Mathematical Problems in Engineering

Volume 2017, Article ID 2936279, 17 pages

https://doi.org/10.1155/2017/2936279

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

^{1}Air Force Engineering University, Xi’an 710038, China^{2}93088 Unit of PLA, Chifeng 024400, China

Correspondence should be addressed to Yi-zhe Chang; moc.qq@954488272

Received 9 May 2017; Accepted 8 August 2017; Published 2 October 2017

Academic Editor: Jean Jacques Loiseau

Copyright © 2017 Yi-zhe Chang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 Section 2, 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 as , , then gradient of function in point can be defined as

It is known from [26] that combat power potential energy of fighter is a function with regard to coordinates of fighters and targets . Calculate the gradient of depicted in [26] on , , and then we can get

Take , for instance; through further derivation, we will obtain

For losing the burden of calculation, , , , are simplified as , , , , where . Substitute simplified , , , into (3); we will derive

and are yielded in the same way as follows:

Descending velocity of fighter in direction of target is got in the light of definition of gradient:

Similarly, descending velocity of target in direction of fighter is given as

###### 2.1.3. Calculation on Relative Superiority

Calculation on relative superiority of fighter with respect to target is expressed aswhere is the descending velocity of fighter in direction to location of target and is the descending velocity of target in direction to location of fighter .

##### 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 superiority .

Consider the assignment problem of fighters to targets: let be the relative superiority, where and . The WTA is formulated as follows: where is the decision value indicating if fighter is assigned to target ; if fighter is assigned to target and if fighter is not assigned to target .

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 Figure 1.