# A New Approach to Weapon-Target Assignment in Cooperative Air Combat

Authors: Yi-zhe Chang; Zhan-wu Li; Ying-xin Kou; Qing-peng Sun; Hai-yan Yang; Zheng-yan Zhao
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

A new approach to solving weapon-target assignment (WTA) problem is proposed in this paper. Firstly, relative superiority that lays the foundation for assignment is calculated based on the combat power energy of the fighters. Based on the relative superiority, WTA problem is formulated. Afterwards, a hybrid algorithm consisting of improved artificial fish swarm algorithm (AFSA) and improved harmony search (HS) is introduced and furthermore applied to solve the assignment formulation. Finally, the proposed approach is validated by eight representative benchmark functions and two concrete cooperative air combat examples. The results show that the approach proposed in this paper achieves good performances in solving WTA problem in cooperative air combat.


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## Body

## 1. Introduction

Weapon-target assignment (WTA) refers to an assignment of defensive weapons to engage or counter identified threats. The primary concern is minimizing the total expected survivability of the targets [1, 2].Weapons are assigned to threats based on the detection outcomes within previous stages. Considering the short time of air combat, the problem must be solved as close to real time as possible. Another characteristic is quality of the derived solution, which causes critical effects on the following combat deployment ‎[3].WTA is regarded as a crucial part in operation; therefore the achievements on that are rich. In the matter of problem formulation, multiple factors such as relative distance, relative angle, and relative velocity are taken into consideration. In domain of algorithms, several sophisticated search and heuristic algorithms have been proposed, such as genetic algorithms [4–7], simulated annealing ‎[8], discrete particle swarm optimizations [9–12], permutation and tabu search heuristics‎ [13], and other algorithms [14–16].Concerned with the characteristics and developing tendency about air combat, an approach to WTA in cooperative air combat is proposed in this paper. Within the approach, relative superiority between fighters and targets, acting as the foundation of the formulation, is calculated in terms of combat power potential, and a hybrid algorithm combined with improved artificial fish swarm algorithm with improved harmony search for solving the formulation is introduced.The concept of combat power potential is firstly introduced by Zhou et al. ‎[17]. As described in ‎[17], air combat space can be seen as a potential consisting of infinite points and each point is under effect by fighters. The effect, generated by fighters and equipment carried by them, is quantitatively defined as potential energy called combat power potential energy. Combat power potential energy reflects the impact on each point inside the air combat space by fighters.The artificial fish swarm algorithm (AFSA) simulating the behavior of a fish inside water was proposed by Li‎ et al. [18] and has already been applied in engineering contexts [19–24]. The main artificial fish behaviors are random, chasing, swarming, and searching, which allows the algorithm to act better in global searching but poor in local searching.Harmony search, mimicking the improvisation process of music players, is originated by Geem‎ et al. [25]. The HS performs excellently in quality of the solutions but with low search speed and is strongly dependent on initial solution.This paper is organized as follows: In Section2, the formulation of WTA is introduced. The design of hybrid algorithm is studied in Section 3. Section 4 presents the results of simulation. Section 5 concludes the paper.


## 2. WTA Problem Formulation

### 2.1. Analysis on Relative Superiority

Superiority relation acts as the critical ingredient in WTA formulation. In this paper, superiority relation is formulated in accordance with the descending velocity of combat power potential energy possessed by one side towards the other side.


#### 2.1.1. Combat Power Potential Energy Model of Fighter

Due to the limitation on pages, combat power potential energy of fighter can be calculated by model of which in ‎[26].


#### 2.1.2. Calculation on Descending Velocity

We get to know from the potential theory that potential energy will be descending while distance is getting larger. As a kind of potential, combat power potential has the same characteristics, which means combat power potential energy generated by fighters descends with more restriction, such as distance limits and angle limits. For calculating descending velocity, gradient is introduced in the paper. Assuming a function described asf=fP, P=(x,y,z), then gradient of function f in point P can be defined as(1)gradfP=∇f=∂f∂x,∂f∂y,∂f∂z.It is known from‎ [26] that combat power potential energy of fighter is a function with regard to coordinates of fighters and targets (x,y,z,xT,yT,zT). Calculate the gradient of e depicted in ‎[26] on xT, yT, zT and then we can get(2)∂exT,yT,zT∂xT=∂eDxT,yT,zT∂xTω1eWxT,yT,zT+ω2eExT,yT,zT+eDxT,yT,zTω1∂eWxT,yT,zT∂xT+ω2∂eExT,yT,zT∂xT,∂exT,yT,zT∂yT=∂eDxT,yT,zT∂yTω1eWxT,yT,zT+ω2eExT,yT,zT+eDxT,yT,zTω1∂eWxT,yT,zT∂yT+ω2∂eExT,yT,zT∂yT,∂exT,yT,zT∂zT=∂eDxT,yT,zT∂zTω1eWxT,yT,zT+ω2eExT,yT,zT+eDxT,yT,zTω1∂eWxT,yT,zT∂zT+ω2∂eExT,yT,zT∂zT.Take∂exT,yT,zT/∂xT, for instance; through further derivation, we will obtain(3)∂exT,yT,zT∂xT=∂ln⁡1+ADRxT,yT,zT+ADIRxT,yT,zT∂xT·ω1ln1+AMxT,yT,zT+ω2ln⁡1+ADISxT,yT,zT+ln⁡1+ADRxT,yT,zT+ADIRxT,yT,zT·ω1∂ln⁡1+AMxT,yT,zT∂xT+ω2∂ln1+ADISxT,yT,zT∂xT.For losing the burden of calculation,ADRxT,yT,zT, ADIRxT,yT,zT, AMxT,yT,zT, ADISxT,yT,zT are simplified as ADRxT,yT,zT=C1/rxT,yT,zT, ADIRxT,yT,zT=C2/rxT,yT,zT, AMxT,yT,zT=C3/rxT,yT,zT, ADISxT,yT,zT=C4/rxT,yT,zT, where rxT,yT,zT=(xA-xT)2+(yA-yT)2+(zA-zT)2. Substitute simplified ADRxT,yT,zT, ADIRxT,yT,zT, AMxT,yT,zT, ADISxT,yT,zT into (3); we will derive(4)∂exT,yT,zT∂xT=∂ln⁡1+C1/rxT,yT,zT+C2/rxT,yT,zT∂xT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1rxT,yT,zT+C2rxT,yT,zT·ω1∂ln⁡1+C3/rxT,yT,zT∂xT+ω2∂ln⁡1+C4/rxT,yT,zT∂xT=∂ln⁡1+C1/rxT,yT,zT+C2/rxT,yT,zT∂rxT,yT,zT·∂rxT,yT,zT∂xT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1rxT,yT,zT+C2rxT,yT,zT·ω1·∂ln⁡1+C3/rxT,yT,zT∂rxT,yT,zT·∂rxT,yT,zT∂xT+ω2·∂ln⁡1+C4/rxT,yT,zT∂rxT,yT,zT·∂rxT,yT,zT∂xT=rxT,yT,zTrxT,yT,zT+C1+C2·-C1+C2r2xT,yT,zT·2xA-xT2rxT,yT,zT·-1·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1rxT,yT,zT+C2rxT,yT,zT·ω1·rxT,yT,zTrxT,yT,zT+C3·-C3r2xT,yT,zT·2xA-xT2rxT,yT,zT·-1+ω2·rxT,yT,zTrxT,yT,zT+C4·-C4r2xT,yT,zT·2xA-xT2rxT,yT,zT·-1=xA-xT·C1+C2rxT,yT,zT+C1+C2·r2xT,yT,zT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln1+C1+C2rxT,yT,zT·ω1·xA-xT·C3rxT,yT,zT+C3·r2xT,yT,zT+ω2·xA-xT·C4rxT,yT,zT+C4·r2xT,yT,zT.∂
e
x
T
,
y
T
,
z
T
/
∂
y
T and ∂exT,yT,zT/∂zT are yielded in the same way as follows:(5)∂exT,yT,zT∂yT=yA-yT·C1+C2rxT,yT,zT+C1+C2·r2xT,yT,zT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1+C2rxT,yT,zT·ω1·yA-yT·C3rxT,yT,zT+C3·r2xT,yT,zT+ω2·yA-yT·C4rxT,yT,zT+C4·r2xT,yT,zT,∂exT,yT,zT∂zT=zA-zT·C1+C2rxT,yT,zT+C1+C2·r2xT,yT,zT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1+C2rxT,yT,zT·ω1·zA-zT·C3rxT,yT,zT+C3·r2xT,yT,zT+ω2·zA-zT·C4rxT,yT,zT+C4·r2xT,yT,zT.Descending velocity of fighterA in direction of target T is got in the light of definition of gradient: (6)vdxA,yA,zAxT,yT,zT=gradexT,yT,zT=∂exT,yT,zT∂xT2+∂exT,yT,zT∂yT2+∂exT,yT,zT∂zT2.Similarly, descending velocity of targetT in direction of fighter A is given as(7)vdxT,yT,zTxA,yA,zA=gradexA,yA,zA=∂exA,yA,zA∂xA2+∂exA,yA,zA∂yA2+∂exA,yA,zA∂zA2.


#### 2.1.3. Calculation on Relative Superiority

Calculation on relative superiority of fighteri with respect to target j is expressed as(8)pij=vdxAi,yAi,zAixTj,yTj,zTjvdxAi,yAi,zAixTj,yTj,zTj+vdxTj,yTj,zTjxAi,yAi,zAi,where vdxAi,yAi,zAixTj,yTj,zTj is the descending velocity of fighter i in direction to location of target j and vdxTj,yTj,zTjxAi,yAi,zAi is the descending velocity of target j in direction to location of fighter i.


### 2.2. WTA Formulation

In the paper, the problem is formulated as a combinatorial optimization problem, with the objective of minimizing the total lethality of the targets. The fundamental problem variable is relative superioritypij.Consider the assignment problem ofm fighters to n targets: let pij∈[0,1] be the relative superiority, where i∈{1,2,…,m} and j∈{1,2,…,n}. The WTA is formulated as follows: (9)J=min∑j=1n∏i=1m1-pijxijs.t.∑i=1mxij≥1,∑j=1nxij≤1,where xij is the decision value indicating if fighter i is assigned to target j; xij=1 if fighter i is assigned to target j and xij=0 if fighter i is not assigned to target j.In the above formulation, we minimize the sum of the weighted cumulative probabilities of threat lethality while ensuring that all constraints are satisfied. Two constraints, respectively, denote that each target must be assigned by one fighter and each fighter can be assigned to one target at most.


## 2.1. Analysis on Relative Superiority

Superiority relation acts as the critical ingredient in WTA formulation. In this paper, superiority relation is formulated in accordance with the descending velocity of combat power potential energy possessed by one side towards the other side.


### 2.1.1. Combat Power Potential Energy Model of Fighter

Due to the limitation on pages, combat power potential energy of fighter can be calculated by model of which in ‎[26].


