Discrete Dynamics in Nature and Society

Volume 2018, Article ID 4674920, 17 pages

https://doi.org/10.1155/2018/4674920

## A Hybrid Discrete Grey Wolf Optimizer to Solve Weapon Target Assignment Problems

^{1}College of Systems Engineering, National University of Defense Technology, Changsha 410073, China^{2}Allsim Technology Inc., Changsha 410073, China

Correspondence should be addressed to Jun Wang; nc.ude.tdun@gnaw_rehsif

Received 17 June 2018; Revised 3 October 2018; Accepted 29 October 2018; Published 7 November 2018

Academic Editor: Pasquale Candito

Copyright © 2018 Jun Wang 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

We propose a hybrid discrete grey wolf optimizer (HDGWO) in this paper to solve the weapon target assignment (WTA) problem, a kind of nonlinear integer programming problems. To make the original grey wolf optimizer (GWO), which was only developed for problems with a continuous solution space, available in the context, we first modify it by adopting a decimal integer encoding method to represent solutions (wolves) and presenting a modular position update method to update solutions in the discrete solution space. By this means, we acquire a discrete grey wolf optimizer (DGWO) and then through combining it with a local search algorithm (LSA), we obtain the HDGWO. Moreover, we also introduce specific domain knowledge into both the encoding method and the local search algorithm to compress the feasible solution space. Finally, we examine the feasibility of the HDGWO and the scalability of the HDGWO, respectively, by adopting it to solve a benchmark case and ten large-scale WTA problems. All of the running results are compared with those of a discrete particle swarm optimization (DPSO), a genetic algorithm with greedy eugenics (GAWGE), and an adaptive immune genetic algorithm (AIGA). The detailed analysis proves the feasibility of the HDGWO in solving the benchmark case and demonstrates its scalability in solving large-scale WTA problems.

#### 1. Introduction

As a classic military operational problem, the purpose of weapon target assignment (WTA) is to find an optimal or a satisfactory assignment solution, which determines the target attacked by each weapon, so as to maximize the total damage expectancy of hostile targets or minimize the loss expectancy of own-force assets. Since first proposed in the 1950s to support operation planning, command officers training, and weapon selection and acquisition [1], the WTA problem has attracted the attention of relevant researchers for decades. From the mathematical point of view, it is essentially an NP-complete problem [2] with nonlinear objective functions, discrete decision variables and multiple constraints. These intricate features indicate that seeking out the optimal solution for small-scale WTA problems is relatively realistic, but becomes impracticable for large-scale ones. This means in such circumstances searching for a satisfactory or suboptimal solution may be more efficient.

In the WTA study, the process of solving WTA problems can generally be divided into two phases, establishing a WTA model, and finding an optimal or a suboptimal solution for the model through an appropriate algorithm. In modeling, a variety of factors need to be considered, such as the type and quantity of equipment involved, the combat capability of each equipment, the characteristics of the battlefield, and the focus of the commander. In addition, for the convenience of modeling, certain assumptions may be made. Conventional WTA models include up to three types of entities, platforms, weapons, and targets. However, considering that weapons are increasingly dependent on the information provided by sensors in modern warfare, Bogdanowicz et al. [3, 4] believed that sensors should be considered in WTA models and established the sensor-WTA (S-WTA) model for the first time. They simplified the S-WTA problem through the sensor/target/weapon decomposition to the following scenario, each sensor-weapon pair can be assigned to at most one target, and each target can be engaged by at most one sensor-weapon pair. Furthermore, through the sensor/weapon/target augmentation, they translated the derived problem into a symmetric optimization problem with an input consisting of the same numbers of sensors, weapons, and targets, along with the benefit matrix, and presented the Swt-opt algorithm derived from the auction algorithm to optimally assign sensors and weapons to targets. Since then, the S-WTA problem has been successively studied by some scholars. Li et al. [5, 6] proposed an improved Swt-opt algorithm to solve the same problem by combining the consensus algorithm, so that the shortcoming of the Swt-opt algorithm, i.e., highly depending on perfect network topologies, can be overcome. Later, they put forward a decentralized cooperative auction algorithm for a similar S-WTA problem where the number of targets that can be struck varied in different operational phases [7]. Chen et al. [8] proposed a particle swarm optimization based on genetic operators to solve the S-WTA problem where each sensor can guide only one weapon once and each target can be engaged by multiple weapons once. Mu et al. [9] proposed a multiscale quantum harmonic oscillator algorithm to solve the S-WTA problem in intelligent minefields considering the probability of detection and killing. Xin et al. [10] constructed an efficient marginal-return-based heuristic to solve the S-WTA problem considering the situation where multiple sensors/weapons can be assigned to one target, but each sensor can detect only one target at the same time and each weapon can shoot only one target simultaneously. The proposed heuristic exploited the marginal return of each sensor-weapon-target triplet and dynamically updated the threat value of all targets. It relied only on simple lookup operations to choose each assignment triplet, thus resulting in very low computational complexity.

