CyberPhysical Mobile Computing, Communications, and Sensing for Industrial Internet of Things
View this Special IssueResearch Article  Open Access
Liyuan Deng, Ping Yang, Weidong Liu, Lina Wang, Sifeng Wang, Xiumei Zhang, "NAAMMOEA/DBased Multitarget Firepower Resource Allocation Optimization in Edge Computing", Wireless Communications and Mobile Computing, vol. 2021, Article ID 5579857, 14 pages, 2021. https://doi.org/10.1155/2021/5579857
NAAMMOEA/DBased Multitarget Firepower Resource Allocation Optimization in Edge Computing
Abstract
In the edge environment, the multiobjective evolutionary algorithm based on decomposition (MOEA/D) has been widely used in the research of multitarget firepower resource allocation. However, as the MOEA/D algorithm uses a fixed neighborhood update mechanism, it is impossible to rationally allocate computing resources based on the difficulty of each subproblem optimization, which results in some problems such as reduced population evolution efficiency and poor evolution quality during the calculation process. In order to solve these problems, a decision mechanism for subproblems and population evolution stages is designed, and on this basis, a MOEA/D algorithm based on the neighborhood adaptive adjustment mechanism is proposed to adapt to the edge environment. The optimization model of multiobjective firepower resource allocation based on the maximization of damage effect and the minimization of strike cost is constructed and solved. Using the ZDT series of test functions for comparative experiments, the simulation results show that the proposed algorithm can balance the distribution and convergence of population evolution and obtain satisfactory optimization results.
1. Introduction
In the edge environment, due to the limited computing resources of edge clients, the allocation of firepower resources based on factors such as battlefield situation, weapon performance, and combat objectives reasonably deploying and allocating various types and quantities of weapons and equipment to obtain the best combat effect is an important part of combat planning [1]. The firepower resource allocation optimization problem in edge environment usually constructs a singleobjective firepower resource allocation optimization model based on the damage probability objective function, using heuristic genetic algorithm [2], simulated annealing genetic algorithm [3], particle swarm algorithm [4], and ant colony algorithm to solve the model. In practical problems, the objective function that only considers the single factor of damage probability is obviously not realistic. Literature [5] establishes interception benefit maximization and loss minimization models and used multiobjective quantum behavior particle swarm algorithm with a single/dual potential trap to solve the model. Literature [6] uses a genetic algorithm based on reference point nondominated sorting to solve the optimization problem of multispacebased ground strike weapon multitarget firepower resource allocation. Literature [7] uses the multitarget discrete particle swarmgravity search algorithm (MODPSOGSA) to achieve the solution of the multitarget allocation model of coordinated air combat weapons. The decompositionbased multiobjective evolutionary algorithm decomposes the highdimensional and complex multiobjective optimization problem into multiple singleobjective subproblems by referring to the decomposition strategy in mathematical programming and optimizes the subproblems separately. It has the advantages of high algorithm efficiency and simple operation [8, 9].
The MOEA/D algorithm has been used in the study of multitarget firepower resource allocation in edge environment. Literature [10] comprehensively considers the influence of factors such as weapon type, target number, and damage probability and uses the MOEA/D algorithm as the framework to construct the WMOM/D algorithm for solving the multitarget fire distribution model. Simulation experiments prove that the WMOM/D algorithm has the advantage of solving the problem of smallscale fire distribution. Literature [11] applies the MOEA/D algorithm to the multiobjective fire optimization problem of aircraft carrier formation antisubmarine warfare and proposes the GDMOEA/D algorithm combining differential evolution and Gaussian mutation operation, which greatly improves the speed of solving the problem and the quality of the solution. Literature [12] integrates the MOEA/D algorithm with the multilevel coevolutionary algorithm and uses the multilevel cooperative MOEA/D algorithm to solve the multiobjective optimization model of the joint fire strike target assignment problem. The simulation experiment proves that the algorithm has good convergence and uniformity.
However, because the MOEA/D algorithm uses a fixed neighborhood update mechanism, the ability to reasonably allocate computing resources is low especially in the edge environment with limited computing resources. So the problems such as reduced population evolution efficiency and poor evolution quality will occur in the calculation process.
To this end, this paper considers the impact of subproblems and the degree of population evolution on the performance of the algorithm, designs the decision mechanism for subproblems and population evolution stages, and proposes a MOEA/D algorithm based on the neighborhood adaptive adjustment mechanism. Compared with the traditional MOEA/D algorithm, the NAAMMOEA/D algorithm can better balance the convergence and distribution and improve the quality of the solution.
In the simulation experiment, the NAAMMOEA/D algorithm was compared with the MOEA/D algorithm, the MOEA/DDE algorithm, and the NSGAIII algorithm. The algorithm running time was reduced by 82.1%, 108.1%, and 153.6%, respectively; the GD value was reduced by 84%, 59%, and 35%, respectively; and the IGD value of the algorithm was reduced by 75%, 56%, and 40%, respectively.
