Many-to-many target distribution method based on improved dung beetle algorithm

Abstract

The invention relates to a many-to-many target distribution method based on an improved dung beetle algorithm. The method comprises the following steps: constructing a multi-missile-to-multi-target distribution model; optimizing the dung beetle optimization algorithm to construct an improved dung beetle optimization algorithm; and solving the multi-missile-to-multi-target distribution model based on an improved dung beetle optimization algorithm to obtain an optimal target distribution scheme. According to the method, the distribution of the initial population is optimized through Piecewise mapping, and the diversity of the population is increased. A parameter R in the algorithm is improved and adjusted into a parabola form to enlarge the area search range, so that the local premature convergence is slowed down, and the probability of falling into local optimum is reduced. And the Levy flight strategy is fused to balance the global exploration capability of the algorithm.

Description

Technical Field

[0001] This invention relates to the field of weapon target allocation technology, and specifically to a many-to-many target allocation method based on an improved dung beetle algorithm. Background Technology

[0002] Target allocation is a core decision-making problem in terminal air defense operations. The quality of target allocation directly determines whether enemy targets can be effectively intercepted and destroyed, and also relates to whether the combat effectiveness of air defense missile systems can be fully realized. Faced with increasingly complex battlefield situations and continuously expanding air attack scales, relying solely on manual target allocation is no longer sufficient to meet modern air defense needs. This approach not only fails to fully exploit the combat potential of air defense missile systems but may also lead to missed opportunities due to decision-making delays, ultimately resulting in the failure of the air defense mission. Therefore, how to quickly and efficiently generate interception plans based on real-time combat situations in complex, dynamic, and highly contested battlefield environments, and maximize the overall combat effectiveness of air defense missile systems, has become a core problem that urgently needs to be solved in current air defense combat systems. This problem is a typical NP-complete problem, characterized by multiple parameters and constraints, and can be reduced to a class of optimal combinatorial optimization problems, namely, finding the conditional extrema of the objective function on a finite set of solutions. Such problems typically contain a large number of local extrema, resulting in high solution complexity.

[0003] Currently, existing dung beetle optimization algorithms are prone to getting trapped in local optima when dealing with complex dynamic weapon target allocation problems, and also have limited global search capabilities. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the present invention aims to provide a many-to-many target allocation method based on an improved dung beetle algorithm.

[0005] To achieve the above objectives, the technical solution adopted by this invention is: a many-to-many target allocation method based on an improved dung beetle algorithm, comprising the following steps: S1. Construct a multi-missile-to-multi-target allocation model; S2. Optimize the dung beetle optimization algorithm to construct an improved dung beetle optimization algorithm; S3. Solve the multi-missile-to-multi-target allocation model based on the improved dung beetle optimization algorithm to obtain the optimal target allocation scheme.

[0006] Furthermore, step S1 includes calculating the threat index of the target point, calculating the basic strike effectiveness of the missile against the target point, and calculating the final combat effectiveness of the missile against the target point.

[0007] Furthermore, the formula for calculating the threat index of a target point is as follows: T = w t1 ·T a +w t2 ·Tv +w t3 ·T h +w t4 ·T r In the formula, T a As an angular threat factor, T v For speed threat factor, T h As a high-threat factor, T r As distance threat factors, the weights of each threat factor satisfy the following requirement: In the formula w ti The weight values ​​for each threat factor are determined using the analytic hierarchy process (AHP).

[0008] Furthermore, the formula for calculating the basic strike effectiveness of a missile against a target point is as follows: P = w f1 ·f(θ)+w f2 ·f(d)+w f3 ·f(v)+w f4 ·f(a) In the formula, f(θ) represents the effectiveness of the missile strike on the target due to the angle, f(d) represents the effectiveness of the missile strike on the target due to the range, f(v) represents the effectiveness of the missile strike on the target due to the velocity, and f(a) represents the effectiveness of the missile strike on the target due to the normal acceleration. The weights of each effectiveness factor satisfy: In the formula w fi These are the weight values ​​for each performance factor determined using the analytic hierarchy process (AHP).

