]←Am[U AV ]-1;

9if Am == 0 then

10 break;

11 else if T ar get is a high-value target and Am 6=0 then

12 Randomly select a UAV from AU , and let Gene ←[T arg et,A2,UAV ];

13 Chromosome ←Chromosome ∪Gene;

14 Am[UAV ]←Am[U AV ]-1;

15 if Am == 0 then

16 break;

17 Randomly select a UAV from CV U , and let Gene ←[Tar get,V,UAV ];

18 Chromosome ←Chromosome ∪Gene;

19 Delete T arget from T S ;

20 if length [T S]==0 then

21 break;

22 else if length [T S ]6=0then

23 Randomly select a target from T S;

set of UAVs with residual attack capability corresponding to Fi. And RTidenotes the set of

targets that are not attacked in Fi,i∈I2. The cases of suﬃcient and insuﬃcient ammunition

are discussed separately in the following.

4.2.1. Crossover operator for suﬃcient ammunition

Obviously,

i=1

AMi≥NT+Lindicates that ammunition is suﬃcient. Two intersection

points p1, p2∈INall are chosen arbitrarily, and crossover operations are performed on the

genes between the two intersection points of F1and F2.

i=1

AMi> NT+L, then RMi6=∅,RTi=∅,i∈I2. For the genes between p1and p2in

F1and F2,Ssc

1and Ssc

2are denoted the set of UAVs performing Cor Vtasks, Scm

1and Scm

are denoted the set of UAVs performing attack task, respectively. The crossover operation

on F1is shown in Algorithm 2. Similarly, the crossover operation on chromosome F2can be

obtained.

Algorithm 2: Crossover operator 1

Input:p1,p2,F1,F2

Output: Crossover oﬀspring

1i←p1−1;

2while i≤p2−1do

3if The task in F1[i]is task Athen

4if RM1∩Scm

26=∅and RM1∩Scm

2− {F1[i][0]} 6=∅then

5Randomly select a UAV from RM1∩Scm

2, and update RM1;

6else if RM1∩Scm

2=∅or RM1∩Scm

2− {F1[i][0]}=∅then

7Randomly select a UAV from RM1− {F1[i][0]}, and update RM1;

8if The task in F1[i]is task Cor Vthen

9if Ssc

26=∅then

10 Randomly select a UAV from Ssc

11 else if Ssc

2=∅then

12 Randomly select a UAV from the UAVs of types sand cthat do not belong to Ssc

13 Update the UAV information in F1[i], and i←i+ 1;

i=1

AMi=NT+L, then RMi=∅,RTi=∅,i∈I2. Algorithm 2is not suitable for

this situation, therefore, the crossover operation for this situation is achieved by exchanging

the internal information of chromosomes. The speciﬁc steps are shown in Algorithm 3.

To illustrate the crossover operator, the following example of 3 UAVs to execute 2 targets

is considered. For simplicity of description, the UAVs are U1,U2, and U4in Table 2, and the

targets are T1and T3in Table 3. Obviously, for this case, Algorithm 2should be adopted.

Assume p1= 3, p2= 5, and two fathers F1,F2of crossover operation are shown in Fig. 5. Then

we can get RM1={U2, U4},Ssc

1={U1},Scm

1={U4},RM2={U2, U4},Ssc

2={U1, U2},

Scm

2={U2}. For Gene3 and Gene4 of F1, the UAVs need to be randomly selected from

Ssc

2due to Ssc

26=∅. We suppose that the selected UAVs are all U2. For Gene5 of F1,

Algorithm 3: Crossover operator 2

Input:p1,p2,F1,F2

Output: Crossover oﬀspring

1i←p1−1;

2while i≤p2−1do

3T arget1←F1[i][0];

4T arget2←F2[i][0];

5if T arget16=T ar get2then

6Exchange gene information of T arget1 and T arg et2 in F1;

7if There is only one high-value target between T arget1and T arg et2then

8if T arget1is a high-value target then

9ht ←T arget1, lt ←T ar get2;

10 else if T ar get1is a low-value target then

11 ht ←T arget2, lt ←T ar get1;

12 Randomly select a attack gene of lt;

13 Replace the target of the selected gene with ht;

14 l←The number of genes updated between p1and p2, and i←i+l;

p1p2p1p2

F1F2

Gene3 Gene4 Gene5 Gene3 Gene4 Gene5

Fig. 5. The parents of crossover operation

RM1∩Scm

2={U2} 6=∅,RM1∩Scm

2− {U2}=∅. Therefore, the UAV needs to be randomly

selected from RM1− {U2}={U4}, and the UAV information of Gene5 of F1does not change.

The new Gene3-Gene5 of F1is shown in Fig. 6. Similarly, for Gene3-Gene5 of F2, UAVs need

to be randomly selected from RM2∩Scm

1, the UAVs of types sand cthat do not belong to

Ssc

2and Ssc

1. Suppose U4,U2, and U1are UAVs that are selected. The updated Gene3-Gene5

after the crossover operation is shown in Fig. 6. The oﬀspring chromosomes of F1and F2are

shown in Fig. 7.

Gene3 Gene4 Gene5

Fig. 6. The updated genes obtained after the crossover operation

Ospring1 O spring2

Fig. 7. The oﬀspring chromosomes of F1and F2

4.2.2. Crossover operator for insuﬃcient ammunition

In this case,

i=1

AMi< NT+L. If there are high-value targets, the dimension of the

chromosome depends on the number of ammunition and high-value targets. There are two

diﬃculties in constructing the crossover operator: 1) The dimensions of the chromosomes are

diﬀerent; 2) RM1=RM2=∅.

The crossover method of replacing the UAV is not suitable for this situation. Considering

that there are targets that have not been performed, the construction of the crossover operator

in this case is based on the idea of replacing targets. Let P T1and P T2denote the set of targets

in F1and F2, respectively. The crossover operation on F1is shown in Algorithm 4. Similarly,

the crossover operation on F2can be obtained.

The procedure of Algorithm 4is explained based on the example of 3 UAVs executing 6

targets. The UAVs are U1,U2and U3in Table 2, and the targets are Ti(i∈I4) in Table

3,T5(MT5={C, A1, V }) and T6(MT6={C, A1, V }). Suppose F1and F2are shown

in Fig. 8. Because min{length(F1), length(F2)}= 10, the p1and p2need to be arbitrarily

selected from [1,10]. Suppose that p1= 5 and p2= 8. It can be seen from Fig. 8that the

genes between p1and p2are Gene5-Gene8, but based on Algorithm 4, the genes that actually

Algorithm 4: Crossover operator 3

Input:p1,p2,F1,F2

Output: Crossover oﬀspring

1i←p1−1;

2while i≤p2−1do

3ˆ

T←The target in chromosome[i];

4if RT1∩P T26=∅then

5Randomly select a target ˜

T∈RT1∩P T2;

6else if RT1∩P T2=∅then

7Randomly select a target ˜

T∈RT1;

8if ˆ

Tis a low-value target then

9if ˜

Tis a low-value target then

10 For genes whose target is ˆ

T, replace their targets with ˜

11 else if ˜

Tis a high-value target then

12 Randomly select a low-value target ˇ

T∈F1,ˇ

T6=ˆ

T, and delete its Cand Vtasks;

13 For genes whose target is ˆ

Tor ˇ

T, replace targets with ˜

14 else if ˆ

Tis a high-value target then

15 if ˜

Tis a high-value target then

16 For genes whose target is ˆ

T, replace their targets with ˜

17 else if ˜

Tis a low-value target then

18 Randomly select a low-value target ˇ

T∈RT1,ˇ

T6=˜

19 For genes whose target is ˆ

T, replace their targets with ˜

20 Randomly select an attack gene of target ˜

T, replace the target of this gene with ˇ

T, and

add the genes of the Cand Vtasks of ˇ

21 Update RT1and P T1;

22 j←The number of genes updated between p1and p2;

23 i←i+j;

need to be changed in F1and F2are Gene4-Gene10 and Gene4-Gene9, respectively. For F1

and F2,RT1={T1, T3, T6},P T1={T2, T4, T5},RT2={T3, T4},P T2={T1, T2, T5, T6}.

p1p2

Gene5 Gene6 Gene7 Gene8

p1p2

Gene5 Gene6 Gene7 Gene8

Fig. 8. The parents of crossover operation

For Gene4-Gene6 of F1, randomly select a target from RT1∩P T2={T1, T6}, and let

the chosen target is T1. Based on Algorithm 4, the low-value target T5of Gene4-Gene6 can

be directly replaced with T1, as shown in Fig. 9. The updated RT1is RT1={T3, T5, T6}.