The main innovations of this article are summarized as follows: (1)Aiming at the defects of the traditional MOEA/D algorithm’s fixed neighborhood update mechanism in solving the multiobjective fire resource allocation problem, a MOEA/D algorithm based on the neighborhood adaptive adjustment mechanism is proposed, which greatly improves the efficiency and quality in edge(2)A method for judging the evolution stage of the population based on the attribution of the weight vector and the degree of evolution of the subproblems is proposed, which provides a reliable basis for judging the evolution state of the population(3)Based on the population evolution stage judgment method, a neighborhood adaptive adjustment mechanism is constructed and used in the MOEA/D algorithm to improve the convergence and distribution of the algorithm
The organizational structure of the paper is as follows:
Firstly, the related work is discussed in Section 2. Then, the optimization model of firepower resource allocation in edge environment is established in Section 3.1, the construction and decomposition of subproblems are discussed in Section 3.2.1, the shortcomings of traditional MOEA/D algorithm are analyzed in Section 3.2.2, and a mechanism for judging population evolution state is proposed in Section 3.2.3. The neighborhood adaptive adjustment mechanism is proposed in Section 3.2.4 and the steps of the NAAMMOEA/D algorithm are summarized in Section 3.2.5. Finally, the simulation experiment is carried out in Section 4, and the performance of the algorithm is tested.
2. Related Work
In order to improve the performance of traditional MOEA/D algorithms in edge environment, researchers have proposed a variety of improved algorithms. The MOEA/DDE algorithm proposed in literature [13] uses a difference operator instead of an evolution operator to enrich the diversity of the population, but the difference operator used by the algorithm is only applicable to a population of a specific size. The MOEA/DDRA algorithm proposed in literature [14] allocates corresponding computing resources according to the complexity of specific problems and improves the performance of the algorithm by dynamically adjusting resource allocation; however, the proposed resource allocation criteria also have certain limitations. The MOEA/DGL algorithm proposed in literature [15] embeds the grouping and statistical learning mechanism in the traditional MOEA/D algorithm, which prevents the population from falling into local optimization and improves the diversity of the population, but the overall performance improvement of the algorithm is not significant. The CDMOEA/DDE algorithm proposed in literature [16] controls the operation process of the algorithm by formulating control parameters and balances the performance of a multiobjective optimization problem solving and adaptive ability; however, the algorithm has a certain randomness in the value of the control parameter and does not have universal applicability.
In addition, the researchers have proposed many specific improvement measures for the shortcomings of the fixed neighborhood update mechanism of the MOEA/D algorithm in solving multiobjective optimization problems especially in the edge conditions with limited computing resources; however, the article does not elaborate on the mechanism of how the neighborhood size affects the performance of the MOEA/D algorithm. Literature [17] points out that the size of the neighborhood will have an important impact on the performance of the MOEA/D algorithm, which provides important research directions for subsequent researchers. Literature [18] believes that different multiobjective optimization problems require different neighborhood sizes, and that the same multiobjective optimization problem also requires different neighborhood sizes at different stages of the algorithm, and proposes the ENSMOEA/D algorithm with neighborhood adaptive adjustment capability; however, the ENSMOEA/D algorithm may fall into local optimization in the later stage of operation. The ADEMO/DENS algorithm proposed in literature [19] combines the adaptive differential evolution algorithm with the variable neighborhood decomposition method to achieve the optimization of the algorithm. The MOEA/DAGR algorithm proposed in literature [20] introduces an adaptive global replacement strategy in the neighborhood update method, which makes up for the shortcomings of the traditional MOEA/D algorithm in terms of global search capabilities. The MOEA/DNMO algorithm proposed in literature [21] combines mutation strategies with different characteristics and neighborhoods of different sizes to select the best evolutionary combination to ensure the convergence of the algorithm while maintaining the diversity of the algorithm. The algorithms proposed in literature [19], literature [20], and literature [21] have all made improvements to the fixed field, but they all have certain limitations in application.
Although the current improved methods for fixed neighborhoods have improved the performance of traditional MOEA/D algorithms, the neighborhood adaptive strategies used by these algorithms do not consider the impact of population evolution on neighborhoods. Literature [22] proposes a neighborhood adaptive adjustment mechanism based on population evolution stage and individual fitness value, so that every individual has a corresponding neighborhood value at different evolution stages, but its neighborhood adjustment method does not consider the evolution status of the subproblems. Although the MOEA/DANS algorithm proposed in literature [23] adopts the ANS mechanism that adaptively adjusts the size of the neighborhood according to the evolution state of the population and subproblems, it can balance the convergence and distribution of population evolution, but it does not give a clear method on the statistical evolution of the number of better subquestions.
3. Method
3.1. Optimization of Fire Resource Allocation Model
The multitarget firepower resource allocation optimization problem in the edge environment can be described as follows: on the basis of satisfying the maximum damage effect and the minimum combat cost, determine the number of various weapons and equipment used to strike specific targets to obtain a feasible combat plan.
