Weapon-target assignment (WTA) is critical to command and decision making in modern battlefields and is a typical nondeterministic polynomial complete problem. To solve WTA problems with multiple optimization objectives, a multipopulation coevolution-based multiobjective particle swarm optimization (MOPSO) algorithm is proposed to realize the rapid search for the globally optimal solution. The algorithm constructs a master-slave population coevolution model. Each slave population corresponds to an objective function and is used to search for noninferior solutions. The master population receives all the noninferior solutions from the slave populations, repairs the gaps between the noninferior solutions, and generates a relatively optimal Pareto optimal solution set. In addition, to accelerate the slave populations searching for noninferior solutions and master population repairing the gaps between noninferior solutions, the particle velocity update method is improved. The simulation results show that the proposed algorithm has higher computational efficiency and achieves better solutions than existing algorithms capable of providing a good solution. The method is suitable for rapidly solving multiobjective WTA (MOWTA) problems.
Weapon-target assignment (WTA) [
The multiobjective evolutionary algorithm based on Pareto theory [
The PSO algorithm is easily implemented and has a high optimization speed, also obtaining satisfactory results in the optimization of the MOWTA problem. However, the PSO algorithm is prone to premature convergence and becoming trapped in locally optimal solutions, mainly caused by a rapid loss of population diversity. To maintain population diversity, researchers have introduced operations such as random mutation [
To quickly optimize the MOWTA problem, considering the advantages and disadvantages of the PSO algorithm and multipopulation cooperation strategy, a multipopulation coevolution MOPSO (MPC-MOPSO) algorithm is proposed. The MPC-MOPSO algorithm constructs a master-slave population coevolution model. A slave population corresponds to an objective function for searching noninferior solutions and speeding up the search for noninferior solutions by randomly selecting several noninferior solutions from other populations to replace particles with lower fitness values in the population. The master population receives all the noninferior solutions from the slave populations, repairs the gaps between noninferior solutions, and generates a relatively optimal Pareto optimal solution set. Through the master-slave population coevolution model, the master and slave populations can simultaneously develop different search areas of the problem. This not only maintains the good diversity of the population but also achieves the efficient coordination and global search of multiple populations. Additionally, to accelerate the process by which the populations search for noninferior solutions, repair the gaps between the noninferior solutions, and further improve the global search speed of the MPC-MOPSO algorithm, a multidirectional competition factor is introduced into the particle velocity update to guide the particles to competitively search in multiple directions. Simulation results show that, compared with the improved MPACO algorithm proposed by Li et al. [
In modern and future battlefields, the benefit of assigning smart weapons to targets heavily relies on the assignment of sensors; thus, the sensor-weapon-target assignment (SWTA) problem is derived from the WTA problem. To effectively solve the SWAT problem, Bogdanowicz et al. [
The remainder of this study is organized as follows. In Section
To correctly construct the MOWTA model, the following assumptions are made:
(1) The probability of weapon damage to the target is the comprehensive damage probability, considering the weapon’s penetration probability, target hit probability, target damage probability, etc.
(2) The sequence of a target strike and the maximum projection ability of a wave of strikes are not considered.
(3) When multiple weapons strike a target, the damage probability of each weapon for the target is unchanged, and each weapon does not affect each other; that is, each strike is independent of each other.
Based on the above assumptions, the MOWTA problem can be described as follows:
In contrast to a single-objective optimization problem, the multiobjective optimization (MOO) problem does not have a unique optimal solution; its optimal solution becomes a set that may contain an infinite number of solutions that do not dominate one another (i.e., a Pareto optimal solution set). An MOO problem can be described as follows:
The MOPSO algorithm is the name of the PSO algorithm used to solve MOO problems. The MOPSO and PSO algorithms update the particle velocity and location in the same way. The PSO algorithm updates the velocity and location of a particle through two “guides”: the optimal solution that the particle has found (i.e., the individual guide
The MOPSO and PSO algorithms also differ in several respects: first, the update and selection methods of the individual and global guides are different. In addition, the solution obtained by the MOPSO algorithm is a Pareto optimal solution set. Moreover, the MOPSO algorithm requires a storage set for storing the noninferior solutions found by the population to be established. Numerous researchers have conducted in-depth studies on the updating and selection of individual and global guides, the establishment of a storage set, and the parameter settings in the MOPSO algorithm and have obtained some results [
To maintain high population diversity and perform a rapid and global search for the Pareto optimal solutions of the MOWTA problem, the MPC-MOPSO algorithm constructs a master-slave population coevolution model, which allows the algorithm to simultaneously develop various search regions for the problem and perform an efficient multipopulation cooperative global search. Additionally, to accelerate the process in which the populations search for noninferior solutions, repair the gaps between the noninferior solutions, and further improve the global search speed of the MPC-MOPSO algorithm, a multidirectional competition factor is introduced into the particle velocity update to guide the particles to competitively search in multiple directions.
