Research on Warship System Resilience Based on Intelligent Recovery with Improved Ant Colony Optimization

Abstract

Faced with complex, ever-changing battlefield environments and diverse attacks, enabling warship combat systems to recover rapidly and effectively after damage is key to enhancing resilience and sustained combat capability. We construct a representative naval battle scenario and propose an integrated Attack-Defense-Recovery Strategy (ADRS) grounded in warship system models for different attack types. To address high parameter sensitivity, weak initial pheromone feedback, suboptimal solution quality, and premature convergence in traditional ant colony optimization (ACO), we introduce three improvements: (i) grid-search calibration of key ACO parameters to enhance global exploration, (ii) a non-uniform initial pheromone mechanism based on the wartime importance of equipment to guide early solutions, and (iii) an ADRS-consistent state-transition rule with group-based starting points to prioritize high-value equipment during the search. Simulation results show that the improved ACO (IACO) outperforms classical ACO in convergence speed and solution optimality. Across torpedo, aircraft/missile, and UAV scenarios, ADRS-ACO improves over GRS-ACO by 7.2%, 0.3%, and 5.5%, while ADRS-IACO achieves gains of 34.9%, 17.1%, and 16.7% over GRS-ACO and 25.9%, 16.7%, and 10.6% over ADRS-ACO. Overall, ADRS-IACO consistently delivers the best solutions. In high-intensity, high-damage torpedo conditions, ADRS-IACO demonstrates superior path planning and repair scheduling, more effectively identifying critical equipment and allocating resources. Moreover, under multi-wave combat, coupling with ADRS effectively reduces cumulative damage and substantially improves overall warship-system resilience.

1. Introduction

With the rapid development of a new round of world technological revolution and military transformation, intelligent warfare is rapidly becoming an important direction for the future [1]. The sea battlefield is one of the important confrontation situations in military, with complex and varied forms involving scenes in the air, on the water, and underwater.

In the face of complex battlefields and sustained attacks, warship systems must have the ability to complete damage assessment, rapid repair, and restore combat functions in a very short period of time. Therefore, system resilience is an important indicator to measure this ability [2]. System resilience refers to the ability to anticipate, counteract, absorb, react, adapt, and recover from external disturbances or changes [3,4,5]. Building a highly resilient system is not only a key principle in modern equipment design and use, but also a prerequisite for maintaining sustained combat capability in multi waves and uncertain confrontations.

In recent years, although deep learning and reinforcement learning have been applied to intelligent maintenance and recovery, there are still limitations in warship combat scenarios: (1) those algorithms rely on a large number of high-quality data, while the actual combat and exercise data are scarce, which is difficult to meet the training needs; (2) the battlefield situation is changeable, so it is necessary to quickly generate recovery strategies. However, the model based on offline training cannot cover all situations and cannot respond in real time; (3) in contrast, ACO has the advantages of simple structure, high efficiency and strong adaptability, which is more suitable for the warship system recovery in situations including high real-time, complexity and data shortage.

Recent studies have shown that ACO [6,7,8] is widely used in path optimization, scheduling, and system recovery. Zhou et al. [9] conducted simulation experiments on the flight path of unmanned aerial vehicles using an improved ACO to find safe and effective routes, achieving accurate identification of insulators and improving the efficiency of unmanned aerial vehicle inspections. Guo et al. [10] proposed a schedulable aware routing algorithm (SAR) based on an improved ACO for non-waiting scheduling of tasks in time sensitive networks. The experiment showed that the algorithm had a significantly better scheduling success rate than traditional methods under different network loads, indicating that the improved ACO has strong solving ability under complex constraint conditions. Ikhar et al. [11] proposed an intrusion detection method that combines VAE and improved ACO, which optimizes decision boundaries through ACO and improves the accuracy of anomaly detection. The experiment showed that this method is superior to traditional methods, reflecting its potential for application in network security and system resilience improvement.

In recent years, research on intelligent maintenance of warship systems has made positive progress. Reference [12] proposed the Smart Maintenance Decision Support Systems (SMDSSs) based on big data for optimizing maintenance strategies and lifecycle management. Reference [13] constructed a transformation framework for intelligent maintenance factories, expanding the application of intelligent manufacturing in the warship service phase. Reference [14] proposed the HIRAL method that integrates AHP, DEA, and target feasibility analysis, achieving reliability optimization in component level. Reference [15] constructed an attack tree model of warship system, identified and validated 20 network attack paths, and improved the systematic modeling of maritime network security. Although there have been significant research results on intelligent maintenance [12,13] of warship systems [14,15], there are still many shortcomings: ① when warships are under different types of attacks, intelligent maintenance for the comprehensive importance of warship equipment has not been considered; ② in the intelligent maintenance algorithm, all damaged equipment have the same importance, different wartime importance are not assigned based on the different type of attack, and attack and defense attributes of different equipment are not distinguished, thus failing to achieve better wartime intelligent maintenance and recovery effects; ③ Most studies focus on the system resilience process under single interference and attack, with less consideration on the complete system resilience and recovery process under multi attack waves.

Therefore, this paper considers the recovery problem of ships subjected to multiple waves of enemy attacks during combat and summarizes it as a parallel maintenance problem with multiple constraints. The Attack-Defense-Recovery Strategy (ADRS) idea is proposed, and the wartime importance of equipment under different attack types is analyzed. The initial pheromone concentration of the ACO is improved by non-uniform distribution to enhance the initial pheromone concentration feedback of the algorithm. Finally, a search rule for starting point classification is proposed based on the ADRS idea, which strengthens the state transition probability of combat-capable equipment pairs and studies the resilience of ship systems.

The structure of this paper is arranged as follows: Section 1 introduces the research background, motivation, and existing limitations in warship system resilience and intelligent maintenance. Section 2 presents the basic methods used in this study, including the definition of resilience metrics and the modeling of the ADRS. Section 3 develops a comprehensive modeling framework for multi-wave attack–defense scenarios, covering the modeling of attack types, defense systems, damage effects, and recovery processes. Section 4 describes the design of the IACO and elaborates on its implementation for repair path optimization under multi-wave combat conditions. Section 5 conducts simulation experiments under various attack scenarios—including torpedo strikes, aircraft/missile attacks, and UAV swarm attacks to compare algorithm performance and evaluate system resilience outcomes. Finally, Section 6 concludes the paper and summarizes the research results.

2. Fundamental Approaches

2.1. Quantification of Resilience

In the field of systems engineering, resilience is usually evaluated by quantitative methods. The following section introduces a representative generic metric, the quotient resilience model.

The quotient resilience model takes the ratio between recovery performance and performance degradation as the measurement standard of the system resilience. The model provides a static resilience evaluation method at a specific time, and its analysis process is based on discrete performance. This model is also called “non-time varying resilience model” [16], and the specific form is shown in Formula (1).

$$R\left(t\right)={\displaystyle \frac{Recovery\left(t\right)}{Loss\left(t_d\right)}}$$

(1)

In this equation, $R\left(t\right)$ represents the system’s resilience at time $t$, $Recovery\left(t\right)$ denotes the system’s recovered performance at that time, and $Loss\left(t_d\right)$ refers to the performance degradation experienced at the disturbance time$t_d$.

This paper adopts the quotient resilience model to quantitatively analyze system resilience under multi-wave attacks in different combat conditions.

2.2. Attack-Defense-Recovery Strategy

In the engineering domain, the recovery strategy has a direct impact on efficiency. Wang et al. [17] addressed power supply interruptions in distribution networks caused by extreme events, internal faults, and scheduled maintenance. They proposed a fault recovery strategy that accounts for multiple failure scenarios. Simulation results demonstrated that the proposed strategy significantly improved both recovery efficiency and power supply reliability. Wan et al. [18] focused on the decline in flight punctuality due to abnormal operations and developed an evaluation method for abnormal flight recovery strategies tailored to airline operations control centers. Validation results showed that this method effectively enhances the executability of decision-making and provides flight dispatchers with a scientific and practical tool for strategy assessment.

Recovery strategies for warship systems also vary based on mission objectives [19]. Maintenance tasks [20,21,22] in wartime scenarios are generally with strict and real-time conditions. This paper studies and discusses the optimal maintenance strategy [23] of warships facing multiple waves of attacks, then proposed the Attack-Defense-Recovery Strategy (ADRS).

The core idea of this recovery strategy is as follows:

Facing the characteristics of heavy maintenance tasks and complex environment in multi-wave attack and defense operations [24,25,26], it is necessary to dynamically adjust the importance of equipment according to different attack scenarios. In the case of short attack intervals in each round, the system evaluates the importance of each equipment in real time under the current wartime attack and accurately determines the priority equipment for maintenance in the current situation.

At the same time, the system classifies the equipment according to their functions, prioritizes the key equipment with combat functions to enter the maintenance sequence, and strengthens the active combat and defense capability. It also defines the wartime importance of different equipment types in response to different attack modes. Based on the calculation results of the recovery algorithm, maintenance priority is more inclined to the equipment that can effectively respond to the next wave of attacks, so as to reduce system damage in multiple rounds of attacks and improve the overall survivability and system flexibility.