For Gene7-Gene10 of F1, randomly select a target from RT1∩P T2={T5, T6}, and assume

that low-value target T6is chosen. Based on the Algorithm 4, three steps are required. First,

replace T4in Gene7-Gene10 with T6. Then, randomly delete one of the attack task genes of

T6, and assume that the deleted gene is Gene3. And update the set RT1. Finally, a target

with only one attack task is randomly selected from RT1. According to the information of

RT1, the selected target can only be T5. The genetic information of Cand Vtasks of T5also

needs to be added. Its worth mentioning that the UAVs that perform Cand Vtasks of T5

are randomly selected.

Gene7 Gene8 Gene9 Gene10

Randomly select a

target to replace T5

Randomly select a

target to replace T4.

Randomly delete

a gene of task A

of T6.

Randomly add the

genetic information

of a target with a

single attack task.

Gene5 Gene6

Gene4

Fig. 9. The crossover process of F1genes

For Gene4-Gene6 of F2, randomly select a target from RT2∩P T1={T4}to replace T6.

Then, randomly select a target from {T1, T2, T5}, and suppose that the chosen target is T2.

And add the attack task information of T2to T4. For Gene7-Gene9 of F2, randomly select a

target from RT2∩P T1={T2, T3, T6}to replace T5. Suppose that the selected target is T2.

The process of crossover operation is shown in Fig. 10. The crossover oﬀspring of F1and F2

are shown in Fig. 11.

Randomly select a

target to replace T6

Gene5 Gene6 Gene7 Gene8 Gene9

Gene4

Gene2 Gene3

Gene1

The crossover

opera on of Gene 4-

6 is completed T4

Randomly select

a target to

replace T5

Randomly select a

target, and only

retain the gene of

its a ack task

Fig. 10. The crossover process of F2genes

Ospring1

Ospring2

Fig. 11. The oﬀspring chromosomes of F1and F2

4.3. Mutation operator

The mutation operator has shown signiﬁcant inﬂuence on the algorithm performance (Zuo

et al.,2021). It has the ability to search locally and maintain the diversity of the distribution.

Considering that the dimension of chromosomes is not ﬁxed, diﬀerent mutation operators are

given for various situations. Let Pmdenotes the probability of mutation. If the probability

obtained is less than Pm, then crossover oﬀspring O1needs to be mutated. We select a

gene segment in O1, and perform mutation operation on the gene segment. Let q1and q2

respectively denote the start and end point of the gene fragment. CT denotes the set of

low-value targets in O1. Let Gcv and Garespectively denote the set of C, V and Atask genes

in O1.¯

RM1and ¯

RT 1respectively denote the set of UAVs with remaining ammunition and

unassigned targets in O1.

The mutation operator is given in Algorithm 5. Take oﬀspring1 in Fig. 7and oﬀspring2

in Fig. 11 as examples to illustrate the mutation operator. The chromosomes and mutation

points are shown in Fig. 12.

For oﬀspring1 in Fig. 12, it is the case of

i=1

AMi> NT+L, so the 1-9 lines in Algorithm

5are used. Let the parameters in Algorithm 5are CV U ={U1, U2},¯

RM1={U2, U4}. Since

Gene2 is a gene for attack task, we randomly select a UAV from ¯

RM1− {O1[i][2]}, in this

situation, is U2. And the updated ¯

RM1={U4}.Gene3 is a gene of Vtask, Gene4 is a gene

of Ctask, so the selected UAVs are all U1. The gene fragment after mutation in oﬀspring1 is

shown in Fig. 13.

Algorithm 5: Mutation operator

Input:q1,q2,O1

Output: Crossover oﬀspring

1i←q1−1;

2while i≤q2−1do

3if

i=1

AMi> NT+Lthen

4if O1[i][1] is Cor Vtask then

5Randomly select Um∈CV U − {O1[i][2]};

6else if O1[i][1] is Atask then

7Randomly select Um∈¯

RM1− {O1[i][2]};

8Update ¯

RM1;

9Replace the UAV information of O1[i] with Um;

10 else if

i=1

AMi< NT+Lthen

11 Randomly select a target Tm∈¯

RT 1;

12 Find all the genes of target O1[i][0];

13 Replace the target information of these genes with Tm;

14 if Tmis a high-value target, but O1[i][0] is not then

15 Randomly delete a target from CT − {O1[i][0]};

16 Allocate ammunition to Tm;

17 else if Tmis a low-value target, but O1[i][0] is not then

18 Randomly delete one of the attack genes of Tm;

19 else if

i=1

AMi=NT+Lthen

20 if O1[i][1] is Cor Vtask then

21 Randomly select a gene from Gcv − {O1[i]};

22 else if O1[i][1] is Atask then

23 Randomly select a gene from Ga− {O1[i]};

24 Exchange the UAV information of O1[i] and the selected gene;

25 j←The number of genes updated between q1and q2;

26 i←i+j;

Ospring1 in Fig.7

q1q2

Ospring2 in Fig.11

q1q2

Gene6 Gene7 Gene8 Gene9

Gene2 Gene3 Gene4

Fig. 12. Chromosomes of mutation operation

Randomly select a

UAV

Update the Gene2

informa on with

this selected UAV

Gene2 Gene3 Gene4

Randomly select a

UAV and update the

Gene3

Randomly select a

UAV and update

the Gene4

Gene2 Gene3 Gene4

Fig. 13. The mutation process of gene fragment of oﬀspring1

For oﬀspring2 in Fig. 12, it is the case of

i=1

AMi< NT+L, so the 10-18 lines in

Algorithm 5are used. The gene fragment should be Gene6−Gene9, but the actual gene

fragment that need to be mutated is Gene5−Gene10. For Gene5−Gene7, randomly select a

target from ¯

RT 1={T3, T5, T6}, and suppose it is T6. Then the updated ¯

RT 1={T2, T3, T5}.

For Gene8−Gene10, we randomly select a target from ¯

RT 1, and suppose it is T3. Then, the

gene fragment after mutation in oﬀspring2 is shown in Fig. 14. The chromosomes obtained

after mutation are shown in Fig. 15.

Gene6 Gene7 Gene8 Gene9

Gene5 Gene10

Randomly select a

target and suppose it is

Update the Gene5-7

Update the set

Gene6 Gene7 Gene8 Gene9

Gene5 Gene10

Randomly select a

target and suppose it is

Update the Gene8-9

Update the set

Gene6 Gene7 Gene8 Gene9

Gene5 Gene10

Fig. 14. The mutation process of gene fragment of oﬀspring2

Chromosome1 Chromosome2

Fig. 15. The chromosomes obtained by mutation

4.4. Deadlock and the method to unlock

4.4.1. Deadlock of chromosome

All chromosomes in the generated initial population satisfy constraints. However, the

chromosomes obtained after crossover and mutation operations cannot be guaranteed to sat-

isfy constraints (6). If constraints (6) are not satisﬁed, then some UAVs will be in an inﬁnite

waiting state. This phenomenon is called deadlock, and it can lead to infeasible solutions (Jia

et al.,2018). There are three cases of deadlock, which are called self-locking, interlocking and

hybrid deadlock.

