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Hindawi Wireless Communications and Mobile Computing Volume 2021, Article ID 5579857, 14 pages https://doi.org/10.1155/2021/5579857 Research Article NAAM-MOEA/D-Based Multitarget Firepower Resource Allocation Optimization in Edge Computing 1 1 1 2 2 3 Liyuan Deng, Ping Yang, Weidong Liu, Lina Wang, Sifeng Wang, and Xiumei Zhang Xi’an Research Institute of High-Technology, China School of Computer Science, Qufu Normal University, China School of Computer Science and Software Engineering, University of Science and Technology Liaoning, China Correspondence should be addressed to Xiumei Zhang; aszxm2002@126.com Received 11 January 2021; Revised 9 February 2021; Accepted 27 February 2021; Published 22 March 2021 Academic Editor: Mohammad R. Khosravi 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. 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 the model. In practical problems, the objective function that only considers the single factor of damage probability is obviously not realistic. Literature [5] establishes inter- In the edge environment, due to the limited computing ception beneﬁt maximization and loss minimization resources of edge clients, the allocation of ﬁrepower resources based on factors such as battleﬁeld situation, models and used multiobjective quantum behavior particle swarm algorithm with a single/dual potential trap to solve weapon performance, and combat objectives reasonably the model. Literature [6] uses a genetic algorithm based on deploying and allocating various types and quantities of reference point nondominated sorting to solve the optimi- weapons and equipment to obtain the best combat eﬀect is zation problem of multispace-based ground strike weapon an important part of combat planning [1]. The ﬁrepower resource allocation optimization problem in edge environ- multitarget ﬁrepower resource allocation. Literature [7] uses the multitarget discrete particle swarm-gravity search ment usually constructs a single-objective ﬁrepower resource algorithm (MODPSO-GSA) to achieve the solution of the allocation optimization model based on the damage proba- multitarget allocation model of coordinated air combat bility objective function, using heuristic genetic algorithm weapons. The decomposition-based multiobjective evolu- [2], simulated annealing genetic algorithm [3], particle swarm algorithm [4], and ant colony algorithm to solve tionary algorithm decomposes the high-dimensional and 2 Wireless Communications and Mobile Computing (2) A method for judging the evolution stage of the pop- complex multiobjective optimization problem into multiple single-objective subproblems by referring to the decomposi- ulation based on the attribution of the weight vector tion strategy in mathematical programming and optimizes and the degree of evolution of the subproblems is proposed, which provides a reliable basis for judging the subproblems separately. It has the advantages of high algorithm eﬃciency and simple operation [8, 9]. the evolution state of the population The MOEA/D algorithm has been used in the study of (3) Based on the population evolution stage judgment multitarget ﬁrepower resource allocation in edge environ- method, a neighborhood adaptive adjustment mech- ment. Literature [10] comprehensively considers the inﬂu- anism is constructed and used in the MOEA/D algo- ence of factors such as weapon type, target number, and rithm to improve the convergence and distribution of damage probability and uses the MOEA/D algorithm as the algorithm the framework to construct the WMOM/D algorithm for solving the multitarget ﬁre distribution model. Simulation The organizational structure of the paper is as follows: experiments prove that the WMOM/D algorithm has the Firstly, the related work is discussed in Section 2. Then, advantage of solving the problem of small-scale ﬁre distri- the optimization model of ﬁrepower resource allocation in bution. Literature [11] applies the MOEA/D algorithm to edge environment is established in Section 3.1, the construc- the multiobjective ﬁre optimization problem of aircraft tion and decomposition of subproblems are discussed in Sec- carrier formation antisubmarine warfare and proposes tion 3.2.1, the shortcomings of traditional MOEA/D the GD-MOEA/D algorithm combining diﬀerential evolu- algorithm are analyzed in Section 3.2.2, and a mechanism tion and Gaussian mutation operation, which greatly for judging population evolution state is proposed in Section improves the speed of solving the problem and the quality 3.2.3. The neighborhood adaptive adjustment mechanism is of the solution. Literature [12] integrates the MOEA/D proposed in Section 3.2.4 and the steps of the NAAM- algorithm with the multilevel coevolutionary algorithm MOEA/D algorithm are summarized in Section 3.2.5. Finally, and uses the multilevel cooperative MOEA/D algorithm the simulation experiment is carried out in Section 4, and the to solve the multiobjective optimization model of the joint performance of the algorithm is tested. ﬁre strike target assignment problem. The simulation experiment proves that the algorithm has good conver- gence and uniformity. 