In order to verify the effect of this strategy, this paper introduces the general recovery strategy (GRS) as a compare. GRS adopts static and fixed equipment importance, which is not adjusted with the change in enemy attack modes. The specific performance is that the importance of all equipment in the system remains unchanged during the recovery process, without distinguishing between equipment types and combat tasks, which belongs to a single recovery mechanism without adaptability.

3. Multi-Wave Attack–Defense Modeling

3.1. Multi-Wave Attack Modeling

To simulate multiple modes of typical attack, this paper comprehensively considers the explosive strength and interception difficulty of weapons, selects three representative attack modes, and constructs a multi-wave simulation model according to the attack threat. There are differences in explosion yield, attack direction, and spatial distribution of landing points in different attacks. Based on this, a multi-wave attack matrix is constructed.

To capture diverse combat dynamics, the simulation incorporates three representative attack modes, each reflecting distinct operational characteristics. Underwater torpedo attacks embody heavy, high-threat strikes launched from beneath the ship and are the most challenging to intercept. High-altitude aircraft or missile attacks constitute medium-intensity threats approaching from above, with comparatively lower interception difficulty. Low-altitude UAV swarm attacks simulate continuous, high-frequency harassment, leveraging large numbers of small targets to overwhelm defensive responses.

In the simulation scenario, an unidentified type of attack is initiated every 30 min, resulting in a total of mmm waves, with its attack sequence represented by $T$. The formulation of $T$ is provided in Formula (2).

$$T=\{\left(n_{ij},tn{t}_{ij},x_{ij},y_{ij},z_{ij}\right)\mid i=1,\dots ,m;j=1,\dots ,q\}.$$

(2)

In Formula (2), the multi-wave attack matrix $T$ is defined, consolidating all attack events over msuccessive waves. Each event is represented by a five-dimensional tuple $\left(n_{ij},tn{t}_{ij},x_{ij},y_{ij},z_{ij}\right)$, where $n_{ij}$ denotes the index of the $j$-th attack unit in the $i$-th wave, tnti indicates its explosive power, and $\left(x_{ij},y_{ij},z_{ij}\right)$ specifies the three-dimensional impact coordinates. Hence, $T$ provides a structured dataset that captures both the spatial distribution and threat characteristics for all attack units in the multi-wave scenario.

3.2. Defense System Representation

Warship defense system is the key system to ensure the effective interception and attack of ships against enemy attack in wartime. In this paper, the defense system model is constructed from the aspects of equipment composition, equipment importance, and combat function.

(1)

Equipment information

In order to more accurately simulate the shipborne system environment, the length, width, and height of the ship body are set as basic parameters, and a three-dimensional model containing 50 equipment is constructed accordingly. All equipment is divided into basic equipment, combat equipment, and auxiliary equipment according to their functions. The classification is presented in Appendix A 

Table A1. Classification of equipment types.

Equipment Index	Equipment Type	Main Tasks and Characteristics of the Equipment
0–9	Base equipment	Responsible for fundamental system functions such as warship power systems, with consistently high importance across all three combat scenarios.
10–14	Radar (Combat equipment)	Responsible for detecting high-altitude aircraft, missiles, and low-altitude. UAVs. Highly important in detecting both high- and low-altitude aerial threats.
15–19	Sonar detection (Combat equipment)	Responsible for detecting underwater torpedoes and mines. Critically important in underwater threat detection.
20–24	Anti-aircraft missile (Combat equipment)	Responsible for intercepting high-altitude aircraft and missiles. Highly important in defending against aerial attacks.
25–29	Near defense artillery (Combat equipment)	Responsible for intercepting low-altitude UAV swarms. Highly important in defending against UAV attacks.
30–34	Antisubmarine missile (Combat equipment)	Responsible for intercepting underwater torpedoes. Critically important in countering underwater threats.
35–49	Secondary equipment	Responsible for less critical tasks and lower repair priority during combat. Relatively low importance across all three combat scenarios.

.

(2)

Combat equipment capability

Combat equipment mainly includes detection and interception, whose performance directly determines the system’s ability to recognize and intercept enemy targets, and then affects the overall combat effect. The equipment capability is quantified by the floating-point value in the [0, 1] interval. The higher the value, the stronger the performance. The specific capacity values of each piece of equipment are listed in Appendix A 

Table A2. Capabilities of combat equipment.

Equipment Types and Tasks	Equipment Index	Equipment Ability
Radar detection—used for detecting high-altitude aircraft, missiles, and low-altitude UAVs	10	0.85
	11	0.82
	12	0.88
	13	0.80
	14	0.86
Sonar detection—used for detecting underwater torpedoes	15	0.50
	16	0.55
	17	0.60
	18	0.70
	19	0.65
Long-range air defense missile—used for intercepting high-altitude aircraft and missiles	20	0.77
	21	0.78
	22	0.82
	23	0.88
	24	0.85
Close-in weapon system—used for intercepting low-altitude UAV swarms	25	0.88
	26	0.76
	27	0.86
	28	0.89
	29	0.85
Anti-submarine missile—used for intercepting underwater torpedoes	30	0.56
	31	0.58
	32	0.48
	33	0.54
	34	0.42

.

(3)

Equipment maintenance information

Each equipment contains key maintenance information such as space location, type of required spare parts, and maintenance time. The relevant data are listed in

Table A3. Maintenance information of equipment.

Equipment Index	X (m)	Y (m)	Z (m)	Spare Part A (Units)	Spare Part B (Units)	Spare Part C (Units)	Repair Time (s)
0	30.4	46.9	50.2	1	2	5	40.0
1	46.8	46.8	38.4	1	1	6	62.0
2	80.6	46.9	44.6	2	3	4	86.0
3	106.4	28.1	29.0	2	2	5	44.0
4	106.4	61.9	22.7	3	2	4	48.0
5	115.6	61.9	10.5	2	3	3	54.0
6	149.0	30.0	4.9	1	1	3	62.0
7	151.0	69.4	29.9	2	2	2	60.0
8	182.4	75.0	41.6	3	1	4	38.0
9	182.4	61.9	59.8	3	1	4	38.0
10	182.4	5.6	30.6	2	2	3	76.0
11	185.4	66.6	8.9	2	1	3	66.0
12	187.0	71.3	39.9	1	1	2	46.0
13	194.6	66.6	10.1	1	2	2	56.0
14	205.2	62.9	9.9	1	1	2	46.0
15	205.2	66.6	50.5	3	2	3	48.0
16	205.2	59.1	16.6	2	2	3	76.0
17	205.0	59.0	36.1	2	3	3	54.0
18	220.4	5.6	45.1	3	1	2	54.0
19	240.2	69.4	17.9	3	2	2	64.0
20	253.3	26.3	54.4	2	3	3	54.0
21	273.6	61.9	12.2	2	3	3	54.0
22	276.6	61.9	51.1	2	2	2	60.0
23	281.2	61.9	53.2	2	3	4	54.0
24	285.8	61.9	26.1	2	2	5	44.0
25	290.4	15.0	46.6	2	3	5	54.0
26	295.0	15.0	13.4	2	3	2	70.0
27	149.0	46.9	39.1	1	3	5	50.0
28	148.0	38.0	16.5	1	2	3	72.0
29	133.8	13.1	2.5	2	1	3	66.0
30	155.1	13.1	39.1	1	2	3	72.0
31	171.8	13.1	38.4	2	3	3	86.0
32	199.1	5.6	20.0	2	2	3	76.0
33	214.4	13.1	36.1	3	3	3	90.0
34	222.0	5.6	45.1	2	3	3	54.0
35	226.6	13.1	36.7	1	2	3	40.0
36	252.4	5.6	12.2	2	1	2	50.0
37	223.4	66.6	40.0	1	2	2	56.0
38	223.4	63.1	41.4	3	2	3	48.0
39	229.5	66.6	34.7	3	1	3	38.0
40	235.6	70.1	17.8	2	2	2	60.0
41	235.6	63.1	43.9	3	1	4	38.0
42	152.0	69.4	9.5	2	1	2	50.0
43	174.8	60.0	9.4	2	3	4	54.0
44	187.0	60.0	5.9	1	1	3	62.0
45	199.1	60.0	29.5	2	3	3	86.0
46	211.3	60.0	50.0	3	3	3	90.0
47	223.4	60.0	53.9	1	2	4	40.0
48	255.4	67.5	3.4	1	1	4	30.0
49	264.5	18.8	2.1	3	1	3	70.0

of Appendix A.

(4)

Equipment importance

(a)

Definition and distribution of importance

The recovery ability of the system after being attacked depends on the effective repair of key equipment in the maintenance process. No matter which specific recovery strategy is adopted, the repair of the equipment should be carried out according to priority under multiple constraints, and the importance of the equipment is the core guidance.

The Attack-Defense-Recovery Strategy (ADRS) proposed in this paper aims to reduce the damage of the system under multi-wave attacks and emphasizes the priority to repair the key equipment of the operation, so as to maintain the detection and attack ability of the warship. In order to adapt to different combat scenarios, this paper sets the wartime importance level for each equipment to guide the adaptive maintenance sequencing and resource allocation.

