Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 9897153, 13 pages

https://doi.org/10.1155/2017/9897153

## MOQPSO-D/S for Air and Missile Defense WTA Problem under Uncertainty

School of Air and Missile Defense, Air Force Engineering University, Xi’an 710051, China

Correspondence should be addressed to Hao Xu; moc.361@oahuxdgk

Received 16 December 2016; Revised 5 November 2017; Accepted 12 November 2017; Published 14 December 2017

Academic Editor: Erik Cuevas

Copyright © 2017 Hao Xu 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

Aiming at the shortcomings of single objective optimization for solving weapon target assignment (WTA) and the existing multiobjective optimization based WTA method having problems being applied in air and missile defense combat under uncertainty, a fuzzy multiobjective programming based WTA method was proposed to enhance the adaptability of WTA decision to the changes of battlefield situation. Firstly, a multiobjective quantum-behaved particle swarm optimization with double/single-well (MOQPSO-D/S) algorithm was proposed by adopting the double/single-well based position update method, the hybrid random mutation method, and the two-stage based guider particles selection method. Secondly, a fuzzy multiobjective programming WTA model was constructed with consideration of air and missile defense combat’s characteristics. And, the uncertain WTA model was equivalently clarified based on the necessity degree principle of uncertainty theory. Thirdly, with particles encoding and illegal particles adjusting, the MOQPSO-D/S algorithm was adopted to solve the fuzzy multiobjective programming based WTA model. Finally, example simulation was conducted, and the result shows that the WTA model constructed is rational and MOQPSO-D/S algorithm is efficient.

#### 1. Introduction

In information warfare, air attack is a highly integrated operation form. Defenders need to carry out the task of antiaerodynamic targets and antiballistic missiles simultaneously [1]. As one of the key phases in air and missile defense combat, weapon target assignment (WTA) plays a significant role on operational effectiveness. Therefore, study on air and missile defense WTA problem is of great significance.

Given the importance, WTA has been studied with application of Lagrange relaxation algorithm [2], ant colony algorithm [3], genetic algorithm [4], clone selection algorithm [5], and particle swarm optimization algorithm [6]. Most of these researches construct WTA model based on single objective optimization with the objective function of maximizing the kill probability to enemy. They need to allocate all the firepower without consideration of the sequent operation need and cannot fit with the operational practice. In order to avoid allocating all firepower, a WTA model was constructed with the objective function of maximizing effectiveness-cost ratio in literature [6]. However, the adaptability of WTA decision, based on this method, to the changes of battlefield situation is still weak. Because this method is still a single objective optimization method, when conducting this method, the decision-making preferences need to be provided beforehand and only one WTA alternative can be given. Aiming at overcoming these weak points, the multiobjective optimization method was introduced into studying WTA problem [7–9]. With consideration of every objective, the Pareto solution set, obtained by multiobjective optimization method, can provide a series of WTA alternatives for the commander. And the final WTA decision will be made by choosing one from the alternatives. These studies [7–9] could provide useful reference for research on air and missile defense WTA problem. However, they cannot be applied to air and missile defense combat, which has the characteristic of multiweapon composite disposition, directly. Moreover, with the growing complexity of battlefield, the certain WTA method [3–9] can hardly be applied to air and missile defense combat under uncertainty.

In order to address the issues above, the WTA model for air and missile defense with uncertain threat information is constructed based on fuzzy multiobjective programming, with consideration of the air and missile defense combat’s characteristics. Due to the superior performance of quantum-behaved particle swarm optimization algorithm, a multiobjective quantum-behaved particle swarm optimization with double/single-well algorithm is proposed for solving the WTA model.

#### 2. Basic Theory

##### 2.1. Concepts of Multiobjective Optimization

*Definition 1 (multiobjective optimization problem). *Taking the minimizing problem as an example, a multiobjective optimization problem can be defined as follows:In (1), is objective function, is the decision vector, and is the decision space defined by a series of constraints.

*Definition 2 (Pareto dominance). *Let and be two feasible solutions in Ω; ifthen Pareto dominates , denoted by .

*Definition 3 (Pareto optimal solution). *Let ; if , meeting , then would be defined as a Pareto optimal solution. And the set would be defined as Pareto optimal solution set.

*Definition 4 (Pareto front). *The region which consisted of the objective values, corresponding to all solutions in , is defined as Pareto front.

##### 2.2. QPSO Algorithm

Quantum-behaved particle swarm optimization (QPSO) algorithm [10, 11], compared with the particle swarm optimization algorithm, has many advantages, such as faster convergence rate, fewer control parameters, and better global convergence. It has attracted much attention of scholars [12, 13].

QPSO algorithm is a quantum mechanics based optimization algorithm. It is assumed that the motion state of the particles in the optimal space relative to the attractor can be described by the wave function in potential well. And is described as follows:In (3), is the characteristic length of potential well.