The phenomenon of self-locking means that when a UAV performs multiple tasks of the

same target, the order of execution of the tasks does not satisfy the constraint (6). A gene

fragment is given in Fig. 16 (a) to illustrate the self-locking. The phenomenon of interlocking

Sequence of tasks

performed by U1

Priorities of tasks of T3

Sequence of tasks

performed by U1

Priorities of tasks of T2

Sequence of tasks

performed by U2

Sequence of tasks

performed by U3

Priorities of tasks of T1

(a) The example of self-locking (b) The example of interlocking

Fig. 16. Examples of locked chromosomes

means that multiple UAVs and targets are locked at the same time, as shown in the example

in Fig. 16 (b). Obviously, the order in which U1and U2perform tasks is opposite to the

priorities of the tasks of T1and T2, which will cause UAVs to be in a state of waiting for

inﬁnite time. This is the interlock state. Hybrid deadlock is a state that combines self-locking

and interlocking.

4.4.2. Unlocking method

In order to unlock the chromosomes, Deng et al. (2014) proposes a transposition opera-

tion based on the directed graph. Because hybrid deadlock is very complicated, the simpler

self-locking situation is solved ﬁrst, which can reduce the complexity of locking. However,

the interlock is resolved by randomly adjusting the assignment and sequence of tasks. And

judging whether the chromosome is locked and unlocking the chromosome are two completely

independent steps. When the scale of the chromosome is large, this unlocking method is not

eﬃcient.

In order to improve the eﬃciency of unlocking, a fast positioning unlocking method (F-

PUM) is proposed in this section. FPUM has two notable features. One feature is that it can

judge whether the chromosome is locked and unlock the chromosome at the same time, and

the other one is that it can accurately unlock the chromosome. First, transform the chromo-

some into the form of Fig. 4. Then, use lines 4-16 of Algorithm 6to determine whether the

chromosome is locked. Finally, if the chromosome is locked, use lines 17-30 of Algorithm 6to

unlock the chromosome at the locked points.

The deadlock examples in Fig. 16 are used to illustrate the unlocking method. For the

deadlock situation in Fig. 16 (a), MU1={TC

2, T A1

1, T V

3, T A1

3}. According to Algorithm 6, the

order of TV

3and TA1

3in MU1needs to be exchanged to unlock the self-locking situation, as

shown in Fig. 17. For the deadlock situation in Fig. 16 (b), MU1={TV

2, T A1

1, T C

3},MU2=

Exchange the order of

selected genes T4

Self-locking Unlocked form

Fig. 17. The unlocking process of the self-locking situation

{TV

1, T C

2, T A1

3},MU3={TC

1, T V

3}. The Ctask of T1can be performed. Update MU3and

MT1, then MU3={TV

3},¯

MT1={TC

1}. At this time, U1,U2and U3are in a state of waiting

for an inﬁnite time. According to the rules of Algorithm 6, the order of tasks in task sets of

MUi(i∈I3) is changed. The updated MU1,MU2and MU3are MU1={TA1

1, T V

2, T C

3},MU2=

{TC

2, T V

1, T A1

2, T A1

3},MU3={TV

3}, respectively. Obviously, the chromosome is unlocked. The

detailed process is shown in Fig. 18.

Exchange the

order of selected

genes

Interlocking Unlocked form

Fig. 18. The unlocking process of the interlocking situation

4.5. Improved genetic algorithm for MOCTAP

Based on the construction methods of the chromosomes and genetic operators and the

FPUM, the process of MOCTAP-GA is given as follows.

Algorithm 7 (MOCTAP-GA)

Step 1: Set the values of algorithm parameters, including the population size N, the crossover

probability Pc, the mutation probability Pm, and the maximum number of evolutions Gmax.

Step 2: Randomly generate an initial population Pg(g= 0) of size Nby Algorithm 1.

Algorithm 6: FPUM

Input: Chromosome

Output: Unlocked chromosome

1A←Empty array, A[i]←The task set MUi+1 of Ui+1 ,i∈Iau−1, where au is the number of allocated

UAVs;

2j←0, ¯

MTi←∅,i∈INT;

3while j < DIM do

4j1←0, n←0;

5for l= 0;l < length(A);l+ + do

6if A[l][0] is Ctask then

7¯

MTl←¯

MTl∪A[l][0], n←n+ 1;

8else if A[l][0] is Atask then

9If ¯

MTlcontains C, then ¯

MTl←¯

MTl∪A[l][0], and n←n+ 1, otherwise, n←n;

10 else if A[l][0] is Vtask then

11 If ¯

MTlcontains Cand A, then ¯

MTl←¯

MTl∪A[l][0], and n←n+ 1, otherwise, n←n;

12 A[l]←A[l]− {A[l][0]};

13 if length(A[l]) = 0 then

14 A←A− {A[l]};

15 l←l+ 1;

16 j1←j1+n;

17 if n= 0 then

18 for l= 0;l < length(A);l+ + do

19 if A[l][0] is Atask then

20 for m= 1;m < length(A);l+ + do

21 if A[l][m]is Ctask then

22 Exchange the order of A[l][m] and A[l][0];

23 Exchange the order of genes corresponding to A[l][m] and A[l][0] in

chromosome;

24 break;

25 else if A[l][0] is Vtask then

26 for m= 1;m < length(A);l+ + do

27 if A[l][m]is Cor Atask then

28 Exchange the order of A[l][m] and A[l][0];

29 Exchange the order of genes corresponding to A[l][m] and A[l][0] in

chromosome;

30 break;

31 j←j+j1;

Step 3: Calculate the values of ﬁtness functions for each chromosome of population Pg.

Step 4: Generate a descendant population of size N.

Step 4.1: Use roulette algorithm to select parent chromosomes for crossover operation

from population Pg.

Step 4.2: According to the battle situation, select the appropriate crossover operator from

the Algorithm 2–4. Generate a random number r1∈[0,1]. If r1< Pc, then cross over the

chromosomes and go to Step 4.3, otherwise, go to Step 4.1.

Step 4.3: Generate a random number r2∈[0,1]. If r2< Pm, then mutate the crossover

oﬀspring by Algorithm 5.

Repeat Steps 4.1-4.3 to get a population Sgof size N.

Step 5: Use Algorithm 6to judge whether the chromosomes in population Sgare locked,

and unlock the locked chromosomes.

Step 6: Combine Pgwith Sgto get a new population Qgof size 2N.

Step 7: Calculate the value of the ﬁtness functions and the crowding distance of chromosomes

in Qg.

Step 8: Select Nchromosomes from Qgto form a new population Pqbased on the fast

non-dominated sorting approach and the elite-preserving strategy.

Step 9: Let g=g+ 1, q=g.

Step 10: If g < Gmax , go to Step 3. Otherwise, sort Pgby using the fast non-dominated

sorting approach, then obtain the Pareto solution set. Select a solution from the Pareto

solution set. Output Pareto solution set and the selected solution.

In MOCTAP-GA, the process of selecting individuals from current individuals to form

a new population in each iteration is based on the fast non-dominated sorting approach

(Deb et al.,2002). The time complexity of the algorithm is governed by the non-dominated

sorting part of the algorithm. Therefore, the algorithm’s time complexity for each iteration

is O(MN 2), where Mis the number of objectives and Nis the population size. In order to

understand the speciﬁc calculation process of the algorithm more clearly, the ﬂow chart of

MOCTAP-GA is shown in Fig. 19.

5. Numerical experiments

In this section, numerical experiments are conducted to analyze the characteristics of

the model of MOCTAP and the performance of the proposed algorithm MOCTAP-GA. The

algorithm proposed in this paper is suitable for solving multi-objective optimization problems

of task assignment in various battle conditions, including suﬃcient ammunition resources and

insuﬃcient ammunition resources. Sections 5.1-5.3 give examples of all battle situations, and

analyze the performance of the proposed algorithm based on these examples. Sections 5.4 gives

the analysis of the eﬀectiveness of the proposed unlocking method. Numerical experiments are

implemented by using Python3.8 on a computer with Intel Core i5-10210U CPU @ 1.60GHz

2.11GHz and 4.00GB RAM.