2. Related Work However, because the MOEA/D algorithm uses a ﬁxed neighborhood update mechanism, the ability to reasonably In order to improve the performance of traditional MOEA/D allocate computing resources is low especially in the edge algorithms in edge environment, researchers have proposed a environment with limited computing resources. So the prob- variety of improved algorithms. The MOEA/D-DE algorithm lems such as reduced population evolution eﬃciency and proposed in literature [13] uses a diﬀerence operator instead poor evolution quality will occur in the calculation process. of an evolution operator to enrich the diversity of the popu- To this end, this paper considers the impact of subprob- lation, but the diﬀerence operator used by the algorithm is lems and the degree of population evolution on the perfor- only applicable to a population of a speciﬁc size. The mance of the algorithm, designs the decision mechanism MOEA/D-DRA algorithm proposed in literature [14] allo- for subproblems and population evolution stages, and pro- cates corresponding computing resources according to the poses a MOEA/D algorithm based on the neighborhood complexity of speciﬁc problems and improves the perfor- adaptive adjustment mechanism. Compared with the tradi- mance of the algorithm by dynamically adjusting resource tional MOEA/D algorithm, the NAAM-MOEA/D algorithm allocation; however, the proposed resource allocation criteria can better balance the convergence and distribution and also have certain limitations. The MOEA/D-GL algorithm improve the quality of the solution. proposed in literature [15] embeds the grouping and statisti- In the simulation experiment, the NAAM-MOEA/D cal learning mechanism in the traditional MOEA/D algo- algorithm was compared with the MOEA/D algorithm, the rithm, which prevents the population from falling into local MOEA/D-DE algorithm, and the NSGA-III algorithm. The optimization and improves the diversity of the population, algorithm running time was reduced by 82.1%, 108.1%, and but the overall performance improvement of the algorithm 153.6%, respectively; the GD value was reduced by 84%, is not signiﬁcant. The CD-MOEA/D-DE algorithm proposed 59%, and 35%, respectively; and the IGD value of the algo- in literature [16] controls the operation process of the algo- rithm was reduced by 75%, 56%, and 40%, respectively. rithm by formulating control parameters ∂ and balances the The main innovations of this article are summarized as performance of a multiobjective optimization problem solv- follows: ing and adaptive ability; however, the algorithm has a certain randomness in the value of the control parameter ∂ and does (1) Aiming at the defects of the traditional MOEA/D not have universal applicability. algorithm’s ﬁxed neighborhood update mechanism In addition, the researchers have proposed many spe- in solving the multiobjective ﬁre resource allocation ciﬁc improvement measures for the shortcomings of the problem, a MOEA/D algorithm based on the neigh- ﬁxed neighborhood update mechanism of the MOEA/D borhood adaptive adjustment mechanism is pro- algorithm in solving multiobjective optimization problems posed, which greatly improves the eﬃciency and especially in the edge conditions with limited computing quality in edge resources; however, the article does not elaborate on the Wireless Communications and Mobile Computing 3 Set the target set of the enemy’s combat system as D = f mechanism of how the neighborhood size aﬀects the per- formance of the MOEA/D algorithm. Literature [17] D , D , ⋯, D g; D represents the i-th target. There are a 1 2 M i points out that the size of the neighborhood will have an total of N types of weapons available for use. important impact on the performance of the MOEA/D B = fB , B , ⋯, B g, and B represents the j-th types of 1 2 N j algorithm, which provides important research directions weapons. If there is a total of M class to choose from the j for subsequent researchers. Literature [18] believes that 1 2 3 -th type of weapons, then B = fB , B , B , ⋯, B g. Select j j j j j diﬀerent multiobjective optimization problems require dif- the j-th weapon in the weapon set B to strike the i-th target ferent neighborhood sizes, and that the same multiobjec- in the target set D; the probability of the target being tive optimization problem also requires diﬀerent destroyed is p and the cost of each use of the j-th weapon ij neighborhood sizes at diﬀerent stages of the algorithm, is C . Suppose that the damage ability of the j-th type of and proposes the ENS-MOEA/D algorithm with neighbor- weapons to target D is hood adaptive adjustment capability; however, the ENS- MOEA/D algorithm may fall into local optimization in the later stage of operation. The ADEMO/D-ENS algo- P =1 − 1 − m p : ð1Þ ij ij rithm proposed in literature [19] combines the adaptive n=1 diﬀerential evolution algorithm with the variable neighbor- hood decomposition method to achieve the optimization Among them, only when the j-th type of weapons of class of the algorithm. The MOEA/D-AGR algorithm proposed n weapon is used to strike target D , there is m =1; other- i ij in literature [20] introduces an adaptive global replace- wise, m =0. The purpose of ﬁrepower resource allocation ment strategy in