The corresponding probability density function is , and the probability distribution function is .

Suppose , and let ; the position of the particle relative to the attractor can be obtained as .

And the absolute position of the particle in the optimal space is . Therefore, the position update formula of particle can be obtained as follows:In (4),* t* is the current generation, . If , “” would be chosen as “+”, or it would be chosen as “−”. and are as follows:In (5), , is the individual best position of particle , and is the global best position. In (6), is the expansion-constriction factor.

##### 2.3. QPSO with Double-Well Algorithm

Based on the double-*δ* potential well quantum model, a QPSO with double-well algorithm is proposed by Xu et al. [14] to resolve the problem of population diversity decline due to the fast convergence of QPSO algorithm. In this algorithm, the motion of particles is simulated as in a double-well space. The double-well based position update formula of particle can be defined as follows:In (7), (*t*) is the distance coefficient between two *δ* potential wells, and is a random number uniformly distributing over . If , “” would be chosen as “+”, or it would be chosen as “−”. The two attractors and are defined as (8), and the characteristic length is defined as (9).

In (8), , , is the individual best position, and and are two guider particles.

#### 3. MOQPSO-D/S Algorithm

Given the superior performance, the QPSO algorithm is introduced to solve the multiobjective problems. And the multiobjective quantum-behaved particle swarm optimization algorithm (MOQPSO) [14–18] is put forward. However, the requirement for solving a multiobjective optimization problem is different from solving a single objective optimization problem. It requires that the diversity of solutions should be good, the distribution of solutions should be uniform, and the distance between solution and the true Pareto front should be as close as possible. The fast convergence performance of QPSO would lead to the premature convergence of MOQPSO. And the global convergence to an optimal solution of QPSO would go against the diversity and distribution of solutions obtained by MOQPSO. In order to improve the performance of MOQPSO algorithm to solve the multiobjective programming problem, a multiobjective quantum-behaved particle swarm optimization with double/single-well (MOQPSO-D/S) algorithm was proposed by combining the QPSO algorithm and the QPSO with double-well algorithm. In this algorithm, a double/single-well based position update method is adopted to balance the relationship of solutions’ diversity, convergence precision, and convergence rate. A hybrid random mutation method is also adopted to avoid premature convergence and improve the convergence precision. A two-stage based guider particles selection method is applied to this algorithm to improve the solution distribution uniformity and convergence rate.

##### 3.1. Double/Single-Well Based Position Update Method

The so-called double/single-well based position update method includes two aspects. On one hand, in order to improve solutions’ diversity and avoid premature convergence, the particle position is updated based on QPSO with double-well algorithm in the early optimization stage. On the other hand, in order to improve convergence precision and convergence rate, the particle position is updated based on QPSO algorithm in the final optimization stage. The double/single-well based position update formula is as follows:In (10), is the maximum generation. If , “” would be chosen as “+”, or it would be chosen as “−”. The characteristic length is defined as in the following equation.

##### 3.2. Hybrid Random Mutation Method

In order to avoid premature convergence and improve convergence precision, a hybrid random mutation method is proposed. The so-called hybrid random mutation is to conduct uniform random mutation in the early optimization stage of the algorithm and to conduct Gaussian random mutation in the final optimization stage. Particles can traverse on the optimal space with equal probability completely by conducting uniform random mutation, which can improve the diversity of solution and avoid premature convergence. Particles can move around the original position by conducting Gaussian mutation so as to avoid lowering optimization efficiency caused by the too large mutation range in the final optimization stage and improve convergence precision. The detail of hybrid random mutation is as follows:In (12), rand is a random number uniformly distributing on , is optimization range, Rand(•) is the uniform random mutation operator, and Gaussian(•) is the Gaussian random mutation operator. is the mutation probability, which is defined as follows:

##### 3.3. Two-Stage Based Guider Particles Selection Method

In MOQPSO-D/S algorithm, the guider particles for each particle should be selected during iteration. The selection of guider particles from the external archive, with application of the roulette method [17], is not conducive to the distribution uniformity of solution, due to its strong randomicity. In order to address the issue above, a two-stage based guider particles selection method is proposed. The so-called two-stage based guider particles selection method includes the following two steps. Firstly, particles in the external archive should be sorted in descending order according to their crowding distance, and the 10% of particles with the biggest crowding distance is preliminary selected. Secondly, the particle with the closest sigma value to particle is selected as the guider particle for . The calculation method of the crowding distance for each particle in external archive can be referred to literature [19]. The calculation method of sigma value [20] for each particle is shown as in (14), and the detail of guider particle selection by adopting sigma value method is shown in Figure 1. When the particle position is updated based on QPSO with double-well, the second guider particle should be selected. The neighbor particle, with the larger distance to , will be selected as the second guider particle . The detail of this method for selection is shown in Figure 2.