# Genetic Algorithm-Based Variable Value Control Method for Solving the Ground Target Attacking Weapon-Target Allocation Problem

Authors: Chao Wang; Guangyuan Fu; Daqiao Zhang; Hongqiao Wang; Jiufen Zhao
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Key ground targets and ground target attacking weapon types are complex and diverse; thus, the weapon-target allocation (WTA) problem has long been a great challenge but has not yet been adequately addressed. A timely and reasonable WTA scheme not only helps to seize a fleeting combat opportunity but also optimizes the use of weaponry resources to achieve maximum battlefield benefits at the lowest cost. In this study, we constructed a ground target attacking WTA (GTA-WTA) model and designed a genetic algorithm-based variable value control method to address the issue that some intelligent algorithms are too slow in resolving the problem of GTA-WTA due to the large scale of the problem or are unable to obtain a feasible solution. The proposed method narrows the search space and improves the search efficiency by constraining and controlling the variable value range of the individuals in the initial population and ensures the quality of the solution by improving the mutation strategy to expand the range of variables. The simulation results show that the improved genetic algorithm (GA) can effectively solve the large-scale GTA-WTA problem with good performance.


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## Body

## 1. Introduction

WTA [1–5] is a critical link in the operational command that directly affects the operational process and outcome and is thus an important military issue that has attracted the attention of military powers worldwide. Many achievements have been made in the study of WTA, mostly regarding the study of air defense WTA (AD-WTA), which addresses air defense interception [3–15], whereas the GTA-WTA problem, which addresses ground targets, has rarely been studied [16–18]. Compared with air defense interception, ground targets are diverse and suitable for many weaponry types, so the scale and complexity of the WTA problem are greater, requiring the use of more efficient algorithms. The use of superior firepower against key enemy ground targets is an important means to quickly win a war. In recent local wars, the US military greatly weakened the enemy combat forces and quickly won the war by attacking and destroying numerous key enemy ground targets. The combating opportunity is fleeting, and weaponry resources are limited, so the failure to quickly and appropriately allocate weaponry resources may cause not only a waste of weapons but also missed combat opportunity. Therefore, it is necessary and imperative to study the GTA-WTA problem and design a highly efficient optimization algorithm.Regarding the study of the algorithm, based on the genetic algorithm, Yang Shanliang [9] solved the AD-WTA problem very well by optimizing the initial population coding mechanism and adopting elite selection and the improved algorithm of dynamic genetic operators. Zhang Jiao [12] proposed a particle coding scheme that specifically addresses the multiconstraint AD-WTA problem and the fitness function of the model, solved the problem of initializing the integer domain of the particles, and improved the iteration efficiency and the optimization capability of the particle swarm optimization (PSO) algorithm. Liu Shuangshuang [17] proposed an improved bat algorithm that integrates the thinking of niche elimination for the GTA-WTA problem, which improved the quality of the solution but failed to effectively improve the computational efficiency. Wang Shunhong [18] improved the particle initialization and weight coefficient selection methods of the PSO algorithm, which improved the computational efficiency and the solution quality, but used a fixed number of iterations as a stage when decreasing the weighting coefficient while not considering the scale of the problem and the quality of the solution at each stage, making the method suitable only for the GTA-WTA problem on a specific scale.The GTA-WTA problem is vastly different from the AD-WTA problem. In the latter case, the types and quantities of the weapons used to intercept air targets are limited, as are those of the target, and the number of weapons used to intercept a single airborne target is usually one or two. Thus, the destruction of the target by the weapon belongs to the 0-1 destruction; it is unnecessary to consider the cumulative damage effect on the target by each weapon, and the optimization principle is that the lower the loss of weaponry resources or the lower the number of remaining enemy targets is, the better [3]. However, in the GTA-WTA problem, the types and number of ground targets are plentiful, and the types and quantities of the attacking weapons are also plentiful; because of the complexity of ground targets, a large weaponry quantity is often used to strike a single ground target, and as the weapons are hitting the target, the cumulative damage effect needs to be considered, in which the optimization principle is to meet the expected target destruction requirement with the minimum weapon consumption value. Thus, the computational scale and the solution space of the GTA-WTA problem are much greater than those of the AD-WTA problem, making it difficult for the existing methods for solving the AD-WTA problem to meet the needs of solving the GTA-WTA problem. The existing solutions to the GTA-WTA problem have an array of issues, such as a too long algorithm search time or a solution meeting the constraints being unobtainable because the number of dimensions is too large, which are not allowed in the actual combat plan and decision. Therefore, it is necessary to design a new method to solve the GTA-WTA problem.In this study, to obtain the solution to the GTA-WTA problem, we constructed an GTA-WTA model and proposed a genetic algorithm-based variable value control method in which the concept of damage contribution is first defined; then, the damage contribution is used to restrain the amount of each type of weapon when striking each target, and the total weapon amount when striking a single target is minimized to decrease the variable initialization space and rationalize the variable value to expedite the search for feasible solutions. Then, the variation range of the variables is expanded, and the mutation strategy is changed to eliminate the problem of missing solution space caused by the narrowing of the variable value range and expedite the convergence of the algorithm. Finally, a simulation and a comparison of examples were performed to verify the correctness and superiority of the proposed method.The remainder of this paper is organized as follows. In Section2, the process of the GTA-WTA is analyzed and the GTA-WTA mode is constructed based on some assumptions. In Section 3, a variable control method for GA is proposed and its feasibility is analyzed. The results of employing different algorithms to solve the GTA-WTA problems are presented and analyzed in Section 4. Section 5 concludes this paper.


## 2. GTA-WTA Problem

### 2.1. Process of GTA-WTA

The GTA-WTA mainly includes two stages. The first stage is the weapon and target matching, that is, analyzing whether the damage mechanism and guidance of weapons are applicable, and the targets can be covered under the attack distance of weapons, and the environment around the target meets the attack conditions of weapons, then selecting weapons suitable for combating targets. The second stage is allocation optimization, that is, comprehensively considering the total amount of weapons and then assigning appropriate weapons to the targets in an optimal cost-effective ratio to achieve the desired damage for all targets.