[0009] Furthermore, the formula for calculating the final combat effectiveness of a missile against a target point is as follows: E = k e1 ·K+k e2 ·G Where K is the target value, G is the missile's effectiveness against the target point, and k e1 ,k e2 The weighting coefficients, which are greater than zero, can be given by expert experience. The formula for calculating the missile's strike effectiveness against the target point is as follows: G = P·T·(1-k) g ·D c ) In the formula, P represents the missile's basic strike effectiveness against the target point, T represents the threat index of the target point, and D... c The defensive capabilities of a target point can be assessed by experts based on different types of targets. g This is the coefficient affecting defensive capabilities, and it is 0. <k g <1.

[0010] Furthermore, step S2 includes the following steps: S21. Use Piecewise mapping to change the initial population position; S22. The parameter R in the dung beetle optimization algorithm is improved to a parabolic form; S23. After the dung beetle's position is updated, use the Levi's flight strategy to update the individual's position again.

[0011] Furthermore, in step S21, the calculation formula for the Piecewise mapping is as follows: Among them, P x It is a parameter of the mapping function, and its value range is (0, 0.5).

[0012] Furthermore, in step S22, the parameter R in the dung beetle optimization algorithm is improved into a parabolic calculation formula as follows: Where t is the current iteration number, and Γ is the maximum number of iterations allowed by the optimization algorithm.

[0013] Furthermore, in step S23, the calculation formula for Levy's flight strategy is as follows: x i (t+1)=x i (t)+Levy(β) Where α is the step size coefficient, and Levy(β) is the step size of the Levy random path.

[0014] The beneficial effects of this application are as follows: 1. This application uses Piecewise mapping to optimize the distribution of the initial population and increase population diversity. The parameter R in the algorithm is improved by adjusting it to a parabolic form to expand the search area, thereby slowing down premature local convergence and reducing the probability of getting trapped in local optima. Furthermore, the Lévy flight strategy is incorporated to balance the algorithm's global exploration capability.

[0015] 2. To address the high real-time requirements of dynamic weapon target allocation algorithms, a multi-missile-to-multi-target allocation model is constructed. This model includes the threat index of the target point, the basic strike effectiveness of the missile against the target point, and the final combat effectiveness of the missile against the target point, thereby reflecting the threat information of the attacking side in real time. Attached Figure Description

[0016] Figure 1 This is a flowchart of the improved dung beetle optimization algorithm of the present invention.

[0017] Figure 2 To improve the R parameter curve.

[0018] Figure 3 This is a simulation diagram of five aircraft striking four targets according to the present invention.

[0019] Figure 4 This is a simulation diagram of 10 aircraft of the present invention striking 4 targets.

[0020] Figure 5 This is a simulation diagram of 10 aircraft striking 8 targets according to the present invention.

[0021] Figure 6 This is a simulation diagram of 15 aircraft of the present invention striking 8 targets. Detailed Implementation

[0022] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0023] This invention provides a many-to-many target allocation method based on an improved dung beetle algorithm, comprising the following steps: S1. Construct a multi-missile-to-multi-target allocation model; S2. Optimize the dung beetle optimization algorithm to construct an improved dung beetle optimization algorithm; S3. Solve the multi-missile-to-multi-target allocation model based on the improved dung beetle optimization algorithm to obtain the optimal target allocation scheme.

[0024] Step S1 includes calculating the threat index of the target point, calculating the basic strike effectiveness of the missile against the target point, and calculating the final combat effectiveness of the missile against the target point.

[0025] This application constructs a confrontational multi-missile target allocation battlefield environment, in which the attacking side consists of fighter jets posing a substantial threat to the defending side. The defending side, acting as a weapon launch platform, can engage the attacking side. Addressing the high real-time requirements of Dynamic Weapon Target Allocation (DWTA) algorithms, a scientifically sound evaluation standard is constructed to reflect the attacking side's threat information in real time. A threat index is established based on angle information, speed, altitude, and relative distance.

[0026] The threat index of a target point is determined by the angle threat factor T. a Speed ​​threat factor T v High threat factor T h Distance Threat Factor T r The composition and calculation formulas for each threat factor are as follows.

[0027] The angle threat factor refers to the potential attack threat posed by the nose direction of an attacking aircraft to the defending aircraft during air combat. The formula for calculating the angle threat factor is as follows: In the formula, q is the angle between the attacking aircraft's velocity direction and the target line; Θ is half of the enemy radar's detection angle range. When the enemy enters the detection range, the angular threat value changes linearly.

