# Improved Ant Colony Optimization for Weapon-Target Assignment

Authors: Xinwu Hu; Pengcheng Luo; Xiaonan Zhang; Jun Wang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Weapon-target assignment (WTA) which is crucial in cooperative air combat explores assigning weapons to targets with the objective of minimizing the threats from those targets. Based on threat functions, there are four WTA models constrained by the payload and other tactical requirements established. The improvements of ant colony optimization are integrated with respect to the rules of path selection, pheromone update, and pheromone concentration interval, and algorithm AScomp is proposed based on the elite strategy of ant colony optimization (ASrank). We add garbage ants to ASrank; when the pheromone is updated, the elite ants are rewarded and the garbage ants are punished. A WTA algorithm is designed based on the improved ant colony optimization (WIACO). For the purpose of demonstration of WIACO in air combat, a real-time WTA simulation algorithm (RWSA) is proposed to provide the results of average damage, damage rate, and kill ratio. The following conclusions are drawn: (1) the third WTA model, considering the threats of both sides and hit probabilities, is the most effective among the four; (2) compared to the traditional ant colony algorithm, the WIACO requires fewer iterations and avoids local optima more effectively; and (3) WTA is better conducted when any fighter is shot down or any fighter’s missiles run out than along with the flight.


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## Body

## 1. Introduction

Weapon-target assignment (WTA) is a dynamic multivariable and multiconstraint problem, which is characterized by antagonism, initiative, and uncertainty. So far, there are bulks of studies on the solution to WTA, such as the use of genetic algorithm (GA), simulated annealing (SA), and particle swarm optimization (PSO) algorithm. Additionally, many scholars use ant colony optimization.In [1, 2], GA was used to solve the problem. In GA, a population of individuals, which encode the problem solutions, are manipulated according to their fitness values through genetic operators, such as reproduction, mutation, and crossover. GA has delightful global searching ability and can find all feasible solutions. However, its local searching ability is poor; to be exact, it is prone to premature convergence. Moreover, it is time consuming as well. Reference [3, 4] used SA in WTA. SA starts from a high initial temperature, and as the temperature falls down, the global optimal solution is found randomly; even when the searching falls into a local optimal solution, SA has a probability to jump out and eventually goes to the global optimum. But this algorithm converges slowly and takes time. PSO was applied to WTA in [5, 6]. PSO builds a swarm intelligence model, initializes a set of random solutions, and searches for the optimal solution by iterations. PSO is simple to implement and converges fast, but it is easy to fall into local optima.In addition, reference [7, 8] developed a novel multiobjective optimization method based on the evolutionary game theory in real-time WTA. Darryl et al. [9] proved that the dynamic programming method could also solve WTA problem. Liang et al. [10] presented an objective optimization approach based on clonal selection algorithm to solve the problems of WTA in warship formation antiaircraft application. Based on the auction algorithm, Fei et al. [11] brought forward a new distributed multiaircraft cooperative fire assignment method. Considering the complexity and strict time constraints, Sahin et al. [12] proposed a fuzzy decision method to aid commanders in making decisions for WTA.As is known, ant colony optimization has the characteristics of distributed computing, self-organization, and positive feedback. The complicated WTA process in air combat can be mapped to ant foraging behavior.The ant system was first proposed by Dorigo [13] in 1991 in his doctoral thesis. In 1994, Lumer and Faita [14] took the idea of ant colony clustering to data analysis and proposed the LF algorithm. Considering the good performance of ant colony optimization in solving discrete combinatorial optimization problems, Lee et al. [15] first applied it to WTA in 2002. The basic ant colony algorithm was applied to the target assignment problem of the air defense C3I systems by Huang et al. [16] in 2005. In recent years, ant colony optimization was widely used. For example, in 2013, Olmo et al. [17] used it in the research on association task of data mining, and the results obtained were very exciting. In 2015, Ariyasingha et al. [18] analyzed the performance of multiobjective ant colony optimization for the traveling salesman problem and concluded that the algorithm performed better in problems with more than two objectives and its performance depended slightly on the number of objectives, iterations, and ants. Lately, in 2017, Li et al. [19] designed a biobjective WTA optimization model which maximized the expected damage of the enemy and minimized the cost of missiles; a modified Pareto ant colony optimization algorithm was used in the solution, which produced better results than two multiobjective optimization algorithms NSGA-II and SPEA-II.Generally, WTA in air combat aims to minimize the threat from opponent fighters, in other words, to maximize the defused threat, constrained by the payload and other tactical requirements. This paper focuses on the WTA modeling, solution, and simulation in air combat scenario. Based on threat functions, four WTA models are established. Among them, Model3 is proposed for the first time considering the threats of both sides and hit probabilities. For the solution to the WTA models, a WTA algorithm based on improved ant colony optimization (WIACO) is designed, integrating the improvements of traditional ant colony optimization with respect to the rules of path selection, pheromone update, and pheromone concentration interval, and proposes algorithm AScomp. Through a comparative experiment, it is concluded that WIACO requires fewer iterations than traditional ant colony algorithm, and it avoids local optima more effectively. Furthermore, in order to demonstrate and exemplify the effectiveness of WIACO in air combat, a real-time WTA simulation algorithm (RWSA) is presented to simulate real-time WTA in air combat, with the results of average damage, damage rates, and kill ratios.The paper is organized as follows: Section2 discusses the air combat threat functions and four WTA models. In Section 3, the WIACO algorithm is introduced, and then in Section 4 a comparative analysis is offered. The RWSA algorithm, three experiments, and result analysis are given in Section 5. Finally, the conclusions of this paper are drawn in Section 6.


## 2. Weapon-Target Assignment Model

The objective of WTA is to maximize the expected impact on the opponents and to minimize the risk we face [20] in terms of threat. In this case, we need to measure the threat in air combat first.


### 2.1. Air Combat Threat

It is assumed that the red side hasN fighters and the blue side has K fighters and that the red side’s early warning aircraft can accurately identify the model, speed, spatial position, and other basic information of the blue fighters. This section uses the air combat situation in reference [21] as shown in Figure 1 for threat modeling.Figure 1
Air combat situation diagram.In Figure1, Ri is the ith red fighter, Bj is the jth blue fighter, XRi is the direction of Ri, VRi is the speed of Ri, εji is the off-axis angle of Bj relative to Ri, Dij is the distance between Ri and Bj, and the other parameters are defined similarly.


#### 2.1.1. Angle Threat Function

The angle threat function [22] is given as follows:(1)Sij1=εij-εji180°where Sij1 is the angle threat of Bj to Ri, with 0°≤εji,εij,≤180° and -1≤ Sij1≤1. In particular, when the Ri off-axis angle εij is 180° and the Bj off-axis angle εji is 0°, that is, when Bj chases Ri from behind, the angle threat of Bj to Ri is 1.


#### 2.1.2. Distance Threat Function

The distance threat function [23] is given as follows:(2)Sij2=1Dij≤Tab1-Dij-TabLrb-TabTab<Dij≤Lrb0Lrb<Dijwhere Tabrepresents the missile range of the blue fighter and Lrb is the maximum detection range of the blue radar. When the red fighter is within the blue attack range, the distance threat of Bj to Ri takes the maximum value of 1; when the red fighter is out of the detection range of blue, the distance threat takes the minimum value of 0.


#### 2.1.3. Speed Threat Function

The speed threat function [22] is given as follows:(3)Sij3=1VRi≤0.5·VBj1.5-VRiVBj0.5·VBj<VRi≤1.4·VBj01.4·VBj<VRiwhere VRi and VBj are the speeds of the red and blue fighters, respectively. The greater the VRi than the VBj, the smaller the threat of Bj to Ri.