### 2.2. GTA-WTA Model

There are many factors involved in the weapon and target matching stage, and the analysis is very complicated, but the results are only “suitable” or “not suitable” which is no need to optimize. Therefore, the GTA-WTA model is mainly designed for the allocation optimization stage. To correctly construct the GTA-WTA model, the following assumptions are made:(1)
The damage probability of a weapon for the target is a comprehensive damage probability, considering the weapon’s penetration probability, target hit probability, target damage probability, etc.(2)
The comprehensive damage probabilities of weapons of the same type are identical.(3)
The sequence of a target strike and the maximum projection ability of a wave of strikes are not considered.Based on the above assumptions, the GTA-WTA problem can be described as follows:N types of weapons are used to attack M types of ground targets, and the damage coefficients of the M types of ground targets are C1,C2,…,CM. The quantities of the N types of weapons are N1,N2,…,NN, with values of V1,V2,…,VN, respectively; the quantities of the M types of ground targets are M1,M2,…,MM, respectively, and the comprehensive damage probability of the ith type of weapon against the jth type of target is pij,i=1,2,…,N,j=1,2,…,M. The optimal WTA scheme is to use a minimal weapon consumption value to meet the target damage requirement. Assuming that the number of the ith type of weapon against the kth target of the jth type of target is mijk,k=1,2,…,Mj, and the weapon consumption value is V, the GTA-WTA model is as follows:(1)minV=∑i=1i=N∑j=1j=M∑k=1k=Mjmijk∗Vi(2)s.t.∑k=1k=Mj1-∏i=1i=N1-pijmijk≥Cj∑j=1j=M∑k=1k=Mjmijk≤Nimijk≥0


## 2.1. Process of GTA-WTA

The GTA-WTA mainly includes two stages. The first stage is the weapon and target matching, that is, analyzing whether the damage mechanism and guidance of weapons are applicable, and the targets can be covered under the attack distance of weapons, and the environment around the target meets the attack conditions of weapons, then selecting weapons suitable for combating targets. The second stage is allocation optimization, that is, comprehensively considering the total amount of weapons and then assigning appropriate weapons to the targets in an optimal cost-effective ratio to achieve the desired damage for all targets.


## 2.2. GTA-WTA Model

There are many factors involved in the weapon and target matching stage, and the analysis is very complicated, but the results are only “suitable” or “not suitable” which is no need to optimize. Therefore, the GTA-WTA model is mainly designed for the allocation optimization stage. To correctly construct the GTA-WTA model, the following assumptions are made:(1)
The damage probability of a weapon for the target is a comprehensive damage probability, considering the weapon’s penetration probability, target hit probability, target damage probability, etc.(2)
The comprehensive damage probabilities of weapons of the same type are identical.(3)
The sequence of a target strike and the maximum projection ability of a wave of strikes are not considered.Based on the above assumptions, the GTA-WTA problem can be described as follows:N types of weapons are used to attack M types of ground targets, and the damage coefficients of the M types of ground targets are C1,C2,…,CM. The quantities of the N types of weapons are N1,N2,…,NN, with values of V1,V2,…,VN, respectively; the quantities of the M types of ground targets are M1,M2,…,MM, respectively, and the comprehensive damage probability of the ith type of weapon against the jth type of target is pij,i=1,2,…,N,j=1,2,…,M. The optimal WTA scheme is to use a minimal weapon consumption value to meet the target damage requirement. Assuming that the number of the ith type of weapon against the kth target of the jth type of target is mijk,k=1,2,…,Mj, and the weapon consumption value is V, the GTA-WTA model is as follows:(1)minV=∑i=1i=N∑j=1j=M∑k=1k=Mjmijk∗Vi(2)s.t.∑k=1k=Mj1-∏i=1i=N1-pijmijk≥Cj∑j=1j=M∑k=1k=Mjmijk≤Nimijk≥0


## 3. Variable Value Control Method of GA

The basic idea of GA is derived from genetic evolution of biology, simulating the evolution process of the survival of the fittest in natural biological group which can effectively deal with various optimization problems [19], and its improved algorithm has been widely used in WTA problem. The genetic evolution of biological population begins with the initial population whose pros and cons directly affect the speed and direction of individual evolution, and the initial population of GA will soon evolve into the optimal individual if it is relatively close to the optimal individual. Therefore, we can design a variable value control method for GA which effectively improves the quality and computational efficiency of the solution by controlling the value range of variables to let initial population individuals be generated in the feasible solution space.


### 3.1. Significance of the Variable Value Control

The GTA-WTA problem is an NP-complete problem [20] in which the algorithm search space increases exponentially with the number of variables or the expansion of the range of variable values. In the GTA-WTA problem, the number of variables is fixed and unchangeable, with a value of the product of the number of weapon types and the number of targets; however, the value range of the variables is uncertain because the GTA-WTA generally does not limit the amount of each type of weapon used on each target but only limits the available amount of each type of weapon. Thus, in solving the problem, the weapon usage range, which is used as the value range of the variables, is uncertain and changeable. If the value range of the variables is not controlled, i.e., the quantity of each weapon type is taken as the upper bound of the value range of the variables, then the number of WTA schemes generated within the search space will be extremely large.We assume that there areN types of weapons, with quantities of N1,N2,…,NN, and M types of targets, with quantities of M1,M2,…,MM. If the quantity of each weapon type is taken as the upper bound of the variables, then the weapon amount (mij) of the ith type of weapon used to strike any target of the jth type of target is(3)mij∈0,Ni,mijisintegerAs shown in formula (3), when striking the jth type of target with the ith type of weapon, the variables mij can have Ni+1 possible values.For thejth type of target, when the N types of weapons strike the kth target, there are N variables, which can be viewed as a vector, and, then, in the search space formed by the value range of the N variables, the number of vectors (njk) that can be generated is(4)njk=∏i=1i=NNi+1When the N types of weapons strike all the targets of the jth type of target, there areN∗Mj variables, and, then, in the search space formed by the value range of the variables, the number of vectors (nj) that can be generated is(5)nj=∏k=1k=Mjnjk=∏i=1i=NNi+1MjWhen the N types of weapons strike all the targets of M types of targets, there are ∏j=1j=MN∗Mj variables, and, then, in the search space formed by the value range of the variables, the number of vectors (n) that can be generated is(6)n=∏j=1j=Mnj=∏j=1j=M∏k=1k=Mjnjk=∏i=1i=NNi+1∑j=1j=MMjIn formula (6), the value of n (the number of vectors) is the number of all possible WTA schemes when N types of weapons strike all the targets of M types of targets.As shown in formula (6), as the value range of the variables expands, the number of GTA-WTA schemes increases drastically. However, the number of feasible solutions for the GTA-WTA problem is fixed and only accounts for a small portion of the total WTA schemes. In addition, most of the feasible solutions are distributed in a certain region of the search space, and the distribution region and the number of feasible solutions are not affected by the value range of the variables. Therefore, if the search space can be sharply reduced while still containing the space in which the feasible solutions are located, then a fast search of feasible solutions is then ensured.For example, if 49 weapons of the same type strike 10 targets of the same type and if the total amount of weapons is taken as the upper bound of the value range of the variables, thenN1=49 and M1=10, and, based on formula (6), the number of WTA schemes (n(1)) is(7)n1=∏i=1i=1Ni+1∑j=1j=1Mj=N1+1M1=5010If the value range of the variables is controlled and the upper value limit of the variables is 9, then N1=9 and M1=10, and, based on formula (6), the number of WTA schemes (n(2)) is(8)n2=N1+1M1=1010From the number of WTA schemes (n(1) and n(2)) solved using formulas (7) and (8), the ratio (α) of WTA schemes within the search space without the control of the value range of the variables to that within the search space with the control of the value range of the variables is(9)α=n1n2=50101010=510:1Obviously, after reducing the upper bound of the value range of the variables, the search space is sharply reduced, and the number of WTA schemes within the search space is also drastically reduced. In addition, the value range of the variables after the reduction is generally reasonable, and the search space formed by the value range of the variables can contain the space of the vast majority of feasible solutions, indicating that the reasonable control of the value range of the variables can ensure a fast search for feasible solutions.