Let ¯pdenotes the number of non-dominated solutions on the Pareto-optimal front end,

Start

Maximum number of it-

erations Gmax, Popula-

tion size N, and g= 0

Initialize the

population

Calculate ﬁt-

ness function

Selection, crossover,

and mutation op-

erations. Get the

oﬀspring popu-

lation of size N

Locked?

Unlock

Combine the par-

ent population

and the oﬀspring

population into a

population of size 2N

Fast non-

dominated sort

Calculate the

crowded distance

of individuals

Select Nchro-

mosomes to form

a new parent

population

g=g+ 1

g < Gmax?

Fast non-

dominated sort

Select a solution

from the Pareto

solution set

The Pareto solution set

and the chosen solution

End

Yes

Yes No

Fig. 19. The ﬂow chart of MOCTAP-GA

and NDi= (Fi

1, F i

2) (i∈I¯p) denotes the ith non-dominated solution. In order to facilitate

decision makers to select a solution from the Pareto solution set, the following selection rule

is given. First, calculate Si=α1Fi

1+α2Fi

2(i∈I¯p), where α1and α2are weights and

α1+α2= 1. The value of the weights is related to the decision maker’s preference for the

objective functions. Second, sort the Si(i∈I¯p). Finally, select Siwith the smallest value,

and its corresponding solution needs to be chosen by the decision maker.

5.1. Scenario I: total ammunition >total number of attack tasks

In this case, we consider 5 UAVs to execute 5 targets. The details of the UAVs and targets

are shown in Tables 4and 5. The probabilities of death and success are shown in Table 6.

Table 4. The information of UAVs

Type AMiValue Type AMiValue

U1c 3 3.2 U4s 0 2.5

U2m 5 3.0 U5s 0 2.5

U3c 3 3.0 - - - -

Table 5. The information of targets

Type Task set Value Type Task set Value

T1High value {C, A1, A2, V }2.5 T4Low value {C, A1, V }1.5

T2High value {C, A1, A2, V }2.0 T5Low value {C, A1, V }1.9

T3Low value {C, A1, V }1.7 - - - -

Table 6. The probabilities of tasks being successfully executed and the probability of survival

of UAVs

T1T2T3T4T5

C A V C A V C A V C A V C A V

KI k

1j0.18 0.48 0.07 0.11 0.45 0.1 0.04 0.16 0.06 0.14 0.24 0.07 0.18 0.1 0.05

SV k

1j0.82 0.7 0.97 0.92 0.78 0.93 0.96 0.93 0.95 0.98 0.96 0.97 0.82 0.95 1.0

KI k

2j0.15 0.37 0.1 0.01 0.35 0.04 0.1 0.2 0.07 0.14 0.14 0.05 0.08 0.23 0.02

SV k

2j0.96 0.72 0.94 0.81 0.81 0.98 0.91 1.0 0.95 1.0 0.94 0.92 0.96 0.97 0.97

KI k

3j0.2 0.47 0.05 0.02 0.38 0.09 0.01 0.27 0.01 0.1 0.23 0.08 0.1 0.23 0.06

SV k

3j0.83 0.8 0.94 0.82 0.79 0.96 0.92 0.98 0.94 0.87 0.97 0.92 0.93 0.92 0.91

KI k

4j0.15 0 0.04 0.09 0 0.1 0.0 0 0.05 0.12 0 0.01 0.19 0 0.01

SV k

4j0.93 0 0.95 0.99 0 0.9 0.94 0 0.99 0.85 0 0.98 0.83 0 0.92

KI k

5j0.01 0 0.09 0.17 0 0.05 0.13 0 0.01 0.15 0 0.02 0.03 0 0.09

SV k

5j0.81 0 0.91 0.8 0 0.96 0.9 0 0.98 0.96 0 0.94 0.94 0 0.98

To verify the eﬀectiveness of the proposed algorithm, MOCTAP-GA is compared with

Multi-objective Particle Swarm Optimization (MOPSO) and Multi-objective Grey Wolf Op-

timization (MOGWO), two unlocking methods (i.e. FPUM and transposition operation (TO))

are used in MOCTAP-GA, MOPSO and MOGWO, respectively. In MOCTAP-GA, the

crossover probability is Pc= 0.8, and the mutation probability is Pm= 0.2. In MOPSO,

the values of the parameters are ω= 0.7298, c1= 1.49618, c2= 1.49618. And in MOGWO,

the values of the parameters are r1=r2=1

3. For the values of the population size Nand

the maximum number of iterations G, two cases are considered, which are N= 80, G= 50

and N= 100, G= 200, respectively. The performance of three algorithms using diﬀeren-

t unlocking methods in the small-scale case (N= 80, G= 50) is compared, and in order

to highlight the performance of MOCTAP-GA, the performance of three algorithms using

diﬀerent unlocking methods in the large-scale case (N= 100, G= 200) is compared.

The comparison results of the Pareto-optimal front under two scales (i.e. N= 80, G= 50;

N= 100, G= 200) obtained by the three algorithms are shown in Fig. 20(a) and Fig. 22(a),

respectively. From the comparison results, it can be concluded that the solutions of MOPSO

and MOGWO are dominated by the solutions of MOCTAP-GA, that is, the results obtained

by MOCTAP-GA are better than those obtained by MOPSO and MOGWO. At the same

time, it can be seen from the comparison that the performance of the algorithm using FPUM

is better than the algorithm using TO. The MOCTAP-GA using FPUM has advantages in

both calculation results and CPU runtime than using TO. More detailed comparison results

between diﬀerent algorithms are given in Table 7. The convergence curves of the objective

functions in the iterations of the three algorithms using diﬀerent unlocking methods are shown

in Fig. 21 and Fig. 23, respectively. Based on the given solution selection strategy, the 9th

solution (5.4921, 3.6313) in Fig. 20(a) and the 15th solution (5.4802, 3.6313) in Fig. 22(a) are

the non-dominated solutions selected from the obtained Pareto-optimal in these two cases.

The corresponding assignment schemes are given in Table 8, respectively.

To verify the computational eﬃciency of MOCTAP-GA using FPUM, 15 experiments

in two scales were performed based on three algorithms using diﬀerent unlocking methods.

The CPU runtime of the experiments is shown in Fig. 20(b) and Fig. 22(b), and the average

CPU runtime of the algorithms is shown in Table 8. Obviously, compared with MOPSO using

FPUM and TO and MOCTAP-GA using TO, the CPU runtime of MOCTAP-GA using FPUM

is the shortest, which proves that its eﬃciency is the best. Because the individuals in MOPSO

are encoded as binary matrices, the large scale of the problem results in long CPU runtime for

pbest and gbest. This is the main reason for the long CPU runtime of MOPSO. Compared with

MOGWO, the CPU runtime of MOCTAP-GA is slightly longer, but the calculation result of

MOCTAP-GA is better than that of MOGWO. Therefore, MOCTAP-GA works better than

MOGWO for CTAP that require high-quality solutions.