[0028] The formula for calculating the speed threat factor is as follows: In the formula, V i V represents the speed at which the missile is launched. j Indicates the speed of the attacking side.

[0029] The formula for calculating the high threat factor is as follows: In the formula, h is the height of the target point above the ground.

[0030] The formula for calculating the distance threat factor is as follows: In the formula, d ij d represents the straight-line distance between the missile and the attacking force. lmin It is the inescapable distance of the enemy's weapons.

[0031] Each threat factor plays a different role in the comprehensive threat assessment decision-making process in a confrontational environment. The analytic hierarchy process (AHP) is used to evaluate the role of each threat factor. The weights of each threat factor satisfy the following formula: In the formula, w ti The weight values ​​for each threat factor are determined using the analytic hierarchy process (AHP).

[0032] The final threat index of the target point is: T = w t1 ·T a +w t2 ·T v +w t3 ·T h +w t4 ·T r In the formula, T a As an angular threat factor, T v For speed threat factor, T h As a high-threat factor, T r This is a distance threat factor.

[0033] The basic threat level of a target point is an important basis for assessing its danger level, and when selecting a target, a missile must also comprehensively consider its effect on the target point. The basic effectiveness of a missile on a target point mainly depends on the following four factors: angle, distance, velocity, and normal acceleration.

[0034] The formula for calculating the effect of angle on the effectiveness of missile strike on the target is as follows: In the formula, θ is the angle between the missile's velocity direction and the line connecting them. If the angle is too large, the missile will not be able to hit the target point.

[0035] The formula for calculating the effectiveness of a missile in striking a target based on distance is as follows: In the formula, d is the distance between the missile and the target point. Both excessively large and excessively small distances between the missile and the target point will reduce the accuracy of the missile in hitting the target.

[0036] The formula for calculating the effect of velocity on the effectiveness of a missile in striking a target is as follows: In the formula, v is the ratio of the missile's velocity to the target's velocity. When the ratio is less than 1, the missile is unlikely to hit the target; only when the ratio is greater than 1 is the missile's speed high enough to have a chance of hitting the target.

[0037] The formula for calculating the effectiveness of normal acceleration in missile strikes is as follows: In the formula, 'a' represents the ratio of the missile's normal acceleration to the target's normal acceleration. When the missile's rate of change of velocity is less than the target's rate of change of velocity, the missile will not hit the target. As the ratio increases, the probability of the missile hitting the target increases.

[0038] Based on the above conditions, the basic strike effectiveness of a missile against a target is as follows: P = w f1 ·f(θ)+w f2 ·f(d)+w f3 ·f(v)+w f4 ·f(a) In the formula, f(θ) represents the effectiveness of the missile strike on the target due to the angle, f(d) represents the effectiveness of the missile strike on the target due to the range, f(v) represents the effectiveness of the missile strike on the target due to the velocity, and f(a) represents the effectiveness of the missile strike on the target due to the normal acceleration. The weights of each effectiveness factor satisfy: In the formula w fi These are the weight values ​​for each performance factor determined using the analytic hierarchy process (AHP).

[0039] The effectiveness of a missile strike on a target is also related to the defensive capabilities of the target point, which are as follows: D c =w d1 ·B1+w d2 ·B2+wd3 ·B3 Among them, B1, B2, and B3 represent the target detection capability, electronic jamming capability, and interception capability, respectively. d1 +w d2 +w d3 If the value is 1, then the missile's effectiveness against the target is: G = P·T·(1-k) g ·D c ) Where P represents the missile's basic strike effectiveness against the target point, T represents the threat index of the target point, and D... c For the defensive capabilities of the target point, k g This is the coefficient affecting defensive capabilities, and it is 0. <k g <1.

[0040] Furthermore, different target points have different values ​​and different impacts on combat. The formula for calculating the final combat effectiveness of a missile against a target point is as follows: E = k e1 ·K+k e2 ·G Where K is the target value, G is the missile's effectiveness against the target point, and k e1 ,k e2 Weighting coefficients that are greater than zero can be given by expert experience.