Set the target set of the enemy’s combat system as ; represents the th target. There are a total of types of weapons available for use.
, and represents the th types of weapons. If there is a total of class to choose from the th type of weapons, then . Select the th weapon in the weapon set to strike the th target in the target set ; the probability of the target being destroyed is and the cost of each use of the th weapon is . Suppose that the damage ability of the th type of weapons to target is
Among them, only when the th type of weapons of class weapon is used to strike target , there is ; otherwise, . The purpose of firepower resource allocation is to maximize the damage effect under limited conditions. It is necessary to consider the priority of attacking the targets with high importance. Therefore, the calculation model of damage capability can be defined as
Among them, is the importance of the th target.
In addition, the minimum operational cost calculation model is defined as follows:
The constraints of the model are as follows: (1)Damage lower bound constraint: if the target is to be destroyed to a certain extent so that it will lose certain combat capability, it is necessary to reach its damage lower bound. If the damage lower bound of target is defined as , then(2)Constraints on the number and types of weapons used: it is stipulated that one weapon can only attack one target at most:
It is stipulated that one type of weapon can only attack one type of target:
In summary, the multiobjective firepower resource allocation optimization model can be defined as
3.2. Detailed Introduction of NAAMMOEA/D Algorithm
3.2.1. Construction and Decomposition of Subproblem
The core of constructing the subproblem of the MOEA/D algorithm is to construct the weight vector of an objective function subproblem. Suppose the weight vector of the subproblem of the objective function is
In the formula, is the number of subproblems after decomposition, .
The core of the MOEA/D algorithm is the decomposition operation, usually using aggregate functions to decompose the multiobjective constraint problem into singleobjective subproblems. Commonly used decomposition methods are weighted sum method, Chebyshev method, and boundary crossing method based on penalty. This paper adopts the Chebyshev method, and its decomposition principle is
Among them, is the weight vector corresponding to the subproblem . is the ideal point. is the th objective function, and is the th component of the weight vector . is the th component of the ideal point .
The singleobjective optimization function of the th subproblem of objective function constructed by the Chebyshev method can be expressed as follows:
In the formula, is the th objective function, is the reference vector, is the th component of the reference vector , is the weight vector, and is the th component of the weight vector .
3.2.2. Defects of Traditional MOEA/D Algorithm
The MOEA/D algorithm maintains the power of population evolution from the update strategy of the neighborhood. The parent gene of an individual comes from the neighborhood, and it adopts a coevolution model based on neighborhood update. The evolution of an individual is carried out on the basis of the neighborhood. While evolving by itself, it drives the evolution of other neighborhoods by optimizing other individuals in the neighborhood. The MOEA/D algorithm uses a fixed neighborhood strategy. For different subproblems, the MOEA/D algorithm divides it into a neighborhood of the same size. In fact, the computational complexity of each subproblem in the objective function is different. The subproblems have different requirements for the size of the neighborhood at different stages. The size of the neighborhood has a very important impact on the evolution of the subproblems. When the size of the neighborhood is large, the probability of other individuals in the neighborhood being replaced by offspring individuals increases, and the population convergence speeds up, but the distribution of the population will become worse as the neighborhood size increases, making it easy for the algorithm to fall into local find the best. When the size of the neighborhood is small, the probability of other individuals in the neighborhood being replaced by offspring individuals decreases, the population convergence speed slows, the algorithm convergence decreases, and the overall evolution speed of the population decreases accordingly.
3.2.3. Judging Mechanism of Population Evolution State
From the previous analysis, we can see that in the MOEA/D algorithm, subproblems and populations have different requirements for neighborhood size at different evolution stages. Then, how to judge the evolution state of the population and whether it can find a mechanism that can effectively evaluate the evolution stage of the population is the core problem that the new algorithm needs to solve.
Some scholars propose to use the individual density of subproblems to assess the degree of population evolution. The individual density of the subproblem is equivalent to the number of individuals in the subinterval. If the individual density of the subproblem is smaller, the surrounding individuals are denser, the better the degree of evolution of the individual is, and the greater the probability of the problem being solved. If the individual density of the subproblem is smaller, the surrounding individuals are sparser, then the degree of evolution of the individual is smaller, and the problem is less likely to be solved.
Some scholars propose that if the distance between the subproblem and a certain solution in space is used as the evaluation criterion, if the distance between them is relatively close, it can be judged that the solution belongs to the subproblem. In the spatial coordinate system, the solution corresponds to the individual in the coordinate system, and the subproblem corresponds to the weight vector. Therefore, the problem of determining the attribution of the solution can be transformed into the problem of finding the distance between the weight vector and the individual.