#### 2.1.4. Ability Threat Function

In this paper, we embrace the air combat capability formula in [24] given as follows:(4)C=ln⁡B+ln⁡∑A1+1+ln⁡∑A2ε1ε2ε3ε4where B is the maneuverability parameter of the fighter; A1 is the fire attack capability parameter; A2 is the radar detection capability parameter; ε1 is the pilot's control capability coefficient; ε2 is the fighter survivability coefficient; ε3 is the fighter range coefficient; and ε4 is the Electronic Counter Measures (ECM) capability coefficient. The ability threat function is defined as [22]:(5)Sij4=1CjCi≥1.50.751.5≥CjCi>10.5CjCi=10.251>CjCi≥0.30CjCi<0.3For example, supposing the air combat capability ofR1 is 17.9, if the capabilities of B1, B2, and B3 are 15.8, 18.8, and 17.9, the ability threat degrees of blue to red are 0.25, 0.75, and 0.5, respectively.As a combination of the above threat functions, the threat degree ofBj to Ri is(6)Sij=ω1Sij1+ω2Sij2+ω3Sij3+ω4Sij4where0<ωi<1  (i = 1,2,3,4) is the weight, Σωi = 1.The overall threat degree ofBj to all the red fighters is(7)Sj=∑iSij


### 2.2. Weapon-Target Assignment

LetRi carry Mi missiles and Ej the number of missiles assigned to Bj:(8)Ej=1Sj≤121<Sj≤33Sj>3Four WTA models are presented in this subsection.Model 1.
(9) m a x  ∑ i ∑ j x i j S i j s . t  ∑ j x i j ≤ M i ,  i = 1,2 , … N ∑ i x i j ≤ E j ,  i = 1,2 , … K k = ∑ i ∑ j x i j = min ⁡ ∑ i M i , ∑ j E jwhere xij, an integer from 0 to 3, represents the missile number of Ri assigned to attack Bj; Sij is the threat of Bj to Ri; k is the number of red missiles actually used to attack the target; N is the number of red fighters; and K is the number of blue fighters. The first constraint indicates that the number of missiles launched by each red fighter cannot exceed its carrying capacity; the second constraint implies that the number of missiles assigned to each blue fighter cannot exceed the value obtained in formula (8); the third constraint indicates that the number of fired missiles takes the smaller value between the total number of missiles carried by the red side and the total number of missiles assigned to the blue side.Model 2.
(10) m a x ∑ j S j 1 - ∏ i 1 - p i j x i j
Model4 was first proposed in reference [25]. The constraints are the same as in Model 1, and pij is the probability that Ri hits Bj, which is calculated with the two-step adjudication model [26] in this paper. Compared to Model 1, the objective here takes the hit probability of the blue fighters into account; the larger the pij, the greater the threat.Model 3.
(11) max ⁡ ∑ i ∑ j S i j + S j i × p i j × x i j
The constraints are the same as in Model1, and Sji is the threat of Ri to Bj. Model 3 considers the blue side threat while also considering the advantages of the red side for the blue side. The objective signifies that red missiles will be preferentially assigned to the blue fighters which pose high threat to red, get high threat from red, and have high hit probability.Model 4.
(12) max ⁡ ∑ j S j P j ∑ i x i j + M ∑ j min ⁡ 0 , P j - P d j (13) P j = 1 - ∏ i 1 - p i j x i j
Model4 was first proposed in reference [27], where Pj is the overall hit probability of the red fighter on Bj; Pdj is the threshold of hit probability of Bj; and M is the penalty factor. Model 4 calculates the average hit probability for each missile, maximizing the threat of each missile resolution. At the same time, the minimum hit probability requirement is set for each blue fighter, and if Pj is lower than Pdj, it will be punished. The objective here is to maximize the average hit probability, considering the blue fighter’s hit probability threshold.


## 2.1. Air Combat Threat

It is assumed that the red side hasN fighters and the blue side has K fighters and that the red side’s early warning aircraft can accurately identify the model, speed, spatial position, and other basic information of the blue fighters. This section uses the air combat situation in reference [21] as shown in Figure 1 for threat modeling.Figure 1
Air combat situation diagram.In Figure1, Ri is the ith red fighter, Bj is the jth blue fighter, XRi is the direction of Ri, VRi is the speed of Ri, εji is the off-axis angle of Bj relative to Ri, Dij is the distance between Ri and Bj, and the other parameters are defined similarly.


### 2.1.1. Angle Threat Function

The angle threat function [22] is given as follows:(1)Sij1=εij-εji180°where Sij1 is the angle threat of Bj to Ri, with 0°≤εji,εij,≤180° and -1≤ Sij1≤1. In particular, when the Ri off-axis angle εij is 180° and the Bj off-axis angle εji is 0°, that is, when Bj chases Ri from behind, the angle threat of Bj to Ri is 1.


### 2.1.2. Distance Threat Function

The distance threat function [23] is given as follows:(2)Sij2=1Dij≤Tab1-Dij-TabLrb-TabTab<Dij≤Lrb0Lrb<Dijwhere Tabrepresents the missile range of the blue fighter and Lrb is the maximum detection range of the blue radar. When the red fighter is within the blue attack range, the distance threat of Bj to Ri takes the maximum value of 1; when the red fighter is out of the detection range of blue, the distance threat takes the minimum value of 0.


### 2.1.3. Speed Threat Function

The speed threat function [22] is given as follows:(3)Sij3=1VRi≤0.5·VBj1.5-VRiVBj0.5·VBj<VRi≤1.4·VBj01.4·VBj<VRiwhere VRi and VBj are the speeds of the red and blue fighters, respectively. The greater the VRi than the VBj, the smaller the threat of Bj to Ri.


### 2.1.4. Ability Threat Function

In this paper, we embrace the air combat capability formula in [24] given as follows:(4)C=ln⁡B+ln⁡∑A1+1+ln⁡∑A2ε1ε2ε3ε4where B is the maneuverability parameter of the fighter; A1 is the fire attack capability parameter; A2 is the radar detection capability parameter; ε1 is the pilot's control capability coefficient; ε2 is the fighter survivability coefficient; ε3 is the fighter range coefficient; and ε4 is the Electronic Counter Measures (ECM) capability coefficient. The ability threat function is defined as [22]:(5)Sij4=1CjCi≥1.50.751.5≥CjCi>10.5CjCi=10.251>CjCi≥0.30CjCi<0.3For example, supposing the air combat capability ofR1 is 17.9, if the capabilities of B1, B2, and B3 are 15.8, 18.8, and 17.9, the ability threat degrees of blue to red are 0.25, 0.75, and 0.5, respectively.As a combination of the above threat functions, the threat degree ofBj to Ri is(6)Sij=ω1Sij1+ω2Sij2+ω3Sij3+ω4Sij4where0<ωi<1  (i = 1,2,3,4) is the weight, Σωi = 1.The overall threat degree ofBj to all the red fighters is(7)Sj=∑iSij


## 2.1.1. Angle Threat Function

The angle threat function [22] is given as follows:(1)Sij1=εij-εji180°where Sij1 is the angle threat of Bj to Ri, with 0°≤εji,εij,≤180° and -1≤ Sij1≤1. In particular, when the Ri off-axis angle εij is 180° and the Bj off-axis angle εji is 0°, that is, when Bj chases Ri from behind, the angle threat of Bj to Ri is 1.


## 2.1.2. Distance Threat Function

The distance threat function [23] is given as follows:(2)Sij2=1Dij≤Tab1-Dij-TabLrb-TabTab<Dij≤Lrb0Lrb<Dijwhere Tabrepresents the missile range of the blue fighter and Lrb is the maximum detection range of the blue radar. When the red fighter is within the blue attack range, the distance threat of Bj to Ri takes the maximum value of 1; when the red fighter is out of the detection range of blue, the distance threat takes the minimum value of 0.


## 2.1.3. Speed Threat Function

The speed threat function [22] is given as follows:(3)Sij3=1VRi≤0.5·VBj1.5-VRiVBj0.5·VBj<VRi≤1.4·VBj01.4·VBj<VRiwhere VRi and VBj are the speeds of the red and blue fighters, respectively. The greater the VRi than the VBj, the smaller the threat of Bj to Ri.


