^{1}

^{2}

^{1}

^{1}

^{1}

^{2}

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.

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 [

In addition, reference [

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 [^{3}I systems by Huang et al. [

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, Model

The paper is organized as follows: Section

The objective of WTA is to maximize the expected impact on the opponents and to minimize the risk we face [

It is assumed that the red side has

Air combat situation diagram.

In Figure

The angle threat function [

The distance threat function [

The speed threat function [

In this paper, we embrace the air combat capability formula in [

For example, supposing the air combat capability of

As a combination of the above threat functions, the threat degree of

where

The overall threat degree of

Let

Four WTA models are presented in this subsection.

Model

The constraints are the same as in Model

Model

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 (

where

_{ k } represents the set of available targets. As the search progresses,_{k} is getting smaller.

When

(1) In MMAS, to avoid local stagnation in searching, the pheromone concentration interval is set as follows [

(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:

Considering the rules of path selection, pheromone update, and pheromone concentration interval, 24 sets of algorithms are obtained, as shown in Table

Algorithm set.

| | |
---|---|---|

_{ 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 of | Smooth the concentration | |

| ||

Punish rule of |

Algorithm

Test results of Algorithm set.

| | | | | | |
---|---|---|---|---|---|---|

| 21.36 | 19.60 | 19.03 | 14.43 | 4.43 | 16 |

| 15.27 | 15.45 | 15.43 | 19.77 | 15.96 | 20.05 |

| ||||||

| | | | | | |

| ||||||

| 23 | 4.30 | 18.53 | 12.90 | 3.03 | 12.56 |

| 19.68 | 14.35 | 19.983 | 20.10 | 13.17 | 20.03 |

| ||||||

| | | | | | |

| ||||||

| 5.66 | 6.10 | 6.23 | 17.36 | 16.86 | 15.76 |

| 14.23 | 14.22 | 14.21 | 18.12 | 19.52 | 17.98 |

| ||||||

| | | | | | |

| ||||||

| 23.10 | 15 | 22.33 | 16.23 | 9.76 | 7.13 |

| 18.61 | 18.40 | 18.49 | 19.75 | 17.99 | 20.15 |

1. _{max} (The largest iteration number)

2.

3.

4.

5.

6.

7. _{b} (Number of attackable blue fighters)

8.

9.

10.

11.

12.

13._{k} minus the assigned target;

14.

15.

16.

17._{rl} (The remaining amount of red missiles)

18.

19.

20.

21._{k} minus the assigned target;

22.

23.

24.

25.

26.

27.

28.

29.

30.

1.

2.

3._{r}

4. _{b}

5. _{ij}_{i} (_{ij} is the distance between two sides, _{i} is the range of the red fighters)

^{r}, ^{b};

6.

7.

8.

9.

11. _{r}

12. _{k}^{r};

13. _{k}^{r}^{r}

14.

15.

16.

17. _{b}

18. _{k}^{b};

19. _{k}^{b}^{b}

20.

21.

22.

23

24.

25.

26

37

As can be seen from Table

Algorithm 24 selects rules of random selections of AS, punishes the rule of

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 Figure

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 [

The ants move according to the following rules:

Pseudocode of the WIACO algorithm is shown as in algorithm

The parameters of the WTA model called by WIACO algorithm are discussed with a control variable experiment in this subsection, including the

Taking Model

Model parameter values.

| | | | |
---|---|---|---|---|

| 1.5 | 1.5 | 1.5 | 1.5 |

| 3 | 2 | 3 | 2 |

| 0.4 | 0.7 | 0.7 | 0.6 |

| 0.7 | 0.5 | 0.4 | 0.6 |

| 0.6 | 0.7 | 0.7 | 0.5 |

Parameter variation curves of Model

The obtained parameters are substituted into the models for the experimental analysis, and the results of one of these experiments are as in Figure

Model convergence curves.

Repeat the experiment 100 times and take the average for analysis. The results are presented in Table

Statistical analysis of the model results.

| | | | |
---|---|---|---|---|

| 17.2896 | 17.2831 | 17.2859 | 0.0021 |

| 12.2186 | 11.8368 | 12.1267 | 0.0109 |

| 22.1207 | 18.6858 | 20.4573 | 0.5852 |

| 6.08764 | 5.8862 | 6.0533 | 0.0024 |

It is assumed that the red side adopts WIACO algorithm with WTA Model

According to Table

Red and blue parameter sets.

| | |
---|---|---|

| 8 | 8 |

| 4 | 4 |

| 340 | 340 |

| 80 | 90 |

| 120 | 120 |

| 25.5 | 25.65 |

| 2761 | 2832 |

| 1514 | 1553 |

| 0.9 | 0.95 |

| 0.915 | 0.995 |

| 1.14 | 1.14 |

| 1.05 | 1.1 |

Based on reference [

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 Figure

Traditional and improved algorithm convergence curves.

The WIACO convergence is shown in Figure

Improved algorithm assignment.

| | |
---|---|---|

| Blue1 | 0 |

| Blue3 | 0 |

| Blue5 | 1 |

| Blue5 | 1 |

| Blue4 | 1 |

| Blue4 | 3 |

| — | 4 |

| Blue6 | 0 |

The 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.