This paper proposes a mechanism for evaluating the evolutionary stage of a population (see Algorithm 1):

3.2.4. Neighborhood Adaptive Adjustment Mechanism
In order to meet the needs of balancing the convergence and distribution of the MOEA/D algorithm, according to the population evolution state judgment mechanism in Section 3.2.3, this paper proposes a neighborhood strategy that adaptively adjusts the population size based on the different evolution stages of the population, which can also be called a neighborhood adaptive adjustment mechanism (NAAM) as in Algorithm 2.

The setting adjustment formula is as follows:
3.2.5. The Framework of NAAMMOEA/D Algorithm
The framework of the NAAMMOEA/D algorithm can be described as in Algorithm 3:

4. Experiment and Simulation
4.1. Example Analysis of Algorithm
There are 4 types of weapons to strike at 4 targets in the enemy’s combat system. Combining the content of Section 3.1, we assume that the model satisfies various constraints, and the model parameters are given in Table 1.

MATLAB 2020 is selected to write the algorithm program. The running environment is a Windows 7 R64bit operating system, 4 GB memory, Intel Pentium processor. The NAAMMOEA/D algorithm, MOEA/D algorithm, MOEAD/DDE algorithm, and NSGAIII algorithm are selected for the simulation operation.
Figure 1 counts the number of various weapons used by the four algorithms. It can be seen from Figure 1 that the NSGAIII algorithm uses 7 W1 weapons, which is more than the MOEA/D algorithm and the MOEA/DDE algorithm; both algorithms use 5 W1 weapons, and the NAAMMOEA/D algorithm uses 4 W1 weapons. The NSGAIII algorithm uses 6 W2 weapons; the MOEA/DDE algorithm and the NAAMMOEA/D algorithm use 4 and 2 W2 weapons, respectively; while the MOEA/D algorithm uses the least number of W2 weapons and only one is used. The NAAMMOEA/D algorithm and the NSGAIII algorithm both use 5 W3 weapons, which is more than the MOEA/D algorithm and the MOEA/DDE algorithm. Both algorithms use 4 W3 weapons. The MOEA/D algorithm uses 8 W4 weapons, which is more than the NSGAIII algorithm. The NAAMMOEA/D algorithm and the MOEA/D algorithm use 3 and 2 W4 weapons, respectively.
Figure 2 counts the total number of weapons used by the four algorithms. It can be seen from Figure 2 that the NSGAIII algorithm uses the largest number of weapons, using 23 weapons in total. The MOEA/D algorithm and the MOEA/DDE algorithm use 18 and 15 weapons, respectively, and the NAAMMOEA/D algorithm uses the least amount of weapons—only 14 weapons are used.
Figure 3 compares the total cost of weapon use of the four algorithms. It can be seen from Figure 3 that the NSGAIII algorithm costs the most weapons, with a total cost of 155, followed by the MOEA/DDE algorithm, with a total cost of 113, while the MOEA/D algorithm and NAAMMOEA/D algorithm had the least weapon use cost, costing 99 and 92, respectively.
Figure 4 compares the computing time of the four algorithms. It can be seen from Figure 4 that the NAAMMOEA/D algorithm has the least computing time, which takes only 12.3 s, and the NSGAIII algorithm has the most computing time, which takes 30 s. The computing times of the MOEA/D algorithm and the MOEA/DDE algorithm are, respectively, 22.4 s and 25.6 s.
The statistics of firepower resource allocation obtained through simulation calculation are shown in Table 2.

It can be seen from Table 2 that the number of weapons used and the total cost obtained by the NAAMMOEA/D algorithm are better than those of the other three algorithms. The number of weapons used by the MOEA/DDE algorithm is close to the number of weapons used by the NAAMMOEA/D algorithm, but the total cost is about 23% higher. The total cost calculated by the MOEA/DDE algorithm is close to the total cost calculated by the NAAMMOEA/D algorithm, but 4 more weapons are used. The number of weapons used and the total cost obtained by the NSGAIII algorithm are significantly more than those of the other three algorithms, indicating that the algorithm has the worst performance. In addition, the running time of the NAAMMOEA/D algorithm is 12.3 s, which is reduced by 82.1%, 108.1%, and 153.6% compared with the MOEA/D algorithm, the MOEA/DDE algorithm, and the NSGAIII algorithm, respectively, indicating that the NAAMMOEA/D algorithm has obvious advantages in computing speed.
4.2. Performance Test of the Algorithm
In order to verify the performance of the NAAMMOEA/D algorithm, ZDT series of test functions are selected to test the performance of the NAAMMOEA/D algorithm with the MOEA/D algorithm, MOEA/DDE algorithm, and NSGAIII algorithm.
In order to ensure the fairness and rationality of the algorithm evaluation, the population size and initial neighborhood size of the four algorithms are set to the same (population size , initial neighborhood ). All algorithms adopt simulated binary crossover (crossover probability ) and polynomial mutation (mutation probability , is the dimension of decision variables). Each algorithm runs 20 times independently, and the evaluation times are set to 10000. Inverse generation distance (IGD) and generation distance (GD) were used as evaluation indexes. Each test function is run 20 times independently and averaged every 10 generations. The variation curve of GD with the number of iterations (0500 generations) of the algorithm is shown in Figure 1.