The equipment importance is quantified as an integer value of 1 to 6, covering the basic importance and scenario specific weight under different combat backgrounds. In highly task-related scenarios, equipment is rated as highly important (Level 5–6), medium level (3–4) in moderately relevant scenarios, and low importance (Level 1–2). Detailed importance values for each piece of equipment under various attack scenarios are provided in Appendix A 

Table A4. Equipment importance matrix.

Equipment Index	Basic Importance	Aircraft/Missile	Unmanned Aerial Vehicle	Torpedo
0	5	5	5	6
1	4	6	5	5
2	4	6	6	6
3	4	6	6	6
4	5	5	5	5
5	6	6	5	6
6	4	5	5	5
7	6	6	6	5
8	3	5	6	5
9	5	5	6	6
10	3	5	6	1
11	5	6	6	1
12	2	6	5	1
13	3	6	5	2
14	2	5	6	2
15	3	1	2	6
16	3	1	2	6
17	2	2	1	6
18	1	1	1	6
19	2	1	2	5
20	1	6	3	1
21	3	5	2	1
22	3	5	1	2
23	3	6	2	1
24	3	6	3	1
25	3	2	5	1
26	3	2	6	1
27	4	2	6	1
28	2	2	5	1
29	1	3	6	1
30	2	1	2	6
31	2	2	1	6
32	1	1	1	5
33	2	1	1	6
34	1	1	1	6
35	3	2	2	2
36	3	3	2	3
37	5	4	3	4
38	5	1	2	3
39	5	2	2	1
40	4	3	3	2
41	5	2	4	2
42	4	2	2	3
43	4	3	1	2
44	4	3	2	4
45	5	2	1	2
46	6	2	2	4
47	4	3	4	3
48	4	3	2	4
49	4	2	3	1

.

(b)

Equipment wartime importance calculation

Equipment wartime importance is the key basis to measure maintenance priority in the ADRS strategy, and it is also the core parameter of algorithm optimization. This indicator will be dynamically adjusted according to the current combat environment, reflecting the adaptability of each equipment to the combat task. Equipment with high importance indicates that it has strong support for the current situation and helps the system resist enemy attacks. On the contrary, low importance indicates that the equipment has limited role in the current mission or lacks combat capability. The specific calculation rules of equipment wartime importance, ${Wartime\_Z}_i,$ are shown in Formula (3):

$${Wartime\_Z}_i={\displaystyle \frac{\sum _{k=0}^s{Z}_i\left(typ{e}_k\right)}{s}}$$

(3)

where $Z_i$ is the importance matrix of equipment$i$,$ty{pe}_k$ represents the attack type of attack $k$,$\mathrm{a}\mathrm{n}\mathrm{d}s$ is the number of attacks in the current wave.

3.3. Damage Modeling Framework

Following an enemy attack, system equipment suffers damage. To describe the degree of damage sustained by the equipment under each wave of attack, we introduce the system equipment damage level matrix, $E_{Destroy}$.

$$E_{Destroy}=\left(\begin{array}{c}\sum _{i\in S}Destro{y}_{(k,i)}\\ ⋮\\ \sum _{i\in S}Destro{y}_{(m,i)}\end{array}\right)$$

(4)

In the above Formula (4), $Destro{y}_{(k,i)}$ represents the degree of damage caused by the enemy attack $i$ on the equipment $k$,$\mathrm{S}$ represents the collection of enemy attacks that have successfully penetrated in the current wave, $k$ represents the first equipment coordinate of the equipment, and m represents the last equipment coordinate of the equipment.

The destructive effects of the shock waves were evaluated with the explosion impact model, the explosion-shock model, described in Ref. [16]. The overpressure criterion states that when the overpressure of the shock wave reaches a certain value, it will cause certain damage and destruction to the target. Referring to the empirical formula of M.A. Saldovsky, the air shock wave overpressure ΔP is given as Formula (5):

$$∆P=1.1{\displaystyle \frac{1}{R\prime }}+4.3{\displaystyle \frac{1}{{R\prime }^2}}+14{\displaystyle \frac{1}{{R\prime }^3}}$$

(5)

$$R^{\prime }={\displaystyle \frac{R}{\sqrt[3]{W_{TNT}}}}$$

(6)

where $∆P$ is the overpressure value of the air shock wave (105 Pa), $R^{\prime }$is the equivalent distance (m), $R$ is the distance from the equipment to the explosion center (m), and $WTNT$ is the $TNT$ equivalent of the attack weapon (kg). According to the relationship between shock wave overpressure and damage radius, it can be concluded that, under the same explosive yield, the farther away from the explosion center, the less affected the equipment are by the shock wave. Convert the shock wave overpressure $∆P$ experienced by the system equipment into damage level and refine the damage situation of the equipment. Please refer to

Table 1. Damage assessment.

Overpressure ΔP (MPa)	Damage Level	Damage Level
<0.008	0	No damage
0.008~0.02	1	Slight damage (Level I)
0.02~0.13	2	Minor damage (Level II)
0.13~0.74	3	Moderate damage (Level III)
0.74~1.2	4	Severe damage (Level IV)
>1.2	5	Complete destruction

for specific descriptions.

3.4. Recovery Objective Formulation

The optimization objective of the recovery problem is to maximize the total importance of the recovered equipment under multiple constraints. On this basis, in order to meet the priority repair strategy of combat equipment according to the ADRS, different weight calculation measures are taken for the importance of different types of equipment, which can be expressed as Formulas (7) and (8). The optimal solution of the recovery problem is the maintenance path that makes f obtain the maximum value.

$$\mathit{max}\left(f\right)={\sum }_{k=1}^S{\sum }_{i=m}^r{Z}_{ik}$$

(7)

$$Z_{ik}=\left\{\begin{array}{c}1.1\ast Z_{ik}ifi\in PowerPoints\\ 0.9\ast Z_{ik}else\end{array}\right.$$

(8)

where$z_{ik}$ represents the importance of the equipment $i$ repaired by the k-th maintainer, $k$ represents the maintainer ID, $i$ represents the serial number of the equipment, $m$ represents the maintenance start equipment of maintainer $k$, $r$ represents the maintenance end equipment of maintainer $k$, S represents the number of maintainers, and $PowerPoints$ represents the combat equipment in the current environment.

In ant colony optimization (ACO), pheromone-related parameters determine the state-transition probability and pheromone update, thus influencing the algorithm throughout its execution. Consequently, how to set these parameters is a key direction for improving ACO. Traditional empirical tuning suffers from randomness and weak evidential support, making it difficult to compare parameter combinations effectively. To address this, we adopted a three-step procedure to determine the parameters:

(1) Initial settings. Based on the empirical relation $n/m\approx 1.5$ (where $n$ is the number of cities, and $m$ is the number of ants), we determine $m$ and obtain an approximate value of the pheromone constant $Q$ via coarse tuning. This yields the initial configuration $m$= 50 and $Q$= 20.

(2) Grid search for core parameters. Because $\alpha$, $\beta$, and $\rho$ have the greatest impact on performance, we use a grid search to determine their optimal combination. To reduce confounding factors, we construct a single-wave maintenance scenario (multi-wave scenarios feature large variability in battlefield conditions and damaged nodes, making control difficult). The objective function is based on ADRS and is defined for a multi-depot MTSP without returning to the depots (see Formula (7)). The parameter search space is:

$$\alpha \in \left[\mathrm{0,5}\right], \beta \in \left[\mathrm{0.1,3}\right], \rho \in \left[\mathrm{0.1,1}\right], \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}\hspace{0.25em}\mathrm{s}\mathrm{i}\mathrm{z}\mathrm{e}0.1.$$

(3) Result screening and finalization. First, in accordance with the ADRS concept, we retain only solutions whose coverage of maintenance task nodes is at least 80%. We then rank the remaining solutions by the objective value and select the parameter set associated with the best solution. The final parameters are:

$$\alpha =4.0, \beta =0.4, \rho =0.9.$$

In summary, our parameter-setting workflow starts from interpretable empirical initialization and, through systematic grid search under a controlled scenario together with an ADRS-based objective, obtains parameter settings that are more persuasive and reproducible than purely empirical choices.

4. Improved Ant Colony Optimization

Despite the advantages of ant colony optimization (ACO), including its ease of integration, powerful search capability, and notable robustness, the algorithm still suffers from inherent drawbacks that warrant further attention:

(1)

ACO relies too much on parameter settings.

(2)

The equipment transition guidance in the early stage of the algorithm is weak.

(3)

ACO tends to mature prematurely and fall into local optimality.

According to the shortcomings mentioned above, this paper improves the ACO from these three directions:

(1)

Because the parameters related to pheromone control the state transition probability, pheromone increment, etc., and are always present throughout the algorithm. In view of the uncertainty of the hyperparameters of the ACO, this paper adopts the grid search method to determine more reasonable values of $\alpha$, $\beta$, and $\rho$. According to the rules of thumb, it is determined that the value range of $\alpha$ is [0, 5], the value range of $\beta$ is [0.1, 3], the search range of $\rho$ is [0.1, 1], and the step size is 0.1.