[0041] This application optimizes the dung beetle optimization algorithm to construct an improved dung beetle optimization algorithm.

[0042] The Dung Beetle Optimization Algorithm (DBO) is a novel swarm intelligence-based optimization algorithm proposed by Xue et al. in 2023. By simulating different survival strategies of dung beetles, the DBO algorithm balances global exploration and local exploitation, exhibiting fast convergence speed and high solution accuracy, making it suitable for missile target allocation. Based on various behaviors, the entire dung beetle population can be divided into rolling dung beetles, egg-incubating dung beetles, small dung beetles, and thieving dung beetles.

[0043] The rolling dung beetle is responsible for rolling its dung ball according to the orientation of celestial bodies (especially the sun, moon, and polarized light) within the population, ensuring it moves as straight as possible to avoid competition with other dung beetles. The rolling dung beetle's position update strategy can be divided into two categories: obstacle-free update strategy and obstacle-based update strategy. When the dung beetle encounters no obstacles in its search space, its position update strategy can be described as follows: x i (t+1)=x i (t)+α×k×x i (t-1)+b×Δx Δx=|x i (t)-X w | Where t represents the current iteration number, x i (t) represents the position information of the i-th dung beetle in the t-th iteration, k∈(0,0.2] represents the deflection coefficient, b represents a constant value belonging to (0,1), α is the natural coefficient, assigned a value of -1 or 1, X w This represents the worst-case position globally, and Δx is used to simulate changes in light intensity.

[0044] When a dung beetle encounters an obstacle and cannot move forward, it needs to reposition itself by dancing to find a new route. The position update at this point is determined by the tangent function. Once the dung beetle successfully determines a new direction, it continues rolling the ball. The dung beetle's position update equation is as follows: x i (t+1)=x i (t)+tanθ|x i (t)-x i (t-1)| Where θ is the deflection angle, and its value ranges from [0, π]. |x i (t)-x i (t-1)| represents the difference between the position of the i-th dung beetle in the t-th iteration and its position in the (t-1)-th iteration. When θ = 0, π / 2, π, tanθ is meaningless, and the dung beetle's position remains unchanged.

[0045] In nature, dung balls are rolled by dung beetles to safe areas. To provide a safe environment for their offspring, the boundary selection strategy simulating the female dung beetle's choice of where to lay eggs is as follows: Lb * =max(X * ×(1-R),Lb) Ub * =min(X) * ×(1+R),Ub) R = 1 - t / T max In the equation, X * Lb represents the current local optimum. * Ub represents the lower boundary of the spawning area. * This represents the upper bound of the spawning region. Lb and Ub represent the lower and upper bounds of the search space of the optimization problem. T max This indicates the maximum number of iterations.

[0046] Once a safe location is found, the female dung beetle will lay her eggs in that area. In the dung beetle algorithm, each female lays only one egg per iteration. The boundary of the egg-laying area is dynamically changed, mainly determined by the R value. The iterative update of the egg's location is as follows: x i (t+1)=X *+b1·(x i (t)-Lb * )+b2·(x i (t)-Ub * ) In the formula, x i (t) represents the position of the i-th dung beetle in the i-th iteration, and b1 and b2 represent two independent random variables of size 1×D, where D is the dimension of the optimization problem. The position of the egg is restricted to the oviposition region.

[0047] After the breeding balls successfully hatch, they will soon grow into adults and search for food; we call them dung beetles. Therefore, it is necessary to establish an optimal foraging area to guide the dung beetles to find food. The foraging area also utilizes a dynamic boundary strategy, with the boundary defined as follows: X b Lb represents the optimal global feeding position. b and Ub b The position update definition for the dung beetle is: (The lower and upper bounds of the optimal feeding area are used as the basis for the position update definition.) x i (t+1)=x i (t)+C1(x i (t)-Lb b )+C2(x i (t)-Ub b ) Where, x i (t) represents the position information of the t-th dung beetle in the i-th iteration, C1 represents a random number that follows a normal distribution, and C2 represents a random vector belonging to (0,1).