### 2.1.2. Calculation on Descending Velocity

We get to know from the potential theory that potential energy will be descending while distance is getting larger. As a kind of potential, combat power potential has the same characteristics, which means combat power potential energy generated by fighters descends with more restriction, such as distance limits and angle limits. For calculating descending velocity, gradient is introduced in the paper. Assuming a function described asf=fP, P=(x,y,z), then gradient of function f in point P can be defined as(1)gradfP=∇f=∂f∂x,∂f∂y,∂f∂z.It is known from‎ [26] that combat power potential energy of fighter is a function with regard to coordinates of fighters and targets (x,y,z,xT,yT,zT). Calculate the gradient of e depicted in ‎[26] on xT, yT, zT and then we can get(2)∂exT,yT,zT∂xT=∂eDxT,yT,zT∂xTω1eWxT,yT,zT+ω2eExT,yT,zT+eDxT,yT,zTω1∂eWxT,yT,zT∂xT+ω2∂eExT,yT,zT∂xT,∂exT,yT,zT∂yT=∂eDxT,yT,zT∂yTω1eWxT,yT,zT+ω2eExT,yT,zT+eDxT,yT,zTω1∂eWxT,yT,zT∂yT+ω2∂eExT,yT,zT∂yT,∂exT,yT,zT∂zT=∂eDxT,yT,zT∂zTω1eWxT,yT,zT+ω2eExT,yT,zT+eDxT,yT,zTω1∂eWxT,yT,zT∂zT+ω2∂eExT,yT,zT∂zT.Take∂exT,yT,zT/∂xT, for instance; through further derivation, we will obtain(3)∂exT,yT,zT∂xT=∂ln⁡1+ADRxT,yT,zT+ADIRxT,yT,zT∂xT·ω1ln1+AMxT,yT,zT+ω2ln⁡1+ADISxT,yT,zT+ln⁡1+ADRxT,yT,zT+ADIRxT,yT,zT·ω1∂ln⁡1+AMxT,yT,zT∂xT+ω2∂ln1+ADISxT,yT,zT∂xT.For losing the burden of calculation,ADRxT,yT,zT, ADIRxT,yT,zT, AMxT,yT,zT, ADISxT,yT,zT are simplified as ADRxT,yT,zT=C1/rxT,yT,zT, ADIRxT,yT,zT=C2/rxT,yT,zT, AMxT,yT,zT=C3/rxT,yT,zT, ADISxT,yT,zT=C4/rxT,yT,zT, where rxT,yT,zT=(xA-xT)2+(yA-yT)2+(zA-zT)2. Substitute simplified ADRxT,yT,zT, ADIRxT,yT,zT, AMxT,yT,zT, ADISxT,yT,zT into (3); we will derive(4)∂exT,yT,zT∂xT=∂ln⁡1+C1/rxT,yT,zT+C2/rxT,yT,zT∂xT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1rxT,yT,zT+C2rxT,yT,zT·ω1∂ln⁡1+C3/rxT,yT,zT∂xT+ω2∂ln⁡1+C4/rxT,yT,zT∂xT=∂ln⁡1+C1/rxT,yT,zT+C2/rxT,yT,zT∂rxT,yT,zT·∂rxT,yT,zT∂xT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1rxT,yT,zT+C2rxT,yT,zT·ω1·∂ln⁡1+C3/rxT,yT,zT∂rxT,yT,zT·∂rxT,yT,zT∂xT+ω2·∂ln⁡1+C4/rxT,yT,zT∂rxT,yT,zT·∂rxT,yT,zT∂xT=rxT,yT,zTrxT,yT,zT+C1+C2·-C1+C2r2xT,yT,zT·2xA-xT2rxT,yT,zT·-1·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1rxT,yT,zT+C2rxT,yT,zT·ω1·rxT,yT,zTrxT,yT,zT+C3·-C3r2xT,yT,zT·2xA-xT2rxT,yT,zT·-1+ω2·rxT,yT,zTrxT,yT,zT+C4·-C4r2xT,yT,zT·2xA-xT2rxT,yT,zT·-1=xA-xT·C1+C2rxT,yT,zT+C1+C2·r2xT,yT,zT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln1+C1+C2rxT,yT,zT·ω1·xA-xT·C3rxT,yT,zT+C3·r2xT,yT,zT+ω2·xA-xT·C4rxT,yT,zT+C4·r2xT,yT,zT.∂
e
x
T
,
y
T
,
z
T
/
∂
y
T and ∂exT,yT,zT/∂zT are yielded in the same way as follows:(5)∂exT,yT,zT∂yT=yA-yT·C1+C2rxT,yT,zT+C1+C2·r2xT,yT,zT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1+C2rxT,yT,zT·ω1·yA-yT·C3rxT,yT,zT+C3·r2xT,yT,zT+ω2·yA-yT·C4rxT,yT,zT+C4·r2xT,yT,zT,∂exT,yT,zT∂zT=zA-zT·C1+C2rxT,yT,zT+C1+C2·r2xT,yT,zT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1+C2rxT,yT,zT·ω1·zA-zT·C3rxT,yT,zT+C3·r2xT,yT,zT+ω2·zA-zT·C4rxT,yT,zT+C4·r2xT,yT,zT.Descending velocity of fighterA in direction of target T is got in the light of definition of gradient: (6)vdxA,yA,zAxT,yT,zT=gradexT,yT,zT=∂exT,yT,zT∂xT2+∂exT,yT,zT∂yT2+∂exT,yT,zT∂zT2.Similarly, descending velocity of targetT in direction of fighter A is given as(7)vdxT,yT,zTxA,yA,zA=gradexA,yA,zA=∂exA,yA,zA∂xA2+∂exA,yA,zA∂yA2+∂exA,yA,zA∂zA2.


### 2.1.3. Calculation on Relative Superiority

Calculation on relative superiority of fighteri with respect to target j is expressed as(8)pij=vdxAi,yAi,zAixTj,yTj,zTjvdxAi,yAi,zAixTj,yTj,zTj+vdxTj,yTj,zTjxAi,yAi,zAi,where vdxAi,yAi,zAixTj,yTj,zTj is the descending velocity of fighter i in direction to location of target j and vdxTj,yTj,zTjxAi,yAi,zAi is the descending velocity of target j in direction to location of fighter i.


## 2.1.1. Combat Power Potential Energy Model of Fighter

Due to the limitation on pages, combat power potential energy of fighter can be calculated by model of which in ‎[26].


## 2.1.2. Calculation on Descending Velocity

We get to know from the potential theory that potential energy will be descending while distance is getting larger. As a kind of potential, combat power potential has the same characteristics, which means combat power potential energy generated by fighters descends with more restriction, such as distance limits and angle limits. For calculating descending velocity, gradient is introduced in the paper. Assuming a function described asf=fP, P=(x,y,z), then gradient of function f in point P can be defined as(1)gradfP=∇f=∂f∂x,∂f∂y,∂f∂z.It is known from‎ [26] that combat power potential energy of fighter is a function with regard to coordinates of fighters and targets (x,y,z,xT,yT,zT). Calculate the gradient of e depicted in ‎[26] on xT, yT, zT and then we can get(2)∂exT,yT,zT∂xT=∂eDxT,yT,zT∂xTω1eWxT,yT,zT+ω2eExT,yT,zT+eDxT,yT,zTω1∂eWxT,yT,zT∂xT+ω2∂eExT,yT,zT∂xT,∂exT,yT,zT∂yT=∂eDxT,yT,zT∂yTω1eWxT,yT,zT+ω2eExT,yT,zT+eDxT,yT,zTω1∂eWxT,yT,zT∂yT+ω2∂eExT,yT,zT∂yT,∂exT,yT,zT∂zT=∂eDxT,yT,zT∂zTω1eWxT,yT,zT+ω2eExT,yT,zT+eDxT,yT,zTω1∂eWxT,yT,zT∂zT+ω2∂eExT,yT,zT∂zT.Take∂exT,yT,zT/∂xT, for instance; through further derivation, we will obtain(3)∂exT,yT,zT∂xT=∂ln⁡1+ADRxT,yT,zT+ADIRxT,yT,zT∂xT·ω1ln1+AMxT,yT,zT+ω2ln⁡1+ADISxT,yT,zT+ln⁡1+ADRxT,yT,zT+ADIRxT,yT,zT·ω1∂ln⁡1+AMxT,yT,zT∂xT+ω2∂ln1+ADISxT,yT,zT∂xT.For losing the burden of calculation,ADRxT,yT,zT, ADIRxT,yT,zT, AMxT,yT,zT, ADISxT,yT,zT are simplified as ADRxT,yT,zT=C1/rxT,yT,zT, ADIRxT,yT,zT=C2/rxT,yT,zT, AMxT,yT,zT=C3/rxT,yT,zT, ADISxT,yT,zT=C4/rxT,yT,zT, where rxT,yT,zT=(xA-xT)2+(yA-yT)2+(zA-zT)2. Substitute simplified ADRxT,yT,zT, ADIRxT,yT,zT, AMxT,yT,zT, ADISxT,yT,zT into (3); we will derive(4)∂exT,yT,zT∂xT=∂ln⁡1+C1/rxT,yT,zT+C2/rxT,yT,zT∂xT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1rxT,yT,zT+C2rxT,yT,zT·ω1∂ln⁡1+C3/rxT,yT,zT∂xT+ω2∂ln⁡1+C4/rxT,yT,zT∂xT=∂ln⁡1+C1/rxT,yT,zT+C2/rxT,yT,zT∂rxT,yT,zT·∂rxT,yT,zT∂xT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1rxT,yT,zT+C2rxT,yT,zT·ω1·∂ln⁡1+C3/rxT,yT,zT∂rxT,yT,zT·∂rxT,yT,zT∂xT+ω2·∂ln⁡1+C4/rxT,yT,zT∂rxT,yT,zT·∂rxT,yT,zT∂xT=rxT,yT,zTrxT,yT,zT+C1+C2·-C1+C2r2xT,yT,zT·2xA-xT2rxT,yT,zT·-1·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1rxT,yT,zT+C2rxT,yT,zT·ω1·rxT,yT,zTrxT,yT,zT+C3·-C3r2xT,yT,zT·2xA-xT2rxT,yT,zT·-1+ω2·rxT,yT,zTrxT,yT,zT+C4·-C4r2xT,yT,zT·2xA-xT2rxT,yT,zT·-1=xA-xT·C1+C2rxT,yT,zT+C1+C2·r2xT,yT,zT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln1+C1+C2rxT,yT,zT·ω1·xA-xT·C3rxT,yT,zT+C3·r2xT,yT,zT+ω2·xA-xT·C4rxT,yT,zT+C4·r2xT,yT,zT.∂
e
x
T
,
y
T
,
z
T
/
∂
y
T and ∂exT,yT,zT/∂zT are yielded in the same way as follows:(5)∂exT,yT,zT∂yT=yA-yT·C1+C2rxT,yT,zT+C1+C2·r2xT,yT,zT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1+C2rxT,yT,zT·ω1·yA-yT·C3rxT,yT,zT+C3·r2xT,yT,zT+ω2·yA-yT·C4rxT,yT,zT+C4·r2xT,yT,zT,∂exT,yT,zT∂zT=zA-zT·C1+C2rxT,yT,zT+C1+C2·r2xT,yT,zT·ω1ln⁡1+C3rxT,yT,zT+ω2ln⁡1+C4rxT,yT,zT+ln⁡1+C1+C2rxT,yT,zT·ω1·zA-zT·C3rxT,yT,zT+C3·r2xT,yT,zT+ω2·zA-zT·C4rxT,yT,zT+C4·r2xT,yT,zT.Descending velocity of fighterA in direction of target T is got in the light of definition of gradient: (6)vdxA,yA,zAxT,yT,zT=gradexT,yT,zT=∂exT,yT,zT∂xT2+∂exT,yT,zT∂yT2+∂exT,yT,zT∂zT2.Similarly, descending velocity of targetT in direction of fighter A is given as(7)vdxT,yT,zTxA,yA,zA=gradexA,yA,zA=∂exA,yA,zA∂xA2+∂exA,yA,zA∂yA2+∂exA,yA,zA∂zA2.


## 2.1.3. Calculation on Relative Superiority

Calculation on relative superiority of fighteri with respect to target j is expressed as(8)pij=vdxAi,yAi,zAixTj,yTj,zTjvdxAi,yAi,zAixTj,yTj,zTj+vdxTj,yTj,zTjxAi,yAi,zAi,where vdxAi,yAi,zAixTj,yTj,zTj is the descending velocity of fighter i in direction to location of target j and vdxTj,yTj,zTjxAi,yAi,zAi is the descending velocity of target j in direction to location of fighter i.


## 2.2. WTA Formulation

In the paper, the problem is formulated as a combinatorial optimization problem, with the objective of minimizing the total lethality of the targets. The fundamental problem variable is relative superioritypij.Consider the assignment problem ofm fighters to n targets: let pij∈[0,1] be the relative superiority, where i∈{1,2,…,m} and j∈{1,2,…,n}. The WTA is formulated as follows: (9)J=min∑j=1n∏i=1m1-pijxijs.t.∑i=1mxij≥1,∑j=1nxij≤1,where xij is the decision value indicating if fighter i is assigned to target j; xij=1 if fighter i is assigned to target j and xij=0 if fighter i is not assigned to target j.In the above formulation, we minimize the sum of the weighted cumulative probabilities of threat lethality while ensuring that all constraints are satisfied. Two constraints, respectively, denote that each target must be assigned by one fighter and each fighter can be assigned to one target at most.


## 3. A Hybrid Algorithm for the Problem

### 3.1. Improvements on AFSA

To improve the performance of the algorithm, several modifications were introduced into the AFSA, including initialization, visual and movement strategy, and leap behavior. Concrete implementations are presented below.


#### 3.1.1. Initialization

Initial solution causes crucial influences on convergence performance of the algorithm. The initial solution is randomly generated in the most existed study, by which it is difficult to get solution of high quality. Therefore a hybrid initialization method involving chaos, information entropy, and opposition-based learning methods is proposed and described in Figure1.Figure 1
Flow of initialization.Assume that there arep artificial fish in the swarm, each of which has q variables. Xrs denotes the variable s within fish r, r∈{1,…,p}, s∈{1,…,q}. The pseudocode for two algorithms in Figure 1 is described in Algorithms 1 and 2.Algorithm 1:Initialization based on logistic chaos and opposition-based learning [27, 28].
(1) Initialize parameter: Set the maximum chaotic iteration
numberingen
(2) for r=1:p do
(3) for s=1:q do
(4) for o=1:ingen do
(5) chos=sin⁡(π⋅cho-1,s) % Update the chaotic factor
(6) end for
(7) Xos=Xmin,s+chos(Xmax,s-Xmin,s) % Generate initial
chaotic solution
(8) OXos=Xmin,s+Xmax,s-Xrs % Generate initial
opposition-based learning solution
(9) end for
(10) end forAlgorithm 2:Initialization based on information entropy ‎[29].
(1) Set the critical entropy H0
(2) H=0
(3) while H<H0
(4)   for r=1:p
(5)    for t=r+1:p
(6)     for s=1:q
(7) Xts=Xmin,s+rand⋅(Xmax,s-Xmin,s)
(8) Prts=1-Xrs-Xts/Asmax-Asmin % Difference ratio
BetweenXrs and Xts, Asmax and Asmin are the maximum
and minimum value of the variables respectively
(9) Hs=∑r=1p∑t=2p(-Prtslog⁡Prts) % Entropy of variable s in
artificial fish swarm
(10) H=1/q∑s=1qHs % Entropy of the whole swarm
(11)    end for
(12)   end for
(13)  end for
(14) end while


#### 3.1.2. Pseudo-Parallel Evolution

In the late period of search, artificial fish will move more slowly and be misled to local optimums. A pseudo-parallel evolution strategy dividing the artificial fish swarm into multiple subswarms is introduced for solving the problem. Each subswarm executes search for optimum independently and exchanges the optimal solution with others at every iteration to maintain the search space and prevent prematurity.