Analyze the operation results and determine the parameters. First, in order to comply with the recovery idea of ADRS, all models with no less than 80% combat equipment in maintenance were screened out. Then, the objective functions of the models were sorted, and the initial parameters corresponding to the optimal solution were selected, that is, $\alpha =4.0$, $\beta =0.4$, $\rho =0.9$.

(2)

To solve the problem of weak guidance in the initial stage of the ACO, we analyze from the setting of the initial pheromones and reasonably distribute the initial pheromones to speed up the algorithm. Therefore, in this paper, the “non-uniform initial pheromone” is set at the beginning of the ACO, so that the initial optimization of the ants will have directionality and speed up the convergence. According to the recovery idea of ADRS in Section 2.2, if we want to minimize damage to the system, we should give priority to repairing equipment with combat capability. So that the system can take the initiative in multi-wave operations as soon as possible. Therefore, it is necessary to prioritize the repair of combat equipment in the current combat scenario. The uneven pheromone distribution mode is adopted according to the wartime importance of equipment, which makes the initial pheromone concentration of equipment with higher importance, and increases the transfer probability to equipment with high importance. The calculation formula of equipment non-uniform pheromone concentration matrix is as Formula (9):

$${\tau }_{ij}^0=1+0.1·{\mathrm{W}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\_\mathrm{Z}}_j-{0.1·\mathrm{W}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\_\mathrm{Z}}_i$$

(9)

In Formula (9), ${\tau }_{ij}^0$ represents the initial pheromone concentration from equipment $i$ to equipment $j$, and ${\mathrm{W}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\_\mathrm{Z}}_i$ represents the wartime importance of equipment $i$. The purpose of subtracting ${0.1·\mathrm{W}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\_\mathrm{Z}}_i$ from the formula is to reduce the transition from high-importance equipment to low-importance equipment.

(3)

In view of the problem that the ACO is prone to falling into local optimality, this paper proposes an improved strategy of “classification of starting points within the group”. The core idea of the improvement is still based on ADRS, giving priority to maintenance of combat equipment to ensure the combat capability of the system, reducing the degree of damage in subsequent waves, and thus alleviating maintenance pressure.

The improved “classification of starting points within the group” strategy is specifically described as follows: First, determine the combat-type equipment ($PowerPoints$) that are suitable for the current environment based on the combat scenario. When the ants in the group begin their initial departure, one equipment is randomly selected from the $PowerPoints$ as the starting point. When an ant reaches the stopping condition, but there is still equipment that needs repair, a second ant is dispatched to continue the repair task. The second ant will inherit the taboo list of the first ant’s repaired equipment. Meanwhile, the second ant’s starting equipment is still prioritized to be randomly chosen from the $PowerPoints$. If no $PowerPoints$ are available for selection, meaning all $PowerPoints$ in the current wave have been repaired, then the initial equipment will be randomly selected from the remaining unprocessed equipment. This procedure is followed for subsequent ants in the group, continuing until all damaged equipment is repaired or the maximum number of ants in the group is reached, thereby ending the repair process for the current wave.

(4)

The multi-wave combat process of IACO is described as:

Start

Initialize algorithm parameters and data (e.g., equipment positions, task importance, iteration settings, etc.)

For wave = 1 to totalWaveNum

For each antGroup (1 to totalGroupNum)

Generate ant group population;

For each ant in antGroup:

Inherit taboo list;

Generate initial repair equipment using classification rules;

Compute movement time $t=d/v$ from last wave;

While repair task is not completed:

Select next repair equipment based on state transition table;

If repair constraints are satisfied:

Perform repair;

Update spare parts status and taboo list;

Else:

Generate sub-feasible solution;

End If

End While

End For

Combine sub-feasible solutions into complete feasible solution;

Evaluate solution quality (using Formula (9));

If current solution better than global best:

Update global best;

End If

End For

Update pheromone matrix using all current ant solutions;

If max iteration not reached:

Continue to next iteration;

End If

End For

Output global best repair path;

End

5. Algorithmic Performance Comparison

Experiments were conducted in Python 3.10 (conda) on a Windows 11 workstation equipped with a Ryzen 9 7945HX CPU, 32 GB RAM, and an RTX 4060 GPU.

In order to intuitively display the advantages of ADRS and the improvement of the IACO, this section quantitatively analyzes its advantages by comparing various algorithms under different combat scenarios. The combat scenarios are divided into three categories: torpedo attack, aircraft/missile attack, and UAV group attack. The specific information of the enemy attack has been given in Section 3.1. There are five waves of attacks in different combat scenarios, and each wave contains several enemy attacks of the same type but with different damage and impact positions.

When the first attack wave is over, each equipment in the system will be damaged to different degrees according to the explosive yield and impact position of the attack. After the algorithm calculates the maintenance path, it will resist the next wave of attack. By observing the final objective function values of different algorithms, we can judge whether the IACO can give better maintenance. By observing the damage of the system after the next attack, we can judge whether the maintenance of the previous attack can effectively reduce the damage of the system, or observe whether the overall damage of the system has been reduced in five wave attacks, to determine whether the IACO has optimized the system resilience and whether ADRS is effective in multi wave attacks.

In order to intuitively show the advantages of ADRS and the improvement of the IACO, three groups of comparative experiments are proposed, respectively:

(1)

Multi-wave combat simulation based on generic recovery strategy (GRS) and traditional ant colony algorithm (GRS-ACO )

(2)

Multi-wave combat simulation based on ADRS and traditional ant colony algorithm (ADRS-ACO)

(3)

Multi-wave combat simulation based on ADRS and IACO (ADRS-IACO)

At the beginning of the IACO, each group of ant colony will initialize the pheromone concentration matrix between equipment. Figure 1 shows the heat map of the initial pheromone concentration with uniform distribution, and Figure 2 shows the heat map of the initial pheromone concentration with non-uniform distribution in various combat scenarios.

As shown in Figure 1, the uniformly distributed pheromone concentration matrix results in equal initial equipment transition probabilities. While this approach increases randomness in solution searching, it also delays convergence toward the optimal solution. The non-uniform pheromone concentration matrix with heat map as Figure 2, calculated according to Equation (9), increases the transition probability from equipment with lower importance to equipment with higher importance, guiding ants toward more optimal paths. Simultaneously, it reduces the probability of transitioning from high-importance equipment to lower-importance equipment, thus preventing ants from deviating from optimal choices.

In subsequent combat simulations, the ACO utilizes the uniformly distributed initial pheromone concentration matrix shown in Figure 1, whereas the IACO employs the initial pheromone concentration matrix with non-uniformly distribution depicted in Figure 2.

5.1. Simulation Under Torpedo Attack Scenarios

This subsection simulates the resilience and analysis recovery performance of three comparative groups based on ant colony algorithms under a five-wave torpedo attack scenario.

(1)

Simulation results and analysis under the first wave of torpedo attack

Prior to the enemy’s attack, all equipment within the naval system is fully operational. After the first wave of torpedo attack, the system suffered extensive damage, causing nearly all equipment to fail and leading to a severe maintenance challenge. The final repair sequences for the three comparative groups in this wave are presented in Figure 3.

Figure 3a presents the repair and recovery paths obtained under constraints using the GRS-ACO. The dark green, light blue, red, light green, and yellow lines represent the maintenance routes of the five ants in the group. Among these, the combat equipment associated with the current torpedo combat scenario includes equipment 15 and 17.

Figure 3b presents the repair and recovery paths obtained using the ADRS-ACO. The combat equipment associated with the current torpedo combat scenario includes equipment 33, 18, 34, 17, 16, 32, 31, and 15.

Figure 3c presents the repair and recovery paths obtained using the ADRS-IACO, in which all combat equipment associated with the current torpedo combat scenario are successfully repaired.

Figure 4 presents the convergence process of the objective function during the algorithm iterations of the three comparative groups under the first torpedo attack.

As shown in Figure 4 and

Table 2. Comparison of algorithm convergence, optimization effectiveness, and improvement under the first wave of torpedo attack.

Algorithm	Iterations to Converge	Final Solution	Increase over GRS-ACO	Increase over ADRS-ACO
GRS-ACO	4	79.3	-	-
ADRS-ACO	21	85.9	+6.6 (+8.3%)	-
ADRS-IACO	3	107.0	+27.7 (34.9%)	+21.1 (24.6%)

, the light-blue line represents the iterative process of the objective function of the GRS-ACO, converging to a final solution of 79.3 at the 4th iteration. The light-red line represents the iterative process of the ADRS-ACO, converging to a Final Solution of 85.9 at the 21st iteration. The light-purple line illustrates the iterative process of the ADRS-IACO, converging to a Final Solution of 107 at the 3rd iteration.

(2)

Simulation results and analysis under the second wave of torpedo attack

Figure 5a presents the repair and recovery paths obtained under constraints using the GRS-ACO. The combat equipment associated with the current torpedo combat scenario includes equipment 15, 16, 17, and 18.