As shown in Figure 5(a), the NAAMMOEA/D algorithm tends to be stable on the test function ZDT1, and the convergence speed is slower than the NSGAIII algorithm and faster than the MOEA/D algorithm and the MOEA/DDE algorithm.
(a) Running results on ZDT1 test function
(b) Running results on ZDT2 test function
(c) Running results on ZDT3 test function
(d) Running results on ZDT4 test function
As shown in Figure 5(b), on the test function ZDT2, the convergence speed of the NAAMMOEA/D algorithm is faster than that of the MOEA/DDE algorithm and the NSGAIII algorithm. Although it is slightly slower than the MOEA/D algorithm, the population degradation degree of the MOEA/D algorithm is higher than that of the NAAMMOEA/D algorithm.
As shown in Figure 5(c), on the test function ZDT3, the NAAMMOEA/D algorithm converges faster than the other algorithms.
As shown in Figure 5(d), on the test function ZDT4, the NAAMMOEA/D algorithm has a faster population convergence speed due to the advantages of the adaptive neighborhood adjustment mechanism adopted, and the algorithm convergence performance is significantly better than the MOEA/D algorithm, MOEAD/DDE algorithm, and NSGA.
Therefore, the NAAMMOEA/D algorithm not only ensures that the algorithm has a faster convergence rate but also solves the population degradation problem that occurs during the algorithm operation and ensures the stability of the algorithm operation, so that the algorithm can have more resources to improve the diversity of the population.
As shown in Figure 6, comparing the IGD box plots of various algorithms on the ZDT series test functions in the comparison Table 3, we can see that the NAAMMOEA/D algorithm’s mean, minimum, median (at the position of the red line in the figure), and interquartile range (key indicators such as box length) are lower than those of the MOEA/D algorithm, MOEA/DDE algorithm, and NSGAIII algorithm. The probability and size of the abnormal value of the NAAMMOEA/D algorithm are also lower than those of the other three algorithms, which show that the stability and quality of the NAAMMOEA/D algorithm is higher.
(a) IGD on ZDT1 test function
(b) IGD on ZDT2 test function
(c) IGD on ZDT3 test function
(d) IGD on ZDT4 test function

On the test functions ZDT1 and ZDT2, the comprehensive performance of the NAAMMOEA/D algorithm is slightly better than that of the MOEA/D algorithm and significantly better than that of the MOEA/DDE algorithm and the NSGAIII algorithm. On the test functions ZDT3 and ZDT4, the comprehensive performance of the NAAMMOEA/D algorithm is significantly better than that of the MOEA/D algorithm, the MOEA/DDE algorithm, and the NSGAIII algorithm. This is because there are many discontinuous regions in the target space of test function ZDT3. These regions adopt the fixed neighborhood setting method, but do not use the adaptive neighborhood allocation strategy to reasonably allocate the algorithm, which leads to the waste of algorithm resources and the slowdown of population evolution speed.
Figure 7 shows the comparison of the Pareto front and the ideal Pareto front obtained by the four algorithms on the ZDT test function. Among them, the red meter character represents the ideal PF, and the blue circle represents the optimal solution of the Pareto frontier obtained by the various algorithms.
(a) NAAMMOEA/D on ZDT1 test function
(b) MOEA/D on ZDT1 test function
(c) MOEA/DDE on ZDT1 test function
(d) NSGAIII on ZDT1 test function
(e) NAAMMOEA/D on ZDT3 test function
(f) MOEA/D on ZDT3 test function
(g) MOEA/DDE on ZDT3 test function
(h) NSGAIII on ZDT3 test function
(i) NAAMMOEA/D on ZDT4 test function
(j) MOEA/D on ZDT4 test function
(k) MOEA/DDE on ZDT4 test function
(l) NSGAIII on ZDT4 test function
On the test function ZDT3, the improved MOEA/D solution set is more evenly distributed on the ideal Pareto front. In the other three algorithms, some leading edges are not completely found, and the solution set is missing to a certain extent. Among them, the MOEA/D algorithm and the MOEA/DDE algorithm have a little poor distribution of solution set, while the NSGAIII algorithm has the least distribution. This is because the other algorithms spend limited computing resources in the discrete region of test function ZDT3 and produce too many nondominated solutions, which hinders the evolution of the population.
On the test function ZDT4, the NAAMMOEA/D algorithm has converged to the ideal, while the other algorithms have fallen into the local optimization state to varying degrees. It can be seen that the NAAMMOEA/D algorithm has more advantages in reasonable allocation of computing resources and can better ensure the convergence of the algorithm.