### 3.2. Determining the Value Range Based on the Damage Contribution

Variable value control method improves the search efficiency by performing a series of operations on the variable value range, that is, narrowing the algorithm search space in the initialization process of GA to shorten the calculating time and amplifying the variable value range in the mutation process to ensure have a reasonable result. For GA to solve the GTA-WTA problem, the variable is the number of each type of weapon acting on each target, and the variable value range is the number of intervals that each type of weapon may act on each target. Variable value control method consists of three parts: one is to constrain the maximum number of weapons acting on each target; the other is to limit the total amount of weapons acting on each target; the third is to expand the variable variation interval and change the mutation strategy.For the same target,n ammunitions are usually used for striking to achieve the expected damage effectiveness. For the same type of ammunition, as the target is being damaged, each ammunition contributes different damage values, which can be called the damage contribution, denoted u. If the damage probability of the ith type of ammunition against the jth type of target is pij, then the damage contribution of the nth ammunition (uijn) is(10)uijn=1-pijn-1pijThe ratio of the damage contribution of the nth ammunition to the value of the ammunition is defined as the damage contribution per unit value, denoted as μijn. If the value of the ith type of ammunition is vi, then(11)μijn=uijnvi=1-pijn-1pijviObviously, for the same target, when choosing ammunition, the higher the damage contribution per unit value is, the more appropriate the ammunition type is.


#### 3.2.1. Constraining the Quantity of Each Type of Weapon against Each Target

When striking the same target, the quantity of weapons is limited, and the use of a large quantity of weapons to strike the same target is generally avoided. According to formula (10), the greater the quantity (n) of weapons used in the strike is, the lower its damage contribution (uijn) will be, which gradually approaches 0. When uijn approaches 0, it indicates that the strike with the nth weapon on the target essentially causes no damage, and the strike with the ith type of weapon against the jth type of target should be stopped; at this time, the quantity of weapons consumed can be viewed as the upper bound of the weapon quantity range of the ith type of weapon against each of the jth type of target. Assuming that when the value of uijn is less than the preset minimal damage contribution (αj) of a given weapon for the jth type of target, the strike on the target is stopped and then n must satisfy(12)n=1,pij=0uijn<αj≤uijn-1,0<pij<1Thus, under the constraint of the damage contribution, the upper bound (tij) of the weapon quantity range of the ith type of weapon against the jth type of target is(13)tij=n-1The value of tij depends on the damage contribution value αj; the lower the value of αj is, the higher the value of tij will be, which can even be higher than the weapon quantity (m) required for the ith type of weapon to destroy the jth type single target. Moreover, if the weapon quantity exceeds m, it is deemed excessive use, and the tij value needs to be decreased. If the damage requirement for the jth type of target is Cj, then m satisfies(14)m=0,pij=01-1-pijm-1<Cj≤1-1-pijm,0<pij<1To prevent the value of tij from exceeding m due to an undersized αj, under the constraint of m, the value of tij is(15)tij=min⁡tij,m


#### 3.2.2. Constraining the Total Number of Weapons against Each Target

Different types of weapons lead to varying damage probabilities, but there is the situation in which multiple types of weapons are suitable for attacking the same target. In this case, in the population initialization of GA, each type of applicable weapon is randomly given a quantity to be used to strike against the target, which is very unlikely to be 0, so the number of weapons used to strike the target may be too large, resulting in the expenditure of a large amount of time in the subsequent optimization process to lower the number into a reasonable range. Obviously, the total number of weapons for a certain target of thejth type of target should not exceed the upper bound of the weapon usage range that causes the minimal damage probability on the jth type of target. If there are N types of ammunition suitable for striking the jth type of target, then the maximum value (Tj) of the total number of weapons used to strike the jth type of target is(16)Tj=max⁡tijAfter the maximum value (Tj) of the total number of the weapons for striking the target is determined, it is necessary to process the quantities of N types of weapon used to strike the target (m1j,m2j,…,mNj) in the algorithm to make it satisfy(17)∑i=1i=Nmij≤Tj


