With the advancement of combat equipment technology and combat concepts, new requirements have been put forward for air defense operations during a group target attack. To achieve high-efficiency and low-loss defensive operations, a reasonable air defense weapon assignment strategy is a key step. In this paper, a multi-objective and multi-constraints weapon target assignment (WTA) model is established that aims to minimize the defensive resource loss, minimize total weapon consumption, and minimize the target residual effectiveness. An optimization framework of air defense weapon mission scheduling based on the multi-objective artificial bee colony (MOABC) algorithm is proposed. The solution for point-to-point saturated attack targets at different operational scales is achieved by encoding the nectar with real numbers. Simulations are performed for an imagined air defense scenario, where air defense weapons are saturated. The non-dominated solution sets are obtained by the MOABC algorithm to meet the operational demand. In the case where there are more weapons than targets, more diverse assignment schemes can be selected. According to the inverse generation distance (IGD) index, the convergence and diversity for the solutions ofthe non-dominated sorting genetic algorithm III (NSGA-III) algorithm and the MOABC algorithm are compared and analyzed. The results prove that the MOABC algorithm has better convergence and the solutions are more evenly distributed among the solution space.
In the modern complex battlefield environment, with the emergence of high-tech and high-speed aerial weapon platforms, intelligent decision-making and scheduling of air defense weapon resources are particularly important. In air defense and anti-missile operations, weapon target assignment (WTA) is an important issue that needs to be resolved in the battlefield fire attack decision [
The intelligent optimization algorithm is an effective method to solve the NP-hard problems, such as the immune-based ant colony optimization (ACO) algorithm [
The targets move fast and can only be destroyed in a short time window. For air defense operations, the requirement of timeliness is high. Considering the short time window of the weapon’s striking, it is still necessary to further improve the real-time performance. The computational efficiency and global convergence of NSGA-III algorithm for WTA solving are insufficient. In addition, the actual data samples for combat are small, and it is difficult to obtain an effective decision model through neural network training. The rationality of input simulated data training cannot be verified.
The artificial bee colony (ABC) algorithm, proposed by Turkey scholar Karaboga in 2005, is a swarm intelligence optimization method simulating bees’ foraging behaviors [
To summarize, the contributions of this work are presented in the following list:
The effectiveness of air defense operations is evaluated through indicators including defensive resource loss, total weapon consumption, and target residual effectiveness based on the characteristics of the battlefield environment and operational tasks. A 0–1 integer-constrained multi-objective WTA optimization model is developed.
The MOABC algorithm for air defense weapon target assignment is presented. One-to-one and many-to-one scenarios weapon and target mapping relationships are established by cleverly designed coding rules, respectively. The effectiveness of the algorithm is verified by establishing Archive update non-dominated solutions.
Recently, as one type of the most effective algorithms for solving non-deterministic polynomial hard (NP-hard) problems like MOPs, multi-objective artificial bee colony (MOABC) algorithm have been proved successfully in practice and demonstrated powerful competitiveness. due to its ease of implementation, it has been successfully applied to many real-world problems, such as industrial systems, image processing, and so on. Compared with other evolutionary algorithms, such as particle swarm optimization (PSO) and NSGA, ABC shows strong global search ability. Zhao et al. [
The MOABC algorithm has unique advantages for solving mission planning and intelligent decision-making in the military field, but its related research in the air defense combat WTA problem is rare. Mao et al. [
At present, there is no research on solving multi-objective air defense WTA with MOABC. Inspired by the above research, we design multi-objective-driven WTA models and use the MOABC algorithm for the optimization of air defense WTA schemes to meet the requirements of rapidity and accuracy.
The multi-objective programming (MOP) is composed of multiple objective functions, decision variables, and constraints, and there are generally conflicting relationships between objective functions. The model can be described as
In
The solution of multi-objective optimization is not unique, but a set of solutions. Pareto dominance is the most common multi-objective solving mechanism. In the case of multiple objective functions, there may be conflicts and incomparable phenomena between objective functions. For example, a solution that is best for one objective function may be the worst for another. While improving any objective function, these solutions called the Pareto solutions will inevitably weaken at least one other objective function, which is also called non-dominated solutions. The set of optimal solutions for multi-objective functions is called the Pareto optimal set. The optimal set forms a Pareto front in the solution space. Pareto proposed the concept of a non-dominated set of multi-objective solutions in 1986, whose basic definition is as follows.