5.2. Scenario II: total ammunition =total number of attack tasks

In this section, an example involving 8 UAVs and 12 targets is considered, which contains

5 high-value targets. For the convenience of description, the information of Ui(i∈I5)

and Tj(j∈I5) is consistent with the information in Table 4-6, and the information of Ui

(i∈ {6,7,8}) and Tj(j∈ {6,7,· · · ,12}) is shown in Table 9and Table 10. Table 11 shows

(a) Pareto-optimal front (b) CPU runtime

Fig. 20. N= 80 and G= 50

Table 7. Information about the results calculated by the algorithms

Algorithm N, G(Size) Unlocking method J3min J2min CPU runtime (s)

MOCTAP-GA

80, 50 FPUM 5.403 (5.403,5.694) 4.900 (5.973,4.900) 4.845

TO 5.437 (5.437,5.796) 5.208 (6.258,5.208) 6.877

100, 200 FPUM 5.236 (5.236,4.099) 3.600 (5.652,3.600) 27.137

TO 5.368 (5.368,4.464) 4.171 (5.540,4.171) 30.959

MOPSO

80, 50 FPUM 5.537 (5.537,6.342) 4.787 (6.182,4.787) 35.35

TO 5.514 (5.514,6.490) 5.457 (6.080,5.457) 49.616

100, 200 FPUM 5.209 (5.209,4.991) 3.941 (5.802,3.941) 1055.985

TO 5.280 (5.280,5.505) 4.912 (5.852,4.912) 1327.714

MOGWO

80, 50 FPUM 5.662 (5.662,6.704) 5.657 (6.778,5.657) 4.001

TO 5.904 (5.904,6.648) 5.695 (6.508,5.695) 14.735

100, 200 FPUM 5.795 (5.795,6.131) 5.708 (6.461,5.780) 22.054

TO 5.699 (5.699,6.569) 5.604 (6.449,5.604) 23.3

Table 8. The task assignment plans of the selected solutions

N, G(Size) Solution UAV Task set UAV Task set

80, 50 (5.4941,5.2658)

U1MU1={T2

5, T 3

5}U2MU2={T2

2, T 2

3, T 2

U3MU3={T1

2, T 2

2, T 3

3, T 2

4, T 2

1, T 3

1}U4MU4={T1

3, T 1

4, T 3

U5MU5={T1

5, T 1

100, 200 (5.4941,5.2658)

U1MU1={T1

4, T 2

2}U2MU2={T2

1, T 2

3, T 2

U3MU3={T1

2, T 2

4}U4MU4={T3

1, T 1

3, T 3

5, T 3

U5MU5={T1

1, T 3

3, T 1

5, T 3

Fig. 21. The curves of changes in the values of J2and J3(N= 80, G= 50)

(a) Pareto-optimal front (b) CPU runtime

Fig. 22. N= 100 and G= 200

the survival probability of UAVs and the probability of the tasks being successfully executed.

In MOCTAP-GA, the parameters are N= 100, G= 200, Pc= 0.8 and Pm= 0.2.

The obtained Pareto-optimal front is shown in Fig. 24(a). According to the given method

for selecting solution from the Pareto-optimal front, the 9th non-dominated solution in Fig. 24(a)

is selected, and its task assignment scheme is shown in Table 12. In order to obtain the CPU

runtime of MOCTAP-GA for this scale experiment, 15 numerical experiments were carried

out for this example. The CPU runtime is shown in Fig. 24(b). In order to analyze the con-

vergence of algorithm, the optimal values of objective functions in each iteration are recorded.

During the entire iteration process, the changes of the minimum value of J2and J3in each

iteration are shown in Fig. 25. It can be observed that the values of both functions tend to

Table 9. The information of Ui(i∈ {6,7,8})

Type AMiValue Type AMiValue

U6s 0 2.5 U8m 4 3.0

U7c 2 2.9 - - - -

Fig. 23. The curves of changes in the values of J3and J2(N= 100, G= 200)

Table 10. The information of Tj(j∈ {6,7,· · · ,12})

Type Task set Value Type Task set Value

T6Low value {C, A1, V }1.8 T10 Low value {C, A1, V }1.9

T7High value {C, A1, A2, V }2.3 T11 High value {C, A1, A2, V }2.4

T8Low value {C, A1, V }1.7 T12 Low value {C, A1, V }1.8

T9High value {C, A1, A2, V }2.2 - - - -

converge as the number of evolved generations increases.

The main contribution and application value of this paper is that the constructed mis-

sion assignment model takes into account the conditional probabilities and the corresponding

algorithms designed on the consideration of various risks in realistic scenarios, i.e., the risk

of the UAV being destroyed by the enemy and the possibility of the mission not being com-

pleted. Ignoring these objective factors can make the obtained solution not a Pareto solution

or even very worse in a realistic scenario. In order to illustrate the necessity of considering

both risks, we compare it with two extreme cases. In the ﬁrst case, only the risk of mission

failure is considered and the risk of UAV being destroyed is ignored. In the second case, the

risk of destroying the UAV is considered without considering the risk of not completing the

task. The initial populations for the comparison experiments are identical, N= 100 and

G= 200. The Pareto-optimal fronts of the populations obtained at the 200th iteration for

each of these two cases are obtained according to the constructed model using conditional

probabilities. Then, the obtained Pareto-optimal fronts were compared, and the results are

shown in Fig. 26. It can be seen that the non-dominated solutions obtained using the method

of this paper dominate the Pareto-optimal fronts obtained in these two cases.

5.3. Scenario III: total ammunition <total number of attack tasks

In the case of insuﬃcient ammunition, a large-scale example of 32 UAVs performing tasks

on 42 targets is considered in this subsection. Among them, there are 24 high-value targets

and 18 low-value targets. The information of UAVs and targets are given in Tables 13 and

Table 11. The probabilities of tasks being successfully executed and the probabilities of