In solving the MOWTA problem, the PSO algorithm first needs to encode its particles. The coding method should not only represent the solution of the problem and attempt to satisfy the constraints set in the model but also reduce the length of the code as much as possible. For this purpose, an integer coding method based on the attacking target was designed. The coding structure is shown in Figure
Diagram of the coding structure for the MOWTA problem.
A principal factor that leads the PSO and MOPSO algorithm towards premature convergence and their low global search efficiency is described as follows. The search space contains multiple local optimal regions. The populations are guided by a rule that they tend to evolve in the direction of the optimal particle in the populations. Under this rule, the particles in the populations rapidly exchange information and tend to move in a single direction. Consequently, the populations rapidly converge to a local optimal region, and the population diversity rapidly decreases, resulting in premature convergence. If the region does not contain the globally optimal solution, the algorithm becomes trapped in local optima. The algorithm requires a substantial amount of time to jump out of local optimal solutions, resulting in low global search efficiency.
If the particles can also evolve in the directions of the optimal particles in their locally optimal regions while tending to evolve in the direction of the optimal particle, it is possible to prevent the populations from rapidly accumulating in a certain locally optimal region, thereby maintaining satisfactory population diversity, effectively preventing the algorithm from becoming trapped in locally optimal solutions and increasing the algorithm’s global search speed. Hence, the particle update method is improved. A multidirectional competition factor is introduced into the particle velocity update to guide the particles to competitively search in multiple directions. The particles may still evolve in the directions of the optimal particles in their locally optimal regions even while evolving in the direction of the optimal solution. The improvement of the particle update method includes two parts: determination of the local guides and introduction of a multidirectional competition factor.
It is difficult to obtain the locally optimal regions in the search space by mathematical operations. Instead, clustering techniques can rapidly obtain more accurate locally optimal regions. Zhang et al. [
To prevent premature convergence and low global search efficiency caused by the tendency of particles to evolve in a single direction (the direction of the optimal particle in a population), a multidirectional competition factor is introduced into the particle velocity update to allow particles to evolve in the directions of the optimal particles within their locally optimal regions. This technique can realize a multidirectional competitive search by the particles, effectively preventing the algorithm from experiencing premature convergence or becoming trapped in local optima and thus improving the global search efficiency. If
The MPC-MOPSO algorithm constructs a master-slave population coevolution model, which includes a master population and multiple slave populations. The number of slave populations is determined by the number of objective functions. One slave population corresponds to one objective function. By implementing the PSO algorithm containing a multidirectional competition factor, the noninferior solutions of the objective function can be searched. The master population receives all noninferior solutions from the slave populations, repairs the gaps between the noninferior solutions by implementing the MOPSO algorithm containing a multidirectional competition factor, and generates a relatively optimal Pareto optimal solution set. Figure
The master-slave population coevolution model.
For an MOO problem with
The slave populations operate in parallel and independently. Since the rates of evolution of the slave populations are different, some slave populations will inevitably have converged while others will have not yet converged. For slave populations that have converged, new noninferior solutions will not be generated due to the loss of population diversity. To avoid this deficiency and increase the rate of convergence of the slave populations that have not converged, a cooperative search strategy for the slave populations is proposed as follows. After each iteration of a slave population, several noninferior solutions are randomly selected from other slave populations to replace the particles with lower fitness values. The cooperative search strategy for the slave populations not only avoids the situation whereby some slave populations cannot generate new noninferior solutions due to rapid convergence but also enables the slave populations to assist one another and jointly search for noninferior solutions of single-objective functions.