For a multi-objective programming problem, its variable feasible region is denoted by
In this paper, the optimization goal is to minimize the target residual effectiveness, minimize the defensive resource loss, and minimize the weapon resource consumption, which is also in line with the experience of battlefield commanders. Then the air defense WTA model is constructed to quickly form a strike decision plan. The specific combat scenario is an air defense combat mission. Suppose we have
The deployed firepower resources of
Defensive resource loss
The purpose of air defense weapons is to destroy aerial targets to secure defense resources from attack. However, in actual air defense operations, the enemy air combat units will attack or even destroy important defense resources, so combat losses are inevitable. Assuming that each aerial target has penetration capability, the target’s ability to strike and destroy is positively correlated to the weapon’s damage probability and the target’s threat level.
In
Total weapon consumption
Assuming that each surface-to-air weapon can only hit one target when weapons attack the opponent’s targets, the total weapon consumption is shown in
Target residual effectiveness
The target residual effectiveness is an important indicator to measure the strike and destroy the capability of surface-to-air weapons. The higher the damage probability of the surface-to-air weapon, the lower the target’s residual effectiveness. In the adversarial battlefield scenario, the damage probability of air defense weapons is used to characterize the damage capability, and the target residual effectiveness function is designed as
To efficiently destroy enemy targets, protect our battlefield defensive resources from attacks, and reduce weapon consumption, the MOP model of WTA shown in
Constraints in
The multi-objective artificial bee colony algorithm (ABC) is a swarm intelligence optimization algorithm based on the nectar-collecting mechanism of bees. The ABC algorithm has the advantages of few parameters, fast convergence, and simple operation. The description of the ABC algorithm is as follows: The candidate solution of the problem is regarded as a nectar source, and bees are divided into three types according to the division of labor, including employed bees, onlooker bees, and scout bees. Employed bees are responsible for the initial search for nectar source and sharing information. Onlooker bees are responsible for staying in the hive to collect nectar according to the information provided by the employed bees. After the original nectar source is abandoned, the scout bees randomly search for a new nectar source to replace the original one. After initializing the bee colony and nectar source, the three operations, including the phase of employed bees, onlooker bees, and scout bees, are performed repeatedly to find the optimal solution.
Initialize the bee colony
Initialize the algorithm parameters and randomly generate a certain amount of nectar source
In
Employed bees phase
Employed bees find new nectar source according to
In
Onlooker bees phase
After the employed bees find a new source of nectar, the employed bees select onlooker bees to follow based on a roulette strategy and then track and develop the nectar source. Like employed bees, a new nectar source location is locally searched according to
Scout bees phase
If the nectar source has not been updated when the number of searches reaches the upper limit of
In order to apply the artificial bee colony algorithm to multi-objective optimization, it is necessary to improve the original algorithm and set up an external file-handling mechanism. The high-quality solutions generated in each iteration are stored in the Archive [
The solution generated during the iteration is discarded if it is dominated by the solution in the Archive.
If the new solution dominates one or more solutions in the Archive, the solutions dominated in the Archive are replaced by the new solution.
All the solutions in the Archive are independent of each other, and a new solution is added to the Archive.
The addition of new non-dominated solutions to the Archive can effectively maintain group diversity. When the number of solutions in the Archive reaches the limit, similar individuals need to be eliminated according to the dominance level and crowdedness distance to satisfy the Archive space.
Calculate the objective function
Calculate the crowdedness distance
In
If the number of solutions in the Archive exceeds the limit, the solution with a higher degree of congestion is preferred, that is, the solution with a smaller
In this paper, real number vectors are used to encode individuals in the population. For a scenario where
Generally, multi-objective optimization algorithms only focus on the domain of variables, in other words, all dimensions of the solution are within the boundary. However, in the process of searching for new nectar source, infeasible solutions that do not meet the constraints may be generated. In this regard, it is necessary to adopt a repair strategy for individuals who do not meet the constraints to ensure the feasibility of the solution. There are two principles of repair. 1. The decimals of each dimension are different. 2. The integers contain the number of all targets.
For example, the position of the nectar source after one update is represented by the vector in
If the integers of two or more elements in
If there are as many weapons as targets, select elements with the equivalent integer, and change the integers of these elements to the serial numbers of unassigned targets.
If there are more weapons than targets and there are unassigned targets, the repairing strategy is the same as 1, otherwise, no repair is required.
Based on the above principle, the pseudo-code of the MOABC algorithm is as follows.
The experiment in this paper is conducted on the Windows 10 operating system with the main frequency of 2.9 GHz, and 8 GB of memory. Suppose that several enemy targets attack our important ground defensive resources. To implement effective defensive measures, we deploy several air defense weapons to strike and destroy opponents’ targets. Target types include bombers, fighter planes, electronic jammers, unmanned jammers, air-to-surface missiles, etc. Due to the confidentiality of real battlefield data, we simulated the data based on operational scenarios. For the construction of the WTA model, two operational scenarios are designed in the experiment. ① Weapon-target one-to-one assignment. ② Weapon-target many-to-one assignment.