survival of UAVs

T1T2T3T4T5

C A V C A V C A V C A V C A V

KI k

6j0.15 0 0.01 0.09 0 0.06 0.23 0 0.08 0.0 0 0.05 0.05 0 0.18

SV k

6j0.9 0 0.99 0.78 0 0.83 0.94 0 0.95 0.82 0 0.91 0.78 0 0.93

KI k

7j0.16 0.17 0.02 0.1 0.23 0.08 0.08 0.45 0.16 0.14 0.3 0.07 0.16 0.26 0.02

SV k

7j0.71 0.96 0.99 0.79 0.84 0.83 0.78 0.97 0.9 0.97 0.97 0.93 0.78 0.94 0.93

KI k

8j0.13 0.14 0.08 0.07 0.44 0.03 0.06 0.49 0.19 0.24 0.42 0.14 0.01 0.49 0.07

SV k

8j0.84 0.84 0.98 0.95 0.81 0.84 0.72 0.85 0.89 0.81 0.76 0.83 0.87 0.79 0.99

T6T7T8T9T10

C A V C A V C A V C A V C A V

KI k

1j0.18 0.44 0.0 0.03 0.21 0.03 0.06 0.35 0.07 0.18 0.12 0.06 0.17 0.49 0.18

SV k

1j0.72 0.74 0.91 0.85 0.97 0.82 0.74 0.96 0.94 0.93 0.85 0.93 0.79 0.75 0.83

KI k

2j0.11 0.17 0.15 0.06 0.22 0.02 0.02 0.33 0.11 0.12 0.27 0.06 0.18 0.49 0.04

SV k

2j0.96 0.94 0.83 0.93 0.74 0.84 0.81 0.95 0.93 0.82 0.93 0.92 0.83 0.79 0.92

KI k

3j0.05 0.1 0.1 0.07 0.49 0.1 0.2 0.18 0.12 0.18 0.29 0.08 0.19 0.45 0.01

SV k

3j0.77 0.85 0.95 0.79 0.83 0.86 0.81 1.0 0.89 0.98 0.82 0.92 0.97 0.9 0.81

KI k

4j0.01 0 0.12 0.03 0 0.04 0.3 0 0.04 0.16 0 0.04 0.08 0 0.16

SV k

4j0.8 0 0.9 0.89 0 0.96 0.75 0 0.96 0.82 0 0.89 0.72 0 0.88

KI k

5j0.22 0 0.11 0.01 0 0.08 0.01 0 0.17 0.15 0 0.08 0.23 0 0.04

SV k

5j0.95 0 0.95 0.82 0 0.97 0.77 0 0.81 0.91 0 0.84 0.72 0 0.92

KI k

6j0.03 0 0.06 0.14 0 0.05 0.06 0 0.19 0.17 0 0.09 0.23 0 0.0

SV k

6j0.98 0 0.94 0.73 0 0.85 0.91 0 0.8 0.77 0 0.8 0.97 0 0.83

KI k

7j0.05 0.46 0.05 0.17 0.36 0.03 0.05 0.11 0.08 0.17 0.34 0.04 0.06 0.44 0.08

SV k

7j0.71 0.76 0.94 0.95 0.86 0.87 0.82 0.93 0.82 0.87 0.91 0.88 0.98 0.86 0.93

KI k

8j0.18 0.35 0.12 0.05 0.26 0.1 0.09 0.4 0.06 0.01 0.38 0.07 0.29 0.2 0.2

SV k

8j0.89 0.74 0.9 0.71 0.94 1.0 0.99 0.78 0.97 0.94 0.8 0.86 0.75 0.98 0.94

T11 T12 T11 T12

C A V C A V C A V C A V

KI k

1j0.09 0.37 0.06 0.27 0.12 0.09 U5

KI k

5j0.18 0 0.07 0.29 0 0.11

SV k

1j0.75 0.8 0.98 0.74 0.71 0.84 S V k

5j0.88 0 0.95 0.92 0 0.89

KI k

2j0.16 0.32 0.08 0.27 0.26 0.06 U6

KI k

6j0.14 0 0.03 0.16 0 0.04

SV k

2j0.82 0.91 0.81 0.9 0.95 0.88 S V k

6j0.99 0 0.96 0.94 0 0.94

KI k

3j0.12 0.26 0.01 0.0 0.35 0.18 U7

KI k

7j0.12 0.44 0.06 0.07 0.2 0.02

SV k

3j0.81 0.93 0.87 0.75 0.99 0.95 SV k

7j0.77 0.9 0.83 0.87 0.96 0.98

KI k

4j0.16 0 0.07 0.03 0 0.08 U8

KI k

8j0.08 0.36 0.07 0.1 0.44 0.13

SV k

4j0.95 0 0.97 0.73 0 0.91 S V k

8j0.79 0.84 0.84 0.97 0.78 0.98

(a) Pareto-optimal front (b) CPU runtime

Fig. 24. Pareo-optimal front and CPU runtime

Table 12. The task assignment plans of the selected solutions

Solution UAV Task set UAV Task set

(14.1019,10.1648)

U1MU1={T2

12, T 3

7, T 2

5, T 3

3, T 3

6, T 2

9, T 3

9}U2MU2={T2

7, T 2

4, T 2

3, T 2

2, T 2

11}

U3MU3={T2

10, T 1

12, T 1

4, T 2

6, T 1

11, T 3

11, T 2

9}U4MU4={T3

5, T 3

4, T 1

3, T 1

6, T 3

2, T 1

U5MU5={T3

12, T 1

7, T 1

5, T 1

1}U6MU6={T1

8, T 1

U7MU7={T1

10, T 3

10, T 2

1, T 3

1, T 2

8, T 3

8}U8MU8={T2

7, T 2

1, T 2

2, T 2

11}

14. Given the large size of the problem, this leads to a huge amount of probabilistic in-

formation involved, which can be found in the URL https://github.com/gaoxh-github/

parameters-values-of-problem-. The parameters in MOCTAP-GA are N= 100, G= 200,

Pc= 0.8 and Pm= 0.2.

The obtained Pareto-optimal front and the CPU runtime of 15 experiments are shown in

Fig. 27. The selected solution for parameter α1=α2= 0.5 is the 5th non-dominated solution.

The task assignment scheme corresponding to the selected solution is shown in Table 15. In

order to clearly show the computational eﬃciency of MOCTAP-GA, the CPU runtime of

MOCTAP-GA for 9 examples of diﬀerent sizes is given, and each case is calculated 15 times.

Fig. 25. The curves of changes in the values of J2and J3

Fig. 26. Experimental comparisons that demonstrate the practical value of the method

Table 13. The information of UAVs for Scenario III

UAV Type AMiValue UAV Type AMiValue UAV Type AMiValue

U1m 3 3.2 U12 s 0 2.5 U23 c 2 3.2

U2c 2 3.2 U13 m 4 3.3 U24 s 0 2.6

U3c 2 3.1 U14 c 3 3.1 U25 m 3 3.1

U4c 2 3.1 U15 c 2 3.0 U26 m 4 3.3

U5s 0 2.7 U16 s 0 2.8 U27 s 0 2.7

U6s 0 2.9 U17 m 3 3.1 U28 c 2 3.1

U7s 0 2.8 U18 s 0 2.5 U29 m 3 3.1

U8c 3 3.0 U19 c 2 2.9 U30 m 4 3.2

U9m 4 3.3 U20 s 0 2.8 U31 m 3 3.1

U10 s 0 2.9 U21 m 4 3.2 U32 s 0 2.8

U11 s 0 2.7 U22 m 4 3.3 - - - -

Table 14. The information of targets for Scenario III

Target Task set Value Target Task set Value Target Task set Value

T1{C, A1, A2, V }2.2 T15 {C, A1, V }1.9 T29 {C, A1, A2, V }2.0

T2{C, A1, V }1.7 T16 {C, A1, V }1.7 T30 {C, A1, V }1.5

T3{C, A1, A2, V }2.4 T17 {C, A1, A2, V }2.5 T31 {C, A1, A2, V }2.4

T4{C, A1, A2, V }2.4 T18 {C, A1, A2, V }2.3 T32 {C, A1, A2, V }2.1

T5{C, A1, V }1.7 T19 {C, A1, V }1.6 T33 {C, A1, A2, V }2.2

T6{C, A1, A2, V }2.2 T20 {C, A1, A2, V }2.1 T34 {C, A1, A2, V }2.0

T7{C, A1, V }1.5 T21 {C, A1, A2, V }2.0 T35 {C, A1, V }1.7

T8{C, A1, A2, V }2.5 T22 {C, A1, V }1.5 T36 {C, A1, A2, V }2.2

T9{C, A1, A2, V }2.0 T23 {C, A1, V }1.7 T37 {C, A1, A2, V }2.4

T10 {C, A1, A2, V }2.1 T24 {C, A1, V }1.7 T38 {C, A1, V }1.8

T11 {C, A1, A2, V }2.0 T25 {C, A1, A2, V }2.4 T39 {C, A1, V }1.8

T12 {C, A1, V }1.9 T26 {C, A1, A2, V }2.4 T40 {C, A1, A2, V }2.1

T13 {C, A1, V }1.5 T27 {C, A1, A2, V }2.4 T41 {C, A1, V }1.6

T14 {C, A1, A2, V }2.2 T28 {C, A1, V }1.6 T42 {C, A1, V }1.8

Table 15. The task assignment of the 5th non-dominated solution in Fig. 27(a)