### 3.3. Amplifying the Value Range of Variables through Directional Variation

The value range of individual variable in the population becomes smaller by constraining the number of weapons acting on each target and the total number of weapons acting on each target, and this variable value range constraint is generally reasonable. But when the damage probability of a certain type of weapon for a certain type of target is low, the number of weapons used to strike a single target of that target type under the constraint may not be sufficient to achieve the required level of damage, resulting in no feasible solutions within the constrained value range. Assuming that theith type of weapon is used to strike against the jth type of target, the maximum number (tij) of the ith type of weapon for each jth type of target can be calculated using formula (15), and, then, the maximum degree of damage (βij) on the jth type of target by the weapon is(18)βij=1-1-pijtijWhen 0 < pij < 1, from formulas (10) and (12), we have (19)1-pijtijpij<αj≤1-pijtij-1pijThat is,(20)1-pijtij<αjpij≤1-pijtij-1Through simultaneous use of formulas (18) and (20), we have(21)βij≤1-1-pijαjpijTherefore,(22)βij=0,pij=0βij≤1+αj-αjpij,0<pij<1In formula (19), 0 < pij < 1 represents the fact that the ith type of weapon can be used to strike the jth type of target, and given that the damage requirement for the jth type of target is Cj, then when Cj>1+αj-αj/pij,Cj>βij, indicating that the maximum number of the ith type of weapon (tij) cannot meet the requirement on the damage to the jth type of target, and no feasible solutions are present within the constrained value range of the variables; thus, it is necessary to expand the value range of the variables.The expansion on the value range of the variables in the mutation process allows the variables to take any value within the actual number of weapons of each type in the process to ensure that a feasible solution is found, which increases the search space and prolongs the search for a feasible solution. To avoid this situation, we adopted the dynamic directional mutation. Assuming the required damage level for a certain type of target isalf0, the actual damage level of this type of target among the individuals in the population is alf, the value of a variable that belongs to this type of target is pop(i,j), and the actual amount of the type of the weapon corresponding to the variable isbound, then the operation in the mutation process is as shown in Figure 1.Figure 1
Mutation operation flow of GA.


### 3.4. GA Flow Based on the Variable Value Control Method

The procedure for the application of the variable value range control method to GA to solve the GTA-WTA problem is as follows.(1) Constrain the number of weapons of each type used on each target. The contribution of each weapon is used to constrain the number of weapons of each type used on each target to control the value range of the variables in the initial population.(2) Generate the initial population, which is initialized in the set value range of the variables.(3) Constrain the total number of weapons used on each target, in which the quantity of the single type of weapon that has the maximum number for a certain target (max⁡(tij)) is used as the maximum quantity of all weapons for the given target, designated Tj.(4) Control the value of the variables in the initial population. The total number ofN types of weapons used to strike the same target (T1j) is calculated. If Tj < T1j, then the quantity (mij) of a type of weapon used to strike the target is randomly chosen, and if mij > 0, mij = mij-1. If T0j < T1j, go to step (4); otherwise, go to step (5).(5) Calculate the fitness value of the individuals of the population, and perform replication and crossover operations.(6) Perform the directional mutation operation, as shown in Figure1.(7) Terminate the iteration. Determine whether the requirement on the number of iterations is met and whether the requirement on the target damage is met, and if both requirements are met, the iteration is terminated; otherwise, go to step (5).


### 3.5. Feasibility Analysis

The variable control method includes two aspects: in the initialization phase, the variable value range is narrowed, so the search space is reduced to improve the search efficiency; in the mutation iteration process, the weapon usage amount is fine-tuned if necessary to ensure the quality of the solution and avoid the premature convergence issue.Because in striking a certain ground target, an optimal combination of multitype weapons can be used to make the comprehensive damage value meet the expected damage requirement, in the process of population initialization, the appropriate constraint on the number of a particular type of weapon and the total number of all types of weapons for a given target according to the expected damage requirement and the damage contribution of each type of weapon can effectively narrow the search space and improve the search efficiency.In the mutation process, the difference between the actual comprehensive damage value of the weapons on the target and the expected damage requirement is used to randomly determine whether it is necessary to increase or decrease the quantity of weapon usage. At this time, the quantity of weapon usage is no longer constrained by the damage contribution. A difference greater than 0 means that the weapon usage is too high and it is necessary to randomly decrease the usage amount of a certain type of weapon; a difference less than 0 means that the weapon usage is too low and it is necessary to randomly increase the usage amount of a certain type of weapon. Thus, it is ensured that, for each target, the expected damage requirement is met; i.e., the solution is feasible; if the total weapon value consumption is minimized, a better solution is obtained.Because real number encoding is adopted, in the mutation process, whether it is to increase or decrease the usage amount of a certain type of weapon, only one type of weapon is increased by 1 or decreased by 1, rather than randomly changing the value of weapon usage, which avoids the issue of prematurity very well through repeated trials and appropriate control of the mutation probability.


## 3.1. Significance of the Variable Value Control

The GTA-WTA problem is an NP-complete problem [20] in which the algorithm search space increases exponentially with the number of variables or the expansion of the range of variable values. In the GTA-WTA problem, the number of variables is fixed and unchangeable, with a value of the product of the number of weapon types and the number of targets; however, the value range of the variables is uncertain because the GTA-WTA generally does not limit the amount of each type of weapon used on each target but only limits the available amount of each type of weapon. Thus, in solving the problem, the weapon usage range, which is used as the value range of the variables, is uncertain and changeable. If the value range of the variables is not controlled, i.e., the quantity of each weapon type is taken as the upper bound of the value range of the variables, then the number of WTA schemes generated within the search space will be extremely large.We assume that there areN types of weapons, with quantities of N1,N2,…,NN, and M types of targets, with quantities of M1,M2,…,MM. If the quantity of each weapon type is taken as the upper bound of the variables, then the weapon amount (mij) of the ith type of weapon used to strike any target of the jth type of target is(3)mij∈0,Ni,mijisintegerAs shown in formula (3), when striking the jth type of target with the ith type of weapon, the variables mij can have Ni+1 possible values.For thejth type of target, when the N types of weapons strike the kth target, there are N variables, which can be viewed as a vector, and, then, in the search space formed by the value range of the N variables, the number of vectors (njk) that can be generated is(4)njk=∏i=1i=NNi+1When the N types of weapons strike all the targets of the jth type of target, there areN∗Mj variables, and, then, in the search space formed by the value range of the variables, the number of vectors (nj) that can be generated is(5)nj=∏k=1k=Mjnjk=∏i=1i=NNi+1MjWhen the N types of weapons strike all the targets of M types of targets, there are ∏j=1j=MN∗Mj variables, and, then, in the search space formed by the value range of the variables, the number of vectors (n) that can be generated is(6)n=∏j=1j=Mnj=∏j=1j=M∏k=1k=Mjnjk=∏i=1i=NNi+1∑j=1j=MMjIn formula (6), the value of n (the number of vectors) is the number of all possible WTA schemes when N types of weapons strike all the targets of M types of targets.As shown in formula (6), as the value range of the variables expands, the number of GTA-WTA schemes increases drastically. However, the number of feasible solutions for the GTA-WTA problem is fixed and only accounts for a small portion of the total WTA schemes. In addition, most of the feasible solutions are distributed in a certain region of the search space, and the distribution region and the number of feasible solutions are not affected by the value range of the variables. Therefore, if the search space can be sharply reduced while still containing the space in which the feasible solutions are located, then a fast search of feasible solutions is then ensured.For example, if 49 weapons of the same type strike 10 targets of the same type and if the total amount of weapons is taken as the upper bound of the value range of the variables, thenN1=49 and M1=10, and, based on formula (6), the number of WTA schemes (n(1)) is(7)n1=∏i=1i=1Ni+1∑j=1j=1Mj=N1+1M1=5010If the value range of the variables is controlled and the upper value limit of the variables is 9, then N1=9 and M1=10, and, based on formula (6), the number of WTA schemes (n(2)) is(8)n2=N1+1M1=1010From the number of WTA schemes (n(1) and n(2)) solved using formulas (7) and (8), the ratio (α) of WTA schemes within the search space without the control of the value range of the variables to that within the search space with the control of the value range of the variables is(9)α=n1n2=50101010=510:1Obviously, after reducing the upper bound of the value range of the variables, the search space is sharply reduced, and the number of WTA schemes within the search space is also drastically reduced. In addition, the value range of the variables after the reduction is generally reasonable, and the search space formed by the value range of the variables can contain the space of the vast majority of feasible solutions, indicating that the reasonable control of the value range of the variables can ensure a fast search for feasible solutions.