## 2.1.4. Ability Threat Function

In this paper, we embrace the air combat capability formula in [24] given as follows:(4)C=ln⁡B+ln⁡∑A1+1+ln⁡∑A2ε1ε2ε3ε4where B is the maneuverability parameter of the fighter; A1 is the fire attack capability parameter; A2 is the radar detection capability parameter; ε1 is the pilot's control capability coefficient; ε2 is the fighter survivability coefficient; ε3 is the fighter range coefficient; and ε4 is the Electronic Counter Measures (ECM) capability coefficient. The ability threat function is defined as [22]:(5)Sij4=1CjCi≥1.50.751.5≥CjCi>10.5CjCi=10.251>CjCi≥0.30CjCi<0.3For example, supposing the air combat capability ofR1 is 17.9, if the capabilities of B1, B2, and B3 are 15.8, 18.8, and 17.9, the ability threat degrees of blue to red are 0.25, 0.75, and 0.5, respectively.As a combination of the above threat functions, the threat degree ofBj to Ri is(6)Sij=ω1Sij1+ω2Sij2+ω3Sij3+ω4Sij4where0<ωi<1  (i = 1,2,3,4) is the weight, Σωi = 1.The overall threat degree ofBj to all the red fighters is(7)Sj=∑iSij


## 2.2. Weapon-Target Assignment

LetRi carry Mi missiles and Ej the number of missiles assigned to Bj:(8)Ej=1Sj≤121<Sj≤33Sj>3Four WTA models are presented in this subsection.Model 1.
(9) m a x  ∑ i ∑ j x i j S i j s . t  ∑ j x i j ≤ M i ,  i = 1,2 , … N ∑ i x i j ≤ E j ,  i = 1,2 , … K k = ∑ i ∑ j x i j = min ⁡ ∑ i M i , ∑ j E jwhere xij, an integer from 0 to 3, represents the missile number of Ri assigned to attack Bj; Sij is the threat of Bj to Ri; k is the number of red missiles actually used to attack the target; N is the number of red fighters; and K is the number of blue fighters. The first constraint indicates that the number of missiles launched by each red fighter cannot exceed its carrying capacity; the second constraint implies that the number of missiles assigned to each blue fighter cannot exceed the value obtained in formula (8); the third constraint indicates that the number of fired missiles takes the smaller value between the total number of missiles carried by the red side and the total number of missiles assigned to the blue side.Model 2.
(10) m a x ∑ j S j 1 - ∏ i 1 - p i j x i j
Model4 was first proposed in reference [25]. The constraints are the same as in Model 1, and pij is the probability that Ri hits Bj, which is calculated with the two-step adjudication model [26] in this paper. Compared to Model 1, the objective here takes the hit probability of the blue fighters into account; the larger the pij, the greater the threat.Model 3.
(11) max ⁡ ∑ i ∑ j S i j + S j i × p i j × x i j
The constraints are the same as in Model1, and Sji is the threat of Ri to Bj. Model 3 considers the blue side threat while also considering the advantages of the red side for the blue side. The objective signifies that red missiles will be preferentially assigned to the blue fighters which pose high threat to red, get high threat from red, and have high hit probability.Model 4.
(12) max ⁡ ∑ j S j P j ∑ i x i j + M ∑ j min ⁡ 0 , P j - P d j (13) P j = 1 - ∏ i 1 - p i j x i j
Model4 was first proposed in reference [27], where Pj is the overall hit probability of the red fighter on Bj; Pdj is the threshold of hit probability of Bj; and M is the penalty factor. Model 4 calculates the average hit probability for each missile, maximizing the threat of each missile resolution. At the same time, the minimum hit probability requirement is set for each blue fighter, and if Pj is lower than Pdj, it will be punished. The objective here is to maximize the average hit probability, considering the blue fighter’s hit probability threshold.


## 3. Improvements of the Ant Colony Optimization for Weapon-Target Assignment

To solve the problem of WTA, some researches tried to make improvements on traditional ant colony optimization with respect to the rules of path selection, pheromone update, and pheromone concentration interval. We consider basic ant colony algorithm (ACO), ant system (AS), elitist-rank ACO algorithm (ASrank), and max-min ant system (MMAS) and improved the elite strategy in ASrank, named AScomp. We add garbage ants to the algorithm of ASrank; when the pheromone is updated, the elite ants are rewarded and the garbage ants are punished, as seen in the following sections (Sections 3.1–3.3). Section 3.4 integrates these improvements in the application of air combat WTA and proposes a WTA algorithm based on the integration.


### 3.1. Path Selection Rule

( 1 ) In the basic ACO, the path is selected according to the probability Pij, calculated as follows:(14)pis=τi,sα·ηi,sβ∑s∈Jkiτi,sα·ηi,sβ,s∈allowedk0,s∉allowedk( 2 ) To avoid search stagnation, the path selection in AS [28–30] uses a combination of deterministic and random selections and dynamically adjusts the state transition probability in searching. The specific path selection rules are as follows:(15)j=arg⁡maxs∈Jki⁡τi,sα·ηi,sβ,q≤q0Jq>q0where i represents the ith red fighter and j, s, J are indices of blue fighters, respectively; J is determined by Pij, the same as the ACO,whereτ(i, s) is the pheromone concentration on the path between the current missile i and the assigned position s.η(i, s) is a heuristic function; the greater the threat of the opponent, the greater the probability of launching missiles.α is the information heuristic factor used to measure the influence of pheromones on the path.β is the expected heuristic factor used to measure the influence of the threat degree.allowed k represents the set of available targets. As the search progresses,allowedk is getting smaller.J k (i) is the set of nodes that the kth ant needs to access after node i has been accessed.q is a uniform distributed random number in [0,1], and q0 is a constant, 0≤q0≤1.p i s is the probability of selecting J.Whenq is less than or is equal to q0, the path with the highest pheromone concentration is selected. When q is greater than q0, the selection probability of each node is obtained by formula (14).


### 3.2. Pheromone Update Rule

( 1 ) In the ASO, the pheromone is updated after all ants have iterated, and the concentration of all passing paths is updated, which is called partial update:(16)τi,j=1-ρ·τi,j+ρ·∑k=1mΔτi,jk(17)Δτi,jk=QLkifantkpassespathi,j0otherswhere △τ(i, j) is the pheromone concentration increment on the optimal path (i, j);ρis pheromone volatilization rate,ρ∈ (0,1); Q is a constant used to regulate the pheromone concentration; and Lk is the path length of the ant k.( 2 ) In AS, both partial update and global update are performed; global update is the same as formula (16). Only the pheromone on the optimal path is updated in the global update [31, 32] to enhance the effect of positive feedback. The update rules are as follows:(18)τi,j=1-ρ·τi,j+ρ·Δτi,j∗(19)Δτi,j∗=1Lgbifi,j∗istheoptimalpath0otherswhere △τi,j∗ is the pheromone concentration increment on the optimal path (i, j) and Lgb is the shortest path length of the current cycle.( 3 ) Traditional algorithms may lead to the elimination of the most adapted ants, and ASrank is proposed to preserve them [33]. ASrank can find better solutions and find these solutions for a shorter period. Set σ elite ants; the global update rules for ASrank are as follows:(20)τi,j=1-ρ·τi,j+ρ·∑l=1σσ-l+1·Δτi,jk(21)Δτi,jk=1Lgb+Lkifantkiseliteant0others( 4 ) Next, the ASrank is improved to make it easier to find the optimal path. Garbage ants are added to the algorithm ASrank. When the pheromone is updated, the elite ants are rewarded while punishing the garbage ants. To prevent the penalty from causing the pheromone concentration to be too low, the penalty factor ω is set. The improved AS formula is as follows (named AScomp):(22)τi,j=1-ρ·τi,j+ρ·∑l=1σσ-l+1·Δτi,jk-∑l=1γγ-l+1·Δτi,js(23)Δτi,js=ωLgb-Lsifantsisgarbageant0otherswhere △τi,jk is the same as formula (21) and γ is the number of ant garbage ants.


### 3.3. Pheromone Concentration Interval Rule

(1) In MMAS, to avoid local stagnation in searching, the pheromone concentration interval is set as follows [34]:(24)τmax=1λ11-ρ·Lgb(25)τmin=τmaxtλ2where λ1, λ2 are two constants used to regulate the pheromone concentration.(2) Another way to adjust pheromone is to smooth the concentration. By increasing the probability of selecting low pheromone paths, the ability to explore new solutions can be improved:(26)τ∗i,j=τi,j+δτmax-τi,jwhere τ∗(i, j) is the amount of pheromone after smoothing; δ is a constant used to regulate the concentration; and τmax is the same as formula (24).