UAV Task set UAV Task set

U1MU1={T2

1, T 2

21, T 2

24}U2MU2={T3

16, T 3

1, T 2

4, T 3

20, T 2

27}

U3MU3={T1

16, T 1

15, T 1

9, T 2

9, T 3

30, T 3

29, T 2

3, T 3

7, T 3

22, T 1

5}U4MU4={T2

26, T 2

23, T 3

27}

U5MU5={T1

1, T 1

14}U6MU6={T1

19, T 1

18, T 3

15, T 3

31, T 1

41, T 3

17}

U7MU7={T1

3, T 1

17}U8MU8={T3

19, T 3

40, T 1

13, T 3

13, T 1

31, T 2

14, T 2

28}

U9MU9={T2

40, T 2

30, T 2

31, T 2

3}U10 MU10 ={T1

21, T 1

6, T 3

6, T 1

25}

U11 MU11 ={T1

23, T 1

29, T 1

24, T 1

27, T 3

5}U12 MU12 ={T3

9, T 1

42}

U13 MU13 ={T2

29, T 2

36, T 2

27, T 2

6}U14 MU14 ={T3

26, T 1

7, T 1

8, T 1

28, T 3

24, T 2

32, T 2

25, T 2

17}

U15 MU15 ={T2

16, T 2

15, T 3

42}U16 MU16 ={T3

25}

U17 MU17 ={T2

4, T 2

37, T 2

20}U18 MU18 ={T1

26, T 1

4, T 3

37, T 1

30, T 3

41, T 101}

U19 MU19 ={T1

38, T 3

21, T 1

37, T 1

33, T 2

35, T 2

6, T 3

10, T 1

32}U20 MU20 ={T3

33, T 3

35, T 3

14, T 3

28, T 1

22}

U21 MU21 ={T2

20, T 2

36, T 2

8, T 2

22}U22 MU22 ={T2

37, T 2

5, T 2

10, T 2

32}

U23 MU23 ={T2

8, T 2

10}U24 MU24 ={T3

2, T 3

3, T 3

36}

U25 MU25 ={T2

1, T 2

33, T 2

7}U26 MU26 ={T2

18, T 2

38, T 2

33, T 2

41}

U27 MU27 ={T1

12, T 3

12}U28 MU28 ={T3

23, T 1

2, T 2

18, T 3

4, T 1

36, T 3

8, T 2

12, T 3

32}

U29 MU29 ={T2

26, T 2

19, T 2

9}U30 MU30 ={T2

2, T 2

13, T 2

29, T 2

14}

U31 MU31 ={T2

40, T 2

21, T 2

42}U32 MU32 ={T1

40, T 1

35, T 1

20}

In these 9 examples, the variation rule for the number of UAVs is that each scale example

has 4 UAVs more than its previous scale example, and the variation rule for the number of

targets is that each scale example has 5 targets more than its previous scale example. The

CPU runtime and the average CPU runtime of MOCTAP-GA for these 9 problems are shown

in Fig. 28 and Fig. 29.

(a) Pareto-optimal front (b) CPU runtime

Fig. 27. Pareo-optimal front and CPU runtime

5.4. Test of the genetic operators

The various constraints of CTAP and the special meaning of chromosome genes make the

comparison of the crossover operator constructed in this paper with other crossover operators

less meaningful. Therefore, to test the eﬀect of the crossover operator part on the performance

of the algorithm, for a given same initial population, diﬀerent crossover probabilities are

considered to observe the change curve of the objective function values. Taking the 8 UAVs

Fig. 28. CPU runtime for 9 problems of diﬀerent sizes

Fig. 29. Average CPU runtime of MOCTAP-GA for 9 problems of diﬀerent sizes

and 12 targets in Section 5.2 as an example, the change curves of the objective function values

for 5 crossover probabilities (Pc={0.5,0.6,· · · ,0.9}) are shown in Fig. 30, from which it can

be seen that the algorithm with a crossover probability of 0.8 has the best performance for

the objective function.

Fig. 30. The curves of changes in the values of J2and J3under diﬀerent crossover probabilities

The mutation operator is an auxiliary for generating new individuals in the algorithm.

It has local search capability and can increase the diversity of the population. To verify

the eﬀectiveness of the constructed mutation operator. Three diﬀerent groups of numerical

simulations with diﬀerent initial populations were carried out for the example in Section 5.2.

Each simulations includes experiments with and without the mutation operator, while the

initial population of numerical experiments in the same group is the same. The obtained

Pareto-optimal front is shown in Fig. 31, and the curve of the objective function is shown

in Fig. 32. It can be seen that the solution of the Pareto-optimal front obtained by the

algorithm using the constructed genetic operator dominates the solution obtained by the

algorithm without the mutation operator in the three groups of numerical simulations. Also,

the curves of the objective function of the algorithm using the mutation operator are better

than those of the algorithm without the mutation operator.

Fig. 31. The Pareto-optimal fronts of 3 groups of simulation experiments

Fig. 32. The curves of changes in the values of J2and J3for the 3 groups of simulation

experiments

5.5. Test of the FPUM

The processing of a chromosome obtained through crossover and mutation operations is

divided into two stages. The ﬁrst stage is to determine whether the chromosome is locked.

If the chromosome is locked, then the second stage is to unlock the chromosome. Deng

et al. (2014) proposes the transposition operation method, which randomly reorders the

genes of chromosome to achieve the purpose of unlocking chromosome. However only the

unlocking stage is considered in the transposition operation, and the random selection of

genes is performed on the entire chromosome. Although transposition operation method

guarantees randomness, the probability of chromosome being unlocked after each operation is

low. In this paper, the FPUM combines the judgment stage and the unlocking stage, and the

chromosome is unlocked according to the given rules at the deadlock point. The unlocking

principle of FPUM can preserve randomness while reducing unlocking time.

The eﬃciency of these two methods is compared by taking the case of 8 UAVs performing

tasks on 12 targets as an example. Among all targets, 5 high value targets are included. First,

100 chromosomes are randomly generated without considering timing constraints. Then, the

two methods were used to unlock these chromosomes for 15 unlocking experiments. The

average CPU runtime for FPUM and the transposition operation is 0.2442sand 66.3289s,

respectively. The CPU runtime comparison of unlocking is shown in the Fig. 33. As can be

seen from the Fig. 33, the unlocking eﬃciency of FPUM is higher than that of the transposition

operation in (Deng et al.,2014). There are two main reasons for the high unlocking eﬃciency

of FPUM. One is that the two stages are combined together, and the other is that the

chromosome is unlocked directly at the deadlock point.

In order to obtain the variation of the unlocking time of the algorithm for problems of

diﬀerent sizes, 9 examples in Section 5.3 are utilized to test the two unlocking algorithms.

In each problem, 100 chromosomes are randomly generated, and these two algorithms are

used to unlock chromosomes 15 times. The unlocking time of FPUM and transposition

operation are shown in Fig. 34. The unlocking eﬃciency of FPUM is much higher than that

of the transposition operation. It can be seen from Fig. 34 that when the problem becomes

Fig. 33. Comparison of time for two unlocking algorithms

complicated, the rate of change of the unlocking time of FPUM is smaller than that of the

transposition operation method. That means that the problem size has a greater impact on

the transposition operation than on the FPUM.

(a) FPUM (b) Transposition operation

Fig. 34. The unlocking times of these two algorithms

6. Conclusions

In this paper, we study MOCTAP for multiple UAVs by considering potential operational

risks. By introducing conditional probabilities for the risk of UAV destruction and mission

failure, we construct a novel mission assignment model for MOCTAP. Based on the idea of

evolutionary algorithms, we construct chromosome coding patterns and genetic operators for

diﬀerent ammunition cases to design a multi-objective optimization algorithm for MOCTAP.

For the case of chromosome deadlock, a fast localization unlocking method is presented, which

can eﬀectively unlock chromosomes at the locked points while maintaining the population

randomness. The results of numerical experiments demonstrate the validity of the algorithm

and the practicality of the model, as well as the use of this unlocking method signiﬁcantly

improves the optimization eﬃciency of the proposed multi-objective optimization algorithm.

The modeling ideas and algorithms can also be transferred to other areas of MOCTAP

that have a similar structure to our work. Moreover, the FPUM construction principles have

a certain generality that can be used to construct unlocking methods that are applicable

to diﬀerent problems. Considering the actual battleﬁeld environment, the feasible path of

the UAV is one of the factors that aﬀects the mission assignment of the UAV between two

target locations. Obstacles are usually present in large operational areas, such as enemy radar

zones, mountains, etc. The probability of UAV detection and attack is signiﬁcantly higher

when passing through the enemy’s radar zone. Therefore, it is meaningful to study the task

allocation problem based on trajectory planning when there are obstacles in the environment,

which is the focus of our further research.