Algorithm parameter settings: Population size 2
It is supposed that there are 10 enemy assault targets, 10 air defense weapons, and 6 important ground resources on the battlefield. The target damage probability, target threat level, defensive resource value, target value, and weapon consumption are shown in
ID | t_{1} | t_{2} | t_{3} | t_{4} | t_{5} | t_{6} | t_{7} | t_{8} | t_{9} | t_{10} |
---|---|---|---|---|---|---|---|---|---|---|
w_{1} | 0.72 | 0.13 | 0.70 | 0.02 | 0.15 | 0.42 | 0.80 | 0.28 | 0.75 | 0.80 |
w_{2} | 0.11 | 0.54 | 0.36 | 0.15 | 0.80 | 0.49 | 0.59 | 0.14 | 0.80 | 0.75 |
w_{3} | 0.40 | 0.64 | 0.87 | 0.99 | 0.16 | 0.56 | 0.37 | 0.77 | 0.20 | 0.76 |
w_{4} | 0.82 | 0.50 | 0.06 | 0.76 | 0.96 | 0.21 | 0.84 | 0.15 | 0.71 | 0.75 |
w_{5} | 0.83 | 0.93 | 0.08 | 0.02 | 0.15 | 0.55 | 0.33 | 0.32 | 0.55 | 0.59 |
w_{6} | 0.21 | 0.39 | 0.47 | 0.22 | 0.09 | 0.06 | 0.93 | 0.33 | 0.09 | 0.15 |
w_{7} | 0.80 | 0.32 | 0.58 | 0.53 | 0.44 | 0.35 | 0.85 | 0.69 | 0.16 | 0.66 |
w_{8} | 0.99 | 0.05 | 0.02 | 0.83 | 0.78 | 0.02 | 0.98 | 0.03 | 0.65 | 0.95 |
w_{9} | 0.37 | 0.52 | 0.30 | 0.03 | 0.18 | 0.30 | 0.90 | 0.68 | 0.84 | 0.23 |
w_{10} | 0.71 | 0.81 | 0.15 | 0.60 | 0.06 | 0.61 | 0.04 | 0.32 | 0.92 | 0.13 |
ID | t_{1} | t_{2} | t_{3} | t_{4} | t_{5} | t_{6} | t_{7} | t_{8} | t_{9} | t_{10} |
---|---|---|---|---|---|---|---|---|---|---|
d_{1} | 0.64 | 0.91 | 0.88 | 0.48 | 0.47 | 0.22 | 0.43 | 0.29 | 0.66 | 0.74 |
d_{2} | 0.27 | 0.38 | 0.21 | 1.00 | 0.54 | 0.27 | 0.23 | 0.63 | 0.13 | 0.69 |
d_{3} | 0.43 | 0.45 | 0.85 | 0.45 | 0.84 | 0.61 | 0.63 | 0.83 | 0.65 | 0.21 |
d_{4} | 0.83 | 0.63 | 0.72 | 0.70 | 0.42 | 0.26 | 0.11 | 0.29 | 0.28 | 0.66 |
d_{5} | 0.77 | 0.64 | 0.67 | 0.87 | 0.04 | 0.27 | 0.36 | 0.45 | 0.33 | 0.33 |
d_{6} | 0.52 | 0.93 | 0.83 | 0.63 | 0.81 | 0.21 | 0.96 | 0.20 | 0.11 | 0.95 |
ID | l_{1} | l_{2} | l_{3} | l_{4} | l_{5} | l_{6} |
---|---|---|---|---|---|---|
8 | 7 | 2 | 4 | 5 | 8 |
ID | v_{1} | v_{2} | v_{3} | v_{4} | v_{5} | v_{6} | v_{7} | v_{8} | v_{9} | v_{10} |
---|---|---|---|---|---|---|---|---|---|---|
4 | 1 | 5 | 1 | 7 | 7 | 10 | 5 | 7 | 6 |
ID | c_{1} | c_{2} | c_{3} | c_{4} | c_{5} | c_{6} | c_{7} | c_{8} | c_{9} | c_{10} |
---|---|---|---|---|---|---|---|---|---|---|
0.5 | 0.1 | 0.8 | 0.6 | 0.4 | 0.2 | 0.4 | 0.7 | 0.9 | 0.3 |
The distribution plan is solved by the MOABC algorithm. In
The MOABC algorithm has good convergence. Since the non-dominated sorting algorithm has a strict selection principle, the Archive has a forced ability to reserve better solutions. In
Defensive resource loss | Total weapon consumption | Target residual effectiveness | |
---|---|---|---|
1 | 40.94 | 4.90 | 9.07 |
2 | 34.03 | 4.90 | 9.32 |
3 | 25.41 | 4.90 | 9.39 |
w_{1} | w_{2} | w_{3} | w_{4} | w_{5} | w_{6} | w_{7} | w_{8} | w_{9} | w_{10} | |
---|---|---|---|---|---|---|---|---|---|---|
1 | t_{10} | t_{2} | t_{3} | t_{5} | t_{6} | t_{7} | t_{4} | t_{1} | t_{8} | t_{9} |
2 | t_{1} | t_{6} | t_{3} | t_{5} | t_{2} | t_{7} | t_{4} | t_{10} | t_{8} | t_{9} |
