A modified crow search algorithm for the weapon-target assignment problem

The Weapon-Target Assignment (WTA) problem is one of the most important optimization problems in military operation research. In the WTA problem, assets of defense aim the best assignment of each weapon to target for decreasing expected damage directed by the offense. In this paper, Modified Crow Search Algorithm (MCSA) is proposed to solve the WTA problem. In MCSA, a trial mechanism is used to improve the quality of solutions using parameter LIMIT. If the solution is not improved after a predetermined number of iterations, then MCSA starts with a new position in the search space. Experimental results on the different sizes of the WTA problem instances show that MCSA outperforms CSA in all problem instances. Also, MCSA achieved better results for 11 out of 12 problem instances compared with four state-of-the-art algorithms. The source codes of MCSA for the WTA are publicly available at http://www.3mrullah.com/MCSA.html


Introduction
Weapon-Target Assignment (WTA) problem is one of the most important optimization problems in military operation research. The WTA problem has two versions as the static weapon-target assignment problem (SWTA) and the dynamic weapon-target assignment problem (DWTA). The main difference between the SWTA and the DWTA is the timing of launching weapons to targets. In the DWTA, the launching of weapons is performed asynchronously, however in the SWTA, all weapons are launching at the same time and only once [1]. In the WTA problem, the aim is to minimize the damage caused by attacks of the targets. Hence, assets of the defense aim the best assignments for minimal damage after the engagement. Several exact and approximation algorithms [2][3][4] have recently involved in solving the WTA problem. Since the WTA is an NP-complete problem [5], exact algorithms can not solve large-scale WTA problems in polynomial time. To overcome this problem, metaheuristic algorithms are presented to solve the WTA problem. Metaheuristic algorithms provide a valid solution in a reasonable time [6]. In recent years, metaheuristic algorithms for solving optimization and engineering problems have attracted much attention in the literature. The development of nature-inspired metaheuristic algorithms has increased rapidly in the last decades [7]. These algorithms have good ability to solve global optimization problems even it is complex or high dimensional. The strategy of metaheuristic algorithms is to obtain a solution in a reasonable time for optimization problems which are naturally intricate and very hard to solve. This strategy is built on two main features: exploration and exploitation. In the exploration stage, the algorithm attempts to find a new solution in the search space. In the exploitation stage, the algorithm searches for the neighborhood of the highest quality solution so far to get better solutions. The balance of these two stages is highly important for the algorithm to be successful. The Crow Search Algorithm (CSA) [8] is a populationbased metaheuristic algorithm inspired by the behavior of crows, has a good exploration and exploitation for optimization problems. Many metaheuristic algorithms have been proposed for the WTA problem. Şahin and Leblebicioğlu [9] presented a Hierarchical Fuzzy Decision Maker method to achieve the best assignment for improving performance on the battlefield. The proposed method increased the approximation performance in comparison to exact and optimal methods. Wang et al. [10] developed a Grey Wolf Optimizer which is the popular population-based algorithm in recent years, to solve the WTA problem. The problem was addressed as a binary problem and the algorithm was modified to a discrete method. According to results, Grey Wolf Optimizer resulted in good quality solutions for smallscale problems and proved that it is competitive for large-scale problems. Li et al. [11] have presented an Ant Colony Optimization for bi-objective the WTA problem. In their study, an optimization model for the WTA is designed which maximizes the expected damage of the enemy (first objective) and minimizes the cost of missiles (second objective). Due to the biobjective model of the WTA, Ant Colony Optimization is modified to get a set of Pareto solutions. According to simulation results, the modified algorithm improved the performance of the pure one and produced better solutions. Sonuc et al. [12] have worked on a Simulated Annealing algorithm to solve the SWTA problem on GPU. The aim of the study was to obtain better solutions with less computational time compared to the solution of the serial algorithm. Computational results on problem instances have shown that the parallel algorithm was 250 times faster than a single-core CPU and improved the quality of solutions. Zhang et al. [13] have developed a hybrid method using Ant Colony Optimization and Genetic Algorithm to obtain fast convergence speed for the WTA problems. Implementation of Artificial Bee Colony algorithm which is inspired by intelligent behavior of honey bees, was proposed for solving the SWTA problem by Durgut et al. [14]. In the study, three local search operators were discussed and according to the results, the swap operator emerged as more effective than insertion and inversion operators. Kutucu et al. [15] presented a hybrid method with Artificial Bee Colony and Simulated Annealing for the SWTA. According to results on benchmark problems, the proposed algorithm was competitive and satisfactory compared to other metaheuristic algorithms for the WTA. To improve the ability of Ant Colony Optimization, an immune system based algorithm was developed to solve the WTA by Lee et al. [16]. According to the comparison results, the proposed algorithm has improved searching performance. Hu et al. [17] improved Ant Colony Optimization in the viewpoints of selection, updating and concentration interval and applied it to the WTA problem. The advantages of the proposed algorithm were faster convergence and better avoidance from local optima. Tokgöz et al. [18] presented combinatorial optimization techniques for WTA problems. Several heuristic algorithms were selected and applied to the WTA and the results proved that Variable Neighborhood Search and Simulated Annealing obtained better solutions than other algorithms. Li et al. [19] developed a decompositionbased evolutionary algorithm for multiobjective SWTA. According to experiments, the proposed method was effective and promising on generated scenarios. Also, real-time heuristics using Construction Heuristic, Quiz Problem Search Heuristic and Greedy Branch and Bound Heuristic, was presented by Kline et al. [20]. All three heuristics were used for comparison with existing heuristics in literature and the results outlined that the computational costs of the proposed methods are less expensive than the existing ones. Hocaoglu [21] aims to generate a model for air defense. The model answers to the question that is how many missiles are necessary to eliminate attacking from the offense. The model gives a better and faster than the Simulated Annealing algorithm. This paper aims to improve the quality of solutions for the SWTA problem using a modified crow search algorithm (MCSA). MCSA is a population-based algorithm and obtained better solutions in less time compared to Simulated Annealing [1] which is an iterative heuristic algorithm. Besides, one agent searches a new solution in the search space for each iteration hence Simulated Annealing has a poor exploration compared to population-based metaheuristics. Also, MCSA was compared with the state-of-the-art algorithms and the experimental results were revealed that MCSA was improved quality of results in 6 of 12 problems. The rest of this paper is organized as follows. In Section 2, the model of the SWTA problem is illustrated and the formulation of the problem is presented. In Section 3, nature-inspired CSA is introduced. In Section 4, MCSA based on a trial mechanism is proposed. Experimental results on the WTA problems are presented to demonstrate the performance of improved CSA in Section 5. Finally, conclusion and future works are described in Section 6.