CRediT authorship contribution statement

Xiaohua Gao : Writing–original draft, Writing–review & editing, Software, Methodology,

Data curation. Lei Wang : Methodology, Supervision, Investigation. Xinyong Yu : Soft-

ware, Formal analysis. Xichao Su : Conceptualization, Data curation. Yu Ding : Super-

vision, Validation. Chen Lu : Supervision, Formal analysis. Haijun Peng : Investigation,

Methodology. Xinwei Wang : Validation, Writing–review & editing, Methodology, Funding

acquisition, Project administration.

Declaration of competing interest

The authors declare that they have no known competing ﬁnancial interests or personal

relationships that could have appeared to inﬂuence the work reported in this paper.

Acknowledgements

This work was supported by the National Key Research and Development Plan (2021YF-

B3302501); the National Natural Science Foundation of China (12102077, 12161076); and the

Fundamental Research Funds for the Central Universities (DUT22LAB305, DUT22RC(3)010).

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Jul 2022

In realistic decision-making environments, human behaviors bring influences to various decision-making procedures, and classic multi-attribute decision-making approaches based on utility decisions own some deviations from actual situations. The behavioral decision theory modifies classic decision-making theories to make the new method more applicable. Regret theory, as one of the important components in behavioral decision theories, has been widely used in theories and applications. Based on regret theory, we establish a generalized three-way decision method on incomplete multi-scale decision information systems with interval fuzzy numbers. First, we select an incomplete optimal sub-system for the incomplete multi-scale information system, and convert the multi-scale evaluation information into an interval fuzzy number by using a linguistic term set. Second, based on the probability distribution of evaluation values and trade-off factors, we propose a target-dependent approximation estimation method for the incomplete interval fuzzy sub-system. Then, the regret-rejoicing preferences between objects are obtained. Finally, a tri-partition and ranking method based on a max-bipartition and preference index is established. Moreover, the incompleteness experiments show that the decision-making results of our method can still maintain more than 97% consistency in an incomplete information system with a missing rate of at most 20%. In addition, the parameter analysis shows that the proposed method maintains an accuracy rate of more than 98% and a classification error rate of no more than 0.6%.

Regret-Theoretic Multiattribute Decision-Making Model Using Three-Way Framework in Multiscale Information Systems

Article

May 2022

The existing multiattribute decision-making (MADM) methods on multiscale information systems (MSISs) are generally studied from the utility point of view, which may cause two problems: 1) the objects are strictly classified into good or bad, which may lead to misclassification and 2) the risk attitude and psychological behaviors of decision makers are difficult to be reflected. In light of this, this article proposes a wide three-way decision (3WD) model on an MSIS, which combines 3WD theory and regret theory and can precisely make up for these two shortcomings. First, by virtue of regret theory, an outranking relation on the comprehensive MSIS is constructed according to the regret-rejoicing index. Second, objects in the outranking relation are classified into three different domains by a clustering method. In each domain, the ranking of objects can be calculated by using the relative closeness coefficient. Finally, we use the cases in the database to simulate the experiment to verify the decision-making effect of the proposed model. Comparative analysis and experimental analysis also show the effectiveness, superiority, and stability of the proposed model.

Regret Theory-Based Three-Way Decision Model in Hesitant Fuzzy Environments and Its Application to Medical Decision

Article

May 2022

In real world, a typical decision-making problem in the medical field can be seen as an uncertain hesitant fuzzy multi-attribute decision-making problem when existing experiences of decision-makers are insufficient. A three-way decision model is an effective tool to deal with uncertain decision-making problems, which can realize the classification of objects. However, different psychological behaviors of decision-makers are likely to affect decision-making results in actual decision-making processes. In order to address this challenge, we first propose a regret-theory-based three-way decision model in hesitant fuzzy environments, which mainly combines regret theory with hesitant fuzzy sets to calculate the perceived utility value of objects. Second, since one of the core ingredients for regret theory is the regret-rejoice function, we put forward a new regret-rejoice function that preserves the original indecisive uncertainty in hesitant fuzzy environments. In addition, according to the inherent relationship of the used data, this paper provides a three-way classification method based on the Preference Ranking Organization method for Enrichment Evaluations I method. Finally, the practicability, rationality and superiority of the proposed method are shown via the case analysis and comparative analysis in the medical field.

Multi-objective optimization-based adaptive class-specific cost extreme learning machine for imbalanced classification

Article

May 2022
NEUROCOMPUTING

Imbalanced classification is a challenging task in the fields of machine learning and data mining. Cost-sensitive learning can tackle this issue by considering different misclassification costs of classes. Weighted extreme learning machine (W-ELM) takes a cost-sensitive strategy to alleviate the learning bias towards the majority class to achieve better classification performance. However, W-ELM may not achieve the optimal weights for the samples from different classes due to the adoption of empirical costs. In order to solve this issue, multi-objective optimization-based adaptive class-specific cost extreme learning machine (MOAC-ELM) is presented in this paper. To be Specific, the initial weights are first assigned depending on the class information. Based on that, the representation of the minority class could be enhanced by adding penalty factors. In addition, a multi-objective optimization with respect to penalty factors is formulated to automatically determine the class-specific costs, in which multiple performance criteria are constructed by comprehensively considering the misclassification rate and generalization gap. Finally, ensemble strategy is implemented to make decisions after optimization. Accordingly, the proposed MOAC-ELM is an adaptive method with good robustness and generalization performance for imbalanced classification problems. Comprehensive experiments have been performed on several benchmark datasets and a real-world application dataset. The statistical results demonstrate that MOAC-ELM can achieve competitive results on classification performance.

An Adaptive Clustering-Based Algorithm for Automatic Path Planning of Heterogeneous UAVs

Article

Dec 2021

Due to the high maneuverability and strong adaptability, autonomous unmanned aerial vehicles (UAVs) are of high interest to many civilian and military organizations around the world. Automatic path planning which autonomously finds a good enough path that covers the whole area of interest, is an essential aspect of UAV autonomy. In this study, we focus on the automatic path planning of heterogeneous UAVs with different flight and scan capabilities, and try to present an efficient algorithm to produce appropriate paths for UAVs. First, models of heterogeneous UAVs are built, and the automatic path planning is abstracted as a multi-constraint optimization problem and solved by a linear programming formulation. Then, inspired by the density-based clustering analysis and symbiotic interaction behaviours of organisms, an adaptive clustering-based algorithm with a symbiotic organisms search-based optimization strategy is proposed to efficiently settle the path planning problem and generate feasible paths for heterogeneous UAVs with a view to minimizing the time consumption of the search tasks. Experiments on randomly generated regions are conducted to evaluate the performance of the proposed approach in terms of task completion time, execution time and deviation ratio.

Coverage Path Planning of Heterogeneous Unmanned Aerial Vehicles Based on Ant Colony System

Article

Oct 2021

Unmanned aerial vehicle (UAV) has been extensively studied and widely adopted in practical systems owing to its effectiveness and flexibility. Although heterogeneous UAVs have an enormous advantage in improving performance and conserving energy with respect to homogeneous ones, they give rise to a complex path planning problem. Especially in large-scale cooperative search systems with multiple separated regions, coverage path planning which seeks optimal paths for UAVs to completely visit and search all of regions of interest, has a NP-hard computation complexity and is difficult to settle. In this work, we focus on the coverage path planning problem of heterogeneous UAVs, and present an ant colony system (ACS)-based algorithm to obtain good enough paths for UAVs and fully cover all regions efficiently. First, models of UAVs and regions are built, and a linear programming-based formulation is presented to exactly provide the best point-to-point flight path for each UAV. Then, inspired by the foraging behaviour of ants that they can obtain the shortest path between their nest and food, an ACS-based heuristic is presented to seek approximately optimal solutions and minimize the time consumption of tasks in the cooperative search system. Experiments on randomly generated regions have been organized to evaluate the performance of the new heuristic in terms of execution time, task completion time and deviation ratio.

Conditional probability based multi-objective cooperative task assignment for heterogeneous UAVs

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