[0048] Some dung beetles are called dung-stealing dung beetles; they steal dung balls from other dung beetles, a common phenomenon in nature. b This is the optimal location for a food source. Therefore, we assume X... b This represents the optimal location for competing for food. During the iteration process, the dung beetle's location update is defined as: x i (t+1)=X b +S×g×(|x i (t)-X * |+|x i (t)-X b |) Where, x i (t) represents the position information of the i-th thieving mantis in the t-th iteration, g is a random vector of size 1×D that follows a normal distribution, and S represents a constant value.

[0049] As an intelligent optimization algorithm, the Dung Beetle Algorithm (DBO) is a stochastic search algorithm that essentially generates an initial population through random initialization. However, this process often encounters problems such as uneven distribution of individual dung beetles, weak global exploration ability, low population diversity, and susceptibility to local optima. To address these issues, chaotic mapping is used in the initialization phase of the DBO population to generate a highly diverse initial population. Chaotic mapping is used to generate chaotic sequences, which are random sequences generated by simple deterministic systems and possess characteristics such as randomness, ergodicity, and regularity. Introducing chaotic mapping during population initialization allows individuals to fully utilize information in the solution space, thereby improving global search ability. This application employs Piecewise mapping to change the position of the initial population.

[0050] Piecewise mapping is used to change the initial population position. The segmentation mechanism of piecewise mapping can effectively delay the algorithm from entering the local exploration stage too early, maintain the exploratory vitality of the population in the early stages of iteration, and thus reduce the probability of getting trapped in local optima.

[0051] The formula for calculating the Piecewise mapping is as follows: Among them, P x These are mapping function parameters, as shown in embodiment P of this application. x When the value is 0.4, the dung beetles are relatively evenly distributed.

[0052] Nonlinear control parameters help balance exploration and extraction capabilities. The optimal search area shrinks linearly with the parameter R; however, excessively rapid shrinkage hinders the algorithm's ability to find the global optimum. Therefore, a parabolic shape parameter is proposed to replace parameter R, thereby widening the local optimum region. The update equation for the parabolic adaptive control parameter R is shown below: Where R controls the change in the local optimal region, t is the current iteration number, and Γ is the maximum number of iterations allowed by the optimization algorithm. The curve of the parabolic adaptive parameter R is shown below. Figure 2 As shown.

[0053] After the dung beetle's location is updated, the individual's location is updated again using the Levy flight strategy.

[0054] Aside from optimizing the initialization phase, the position update equations for breeding beetles, foraging beetles, and dung-stealing beetles show a close correlation between their position updates and the current individual optimum. During iteration, this characteristic can lead to premature convergence and being subject to local optima. To address this issue, we apply the Levy flight strategy to the position updates of breeding beetles, foraging beetles, and dung-stealing beetles.

[0055] The Levy flight strategy is a probability distribution with heavy-tailed characteristics. By introducing the steps of taxed flight, dung beetles can randomly roam the search space, thus enhancing the algorithm's global exploration. That is, after updating the original positions, a taxed flight strategy is used again to update the positions, ensuring the diversity of the dung beetle population in the later stages of the algorithm and expanding its search capabilities. The Levy flight strategy is as follows: x i (t+1)=x i (t)+Levy(β) Where Levy(β) is the step size of the Levy random path, which we define as: Where u and v follow a random normal distribution, and Γ(x) = (x-1)! is the gamma function. After incorporating the Levy flight strategy, the position update formulas for the female dung beetle, the baby dung beetle, and the thieving dung beetle are as follows: x i (t+1)=X * +b1·(x i (t)-Lb * )+b2·(x i (t)-Ub * )+α*Levy(β) x i (t+1)=x i (t)+C1(x i (t)-Lb b )+C2(x i (t)-Ub b )+α*Levy(β) x i (t+1)=X b +S×g×(|x i (t)-X * |+|x i (t)-X b |)+α*Levy(β) Where α is the step size coefficient.

[0056] To verify the optimization capability of the dung beetle algorithm in solving the DWTA problem, the air combat environment described in the problem description was used as a background, and referenced... Figure 3-6 As shown, simulations of four scenarios were implemented, namely 5 aircraft attacking 4 targets, 10 aircraft attacking 4 targets, 10 aircraft attacking 8 targets, and 15 aircraft attacking 8 targets. The algorithm performance was systematically analyzed and verified at different scales.

