2.1 Angle performance function

The angle in the speed direction of an interceptor plays an enormous role in evaluating the attack effectiveness between the interceptor and its target. When the interceptor moves towards the target, if the speed direction of the interceptor is closer to the line of sight (LOS) direction, it can effectively attack the target. In turn, when $\eta=0$ holds, the angle advantage of the interceptor for attacking the target is highly effective. Thus, the angle performance function is described as follows:

(1) \begin{equation} f_{\eta}={e^{\left(-\frac{\eta}{a \pi}\right)}}^{2}\end{equation}

where a is a variable parameter that needs to be designed and is related to the relative range R.

2.2 Distance performance function

An interceptor with detection capability has the chance to complete an attack mission only when the target is located within the interceptor detection bound zone. Once the target moves into the interceptor detection blind zone, the interceptor cannot directly detect the target, thus resulting in a failed attack mission. Let the lower and upper bounds of the interceptor’s detection capability be $R_{\min}$ and $R_{\max}$ , respectively. It is assumed that when the relative range between an interceptor and a target is $(R_{\min}+R_{\max})/{2}$ , the detection capability is the strongest, implying the interceptor distance performance is most notable. Thus, the distance performance function is described as follows:

(2) \begin{equation} f_{R}=e^{-\left(\frac{R-R_{0}}{\sigma_{R}}\right)^{2}}\end{equation}

where $R_0=(R_{\min}+R_{\max})/{2}$ , and $\sigma_{R}$ is determined by the detection range, which is associated with a seeker equipped with the interceptor.

2.3 Speed performance function

When attacking a target in a real scenario, an interceptor is generally required to move faster than the target, which means that $V_T<V_M$ holds. Thus, the speed performance function is described as follows:

(3) \begin{equation} f_{V}=1-\frac{V_{T}}{V_{M}}\end{equation}

It can be seen from Equation (3) that when interceptor speed is the same as that of its target, the speed performance function is 0; when the target is stationary, the speed performance function is equal to 1.

2.4 Capture probability performance function

During interceptor formation, each interceptor can exchange and share information with its neighbours through a communication network. As a result, although each interceptor has a detection blind zone restricted by its seeker, a larger range of situational awareness associated with its target can be obtained through the information interaction from other interceptors. Therefore, the communication network enlarges the capture probability of each interceptor. However, there are also inevitable communication delays and packet loss problems in the formation of a communication network, which decreases the capture probability of each interceptor. Thus, it is necessary to comprehensively analyse the capture probability performance of each interceptor, including its own capture probability and the communication network capture probability. The capture probability performance function is described as follows:

(4) \begin{equation} f_c=\left\{\begin{array}{rc} p_{sc} & \\ \\[-7pt] p_{nc} & \\\end{array} \right.\end{equation}

where $p_{sc},(0 \le p_{sc} \le 1)$ denotes the probability that an interceptor captures its target by itself, and $p_{nc},(0 \le p_{nc} \le 1)$ denotes the probability that it captures using the communication network.

2.5 Radiation source matching performance function

Generally, typical targets can be classified into electromagnetic radiation, infrared radiation, and visible light radiation sources according to radiation types. However, different types of detection equipment are required to capture a target with different radiation sources. Similarly, seekers can be classified into radars, infrared imaging and visible light. As a result, when multiple interceptors cooperate to attack their targets, the combination format of the seekers needs to be considered to enhance the cooperative detection and disturbance countermeasure capabilities. In this study, we appropriately simplify radiation source matching performance and stipulate its corresponding performance function as follows:

(5) \begin{equation} f_s=\left\{ \begin{array}{rc} 1 & \\ 0 & \\ \end{array} \right.\end{equation}

where $f_s=1$ denotes that the target radiation source is matched with the seeker, and $f_s=0$ denotes that it is unmatched with the seeker.

Therefore, based on the above-established five performance functions, the fitness function can be described as Equation (6). Moreover, we can ensure the result of the multitarget allocation, optimise the fitness function and realise maximised attack effectiveness.

(6) \begin{equation} F=f_c \left(\rho_1 \,f_{\eta} \,f_R + \rho_2 \,f_{\eta} \,f_V + \rho_3 \,f_s \right)\end{equation}

where $\rho_1$ $\rho_2$ and $\rho_3$ are weight coefficients determined by the influence of each performance function on the comprehensive fitness function. $0 \le\ \rho_{i} \le 1, \left(i=1,2,3 \right)$ and $\sum\limits_{i=1}^{3}\rho_i = 1 $ .