the neighborhood update method, which ij is to maximize the damage eﬀect under limited conditions. makes up for the shortcomings of the traditional It is necessary to consider the priority of attacking the targets MOEA/D algorithm in terms of global search capabilities. with high importance. Therefore, the calculation model of The MOEA/D-NMO algorithm proposed in literature damage capability can be deﬁned as [21] combines mutation strategies with diﬀerent character- istics and neighborhoods of diﬀerent sizes to select the "# M N j best evolutionary combination to ensure the convergence max fxðÞ = 〠〠 ω 1 − 1 − m p : ð2Þ of the algorithm while maintaining the diversity of the i ij ij i=1 j=1 n=1 algorithm. The algorithms proposed in literature [19], lit- erature [20], and literature [21] have all made improve- Among them, ω is the importance of the i-th target. ments to the ﬁxed ﬁeld, but they all have certain In addition, the minimum operational cost calculation limitations in application. model is deﬁned as follows: Although the current improved methods for ﬁxed neigh- borhoods have improved the performance of traditional M N MOEA/D algorithms, the neighborhood adaptive strategies min Cx = 〠〠〠 m C : ð3Þ used by these algorithms do not consider the impact of pop- ðÞ ij i=1 j=1 n=1 ulation evolution on neighborhoods. Literature [22] proposes a neighborhood adaptive adjustment mechanism based on The constraints of the model are as follows: population evolution stage and individual ﬁtness value, so that every individual has a corresponding neighborhood (1) Damage lower bound constraint: if the target is to be value at diﬀerent evolution stages, but its neighborhood destroyed to a certain extent so that it will lose certain adjustment method does not consider the evolution status combat capability, it is necessary to reach its damage of the subproblems. Although the MOEA/D-ANS algorithm lower bound. If the damage lower bound of target i is proposed in literature [23] adopts the ANS mechanism that deﬁned as β , then 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 statisti- P =1 − 1 − m p ≥ β : ð4Þ ij ij j cal evolution of the number of better subquestions. n=1 3. Method (2) Constraints on the number and types of weapons used: 3.1. Optimization of Fire Resource Allocation Model. The it is stipulated that one weapon can only attack one multitarget ﬁrepower resource allocation optimization prob- target at most: lem 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 M weapons and equipment used to strike speciﬁc targets to 〠 m ≤ 1, j =1,2,3, ⋯, N, n =1,2,3, ⋯, M : ð5Þ ij j i=1 obtain a feasible combat plan. 4 Wireless Communications and Mobile Computing It is stipulated that one type of weapon can only attack shev method can be expressed as follows: one type of target: f ðÞ x − Z N te r r i gðÞ μγ j , Z = max γ : ð10Þ n i 1≤i≤m 〠 m ≤ 1, i =1, 2,3, ⋯, M, n =1,2,3, ⋯, M : ð6Þ c ij j j=1 In the formula, f ðxÞ is the i-th objective function, Z is the In summary, the multiobjective ﬁrepower resource allo- reference vector, Z is the i-th component of the reference cation optimization model can be deﬁned as r r vector Z, γ is the weight vector, and γ is the i-th component "# of the weight vector γ . M N j > n max fxðÞ = 〠〠 ω 1 − 1 − m p , > i ij ij 3.2.2. Defects of Traditional MOEA/D Algorithm. The i=1 j=1 > j=1 MOEA/D algorithm maintains the power of population evo- lution from the update strategy of the neighborhood. The > M N j parent gene of an individual comes from the neighborhood, min Cx = 〠〠〠 m C , > ðÞ ij j i=1 j=1 n=1 and it adopts a coevolution model based on neighborhood update. The evolution of an individual is carried out on the s:t: basis of the neighborhood. While evolving by itself, it drives ð7Þ the evolution of other neighborhoods by optimizing other 〠 m ≤ 1, j =1,2,3, ⋯, N, n =1,2,3, ⋯, M , ij j individuals in the neighborhood. The MOEA/D algorithm > i=1 > uses a ﬁxed neighborhood strategy. For diﬀerent subprob- > N lems, the MOEA/D algorithm divides it into a neighborhood 〠 m ≤ 1, i =1,2,3, ⋯, M, n =1,2,3, ⋯, M , ij j of the same size. In fact, the computational complexity of j=1 each subproblem in the objective function is diﬀerent. The j subproblems have diﬀerent requirements for the size of the > Y > P =1 − 1 − m p ≥ β : neighborhood at diﬀerent stages. The size of the neighbor- : i ij ij j n=1 hood has a very important impact on the evolution of the subproblems. When the size of the neighborhood is large, 3.2. Detailed Introduction of NAAM-MOEA/D Algorithm the probability of other individuals in the neighborhood being replaced by oﬀspring individuals increases, and the 3.2.1. Construction and Decomposition of Subproblem. The population convergence speeds up, but the distribution of core of constructing the subproblem of the MOEA/D algo- the population will become worse as the neighborhood size rithm is to construct the weight vector of an objective func- increases, making it easy for the algorithm to fall into local tion subproblem. Suppose the weight vector of the ﬁnd the best. When the size of the neighborhood is small, subproblem of the objective function is the probability of other individuals in the neighborhood being replaced by oﬀspring