Figure 5b presents the repair and recovery paths obtained using the ADRS-ACO. The combat equipment associated with the current torpedo combat scenario includes equipment 17, 15, 19, 16, 34, 18, and 33.

Figure 5c presents the repair and recovery paths obtained using the ADRS-IACO, in which all combat equipment associated with the current torpedo combat scenario are successfully repaired.

Figure 6 presents the convergence processes of the objective function for the three comparative experiments during the current wave.

As shown in Figure 6 and

Table 3. Comparison of algorithm convergence, optimization effectiveness, and improvement under the second wave of torpedo attack.

Algorithm	Iterations to Converge	Final Solution	Increase over GRS-ACO	Increase over ADRS-ACO
GRS-ACO	10	71.2	-	-
ADRS-ACO	17	86.5	+15.3 (+21.5%)	-
ADRS-IACO	2	105.2	+34.0 (+47.8%)	+18.7 (+21.6%)

, the light-blue curve represents the iteration process of the GRS-ACO, whose objective function converges to 71.2 at generation 10. The light-red curve corresponds to the ADRS-ACO, reaching convergence at generation 17 with a value of 86.5. The light-purple curve shows the ADRS-IACO, which converges at generation 2 with an objective-function value of 105.2. These results confirm that ADRS provides effective guidance for optimizing the objective function, and that the IACO, benefiting from a higher initial objective value, achieves superior performance compared with ACO.

(3)

Simulation results and analysis under the third wave of torpedo attack

Figure 7a presents the repair and recovery paths obtained under constraints using the GRS-ACO. The combat equipment associated with the current torpedo combat scenario includes equipment 17 and 19.

Figure 7b presents the repair and recovery paths obtained using the ADRS-ACO, in which all combat equipment associated with the current scenario is successfully repaired.

Figure 7c presents the repair and recovery paths obtained using the ADRS-IACO, also achieving successful repair of all relevant combat equipment.

Figure 8 presents the convergence processes of the objective function for the three comparative groups during the current attack wave.

As shown in Figure 8 and

Table 4. Comparison of algorithm convergence, optimization effectiveness, and improvement under the third wave of torpedo attack.

Algorithm	Iterations to Converge	Final Solution	Increase over GRS-ACO	Increase over ADRS-ACO
GRS-ACO	18	68.5	-	-
ADRS-ACO	2	116.0	+47.5 (+69.3%)	-
ADRS-IACO	0	134.0	+65.5 (+95.6%)	+18 (+15.5%)

, the light-blue curve represents the iteration process of the GRS-ACO, with the objective function converging to 68.5 at generation 18. The light-red curve corresponds to the ADRS-ACO, which converges to 116 at generation 2. The light-purple curve represents the ADRS-IACO, where the objective function reaches 134 from the very beginning. These results clearly demonstrate the superior optimization performance of the ADRS-IACO, including solution and convergence.

In the subsequent fourth and fifth waves of attack, the system based on ADRS remained undamaged due to effective recovery, resulting in relatively light maintenance workloads and allowing the system to gradually recover. In contrast, the GRS-based system, due to previously suboptimal repair strategies, failed to intercept the follow-up enemy attacks. A detailed analysis of system resilience is provided in Section 5.4.

5.2. Simulation Under Aircraft/Missile Attack Scenarios

This subsection simulates the resilience and analysis recovery performance of three comparative groups based on ACO under a five-wave aircraft/missile attack scenario.

(1)

Simulation results and analysis under the first wave of aircraft/missile attacks

Prior to the onset of enemy attacks, all equipment within the naval system was fully operational. Upon the arrival of the first wave of aircraft/missile attacks, the system sustained damage. The damaged equipment assigned for repair in each group are listed below.

Figure 9a presents the repair and recovery paths obtained using the GRS-ACO. The combat equipment associated with the current aircraft/missile combat scenario includes equipment 23, 22, 21, 13, 11, and 14.

Figure 9b presents the repair and recovery paths obtained using the ADRS-ACO, in which all relevant combat equipment is successfully repaired.

Figure 9c presents the repair and recovery paths obtained using the ADRS-IACO, also achieving successful repair of all relevant combat equipment.

Figure 10 presents the convergence processes of the objective function for the three comparative groups under the current attack wave.

As shown in Figure 10 and

Table 5. Comparison of algorithm convergence, optimization effectiveness, and improvement under the first wave of aircraft/missile attack.

Algorithm	Iterations to Converge	Final Solution	Increase over GRS-ACO	Increase over ADRS-ACO
GRS-ACO	30	112.6	-	-
ADRS-ACO	3	112.9	+0.3 (+0.3%)	-
ADRS-IACO	9	131.8	+19.2 (+17.1%)	+18.9 (+16.7%)

, the light-blue curve represents the iteration process of the GRS-ACO, with the objective function reaching 112.6 at generation 30. The light-red curve corresponds to the ADRS-ACO, which converges to 112.9 at generation 3. The light-purple curve represents the ADRS-IACO, with the objective function converging to 131.8 at generation 9. These results clearly indicate that the IACO demonstrates significantly stronger optimization capability than the traditional ACO, enabling the prioritized repair of combat equipment and more effective enhancement of the objective function.

(2)

Simulation results and analysis under the second wave of aircraft/missile attacks

Figure 11a presents the repair and recovery paths obtained using the GRS-ACO, in which all combat equipment is successfully repaired.

Figure 11b presents the repair and recovery paths obtained using the ADRS-ACO, also achieving complete restoration of all combat equipment.

Figure 11c presents the repair and recovery paths obtained using the ADRS-IACO, where all combat equipment associated with the current combat scenario is successfully repaired.

Figure 12 presents the convergence processes of the objective function for the three comparative groups during this attack wave.

As shown in Figure 12 and

Table 6. Comparison of algorithm convergence, optimization effectiveness, and improvement under the second wave of aircraft/missile attack.

Algorithm	Iterations to Converge	Final Solution	Increase over GRS-ACO	Increase over ADRS-ACO
GRS-ACO	22	107.2	-	-
ADRS-ACO	0	131.8	+24.6 (+22.9%)	-
ADRS-IACO	16	147.1	+39.9 (+37.2%)	+15.3 (+11.6%)

, the light-blue curve represents the iteration process of the GRS-ACO, with the objective function reaching 107.2 at generation 22. The light-red curve corresponds to the ADRS-ACO, which reaches 131.8 at the early stage of the iteration. Although it becomes trapped in a local optimum in subsequent iterations, it still maintains a higher objective value compared to the GRS-based model, indicating that ADRS offers improved optimization capability over GRS. The light-purple curve represents the ADRS-IACO, which converges to 147.1 at generation 16, further demonstrating the enhanced optimization capability of the ADRS-based improved recovery system.

In the subsequent third, fourth, and fifth waves of attack, no further damage occurred in any of the three experimental groups, as aircraft/missile attacks were comparatively easier to recover and intercept than torpedoes, and the equipment gradually recovered. A detailed analysis of system resilience is provided in Section 5.4.

5.3. Simulation Under UAV Swarm Attack Scenarios

This subsection simulates the resilience and analysis recovery performance of three comparative groups based on ACO under a five-wave UAV attack scenario.

(1)

Simulation results and analysis under the first wave of UAV swarm attacks

Prior to the onset of enemy attacks, all equipment in the naval system was fully operational. Following the arrival of the first wave of unmanned aerial vehicle (UAV) attacks, the system suffered extensive damage, resulting in the failure of all equipment and presenting a highly challenging repair task.

Figure 13a presents the repair and recovery paths obtained using the GRS-ACO. The combat equipment associated with the current UAV swarm combat scenario includes equipment 14, 12, 11, 13, 27, and 28, totaling six.

Figure 13b presents the repair and recovery paths obtained using the ADRS-ACO. The relevant combat equipment includes equipment 14, 26, 29, 27, 10, and 11, also totaling six.

Figure 13c presents the repair and recovery paths obtained using the ADRS-IACO, in which equipment 27, 10, 29, 11, 14, and 26 are all successfully repaired.

Figure 14 presents the convergence processes of the objective function for the three comparative groups during the current wave of attack.

As shown in Figure 14 and

Table 7. Comparison of algorithm convergence, optimization effectiveness, and improvement under the first wave of UAV swarm attack.

Algorithm	Iterations to Converge	Final Solution	Increase over GRS-ACO	Increase over ADRS-ACO
GRS-ACO	26	112.6	-	-
ADRS-ACO	4	118.8	+6.2 (+5.5%)	-
ADRS-IACO	4	131.4	+18.8 (+16.7%)	+12.6 (+10.6%)

, the light-blue curve represents the iteration process of the GRS-ACO, with the objective function reaching 112.6 at generation 26. The light-red curve corresponds to the ADRS-ACO, which converges to 118.8 at generation 4. The light-purple curve represents the ADRS-IACO, which also converges at generation 4, achieving a higher objective function value of 131.4.