Problem formulation
According to the WTA model, which is a minimization optimization problem, assets of defense aim the best assignment of each weapon to target for decreasing expected damage directed by the offense. Each weapon has a destroying probability for each target and the expected damage for assets of defense is evaluated after engagement in the battlefield. An illustration of the WTA problem is presented in Figure 1.  Table 1 shows the explanation of each symbol for the WTA model. In general, a WTA problem for a defensive mission can be formulated as follows:

The crow search algorithm (CSA)
Crows live in flocks and can follow the other birds and steal the food they have stored in their nests. As a result of this follow-up, they can remember the location of other birds' hiding-place and find it whenever they want. The pseudocode of the CSA, which is inspired by the behavior of crows, is shown in Figure 2. CSA has an easy to implement structure and only needs two parameters. Implementation of CSA for optimization problems is an easy process since it has only two parameters: Awareness Probability (AP) and Flight Length (FL).
According to the strategy of CSA, the crow updates its position in two states. In the first state, each crow (crow i) selects a random crow (crow j) to steal food from its hiding place without being noticed. The decision to follow the selected crow is determined by the parameter AP. If the follow-up is carried out, the new position of the crow is determined according to Eq. (3) using the memory of crow j (mj).
The second state is that crow j recognizes that is being followed by crow i. In this state, the crow moves to a new position in the search space. For the second state, the new position of the crow is defined as follows: .