individuals decreases, the popula- r − 1 N − r tion convergence speed slows, the algorithm convergence φ = , : ð8Þ N − 1 N − 1 decreases, and the overall evolution speed of the population decreases accordingly. In the formula, N is the number of subproblems after 3.2.3. Judging Mechanism of Population Evolution State. decomposition, r =1,2, ⋯, N. From the previous analysis, we can see that in the MOEA/D The core of the MOEA/D algorithm is the decomposition algorithm, subproblems and populations have diﬀerent operation, usually using aggregate functions to decompose requirements for neighborhood size at diﬀerent evolution the multiobjective constraint problem into single-objective stages. Then, how to judge the evolution state of the popula- subproblems. Commonly used decomposition methods are tion and whether it can ﬁnd a mechanism that can eﬀectively weighted sum method, Chebyshev method, and boundary evaluate the evolution stage of the population is the core crossing method based on penalty. This paper adopts the problem that the new algorithm needs to solve. Chebyshev method, and its decomposition principle is Some scholars propose to use the individual density of te r ∗ r ∗ subproblems to assess the degree of population evolution. gðÞ xjφ , Z = maxfg φ jf ðÞ x − Z : ð9Þ i i 1≤i≤m The individual density of the subproblem is equivalent to the number of individuals in the subinterval. If the individual ∗ ∗ ∗ ∗ Among them, φ = fφ , φ , ⋯, φ g is the weight vector density of the subproblem is smaller, the surrounding indi- 1 2 m ∗ ∗ ∗ ∗ corresponding to the subproblem r. Z = fZ ,Z , ⋯, Z g is viduals are denser, the better the degree of evolution of the 1 2 m the ideal point. f ðxÞ is the i-th objective function, and φ is individual is, and the greater the probability of the problem i i the i-th component of the weight vector φ . Z is the i-th being solved. If the individual density of the subproblem is component of the ideal point Z . smaller, the surrounding individuals are sparser, then the The single-objective optimization function of the i-th degree of evolution of the individual is smaller, and the prob- subproblem of objective function constructed by the Cheby- lem is less likely to be solved. Wireless Communications and Mobile Computing 5 i i Input: the threshold d , ω , ε ; JM is the attribution judging mechanism of weight vector; JM is the subproblem evolution degree m m m wei sub judgment mechanism; JM is the population evolution degree judgment mechanism; d is the distance between the weight vector pop wi and the individual; ω is the number of individuals owned by the weight vector; ε represents the number of subproblems with better evolution. Output: the population evolution state 1 Determine the attribution of the weight vector; 2 ford ≤ d , do wi m JM =1; determine that the individual belongs to the weight vector wei else doJM =0 wei 3 Calculate the number of individuals owned by the weight vector: ω = ∑ JM ; i=1 wei 4 Determine the degree of evolution of subproblems; 5 forω ≥ ω , do JM =1 and determine the degree of evolution of subproblems is better sub else doJM =0; sub 6 Calculate the number of subproblems with better evolution: ε = ∑ JM i=1 sub 7 Determine the evolution stage of the population; 8 forε > ε , do JM =2 and determine that the current population evolution degree is too fast, belonging to an overevolution state; pop else forε < ε , do JM =0 and determine that the current population is slowly evolving and belongs to a state of lagging evolution pop else for ε = ε , do JM =1 and determine that the current population has a good evolution speed and belongs to a normal evolutionary state pop 9 end Algorithm 1: Judgment mechanism of population evolution status. Input: the initial population X = fx , x , ⋯, x g; the size of the initial neighborhood corresponding to the subproblem within the pop- 1 2 n ulation T = fT ,T , ⋯, T g; the population initial neighborhood T s s1 s2 sn i Output: the size of the neighborhood corresponding to the subproblem within the population T = fT ,T , ⋯, T g, the current s s1 s2 sn population size T . 1 Initialize: T = T = T . i s 2 evolution 3 forx ∈ Xdo Take x corresponding to the individuals in neighborhood BðiÞ = fi , i , ⋯, i g to perform crossover and mutation opera- i 1 2 T tions to obtain oﬀspring individuals; 4 Determine the degree of evolution of the subproblems and the evolution status of the population according to the mechanism pro- vided in Section 3.2.3; 5 Neighborhood adaptive adjustment 6 forJM =1, do sub The evolution of the previous generation is fast and uses formula (11) to appropriately reduce the current neighborhood size T; else forJM =0, do sub The evolution of the previous generation is slow and uses formula (11) to appropriately increase the current neighborhood size T; 7 forJM =1, do pop The evolution rate of the previous generation population is moderate; else forJM =2, do pop The evolution rate of the previous generation population is fast and uses formula (12) to appropriately reduce the current population size T ; else forJM =0, do pop The evolution rate of the previous generation population is slow and uses formula (12) to appropriately increase the current population size T ; 8 Output T = fT , T , ⋯, T g and T . s s1 s2 sn Algorithm 2: Neighborhood adaptive adjustment mechanism. 