3 | t_{3} | t_{6} | t_{4} | t_{5} | t_{2} | t_{7} | t_{1} | t_{10} | t_{8} | t_{9} |
It is supposed that there are 7 enemy assault targets. Other settings are consistent with operational scenario 1. The enemy’s target damage probability, target threat degree, and target value are shown in
ID | t_{1} | t_{2} | t_{3} | t_{4} | t_{5} | t_{6} | t_{7} |
---|---|---|---|---|---|---|---|
w_{1} | 0.01 | 0.08 | 0.03 | 0.97 | 0.84 | 0.58 | 0.82 |
w_{2} | 0.97 | 0.53 | 0.92 | 0.70 | 0.07 | 0.34 | 0.25 |
w_{3} | 0.64 | 0.38 | 0.33 | 0.73 | 0.02 | 0.97 | 0.28 |
w_{4} | 1.00 | 0.34 | 0.10 | 0.20 | 0.15 | 0.77 | 0.39 |
w_{5} | 0.82 | 0.01 | 0.70 | 0.35 | 0.49 | 0.14 | 0.62 |
w_{6} | 0.09 | 0.21 | 0.03 | 0.46 | 0.78 | 0.10 | 0.94 |
w_{7} | 0.87 | 0.40 | 0.38 | 0.19 | 0.30 | 0.61 | 0.27 |
w_{8} | 0.63 | 0.23 | 0.75 | 0.04 | 0.18 | 0.32 | 0.72 |
w_{9} | 0.51 | 0.86 | 0.88 | 0.68 | 0.09 | 0.77 | 0.41 |
w_{10} | 0.15 | 0.74 | 0.04 | 0.01 | 0.39 | 0.79 | 0.70 |
ID | t_{1} | t_{2} | t_{3} | t_{4} | t_{5} | t_{6} | t_{7} |
---|---|---|---|---|---|---|---|
d_{1} | 0.17 | 0.69 | 0.46 | 0.87 | 0.42 | 0.46 | 0.48 |
d_{2} | 0.23 | 0.36 | 0.49 | 0.70 | 0.28 | 0.07 | 0.48 |
d_{3} | 0.41 | 0.44 | 0.95 | 0.15 | 0.80 | 0.04 | 0.72 |
d_{4} | 0.20 | 0.09 | 0.37 | 0.22 | 0.12 | 0.02 | 0.50 |
d_{5} | 0.54 | 0.20 | 0.14 | 0.26 | 0.61 | 0.94 | 0.58 |
d_{6} | 0.32 | 0.60 | 0.72 | 0.91 | 0.76 | 0.59 | 0.24 |
ID | v_{1} | v_{2} | v_{3} | v_{4} | v_{5} | v_{6} | v_{7} |
---|---|---|---|---|---|---|---|
8 | 5 | 7 | 1 | 9 | 2 | 3 |
Defensive resource loss | Total weapon consumption | Target residual effectiveness | |
---|---|---|---|
1 | 14.39 | 4.10 | 4.30 |
2 | 19.27 | 4.90 | 2.70 |
3 | 44.53 | 2.50 | 15.56 |
4 | 34.05 | 3.20 | 7.87 |
5 | 8.84 | 4.20 | 3.08 |
6 | 12.90 | 4.90 | 2.96 |
7 | 18.44 | 3.50 | 5.04 |
8 | 38.26 | 3.40 | 7.38 |
9 | 23.69 | 3.40 | 7.51 |
10 | 51.62 | 2.60 | 13.74 |
11 | 31.33 | 3.20 | 10.39 |
12 | 14.95 | 4.10 | 3.56 |
13 | 17.74 | 4.00 | 3.40 |
14 | 49.87 | 3.00 | 14.25 |
15 | 44.02 | 3.10 | 18.34 |
w_{1} | w_{2} | w_{3} | w_{4} | w_{5} | w_{6} | w_{7} | w_{8} | w_{9} | w_{10} | |
---|---|---|---|---|---|---|---|---|---|---|
1 | t_{4} | t_{3} | - | t_{1} | t_{2} | t_{5} | t_{6} | t_{7} | t_{2} | t_{7} |
2 | t_{5} | t_{3} | t_{6} | t_{1} | t_{5} | t_{4} | t_{5} | t_{7} | t_{2} | t_{2} |
3 | t_{4} | t_{1} | - | t_{7} | t_{5} | t_{3} | t_{6} | - | - | t_{2} |
4 | t_{5} | t_{7} | t_{6} | - | t_{1} | t_{4} | - | - | t_{3} | t_{2} |
5 | t_{4} | t_{3} | t_{6} | t_{1} | t_{7} | t_{5} | t_{5} | - | t_{2} | t_{7} |
6 | t_{5} | t_{3} | t_{6} | t_{1} | t_{5} | t_{7} | t_{4} | t_{7} | t_{4} | t_{2} |
7 | t_{4} | t_{1} | - | - | t_{5} | t_{5} | t_{6} | t_{7} | t_{3} | t_{2} |
8 | t_{5} | t_{2} | - | t_{6} | t_{1} | t_{7} | t_{3} | - | t_{3} | t_{4} |
9 | t_{4} | t_{1} | - | t_{2} | t_{7} | t_{5} | t_{6} | - | t_{3} | t_{7} |
10 | t_{5} | t_{2} | - | - | t_{4} | t_{7} | t_{1} | t_{6} | - | t_{3} |