After each iteration, the master population receives all the noninferior solutions from the slave populations. Then, the population searches for more and better noninferior solutions by the MOPSO algorithm proposed, thus repairing the gap between the noninferior solutions and generating a good Pareto optimal solution set.
Since the master population continuously receives noninferior solutions from the slave populations, its scale and particle density increase, resulting in a decrease in the optimization speed. Densely distributed particles not only are unfavourable to the maintenance of population diversity but also significantly reduce the search efficiency of the populations. Therefore, it is necessary to eliminate some densely distributed particles.
The more densely the particles are distributed, the closer the particles are to each other, and the “crowding distance” is used to measure the density of the particles. The crowding distance of a particle is the sum of its distances from its two nearest particles. If particles
Figure
Flowchart of the MPC-MOPSO algorithm.
(1) Initialize the slave populations. The number of slave populations is determined by the number of objective functions. First, encode the particles of the population following the method proposed in Section
(2) Search for noninferior solutions. The slave population evolution model is used to search for noninferior solutions to the single-objective functions and move them to the master population.
(3) Generate a Pareto optimal solution set. The master population receives all the noninferior solutions from the slave populations and uses the master population evolution model to repair the gaps between the noninferior solutions. Then, a good Pareto optimal solution set is generated.
(4) Determine whether the termination condition is satisfied. If the termination condition is met, the algorithm stops searching; otherwise, the algorithm returns to Step (2).
The case background is as follows. There are ten different types of weapons available, and twelve targets will be attacked. The number of each type of weapon, the combat value coefficient of each target, and the maximum number of weapons that can be used on each target are shown in Table
Settings of weapons and targets.
Number of each type of weapon | Combat value coefficient of each target | Maximum weapon usage on each target |
---|---|---|
1, 3, 2, 2, 2, 2, 3, 1, 2, 2 | 8, 7, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5 | 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 1 |
Damage probability of a weapon to a target.
Weapon type | Target | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | T10 | T11 | T12 | |
W1 | 0.4 | 0.4 | 0.5 | 0.6 | 0.4 | 0.6 | 0.7 | 0.2 | 0.2 | 0.4 | 0.5 | 0.3 |
W2 | 0.3 | 0.5 | 0.1 | 0.4 | 0.1 | 0.6 | 0.3 | 0.5 | 0.7 | 0.5 | 0.4 | 0.7 |
W3 | 0.2 | 0.3 | 0.6 | 0.6 | 0.5 | 0.3 | 0.5 | 0.1 | 0.4 | 0.3 | 0.6 | 0.4 |
W4 | 0.5 | 0.5 | 0.2 | 0.4 | 0.1 | 0.6 | 0.3 | 0.5 | 0.7 | 0.5 | 0.4 | 0.6 |
W5 | 0.4 | 0.3 | 0.1 | 0.7 | 0.5 | 0.7 | 0.5 | 0.6 | 0.4 | 0.4 | 0.3 | 0.7 |
W6 | 0.5 | 0.4 | 0.7 | 0.2 | 0.3 | 0.2 | 0.6 | 0.3 | 0.2 | 0.6 | 0.1 | 0.3 |
W7 | 0.3 | 0.5 | 0.2 | 0.1 | 0.4 | 0.5 | 0.7 | 0.6 | 0.2 | 0.3 | 0.4 | 0.5 |
W8 | 0.2 | 0.4 | 0.5 | 0.5 | 0.6 | 0.4 | 0.3 | 0.1 | 0.3 | 0.2 | 0.5 | 0.5 |
W9 | 0.5 | 0.4 | 0.6 | 0.3 | 0.4 | 0.5 | 0.3 | 0.7 | 0.2 | 0.1 | 0.3 | 0.4 |
W10 | 0.1 | 0.4 | 0.6 | 0.5 | 0.2 | 0.4 | 0.6 | 0.5 | 0.4 | 0.3 | 0.5 | 0.2 |
Based on the above background, five different cases were established. The weapon types and targets in each case are shown in Table
Weapon types and targets for five cases.