### 3.4. Algorithm Performance Comparison

Considering the rules of path selection, pheromone update, and pheromone concentration interval, 24 sets of algorithms are obtained, as shown in Table1. These algorithms are brought into the WTA problem for performance comparison analysis. The average optimal solution and convergence of the 100 trials are counted.Table 1
Algorithm set.
Path Selection Rule Pheromone Update Rule Pheromone Interval P ij of basic ACO Partial update of ACO None  Random selections of AS Partial and global update of AS MMAS’s interval  Elite ant strategy ofASRank Smooth the concentration  Punish rule ofAScompAlgorithm1 selects Pij of basic ACO, partial update of ACO, and none. Algorithm 2 selects Pij of basic ACO, partial update of ACO, and MMAS’s interval, and other algorithms are selected according to the same method. The test results are as shown in Table 2.Table 2
Test results of Algorithm set.
Algorithm 1 2 3 4 5 6 convergence 21.36 19.60 19.03 14.43 4.43 16 optimal 15.27 15.45 15.43 19.77 15.96 20.05  Algorithm 7 8 9 10 11 12  convergence 23 4.30 18.53 12.90 3.03 12.56 optimal 19.68 14.35 19.983 20.10 13.17 20.03  Algorithm 13 14 15 16 17 18  convergence 5.66 6.10 6.23 17.36 16.86 15.76 optimal 14.23 14.22 14.21 18.12 19.52 17.98  Algorithm 19 20 21 22 23 24  convergence 23.10 15 22.33 16.23 9.76 7.13 optimal 18.61 18.40 18.49 19.75 17.99 20.15Algorithm 1:WIACO (WTA algorithm based on improved ant colony optimization).
Call the threat model; Initialize parameters: 1.While  NC- 1<NCmax (The largest iteration number) 2.For  i=1:m(The number of ants) 3.Put m ants on red missiles randomly; 4.End 5.For  k=1:count (The total number of ants’ movements) 6.Forj=1:m 7.Fori=1:Kb (Number of attackable blue fighters) 8.Generate a random number q; 9.Select path with pseudorandom probability using formula ((14), (15)); 10.Add the target to selected set; 11.Assignable target number minus 1; 12.The number of remaining missiles in red fighters minus 1; 13.allowedk minus the assigned target; 14.End 15.End 16.For  j=1:m 17.For  i=1:Nrl (The remaining amount of red missiles) 18.Generate a random number q; 19.Select path with pseudorandom probability using formula ((14), (15)); 20.Add the target to selected set; 21.allowedk minus the assigned target; 22.End 23.End 24.Determine the path of the maximum threat in this cycle using WTA model; 25.Global update of pheromone using formula ((22), (23)); 26.Update the pheromone interval on the paths using formula (26); 27.NC=NC+1; 28.End 29.End 30.Get the optimal WTA solution.Algorithm 2:RWSA (real-time WTA simulation algorithm).
1.For  time=1:T 2.Call the WIACO algorithm; 3.For  i=1:Nr 4.For  j=1:Nb 5.If  Dij<Ti (Dij is the distance between two sides, Ti is the range of the red fighters) Call the two-step adjudication model to get the probability of damage Pr, Pb; 6.End 7.End 8.End 9. 11.Fork_r=1:Nr 12.Get random number Pkr; 13.IfPkr<Pr 14.Red fighter k_ris shot down; 15.End 16.End 17.For  k_b=1:Nb 18.Get random number Pkb; 19.If  Pkb<Pb 20.Blue fighter k_bis shot down; 21.End 22.End 23Update the air combat situation; 24.If one termination condition is met, end the loop; 25.time=time+1; 26End 37Output simulation results.As can be seen from Table2, the average optimal solution of algorithm 24 is the largest, which is 20.15. So, it can resolve the largest threat, and the average convergence is 7.13. Algorithm 11 has the worst effect, and the optimal solution is only 13.17, which is the easiest to fall the local optimal solution.Algorithm 24 selects rules of random selections of AS, punishes the rule ofAScomp, smooths the concentration, and achieves the best results. Algorithm 24 is used as the research algorithm of this paper.


### 3.5. WIACO Algorithm

When the ant colony optimization is used to solve the WTA problem, the assignment process needs to be modeled with an ant colony network. For example, in Figure2, each red missile is represented by a small node and each red and blue fighter is represented by a big node; the two sides both dispatch two fighters; red fighter 1 carries four missiles and red fighter 2 carries three; blue fighter 1 is assigned two missiles and blue fighter 2 is assigned one missile.Figure 2
Ant colony network of air combat.The ants follow the path from the red nodes to the blue nodes, and then, according to the same strategy, take a virtual path back to the red nodes until the assignment is completed.The number of ants in the population is set to [35]:(27)m=Nrm+∑Ejwhere Nrm is the total number of missiles carried by red fighters and Ej is the number of missiles assigned to Bj. In the beginning of the iteration, m ants are placed randomly on the red missiles, and the initial pheromone concentration on each path is set to 1.The ants move according to the following rules:Rule  1. An ant can only move to a blue fighter whose missile assignment is insufficient; red fighters launch the remaining number of missiles at most.Rule  2. Each ant can only reach one node at a time; that is, each missile can only attack a single target.Rule  3. The ants do not interfere with each other. Ants return to red fighter nodes with the same pseudorandom probability, and the targets are the red fighters who still have missiles.Rule  4. All ants need to update the pheromone by the end of a cycle and generate new pheromones only on the optimal path; pheromone is partially volatile.Pseudocode of the WIACO algorithm is shown as in algorithm1.


## 3.1. Path Selection Rule

( 1 ) In the basic ACO, the path is selected according to the probability Pij, calculated as follows:(14)pis=τi,sα·ηi,sβ∑s∈Jkiτi,sα·ηi,sβ,s∈allowedk0,s∉allowedk( 2 ) To avoid search stagnation, the path selection in AS [28–30] uses a combination of deterministic and random selections and dynamically adjusts the state transition probability in searching. The specific path selection rules are as follows:(15)j=arg⁡maxs∈Jki⁡τi,sα·ηi,sβ,q≤q0Jq>q0where i represents the ith red fighter and j, s, J are indices of blue fighters, respectively; J is determined by Pij, the same as the ACO,whereτ(i, s) is the pheromone concentration on the path between the current missile i and the assigned position s.η(i, s) is a heuristic function; the greater the threat of the opponent, the greater the probability of launching missiles.α is the information heuristic factor used to measure the influence of pheromones on the path.β is the expected heuristic factor used to measure the influence of the threat degree.allowed k represents the set of available targets. As the search progresses,allowedk is getting smaller.J k (i) is the set of nodes that the kth ant needs to access after node i has been accessed.q is a uniform distributed random number in [0,1], and q0 is a constant, 0≤q0≤1.p i s is the probability of selecting J.Whenq is less than or is equal to q0, the path with the highest pheromone concentration is selected. When q is greater than q0, the selection probability of each node is obtained by formula (14).


## 3.2. Pheromone Update Rule

( 1 ) In the ASO, the pheromone is updated after all ants have iterated, and the concentration of all passing paths is updated, which is called partial update:(16)τi,j=1-ρ·τi,j+ρ·∑k=1mΔτi,jk(17)Δτi,jk=QLkifantkpassespathi,j0otherswhere △τ(i, j) is the pheromone concentration increment on the optimal path (i, j);ρis pheromone volatilization rate,ρ∈ (0,1); Q is a constant used to regulate the pheromone concentration; and Lk is the path length of the ant k.( 2 ) In AS, both partial update and global update are performed; global update is the same as formula (16). Only the pheromone on the optimal path is updated in the global update [31, 32] to enhance the effect of positive feedback. The update rules are as follows:(18)τi,j=1-ρ·τi,j+ρ·Δτi,j∗(19)Δτi,j∗=1Lgbifi,j∗istheoptimalpath0otherswhere △τi,j∗ is the pheromone concentration increment on the optimal path (i, j) and Lgb is the shortest path length of the current cycle.( 3 ) Traditional algorithms may lead to the elimination of the most adapted ants, and ASrank is proposed to preserve them [33]. ASrank can find better solutions and find these solutions for a shorter period. Set σ elite ants; the global update rules for ASrank are as follows:(20)τi,j=1-ρ·τi,j+ρ·∑l=1σσ-l+1·Δτi,jk(21)Δτi,jk=1Lgb+Lkifantkiseliteant0others( 4 ) Next, the ASrank is improved to make it easier to find the optimal path. Garbage ants are added to the algorithm ASrank. When the pheromone is updated, the elite ants are rewarded while punishing the garbage ants. To prevent the penalty from causing the pheromone concentration to be too low, the penalty factor ω is set. The improved AS formula is as follows (named AScomp):(22)τi,j=1-ρ·τi,j+ρ·∑l=1σσ-l+1·Δτi,jk-∑l=1γγ-l+1·Δτi,js(23)Δτi,js=ωLgb-Lsifantsisgarbageant0otherswhere △τi,jk is the same as formula (21) and γ is the number of ant garbage ants.