## 3.2. Determining the Value Range Based on the Damage Contribution

Variable value control method improves the search efficiency by performing a series of operations on the variable value range, that is, narrowing the algorithm search space in the initialization process of GA to shorten the calculating time and amplifying the variable value range in the mutation process to ensure have a reasonable result. For GA to solve the GTA-WTA problem, the variable is the number of each type of weapon acting on each target, and the variable value range is the number of intervals that each type of weapon may act on each target. Variable value control method consists of three parts: one is to constrain the maximum number of weapons acting on each target; the other is to limit the total amount of weapons acting on each target; the third is to expand the variable variation interval and change the mutation strategy.For the same target,n ammunitions are usually used for striking to achieve the expected damage effectiveness. For the same type of ammunition, as the target is being damaged, each ammunition contributes different damage values, which can be called the damage contribution, denoted u. If the damage probability of the ith type of ammunition against the jth type of target is pij, then the damage contribution of the nth ammunition (uijn) is(10)uijn=1-pijn-1pijThe ratio of the damage contribution of the nth ammunition to the value of the ammunition is defined as the damage contribution per unit value, denoted as μijn. If the value of the ith type of ammunition is vi, then(11)μijn=uijnvi=1-pijn-1pijviObviously, for the same target, when choosing ammunition, the higher the damage contribution per unit value is, the more appropriate the ammunition type is.


### 3.2.1. Constraining the Quantity of Each Type of Weapon against Each Target

When striking the same target, the quantity of weapons is limited, and the use of a large quantity of weapons to strike the same target is generally avoided. According to formula (10), the greater the quantity (n) of weapons used in the strike is, the lower its damage contribution (uijn) will be, which gradually approaches 0. When uijn approaches 0, it indicates that the strike with the nth weapon on the target essentially causes no damage, and the strike with the ith type of weapon against the jth type of target should be stopped; at this time, the quantity of weapons consumed can be viewed as the upper bound of the weapon quantity range of the ith type of weapon against each of the jth type of target. Assuming that when the value of uijn is less than the preset minimal damage contribution (αj) of a given weapon for the jth type of target, the strike on the target is stopped and then n must satisfy(12)n=1,pij=0uijn<αj≤uijn-1,0<pij<1Thus, under the constraint of the damage contribution, the upper bound (tij) of the weapon quantity range of the ith type of weapon against the jth type of target is(13)tij=n-1The value of tij depends on the damage contribution value αj; the lower the value of αj is, the higher the value of tij will be, which can even be higher than the weapon quantity (m) required for the ith type of weapon to destroy the jth type single target. Moreover, if the weapon quantity exceeds m, it is deemed excessive use, and the tij value needs to be decreased. If the damage requirement for the jth type of target is Cj, then m satisfies(14)m=0,pij=01-1-pijm-1<Cj≤1-1-pijm,0<pij<1To prevent the value of tij from exceeding m due to an undersized αj, under the constraint of m, the value of tij is(15)tij=min⁡tij,m


### 3.2.2. Constraining the Total Number of Weapons against Each Target

Different types of weapons lead to varying damage probabilities, but there is the situation in which multiple types of weapons are suitable for attacking the same target. In this case, in the population initialization of GA, each type of applicable weapon is randomly given a quantity to be used to strike against the target, which is very unlikely to be 0, so the number of weapons used to strike the target may be too large, resulting in the expenditure of a large amount of time in the subsequent optimization process to lower the number into a reasonable range. Obviously, the total number of weapons for a certain target of thejth type of target should not exceed the upper bound of the weapon usage range that causes the minimal damage probability on the jth type of target. If there are N types of ammunition suitable for striking the jth type of target, then the maximum value (Tj) of the total number of weapons used to strike the jth type of target is(16)Tj=max⁡tijAfter the maximum value (Tj) of the total number of the weapons for striking the target is determined, it is necessary to process the quantities of N types of weapon used to strike the target (m1j,m2j,…,mNj) in the algorithm to make it satisfy(17)∑i=1i=Nmij≤Tj