In this study, it is assumed that the numbers of interceptors and targets are $m,m\in \mathbb{N}^{+}$ and $n,n\in \mathbb{N}^{+}$ , respectively $\left(n<m\right)$ . Based on Equation (6), we can obtain the value of the fitness function between an i-th interceptor and a j-th target, thereby constructing the comprehensive matrix between the i-th interceptor and the j-th target as follows:

(7) \begin{equation} \left( \begin{array}{c@{\quad}c@{\quad}c} F_{11} & \cdots & F_{1n}\\ \\[-7pt] \vdots & \vdots & \vdots\\ \\[-7pt] & F_{ij} & \\ \\[-7pt] \vdots & \vdots & \vdots\\ \\[-7pt] F_{m1} & \cdots & F_{mn}\\ \\[-7pt] \end{array} \right)\end{equation}

Therefore, the objective of the multitarget allocation can be described as follows:

(8) \begin{equation} F_{\max}=\sum \limits_{i=1}^m \sum \limits_{j=1}^n F_{ij}X_{ij}\end{equation}

(9) \begin{equation} s.t=\left\{ \begin{array}{rcl} \sum \limits_{j=1}^n X_{ij}=1& ,& \left(i=1,2,\cdots,m\right) \\ \\[-9pt] \sum \limits_{i=1}^m X_{ij}\le T_{j}&, & \left(j=1,2,\cdots,n\right) \\ \\[-9pt] X_{ij}=\left \{0,1\right\}& \end{array} \right.\end{equation}

It can be seen from Equation (9) that there are three restrictions in the objective of the multitarget allocation. The first constraint denotes that each interceptor can only attack a single target. The second constraint denotes that a j-th target can be attacked by $T_j$ at most, which implies that its ammunition need is $T_j$ in turn. $X_{ij}$ denotes the probability that the i-th interceptor will be allocated to the j-th target, which can either be 0 or 1.

3.1 PSO algorithm

The PSO algorithm, derived from the laws of the movement of birds, is a swarm intelligence algorithm. First proposed by James Kennedy and Russell Eherhart [Reference Kennedy and Eberhart21,Reference Eberhart and Kennedy22], it is assumed that each bird is a single particle. Then, the process of a particle searching for the global optimal space is analogous to a bird looking for its favourite food. Firstly, it is necessary to initialise a certain number of randomly moving populations, in which the position of each particle is a feasible solution in a search space. Then, the fitness value of an i-th particle is recorded in the process of iterating, and its optimal value during all iterations before the current position is memorised, called the individual extreme value ( $p_{best}$ ). Simultaneously, other particles exchange information with each other so that an optimal value occurs among all particles in the current iteration process, called the global extreme value ( $g_{best}$ ). Finally, each particle adopts these two extreme values to update the speed of the next iteration, thereby adjusting the position of the particle’s movement to move closer to the optimal point.

Figure 2 illustrates the position change of an i-th particle in the iteration process. $x_i$ and $v_i$ denote its position and speed, respectively. For any particle, their speed is composed of three components: $v_{1i}$ , $v_{2i}$ and $v_{3i}$ . $v_{1i}$ represents the particle speed components, $v_{2i}$ represents the self-recognition learning component of the individual extreme value obtained by the particle in its iterative process and $v_{3i}$ represents the social experience component in the populations.

Figure 2. Diagram of particle motion.

Assuming that in a D-dimensional space, the total number of particles is n. The next iteration of each particle is determined by its experience and the best experience of other particles. Each iteration uses Equations (10)–(11) to update its speed and position.

(10) \begin{equation} \begin{array}{ccc} &v_{id}(t+1)=\omega v_{id}(t)+c_{1} r_{1}\left(p_{i}(t)-x_{id}(t)\right)+c_{2} r_{2}\left(g_{i}(t)-x_{id}(t)\right)\\ \\[-7pt] &v_{i d}(t+1)=v_{\max }, v_{i d}(t+1)>v_{\max } \\ \\[-7pt] &v_{i d}(t+1)=v_{\min }, v_{i d}(t+1)<v_{\min } \end{array}\end{equation}