11 | t_{4} | t_{3} | - | t_{1} | t_{5} | t_{2} | t_{6} | t_{3} | - | t_{7} |
12 | t_{4} | t_{3} | - | t_{1} | t_{5} | t_{5} | t_{6} | t_{3} | t_{2} | t_{7} |
13 | t_{5} | t_{3} | t_{4} | t_{1} | t_{5} | t_{7} | t_{6} | t_{3} | - | t_{2} |
14 | - | t_{1} | - | - | t_{4} | t_{2} | t_{5} | t_{3} | t_{6} | t_{7} |
15 | t_{4} | t_{2} | t_{5} | t_{6} | t_{1} | - | t_{3} | - | - | t_{7} |
In
Nie et al. [
NSGA-III algorithm: population size 2
MOABC algorithm: population size 2
The performance is evaluated by inverse generation distance (
In
Scale index algorithm | 10–10 | 10–7 | ||
---|---|---|---|---|
running time/s | running time/s | |||
NSGA-III | 3.71 | 6.54 | 4.25 | 8.77 |
MOABC | 2.67 | 3.92 | 3.47 | 8.15 |
It is of great military significance to study the issue of WTA in air defense operations. From the perspective of both opposing sides, we evaluated the results of air defense WTA with three indicators, including defensive resource loss, total weapon consumption, and target residual effectiveness. A multi-objective and multi-constrained air defense WTA model is constructed. The MOABC algorithm is used to realize the optimal solution of air defense weapon scheduling at different countermeasure scales, and the efficiency of the algorithm is verified. The performance of MOABC algorithm and NSGA-III algorithm is compared. The former has a lower IGD value of the Pareto solution set than the latter, and the overall quality of the solutions is higher and more uniformly distributed. The running time of the former is less than that of the latter, and the search efficiency of the algorithm is higher. This work provides certain ideas and means for air defense for group targets. Firstly, the applicability of the model has been improved. Multi-objective programming is established to objectively select the optimal assignment scheme, avoiding the commander’s personal preference affecting the decision-making results. Secondly, the scientific and reasonable evaluation indicators are designed to compare and verify the optimization results, and closed-loop research is realized. Finally, the Pareto optimal solution set can be regarded as an important reference to assist decision-making, which may also provide the best plan for decision-making commanders.
The authors are very grateful to the referees for their valuable remarks, which improved the presentation of the paper.
This work was supported by the National Natural Science Foundation of China (71771216).
The authors confirm contribution to the paper as follows: study conception and design: H. Xing, Q. Xing; data collection: H. Xing; analysis and interpretation of results: H. Xing, Q. Xing; draft manuscript preparation: H. Xing. All authors reviewed the results and approved the final version of the manuscript.
The authors confirm that the data supporting the findings of this study are available within the paper.
The authors declare that they have no conflicts of interest to report regarding the present study.