## 3.3. Pheromone Concentration Interval Rule

(1) In MMAS, to avoid local stagnation in searching, the pheromone concentration interval is set as follows [34]:(24)τmax=1λ11-ρ·Lgb(25)τmin=τmaxtλ2where λ1, λ2 are two constants used to regulate the pheromone concentration.(2) Another way to adjust pheromone is to smooth the concentration. By increasing the probability of selecting low pheromone paths, the ability to explore new solutions can be improved:(26)τ∗i,j=τi,j+δτmax-τi,jwhere τ∗(i, j) is the amount of pheromone after smoothing; δ is a constant used to regulate the concentration; and τmax is the same as formula (24).


## 3.4. Algorithm Performance Comparison

Considering the rules of path selection, pheromone update, and pheromone concentration interval, 24 sets of algorithms are obtained, as shown in Table1. These algorithms are brought into the WTA problem for performance comparison analysis. The average optimal solution and convergence of the 100 trials are counted.Table 1
Algorithm set.
Path Selection Rule Pheromone Update Rule Pheromone Interval P ij of basic ACO Partial update of ACO None  Random selections of AS Partial and global update of AS MMAS’s interval  Elite ant strategy ofASRank Smooth the concentration  Punish rule ofAScompAlgorithm1 selects Pij of basic ACO, partial update of ACO, and none. Algorithm 2 selects Pij of basic ACO, partial update of ACO, and MMAS’s interval, and other algorithms are selected according to the same method. The test results are as shown in Table 2.Table 2
Test results of Algorithm set.
Algorithm 1 2 3 4 5 6 convergence 21.36 19.60 19.03 14.43 4.43 16 optimal 15.27 15.45 15.43 19.77 15.96 20.05  Algorithm 7 8 9 10 11 12  convergence 23 4.30 18.53 12.90 3.03 12.56 optimal 19.68 14.35 19.983 20.10 13.17 20.03  Algorithm 13 14 15 16 17 18  convergence 5.66 6.10 6.23 17.36 16.86 15.76 optimal 14.23 14.22 14.21 18.12 19.52 17.98  Algorithm 19 20 21 22 23 24  convergence 23.10 15 22.33 16.23 9.76 7.13 optimal 18.61 18.40 18.49 19.75 17.99 20.15Algorithm 1:WIACO (WTA algorithm based on improved ant colony optimization).
Call the threat model; Initialize parameters: 1.While  NC- 1<NCmax (The largest iteration number) 2.For  i=1:m(The number of ants) 3.Put m ants on red missiles randomly; 4.End 5.For  k=1:count (The total number of ants’ movements) 6.Forj=1:m 7.Fori=1:Kb (Number of attackable blue fighters) 8.Generate a random number q; 9.Select path with pseudorandom probability using formula ((14), (15)); 10.Add the target to selected set; 11.Assignable target number minus 1; 12.The number of remaining missiles in red fighters minus 1; 13.allowedk minus the assigned target; 14.End 15.End 16.For  j=1:m 17.For  i=1:Nrl (The remaining amount of red missiles) 18.Generate a random number q; 19.Select path with pseudorandom probability using formula ((14), (15)); 20.Add the target to selected set; 21.allowedk minus the assigned target; 22.End 23.End 24.Determine the path of the maximum threat in this cycle using WTA model; 25.Global update of pheromone using formula ((22), (23)); 26.Update the pheromone interval on the paths using formula (26); 27.NC=NC+1; 28.End 29.End 30.Get the optimal WTA solution.Algorithm 2:RWSA (real-time WTA simulation algorithm).
1.For  time=1:T 2.Call the WIACO algorithm; 3.For  i=1:Nr 4.For  j=1:Nb 5.If  Dij<Ti (Dij is the distance between two sides, Ti is the range of the red fighters) Call the two-step adjudication model to get the probability of damage Pr, Pb; 6.End 7.End 8.End 9. 11.Fork_r=1:Nr 12.Get random number Pkr; 13.IfPkr<Pr 14.Red fighter k_ris shot down; 15.End 16.End 17.For  k_b=1:Nb 18.Get random number Pkb; 19.If  Pkb<Pb 20.Blue fighter k_bis shot down; 21.End 22.End 23Update the air combat situation; 24.If one termination condition is met, end the loop; 25.time=time+1; 26End 37Output simulation results.As can be seen from Table2, the average optimal solution of algorithm 24 is the largest, which is 20.15. So, it can resolve the largest threat, and the average convergence is 7.13. Algorithm 11 has the worst effect, and the optimal solution is only 13.17, which is the easiest to fall the local optimal solution.Algorithm 24 selects rules of random selections of AS, punishes the rule ofAScomp, smooths the concentration, and achieves the best results. Algorithm 24 is used as the research algorithm of this paper.


## 3.5. WIACO Algorithm

When the ant colony optimization is used to solve the WTA problem, the assignment process needs to be modeled with an ant colony network. For example, in Figure2, each red missile is represented by a small node and each red and blue fighter is represented by a big node; the two sides both dispatch two fighters; red fighter 1 carries four missiles and red fighter 2 carries three; blue fighter 1 is assigned two missiles and blue fighter 2 is assigned one missile.Figure 2
Ant colony network of air combat.The ants follow the path from the red nodes to the blue nodes, and then, according to the same strategy, take a virtual path back to the red nodes until the assignment is completed.The number of ants in the population is set to [35]:(27)m=Nrm+∑Ejwhere Nrm is the total number of missiles carried by red fighters and Ej is the number of missiles assigned to Bj. In the beginning of the iteration, m ants are placed randomly on the red missiles, and the initial pheromone concentration on each path is set to 1.The ants move according to the following rules:Rule  1. An ant can only move to a blue fighter whose missile assignment is insufficient; red fighters launch the remaining number of missiles at most.Rule  2. Each ant can only reach one node at a time; that is, each missile can only attack a single target.Rule  3. The ants do not interfere with each other. Ants return to red fighter nodes with the same pseudorandom probability, and the targets are the red fighters who still have missiles.Rule  4. All ants need to update the pheromone by the end of a cycle and generate new pheromones only on the optimal path; pheromone is partially volatile.Pseudocode of the WIACO algorithm is shown as in algorithm1.


## 4. Examples of Comparative Analysis

### 4.1. Model Parameter Analysis

The parameters of the WTA model called by WIACO algorithm are discussed with a control variable experiment in this subsection, including theα, β, ρ, Q, and q0. Set δ to 0.001 and set ω to 0.1. The number of iterations is 200, and the means of the results are taken.Taking Model3 in Section 2.2 for example, the parameter variation curves of Model 3 are shown in Figure 3. The horizontal axis represents the parameter; the vertical axis represents the maximum defused threat. As we can see in Figure 3, when α = 1.5, β = 3, ρ = 0.7,Q = 0.4, and q0 = 0.7, the curves reach their respective maximum defused threat. Therefore, Model 3 takes the final parameters as α = 1.5, β = 3, ρ = 0.7,Q = 0.4, and q0 = 0.7. Similarly, the final parameters of the other three models are set as in Table 3.Table 3
Model parameter values.
Parameter Model1 Model2 Model3 Model4 α 1.5 1.5 1.5 1.5 β 3 2 3 2 ρ 1 0.4 0.7 0.7 0.6 ρ 2 0.7 0.5 0.4 0.6 q 0 0.6 0.7 0.7 0.5Figure 3
Parameter variation curves of Model3.The obtained parameters are substituted into the models for the experimental analysis, and the results of one of these experiments are as in Figure4. The horizontal axis represents the iterations; the vertical axis represents the maximum defused threat.Figure 4
Model convergence curves.
(a) (b) (c) (d)Repeat the experiment 100 times and take the average for analysis. The results are presented in Table4. As seen in Table 4, the variances of the four models are not large and the iterations converge at the 20th iteration, earlier or later, and that is acceptable.Table 4
Statistical analysis of the model results.
Max Min Mean Var Model1 17.2896 17.2831 17.2859 0.0021 Model2 12.2186 11.8368 12.1267 0.0109 Model3 22.1207 18.6858 20.4573 0.5852 Model4 6.08764 5.8862 6.0533 0.0024