## 3.2.1. Constraining the Quantity of Each Type of Weapon against Each Target

When striking the same target, the quantity of weapons is limited, and the use of a large quantity of weapons to strike the same target is generally avoided. According to formula (10), the greater the quantity (n) of weapons used in the strike is, the lower its damage contribution (uijn) will be, which gradually approaches 0. When uijn approaches 0, it indicates that the strike with the nth weapon on the target essentially causes no damage, and the strike with the ith type of weapon against the jth type of target should be stopped; at this time, the quantity of weapons consumed can be viewed as the upper bound of the weapon quantity range of the ith type of weapon against each of the jth type of target. Assuming that when the value of uijn is less than the preset minimal damage contribution (αj) of a given weapon for the jth type of target, the strike on the target is stopped and then n must satisfy(12)n=1,pij=0uijn<αj≤uijn-1,0<pij<1Thus, under the constraint of the damage contribution, the upper bound (tij) of the weapon quantity range of the ith type of weapon against the jth type of target is(13)tij=n-1The value of tij depends on the damage contribution value αj; the lower the value of αj is, the higher the value of tij will be, which can even be higher than the weapon quantity (m) required for the ith type of weapon to destroy the jth type single target. Moreover, if the weapon quantity exceeds m, it is deemed excessive use, and the tij value needs to be decreased. If the damage requirement for the jth type of target is Cj, then m satisfies(14)m=0,pij=01-1-pijm-1<Cj≤1-1-pijm,0<pij<1To prevent the value of tij from exceeding m due to an undersized αj, under the constraint of m, the value of tij is(15)tij=min⁡tij,m


## 3.2.2. Constraining the Total Number of Weapons against Each Target

Different types of weapons lead to varying damage probabilities, but there is the situation in which multiple types of weapons are suitable for attacking the same target. In this case, in the population initialization of GA, each type of applicable weapon is randomly given a quantity to be used to strike against the target, which is very unlikely to be 0, so the number of weapons used to strike the target may be too large, resulting in the expenditure of a large amount of time in the subsequent optimization process to lower the number into a reasonable range. Obviously, the total number of weapons for a certain target of thejth type of target should not exceed the upper bound of the weapon usage range that causes the minimal damage probability on the jth type of target. If there are N types of ammunition suitable for striking the jth type of target, then the maximum value (Tj) of the total number of weapons used to strike the jth type of target is(16)Tj=max⁡tijAfter the maximum value (Tj) of the total number of the weapons for striking the target is determined, it is necessary to process the quantities of N types of weapon used to strike the target (m1j,m2j,…,mNj) in the algorithm to make it satisfy(17)∑i=1i=Nmij≤Tj


## 3.3. Amplifying the Value Range of Variables through Directional Variation

The value range of individual variable in the population becomes smaller by constraining the number of weapons acting on each target and the total number of weapons acting on each target, and this variable value range constraint is generally reasonable. But when the damage probability of a certain type of weapon for a certain type of target is low, the number of weapons used to strike a single target of that target type under the constraint may not be sufficient to achieve the required level of damage, resulting in no feasible solutions within the constrained value range. Assuming that theith type of weapon is used to strike against the jth type of target, the maximum number (tij) of the ith type of weapon for each jth type of target can be calculated using formula (15), and, then, the maximum degree of damage (βij) on the jth type of target by the weapon is(18)βij=1-1-pijtijWhen 0 < pij < 1, from formulas (10) and (12), we have (19)1-pijtijpij<αj≤1-pijtij-1pijThat is,(20)1-pijtij<αjpij≤1-pijtij-1Through simultaneous use of formulas (18) and (20), we have(21)βij≤1-1-pijαjpijTherefore,(22)βij=0,pij=0βij≤1+αj-αjpij,0<pij<1In formula (19), 0 < pij < 1 represents the fact that the ith type of weapon can be used to strike the jth type of target, and given that the damage requirement for the jth type of target is Cj, then when Cj>1+αj-αj/pij,Cj>βij, indicating that the maximum number of the ith type of weapon (tij) cannot meet the requirement on the damage to the jth type of target, and no feasible solutions are present within the constrained value range of the variables; thus, it is necessary to expand the value range of the variables.The expansion on the value range of the variables in the mutation process allows the variables to take any value within the actual number of weapons of each type in the process to ensure that a feasible solution is found, which increases the search space and prolongs the search for a feasible solution. To avoid this situation, we adopted the dynamic directional mutation. Assuming the required damage level for a certain type of target isalf0, the actual damage level of this type of target among the individuals in the population is alf, the value of a variable that belongs to this type of target is pop(i,j), and the actual amount of the type of the weapon corresponding to the variable isbound, then the operation in the mutation process is as shown in Figure 1.Figure 1
Mutation operation flow of GA.


## 3.4. GA Flow Based on the Variable Value Control Method

The procedure for the application of the variable value range control method to GA to solve the GTA-WTA problem is as follows.(1) Constrain the number of weapons of each type used on each target. The contribution of each weapon is used to constrain the number of weapons of each type used on each target to control the value range of the variables in the initial population.(2) Generate the initial population, which is initialized in the set value range of the variables.(3) Constrain the total number of weapons used on each target, in which the quantity of the single type of weapon that has the maximum number for a certain target (max⁡(tij)) is used as the maximum quantity of all weapons for the given target, designated Tj.(4) Control the value of the variables in the initial population. The total number ofN types of weapons used to strike the same target (T1j) is calculated. If Tj < T1j, then the quantity (mij) of a type of weapon used to strike the target is randomly chosen, and if mij > 0, mij = mij-1. If T0j < T1j, go to step (4); otherwise, go to step (5).(5) Calculate the fitness value of the individuals of the population, and perform replication and crossover operations.(6) Perform the directional mutation operation, as shown in Figure1.(7) Terminate the iteration. Determine whether the requirement on the number of iterations is met and whether the requirement on the target damage is met, and if both requirements are met, the iteration is terminated; otherwise, go to step (5).


## 3.5. Feasibility Analysis

The variable control method includes two aspects: in the initialization phase, the variable value range is narrowed, so the search space is reduced to improve the search efficiency; in the mutation iteration process, the weapon usage amount is fine-tuned if necessary to ensure the quality of the solution and avoid the premature convergence issue.Because in striking a certain ground target, an optimal combination of multitype weapons can be used to make the comprehensive damage value meet the expected damage requirement, in the process of population initialization, the appropriate constraint on the number of a particular type of weapon and the total number of all types of weapons for a given target according to the expected damage requirement and the damage contribution of each type of weapon can effectively narrow the search space and improve the search efficiency.In the mutation process, the difference between the actual comprehensive damage value of the weapons on the target and the expected damage requirement is used to randomly determine whether it is necessary to increase or decrease the quantity of weapon usage. At this time, the quantity of weapon usage is no longer constrained by the damage contribution. A difference greater than 0 means that the weapon usage is too high and it is necessary to randomly decrease the usage amount of a certain type of weapon; a difference less than 0 means that the weapon usage is too low and it is necessary to randomly increase the usage amount of a certain type of weapon. Thus, it is ensured that, for each target, the expected damage requirement is met; i.e., the solution is feasible; if the total weapon value consumption is minimized, a better solution is obtained.Because real number encoding is adopted, in the mutation process, whether it is to increase or decrease the usage amount of a certain type of weapon, only one type of weapon is increased by 1 or decreased by 1, rather than randomly changing the value of weapon usage, which avoids the issue of prematurity very well through repeated trials and appropriate control of the mutation probability.