Through the comparison, we can see that the Pareto frontier solution set obtained by the NAAMMOEA/D algorithm almost uniformly converges to the PF of the ideal Pareto. However, the other three algorithms have different degrees of missing or uneven distribution of solution sets in various test functions. The NAAMMOEA/D algorithm shows some performance advantages when dealing with simple test problems such as ZDT1, but the advantages are not obvious. However, the NAAMMOEA/D can allocate computing resources reasonably and take into account the convergence and distribution of the algorithm due to its flexible neighborhood update strategy when dealing with relatively complex test problems such as ZDT3 and ZDT4.
5. Discussion
In this section, we establish a firepower resource allocation optimization model for edge environment based on given specific data, conduct simulation experiments, and test and evaluate the performance of the algorithm combined with the ZDT series of functions. However, several additional points should be pointed out and further analyzed in detail, which are specified as below. (1)The types of weapons and the number of samples given in Section 4.1 are not large enough (both are 4). Therefore, in the future simulation experiments, we should focus on large sample data sets to verify the performance of the method under the condition of large sample data(2)In Section 4.2, the ZDT series functions are selected to test the performance of the algorithm. The simulation results show that the performance of the NAAMMOEA/D algorithm is better than that of the other three algorithms. However, only one kind of test function verification is not convincing enough, so DLTZ, WFG, and other test functions should be selected to evaluate the algorithm, so as to provide more sufficient reference for the improvement of algorithm performance
6. Conclusion
This paper constructs a multiobjective firepower resource allocation optimization model for edge environment with limited computing resources, based on maximizing damage effect and minimizing combat cost. Aiming at the defects of the traditional MOEA/D algorithm fixed neighborhood update mechanism, a MOEA/D algorithm based on neighborhood adaptive adjustment mechanism is proposed and the model is solved. It can be seen from the simulation experiment that the MOEA/D algorithm based on the neighborhood adaptive adjustment mechanism has significantly improved its stability, convergence, and distribution.
In the next step, current work will continue to be improved by considering security and privacy issues [24–33]. In addition, more complex multiobjective solutions with more context factors [34–41] will be considered.
Abbreviations
MOEA/D:  Multiobjective evolutionary algorithm based on decomposition 
NAAMMOEA/D:  Neighborhood adaptive adjustment mechanismmultiobjective evolutionary algorithm based on decomposition 
MODPSOGSA:  Multiobjective discrete particle swarm optimizationgravitational search algorithm 
WMOM/D:  Weapontarget assignment multiobjective model based on decomposition 
GDMOEA/D:  Gauss mutation and differential evolution based on a multiobjective evolutionary algorithm based on decomposition 
MOEA/DDE:  Multiobjective evolutionary algorithm based on decompositiondifferential evolution 
MOEA/DDRA:  Multiobjective evolutionary algorithm based on decompositiondynamical resource allocation 
ENSMOEA/D:  Ensemble neighborhood sizemultiobjective evolutionary algorithm based on decomposition 
ADEMO/DENS:  Adaptive differential evolution for multiobjective problemsensemble neighborhood size 
MOEA/DAGR:  Multiobjective evolutionary algorithm based on decompositionadaptive global replacement 
MOEA/DNMO:  Multiobjective evolutionary algorithm based on decompositionneighborhood mutation operator 
MOEA/DANS:  Multiobjective evolutionary algorithm based on decompositionadaptive neighborhood strategy 
NSGAIII:  Nondominated sorted genetic algorithmIII. 
Data Availability
The experiment dataset is generated randomly through simulation.
Conflicts of Interest
We declare that there is no conflict of interest regarding this submission.
Authors’ Contributions
Liyuan Deng finished the English writing, review, and editing of the paper. Liyuan Deng, Ping Yang, and Weidong Liu finished the experiments. Lina Wang, Sifeng Wang, and Xiumei Zhang finished the algorithm design.
Acknowledgments
This work was supported by Xi’an Research Institute of HighTechnology.