(11) \begin{equation} \begin{array}{ccc} &x_{i d}(t+1)=x_{i d}(t)+v_{i d}(t)\\ \\[-7pt] &x_{i d}(t+1)=x_{\max }, x_{i d}(t+1)>x_{\max }\\ \\[-7pt] &x_{i d}(t+1)=x_{\min }, x_{i d}(t+1)<x_{\min } \end{array}\end{equation}

where the subscript i denotes the i-th particle. t denotes the number of particle iterations. $\omega$ denotes an inertia weight factor. $c_1,c_2\in\mathbb{R}^{+}$ denotes the acceleration factor. $r_1,r_2\in\mathbb{R}^{+}$ denote two random uniformly disturbed in the interval (0,1). $\left[x_{\min }, x_{\max }\right]$ and $\left[v_{\min }, v_{\max }\right]$ denote the upper and lower bounds of the particle search range, respectively. The speed of the i-th particle is denoted by $v_{i}=\left(v_{i 1}, v_{i 2}, \cdots , v_{i D}\right)$ , and the position of the i-th particle is denoted by $x_{i}=\left(x_{i 1}, x_{i 2}, \cdots , x_{i D}\right)$ . The calculation process of the PSO algorithm is presented in Fig. 3.

Figure 3. PSO algorithm diagram.

Figure 4. Diagram of multitarget allocation based on adaptive SA-PSO algorithm.

3.2 Adaptive improvement of PSO algorithm

Considering the PSO algorithm disadvantage, which converges slowly in optimisation performance, an adaptive improvement is made in this study. The adjustable parameter C(k) is added to the PSO algorithm to adaptively adjust the inertia weight factor $\omega$ and the acceleration factors $c_1$ and $c_2$ . The principle is described as Equation (12).

(12) \begin{equation} \left\{\begin{array}{l}D(k)=\dfrac{\sum_{i=1}^{N} \sqrt{\sum_{j-1}^{D}\left[x_{i}(j)-P_{b}^{i}(j)\right]^{2}}}{N} \\ \\[-9pt] C(k)=\dfrac{D(k)}{\max (D)} \\ \\[-9pt] \omega=C(k) \\ \\[-9pt] c_{1}=2 \times C(k) \\ \\[-9pt] c_{2}=2-c_{1}\end{array}\right.\end{equation}

3.3 Multitarget allocation strategy based on adaptive SA-PSO algorithm

To further improve the performance of the PSO algorithm, a simulated annealing (SA) algorithm is introduced. The SA algorithm is a heuristic algorithm that allows the particle to accept a solution worse than the current solution with a certain probability. Thus, it is possible to break away from a local optimal solution and obtain a global optimal solution, effectively overcoming the disadvantage of the PSO algorithm easily falling into a local extreme point [Reference Wu, Wu and Zhang23,Reference Lu, Yang, Sun, Ding and Zhao24]. In addition, the roulette rules are applied to improve selection of optimal particles. While obtaining a better particle, inferior particles can also be obtained with a certain probability. Then, the particle speed can be adapted to appear from the local extremum and quickly converge to the global optimal point. The calculation process of the adaptive SA-PSO algorithm is presented in Fig. 4. The optimisation strategy for multitarget allocation based on the proposed adaptive SA-PSO algorithm is as follows:

1. a. Initialising the position and speed of each particle in the population

2. b. Calculating the fitness value of each particle, storing the current particle position and particle fitness value in ${p_i}$ , and storing the optimal individual position and fitness value of all ${p_b}$ in ${P_b}$

3. c. Determining the initial temperature $t_0$ :

   (13) \begin{equation} t_{0}=\frac{f\left(p_{b}\right)}{\ln 5}\end{equation}

4. d. Determining the adaptation value of each $p_i$ at the current temperature:

   (14) \begin{equation} T_{F}\left(p_{i}\right)=\frac{\mathrm{e}^{-\left(f\left(p_{i}\right)-f\left(p_{b}\right)\right) / t}}{\sum_{i=1}^{N} \mathrm{e}^{-\left(f\left(p_{i}\right)-f\left(p_{b}\right)\right){/ t}}}\end{equation}

5. e. Applying the roulette rules to determine the global optimal substitute value $p_{b}^{\prime}$ from all $p_i$ and then updating the speed and position of each particle according to Equations (10)–(11)

6. f. Applying an attenuation coefficient method for the temperature-reducing process:

   (15) \begin{equation} t_{k+1}=0.6 t_k\end{equation}

7. g. Halting the optimization process and outputting the results if the termination conditions are met; else, returning to step d to continue the optimization search

Table 1. Initial conditions of the interceptors

Figure 5. Simulation block diagram.