These results clearly demonstrate that the IACO exhibits significantly stronger optimization capability than the ACO, effectively maximizing the objective function while prioritizing the repair of combat equipment.

(2)

Simulation results and analysis under the second wave of UAV swarm attacks

Figure 15a presents the repair and recovery paths calculated using the GRS-based traditional ant colony algorithm, in which the combat equipment relevant to the current UAV swarm combat scenario include equipment 14, 27, 11, 12, and 13, totaling five.

Figure 15b presents the repair and recovery paths derived from the ADRS-based traditional ant colony algorithm, with combat equipment including 29, 10, 14, 11, and 27, also totaling five.

Figure 15c presents the repair and recovery paths computed using the ADRS-based IACO, where the combat equipment includes 29, 14, 11, 27, 26, and 10, totaling six.

These paths reflect the respective algorithms’ ability to prioritize and repair key combat-relevant equipment under the second wave of UAV swarm attacks.

As shown in Figure 16 and

Table 8. Comparison of algorithm convergence, optimization effectiveness, and improvement under the second wave of UAV swarm attack.

Algorithm	Iterations to Converge	Final Solution	Increase over GRS-ACO	Increase over ADRS-ACO
GRS-ACO	12	118.0	0.0	-
ADRS-ACO	24	121.2	+3.2 (+2.7%)	-
ADRS-IACO	5	139.5	+21.5 (+18.2%)	+18.3 (+15%)

, the light-blue curve represents the iteration process of the GRS-ACO, with the objective function reaching 118 at generation 12. The light-red curve corresponds to the ADRS-ACO, which converges to 121.2 at generation 24. Although the objective value remained relatively low during the early stages, it eventually surpassed that of the GRS-ACO, driven by the influence of ADRS in prioritizing equipment with high wartime importance. The light-purple curve represents the ADRS-IACO, which converges at generation 5, with a final objective function value of 139.5.

These results clearly demonstrate that the IACO exhibits significantly enhanced optimization capability compared to the ACO.

In the subsequent third, fourth, and fifth waves of attack, UAV swarm attacks caused only minor damage and were relatively easy to intercept. As the combat equipment gradually recovered, none of the three experimental groups sustained further damage. A detailed analysis of system resilience is provided in Section 5.4.

5.4. System Resilience Evaluation

This section analyzes system resilience from two indicators of system equipment: the overall system effectiveness level and the overall system importance. The resilience indicators undergo corresponding changes due to the process of enemy attacks and friendly repairs during multiple wave operations. The change in system efficiency levels divides each equipment into five efficiency levels, and the impact of damage and maintenance on the indicators is the same for each equipment. The initial total efficiency level of a system with 50 equipment is 250. The total importance of equipment is to treat each equipment differently, and the impact of the loss and recovery of different equipment on the resilience index is determined based on the equipment importance of the current task. Equipment with high importance has a greater impact, while those with low importance have a smaller impact. Regardless of the task, the total importance of all equipment in the system is 171. By analyzing the degree of change in two resilience indicators, the impact of different recovery strategies and algorithms on system resilience is analyzed. The calculation formula for system resilience is shown in Formula (10):

$$R={\displaystyle \frac{{\sum }_{t=0}^{t_d}Recovey\left(t_d\right)}{{\sum }_{t=0}^{t_d}Loss\left(t_d\right)}}$$

(10)

In Formula (10), $Recovey\left(t_d\right)$ denotes the total recovery value of the system at time $t_d$, while ${Loss(t}_d)$ represents the total loss value of the system at the same time.

5.4.1. Torpedo Attack Scenario

Figure 17 depicts the fully elastic process of the system based on the ACO experimental group under five waves of torpedo attacks. A* represents before the *th wave attack, and A*′ represents after the *th wave attack; R* represents before the elastic recovery of the first wave, and R*′ represents after the elastic recovery of the *th wave. From the changes in the indicators of attack A3-A3′ in the third wave of the two graphs, it can be seen that the decrease in the purple and pink lines is smaller than that of the blue line, indicating that ADRS, compared to GRS, can effectively reduce the damage to the system by actively restoring combat nodes. Moreover, the recovery of combat nodes has intercepted subsequent attacks in waves A4 and A5, preventing the system from being damaged. In the fourth wave of maintenance, the system was restored to a complete state, improving its resilience.

Moreover, in the experimental group based on ADRS in the left figure, the purple and pink lines basically overlap. However, in the right figure, the purple line in the maintenance R*-R*′ of each wave is also improved compared to the pink line, indicating that although the IACO and the ACO can repair about the same number of nodes in multi-constraint maintenance tasks, the IACO has stronger optimization ability for the objective function.

To quantify resilience variations during the multi-wave engagement, equipment importance values are substituted into Equation (10). The resulting changes in system resilience for each attack wave are summarized in

Table 9. Comparison of system resilience under torpedo attack scenario.

Experimental Group	Metric	Wave 1	Wave 2	Wave 3	Wave 4	Wave 5
GRS-ACO	Damage	152.0	89.2	75.0	70.0	74.6
	Recovery	78.6	72.8	72.2	73.4	75.0
	Resilience	0.517	0.628	0.707	0.769	0.807
	Resilience Increase	-	-	-	-	-
ADRS-ACO	Damage	157.2	88.0	36.4	0.0	0
	Recovery	79.6	83.2	68.6	50.2	0
	Resilience	0.506	0.664	0.822	1.0	1.0
	Resilience Increase	−2.1%	+5.7%	+16.3%	+30.0%	+23.9%
ADRS-ACO	Damage	157.2	103.8	46.2	0.0	0
	Recovery	98.0	100.4	73.8	35.0	0
	Resilience	0.623	0.76	0.886	1.0	1.0
	Resilience Increase	+20.5%	+21.0%	+25.3%	+30.0%	+23.9%

.

Figure 18 and Table 9 show the comparative results of the system damage, recovery capability, and resilience indicators of three sets of experiments under multi wave torpedo strikes.

① System damage and recovery capability: Compared to GRS-ACO, the two algorithms based on ADRS demonstrated lower system damage values and stronger recovery capabilities in most waves, with particularly notable advantages after the third wave. Specifically, both ADRS-ACO and ADRS-IACO achieved zero system damage after the fourth wave, effectively fending off the fifth wave attack. Among them, ADRS-IACO achieved the highest recovery value of 98.0 in the first wave and zero system damage in the fourth wave, showcasing significantly superior recovery efficiency compared to other algorithm groups. In contrast, GRS-ACO still exhibited significant residual damage in the fourth and fifth waves, with limited recovery effects.

② System resilience and improvement: The system resilience values of the two sets of ADRS algorithms significantly increased with the increase in attack waves, and both achieved complete recovery (Resilience = 1.0) in the fourth wave. Among them, the resilience improvement of ADRS-IACO per wave exceeded 20%, with a maximum of 30.0%, significantly outperforming ACO, indicating its stronger anti-attack capability and recovery stability.

③ Comprehensive comparative analysis: In terms of the total damage value across five wave attacks, the cumulative system importance loss of GRS-ACO reached 460.8, while ADRS-ACO and ADRS-IACO reduced the loss by over 150, respectively, demonstrating significant key node protection effects. In terms of system resilience improvement, ADRS-IACO exhibited a smoother and more continuous optimization trend in each wave, indicating its superior system resilience recovery performance in multi-wave torpedo attack scenarios.

5.4.2. Aircraft/Missile Attack Scenario

Figure 19 depicts the system resilience process of the algorithm experimental group subjected to five waves of aircraft/missile attacks. From the variations in system indicators denoted by A2 to A2′ in the left panel, it is evident that the declines in the purple and pink lines are less pronounced than that of the blue line, suggesting that the ADRS-based system is adept at repairing combat-type nodes. In comparison to GRS, it significantly minimizes system damage and enhances resilience.

In the left figure, the route optimized by the IACO exhibits a less pronounced advantage over the ACO, as indicated by the pink line. This is attributable to the fact that the current attack inflicts less damage compared to torpedo attacks, and intercepting aircraft/missiles poses a lesser challenge than intercepting torpedoes. Consequently, most damaged nodes require restoration to just 2–3 levels of effectiveness, drastically reducing time and spare parts constraints, thereby facilitating easier maintenance tasks. Under equivalent constraints, more nodes can be restored.

However, from the standpoint of system criticality, the advantages of the IACO become apparent in the recovery processes of the second and third waves. This underscores that when restoring a comparable number of nodes, the IACO selects more critical nodes for restoration, thereby maximizing the subsequent safety of the system. This demonstrates that the IACO offers a superior feasible solution compared to the ACO.

A quantitative analysis is performed to assess the variations in resilience indicators of the ship system across multiple combat waves, with node importance incorporated into Equation (10). The resilience fluctuations of the ship system during each combat wave are determined using this equation, as illustrated in

Table 10. Comparison of system resilience under aircraft/missile attack scenarios.