#### 3.1.3. Artificial Fish Moving Strategy

To improve the search ability of the artificial fish, a moving strategy combining maintaining the self-advantage with absorbing the advantage from others is proposed below:(10)Xrgen+1=genmax⁡genXrgen+max⁡gen-genmax⁡genXugen-Xrgen,where gen is the current iteration, max⁡gen is the maximum iteration, Xrgen+1, Xrgen are the position of artificial fish r at iteration gen+1 and gen, respectively, and Xugen is the position of artificial fish u at gen iteration. The first part, gen/max⁡genXrgen, describes artificial fish keeping the self-superiority; the second part inspired by neighborhood search strategy, (max⁡gen-gen)/max⁡genXugen-Xrgen, indicates the absorbance of advantage from another artificial fish u within visual range.


#### 3.1.4. Self-Adaptive Visual Procedure

Visual, which has been deeply studied, determines the convergence speed and accuracy of AFSA. The modifications on visual have been proposed by many researches, as given in (11)~(13).‎ Reference [19] proposed an improved visual formulation: (11)visualgen=visualgen-1·Llow+rand·Lhigh-Llow,where visualgen and visualgen-1 stand for the visual at current iteration gen and previous iteration gen-1. Lhigh and Llow represent the upper and lower bounds of the visual, respectively.The model utilizes a random number to control the visual change and to ensure the overall trend getting lower, whereas, due to stochastic factors existing during the evolution, it is incapable of precise adjustment to the visual.Another modification on visual is presented in the following equation ‎[20]:(12)visualgen=visualgen-1·ybest-y-ybest,where ybest is the global best solution by estimation and y- is the mean solution.In accordance with (12), the mean solution is getting closer to the global best solution by estimation. During the approach to the optimal solution, difference between the current mean solution and the optimal one decreases, as well as the visual scope. However, ybest is hard to estimate, which easily causes the algorithm to fall into local optimal or poor performance in convergence.‎Reference [21] proposed another kind of improvement on visual: (13)visualgen=β-genN·visualfixedα·visualgen-1,α=visualNvisual11/N,where N is the maximum iteration, visual1 is the initial value, visualN is the previously set value at the last iteration, visualfixed is a fixed value, and β is a constant number.In initial iterations, the visual varies in accordance with the upper formulation for ensuring good performance in convergence speed and global searching, thereafter changing by means of the next one for better convergence accuracy. But there are two deficiencies that existed; the switch condition from the upper formulation to the one below is not quite clear and the visual at the last iteration is not easy to obtain.The main purpose is to ensure the artificial fish searching with larger visual within initial iterations and with smaller visual afterwards, although the modified means are quite different such as (11) to (13). Whereas the above improvements on visual are insufficient for the research on the algorithm falling into the local optimum, therefore a modified visual model concerned with iteration and convergence speed is constructed as follows: (14)visualrgen=max⁡visual·Drgen-Drgen-1·e-30gen/max⁡genS,1,Drgen-Drgen-1≠0,visual·Drgen-gen′-Drgen-gen′-1·e-30gen/max⁡genS,Drgen-Drgen-1=0,where gen′ measures the iterations at which the optimal solution stays unchanged, visual is the initial visual, Drgenlast is the solution at the last iteration before Drgen-Drgen-1=0, Drgenlast-1 is the solution at the previous iteration before genlast, and S is a constant number that could be 1, 3, 5, 10.From (14) we can see that adjustment in visual changes one dimension at least and gets smaller as Drgen-Drgen-1 decreases, which helps improve the accuracy of the solutions. When Drgen-Drgen-1=0, the visual obtained by (14) will get larger, by which artificial fish are capable of escaping from local optimal regions.


#### 3.1.5. Improvements on Leap Behavior

For preventing the artificial fish from converging to local optimal solutions, leap behavior forcing the artificial fish to change its location is adopted in certain references [22, 23]. In this paper, artificial fish executes leap behavior when the optimal solution stands still after one iteration. Meanwhile, it is thought that the higher the number of iterations at which optimal solution stagnates is, the higher the extent the optimal solution varies and the higher the probability of leap behavior execution is. Accordingly, improvements on leap behavior probability and leap behavior strategy are designed as follows.The leap behavior is modified as outlined in (15): (15)Xrgen+1=Xrgen+visualrgen·λ·rand,where λ is a bool vector; the quantity of 1 in vector is obtained from (16)θ=max1,q-q·rounde-30gen′/genS·Drgen-gen′Drgen.Equation (16) makes it certain that one dimension at least, q dimension at most in location of artificial fish r, changes by leap behavior at each iteration. The bigger Drgen-gen′/Drgen and gen′/gen are, the more dimension changes. By this way, leap behavior is able to adjust the location of artificial fish dynamically and to achieve the goal of avoiding premature convergence.


### 3.2. Improvement on HS

In this article, improvements on HS contains two domains: harmony memory consideration rate and new harmony generating strategy.


#### 3.2.1. Improvements on Harmony Memory Consideration Rate

Modifications on harmony memory consideration rate have been studied in many researches [30, 31]. Through the reference, it can be concluded that harmony memory consideration rate should be given a small value to increase the diversity of the harmony memory and global search when solutions differ greatly. On the other hand, when the solutions are relatively resembled, harmony memory consideration rate should be raised for performing search more accurately and reducing the time for searching. In this paper, it is considered that iteration has effect on harmony memory consideration rate as solution does; particularly harmony memory consideration rate is supposed to increase gradually at final iterations for acting better in global search. The harmony memory consideration rate is formulated from the perspectives of current iteration and deviations between harmony vectors as follows: (17)HMCRgen=EmingenEmaxgenegen/max⁡gen-1,where Emingen, Emaxgen, respectively, represent the best and worst solution in harmony memory at iteration gen. The harmony memory consideration rate in (17) utilizes dynamic mechanism to select the harmony vector under various circumstances. The larger Emingen/Emaxgen is, the smaller the difference among the harmony vectors in harmony memory is, which indicates that HMCRgen becomes larger to converge to the optimum fastly. Likewise, the larger gen is, the larger HMCRgen is, and the algorithm is able to have a faster convergence to the optimum.


#### 3.2.2. New Harmony Improvisation

New harmony improvisation deciding general performance in optimization is also seen as the key point in modifications on HS. A new method for harmony improvisation in terms of learning strategy is defined in this paper, which is described as follows:(18)usgen=ur1,sgen+minc1,c2·ubest,sgen-ur1,sgen+maxc1,c2·ur2,sgen-ur1,sgen,Dbestgen-Dbestgen-1=0,ur1,sgen+maxc1,c2·ubest,sgen-ur1,sgen+minc1,c2·ur2,sgen-ur1,sgen,Dbestgen-Dbestgen-1≠0,where usgen is the decision variable s in harmony vector generated at iteration gen, ur1,sgen, ur2,sgen, ubest,sgen are the decision variables s in harmony vectors r1, r2 and the optimal harmony vector in harmony memory generated at iteration gen, and c1, c2 are learning coefficients, 0≤c1,c2≤1, c1≠c2.The bigger one betweenc1 and c2 is selected to make sure that the harmony is affected by the neighborhood solution inside the harmony memory and avoid prematurity when Dbestgen-Dbestgen-1=0. Otherwise, the harmony improvisation is more influenced by the optimal solution, which helps in converging to optimum more quickly.


### 3.3. Switch Condition

#### 3.3.1. Switch Condition Analysis

Switch condition acting as a key part of the hybrid algorithm causes a great influence on speed and accuracy of convergence. Switch condition is discussed by two aspects as switching iteration and individual standard error of artificial fish.(1) Switching Iteration. Switching iteration is to make the hybrid algorithm switch from improved AFSA to improved HS at a certain iteration, which is obtained by multiple concrete experiments.(2) Individual Standard Error of Artificial Fish. This factor reflects the deviation among artificial fish in the swarm. It should be switched to the HS for the following search while the individual standard error of artificial fish gets smaller, which means the search space gets smaller. The model of individual standard error of artificial fish, with regard to location and fitness value, is proposed as follows.The standard error of location on artificial fishlse is calculated by(19)lse=1q-1∑s=1qlses2,lses2=1p-1∑r=1pXrs-X-s2,where X-s is the average location of all artificial fish in dimension s. The artificial fish are of high probability to gather in local optimal region while each fish gets closer, as well as lse being smaller.The standard error of fitness value on artificial fishfse is calculated by (20)fse=∑r=1pDrgen-D-gen2p-1,where D-gen is the average fitness value. Likewise, the artificial fish seems to get trapped in the local optimum area while fitness value of each fish is resembled, as well as fse being smaller.From the information mentioned above, the individual standard error of artificial fish is derived by(21)se=φ1·lse+φ2·fses.t.0≤φ1,φ2≤1φ1+φ2=1,where φ1, φ2, respectively, represent the coefficients about lse and fse. The switch to improved HS should be executed for global optimum if se is smaller than the presented value sedef, which means the stand error of fitness value and location fluctuate to a smaller extent; therefore the solution is weak.


#### 3.3.2. Design for Switch Step

For the sake of fast convergence and quality of solutions, it is ruled in this paper that the switch is executed only if any condition is satisfied. Due to the different performances of the algorithms when facing various problems or benchmark functions, it is also regulated that the individual standard error of artificial fish is seen as the prior factor, and the switch flow is given in Figure2.Figure 2
Switch flow for algorithm.It can be seen from Figure2 that individual standard error of artificial fish is investigated at first; switch is executed if satisfied; otherwise checking whether the algorithm is qualified to switch from the perspective of switching iteration, the switch is carried on if satisfied, or else the algorithm runs back to the improved artificial fish swarm algorithm.


### 3.4. Flow for Hybrid Algorithm

The concrete flow for the proposed hybrid algorithm is depicted in Figure3.Figure 3
Flow for the proposed hybrid algorithm.


## 3.1. Improvements on AFSA

To improve the performance of the algorithm, several modifications were introduced into the AFSA, including initialization, visual and movement strategy, and leap behavior. Concrete implementations are presented below.


### 3.1.1. Initialization

Initial solution causes crucial influences on convergence performance of the algorithm. The initial solution is randomly generated in the most existed study, by which it is difficult to get solution of high quality. Therefore a hybrid initialization method involving chaos, information entropy, and opposition-based learning methods is proposed and described in Figure1.Figure 1
Flow of initialization.Assume that there arep artificial fish in the swarm, each of which has q variables. Xrs denotes the variable s within fish r, r∈{1,…,p}, s∈{1,…,q}. The pseudocode for two algorithms in Figure 1 is described in Algorithms 1 and 2.Algorithm 1:Initialization based on logistic chaos and opposition-based learning [27, 28].
(1) Initialize parameter: Set the maximum chaotic iteration
numberingen
(2) for r=1:p do
(3) for s=1:q do
(4) for o=1:ingen do
(5) chos=sin⁡(π⋅cho-1,s) % Update the chaotic factor
(6) end for
(7) Xos=Xmin,s+chos(Xmax,s-Xmin,s) % Generate initial
chaotic solution
(8) OXos=Xmin,s+Xmax,s-Xrs % Generate initial
opposition-based learning solution
(9) end for
(10) end forAlgorithm 2:Initialization based on information entropy ‎[29].
(1) Set the critical entropy H0
(2) H=0
(3) while H<H0
(4)   for r=1:p
(5)    for t=r+1:p
(6)     for s=1:q
(7) Xts=Xmin,s+rand⋅(Xmax,s-Xmin,s)
(8) Prts=1-Xrs-Xts/Asmax-Asmin % Difference ratio
BetweenXrs and Xts, Asmax and Asmin are the maximum
and minimum value of the variables respectively
(9) Hs=∑r=1p∑t=2p(-Prtslog⁡Prts) % Entropy of variable s in
artificial fish swarm
(10) H=1/q∑s=1qHs % Entropy of the whole swarm
(11)    end for
(12)   end for
(13)  end for
(14) end while


### 3.1.2. Pseudo-Parallel Evolution

In the late period of search, artificial fish will move more slowly and be misled to local optimums. A pseudo-parallel evolution strategy dividing the artificial fish swarm into multiple subswarms is introduced for solving the problem. Each subswarm executes search for optimum independently and exchanges the optimal solution with others at every iteration to maintain the search space and prevent prematurity.