### 4.2. Comparison with Traditional Algorithm

It is assumed that the red side adopts WIACO algorithm with WTA Model3.According to Table5 and the air combat capability in formula (4), the air combat capability of each red fighter is Cr = 20.79 and that of each blue fighter is Cb = 21.97.Table 5
Red and blue parameter sets.
Parameter Red Blue Number of fighters 8 8 Number of missiles 4 4 Flight speed(m/s) 340 340 Missile range(km) 80 90 Radar detection distance(km) 120 120 Maneuvering parameter 25.5 25.65 Fire parameter 2761 2832 Detection capability parameter 1514 1553 Manipulation efficiency factor 0.9 0.95 Viability coefficient 0.915 0.995 Voyage coefficient 1.14 1.14 Electronic countermeasure coefficient 1.05 1.1Based on reference [25], assuming the fighter performance has the greatest impact on air combat, we let ω1 = 0.2, ω2 = 0.2, ω3 = 0.2, and ω4 = 0.4 and take the value of λ1=3, λ2=40.The traditional ant colony optimization and the WIACO proposed in this paper are both simulated 100 times, and the best convergence results of the two algorithms are obtained.The convergence of the traditional algorithm is shown in Figure5(a); thereinto, the results fluctuate. After the algorithm goes through the maximum defused threat degree of 26.06, it falls into the local optimum and gets a stable defused threat degree of 25.9. In light of this, the traditional algorithm cannot jump out of local optimum, and it does not produce the global optimal WTA solution.Figure 5
Traditional and improved algorithm convergence curves.
(a) (b)The WIACO convergence is shown in Figure5(b). Figure 5(b) shows that at the 3rd iteration the improved algorithm finds a locally optimal assignment with the defused threat degree of 26.19; then it jumps out of the local optimum quickly and converges to a stable optimal assignment with a maximum defused threat degree of 26.29 at the 12th iteration. Table 6 shows the WTA solution in detail. For example, as we can see on the first row of Table 6, the red fighter 1 launches two missiles to attack the blue fighter 1, one missile to attack blue fighter 2, and one missile to attack the blue fighter 6, and the missiles all run out; on the seventh row, none of the red fighter 7’s missiles are assigned.Table 6
Improved algorithm assignment.
Red fighter Assignment (missiles) Remaining missiles Red1 Blue1(2); Blue2 (1); Blue6 (1) 0 Red2 Blue3(3); Blue6 (1) 0 Red3 Blue5(2); Blue4 (1) 1 Red4 Blue5(1); Blue4 (2) 1 Red5 Blue4(1); Blue7(2) 1 Red6 Blue4(1) 3 Red7 — 4 Red8 Blue6(1); Blue8 (3) 0The above analysis indicates the advantages of the WIACO algorithm, which can provide better solution than traditional algorithm for the WTA. Comparatively speaking, it can be considered that the traditional algorithm takes longer time to convergence, and it is harder to jump out of local optimum.


## 4.1. Model Parameter Analysis

The parameters of the WTA model called by WIACO algorithm are discussed with a control variable experiment in this subsection, including theα, β, ρ, Q, and q0. Set δ to 0.001 and set ω to 0.1. The number of iterations is 200, and the means of the results are taken.Taking Model3 in Section 2.2 for example, the parameter variation curves of Model 3 are shown in Figure 3. The horizontal axis represents the parameter; the vertical axis represents the maximum defused threat. As we can see in Figure 3, when α = 1.5, β = 3, ρ = 0.7,Q = 0.4, and q0 = 0.7, the curves reach their respective maximum defused threat. Therefore, Model 3 takes the final parameters as α = 1.5, β = 3, ρ = 0.7,Q = 0.4, and q0 = 0.7. Similarly, the final parameters of the other three models are set as in Table 3.Table 3
Model parameter values.
Parameter Model1 Model2 Model3 Model4 α 1.5 1.5 1.5 1.5 β 3 2 3 2 ρ 1 0.4 0.7 0.7 0.6 ρ 2 0.7 0.5 0.4 0.6 q 0 0.6 0.7 0.7 0.5Figure 3
Parameter variation curves of Model3.The obtained parameters are substituted into the models for the experimental analysis, and the results of one of these experiments are as in Figure4. The horizontal axis represents the iterations; the vertical axis represents the maximum defused threat.Figure 4
Model convergence curves.
(a) (b) (c) (d)Repeat the experiment 100 times and take the average for analysis. The results are presented in Table4. As seen in Table 4, the variances of the four models are not large and the iterations converge at the 20th iteration, earlier or later, and that is acceptable.Table 4
Statistical analysis of the model results.
Max Min Mean Var Model1 17.2896 17.2831 17.2859 0.0021 Model2 12.2186 11.8368 12.1267 0.0109 Model3 22.1207 18.6858 20.4573 0.5852 Model4 6.08764 5.8862 6.0533 0.0024


## 4.2. Comparison with Traditional Algorithm

It is assumed that the red side adopts WIACO algorithm with WTA Model3.According to Table5 and the air combat capability in formula (4), the air combat capability of each red fighter is Cr = 20.79 and that of each blue fighter is Cb = 21.97.Table 5
Red and blue parameter sets.
Parameter Red Blue Number of fighters 8 8 Number of missiles 4 4 Flight speed(m/s) 340 340 Missile range(km) 80 90 Radar detection distance(km) 120 120 Maneuvering parameter 25.5 25.65 Fire parameter 2761 2832 Detection capability parameter 1514 1553 Manipulation efficiency factor 0.9 0.95 Viability coefficient 0.915 0.995 Voyage coefficient 1.14 1.14 Electronic countermeasure coefficient 1.05 1.1Based on reference [25], assuming the fighter performance has the greatest impact on air combat, we let ω1 = 0.2, ω2 = 0.2, ω3 = 0.2, and ω4 = 0.4 and take the value of λ1=3, λ2=40.The traditional ant colony optimization and the WIACO proposed in this paper are both simulated 100 times, and the best convergence results of the two algorithms are obtained.The convergence of the traditional algorithm is shown in Figure5(a); thereinto, the results fluctuate. After the algorithm goes through the maximum defused threat degree of 26.06, it falls into the local optimum and gets a stable defused threat degree of 25.9. In light of this, the traditional algorithm cannot jump out of local optimum, and it does not produce the global optimal WTA solution.Figure 5
Traditional and improved algorithm convergence curves.
(a) (b)The WIACO convergence is shown in Figure5(b). Figure 5(b) shows that at the 3rd iteration the improved algorithm finds a locally optimal assignment with the defused threat degree of 26.19; then it jumps out of the local optimum quickly and converges to a stable optimal assignment with a maximum defused threat degree of 26.29 at the 12th iteration. Table 6 shows the WTA solution in detail. For example, as we can see on the first row of Table 6, the red fighter 1 launches two missiles to attack the blue fighter 1, one missile to attack blue fighter 2, and one missile to attack the blue fighter 6, and the missiles all run out; on the seventh row, none of the red fighter 7’s missiles are assigned.Table 6
Improved algorithm assignment.
Red fighter Assignment (missiles) Remaining missiles Red1 Blue1(2); Blue2 (1); Blue6 (1) 0 Red2 Blue3(3); Blue6 (1) 0 Red3 Blue5(2); Blue4 (1) 1 Red4 Blue5(1); Blue4 (2) 1 Red5 Blue4(1); Blue7(2) 1 Red6 Blue4(1) 3 Red7 — 4 Red8 Blue6(1); Blue8 (3) 0The above analysis indicates the advantages of the WIACO algorithm, which can provide better solution than traditional algorithm for the WTA. Comparatively speaking, it can be considered that the traditional algorithm takes longer time to convergence, and it is harder to jump out of local optimum.