Experimental Group	Metric	Wave 1	Wave 2	Wave 3	Wave 4	Wave 5
GRS-ACO	Damage	107.8	76.2	0.0	0	0
	Recovery	72.6	67.8	42.6	0	0
	Resilience	0.673	0.763	0.995	1.0	1.0
	Resilience Increase	-	-	-	-	-
ADRS-ACO	Damage	107.8	54.0	0.0	0	0
	Recovery	67.8	68.8	25.2	0	0
	Resilience	0.629	0.844	1.0	1.0	1.0
	Resilience Increase	−6.5%	+10.6%	+0.5%	-	-
ADRS-ACO	Damage	107.8	54.8	0.0	0	0
	Recovery	80.0	63.8	18.8	0	0
	Resilience	0.742	0.884	1.0	1.0	1.0
	Resilience Increase	+10.3%	+15.9%	+0.5%	-	-

.

Figure 20 and Table 10 show the comparative results of system damage, recovery capability, and resilience indicators for three sets of experiments under multi-wave aircraft/missile strikes.

① System damage and recovery capability:

Compared to the GRS-ACO, the two strategies based on ADRS demonstrated superior recovery capabilities and accelerated system repair processes in the first two waves. In the first wave, the system damage values for all three algorithms were 107.8. However, ADRS-IACO exhibited the highest recovery value (80.0), significantly outperforming ADRS-ACO (67.8) and GRS-ACO (72.6). In the second wave, the system damage values for the two ADRS groups (54.0 and 54.8, respectively) were notably lower than that of the GRS-ACO group (76.2). After the third wave, all three algorithms achieved zero system damage, effectively fulfilling the defense mission.

② System elasticity and improvement:

The resilience values of the two ADRS algorithms continued to increase with each attack wave, ultimately reaching full recovery (Resilience = 1.0) in the third wave. Specifically, ADRS-IACO achieved a resilience value of 0.742 in the first wave, significantly surpassing the other two groups. It further increased to 0.884 in the second wave, corresponding to resilience increases of 10.3% and 15.9%, respectively. In contrast, ADRS-ACO performed poorly in the first wave, with a resilience decrease in −6.5%, but improved in the second wave, with a resilience increase of 10.6%.

③ Comprehensive comparative analysis:

From the perspectives of recovery speed and resilience enhancement, ADRS-IACO demonstrated the best performance in the first two waves. It achieved a higher level of recovery and stronger system resilience under the same damage conditions, significantly accelerating the system recovery process. Although all three algorithms can ultimately achieve complete recovery, the improved algorithm exhibited greater invulnerability and repair efficiency in the initial stages of high-altitude strikes, such as those involving aircraft or missiles. This further validates its resilience advantage and adaptability in such high-altitude strike environments.

5.4.3. UAV Swarm Attack Scenario

Figure 21 depicts the system resilience process of an experimental group employing the ACO under the assault of a five-wave UAV swarm. Observing the attack on A3 in the left panel, it is evident that systems grounded on ADRS remained undamaged, whereas those relying on GRS incurred substantial damage. This underscores that, after repairing combat nodes, ADRS-based systems can effectively thwart enemy strikes and mitigate system destruction.

In the left panel, the path of the refined ACO exhibits a less pronounced advantage compared to the pink line representing the conventional ACO. This is attributable to the fact that current UAV swarm attacks inflict less damage compared to torpedo attacks, and intercepting UAV swarms poses fewer challenges than intercepting torpedoes. Consequently, most damaged nodes require restoration to just 2–3 performance levels, thereby reducing time and spare part constraints accordingly and rendering maintenance tasks relatively simpler. Under identical constraints, more nodes can be restored. However, from the vantage point of system importance, the benefits of the refined algorithm become apparent in the restoration of nodes during the first and second waves. This demonstrates that when restoring a comparable number of nodes, the refined algorithm selects superior nodes for restoration, thereby maximizing the subsequent safety of the system. This underscores that the refined ACO offers a superior feasible solution compared to its conventional counterpart.

A quantitative analysis of the metric variations within the ship system across multiple combat waves is conducted, incorporating node importance into Equation (10). Utilizing this equation, the elasticity changes in the ship system during each combat wave are determined, as presented in

Table 11. Comparison of system resilience under UAV swarm strikes scenario.

Experimental Group	Metric	Wave 1	Wave 2	Wave 3	Wave 4	Wave 5
GRS-ACO	Damage	102.2	68.8	65.4	0.0	0
	Recovery	71.8	64.6	65.8	34.2	0
	Resilience	0.703	0.798	0.855	1.0	1.0
	Resilience Increase	-	-	-	-	-
ADRS-ACO	Damage	100.8	70.6	0.0	0	0
	Recovery	71.4	64.2	35.8	0	0
	Resilience	0.708	0.791	1.0	1.0	1.0
	Resilience Increase	+0.7%	−0.8%	+17.0%	-	-
ADRS-ACO	Damage	100.8	71.4	0.0	0	0
	Recovery	79.4	70.4	22.4	0	0
	Resilience	0.788	0.87	1.0	1.0	1.0
	Resilience Increase	+12.1%	+9.0%	+17.0%	-	-

.

Figure 22 and Table 11 show the comparative results of system damage, recovery capability, and resilience indicators for three sets of experiments under multi-wave UAV swarm attacks.

① System damage and recovery capability:

In the first three waves, the system damage values of the three algorithms were relatively similar. The system damage value in the first wave was approximately 100, with slight fluctuations observed in the second wave. After the third wave, the ADRS group achieved zero system damage. Additionally, in the first wave, the ADRS-IACO group attained the highest recovery value of 79.4, outperforming GRS-ACO (71.8) and ADRS-ACO (71.4). In the second wave, ADRS-IACO maintained its recovery advantage at 70.4, while the recovery values of algorithms such as GRS-ACO and ADRS-ACO were relatively lower, indicating that the ADRS-IACO exhibits stronger recovery capabilities and efficiency under continuous attacks. All three algorithm groups achieved zero system damage after the third wave, effectively fulfilling their defensive mandates.

② System elasticity and improvement:

As the attack waves progressed, the resilience values of the three systems continued to climb, ultimately achieving full recovery (Resilience = 1.0) in the fourth wave. Notably, ADRS-IACO demonstrated significant resilience improvements in the first three waves, with resilience values of 0.788, 0.870, and 1.0, corresponding to resilience growth rates of 12.1%, 9.0%, and 17.0%, respectively. In contrast, the ADRS-ACO showed a slight decline (−0.8%) in the second wave but experienced a resilience surge (17.0%) in the third wave. Although GRS-ACO exhibited a positive overall trend, its resilience improvement process was relatively gradual.

③ Comprehensive comparative analysis:

From the perspective of resilience and elasticity evolution trends, ADRS-IACO exhibits superior invulnerability and system recovery performance in the context of high-frequency UAV swarm attacks. In the first three waves, it consistently maintained a high recovery value, accompanied by stable and notable elasticity improvements, indicating that the ADRS-IACO can swiftly repair damaged equipment while possessing the resilience to continuously withstand intensive attacks. In comparison, the ACO lagged in terms of elasticity improvements, with significant fluctuations in recovery levels, further underscoring the robustness and adaptability advantages of ADRS-IACO in such attack environments.

5.4.4. Comparison Across the Three Scenarios

As shown in Figure 23 and

Table 12. Comparison of the algorithm across the three scenarios.

Scene	Waves	Solution			Increase in Solution Compared with GRS-ACO		Increase in Solution Compared with ADRS-ACO
		GRS-ACO	ADRS-ACO	ADRS-IACO	ADRS-ACO	ADRS-IACO	ADRS-IACO
Torpedo	First	79.3	85.0	107.0	+7.2%	+34.9%	+25.9%
Aircraft/Missile	First	112.6	112.9	131.8	+0.3%	+17.1%	+16.7%
UAV	First	112.6	118.8	131.4	+5.5%	+16.7%	+10.6%

, the three algorithms exhibit pronounced performance differences across the attack scenarios. ADRS-IACO consistently attains the best solutions, indicating stronger repair-path optimization and faster system recovery. In the torpedo scenario, ADRS-IACO improves solution quality by 34.9% over GRS-ACO and 25.9% over ADRS-ACO; ADRS-ACO itself improves on GRS-ACO by 7.2%. In the aircraft/missile and UAV scenarios, ADRS-IACO yields improvements of 17.1% and 16.7% over GRS-ACO (16.7% and 10.6% over ADRS-ACO), whereas ADRS-ACO provides only marginal gains, especially in the aircraft/missile case (+0.3%). These results suggest that ADRS-IACO preserves the recovery strengths of ADRS while offering superior path optimization and maintenance scheduling, particularly under severe damage conditions.

Overall, ADRS-IACO outperforms both GRS-ACO and ADRS-ACO across scenarios, supporting its promise as a scalable approach to enhancing naval combat system resilience under multi-wave attacks.

The following results can be obtained from

Table 13. Comparison of three algorithms under 10 sets of data in different scenarios.