References
 L. I. Ping and L. I. Changwen, “Modeling and algorithm of weapon target cooperative fire assignment,” Command Control & Simulation, vol. 37, no. 2, pp. 36–40, 2015. View at: Google Scholar
 J. Zhang, Z. X. Wang, L. Chen, Z. B. Wu, and J. F. Lu, “Modeling and optimization on antiaircraft weapontarget assignment at multiple interception opportunity,” Acta Armamentarii, vol. 35, no. 10, pp. 1644–1650, 2014. View at: Google Scholar
 D. Chaoyang, L. Yao, and W. Qing, “Improved genetic algorithm for solve firepower distribution,” Acta Armamentarh, vol. 37, no. 1, pp. 97–102, 2016. View at: Google Scholar
 C. L. Fan, Q. H. Xing, and M. F. Zheng, “Weapontarget allocation optimization algorithm based on IDPSO,” Systems Engineering and Electronics, vol. 37, no. 2, pp. 336–342, 2015. View at: Google Scholar
 X. Hao, X. Qinghua, and W. Wei, “WTA for air and missile defense based on fuzzy multiobjective programming,” Systems Engineering and Electronics, vol. 40, no. 3, pp. 563–570, 2018. View at: Google Scholar
 L. Qingguo, L. Xinxue, W. Jian, L. Yaxiong, and C. Hao, “Optimization of fire distribution for multiple SGSW based on improved NSGAIII,” Systems Engineering and Electronics, vol. 42, no. 9, pp. 1995–2002, 2020. View at: Google Scholar
 J. J. Gu, J. J. Zhao, J. Yan, and X. Chen, “Cooperative weapontarget assignment based on multiobjective discrete particle swarm optimizationgravitational search algorithm in air combat,” Journal of Beijing University of Aeronautics and Astronautics, vol. 41, no. 2, pp. 252–258, 2015. View at: Google Scholar
 Q. Zhang and H. Li, “MOEA/D: a multiobjective evolutionary algorithm based on decomposition,” IEEE Transaction on Evolutionary Computation, vol. 11, no. 6, pp. 712–731, 2007. View at: Publisher Site  Google Scholar
 S. Zhao, P. Suganthan, and Q. Zhang, “Decompositionbased multiobjective evolutionary algorithm with an ensemble of neighborhood sizes,” IEEE Transactions on Evolutionary Computation, vol. 16, no. 3, pp. 442–446, 2012. View at: Publisher Site  Google Scholar
 Y. Zhang, R. N. Yang, J. L. Zuo, and X. Jing, “Weapontarget assignment based on decompositionbased evolutionary multiobjective optimization algorithms,” Systems Engineering and Electronics, vol. 36, no. 12, pp. 2435–2441, 2014. View at: Google Scholar
 L. Chen and Y. Ma, “Antisubmarine firepower optimization of aircraft carrier formation based on GDMOEA/D algorithm,” Computer Simulation, vol. 35, no. 10, pp. 33–38, 2018. View at: Google Scholar
 C. Hui and Y. Ma, “Model of target assignment in joint fire strike operations,” Journal of Systems Simulation, vol. 30, no. 8, pp. 2942–2949, 2018. View at: Google Scholar
 H. Li and Q. Zhang, “Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGAII,” IEEE Transactions on Evolutionary Computation, vol. 13, no. 2, pp. 284–302, 2009. View at: Publisher Site  Google Scholar
 Z. H. A. N. G. Qingfu, L. I. U. Wudong, and L. I. Hui, “The performance of a new version of MOEA/D on CEC09 unconstrained MOP test instances,” in 2009 IEEE Congress on Evolutionary Computation, pp. 203–208, Washington D.C., USA, 2009. View at: Publisher Site  Google Scholar
 L. Li, D. Liu, and X. Wang, Multiobjective permutation flow shop scheduling problem based on improved MOEA/D algorithm, Computer Integrated Manufacturing Systems, 2020.
 X. Zhou, W. Xuewu, and X. Gu, “MOEA/D based on constrained approach and differential evolution,” in Proceedings of the 38th Chinese Control Conference, pp. 2034–2039, Guangzhou Baiyun International Convention Center, China, 2019. View at: Google Scholar
 H. Ishibuchi, Y. Hitotsuyanagi, N. Tsukamoto, and Y. Nojima, “Use of biased neighborhood structures in multiobjective memetic algorithms,” Soft Computing, vol. 13, no. 89, pp. 795–810, 2009. View at: Publisher Site  Google Scholar
 S.Z. Zhao, P. N. Suganthan, and Q. Zhang, “Decompositionbased multiobjective evolutionary algorithm with an ensemble of neighborhood sizes,” IEEE Transactons on Evolusonary Computation, vol. 16, no. 3, pp. 442–446, 2012. View at: Publisher Site  Google Scholar
 H. Xia, J. Zhuang, and D. Yu, “Combining crowding estimation in objective and decision space with multiple selection and search strategies for multiobjective evolutionary optimization,” IEEE Transactions on Cybernetics, vol. 44, no. 3, pp. 378–393, 2013. View at: Google Scholar
 Z. Wang, Q. Zhang, A. Zhou, M. Gong, and L. Jiao, “Adaptive replacement strategies for MOEA/D,” IEEE Transcations on Cybernetics, vol. 46, no. 2, pp. 474–486, 2016. View at: Publisher Site  Google Scholar
 L. Liu and L. Zheng, “MOEA/D algorithm based on combinational optimization of neighborhood and mutation operator,” Computer Engineering, vol. 43, no. 3, pp. 232–240, 2017. View at: Google Scholar