The WTA problem using MCSA
The WTA problem is a combinatorial optimization problem and each weapon must be assigned to a target. This assignment is represented as a permutation in the problem. Also, this permutation represents a position in the search space for a crow. The aim is finding the best position (permutation) in the search space to minimize the objective function (Eq. (1)). CSA is modified to improve the quality of solutions using a new parameter called LIMIT. If a solution that represents a position in the search space, is not improved by a predetermined number of trials, then a new position is generated. This method is proposed by Karaboga et al. [22,23] for Artificial Bee Colony Algorithm to solve optimization problems. The implementation of MCSA for the SWTA problem is carried out through the following steps: Step 1. Initialization of MCSA parameters. Initialize the parameters: N, itermax, FL, AP and number of non-improved trials LIMIT.
Step 2. Initialize permutation and memory of crows. Randomly generate a permutation for each crow and memorize the initial permutations.
Step 3. Evaluate the objective function. Compute objective function using its permutation for each crow.
Step 4. Generate a new permutation. Generate a new permutation for crow i as follows: Randomly select one other crow (crow j) to use its permutation. Generate a new position using the swap operator (see Figure 3.) for permutation of crow j. Thus, a new permutation of crow i is determined if , j iter j r AP  . This procedure is repeated for all crows. Otherwise, it keeps its current permutation. This procedure is defined as follows:  Step 5. Evaluate the objective function of new permutations.
Compute the objective function of the new permutation for each crow.
If the new objective function value of each crow is less than the memorized one, then update the memory of each crow using: For each crow, the objective function value of the new permutation is computed.
Step 8. Evaluate the objective function and update memory.
Computation of objective function for each crow using its permutation. After computation, update the memory of crows.
Repeat Steps 4-8 until itermax is reached. The flowchart of MCSA is presented in Figure 4.  Figure 4. Flowchart of the modified CSA for solving the WTA problem.

Experimental results
MCSA is tested on 12 problem instances (available at https://doi.org/10.17632/jt2ppwr62p.1) presented in [12]. Dimensions of problem instances are in the range 5 -200 and listed in Table 2. The numerical experiments were performed on a PC with Intel(R) Core(TM) i7-5600U CPU @ 2.60 GHz, with 8.00 GB of RAM, running Windows 8 64-bit operating system. The codes of MCSA and CSA have been written in C under CodeBlocks IDE v17.12.

Comparison MCSA and CSA
Firstly, robustness of MCSA is tested in comparison with the pure CSA by using parameters which are AP = 0.2, FL = 2, N= 20, ITERATION = 1000 and LIMIT = 10 x size of problem (for MCSA only). Figure 5 shows the box plot of 10 independent runs for the problem instances from WTA1 to WTA12 with the aim of comparison between MCSA and CSA. The results show that MCSA outperforms CSA in all problem instances. Also, the box plots show that MCSA converges quickly to the optimal solutions as it has better values and fewer heights compared to CSA.

Comparison of MCSA with the state-of-the-art algorithms
MCSA was compared with four other metaheuristic algorithms for solving the WTA, which are ABC [14], ABC-SA [15], SA [12] and pure CSA. All parameters for the algorithms are given in Table 3. LIMIT parameter for MCSA is selected depending on problem size (see in Table 3) as suggested in [24]. With this tuning, LIMIT increases when the size of the WTA problem is increased.
The results of all metaheuristic algorithms are compared in terms of the best, mean, worst, median, standard deviation (SD) and time (seconds) in Table 4. However, median and SD values are not available for ABC and ABC-SA. The best results for each problem are shown in bold. Overall, MCSA obtained better results compared to other methods for 11 out of 12 problem instances. All algorithms can achieve the same best results for WTA1 and WTA2. The best result is the same on WTA3 and WTA4 for all algorithms except for CSA. Comparing the results obtained by all metaheuristic algorithms it can be inferred that all algorithms except CSA are successful in reaching the optimum of small size problems.   A comparison between MCSA and ABC-SA based on time is presented in Figure 6. Although it is not fair to compare MCSA and ABC-SA as we don't know some parameters and number of function evaluations, the capabilities of the used devices for running these two algorithms are approximately similar. It can be shown that the average run time for MCSA is better than ABC-SA.

Conclusion and future works
This paper proposed a Modified Crow Search Algorithm (MCSA) for solving the static WTA problem. In MCSA, a trial mechanism that starts with a new position in the search space after a predetermined number of trials, has been adapted to the exploration phase. The number of trials defines as a parameter called LIMIT, is adjusted to the size of the problem. With this update, the exploitation stage of CSA is strengthened for combinatorial problems like the WTA. Experimental results of MCSA have been compared with four state-of-the-art algorithms on the WTA problem instances with different dimensions. In each problem, the numbers of the weapons and targets are equal and limited and this limitation occurs the size of the problem. According to the experimental results, MCSA achieved the best results on all problem instances except for only one and outperformed the state-of-the-art algorithms. In future works, MCSA can be combined with single solution based algorithms (hill-climbing, tabu search, simulated annealing, etc.), especially for the second state of CSA. Also, MCSA can be applied to solve dynamic WTA problem or other discrete optimization problems.