## 4. Calculation Example

### 4.1. GTA-WTA Problem Example

For the typical GTA-WTA problem, we describe four different cases under the same background, based on which we make comparisons.Case Background. A total of five different types of weapons are used, and the value coefficient (unit: million) of each type of weapon and the total investment amount in each case are shown in Table 1. Six types of ground targets are to be attacked, and the number of targets in each case is listed in Table 2. The damage probability of each type of weapon to each type of target is shown in Table 3. It is required that the average damage coefficient on each type of target be 0.8, 0.8, 0.9, 0.85, 0.8, and 0.9. Based on the above conditions, we try to solve the best weapon allocation scheme.Table 1
The value and total amount of all types of weapons.
Weapon type Total amount of weapons Value coefficientof weapon Case 1 Case 2 Case 3 Case 4 W1 5 10 15 20 8 W2 5 5 10 15 7 W3 5 10 15 20 6 W4 10 10 20 25 5 W5 5 10 15 20 4Table 2
Target type and quantity.
Target type Target quantity Case 1 Case 2 Case 3 Case 4 T1 2 3 4 5 T2 1 3 4 5 T3 1 3 4 5 T4 1 3 4 5 T5 1 2 4 6 T6 2 2 4 6Table 3
Damage probability of weapon to target.
Target type Weapon type W1 W2 W3 W4 W5 T1 0.85 0.6 0.5 0.45 0.4 T2 0.46 0.35 0.6 0.5 0.35 T3 0.4 0.32 0.6 0.5 0.4 T4 0.8 0.7 0.65 0.5 0.4 T5 0.4 0.3 0.85 0.7 0.6 T6 0.61 0.45 0.75 0.6 0.5


### 4.2. Simulation and Analysis of the GTA-WTA Problem

For the above case, under the same hardware and software condition (win7 x64, Intel (R) Xeon (R) CPU E3-1240 v3 @3.4 GHz, RAM 16 GB), we used MATLAB software to program and simulate the AG-WTA problem using the improved bat algorithm of [17], the improved PSO method of [18], and the method proposed in this study to compare the weapon consumption values and the time consumptions (unit: second) of the three methods. The results are shown in Table 4, in which “number of variables” is the product of the number of weapon types and the number of targets, “minimal weapon consumption value” is the weapon value required to meet the specified damage requirement, “weapon consumption value” is the mean of the weapon consumption values of 20 simulations, and “time consumption” is the mean of the run times of 20 simulations.Table 4
Solution and time consumption of the three improved algorithm.
Case number of variables minimal weapon consumption value variable value control method improved bat algorithm improved PSO method weapon consumption value time consumption weapon consumption value time consumption weapon consumption value time consumption Case 1 40 83 86.2 10.7 89.6 14.3 91.5 12.1 Case 2 80 173 189.5 32.6 197.3 50.9 201.4 47.5 Case 3 120 250 297.3 79.2 314.8 182.7 319.6 173.2 Case 4 160 327 409.5 153.4 438.1 406.6 447.5 425.8Table4 shows that, compared with the improved bat algorithm or the improved PSO method, the proposed optimization method has a lower weapon consumption value and shorter time consumption, and as the number of variables increases, the advantages of the proposed method become more profound.To present and analyze the superiority of the proposed method more visually, the weapon consumption value and the time consumption in Table4 are plotted in Figures 2 and 3, respectively.Figure 2
Optimized weapon consumption value of the three methods.Figure 3
Time consumption of the three methods.Figure2 shows that the weapon consumption value optimized using the proposed method is always less than that of the other two methods; as the number of variables increases, the weapon consumption value difference between the proposed method and each of the other two methods also increases. Figure 3 shows that, as the number of variables increases, the time consumption of each of the three methods also increases, and the slope of the curve becomes steeper; however, the time consumption of the proposed method is always shorter than that of each of the other two methods, and the slope of the proposed method is always smaller than that of each of the other two methods. These results indicate that the optimization performance of the proposed method is better than that of the literature methods [17, 18], and the larger the scale of the problem is, the more profound the superiority is.In short, the proposed method can further improve the computational efficiency and solution quality of the GTA-WTA problem, and the larger the scale of the problem is, the more profound the superiority is.Robustness is another criterion for measuring the pros and cons of an algorithm. The better the robustness of the algorithm, the more reliable the algorithm. Table4 shows the weapon consumption value and time consumption optimized by the three methods, but the robustness of the three methods cannot be verified and demonstrated. The robustness of the three methods can be measured by the standard deviation of weapon consumption value and time consumption. The smaller the standard deviation of weapon consumption value and time consumption, the better the robustness of the method. For a set of variables (y1,y2,…,yn), the standard deviation (σ) is (23)σ=1n∑i=1nyi-y-2,y-=1n∑i=1nyiAccording to formula (23), the standard deviation of weapon consumption value and time consumption optimized by the three methods for the 20 simulations of the 4 cases can be calculated. The results are shown in Table 5. The “σw” and “σt” in Table 5 represent “the standard deviation of weapon consumption value” and “the standard deviation of time consumption”, respectively.Table 5
The standard deviation of weapon consumption value and time consumption.
Case variable value control method improved bat algorithm improved PSO method σ w σ t σ w σ t σ w σ t Case 1 3.5 0.8 4.9 1.2 3.6 1.0 Case 2 8.9 2.0 10.4 3.7 11.5 3.1 Case 3 15.6 4.7 19.5 8.1 20.4 7.6 Case 4 27.4 8.1 31.6 19.6 33.8 22.4Table5 shows that, compared with the improved bat algorithm or the improved PSO method, the proposed optimization method has a lower standard deviation of weapon consumption value and time consumption, indicating that the proposed optimization method has better robustness and is more suitable to optimize the GTA-WTA problem.