[0057] It can be seen that, under different computational scales, for the proposed multi-constraint programming model, the improved dung beetle-based target allocation algorithm can converge quickly in approximately 80 iterations, and provides reasonable target allocation results based on combat effectiveness. Table 1 shows the system performance analysis of the improved dung beetle algorithm in multiple scenarios.

[0058] Table 1 Performance Analysis of the Improved Dung Beetle Algorithm System in Multiple Scenarios

[0059] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes, and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.

Claims

1. A many-to-many target allocation method based on an improved dung beetle algorithm, characterized in that, Includes the following steps: S1. Construct a multi-missile-to-multi-target allocation model; S2. Optimize the dung beetle optimization algorithm to construct an improved dung beetle optimization algorithm; S3. Solve the multi-missile-to-multi-target allocation model based on the improved dung beetle optimization algorithm to obtain the optimal target allocation scheme.

2. The many-to-many target allocation method based on the improved dung beetle algorithm according to claim 1, characterized in that, Step S1 includes calculating the threat index of the target point, calculating the basic strike effectiveness of the missile against the target point, and calculating the final combat effectiveness of the missile against the target point.

3. The many-to-many target allocation method based on the improved dung beetle algorithm according to claim 2, characterized in that, The formula for calculating the threat index of a target point is as follows: T=w t1 ·T a +w t2 ·T v +w t3 ·T h +w t4 ·T r In the formula, T a As an angular threat factor, T v For speed threat factor, T h As a high-threat factor, T r As distance threat factors, the weights of each threat factor satisfy the following requirement: In the formula w ti The weight values ​​for each threat factor are determined using the analytic hierarchy process (AHP).

4. A many-to-many target allocation method based on an improved dung beetle algorithm according to claim 2, characterized in that, The formula for calculating the basic strike effectiveness of a missile against a target point is as follows: P=w f1 ·f(θ)+w f2 ·f(d)+w f3 ·f(v)+w f4 ·f(a) In the formula, f(θ) represents the effectiveness of the missile strike on the target due to the angle, f(d) represents the effectiveness of the missile strike on the target due to the range, f(v) represents the effectiveness of the missile strike on the target due to the velocity, and f(a) represents the effectiveness of the missile strike on the target due to the normal acceleration. The weights of each effectiveness factor satisfy: In the formula, w fi These are the weight values ​​for each performance factor determined using the analytic hierarchy process (AHP).

5. A many-to-many target allocation method based on an improved dung beetle algorithm according to claim 2, characterized in that, The formula for calculating the final combat effectiveness of a missile against a target point is: E=k e1 ·K+k e2 ·G Where K is the target value, G is the missile's effectiveness against the target point, and k e1 ,k e2 All are weighting coefficients that are greater than zero. The formula for calculating the missile's strike effectiveness against a target point is as follows: G=P·T·(1-k g ·D c ) In the formula, P represents the missile's basic strike effectiveness against the target point, T represents the threat index of the target point, and D... c For the defensive capabilities of the target point, k g This is the coefficient affecting defensive capabilities, and it is 0. <k g <1.

6. A many-to-many target allocation method based on an improved dung beetle algorithm according to claim 3, characterized in that, Step S2 includes the following steps: S21. Use Piecewise mapping to change the initial population position; S22. The parameter R in the dung beetle optimization algorithm is improved to a parabolic form; S23. After the dung beetle's position is updated, use the Levi's flight strategy to update the individual's position again.

7. A many-to-many target allocation method based on an improved dung beetle algorithm according to claim 3, characterized in that, In step S21, the calculation formula for the Piecewise mapping is as follows: Among them, P x It is a parameter of the mapping function, and its value range is (0, 0.5).

8. A many-to-many target allocation method based on an improved dung beetle algorithm according to claim 3, characterized in that, In step S22, the parameter R in the dung beetle optimization algorithm is improved into a parabolic calculation formula as follows: Where t is the current iteration number, and Γ is the maximum number of iterations allowed by the optimization algorithm.

9. A many-to-many target allocation method based on an improved dung beetle algorithm according to claim 3, characterized in that, In step S23, the formula for calculating Levi's flight strategy is as follows: x i (t+1)=x i (t)+Levy(β) Where α is the step size coefficient, and Levy(β) is the step size of the Levy random path.

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