Case | C1 | C2 | C3 | C4 | C5 |
---|---|---|---|---|---|
Weapon type | W1-W6 | W1-W8 | W1-W10 | W1-W10 | W1-W10 |
Target | T1-T8 | T1-T8 | T1-T8 | T1-T10 | T1-T12 |
The improved MPACO algorithm proposed by Li et al. [
Table
Simulation time statistics.
Case | Number of variables | The improved MPACO algorithm | The MOPSO-II algorithm | The MPC-MOPSO algorithm | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| | | | | | | | | | | | ||
C1 | 48 | 0.6 | 7.5 | 1.3 | 0.7 | 0.5 | 7.0 | 1.2 | 0.7 | 0.3 | 0.6 | 0.4 | 0.1 |
C2 | 64 | 1.3 | 14.1 | 3.5 | 1.2 | 1.1 | 15.3 | 3.0 | 1.3 | 0.6 | 1.5 | 0.9 | 0.2 |
C3 | 80 | 2.6 | 20.5 | 5.8 | 2.0 | 2.4 | 21.3 | 4.5 | 1.9 | 1.1 | 2.6 | 1.7 | 0.4 |
C4 | 100 | 6.4 | 53.8 | 11.9 | 4.7 | 5.6 | 48.9 | 10.3 | 4.3 | 1.9 | 5.8 | 2.5 | 1.0 |
C5 | 120 | 18.7 | 193.6 | 32.5 | 13.4 | 14.8 | 167.5 | 28.6 | 12.5 | 4.5 | 16.2 | 5.9 | 2.3 |
A comparison of
Figures
Pareto front in Case 1.
Pareto front in Case 2.
Pareto front in Case 3.
Pareto front in Case 4.
Pareto front in Case 5.
Figures
For the targets, if additional weapons are put into the strike sequence, the targets will be assigned more suitable weapons. For the weapons, if additional targets are added to the strike mission, the weapons will choose more suitable targets to attack. Both of the above cases will increase the value damage of targets. To further evaluate the effectiveness, whether the MPC-MOPSO algorithm can assign more suitable weapons to the targets for the above two cases is determined. Tables
The value damage of targets of Case 1 to Case 3.
Case | Weapon consumption | |||||
---|---|---|---|---|---|---|
11 | 12 | 13 | 14 | 15 | 16 | |
C1 | 31.25 | 32.16 | N.A. | N.A. | N.A. | N.A. |
C2 | 32.11 | 33.36 | 34.37 | 35.23 | 35.95 | 36.62 |
C3 | 32.54 | 33.69 | 34.66 | 35.38 | 36.14 | 36.77 |
The value damage of targets of Case 3 to Case 5.
Case | Weapon consumption | |||||
---|---|---|---|---|---|---|
15 | 16 | 17 | 18 | 19 | 20 | |
C3 | 36.14 | 36.77 | N.A. | N.A. | N.A. | N.A. |
C4 | 40.72 | 41.53 | 42.40 | 43.09 | 43.56 | 43.91 |
C5 | 45.08 | 46.04 | 46.92 | 47.67 | 48.35 | 48.96 |
Table
The comparative analysis in Section
To quickly search for the optimal solution of the MOWTA problem, the MPC-MOPSO algorithm is proposed. The MPC-MOPSO algorithm constructs a master-slave population coevolution model and introduces a multidirectional competition factor into the particle velocity update. The simulation results show that the MPC-MOPSO algorithm has higher computational efficiency and obtains better solutions than the improved MPACO algorithm or the MOPSO-II algorithm. The MPC-MOPSO algorithm can significantly increase the optimization speed of the MOWTA problem and generate relatively optimal assignments. Furthermore, the MPC-MOSPO algorithm can be used for rapidly solving large-scale MOWTA problems.
The MPC-MOPSO algorithm can also solve the SWTA problem. Compared with the MRBCH algorithm, which can solve the SWTA problem well, the MPC-MOPSO algorithm performs better in solving the SWTA problem with multiple optimization objectives.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This work was jointly supported by the National Natural Science Foundation for Young Scientists of China (Grants Nos. 61403397 and 61202332) and the Natural Science Foundation of Shanxi Province, China (Grant No. 2015JM6313).