### 3.1.3. Artificial Fish Moving Strategy

To improve the search ability of the artificial fish, a moving strategy combining maintaining the self-advantage with absorbing the advantage from others is proposed below:(10)Xrgen+1=genmax⁡genXrgen+max⁡gen-genmax⁡genXugen-Xrgen,where gen is the current iteration, max⁡gen is the maximum iteration, Xrgen+1, Xrgen are the position of artificial fish r at iteration gen+1 and gen, respectively, and Xugen is the position of artificial fish u at gen iteration. The first part, gen/max⁡genXrgen, describes artificial fish keeping the self-superiority; the second part inspired by neighborhood search strategy, (max⁡gen-gen)/max⁡genXugen-Xrgen, indicates the absorbance of advantage from another artificial fish u within visual range.


### 3.1.4. Self-Adaptive Visual Procedure

Visual, which has been deeply studied, determines the convergence speed and accuracy of AFSA. The modifications on visual have been proposed by many researches, as given in (11)~(13).‎ Reference [19] proposed an improved visual formulation: (11)visualgen=visualgen-1·Llow+rand·Lhigh-Llow,where visualgen and visualgen-1 stand for the visual at current iteration gen and previous iteration gen-1. Lhigh and Llow represent the upper and lower bounds of the visual, respectively.The model utilizes a random number to control the visual change and to ensure the overall trend getting lower, whereas, due to stochastic factors existing during the evolution, it is incapable of precise adjustment to the visual.Another modification on visual is presented in the following equation ‎[20]:(12)visualgen=visualgen-1·ybest-y-ybest,where ybest is the global best solution by estimation and y- is the mean solution.In accordance with (12), the mean solution is getting closer to the global best solution by estimation. During the approach to the optimal solution, difference between the current mean solution and the optimal one decreases, as well as the visual scope. However, ybest is hard to estimate, which easily causes the algorithm to fall into local optimal or poor performance in convergence.‎Reference [21] proposed another kind of improvement on visual: (13)visualgen=β-genN·visualfixedα·visualgen-1,α=visualNvisual11/N,where N is the maximum iteration, visual1 is the initial value, visualN is the previously set value at the last iteration, visualfixed is a fixed value, and β is a constant number.In initial iterations, the visual varies in accordance with the upper formulation for ensuring good performance in convergence speed and global searching, thereafter changing by means of the next one for better convergence accuracy. But there are two deficiencies that existed; the switch condition from the upper formulation to the one below is not quite clear and the visual at the last iteration is not easy to obtain.The main purpose is to ensure the artificial fish searching with larger visual within initial iterations and with smaller visual afterwards, although the modified means are quite different such as (11) to (13). Whereas the above improvements on visual are insufficient for the research on the algorithm falling into the local optimum, therefore a modified visual model concerned with iteration and convergence speed is constructed as follows: (14)visualrgen=max⁡visual·Drgen-Drgen-1·e-30gen/max⁡genS,1,Drgen-Drgen-1≠0,visual·Drgen-gen′-Drgen-gen′-1·e-30gen/max⁡genS,Drgen-Drgen-1=0,where gen′ measures the iterations at which the optimal solution stays unchanged, visual is the initial visual, Drgenlast is the solution at the last iteration before Drgen-Drgen-1=0, Drgenlast-1 is the solution at the previous iteration before genlast, and S is a constant number that could be 1, 3, 5, 10.From (14) we can see that adjustment in visual changes one dimension at least and gets smaller as Drgen-Drgen-1 decreases, which helps improve the accuracy of the solutions. When Drgen-Drgen-1=0, the visual obtained by (14) will get larger, by which artificial fish are capable of escaping from local optimal regions.


### 3.1.5. Improvements on Leap Behavior

For preventing the artificial fish from converging to local optimal solutions, leap behavior forcing the artificial fish to change its location is adopted in certain references [22, 23]. In this paper, artificial fish executes leap behavior when the optimal solution stands still after one iteration. Meanwhile, it is thought that the higher the number of iterations at which optimal solution stagnates is, the higher the extent the optimal solution varies and the higher the probability of leap behavior execution is. Accordingly, improvements on leap behavior probability and leap behavior strategy are designed as follows.The leap behavior is modified as outlined in (15): (15)Xrgen+1=Xrgen+visualrgen·λ·rand,where λ is a bool vector; the quantity of 1 in vector is obtained from (16)θ=max1,q-q·rounde-30gen′/genS·Drgen-gen′Drgen.Equation (16) makes it certain that one dimension at least, q dimension at most in location of artificial fish r, changes by leap behavior at each iteration. The bigger Drgen-gen′/Drgen and gen′/gen are, the more dimension changes. By this way, leap behavior is able to adjust the location of artificial fish dynamically and to achieve the goal of avoiding premature convergence.


## 3.1.1. Initialization

Initial solution causes crucial influences on convergence performance of the algorithm. The initial solution is randomly generated in the most existed study, by which it is difficult to get solution of high quality. Therefore a hybrid initialization method involving chaos, information entropy, and opposition-based learning methods is proposed and described in Figure1.Figure 1
Flow of initialization.Assume that there arep artificial fish in the swarm, each of which has q variables. Xrs denotes the variable s within fish r, r∈{1,…,p}, s∈{1,…,q}. The pseudocode for two algorithms in Figure 1 is described in Algorithms 1 and 2.Algorithm 1:Initialization based on logistic chaos and opposition-based learning [27, 28].
(1) Initialize parameter: Set the maximum chaotic iteration
numberingen
(2) for r=1:p do
(3) for s=1:q do
(4) for o=1:ingen do
(5) chos=sin⁡(π⋅cho-1,s) % Update the chaotic factor
(6) end for
(7) Xos=Xmin,s+chos(Xmax,s-Xmin,s) % Generate initial
chaotic solution
(8) OXos=Xmin,s+Xmax,s-Xrs % Generate initial
opposition-based learning solution
(9) end for
(10) end forAlgorithm 2:Initialization based on information entropy ‎[29].
(1) Set the critical entropy H0
(2) H=0
(3) while H<H0
(4)   for r=1:p
(5)    for t=r+1:p
(6)     for s=1:q
(7) Xts=Xmin,s+rand⋅(Xmax,s-Xmin,s)
(8) Prts=1-Xrs-Xts/Asmax-Asmin % Difference ratio
BetweenXrs and Xts, Asmax and Asmin are the maximum
and minimum value of the variables respectively
(9) Hs=∑r=1p∑t=2p(-Prtslog⁡Prts) % Entropy of variable s in
artificial fish swarm
(10) H=1/q∑s=1qHs % Entropy of the whole swarm
(11)    end for
(12)   end for
(13)  end for
(14) end while


## 3.1.2. Pseudo-Parallel Evolution

In the late period of search, artificial fish will move more slowly and be misled to local optimums. A pseudo-parallel evolution strategy dividing the artificial fish swarm into multiple subswarms is introduced for solving the problem. Each subswarm executes search for optimum independently and exchanges the optimal solution with others at every iteration to maintain the search space and prevent prematurity.


## 3.1.3. Artificial Fish Moving Strategy

To improve the search ability of the artificial fish, a moving strategy combining maintaining the self-advantage with absorbing the advantage from others is proposed below:(10)Xrgen+1=genmax⁡genXrgen+max⁡gen-genmax⁡genXugen-Xrgen,where gen is the current iteration, max⁡gen is the maximum iteration, Xrgen+1, Xrgen are the position of artificial fish r at iteration gen+1 and gen, respectively, and Xugen is the position of artificial fish u at gen iteration. The first part, gen/max⁡genXrgen, describes artificial fish keeping the self-superiority; the second part inspired by neighborhood search strategy, (max⁡gen-gen)/max⁡genXugen-Xrgen, indicates the absorbance of advantage from another artificial fish u within visual range.


## 3.1.4. Self-Adaptive Visual Procedure

Visual, which has been deeply studied, determines the convergence speed and accuracy of AFSA. The modifications on visual have been proposed by many researches, as given in (11)~(13).‎ Reference [19] proposed an improved visual formulation: (11)visualgen=visualgen-1·Llow+rand·Lhigh-Llow,where visualgen and visualgen-1 stand for the visual at current iteration gen and previous iteration gen-1. Lhigh and Llow represent the upper and lower bounds of the visual, respectively.The model utilizes a random number to control the visual change and to ensure the overall trend getting lower, whereas, due to stochastic factors existing during the evolution, it is incapable of precise adjustment to the visual.Another modification on visual is presented in the following equation ‎[20]:(12)visualgen=visualgen-1·ybest-y-ybest,where ybest is the global best solution by estimation and y- is the mean solution.In accordance with (12), the mean solution is getting closer to the global best solution by estimation. During the approach to the optimal solution, difference between the current mean solution and the optimal one decreases, as well as the visual scope. However, ybest is hard to estimate, which easily causes the algorithm to fall into local optimal or poor performance in convergence.‎Reference [21] proposed another kind of improvement on visual: (13)visualgen=β-genN·visualfixedα·visualgen-1,α=visualNvisual11/N,where N is the maximum iteration, visual1 is the initial value, visualN is the previously set value at the last iteration, visualfixed is a fixed value, and β is a constant number.In initial iterations, the visual varies in accordance with the upper formulation for ensuring good performance in convergence speed and global searching, thereafter changing by means of the next one for better convergence accuracy. But there are two deficiencies that existed; the switch condition from the upper formulation to the one below is not quite clear and the visual at the last iteration is not easy to obtain.The main purpose is to ensure the artificial fish searching with larger visual within initial iterations and with smaller visual afterwards, although the modified means are quite different such as (11) to (13). Whereas the above improvements on visual are insufficient for the research on the algorithm falling into the local optimum, therefore a modified visual model concerned with iteration and convergence speed is constructed as follows: (14)visualrgen=max⁡visual·Drgen-Drgen-1·e-30gen/max⁡genS,1,Drgen-Drgen-1≠0,visual·Drgen-gen′-Drgen-gen′-1·e-30gen/max⁡genS,Drgen-Drgen-1=0,where gen′ measures the iterations at which the optimal solution stays unchanged, visual is the initial visual, Drgenlast is the solution at the last iteration before Drgen-Drgen-1=0, Drgenlast-1 is the solution at the previous iteration before genlast, and S is a constant number that could be 1, 3, 5, 10.From (14) we can see that adjustment in visual changes one dimension at least and gets smaller as Drgen-Drgen-1 decreases, which helps improve the accuracy of the solutions. When Drgen-Drgen-1=0, the visual obtained by (14) will get larger, by which artificial fish are capable of escaping from local optimal regions.


## 3.1.5. Improvements on Leap Behavior

For preventing the artificial fish from converging to local optimal solutions, leap behavior forcing the artificial fish to change its location is adopted in certain references [22, 23]. In this paper, artificial fish executes leap behavior when the optimal solution stands still after one iteration. Meanwhile, it is thought that the higher the number of iterations at which optimal solution stagnates is, the higher the extent the optimal solution varies and the higher the probability of leap behavior execution is. Accordingly, improvements on leap behavior probability and leap behavior strategy are designed as follows.The leap behavior is modified as outlined in (15): (15)Xrgen+1=Xrgen+visualrgen·λ·rand,where λ is a bool vector; the quantity of 1 in vector is obtained from (16)θ=max1,q-q·rounde-30gen′/genS·Drgen-gen′Drgen.Equation (16) makes it certain that one dimension at least, q dimension at most in location of artificial fish r, changes by leap behavior at each iteration. The bigger Drgen-gen′/Drgen and gen′/gen are, the more dimension changes. By this way, leap behavior is able to adjust the location of artificial fish dynamically and to achieve the goal of avoiding premature convergence.


## 3.2. Improvement on HS

In this article, improvements on HS contains two domains: harmony memory consideration rate and new harmony generating strategy.


### 3.2.1. Improvements on Harmony Memory Consideration Rate

Modifications on harmony memory consideration rate have been studied in many researches [30, 31]. Through the reference, it can be concluded that harmony memory consideration rate should be given a small value to increase the diversity of the harmony memory and global search when solutions differ greatly. On the other hand, when the solutions are relatively resembled, harmony memory consideration rate should be raised for performing search more accurately and reducing the time for searching. In this paper, it is considered that iteration has effect on harmony memory consideration rate as solution does; particularly harmony memory consideration rate is supposed to increase gradually at final iterations for acting better in global search. The harmony memory consideration rate is formulated from the perspectives of current iteration and deviations between harmony vectors as follows: (17)HMCRgen=EmingenEmaxgenegen/max⁡gen-1,where Emingen, Emaxgen, respectively, represent the best and worst solution in harmony memory at iteration gen. The harmony memory consideration rate in (17) utilizes dynamic mechanism to select the harmony vector under various circumstances. The larger Emingen/Emaxgen is, the smaller the difference among the harmony vectors in harmony memory is, which indicates that HMCRgen becomes larger to converge to the optimum fastly. Likewise, the larger gen is, the larger HMCRgen is, and the algorithm is able to have a faster convergence to the optimum.