## 4.1. GTA-WTA Problem Example

For the typical GTA-WTA problem, we describe four different cases under the same background, based on which we make comparisons.Case Background. A total of five different types of weapons are used, and the value coefficient (unit: million) of each type of weapon and the total investment amount in each case are shown in Table 1. Six types of ground targets are to be attacked, and the number of targets in each case is listed in Table 2. The damage probability of each type of weapon to each type of target is shown in Table 3. It is required that the average damage coefficient on each type of target be 0.8, 0.8, 0.9, 0.85, 0.8, and 0.9. Based on the above conditions, we try to solve the best weapon allocation scheme.Table 1
The value and total amount of all types of weapons.
Weapon type Total amount of weapons Value coefficientof weapon Case 1 Case 2 Case 3 Case 4 W1 5 10 15 20 8 W2 5 5 10 15 7 W3 5 10 15 20 6 W4 10 10 20 25 5 W5 5 10 15 20 4Table 2
Target type and quantity.
Target type Target quantity Case 1 Case 2 Case 3 Case 4 T1 2 3 4 5 T2 1 3 4 5 T3 1 3 4 5 T4 1 3 4 5 T5 1 2 4 6 T6 2 2 4 6Table 3
Damage probability of weapon to target.
Target type Weapon type W1 W2 W3 W4 W5 T1 0.85 0.6 0.5 0.45 0.4 T2 0.46 0.35 0.6 0.5 0.35 T3 0.4 0.32 0.6 0.5 0.4 T4 0.8 0.7 0.65 0.5 0.4 T5 0.4 0.3 0.85 0.7 0.6 T6 0.61 0.45 0.75 0.6 0.5


## 4.2. Simulation and Analysis of the GTA-WTA Problem

For the above case, under the same hardware and software condition (win7 x64, Intel (R) Xeon (R) CPU E3-1240 v3 @3.4 GHz, RAM 16 GB), we used MATLAB software to program and simulate the AG-WTA problem using the improved bat algorithm of [17], the improved PSO method of [18], and the method proposed in this study to compare the weapon consumption values and the time consumptions (unit: second) of the three methods. The results are shown in Table 4, in which “number of variables” is the product of the number of weapon types and the number of targets, “minimal weapon consumption value” is the weapon value required to meet the specified damage requirement, “weapon consumption value” is the mean of the weapon consumption values of 20 simulations, and “time consumption” is the mean of the run times of 20 simulations.Table 4
Solution and time consumption of the three improved algorithm.
Case number of variables minimal weapon consumption value variable value control method improved bat algorithm improved PSO method weapon consumption value time consumption weapon consumption value time consumption weapon consumption value time consumption Case 1 40 83 86.2 10.7 89.6 14.3 91.5 12.1 Case 2 80 173 189.5 32.6 197.3 50.9 201.4 47.5 Case 3 120 250 297.3 79.2 314.8 182.7 319.6 173.2 Case 4 160 327 409.5 153.4 438.1 406.6 447.5 425.8Table4 shows that, compared with the improved bat algorithm or the improved PSO method, the proposed optimization method has a lower weapon consumption value and shorter time consumption, and as the number of variables increases, the advantages of the proposed method become more profound.To present and analyze the superiority of the proposed method more visually, the weapon consumption value and the time consumption in Table4 are plotted in Figures 2 and 3, respectively.Figure 2
Optimized weapon consumption value of the three methods.Figure 3
Time consumption of the three methods.Figure2 shows that the weapon consumption value optimized using the proposed method is always less than that of the other two methods; as the number of variables increases, the weapon consumption value difference between the proposed method and each of the other two methods also increases. Figure 3 shows that, as the number of variables increases, the time consumption of each of the three methods also increases, and the slope of the curve becomes steeper; however, the time consumption of the proposed method is always shorter than that of each of the other two methods, and the slope of the proposed method is always smaller than that of each of the other two methods. These results indicate that the optimization performance of the proposed method is better than that of the literature methods [17, 18], and the larger the scale of the problem is, the more profound the superiority is.In short, the proposed method can further improve the computational efficiency and solution quality of the GTA-WTA problem, and the larger the scale of the problem is, the more profound the superiority is.Robustness is another criterion for measuring the pros and cons of an algorithm. The better the robustness of the algorithm, the more reliable the algorithm. Table4 shows the weapon consumption value and time consumption optimized by the three methods, but the robustness of the three methods cannot be verified and demonstrated. The robustness of the three methods can be measured by the standard deviation of weapon consumption value and time consumption. The smaller the standard deviation of weapon consumption value and time consumption, the better the robustness of the method. For a set of variables (y1,y2,…,yn), the standard deviation (σ) is (23)σ=1n∑i=1nyi-y-2,y-=1n∑i=1nyiAccording to formula (23), the standard deviation of weapon consumption value and time consumption optimized by the three methods for the 20 simulations of the 4 cases can be calculated. The results are shown in Table 5. The “σw” and “σt” in Table 5 represent “the standard deviation of weapon consumption value” and “the standard deviation of time consumption”, respectively.Table 5
The standard deviation of weapon consumption value and time consumption.
Case variable value control method improved bat algorithm improved PSO method σ w σ t σ w σ t σ w σ t Case 1 3.5 0.8 4.9 1.2 3.6 1.0 Case 2 8.9 2.0 10.4 3.7 11.5 3.1 Case 3 15.6 4.7 19.5 8.1 20.4 7.6 Case 4 27.4 8.1 31.6 19.6 33.8 22.4Table5 shows that, compared with the improved bat algorithm or the improved PSO method, the proposed optimization method has a lower standard deviation of weapon consumption value and time consumption, indicating that the proposed optimization method has better robustness and is more suitable to optimize the GTA-WTA problem.


## 5. Conclusion

In this study, we propose a variable control method to improve the genetic algorithm to address the GTA-WTA problem. The proposed method has two advantages. First, it reasonably narrows the search space, thus improving the computational efficiency; second, it improves the mutation strategy to ensure the quality of the solution and effectively avoids the issue of prematurity. Simulations, comparisons and analyses indicate that the proposed method exhibits superiority in terms of solution quality and computational efficiency when addressing the large-scale GTA-WTA problem, and the larger the scale of the problem is, the more profound the superiority is. And the proposed method has better robustness.Notably, the proposed method has obvious advantages in solving the GTA-WTA problem but is not suitable for solving the AD-WTA problem. Further investigation is needed to determine whether it is effective in addressing other complex optimization problems.



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