## 5. Simulation Analysis of WTA in Air Combat

During air combat, the combat situation is constantly changing, and the WTA needs to be continuously updated. This section simulates the complete air combat process, identifies the final effectiveness, and makes relevant experimental analysis.


### 5.1. Simulation Strategy

Assuming the red side performs WTA using the WIACO algorithm, the air combat simulation in this section is based on the following rules:Rule  1. Due to the early warning aircraft, air combat situation is accessed in real time. During the simulation, both sides constantly approach each other. To reflect the randomness and flexibility of the formation and the air combat situation, the fighters move with a random and constant speed under their respective limits. Assuming that both sides are constantly adjusting their direction of flight, the off-axis angles of two side fighters are decreasing [36].Rule  2. Once the blue fighters move into range, the red fighters launch their missiles.Rule  3. The data are put into the two-step adjudication model [26] to calculate the effectiveness.Rule  4. The termination conditions of the simulation are as follows: (1) one side runs out of missiles; (2) all fighters on either side are shot down; and (3) the air combat move beyond the horizon (within 20 km).The simulation steps are as follows:Step 1.
Obtain the initial data of the red and blue sides and initialize the parameters.Step 2.
Call the WIACO algorithm.Step 3.
Determine whether or not the blue fighters are within range of red side. Once they are within range, launch the red missiles and call the two-step adjudication model to assess the effectiveness.Step 4.
Update the air combat situation.Step 5.
Repeat Steps2–4 and output the final effectiveness when one of the termination conditions is reached.In the simulations, the number of red fighters isNr and that of blue fighters is Nb, and the number of simulations is T. The pseudocode of the RWSA algorithm is given as shown in Algorithm 2.


### 5.2. Experiment 1

In experiment 1, both sides dispatch 8 fighters. The parameters used in the two-step adjudication model are shown in Table7.Table 7
Two-step adjudication model parameters.
Parameter Red Blue Command factor 0.6 0.7 Fighter vulnerability 0.45 0.4 Multitarget attack capability 4 4 Sorting ratios 1 1 Allow war to match 0.8 0.8 Fighter radar reflection cross section (m 2 ) 3.2 2 Probability of target discovery 0.7 0.7 Missile score 0.8 0.8 Airborne radar maximum discovery distance (km) 120 120 Airborne missile maximum effective range (km) 80 90Because of the randomness of fighter kill and air combat situation, each simulation result is different, which reflects the uncertainty of air combat. Table8 lists the results of one simulation as an example. As seen in Table 8, by the end of the simulation, 4 red fighters (3,6,7, and 8) are shot down, 4 red fighters (1,2,4, and 5) run out of missiles, 3 blue fighters (1,3, and 7) are shot down, and 2 blue fighters (2 and 4) run out of missiles.Table 8
Simulation results (example).
Red Blue clock 6 Fighters 6 and 8 shot down; fighters 4 and 5 out of missiles Fighters 1 and 7 shot down; Fighter 4 out of missiles  clock 8 Fighter 3 shot down; fighter 2 out of missiles Fighter 3 shot down  clock 15 Fighter 7 shot down; 1 out of missiles Fighter 2 out of missiles  total Shot down: 4; Missiles exhausted: 4 Shot down: 3, Missiles exhausted: 2In this paper, we run a large number of simulations and statistical results with formula ((28)-(30)). Denote the amount of damage of red side in the ith simulation withLir and that of blue side withLib, and the damage rates of red and blue sides,Pr andPb, are calculated with formula ((28), (29)):(28)Pr=∑i=1TLirT×Nr(29)Pb=∑i=1TLibT×NbThe kill ratioW is calculated with formula (23):(30)W=∑i=1TLir∑i=1TLibThe smaller the kill ratio is, the greater the advantage of the red side is in air combat.First, we run 1000 times simulations using Model1, and the relationship between the kill ratio and the number of iterations is shown in Figure 6. Figure 6 shows that, as the number of iterations increases, the kill ratio tends to stabilize. The results of the air combat are shown in Figure 7 and Table 9. From Figure 7 we can see that 4 to 6 red fighters are killed mostly and 2 to 4 blue fighters are killed in most cases. The kill ratio in Table 9 is 1.6921; that is, the blue side is ascendant.Table 9
Model1 simulation results.
Red Blue Average damage 7.3945 4.1864 Damage rate 0.9243 0.5233 Kill ratio 1.7663Figure 6
Kill ratio and iterations curve.Figure 7
Damage distribution.And then additional experiments are conducted using Models2, 3, and 4 (M = 100, Pdj = 0.2), respectively. Among the results of the four simulations seen in Tables 9–12, the kill ratio of Model 3 is the smallest, so Model 3 gets the best effectiveness for WTA in air combat.Table 10
Model2 simulation results.
Red Blue Average damage 7.2821 4.3632 Damage rate 0.9103 0.5454 Kill ratio 1.6690Table 11
Model3 simulation results.
Red Blue Average damage 7.2492 4. 5326 Damage rate 0.9062 0.5665 Kill ratio 1.5993Table 12
Model4 simulation results (M=100, Pdj=0.2).
Red Blue Average damage 7.3215 4. 0854 Damage rate 0.9151 0.5106 Kill ratio 1.7921


### 5.3. Experiment 2

In the air combat simulations, there are two possible options of WTA timing. The first is to reassign at each time step of the simulation so that the WTA can be adjusted according to the real-time air combat situation. However, this increases the requirements of the pilots’ target locating capability. The second option is to reassign when any fighter is shot down or any fighter’s missiles are used up. This option allows the pilot to focus on the attacks on located targets but reduces the ability to adapt to the battlefield.Experiment 2 is carried out using the above two options, where parameters are the same as in experiment 1. The results are shown in Tables13 and 14. As seen in Tables 13 and 14, there is not much difference in the kill ratio between option 1 and option 2, but the red damage rate in option 2 is lower. Therefore, option 2 is more suitable for WTA regarding survivability and economics.Table 13
Option 1 simulation results.
Red Blue Average damage 4.5212 2.8727 Damage rate 0.5652 0.3591 Kill ratio 1.5738Table 14
Option 2 simulation results.
Red Blue Average damage 3.9122 2.4820 Damage rate 0.4890 0.3103 Kill ratio 1.5762


### 5.4. Experiment 3

Given the number of blue fighters, experiment 3 studies how many red fighters should be dispatched. The number of blue fighters is fixed at 8, while the number of red fighters increases from 3 to 18. The parameters are the same as in experiment 1. A total of 1000 simulations are performed to calculate the kill ratio, and the results are shown in Table15 and Figure 8.Table 15
Red fighter number and kill ratio.
Number of red fighters 3 4 5 6 Kill ratio 3.0726 2.6845 2.4587 1.9752  Number of red fighters 7 8 9 10 Kill ratio 1.6872 1.6154 1.5886 1.5754  Number of red fighters 11 12 13 14 Kill ratio 1.5743 1.5159 1.5245 1.5088  Number of red fighters 15 16 17 18 Kill ratio 1.5051 1.4912 1.4921 1.5011Figure 8
Red fighter number and kill ratio curve.In Table15 and Figure 8, as the number of red fighters’ increases, the kill ratio declines; however, the downward trend weakens after 12. This implies that the red side achieves a satisfying combat effectiveness if it dispatches 12 fighters to cope with 8 blue fighters at a kill ratio of about 1.5159.


## 5.1. Simulation Strategy

Assuming the red side performs WTA using the WIACO algorithm, the air combat simulation in this section is based on the following rules:Rule  1. Due to the early warning aircraft, air combat situation is accessed in real time. During the simulation, both sides constantly approach each other. To reflect the randomness and flexibility of the formation and the air combat situation, the fighters move with a random and constant speed under their respective limits. Assuming that both sides are constantly adjusting their direction of flight, the off-axis angles of two side fighters are decreasing [36].Rule  2. Once the blue fighters move into range, the red fighters launch their missiles.Rule  3. The data are put into the two-step adjudication model [26] to calculate the effectiveness.Rule  4. The termination conditions of the simulation are as follows: (1) one side runs out of missiles; (2) all fighters on either side are shot down; and (3) the air combat move beyond the horizon (within 20 km).The simulation steps are as follows:Step 1.
Obtain the initial data of the red and blue sides and initialize the parameters.Step 2.
Call the WIACO algorithm.Step 3.
Determine whether or not the blue fighters are within range of red side. Once they are within range, launch the red missiles and call the two-step adjudication model to assess the effectiveness.Step 4.
Update the air combat situation.Step 5.
Repeat Steps2–4 and output the final effectiveness when one of the termination conditions is reached.In the simulations, the number of red fighters isNr and that of blue fighters is Nb, and the number of simulations is T. The pseudocode of the RWSA algorithm is given as shown in Algorithm 2.