### 3.2.2. New Harmony Improvisation

New harmony improvisation deciding general performance in optimization is also seen as the key point in modifications on HS. A new method for harmony improvisation in terms of learning strategy is defined in this paper, which is described as follows:(18)usgen=ur1,sgen+minc1,c2·ubest,sgen-ur1,sgen+maxc1,c2·ur2,sgen-ur1,sgen,Dbestgen-Dbestgen-1=0,ur1,sgen+maxc1,c2·ubest,sgen-ur1,sgen+minc1,c2·ur2,sgen-ur1,sgen,Dbestgen-Dbestgen-1≠0,where usgen is the decision variable s in harmony vector generated at iteration gen, ur1,sgen, ur2,sgen, ubest,sgen are the decision variables s in harmony vectors r1, r2 and the optimal harmony vector in harmony memory generated at iteration gen, and c1, c2 are learning coefficients, 0≤c1,c2≤1, c1≠c2.The bigger one betweenc1 and c2 is selected to make sure that the harmony is affected by the neighborhood solution inside the harmony memory and avoid prematurity when Dbestgen-Dbestgen-1=0. Otherwise, the harmony improvisation is more influenced by the optimal solution, which helps in converging to optimum more quickly.


## 3.2.1. Improvements on Harmony Memory Consideration Rate

Modifications on harmony memory consideration rate have been studied in many researches [30, 31]. Through the reference, it can be concluded that harmony memory consideration rate should be given a small value to increase the diversity of the harmony memory and global search when solutions differ greatly. On the other hand, when the solutions are relatively resembled, harmony memory consideration rate should be raised for performing search more accurately and reducing the time for searching. In this paper, it is considered that iteration has effect on harmony memory consideration rate as solution does; particularly harmony memory consideration rate is supposed to increase gradually at final iterations for acting better in global search. The harmony memory consideration rate is formulated from the perspectives of current iteration and deviations between harmony vectors as follows: (17)HMCRgen=EmingenEmaxgenegen/max⁡gen-1,where Emingen, Emaxgen, respectively, represent the best and worst solution in harmony memory at iteration gen. The harmony memory consideration rate in (17) utilizes dynamic mechanism to select the harmony vector under various circumstances. The larger Emingen/Emaxgen is, the smaller the difference among the harmony vectors in harmony memory is, which indicates that HMCRgen becomes larger to converge to the optimum fastly. Likewise, the larger gen is, the larger HMCRgen is, and the algorithm is able to have a faster convergence to the optimum.


## 3.2.2. New Harmony Improvisation

New harmony improvisation deciding general performance in optimization is also seen as the key point in modifications on HS. A new method for harmony improvisation in terms of learning strategy is defined in this paper, which is described as follows:(18)usgen=ur1,sgen+minc1,c2·ubest,sgen-ur1,sgen+maxc1,c2·ur2,sgen-ur1,sgen,Dbestgen-Dbestgen-1=0,ur1,sgen+maxc1,c2·ubest,sgen-ur1,sgen+minc1,c2·ur2,sgen-ur1,sgen,Dbestgen-Dbestgen-1≠0,where usgen is the decision variable s in harmony vector generated at iteration gen, ur1,sgen, ur2,sgen, ubest,sgen are the decision variables s in harmony vectors r1, r2 and the optimal harmony vector in harmony memory generated at iteration gen, and c1, c2 are learning coefficients, 0≤c1,c2≤1, c1≠c2.The bigger one betweenc1 and c2 is selected to make sure that the harmony is affected by the neighborhood solution inside the harmony memory and avoid prematurity when Dbestgen-Dbestgen-1=0. Otherwise, the harmony improvisation is more influenced by the optimal solution, which helps in converging to optimum more quickly.


## 3.3. Switch Condition

### 3.3.1. Switch Condition Analysis

Switch condition acting as a key part of the hybrid algorithm causes a great influence on speed and accuracy of convergence. Switch condition is discussed by two aspects as switching iteration and individual standard error of artificial fish.(1) Switching Iteration. Switching iteration is to make the hybrid algorithm switch from improved AFSA to improved HS at a certain iteration, which is obtained by multiple concrete experiments.(2) Individual Standard Error of Artificial Fish. This factor reflects the deviation among artificial fish in the swarm. It should be switched to the HS for the following search while the individual standard error of artificial fish gets smaller, which means the search space gets smaller. The model of individual standard error of artificial fish, with regard to location and fitness value, is proposed as follows.The standard error of location on artificial fishlse is calculated by(19)lse=1q-1∑s=1qlses2,lses2=1p-1∑r=1pXrs-X-s2,where X-s is the average location of all artificial fish in dimension s. The artificial fish are of high probability to gather in local optimal region while each fish gets closer, as well as lse being smaller.The standard error of fitness value on artificial fishfse is calculated by (20)fse=∑r=1pDrgen-D-gen2p-1,where D-gen is the average fitness value. Likewise, the artificial fish seems to get trapped in the local optimum area while fitness value of each fish is resembled, as well as fse being smaller.From the information mentioned above, the individual standard error of artificial fish is derived by(21)se=φ1·lse+φ2·fses.t.0≤φ1,φ2≤1φ1+φ2=1,where φ1, φ2, respectively, represent the coefficients about lse and fse. The switch to improved HS should be executed for global optimum if se is smaller than the presented value sedef, which means the stand error of fitness value and location fluctuate to a smaller extent; therefore the solution is weak.


### 3.3.2. Design for Switch Step

For the sake of fast convergence and quality of solutions, it is ruled in this paper that the switch is executed only if any condition is satisfied. Due to the different performances of the algorithms when facing various problems or benchmark functions, it is also regulated that the individual standard error of artificial fish is seen as the prior factor, and the switch flow is given in Figure2.Figure 2
Switch flow for algorithm.It can be seen from Figure2 that individual standard error of artificial fish is investigated at first; switch is executed if satisfied; otherwise checking whether the algorithm is qualified to switch from the perspective of switching iteration, the switch is carried on if satisfied, or else the algorithm runs back to the improved artificial fish swarm algorithm.


## 3.3.1. Switch Condition Analysis

Switch condition acting as a key part of the hybrid algorithm causes a great influence on speed and accuracy of convergence. Switch condition is discussed by two aspects as switching iteration and individual standard error of artificial fish.(1) Switching Iteration. Switching iteration is to make the hybrid algorithm switch from improved AFSA to improved HS at a certain iteration, which is obtained by multiple concrete experiments.(2) Individual Standard Error of Artificial Fish. This factor reflects the deviation among artificial fish in the swarm. It should be switched to the HS for the following search while the individual standard error of artificial fish gets smaller, which means the search space gets smaller. The model of individual standard error of artificial fish, with regard to location and fitness value, is proposed as follows.The standard error of location on artificial fishlse is calculated by(19)lse=1q-1∑s=1qlses2,lses2=1p-1∑r=1pXrs-X-s2,where X-s is the average location of all artificial fish in dimension s. The artificial fish are of high probability to gather in local optimal region while each fish gets closer, as well as lse being smaller.The standard error of fitness value on artificial fishfse is calculated by (20)fse=∑r=1pDrgen-D-gen2p-1,where D-gen is the average fitness value. Likewise, the artificial fish seems to get trapped in the local optimum area while fitness value of each fish is resembled, as well as fse being smaller.From the information mentioned above, the individual standard error of artificial fish is derived by(21)se=φ1·lse+φ2·fses.t.0≤φ1,φ2≤1φ1+φ2=1,where φ1, φ2, respectively, represent the coefficients about lse and fse. The switch to improved HS should be executed for global optimum if se is smaller than the presented value sedef, which means the stand error of fitness value and location fluctuate to a smaller extent; therefore the solution is weak.


## 3.3.2. Design for Switch Step

For the sake of fast convergence and quality of solutions, it is ruled in this paper that the switch is executed only if any condition is satisfied. Due to the different performances of the algorithms when facing various problems or benchmark functions, it is also regulated that the individual standard error of artificial fish is seen as the prior factor, and the switch flow is given in Figure2.Figure 2
Switch flow for algorithm.It can be seen from Figure2 that individual standard error of artificial fish is investigated at first; switch is executed if satisfied; otherwise checking whether the algorithm is qualified to switch from the perspective of switching iteration, the switch is carried on if satisfied, or else the algorithm runs back to the improved artificial fish swarm algorithm.


## 3.4. Flow for Hybrid Algorithm

The concrete flow for the proposed hybrid algorithm is depicted in Figure3.Figure 3
Flow for the proposed hybrid algorithm.


## 4. Simulation Results

### 4.1. Verification Tests by Benchmark Functions

The algorithms are always tested by benchmark functions. For thoroughly validating the performances of the proposed IAFSA-HIS, other three algorithms including IABHS, IBAFSA, and GDHS are introduced for comparison. Main properties of the selected benchmark functions used in the experiments are shown in Table1.Table 1
Benchmark functions.
Function
Equation
Range
Optimum
Property
Sphere
f
1
x
=
∑
i
=
1
n
x
i
2
-
100,100
0
Unimodal
Quartic
f
2
x
=
∑
i
=
1
n
i
x
i
4
+
r
a
n
d
o
m
0,1
-
1.28,1.28
0
Unimodal
Duadric
f
3
x
=
∑
i
=
1
n
∑
j
=
1
i
x
j
2
-
100,100
0
Unimodal
Rosenbrock
f
4
x
=
∑
i
=
1
n
-
1
x
i
-
1
2
+
100
x
i
+
1
-
x
i
2
2
-
30,30
0
Unimodal
Schwefel
f
5
x
=
-
∑
i
=
1
n
x
i
sin
⁡
x
i
-
500,500
-
12569.5
Multimodal
Rastrigin
f
6
x
=
∑
i
=
1
n
x
i
2
-
10
2
π
x
i
+
10
-
5.12,5.12
0
Multimodal
Griewank
f
7
x
=
1
4000
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
cos
⁡
x
i
i
+
1
-
600,600
0
Multimodal
Ackley
f
8
x
=
-
20
e
-
0.2
∑
i
=
1
n
x
i
2
/
n
-
e
∑
i
=
1
n
cos
⁡
2
π
x
i
/
n
+
e
+
20
-
30,30
0
MultimodalParameters are set as follows:ingen=100, H0=0.3, S=5, visual=1.5, γ1=0.3, γ2=0.5, c1=0.6, c2=0.4, φ1=0.6, φ2=0.4, max⁡gen=100000, and n=30; for other parameters, refer to [24, 30, 32].The algorithms were run 100 times independently to catch their stochastic properties and the results are depicted in Table2 and Figures 4–11.Table 2
Comparison of results with other algorithms.
IAFSA-IHS
IBAFSA
IABHS
GDHS
f
1
Best
3.0303
e
-
218
3.4848
e
-
100
3.8552
e
-
60
0
Mean
3.7514
e
-
220
4.3574
e
-
99
6.8125
e
-
59
0
Worst
9.3558
e
-
221
6.1893
e
-
97
7.2759
e
-
57
0
Std.dev
7.2377
e
-
218
3.5238
e
-
99
8.7153
e
-
59
0
f
2
Best
4.3659
e
-
3
5.7852
e
-
2
1.5798
e
-
4
8.9654
e
-
3
Mean
5.9283
e
-
2
6.0237
e
-
2
2.0721
e
-
2
9.3571
e
-
2
Worst
1.1135
e
-
1
1.3145
e
-
1
5.3492
e
-
2
7.5883
e
-
2
Std.dev
6.3844
e
-
2
9.3561
e
-
2
8.9443
e
-
3
1.0184
e
-
2
f
3
Best
6.901
e
-
3
1.8312
e
+
3
5.7163
e
-
4
3.3694
e
-
2
Mean
5.4431
e
-
2
7.5403
e
+
3
6.7303
e
-
2
5.2391
e
-
1
Worst
3.2967
e
-
1
6.5727
e
+
4
8.1835
e
-
2
2.4749
e
+
0
Std.dev
2.3617
e
-
2
5.4329
e
+
3
4.3516
e
-
3
4.7161
e
-
2
f
4
Best
1.1364
e
+
1
3.5801
e
+
1
1.0075
e
+
1
6.4625
e
+
1
Mean
9.5879
e
+
1
6.0574
e
+
1
9.5855
e
+
1
8.5942
e
+
1
Worst
1.2841
e
+
2
2.9714
e
+
1
3.3575
e
+
1
1.5399
e
+
2
Std.dev
9.9836
e
+
1
3.4769
e
+
1
1.4848
e
+
1
7.6935
e
+
1
f
5
Best
-
1.1821
e
+
4
-
1.1108
e
+
4
-
1.1753
e
+
4
-
1.2024
e
+
4
Mean
-
1.1028
e
+
4
-
9.9931
e
+
3
-
1.0387
e
+
4
-
1.1569
e
+
4
Worst
-
1.0254
e
+
4
-
9.8301
e
+
3
-
9.4539
e
+
3
-
1.0965
e
+
4
Std.dev
-
1.0879
e
+
4
-
1.0256
e
+
4
-
1.0024
e
+
4
-
1.2569
e
+
4
f
6
Best
0
3.1827
e
-
4
1.9847
e
-
3
4.2084
e
-
4
Mean
0
9.5648
e
-
4
9.0955
e
-
3
1.5682
e
-
3
Worst
0
8.1815
e
-
3
6.5276
e
-
1
3.3524
e
-
2
Std.dev
0
3.1348
e
-
4
6.4596
e
-
3
1.2563
e
-
3
f
7
Best
0
6.0477
e
-
10
3.2291
e
-
14
7.7708
e
-
16
Mean
0
7.5628
e
-
9
9.2486
e
-
13
6.2967
e
-
14
Worst
0
9.2466
e
-
8
9.7123
e
-
12
9.1222
e
-
13
Std.dev
0
1.3132
e
-
9
2.2486
e
-
13
1.5422
e
-
14
f
8
Best
4.0107
e
-
16
3.5687
e
-
15
3.0026
e
-
13
1.2627
e
-
15
Mean
7.5975
e
-
16
7.0145
e
-
15
5.2435
e
-
13
3.3275
e
-
15
Worst
3.2354
e
-
15
9.6461
e
-
14
6.4564
e
-
12
9.5619
e
-
15
Std.dev
5.5484
e
-
16
5.5686
e
-
15
1.3967
e
-
13
2.0131
e
-
15Figure 4
Sphere.Figure 5
Quartic.Figure 6
Duadric.Figure 7
Rosenbrock.Figure 8
Schwefel.Figure 9
Rastrigin.Figure 10
Griewank.Figure 11
Ackley.The parameters Best, Mean, Worst, and Std.dev are the best, mean, worst, and stand error of the optimum on average after 100 running times, where Mean reflects the precision of the results obtained by the algorithms within certain iterations and Std.dev reflects the robustness and stability of the algorithms.As seen from Table2 and Figures 4–11, the advantages in terms of best, median, worst, mean, and standard error of IAFSA-HIS are outstanding among the four algorithms in general. Particularly, IAFSA-IHS achieves the theoretical optimum while others fail still the termination in f6 and f7. However, the proposed algorithm performs relatively poor with regard to f1, f4, f5, especially for f1. Compared with GDHS obtaining the theoretical optimum within fewer iterations in f1, IAFSA-IHS fails to get the satisfying solution, which seems to fall into the prematurity as IABHS and IBAFSA do.It can be concluded that IAFSA-IHS exactly performs better on robustness, convergence speed, and precision with the improvements mentioned above synthetically. Takingf6, for instance, as a multimodal function, there exist a lot of local optima surrounding the global optimum which easily misguides the algorithm to stay in the local optimum. For solving the problem, GDHS sets HMCR, PAR, bw in dynamical mode, IBAFSA decodes the artificial fish into binary, sets the trial point, and reinitializes the artificial fish swarm regularly, and IABHS introduces the improvisation strategies with consideration of individual as well as dimension and self-adaptively adjusts HMCR, PAR, bw. All of those improvements are aimed at strengthening the convergence speed and accuracy, just as IAFSA-IHS does by means of the proposed modification in the paper. From the analysis of the result, these improvements indeed have effects as supposed.