Scene	Average Value of the Solution			Variance of the Solution			Average Optimal Number of Iterations			Time Required to Reach the Optimal Solution (Seconds)			Recovery Ratio of Damaged Combat Nodes (First Wave)			Recovery the Importance of Critical Nodes (First Wave)
	GRS-ACO	ADRS-ACO	ADRS-IACO	GRS-ACO	ADRS-ACO	ADRS-IACO	GRS-ACO	ADRS-ACO	ADRS-IACO	GRS-ACO	ADRS-ACO	ADRS-IACO	GRS-ACO	ADRS-ACO	ADRS-IACO	GRS-ACO	ADRS-ACO	ADRS-IACO
Torpedo	66.71	72.55	73.40	0.618	0.274	0.158	89.94	10.44	46.62	2.125	0.268	1.217	78.6%	79.6%	98.0%	100.8	96.8	115.7
Aircraft/Missile	56.98	62.49	63.04	38.969	27.233	29.580	79.42	32.45	44.90	1.983	0.732	1.154	72.6%	67.8%	80.0%	137.7	135.4	143.6
UAV	43.49	44.36	44.00	1.081	1.436	1.098	67.50	14.97	50.47	1.415	0.334	1.036	71.8%	71.4%	79.4%	142.3	143.6	150.6

and

Table 14. Comparison of three algorithms under 20 sets of data in different scenarios.

Scene	Average Value of the Solution			Variance of the Solution			Average Optimal Number of Iterations			Time Required to Reach the Optimal Solution			Recovery Ratio of Damaged Combat Nodes (First Wave)			Recovery the Importance of Critical Nodes (First Wave)
	GRS-ACO	ADRS-ACO	ADRS-IACO	GRS-ACO	ADRS-ACO	ADRS-IACO	GRS-ACO	ADRS-ACO	ADRS-IACO	GRS-ACO	ADRS-ACO	ADRS-IACO	GRS-ACO	ADRS-ACO	ADRS-IACO	GRS-ACO	ADRS-ACO	ADRS-IACO
Torpedo	66.29	72.43	73.36	0.510	0.172	0.145	81.38	20.89	51.54	2.004	0.468	1.293	78.9%	79.2%	99.6%	100.2	99.3	116.1
Aircraft/Missile	53.23	61.61	64.97	33.203	32.347	8.040	79.20	24.59	59.48	1.953	0.570	1.407	72.1%	69.3%	80.5%	137.1	135.9	144.1
UAV	42.61	43.69	43.82	2.459	1.894	1.462	49.88	26.73	31.58	0.990	0.517	0.669	72.0%	71.0%	79.1%	142.8	143.9	149.8

:

(1)

Average solution value. Representative results from Torpedo/10-set show a rise from 66.71 (GRS-ACO) to 72.55 (ADRS-ACO) and 73.40 (ADRS-IACO). The same ordering persists with 20 sets (Torpedo: 66.29 → 72.43 → 73.36) and in the Aircraft/Missile and UAV scenarios, confirming that ADRS contributes mission-aligned gains and that IACO further amplifies them.

(2)

Stability (variance). Variance declines markedly once ADRS is introduced and decreases further with IACO. For Torpedo/10-set, variance drops from 0.618 → 0.274 → 0.158 (GRS-ACO → ADRS-ACO → ADRS-IACO). With 20 sets, it is 0.510 → 0.172 → 0.145. Similar trends appear elsewhere (UAV/20-set: 2.459 → 1.894 → 1.462), indicating that the initialization and transfer probability based on non-uniform pheromone concentration matrix stabilize the search solution.

(3)

Efficiency (iterations and time-to-optimum). ADRS-ACO yields the largest efficiency gains versus GRS-ACO (Torpedo/10-set: iterations 89.94 → 10.44, time 2.125 → 0.268). ADRS-IACO requires more time and iteration to reach the best solution than ADRS-ACO (Torpedo/10-set: 46.62 iterations, 1.217 time units) to support broader exploration, yet remains far faster than GRS-ACO—consistent with a deliberate exploration–exploitation re-balance. ADRS-IACO has achieved better solutions than ADRS-ACO at a very small acceptable time cost and is far superior to GRS-ACO in terms of time consumption and solution quality.

(4)

Recovery capability-oriented ratios. We report first-wave outcomes because they most clearly differentiate algorithms on the recovery ratio of damaged combat nodes and the recovered importance of critical nodes. As subsequent waves proceed, repairs approach completion and inter-method differences diminish, reducing statistical discriminability. Consequently, our analysis focuses on first-wave results.

(i)

Recovery ratio of damaged combat nodes (First Wave). Across all scenarios and dataset sizes, ADRS-IACO attains the highest first-wave repair coverage. For example, in Torpedo/10-set, the recovery ratio reaches 98.0% with ADRS-IACO versus 79.6% (ADRS-ACO) and 78.6% (GRS-ACO); with 20 sets, it further rises to 99.6% (vs. 79.2%/78.9%). Consistent—though smaller—advantages appear in Aircraft/Missile and UAV. These gains reflect group-based multi-starts that broaden early feasible coverage while still steering repairs toward militarily valuable nodes.

(ii)

Recovered importance of critical nodes (First Wave). ADRS-IACO also yields the greatest recovery of aggregated importance among critical nodes. In UAV/10-set, the metric improves from 142.3 (GRS-ACO) and 143.6 (ADRS-ACO) to 150.7 with ADRS-IACO; in UAV/20-set, it achieves 142.8 (vs. 143.9/149.8). The pattern indicates that non-uniform initial pheromones and importance-weighted transitions increase the hit rate on the most consequential assets during the earliest recovery window.

(5)

Robustness to dataset size. All qualitative conclusions persist when the dataset doubles from 10 to 20. ADRS-ACO consistently outperforms GRS-ACO, and ADRS-IACO further improves averages and reduces variance while keeping time-to-solution within acceptable bounds.

Across all scenarios and dataset sizes, ADRS-ACO > GRS-ACO and ADRS-IACO > ADRS-ACO for average outcomes, with a monotonic decrease in variance from GRS-ACO to ADRS-ACO to ADRS-IACO. Table 13 and Table 14 thus substantiate the mechanism underlying Table 12: ADRS injects mission priorities and precedence constraints, yielding stable and efficient improvements; IACO then reshapes ACO’s search dynamics—via non-uniform initial pheromones, importance-weighted state transitions, and group-based multi-starts—to (i) further lower variance, (ii) raise average solution quality, and (iii) achieve these benefits within practical time budgets. Hence, ADRS-IACO’s superiority stems from strategy alignment (ADRS) plus enhanced search dynamics (IACO), rather than preferential parameter tuning.

6. Conclusions and Future Work

This article conducts systematic simulation modeling and analysis research centered around the ACO-based multi-wave combat model. Through parameter optimization, construction of a complete simulation process, and comparative experiments conducted in three representative combat environments, the following key conclusions are drawn:

(1)

ACO parameter optimization: The initial parameters α, β, and ρ have a significant impact on the performance of ACO. By combining empirical analysis with grid search, near-optimal parameter values were determined, laying the foundation for improving search efficiency.

(2)

Effectiveness of recovery strategy: Compared to the GRS, the integrated ADRS significantly enhances the system’s offensive and defensive capabilities under various attack scenarios, effectively reducing the cumulative damage to the system under continuous attack waves.

(3)

Algorithm performance comparison: Compared to the ACO, IACO demonstrates stronger global search capabilities, achieving higher-quality solutions and accelerating convergence speed. Although all algorithms tend to converge to local optima, ADRS can effectively guide the optimization direction of the objective function. Furthermore, the IACO starts searching from a higher initial value, outperforming other algorithms overall.

(4)

Synergistic advantages of ADRS and IACO: Compared to GRS-ACO and ADRS-ACO, ADRS-IACO not only minimizes system damage to the greatest extent, but also significantly accelerates the process of system resilience recovery.

(5)

ADRS-IACO demonstrates superior path planning and maintenance scheduling capabilities in high-intensity, high-damage torpedo attack scenarios. It can more effectively identify key maintenance nodes and guide resource allocation reasonably. In terms of search efficiency and recovery decision-making, it significantly outperforms GRS-ACO and ADRS-ACO, fully reflecting its adaptability and system recovery capabilities in extreme battlefield environments.

In summary, under the background of multi-wave combat, the ship combat system adopting ADRS-IACO demonstrates higher system resilience and stronger recovery capability, which is significantly superior to other comparative strategies.

Future work will be promoted on the following aspects:

(1)

Multi-domain joint operations. Extend ADRS-IACO to multi-domain scenarios—undersea, surface, low-altitude, and high-altitude—where heterogeneous weapons operate simultaneously.

(2)

Bilateral attack–defense with game-theoretic strategy. Build a blue–red game in which both sides can attack, defend, and repair; compute equilibrium policies under joint path–maintenance optimization, compare damage–recovery trajectories and strategy stability across countermeasures, and incorporate adaptive opponent models to capture dynamic adversaries.

(3)

Algorithmic extensions. Investigate additional algorithms within the ADRS framework and benchmark their performance against IACO.

(4)

Data-driven calibration and validation. When public real-world datasets become available, calibrate model parameters and refine experiments accordingly, with sensitivity and robustness analyses.