6 Wireless Communications and Mobile Computing Input: optimal model of multiobjective ﬁre resource allocation; termination criteria; population size N; population crossover proba- bility p ; probability of population variation p ; c m Output: optimal plan for ﬁrepower resource allocation 1 Initialize 2 EP = ∅ 3 Initialize population individuals x , x , ⋯, x ; 1 2 N 4 Generate weight vector φ , φ ,…, φ ; 1 2 N 5 Calculate the Euclidean distance between any two weight vectors. For each weight vector, ﬁnd the T nearest weight vectors to form its neighborhood. i =1,2, ⋯, N and BðiÞ = fi , i , ⋯, i g. Among them, φ , φ ,…, φ are the T weight vectors closest to φ ; 1 1 T i i i i 1 2 T 6 Initialize the ideal point Z = ðZ , Z , ::, Z Þ ; 1 2 m 7 Evolution 8 fori =1, 2, ⋯, N, do 9 Crossover and mutation: randomly select two individuals in BðiÞ to perform crossover and mutation operations to obtain oﬀ- spring individual y; 10 Update ideal point Z 11 for eachj =1, 2, ⋯, m. ifZ < f ðyÞ, thenZ = f ðyÞ j j j j end 12 Set adaptive neighborhood 13 Judge the subproblems and the evolution status of the population through the criteria provided in Section 2.1; 14 Use the method provided in Section 2.2 to obtain the population neighborhood T and the neighborhood T corresponding to the subproblem; 15 Update neighborhood BðiÞ for eachj ∈ BðiÞ ωs ωs j ifg ðy jλ , zÞ ≤ g ðx jλ , zÞ then j j j j x = y′, FV = Fðy ′Þ end end 16 Remove all individuals dominated by Fðy Þ in EP and add individuals not dominated to EP at the same time; 17 end 18 Stop operation 19 After the algorithm evolves to the maximum algebra G max, it stops and outputs the optimal solution. If the stopping condition is not met, it returns to Step 7. Algorithm 3: The framework of NAAM-MOEA/D algorithm. Table 1: Optimization model parameters of ﬁrepower allocation This paper proposes a mechanism for evaluating the evo- resources. lutionary stage of a population (see Algorithm 1): Project W1 W2 W3 W4 Importance T1 0.82 ——— 0.22 3.2.4. Neighborhood Adaptive Adjustment Mechanism. In T2 — 0.95 —— 0.31 order to meet the needs of balancing the convergence and T3 —— 0.87 — 0.28 distribution of the MOEA/D algorithm, according to the T4 ——— 0.85 0.19 population evolution state judgment mechanism in Section 3.2.3, this paper proposes a neighborhood strategy that adap- Unit cost 5 10 8 4 — tively adjusts the population size based on the diﬀerent evo- Total number 10 10 10 10 — lution stages of the population, which can also be called a neighborhood adaptive adjustment mechanism (NAAM) as in Algorithm 2. Some scholars propose that if the distance between the The setting adjustment formula is as follows: 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 sub- "# problem. In the spatial coordinate system, the solution corre- > 1 mT > ∗ i T 1 − ω ,JM =1, ∗ sub sponds to the individual in the coordinate system, and the > T N subproblem corresponds to the weight vector. Therefore, T = ð11Þ "# > θ the problem of determining the attribution of the solution 1 mT ∗ i T 1+ ω ,JM =0, can be transformed into the problem of ﬁnding the distance ∗ sub T N between the weight vector and the individual. Wireless Communications and Mobile Computing 7 12 200 6 100 92 55 5 4 444 0 0 W1 W2 W3 W4 NAAM-MOEA/D MOEA/D MOEA/D-DE NSGA-III NAAM-MOEA/D MOEA/D-DE Figure 3: Total computational cost of the 4 algorithms. MOEA/D NSGA-III Figure 1: Number of weapons of each type used by the 4 algorithms. 25.6 30 25 22.4 12.3 15 5 NAAM-MOEA/D MOEA/D MOEA/D-DE NSGA-III Figure 4: Calculation time spent of the 4 algorithms. Table 2: Statistics of ﬁre resource distribution. NAAM-MOEA/D MOEA/D MOEA/D-DE NSGA-III W1 W2 W3 W4 Total Total Running Figure 2: Total number of weapons used by the 4 algorithms. Project T1 T2 T3 T4 use cost time NAAM- "# 4 2 5 3 14 92 12.3 s > MOEA/D 1 mT i i > T 1 − ω ,JM =2, > pop MOEA/D 5 1 4 8 18 99 22.4 s T N > i < MOEAD/D- 5 4 4 2 15 113 25.6 s ∗ i DE T = T,JM =1, ð12Þ i pop "# NSGA-III 7 6 5 5 23 155 30 s > 1 mT > i T 1+ ω ,JM =0: > i pop T N MOEAD/D-DE algorithm, and NSGA-III algorithm are selected for the simulation operation. 3.2.5. The Framework of NAAM-MOEA/D Algorithm. The Figure 1 counts the number of various weapons used by framework of the NAAM-MOEA/D algorithm can be the four algorithms. It can be seen from Figure 1 that the described as in Algorithm 3: NSGA-III algorithm uses 7 W1 weapons, which is more than the MOEA/D algorithm and the MOEA/D-DE algorithm; 4. Experiment and Simulation both algorithms use 5 W1 weapons, and the NAAM- MOEA/D algorithm uses 4 W1 weapons. The NSGA-III 4.1. Example Analysis of Algorithm. There are 4 types of algorithm uses 6 W2 weapons; the MOEA/D-DE algorithm weapons to strike at 4 targets in the enemy’s combat system. Combining the content of Section 3.1, we assume that the and the NAAM-MOEA/D algorithm use 4 and 2 W2 weapons, respectively; while the MOEA/D algorithm uses model satisﬁes various constraints, and the model parame- ters are given in Table 1. the least number of W2 weapons and only one is used. The NAAM-MOEA/D algorithm and the NSGA-III algorithm MATLAB 2020 is selected to write the algorithm