Both the conventional WTA model and the S-WTA model can be classified as follows. For ease of presentation, the two models will not be distinguished, both called the WTA model. A WTA model is called to be dynamic [10–12], if the operational process is time related. This indicates there are always several operational stages or new situations usually arise during operation. Otherwise, the model is static [3–6, 8, 9, 13–16]. If two or more objectives, such as maximizing the damage expectancy of hostile targets and minimizing the consumption of own ammunition and mission completion time, are considered, the WTA model is multiobjective [17, 18]. If there is only one objective, it is single-objective [3–11, 19]. Depending on the type of combat scenario, the WTA model may be asset-based [11] or target-based [17]. In defensive missions, the asset-based model is often established to minimize the loss expectancy of own-force assets, while in offensive missions, the target-based model is always adopted to maximize the total damage expectancy of hostile targets. In addition to modeling, another research aspect for WTA is to develop algorithms to solve the problem optimally in the least possible time. These algorithms can be generally divided into two categories: exact algorithms and heuristic algorithms. Exact algorithms are developed according to the specific mathematical properties of WTA problems and can gradually eliminate the nonlinearity of the problem through transformation, decomposition, and other processing means [16]. By this way, the model is translated to a linear one, and then classical operations research methods can apply, such as the dynamic programming method [12], the branch and bound method [20, 21], the branch and price method [22], the mixed integer linear program (MILP) algorithm [17], and the Lagrange relaxation method [14]. These methods have demonstrated the feasibility and effectiveness on solving static and dynamic WTA problems but may become difficult to apply when a large number of weapons and targets are involved [23].

With the rapid development of heuristic algorithms, more complex WTA problems are able to be well solved. Most research to date on solving WTA problems by heuristic algorithms either constructs some specific search rules based on the properties of the problem to achieve solutions rapidly or introduces some local search mechanisms into the original algorithms to improve the solution quality. These algorithms, including auction algorithms [3–6, 15], improved genetic algorithms [18, 24–26], clonal selection algorithms [27, 28], particle swarm algorithms [8, 13, 29], tabu search algorithms [30], rule-based constructive heuristic algorithms [10, 31], and other intelligent optimization algorithms, have shown evident advantages over traditional methods in terms of computation time and solution accuracy and however still suffer from some drawbacks, such as easily falling into premature convergence and local optimum [32]. Furthermore, considering the variability of the battlefield environment, decisions always need to be made immediately; that is, WTA problems need to be resolved in a very short time. Therefore, we propose in this paper a hybrid discrete grey wolf optimizer (HDGWO), which is an integration of a discrete grey wolf optimizer (DGWO) with a local search algorithm (LSA). Moreover, in order to enhance the efficiency of the algorithm, we introduce the specific domain knowledge into the encode methods and the LSA.

The rest of this paper is organized as follows. The mathematical model of a typical WTA problem for an offensive mission is formulated in Section 2. Section 3 is a brief introduction to the original grey wolf optimizer (GWO) presented in [33], and Section 4 is a detailed introduction to the proposed HDGWO. In Section 5, we first analyze the feasibility of HDGWO in solving small-scale WTA problems by comparing it with the discrete particle swarm optimization (DPSO) [13], the genetic algorithm with greedy eugenics (GAWGE) [24], and the adaptive immune genetic algorithm (AIGA) [32] and then investigate the scalability of HDGWO by solving ten different scale WTA problems. Finally, we make a conclusion and look ahead to the future research in Section 6.

#### 2. Problem Formulation

In general, a typical WTA problem for an offensive mission can be formulated as the following nonlinear integer programming model [13, 32]: subject toThe symbols in the model are explained in Table 1. For clarity, we also list the symbols employed in the following in this table.