## 5.2. Experiment 1

In experiment 1, both sides dispatch 8 fighters. The parameters used in the two-step adjudication model are shown in Table7.Table 7
Two-step adjudication model parameters.
Parameter Red Blue Command factor 0.6 0.7 Fighter vulnerability 0.45 0.4 Multitarget attack capability 4 4 Sorting ratios 1 1 Allow war to match 0.8 0.8 Fighter radar reflection cross section (m 2 ) 3.2 2 Probability of target discovery 0.7 0.7 Missile score 0.8 0.8 Airborne radar maximum discovery distance (km) 120 120 Airborne missile maximum effective range (km) 80 90Because of the randomness of fighter kill and air combat situation, each simulation result is different, which reflects the uncertainty of air combat. Table8 lists the results of one simulation as an example. As seen in Table 8, by the end of the simulation, 4 red fighters (3,6,7, and 8) are shot down, 4 red fighters (1,2,4, and 5) run out of missiles, 3 blue fighters (1,3, and 7) are shot down, and 2 blue fighters (2 and 4) run out of missiles.Table 8
Simulation results (example).
Red Blue clock 6 Fighters 6 and 8 shot down; fighters 4 and 5 out of missiles Fighters 1 and 7 shot down; Fighter 4 out of missiles  clock 8 Fighter 3 shot down; fighter 2 out of missiles Fighter 3 shot down  clock 15 Fighter 7 shot down; 1 out of missiles Fighter 2 out of missiles  total Shot down: 4; Missiles exhausted: 4 Shot down: 3, Missiles exhausted: 2In this paper, we run a large number of simulations and statistical results with formula ((28)-(30)). Denote the amount of damage of red side in the ith simulation withLir and that of blue side withLib, and the damage rates of red and blue sides,Pr andPb, are calculated with formula ((28), (29)):(28)Pr=∑i=1TLirT×Nr(29)Pb=∑i=1TLibT×NbThe kill ratioW is calculated with formula (23):(30)W=∑i=1TLir∑i=1TLibThe smaller the kill ratio is, the greater the advantage of the red side is in air combat.First, we run 1000 times simulations using Model1, and the relationship between the kill ratio and the number of iterations is shown in Figure 6. Figure 6 shows that, as the number of iterations increases, the kill ratio tends to stabilize. The results of the air combat are shown in Figure 7 and Table 9. From Figure 7 we can see that 4 to 6 red fighters are killed mostly and 2 to 4 blue fighters are killed in most cases. The kill ratio in Table 9 is 1.6921; that is, the blue side is ascendant.Table 9
Model1 simulation results.
Red Blue Average damage 7.3945 4.1864 Damage rate 0.9243 0.5233 Kill ratio 1.7663Figure 6
Kill ratio and iterations curve.Figure 7
Damage distribution.And then additional experiments are conducted using Models2, 3, and 4 (M = 100, Pdj = 0.2), respectively. Among the results of the four simulations seen in Tables 9–12, the kill ratio of Model 3 is the smallest, so Model 3 gets the best effectiveness for WTA in air combat.Table 10
Model2 simulation results.
Red Blue Average damage 7.2821 4.3632 Damage rate 0.9103 0.5454 Kill ratio 1.6690Table 11
Model3 simulation results.
Red Blue Average damage 7.2492 4. 5326 Damage rate 0.9062 0.5665 Kill ratio 1.5993Table 12
Model4 simulation results (M=100, Pdj=0.2).
Red Blue Average damage 7.3215 4. 0854 Damage rate 0.9151 0.5106 Kill ratio 1.7921


## 5.3. Experiment 2

In the air combat simulations, there are two possible options of WTA timing. The first is to reassign at each time step of the simulation so that the WTA can be adjusted according to the real-time air combat situation. However, this increases the requirements of the pilots’ target locating capability. The second option is to reassign when any fighter is shot down or any fighter’s missiles are used up. This option allows the pilot to focus on the attacks on located targets but reduces the ability to adapt to the battlefield.Experiment 2 is carried out using the above two options, where parameters are the same as in experiment 1. The results are shown in Tables13 and 14. As seen in Tables 13 and 14, there is not much difference in the kill ratio between option 1 and option 2, but the red damage rate in option 2 is lower. Therefore, option 2 is more suitable for WTA regarding survivability and economics.Table 13
Option 1 simulation results.
Red Blue Average damage 4.5212 2.8727 Damage rate 0.5652 0.3591 Kill ratio 1.5738Table 14
Option 2 simulation results.
Red Blue Average damage 3.9122 2.4820 Damage rate 0.4890 0.3103 Kill ratio 1.5762


## 5.4. Experiment 3

Given the number of blue fighters, experiment 3 studies how many red fighters should be dispatched. The number of blue fighters is fixed at 8, while the number of red fighters increases from 3 to 18. The parameters are the same as in experiment 1. A total of 1000 simulations are performed to calculate the kill ratio, and the results are shown in Table15 and Figure 8.Table 15
Red fighter number and kill ratio.
Number of red fighters 3 4 5 6 Kill ratio 3.0726 2.6845 2.4587 1.9752  Number of red fighters 7 8 9 10 Kill ratio 1.6872 1.6154 1.5886 1.5754  Number of red fighters 11 12 13 14 Kill ratio 1.5743 1.5159 1.5245 1.5088  Number of red fighters 15 16 17 18 Kill ratio 1.5051 1.4912 1.4921 1.5011Figure 8
Red fighter number and kill ratio curve.In Table15 and Figure 8, as the number of red fighters’ increases, the kill ratio declines; however, the downward trend weakens after 12. This implies that the red side achieves a satisfying combat effectiveness if it dispatches 12 fighters to cope with 8 blue fighters at a kill ratio of about 1.5159.


## 6. Conclusions and Future Directions

In the air combat, the battlefield situation is complex and changeable, and WTA plays a decisive role. In Section2 of this paper, four threat functions are used to evaluate the threat. Meanwhile, four WTA models are set up with different objectives. Among them, Model 3 is proposed for the first time considering the threats of both sides and hit probabilities.In order to solve the WTA problem, WIACO algorithm is presented in Section3 with improvements of the traditional ant colony optimization in the aspects of path selection rule, pheromone updating rule, and pheromone concentration interval rule. The comparative experiment in Section 4 shows that WIACO algorithm which provides the optimal solution for WTA has the advantages of faster convergence and better avoidance from local optima.For the sake of the demonstration and exemplification of WIACO in air combat, four combat simulation rules and RWSA algorithm are designed in Section5. Based on this, this section carries out three simulation experiments. Through experiment 1, we can find that Model 3 gets the minimal kill ratio indicating the largest advantage of red side. Hence, we can conclude that Model 3 gets the best effectiveness for WTA in air combat. In experiment 2, we analyze different WTA timings with results showing that WTA is better conducted when air combat situation changes (i.e., any fighter is shot down or any fighter’s missiles are used up) than along with the flight. Finally, experiment 3 shows that, when the blue dispatches 8 fighters, 12 red fighters shall be dispatched accordingly. When the number of red fighters exceeds 12, the decrease in kill ratio is not obvious, if it is not increasing.In general, from the advantages exemplified by the simulation experiments, it can be concluded that the improved ant colony optimization proposed in this paper can be applied to WTA in air combat.As future work, we intend to apply other intelligent algorithms to the WTA problem and compare it with the improved ant colony algorithm to further explore the best solution for WTA. At the same time, the air combat simulation process needs to be refined. The current simulation hypothesis is relatively simple. The next step is to make the simulation process closer to actual combat and to make the simulation results more practical.



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*Source: 6481635-2018-10-11.xml*