pro- gram. The running environment is a Windows 7 R64-bit both use 5 W3 weapons, which is more than the MOEA/D operating system, 4 GB memory, Intel Pentium processor. algorithm and the MOEA/D-DE algorithm. Both algorithms use 4 W3 weapons. The MOEA/D algorithm uses 8 W4 The NAAM-MOEA/D algorithm, MOEA/D algorithm, Total use The number of weapons used Running time (s) Total cost 8 Wireless Communications and Mobile Computing 1 1 10 10 0 0 10 10 –1 –1 10 10 –2 –2 10 10 –3 –3 10 10 –4 –4 10 10 –5 –5 10 10 –6 –6 10 10 0 100 200 300 400 500 0 100 200 300 400 500 Number of iterations Number of iterations NAAM-MOEA/D MOEA/D-DE NAAM-MOEA/D MOEA/D-DE MOEA/D NSGA-III MOEA/D NSGA-III (a) Running results on ZDT1 test function (b) Running results on ZDT2 test function 1 1 10 10 0 0 10 10 –1 –1 10 10 –2 –2 10 10 –3 –3 10 10 –4 –4 10 10 –5 –5 10 10 –6 –6 10 10 0 100 200 300 400 500 0 100 200 300 400 500 Number of iterations Number of iterations NAAM-MOEA/D MOEA/D-DE NAAM-MOEA/D MOEA/D-DE MOEA/D NSGA-III MOEA/D NSGA-III (c) Running results on ZDT3 test function (d) Running results on ZDT4 test function Figure 5: Variation curve of GD with algorithm iteration number. weapons, which is more than the NSGA-III algorithm. The computing time, which takes 30 s. The computing times of NAAM-MOEA/D algorithm and the MOEA/D algorithm the MOEA/D algorithm and the MOEA/D-DE algorithm use 3 and 2 W4 weapons, respectively. are, respectively, 22.4 s and 25.6 s. Figure 2 counts the total number of weapons used by the The statistics of ﬁrepower resource allocation obtained four algorithms. It can be seen from Figure 2 that the NSGA- through simulation calculation are shown in Table 2. III algorithm uses the largest number of weapons, using 23 It can be seen from Table 2 that the number of weapons used and the total cost obtained by the NAAM-MOEA/D weapons in total. The MOEA/D algorithm and the MOEA/D-DE algorithm use 18 and 15 weapons, respec- algorithm are better than those of the other three algorithms. tively, and the NAAM-MOEA/D algorithm uses the least The number of weapons used by the MOEA/D-DE algorithm amount of weapons—only 14 weapons are used. is close to the number of weapons used by the NAAM- Figure 3 compares the total cost of weapon use of the four MOEA/D algorithm, but the total cost is about 23% higher. algorithms. It can be seen from Figure 3 that the NSGA-III algo- The total cost calculated by the MOEA/D-DE algorithm is rithm costs the most weapons, with a total cost of 155, followed close to the total cost calculated by the NAAM-MOEA/D by the MOEA/D-DE algorithm, with a total cost of 113, while algorithm, but 4 more weapons are used. The number of the MOEA/D algorithm and NAAM-MOEA/D algorithm weapons used and the total cost obtained by the NSGA-III had the least weapon use cost, costing 99 and 92, respectively. algorithm are signiﬁcantly more than those of the other three Figure 4 compares the computing time of the four algo- algorithms, indicating that the algorithm has the worst per- rithms. It can be seen from Figure 4 that the NAAM- formance. In addition, the running time of the NAAM- MOEA/D algorithm has the least computing time, which MOEA/D algorithm is 12.3 s, which is reduced by 82.1%, takes only 12.3 s, and the NSGA-III algorithm has the most 108.1%, and 153.6% compared with the MOEA/D algorithm, GD GD GD GD Wireless Communications and Mobile Computing 9 –3 –3 ×10 ×10 NAAM-MOEA/D MOEA/D MOEA/D-DE NSGA-III NAAM-MOEA/D MOEA/D MOEA/D-DE NSGA-III Algorithm Algorithm (a) IGD on ZDT1 test function (b) IGD on ZDT2 test function –3 –3 ×10 ×10 NAAM-MOEA/D MOEA/D MOEA/D-DE NSGA-III NAAM-MOEA/D MOEA/D MOEA/D-DE NSGA-III Algorithm Algorithm (c) IGD on ZDT3 test function (d) IGD on ZDT4 test function Figure 6: IGD box plot of the algorithm under diﬀerent test functions. Table 3: Comparison of IGD indicators of algorithms on ZDT series functions. NAAM-MOEA/D MOEA/D MOEA/D-DE NSGA-III Test function Mean (std) Mean (std) Mean (std) Mean (std) ZDT1 5.83E-03 (4.73E-04) 5.96E-03 (5.13E-04) 9.57E-03 (1.31E-02) 1.01E-02 (1.65E-02) ZDT2 4.17E-03 (4.85E-04) 4.51E-03 (8.13E-04) 9.33E-03 (1.07E-04) 8.75E-03 (2.27E-03) ZDT3 3.17E-03 (2.05E-04) 8.48E-03 (9.30E-04) 8.73E-03 (5.95E-04) 9.55E-03 (2.68E-04) ZDT4 3.25E-03 (2.72E-04) 8.21E-03 (4.03E-04) 9.12E-03 (7.15E-04) 1.04E-02 (4.52E-03) the MOEA/D-DE algorithm, and the NSGA-III algorithm, probability p =1/n, n is the dimension of decision vari- respectively, indicating that the NAAM-MOEA/D algorithm ables). Each algorithm runs 20 times independently, and has obvious advantages in computing speed. the evaluation times are set to 10000. Inverse generation dis- tance (IGD) and generation distance (GD) were used as eval- uation indexes. Each test function is run 20 times 4.2. Performance Test of the Algorithm. In order to verify the performance of the NAAM-MOEA/D algorithm, ZDT series independently and averaged every 10 generations. The varia- of test functions are selected to test the performance of the tion curve of GD with the number of iterations (0-500 gener- NAAM-MOEA/D algorithm with the MOEA/D algorithm, ations) of the algorithm is shown in Figure 1. MOEA/D-DE algorithm, and NSGA-III algorithm. As shown in Figure 5(a), the NAAM-MOEA/D algo- rithm