### 4.2. Simulation of WTA Problems

#### 4.2.1. Example 1

There is red formation consisting of eight fighters confronting blue formation including six targets. Through calculation by (1)~(8), situation information referring to [17, 33] on both sides is given in Table 3 and Figures 12–14.Table 3
Situation information on both sides.
X/km
Y/km
Z/km
V
x/(km/h)
V
y/(km/h)
V
z/(km/h)
C
1
C
2
C
3
C
4
Information on red formation
1
40
79
5
662
662
0
361.65
43.59
1973.9
225.82
2
44
76
6
662
662
0
3
48
76
5
662
662
0
4
50
70
5
662
662
0
5
54
66
6
662
662
0
6
58
66
5
662
662
0
7
59
60
5
662
662
0
8
62
56
6
662
662
0
Information on blue formation
1
130
170
8
-
641
-
641
-
320
154.99
43.59
1789.7
451.64
2
140
160
7
-
641
-
641
-
320
3
145
155
8
-
641
-
641
-
320
4
150
146
8
-
641
-
641
-
320
5
160
140
7
-
641
-
641
-
320
6
165
130
8
-
641
-
641
-
320Figure 12
Air combat situation.Figure 13
Distribution of combat power energy possessed by both formations in vertical plane.Figure 14
Distribution of combat power energy possessed by both formations in horizontal plane.Descending velocity of each side with respect to the other side is obtained by (6) and (7) and shown in Tables 4 and 5.Table 4
Descending velocity of red formation with regard to blue formation.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.0094
0.0094
0.0098
0.0094
0.0093
0.0095
0.0090
0.0088
T2
0.0093
0.0094
0.0098
0.0095
0.0094
0.0098
0.0093
0.0091
T3
0.0092
0.0093
0.0097
0.0094
0.0095
0.0098
0.0093
0.0092
T4
0.0093
0.0095
0.0099
0.0097
0.0098
0.0102
0.0097
0.0096
T5
0.0086
0.0088
0.0092
0.0091
0.0092
0.0096
0.0093
0.0092
T6
0.0085
0.0088
0.0093
0.0091
0.0093
0.0098
0.0095
0.0095Table 5
Descending velocity of blue formation with regard to red formation.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.0078
0.0079
0.0082
0.0078
0.0077
0.0080
0.0075
0.0073
T2
0.0077
0.0078
0.0082
0.0079
0.0079
0.0082
0.0077
0.0076
T3
0.0076
0.0078
0.0081
0.0079
0.0079
0.0082
0.0078
0.0077
T4
0.0077
0.0079
0.0083
0.0081
0.0081
0.0085
0.0081
0.0080
T5
0.0071
0.0073
0.0077
0.0075
0.0077
0.0080
0.0077
0.0077
T6
0.0071
0.0073
0.0077
0.0076
0.0078
0.0082
0.0079
0.0079Relative superiority of red formation with regard to the blue one,pij, 1≤i≤8, 1≤j≤6, is derived by (8) and is given in Table 6.Table 6
Relative superiority of red formation with regard to the blue one.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.5455
0.5454
0.5449
0.5455
0.5457
0.5452
0.5461
0.5464
T2
0.5456
0.5454
0.5448
0.5454
0.5454
0.5449
0.5456
0.5459
T3
0.5458
0.5455
0.5449
0.5454
0.5454
0.5448
0.5455
0.5457
T4
0.5457
0.5453
0.5447
0.5450
0.5449
0.5443
0.5450
0.5451
T5
0.5468
0.5464
0.5457
0.5460
0.5457
0.5451
0.5457
0.5457
T6
0.5469
0.5464
0.5457
0.5458
0.5455
0.5449
0.5453
0.5453F
i, 1≤i≤8, denotes fighter i. Tj, 1≤j≤6, is target j. ingen=100, H0=0.3, S=5, visual=8, step=4, c1=0.6, c2=0.4, φ1=0.6, φ2=0.4, and max⁡gen=200; other parameters are set similarly as in verification tests.Compared with three algorithms mentioned in verification tests, the optimization process containing four algorithms and the final assignment solution is shown in Figure15 and Table 7.Table 7
Results of four algorithms.
F1
F2
F3
F4
F5
F6
F7
F8
Fitness value
GDHS
T6
T1
T3
T4
T5
T2
T4
T1
2.2308
IABHS
T1
T1
T6
T4
T5
T3
T4
T2
2.2315
IBAFSA
T6
T3
T5
T2
T1
T2
T1
T4
2.2299
IAFSA-IHS
T6
T2
T3
T5
T4
T3
T1
T4
2.2298Figure 15
Optimization figure containing four algorithms.It is obviously observed from Figure15 and Table 7 that solution obtained by IAFSA-IHS gets the lowest fitness value but in slower convergence speed. By comparison, the results derived by GDHS, IBAFSA, and IABHS achieve higher fitness value. GDHS only acts better than IABHS, and the result got by IBAFSA is similar to the one got by IAFSA-IHS. It is concluded that the proposed IAFSA-IHS outperforms the others in solving the example.


#### 4.2.2. Example 2

To test its performance widely, another situation is chosen in simulation; the situation information is shown in Table8 and Figures 16–18.Table 8
Situation information on both sides.
X/km
Y/km
Z/km
V
x/(km/h)
V
y/(km/h)
V
z/(km/h)
C
1
C
2
C
3
Information on red formation
1
50
80
5
828
552
138
361.65
43.59
1973.9
2
51
76
6
828
552
138
3
52
72
5
828
552
138
4
53
69
6
828
552
138
5
56
65
6
552
828
138
6
60
64
5
552
828
138
7
63
63
6
552
828
138
8
68
62
5
552
828
138
Information on blue formation
1
155
143
7
-
402
-
768
-
283
154.99
43.59
1789.7
2
150
144
7
-
402
-
768
-
283
3
140
146
8
-
402
-
768
-
283
4
135
150
7
-
768
-
402
-
283
5
132
160
8
-
768
-
402
-
283
6
130
170
8
-
768
-
402
-
283Figure 16
Air combat situation.Figure 17
Distribution of combat power energy possessed by both formations in vertical plane.Figure 18
Distribution of combat power energy possessed by both formations in horizontal plane.Descending velocity of each side with respect to the other side is gained by (6) and (7) and shown in Tables 9 and 10.Table 9
Descending velocity of red formation with regard to blue formation.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.0104
0.0100
0.0097
0.0095
0.0093
0.0095
0.0096
0.0098
T2
0.0113
0.0110
0.0106
0.0104
0.0102
0.0104
0.0106
0.0109
T3
0.0121
0.0117
0.0114
0.0112
0.0111
0.0114
0.0116
0.0120
T4
0.0118
0.0115
0.0112
0.0110
0.0110
0.0113
0.0115
0.0120
T5
0.0106
0.0104
0.0102
0.0101
0.0100
0.0104
0.0106
0.0111
T6
0.0101
0.0099
0.0097
0.0096
0.0096
0.0099
0.0102
0.0106Table 10
Descending velocity of blue formation with regard to red formation.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.0087
0.0084
0.0081
0.0079
0.0078
0.0079
0.0089
0.0082
T2
0.0095
0.0092
0.0089
0.0087
0.0085
0.0088
0.0089
0.0092
T3
0.0102
0.0099
0.0096
0.0094
0.0093
0.0096
0.0098
0.0101
T4
0.0100
0.0097
0.0095
0.0093
0.0092
0.0095
0.0097
0.0101
T5
0.0089
0.0087
0.0086
0.0084
0.0084
0.0087
0.0089
0.0093
T6
0.0084
0.0083
0.0081
0.0080
0.0080
0.0083
0.0085
0.0089Relative superiority of red formation with regard to the blue one,pij, 1≤i≤8, 1≤j≤6, is derived by (8) and is given in Table 11.Table 11
Relative superiority of red formation with regard to the blue one.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.5440
0.5445
0.5450
0.5453
0.5456
0.5453
0.5452
0.5448
T2
0.5428
0.5432
0.5437
0.5440
0.5442
0.5439
0.5437
0.5433
T3
0.5419
0.5423
0.5427
0.5429
0.5431
0.5427
0.5425
0.5420
T4
0.5422
0.5425
0.5429
0.5431
0.5432
0.5428
0.5425
0.5420
T5
0.5437
0.5439
0.5442
0.5444
0.5445
0.5440
0.5437
0.5431
T6
0.5444
0.5447
0.5449
0.5451
0.5451
0.5446
0.5443
0.5437Parameters remain unchanged as in Example1, and the result can be seen in Figure 19 and Table 12.Table 12
Results of four algorithms.
F1
F2
F3
F4
F5
F6
F7
F8
Fitness value
GDHS
T6
T5
T1
T2
T4
T1
0
T3
2.4894
IABHS
T6
T4
T2
T2
T5
T1
T3
T5
2.2415
IBAFSA
T3
T5
T6
T6
T4
T1
T2
T1
2.2413
IAFSA-IHS
T6
T5
T4
T6
T2
T1
T3
T5
2.2408Figure 19
Optimization figure containing four algorithms.From Figure19 and Table 12, it is concluded that IAFSA-IHS finds the best solution in highest convergence speed among the four algorithms. Besides, the initial solutions generated by IAFSA-IHS are far better than the ones of others. Among the other three algorithms, IBAFSA performs better in fitness value. IABHS presents a slightly inferior performance than the ones above. GDHS gets the worst results in fitness value as well as in convergence speed. From the example mentioned above, the proposed IAFSA-IHS achieves a general better performance in both simulations that indeed demonstrates the effectiveness of IAFSA-IHS for WTA.