 E. Li and R. Chen, “Improved MOEA/D algorithm based on adaptive mutation operator and neighborhood size,” Computer Engneering and Applications, vol. 55, no. 9, pp. 49–55, 2019. View at: Google Scholar
 H. Geng, W. Han, Y. Ding, and S. Zhou, “Improved MOEA/D algorithm based on adaptive neighborhood strategy,” Computer Engineering, vol. 45, no. 5, pp. 161–168, 2019. View at: Google Scholar
 Z. Cai, Z. He, X. Guan, and Y. Li, “Collective datasanitization for preventing sensitive information inference attacks in social networks,” IEEE Transactions on Dependable and Secure Computing, vol. 15, no. 4, pp. 577–590, 2016. View at: Publisher Site  Google Scholar
 Z. Sun, Y. Wang, Z. Cai, T. Liu, X. Tong, and N. Jiang, “A twostage privacy protection mechanism based on blockchain in mobile crowdsourcing,” International Journal of Intelligent Systems, 2021. View at: Publisher Site  Google Scholar
 Y. Xu, J. Ren, Y. Zhang, C. Zhang, B. Shen, and Y. Zhang, “Blockchain empowered arbitrable data auditing scheme for network storage as a service,” IEEE Transactions on Services Computing, vol. 13, no. 2, pp. 289–300, 2020. View at: Publisher Site  Google Scholar
 Z. Cai and X. Zheng, “A private and efficient mechanism for data uploading in smart cyberphysical systems,” IEEE Transactions on Network Science and Engineering, vol. 7, no. 2, pp. 766–775, 2020. View at: Publisher Site  Google Scholar
 L. Qi, C. Hu, X. Zhang et al., “Privacyaware data fusion and prediction with spatialtemporal context for smart city industrial environment,” IEEE Transactions on Industrial Informatics, vol. 17, no. 6, pp. 4159–4167, 2020. View at: Publisher Site  Google Scholar
 T. Liu, Y. Wang, Y. Li, X. Tong, L. Qi, and N. Jiang, “Privacy protection based on stream cipher for spatiotemporal data in IoT,” IEEE Internet of Things Journal, vol. 7, no. 9, pp. 7928–7940, 2020. View at: Publisher Site  Google Scholar
 Z. Cai and Z. He, “Trading private range counting over big IoT data,” in 2019 IEEE 39th International Conference on Distributed Computing Systems (ICDCS), Dallas, TX, USA, 2019. View at: Publisher Site  Google Scholar
 Y. Xu, C. Zhang, G. Wang, Z. Qin, and Q. Zeng, “A blockchainenabled deduplicatable data auditing mechanism for network storage services,” IEEE Transactions on Emerging Topics in Computing, p. 1, 2020. View at: Publisher Site  Google Scholar
 W. Zhong, X. Yin, X. Zhang et al., “Multidimensional qualitydriven service recommendation with privacypreservation in mobile edge environment,” Computer Communications, vol. 157, pp. 116–123, 2020. View at: Publisher Site  Google Scholar
 Q. Liu, Y. Tian, J. Wu, T. Peng, and G. Wang, “Enabling verifiable and dynamic ranked search over outsourced data,” IEEE Transactions on Services Computing, p. 1, 2019. View at: Publisher Site  Google Scholar
 L. Wang, X. Zhang, R. Wang, C. Yan, H. Kou, and L. Qi, “Diversified service recommendation with high accuracy and efficiency,” KnowledgeBased Systems, vol. 204, article 106196, 2020. View at: Publisher Site  Google Scholar
 J. Li, T. Cai, K. Deng, X. Wang, T. Sellis, and F. Xia, “Communitydiversified influence maximization in social networks,” Information Systems, vol. 92, article 101522, 2020. View at: Publisher Site  Google Scholar
 H. Liu, H. Kou, C. Yan, and L. Qi, “Keywordsdriven and popularityaware paper recommendation based on undirected paper citation graph,” Complexity, vol. 2020, Article ID 2085638, 15 pages, 2020. View at: Publisher Site  Google Scholar
 S. Zhang, Q. Liu, and Y. Lin, “Anonymizing popularity in online social networks with full utility,” Future Generation Computer Systems, vol. 72, no. 7, pp. 227–238, 2017. View at: Publisher Site  Google Scholar
 Z. Chunjie, L. Ali, H. Aihua, Z. Zhiwang, Z. Zhenxing, and W. Fusheng, “Modeling methodology for early warning of chronic heart failure based on real medical big data,” Expert Systems with Applications, vol. 151, article 113361, 2020. View at: Publisher Site  Google Scholar
 T. Cai, J. Li, A. S. Mian, R. Li, T. Sellis, and J. X. Yu, “Targetaware holistic influence maximization in spatial social networks,” IEEE Transactions on Knowledge and Data Engineering, p. 1, 2020. View at: Publisher Site  Google Scholar
 Q. Liu, P. Hou, G. Wang, T. Peng, and S. Zhang, “Intelligent route planning on large road networks with efficiency and privacy,” Journal of Parallel and Distributed Computing, vol. 133, pp. 93–106, 2019. View at: Publisher Site  Google Scholar
 Y. Wang, G. Yang, Y. Li, and X. Tong, “A workerselection incentive mechanism for optimizing platformcentric mobile crowdsourcing systems,” Computer Networks, vol. 107, article 107144, 2020. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2021 Liyuan Deng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.