tends to be stable on the test function ZDT1, and the In order to ensure the fairness and rationality of the algo- rithm evaluation, the population size and initial neighbor- convergence speed is slower than the NSGA-III algorithm and faster than the MOEA/D algorithm and the MOEA/D- hood size of the four algorithms are set to the same (population size N = 100, initial neighborhood T = 100). All DE algorithm. algorithms adopt simulated binary crossover (crossover As shown in Figure 5(b), on the test function ZDT2, the convergence speed of the NAAM-MOEA/D algorithm is probability p =0:9) and polynomial mutation (mutation IGD IGD IGD IGD 10 Wireless Communications and Mobile Computing ZDT1 ZDT1 1 1 0.5 0.5 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Objective function 1 Objective function 1 PF PF NAAM-MOEA/D MOEA/D (a) NAAM-MOEA/D on ZDT1 test function (b) MOEA/D on ZDT1 test function ZDT1 ZDT1 1 1 0.5 0.5 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Objective function 1 Objective function 1 PF PF MOEAD-DE NSGA-III (c) MOEA/D-DE on ZDT1 test function (d) NSGA-III on ZDT1 test function ZDT3 ZDT3 1 1 0.5 0.5 0 0 –0.5 –0.5 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Objective function 1 Objective function 1 PF PF MOEA/D NAAM-MOEA/D (e) NAAM-MOEA/D on ZDT3 test function (f) MOEA/D on ZDT3 test function ZDT3 ZDT3 1 1 0.5 0.5 0 0 –0.5 –0.5 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Objective function 1 Objective function 1 PF PF MOEA/D-DE NSGA-III (g) MOEA/D-DE on ZDT3 test function (h) NSGA-III on ZDT3 test function Figure 7: Continued. Objective function 2 Objective function 2 Objective function 2 Objective function 2 Objective function 2 Objective function 2 Objective function 2 Objective function 2 Wireless Communications and Mobile Computing 11 ZDT4 ZDT4 0.5 0.5 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Objective function 1 Objective function 1 PF PF NAAM-MOEA/D MOEA/D (i) NAAM-MOEA/D on ZDT4 test function (j) MOEA/D on ZDT4 test function ZDT4 ZDT4 1 1 0.5 0.5 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Objective function 1 Objective function 1 PF PF NSGA-III MOEA/D-DE (k) MOEA/D-DE on ZDT4 test function (l) NSGA-III on ZDT4 test function Figure 7: Comparison of Pareto front and ideal Pareto front on ZDT test function. faster than that of the MOEA/D-DE algorithm and the NAAM-MOEA/D algorithm are also lower than those of NSGA-III algorithm. Although it is slightly slower than the the other three algorithms, which show that the stability MOEA/D algorithm, the population degradation degree of and quality of the NAAM-MOEA/D algorithm is higher. the MOEA/D algorithm is higher than that of the NAAM- On the test functions ZDT1 and ZDT2, the comprehensive performance of the NAAM-MOEA/D algorithm is slightly bet- MOEA/D algorithm. As shown in Figure 5(c), on the test function ZDT3, the ter than that of the MOEA/D algorithm and signiﬁcantly better NAAM-MOEA/D algorithm converges faster than the other than that of the MOEA/D-DE algorithm and the NSGA-III algorithms. algorithm. On the test functions ZDT3 and ZDT4, the compre- As shown in Figure 5(d), on the test function ZDT4, the hensive performance of the NAAM-MOEA/D algorithm is sig- NAAM-MOEA/D algorithm has a faster population conver- niﬁcantly better than that of the MOEA/D algorithm, the gence speed due to the advantages of the adaptive neighbor- MOEA/D-DE algorithm, and the NSGA-III algorithm. This hood adjustment mechanism adopted, and the algorithm is because there are many discontinuous regions in the target space of test function ZDT3. These regions adopt the ﬁxed convergence performance is signiﬁcantly better than the MOEA/D algorithm, MOEAD/D-DE algorithm, and NSGA. neighborhood setting method, but do not use the adaptive Therefore, the NAAM-MOEA/D algorithm not only neighborhood allocation strategy to reasonably allocate the ensures that the algorithm has a faster convergence rate but algorithm, which leads to the waste of algorithm resources also solves the population degradation problem that occurs and the slowdown of population evolution speed. during the algorithm operation and ensures the stability of Figure 7 shows the comparison of the Pareto front and the the algorithm operation, so that the algorithm can have more ideal Pareto front obtained by the four algorithms on the ZDT resources to improve the diversity of the population. test function. Among them, the red meter character represents As shown in Figure 6, comparing the IGD box plots of the ideal PF, and the blue circle represents the optimal solution various algorithms on the ZDT series test functions in the of the Pareto frontier obtained by the various algorithms. On the test function ZDT3, the improved MOEA/D solu- comparison Table 3, we can see that the NAAM-MOEA/D algorithm’s mean, minimum, median (at the position of the tion set is more evenly distributed on the ideal Pareto front. red line in the ﬁgure), and interquartile range (key indicators In the other three algorithms, some leading edges are not such as box length) are lower than those of the MOEA/D completely found, and the solution set is missing to a certain algorithm, MOEA/D-DE algorithm, and NSGA-III algo- extent. Among them, the