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.

Assuming the red side performs WTA using the WIACO algorithm, the air combat simulation in this section is based on the following rules:

The simulation steps are as follows:

Obtain the initial data of the red and blue sides and initialize the parameters.

Call the WIACO algorithm.

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.

Update the air combat situation.

Repeat Steps

In the simulations, the number of red fighters is

In experiment 1, both sides dispatch 8 fighters. The parameters used in the two-step adjudication model are shown in Table

Two-step adjudication model parameters.

| | |
---|---|---|

| 0.6 | 0.7 |

| 0.45 | 0.4 |

| 4 | 4 |

| 1 | 1 |

| 0.8 | 0.8 |

^{ 2 } | 3.2 | 2 |

| 0.7 | 0.7 |

| 0.8 | 0.8 |

| 120 | 120 |

| 80 | 90 |

Because of the randomness of fighter kill and air combat situation, each simulation result is different, which reflects the uncertainty of air combat. Table

Simulation results (example).

| | |
---|---|---|

| Fighters 6 and 8 shot down; fighters 4 and 5 out of missiles | Fighters 1 and 7 shot down; Fighter 4 out of missiles |

| ||

| Fighter 3 shot down; fighter 2 out of missiles | Fighter 3 shot down |

| ||

| Fighter 7 shot down; 1 out of missiles | Fighter 2 out of missiles |

| ||

| Shot down: 4; Missiles exhausted: 4 | Shot down: 3, Missiles exhausted: 2 |

In this paper, we run a large number of simulations and statistical results with formula ((_{i}^{r} and that of blue side with_{i}^{b}, and the damage rates of red and blue sides,^{r} and^{b}, are calculated with formula ((

The kill ratio

The smaller the kill ratio is, the greater the advantage of the red side is in air combat.

First, we run 1000 times simulations using Model

Model

| | |
---|---|---|

| 7.3945 | 4.1864 |

| 0.9243 | 0.5233 |

| 1.7663 |

Kill ratio and iterations curve.

Damage distribution.

And then additional experiments are conducted using Models

Model

| | |
---|---|---|

| 7.2821 | 4.3632 |

| 0.9103 | 0.5454 |

| 1.6690 |

Model

| | |
---|---|---|

| 7.2492 | 4. 5326 |

| 0.9062 | 0.5665 |

| 1.5993 |

Model _{dj}=0.2).

| | |
---|---|---|

| 7.3215 | 4. 0854 |

| 0.9151 | 0.5106 |

| 1.7921 |

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 Tables

Option 1 simulation results.

| | |
---|---|---|

| 4.5212 | 2.8727 |

| 0.5652 | 0.3591 |

| 1.5738 |

Option 2 simulation results.

| | |
---|---|---|

| 3.9122 | 2.4820 |

| 0.4890 | 0.3103 |

| 1.5762 |

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 Table

Red fighter number and kill ratio.

| 3 | 4 | 5 | 6 |

| 3.0726 | 2.6845 | 2.4587 | 1.9752 |

| ||||

| 7 | 8 | 9 | 10 |

| 1.6872 | 1.6154 | 1.5886 | 1.5754 |

| ||||

| 11 | 12 | 13 | 14 |

| 1.5743 | 1.5159 | 1.5245 | 1.5088 |

| ||||

| 15 | 16 | 17 | 18 |

| 1.5051 | 1.4912 | 1.4921 | 1.5011 |

Red fighter number and kill ratio curve.

In Table

In the air combat, the battlefield situation is complex and changeable, and WTA plays a decisive role. In Section

In order to solve the WTA problem, WIACO algorithm is presented in Section

For the sake of the demonstration and exemplification of WIACO in air combat, four combat simulation rules and RWSA algorithm are designed in Section

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.

Number of red fighters

Number of blue fighters

The

The

The direction of

The speed of

The off-axis angle of

The distance between

The threat of

The missile ranges of the blue fighter

The maximum detection ranges of the blue radar

The speeds of the red fighter

The speeds of the blue fighter

The fire attack capability parameter

The radar detection capability parameter

The maneuverability parameter of the fighter

The air combat capability

The pilot’s control capability coefficient

The fighter survivability coefficient

The fighter range coefficient

The Electronic Counter Measures capability coefficient

The number of missiles of

The number of missiles assigned to

The missile number of

The number of red missiles actually used to attack

The probability that

The overall hit probability of the red fighter on

The threshold of hit probability of

The path selection probability

The pheromone concentration

The heuristic function

The information heuristic factor

The expected heuristic factor

The set of available targets

The set of nodes that the

A random number in

A constant,

The pheromone volatilization rate

A constant used to regulate the pheromone concentration

The path length of the ant

The shortest path length of the current cycle

The number of ant garbage ants

A constant used to regulate the pheromone concentration

A constant used to regulate the concentration

The penalty factor

The total number of missiles carried by red fighters

The amount of damage of red side in the

The amount of damage of blue side in the

The damage rate of red side

The damage rate of blue side

The kill ratio.

The

The authors declare that there are no conflicts of interest regarding the publication of this paper.