## 4.1. Verification Tests by Benchmark Functions

The algorithms are always tested by benchmark functions. For thoroughly validating the performances of the proposed IAFSA-HIS, other three algorithms including IABHS, IBAFSA, and GDHS are introduced for comparison. Main properties of the selected benchmark functions used in the experiments are shown in Table1.Table 1
Benchmark functions.
Function
Equation
Range
Optimum
Property
Sphere
f
1
x
=
∑
i
=
1
n
x
i
2
-
100,100
0
Unimodal
Quartic
f
2
x
=
∑
i
=
1
n
i
x
i
4
+
r
a
n
d
o
m
0,1
-
1.28,1.28
0
Unimodal
Duadric
f
3
x
=
∑
i
=
1
n
∑
j
=
1
i
x
j
2
-
100,100
0
Unimodal
Rosenbrock
f
4
x
=
∑
i
=
1
n
-
1
x
i
-
1
2
+
100
x
i
+
1
-
x
i
2
2
-
30,30
0
Unimodal
Schwefel
f
5
x
=
-
∑
i
=
1
n
x
i
sin
⁡
x
i
-
500,500
-
12569.5
Multimodal
Rastrigin
f
6
x
=
∑
i
=
1
n
x
i
2
-
10
2
π
x
i
+
10
-
5.12,5.12
0
Multimodal
Griewank
f
7
x
=
1
4000
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
cos
⁡
x
i
i
+
1
-
600,600
0
Multimodal
Ackley
f
8
x
=
-
20
e
-
0.2
∑
i
=
1
n
x
i
2
/
n
-
e
∑
i
=
1
n
cos
⁡
2
π
x
i
/
n
+
e
+
20
-
30,30
0
MultimodalParameters are set as follows:ingen=100, H0=0.3, S=5, visual=1.5, γ1=0.3, γ2=0.5, c1=0.6, c2=0.4, φ1=0.6, φ2=0.4, max⁡gen=100000, and n=30; for other parameters, refer to [24, 30, 32].The algorithms were run 100 times independently to catch their stochastic properties and the results are depicted in Table2 and Figures 4–11.Table 2
Comparison of results with other algorithms.
IAFSA-IHS
IBAFSA
IABHS
GDHS
f
1
Best
3.0303
e
-
218
3.4848
e
-
100
3.8552
e
-
60
0
Mean
3.7514
e
-
220
4.3574
e
-
99
6.8125
e
-
59
0
Worst
9.3558
e
-
221
6.1893
e
-
97
7.2759
e
-
57
0
Std.dev
7.2377
e
-
218
3.5238
e
-
99
8.7153
e
-
59
0
f
2
Best
4.3659
e
-
3
5.7852
e
-
2
1.5798
e
-
4
8.9654
e
-
3
Mean
5.9283
e
-
2
6.0237
e
-
2
2.0721
e
-
2
9.3571
e
-
2
Worst
1.1135
e
-
1
1.3145
e
-
1
5.3492
e
-
2
7.5883
e
-
2
Std.dev
6.3844
e
-
2
9.3561
e
-
2
8.9443
e
-
3
1.0184
e
-
2
f
3
Best
6.901
e
-
3
1.8312
e
+
3
5.7163
e
-
4
3.3694
e
-
2
Mean
5.4431
e
-
2
7.5403
e
+
3
6.7303
e
-
2
5.2391
e
-
1
Worst
3.2967
e
-
1
6.5727
e
+
4
8.1835
e
-
2
2.4749
e
+
0
Std.dev
2.3617
e
-
2
5.4329
e
+
3
4.3516
e
-
3
4.7161
e
-
2
f
4
Best
1.1364
e
+
1
3.5801
e
+
1
1.0075
e
+
1
6.4625
e
+
1
Mean
9.5879
e
+
1
6.0574
e
+
1
9.5855
e
+
1
8.5942
e
+
1
Worst
1.2841
e
+
2
2.9714
e
+
1
3.3575
e
+
1
1.5399
e
+
2
Std.dev
9.9836
e
+
1
3.4769
e
+
1
1.4848
e
+
1
7.6935
e
+
1
f
5
Best
-
1.1821
e
+
4
-
1.1108
e
+
4
-
1.1753
e
+
4
-
1.2024
e
+
4
Mean
-
1.1028
e
+
4
-
9.9931
e
+
3
-
1.0387
e
+
4
-
1.1569
e
+
4
Worst
-
1.0254
e
+
4
-
9.8301
e
+
3
-
9.4539
e
+
3
-
1.0965
e
+
4
Std.dev
-
1.0879
e
+
4
-
1.0256
e
+
4
-
1.0024
e
+
4
-
1.2569
e
+
4
f
6
Best
0
3.1827
e
-
4
1.9847
e
-
3
4.2084
e
-
4
Mean
0
9.5648
e
-
4
9.0955
e
-
3
1.5682
e
-
3
Worst
0
8.1815
e
-
3
6.5276
e
-
1
3.3524
e
-
2
Std.dev
0
3.1348
e
-
4
6.4596
e
-
3
1.2563
e
-
3
f
7
Best
0
6.0477
e
-
10
3.2291
e
-
14
7.7708
e
-
16
Mean
0
7.5628
e
-
9
9.2486
e
-
13
6.2967
e
-
14
Worst
0
9.2466
e
-
8
9.7123
e
-
12
9.1222
e
-
13
Std.dev
0
1.3132
e
-
9
2.2486
e
-
13
1.5422
e
-
14
f
8
Best
4.0107
e
-
16
3.5687
e
-
15
3.0026
e
-
13
1.2627
e
-
15
Mean
7.5975
e
-
16
7.0145
e
-
15
5.2435
e
-
13
3.3275
e
-
15
Worst
3.2354
e
-
15
9.6461
e
-
14
6.4564
e
-
12
9.5619
e
-
15
Std.dev
5.5484
e
-
16
5.5686
e
-
15
1.3967
e
-
13
2.0131
e
-
15Figure 4
Sphere.Figure 5
Quartic.Figure 6
Duadric.Figure 7
Rosenbrock.Figure 8
Schwefel.Figure 9
Rastrigin.Figure 10
Griewank.Figure 11
Ackley.The parameters Best, Mean, Worst, and Std.dev are the best, mean, worst, and stand error of the optimum on average after 100 running times, where Mean reflects the precision of the results obtained by the algorithms within certain iterations and Std.dev reflects the robustness and stability of the algorithms.As seen from Table2 and Figures 4–11, the advantages in terms of best, median, worst, mean, and standard error of IAFSA-HIS are outstanding among the four algorithms in general. Particularly, IAFSA-IHS achieves the theoretical optimum while others fail still the termination in f6 and f7. However, the proposed algorithm performs relatively poor with regard to f1, f4, f5, especially for f1. Compared with GDHS obtaining the theoretical optimum within fewer iterations in f1, IAFSA-IHS fails to get the satisfying solution, which seems to fall into the prematurity as IABHS and IBAFSA do.It can be concluded that IAFSA-IHS exactly performs better on robustness, convergence speed, and precision with the improvements mentioned above synthetically. Takingf6, for instance, as a multimodal function, there exist a lot of local optima surrounding the global optimum which easily misguides the algorithm to stay in the local optimum. For solving the problem, GDHS sets HMCR, PAR, bw in dynamical mode, IBAFSA decodes the artificial fish into binary, sets the trial point, and reinitializes the artificial fish swarm regularly, and IABHS introduces the improvisation strategies with consideration of individual as well as dimension and self-adaptively adjusts HMCR, PAR, bw. All of those improvements are aimed at strengthening the convergence speed and accuracy, just as IAFSA-IHS does by means of the proposed modification in the paper. From the analysis of the result, these improvements indeed have effects as supposed.


## 4.2. Simulation of WTA Problems

### 4.2.1. Example 1

There is red formation consisting of eight fighters confronting blue formation including six targets. Through calculation by (1)~(8), situation information referring to [17, 33] on both sides is given in Table 3 and Figures 12–14.Table 3
Situation information on both sides.
X/km
Y/km
Z/km
V
x/(km/h)
V
y/(km/h)
V
z/(km/h)
C
1
C
2
C
3
C
4
Information on red formation
1
40
79
5
662
662
0
361.65
43.59
1973.9
225.82
2
44
76
6
662
662
0
3
48
76
5
662
662
0
4
50
70
5
662
662
0
5
54
66
6
662
662
0
6
58
66
5
662
662
0
7
59
60
5
662
662
0
8
62
56
6
662
662
0
Information on blue formation
1
130
170
8
-
641
-
641
-
320
154.99
43.59
1789.7
451.64
2
140
160
7
-
641
-
641
-
320
3
145
155
8
-
641
-
641
-
320
4
150
146
8
-
641
-
641
-
320
5
160
140
7
-
641
-
641
-
320
6
165
130
8
-
641
-
641
-
320Figure 12
Air combat situation.Figure 13
Distribution of combat power energy possessed by both formations in vertical plane.Figure 14
Distribution of combat power energy possessed by both formations in horizontal plane.Descending velocity of each side with respect to the other side is obtained by (6) and (7) and shown in Tables 4 and 5.Table 4
Descending velocity of red formation with regard to blue formation.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.0094
0.0094
0.0098
0.0094
0.0093
0.0095
0.0090
0.0088
T2
0.0093
0.0094
0.0098
0.0095
0.0094
0.0098
0.0093
0.0091
T3
0.0092
0.0093
0.0097
0.0094
0.0095
0.0098
0.0093
0.0092
T4
0.0093
0.0095
0.0099
0.0097
0.0098
0.0102
0.0097
0.0096
T5
0.0086
0.0088
0.0092
0.0091
0.0092
0.0096
0.0093
0.0092
T6
0.0085
0.0088
0.0093
0.0091
0.0093
0.0098
0.0095
0.0095Table 5
Descending velocity of blue formation with regard to red formation.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.0078
0.0079
0.0082
0.0078
0.0077
0.0080
0.0075
0.0073
T2
0.0077
0.0078
0.0082
0.0079
0.0079
0.0082
0.0077
0.0076
T3
0.0076
0.0078
0.0081
0.0079
0.0079
0.0082
0.0078
0.0077
T4
0.0077
0.0079
0.0083
0.0081
0.0081
0.0085
0.0081
0.0080
T5
0.0071
0.0073
0.0077
0.0075
0.0077
0.0080
0.0077
0.0077
T6
0.0071
0.0073
0.0077
0.0076
0.0078
0.0082
0.0079
0.0079Relative superiority of red formation with regard to the blue one,pij, 1≤i≤8, 1≤j≤6, is derived by (8) and is given in Table 6.Table 6
Relative superiority of red formation with regard to the blue one.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.5455
0.5454
0.5449
0.5455
0.5457
0.5452
0.5461
0.5464
T2
0.5456
0.5454
0.5448
0.5454
0.5454
0.5449
0.5456
0.5459
T3
0.5458
0.5455
0.5449
0.5454
0.5454
0.5448
0.5455
0.5457
T4
0.5457
0.5453
0.5447
0.5450
0.5449
0.5443
0.5450
0.5451
T5
0.5468
0.5464
0.5457
0.5460
0.5457
0.5451
0.5457
0.5457
T6
0.5469
0.5464
0.5457
0.5458
0.5455
0.5449
0.5453
0.5453F
i, 1≤i≤8, denotes fighter i. Tj, 1≤j≤6, is target j. ingen=100, H0=0.3, S=5, visual=8, step=4, c1=0.6, c2=0.4, φ1=0.6, φ2=0.4, and max⁡gen=200; other parameters are set similarly as in verification tests.Compared with three algorithms mentioned in verification tests, the optimization process containing four algorithms and the final assignment solution is shown in Figure15 and Table 7.Table 7
Results of four algorithms.
F1
F2
F3
F4
F5
F6
F7
F8
Fitness value
GDHS
T6
T1
T3
T4
T5
T2
T4
T1
2.2308
IABHS
T1
T1
T6
T4
T5
T3
T4
T2
2.2315
IBAFSA
T6
T3
T5
T2
T1
T2
T1
T4
2.2299
IAFSA-IHS
T6
T2
T3
T5
T4
T3
T1
T4
2.2298Figure 15
Optimization figure containing four algorithms.It is obviously observed from Figure15 and Table 7 that solution obtained by IAFSA-IHS gets the lowest fitness value but in slower convergence speed. By comparison, the results derived by GDHS, IBAFSA, and IABHS achieve higher fitness value. GDHS only acts better than IABHS, and the result got by IBAFSA is similar to the one got by IAFSA-IHS. It is concluded that the proposed IAFSA-IHS outperforms the others in solving the example.