MOEA/D algorithm and the MOEA/D-DE algorithm have a little poor distribution of rithm. The probability and size of the abnormal value of the Objective function 2 Objective function 2 Objective function 2 Objective function 2 12 Wireless Communications and Mobile Computing defects of the traditional MOEA/D algorithm ﬁxed neigh- solution set, while the NSGA-III algorithm has the least dis- tribution. This is because the other algorithms spend limited borhood update mechanism, a MOEA/D algorithm based computing resources in the discrete region of test function on neighborhood adaptive adjustment mechanism is pro- ZDT3 and produce too many nondominated solutions, posed and the model is solved. It can be seen from the which hinders the evolution of the population. simulation experiment that the MOEA/D algorithm based On the test function ZDT4, the NAAM-MOEA/D algo- on the neighborhood adaptive adjustment mechanism has rithm has converged to the ideal, while the other algorithms signiﬁcantly improved its stability, convergence, and have fallen into the local optimization state to varying degrees. distribution. It can be seen that the NAAM-MOEA/D algorithm has more In the next step, current work will continue to be advantages in reasonable allocation of computing resources improved by considering security and privacy issues [24– and can better ensure the convergence of the algorithm. 33]. In addition, more complex multiobjective solutions with Through the comparison, we can see that the Pareto fron- more context factors [34–41] will be considered. tier solution set obtained by the NAAM-MOEA/D algorithm almost uniformly converges to the PF of the ideal Pareto. However, the other three algorithms have diﬀerent degrees Abbreviations of missing or uneven distribution of solution sets in various MOEA/D: Multiobjective evolutionary algorithm test functions. The NAAM-MOEA/D algorithm shows some based on decomposition performance advantages when dealing with simple test prob- NAAM-MOEA/D: Neighborhood adaptive adjustment lems such as ZDT1, but the advantages are not obvious. mechanism-multiobjective evolutionary However, the NAAM-MOEA/D can allocate computing algorithm based on decomposition resources reasonably and take into account the convergence MODPSO-GSA: Multiobjective discrete particle swarm and distribution of the algorithm due to its ﬂexible neighbor- optimization-gravitational search hood update strategy when dealing with relatively complex algorithm test problems such as ZDT3 and ZDT4. WMOM/D: Weapon-target assignment multiobjec- tive model based on decomposition 5. Discussion GD-MOEA/D: Gauss mutation and diﬀerential evolu- tion based on a multiobjective evolu- In this section, we establish a ﬁrepower resource allocation tionary algorithm based on optimization model for edge environment based on given decomposition speciﬁc data, conduct simulation experiments, and test and MOEA/D-DE: Multiobjective evolutionary algorithm evaluate the performance of the algorithm combined with based on decomposition-diﬀerential the ZDT series of functions. However, several additional evolution points should be pointed out and further analyzed in detail, MOEA/D-DRA: Multiobjective evolutionary algorithm which are speciﬁed as below. based on decomposition-dynamical resource allocation (1) The types of weapons and the number of samples ENS-MOEA/D: Ensemble neighborhood size- given in Section 4.1 are not large enough (both are multiobjective evolutionary algorithm 4). Therefore, in the future simulation experiments, based on decomposition we should focus on large sample data sets to verify ADEMO/D-ENS: Adaptive diﬀerential evolution for mul- the performance of the method under the condition tiobjective problems-ensemble neigh- of large sample data borhood size (2) In Section 4.2, the ZDT series functions are selected MOEA/D-AGR: Multiobjective evolutionary algorithm to test the performance of the algorithm. The simula- based on decomposition-adaptive global tion results show that the performance of the replacement NAAM-MOEA/D algorithm is better than that of MOEA/D-NMO: Multiobjective evolutionary algorithm the other three algorithms. However, only one kind based on decomposition-neighborhood of test function veriﬁcation is not convincing enough, mutation operator so DLTZ, WFG, and other test functions should be MOEA/D-ANS: Multiobjective evolutionary algorithm selected to evaluate the algorithm, so as to provide based on decomposition-adaptive more suﬃcient reference for the improvement of neighborhood strategy algorithm performance NSGA-III: Nondominated sorted genetic algo- rithm-III. 6. Conclusion Data Availability This paper constructs a multiobjective ﬁrepower resource allocation optimization model for edge environment with limited computing resources, based on maximizing dam- The experiment dataset is generated randomly through age eﬀect and minimizing combat cost. Aiming at the simulation. Wireless Communications and Mobile Computing 13 Conflicts of Interest [12] C. Hui and Y. 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Wireless Communications and Mobile Computing – Hindawi Publishing Corporation
Published: Mar 22, 2021
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