### 4.2.2. Example 2

To test its performance widely, another situation is chosen in simulation; the situation information is shown in Table8 and Figures 16–18.Table 8
Situation information on both sides.
X/km
Y/km
Z/km
V
x/(km/h)
V
y/(km/h)
V
z/(km/h)
C
1
C
2
C
3
Information on red formation
1
50
80
5
828
552
138
361.65
43.59
1973.9
2
51
76
6
828
552
138
3
52
72
5
828
552
138
4
53
69
6
828
552
138
5
56
65
6
552
828
138
6
60
64
5
552
828
138
7
63
63
6
552
828
138
8
68
62
5
552
828
138
Information on blue formation
1
155
143
7
-
402
-
768
-
283
154.99
43.59
1789.7
2
150
144
7
-
402
-
768
-
283
3
140
146
8
-
402
-
768
-
283
4
135
150
7
-
768
-
402
-
283
5
132
160
8
-
768
-
402
-
283
6
130
170
8
-
768
-
402
-
283Figure 16
Air combat situation.Figure 17
Distribution of combat power energy possessed by both formations in vertical plane.Figure 18
Distribution of combat power energy possessed by both formations in horizontal plane.Descending velocity of each side with respect to the other side is gained by (6) and (7) and shown in Tables 9 and 10.Table 9
Descending velocity of red formation with regard to blue formation.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.0104
0.0100
0.0097
0.0095
0.0093
0.0095
0.0096
0.0098
T2
0.0113
0.0110
0.0106
0.0104
0.0102
0.0104
0.0106
0.0109
T3
0.0121
0.0117
0.0114
0.0112
0.0111
0.0114
0.0116
0.0120
T4
0.0118
0.0115
0.0112
0.0110
0.0110
0.0113
0.0115
0.0120
T5
0.0106
0.0104
0.0102
0.0101
0.0100
0.0104
0.0106
0.0111
T6
0.0101
0.0099
0.0097
0.0096
0.0096
0.0099
0.0102
0.0106Table 10
Descending velocity of blue formation with regard to red formation.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.0087
0.0084
0.0081
0.0079
0.0078
0.0079
0.0089
0.0082
T2
0.0095
0.0092
0.0089
0.0087
0.0085
0.0088
0.0089
0.0092
T3
0.0102
0.0099
0.0096
0.0094
0.0093
0.0096
0.0098
0.0101
T4
0.0100
0.0097
0.0095
0.0093
0.0092
0.0095
0.0097
0.0101
T5
0.0089
0.0087
0.0086
0.0084
0.0084
0.0087
0.0089
0.0093
T6
0.0084
0.0083
0.0081
0.0080
0.0080
0.0083
0.0085
0.0089Relative superiority of red formation with regard to the blue one,pij, 1≤i≤8, 1≤j≤6, is derived by (8) and is given in Table 11.Table 11
Relative superiority of red formation with regard to the blue one.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.5440
0.5445
0.5450
0.5453
0.5456
0.5453
0.5452
0.5448
T2
0.5428
0.5432
0.5437
0.5440
0.5442
0.5439
0.5437
0.5433
T3
0.5419
0.5423
0.5427
0.5429
0.5431
0.5427
0.5425
0.5420
T4
0.5422
0.5425
0.5429
0.5431
0.5432
0.5428
0.5425
0.5420
T5
0.5437
0.5439
0.5442
0.5444
0.5445
0.5440
0.5437
0.5431
T6
0.5444
0.5447
0.5449
0.5451
0.5451
0.5446
0.5443
0.5437Parameters remain unchanged as in Example1, and the result can be seen in Figure 19 and Table 12.Table 12
Results of four algorithms.
F1
F2
F3
F4
F5
F6
F7
F8
Fitness value
GDHS
T6
T5
T1
T2
T4
T1
0
T3
2.4894
IABHS
T6
T4
T2
T2
T5
T1
T3
T5
2.2415
IBAFSA
T3
T5
T6
T6
T4
T1
T2
T1
2.2413
IAFSA-IHS
T6
T5
T4
T6
T2
T1
T3
T5
2.2408Figure 19
Optimization figure containing four algorithms.From Figure19 and Table 12, it is concluded that IAFSA-IHS finds the best solution in highest convergence speed among the four algorithms. Besides, the initial solutions generated by IAFSA-IHS are far better than the ones of others. Among the other three algorithms, IBAFSA performs better in fitness value. IABHS presents a slightly inferior performance than the ones above. GDHS gets the worst results in fitness value as well as in convergence speed. From the example mentioned above, the proposed IAFSA-IHS achieves a general better performance in both simulations that indeed demonstrates the effectiveness of IAFSA-IHS for WTA.


## 4.2.1. Example 1

There is red formation consisting of eight fighters confronting blue formation including six targets. Through calculation by (1)~(8), situation information referring to [17, 33] on both sides is given in Table 3 and Figures 12–14.Table 3
Situation information on both sides.
X/km
Y/km
Z/km
V
x/(km/h)
V
y/(km/h)
V
z/(km/h)
C
1
C
2
C
3
C
4
Information on red formation
1
40
79
5
662
662
0
361.65
43.59
1973.9
225.82
2
44
76
6
662
662
0
3
48
76
5
662
662
0
4
50
70
5
662
662
0
5
54
66
6
662
662
0
6
58
66
5
662
662
0
7
59
60
5
662
662
0
8
62
56
6
662
662
0
Information on blue formation
1
130
170
8
-
641
-
641
-
320
154.99
43.59
1789.7
451.64
2
140
160
7
-
641
-
641
-
320
3
145
155
8
-
641
-
641
-
320
4
150
146
8
-
641
-
641
-
320
5
160
140
7
-
641
-
641
-
320
6
165
130
8
-
641
-
641
-
320Figure 12
Air combat situation.Figure 13
Distribution of combat power energy possessed by both formations in vertical plane.Figure 14
Distribution of combat power energy possessed by both formations in horizontal plane.Descending velocity of each side with respect to the other side is obtained by (6) and (7) and shown in Tables 4 and 5.Table 4
Descending velocity of red formation with regard to blue formation.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.0094
0.0094
0.0098
0.0094
0.0093
0.0095
0.0090
0.0088
T2
0.0093
0.0094
0.0098
0.0095
0.0094
0.0098
0.0093
0.0091
T3
0.0092
0.0093
0.0097
0.0094
0.0095
0.0098
0.0093
0.0092
T4
0.0093
0.0095
0.0099
0.0097
0.0098
0.0102
0.0097
0.0096
T5
0.0086
0.0088
0.0092
0.0091
0.0092
0.0096
0.0093
0.0092
T6
0.0085
0.0088
0.0093
0.0091
0.0093
0.0098
0.0095
0.0095Table 5
Descending velocity of blue formation with regard to red formation.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.0078
0.0079
0.0082
0.0078
0.0077
0.0080
0.0075
0.0073
T2
0.0077
0.0078
0.0082
0.0079
0.0079
0.0082
0.0077
0.0076
T3
0.0076
0.0078
0.0081
0.0079
0.0079
0.0082
0.0078
0.0077
T4
0.0077
0.0079
0.0083
0.0081
0.0081
0.0085
0.0081
0.0080
T5
0.0071
0.0073
0.0077
0.0075
0.0077
0.0080
0.0077
0.0077
T6
0.0071
0.0073
0.0077
0.0076
0.0078
0.0082
0.0079
0.0079Relative superiority of red formation with regard to the blue one,pij, 1≤i≤8, 1≤j≤6, is derived by (8) and is given in Table 6.Table 6
Relative superiority of red formation with regard to the blue one.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.5455
0.5454
0.5449
0.5455
0.5457
0.5452
0.5461
0.5464
T2
0.5456
0.5454
0.5448
0.5454
0.5454
0.5449
0.5456
0.5459
T3
0.5458
0.5455
0.5449
0.5454
0.5454
0.5448
0.5455
0.5457
T4
0.5457
0.5453
0.5447
0.5450
0.5449
0.5443
0.5450
0.5451
T5
0.5468
0.5464
0.5457
0.5460
0.5457
0.5451
0.5457
0.5457
T6
0.5469
0.5464
0.5457
0.5458
0.5455
0.5449
0.5453
0.5453F
i, 1≤i≤8, denotes fighter i. Tj, 1≤j≤6, is target j. ingen=100, H0=0.3, S=5, visual=8, step=4, c1=0.6, c2=0.4, φ1=0.6, φ2=0.4, and max⁡gen=200; other parameters are set similarly as in verification tests.Compared with three algorithms mentioned in verification tests, the optimization process containing four algorithms and the final assignment solution is shown in Figure15 and Table 7.Table 7
Results of four algorithms.
F1
F2
F3
F4
F5
F6
F7
F8
Fitness value
GDHS
T6
T1
T3
T4
T5
T2
T4
T1
2.2308
IABHS
T1
T1
T6
T4
T5
T3
T4
T2
2.2315
IBAFSA
T6
T3
T5
T2
T1
T2
T1
T4
2.2299
IAFSA-IHS
T6
T2
T3
T5
T4
T3
T1
T4
2.2298Figure 15
Optimization figure containing four algorithms.It is obviously observed from Figure15 and Table 7 that solution obtained by IAFSA-IHS gets the lowest fitness value but in slower convergence speed. By comparison, the results derived by GDHS, IBAFSA, and IABHS achieve higher fitness value. GDHS only acts better than IABHS, and the result got by IBAFSA is similar to the one got by IAFSA-IHS. It is concluded that the proposed IAFSA-IHS outperforms the others in solving the example.


## 4.2.2. Example 2

To test its performance widely, another situation is chosen in simulation; the situation information is shown in Table8 and Figures 16–18.Table 8
Situation information on both sides.
X/km
Y/km
Z/km
V
x/(km/h)
V
y/(km/h)
V
z/(km/h)
C
1
C
2
C
3
Information on red formation
1
50
80
5
828
552
138
361.65
43.59
1973.9
2
51
76
6
828
552
138
3
52
72
5
828
552
138
4
53
69
6
828
552
138
5
56
65
6
552
828
138
6
60
64
5
552
828
138
7
63
63
6
552
828
138
8
68
62
5
552
828
138
Information on blue formation
1
155
143
7
-
402
-
768
-
283
154.99
43.59
1789.7
2
150
144
7
-
402
-
768
-
283
3
140
146
8
-
402
-
768
-
283
4
135
150
7
-
768
-
402
-
283
5
132
160
8
-
768
-
402
-
283
6
130
170
8
-
768
-
402
-
283Figure 16
Air combat situation.Figure 17
Distribution of combat power energy possessed by both formations in vertical plane.Figure 18
Distribution of combat power energy possessed by both formations in horizontal plane.Descending velocity of each side with respect to the other side is gained by (6) and (7) and shown in Tables 9 and 10.Table 9
Descending velocity of red formation with regard to blue formation.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.0104
0.0100
0.0097
0.0095
0.0093
0.0095
0.0096
0.0098
T2
0.0113
0.0110
0.0106
0.0104
0.0102
0.0104
0.0106
0.0109
T3
0.0121
0.0117
0.0114
0.0112
0.0111
0.0114
0.0116
0.0120
T4
0.0118
0.0115
0.0112
0.0110
0.0110
0.0113
0.0115
0.0120
T5
0.0106
0.0104
0.0102
0.0101
0.0100
0.0104
0.0106
0.0111
T6
0.0101
0.0099
0.0097
0.0096
0.0096
0.0099
0.0102
0.0106Table 10
Descending velocity of blue formation with regard to red formation.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.0087
0.0084
0.0081
0.0079
0.0078
0.0079
0.0089
0.0082
T2
0.0095
0.0092
0.0089
0.0087
0.0085
0.0088
0.0089
0.0092
T3
0.0102
0.0099
0.0096
0.0094
0.0093
0.0096
0.0098
0.0101
T4
0.0100
0.0097
0.0095
0.0093
0.0092
0.0095
0.0097
0.0101
T5
0.0089
0.0087
0.0086
0.0084
0.0084
0.0087
0.0089
0.0093
T6
0.0084
0.0083
0.0081
0.0080
0.0080
0.0083
0.0085
0.0089Relative superiority of red formation with regard to the blue one,pij, 1≤i≤8, 1≤j≤6, is derived by (8) and is given in Table 11.Table 11
Relative superiority of red formation with regard to the blue one.
F1
F2
F3
F4
F5
F6
F7
F8
T1
0.5440
0.5445
0.5450
0.5453
0.5456
0.5453
0.5452
0.5448
T2
0.5428
0.5432
0.5437
0.5440
0.5442
0.5439
0.5437
0.5433
T3
0.5419
0.5423
0.5427
0.5429
0.5431
0.5427
0.5425
0.5420
T4
0.5422
0.5425
0.5429
0.5431
0.5432
0.5428
0.5425
0.5420
T5
0.5437
0.5439
0.5442
0.5444
0.5445
0.5440
0.5437
0.5431
T6
0.5444
0.5447
0.5449
0.5451
0.5451
0.5446
0.5443
0.5437Parameters remain unchanged as in Example1, and the result can be seen in Figure 19 and Table 12.Table 12
Results of four algorithms.
F1
F2
F3
F4
F5
F6
F7
F8
Fitness value
GDHS
T6
T5
T1
T2
T4
T1
0
T3
2.4894
IABHS
T6
T4
T2
T2
T5
T1
T3
T5
2.2415
IBAFSA
T3
T5
T6
T6
T4
T1
T2
T1
2.2413
IAFSA-IHS
T6
T5
T4
T6
T2
T1
T3
T5
2.2408Figure 19
Optimization figure containing four algorithms.From Figure19 and Table 12, it is concluded that IAFSA-IHS finds the best solution in highest convergence speed among the four algorithms. Besides, the initial solutions generated by IAFSA-IHS are far better than the ones of others. Among the other three algorithms, IBAFSA performs better in fitness value. IABHS presents a slightly inferior performance than the ones above. GDHS gets the worst results in fitness value as well as in convergence speed. From the example mentioned above, the proposed IAFSA-IHS achieves a general better performance in both simulations that indeed demonstrates the effectiveness of IAFSA-IHS for WTA.


## 5. Conclusion and Future Works

A new approach to weapon-target assignment in cooperative air combat is proposed. Firstly, assignment model is constructed based on the concept of combat power potential. On the other hand, a hybrid algorithm combining improved AFSA with improved HS is proposed. Finally, the algorithm is tested by eight benchmark functions and two concrete WTA problems in air combat; the results validate the superiority of the proposed approach.In the future, we will pay more attention to details about the combat power potential energy model of fighter in order to accurately reflect the practical cooperative air combat.



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*Source: 2936279-2017-10-02.xml*