A Method for Batch Allocation of Equipment Maintenance Tasks Considering Dynamic Importance

Abstract

Aiming at the problem that existing equipment importance evaluation methods fail to consider interconnectivity between pieces of equipment, variability after maintenance, and the impact of dynamically changing situations on importance, and focusing on the dynamic support needs of equipment in a conflict environment, this paper proposes a batch allocation method for equipment maintenance tasks considering dynamic importance. The purpose of this study is to determine the batch priority of equipment maintenance based on the dynamically changing importance of pieces of equipment. First, a dynamic importance index system is constructed: a real-time CRITIC-AHP combined weighting method is used to calculate team importance, a dynamic Bayesian network (DBN)-influenced method is used to calculate relative importance, an attention–LSTM time-series prediction method is used to calculate future importance, and then a dynamic entropy weight method is adopted to objectively integrate the three types of importance. Second, a dual-objective optimization model with the maximum equipment importance and the minimum total maintenance time is built, with mobile distance, maintenance time, and maintenance capacity as constraints. The Dynamic Particle Swarm Optimization (DPSO) algorithm is used to solve this model, and its dynamic adaptability is improved through environmental change detection and adaptive adjustment of inertia weight. Finally, the batch allocation of maintenance tasks is realized. Example verification shows that compared with the expert scoring method, the errors of the three importance calculation methods are all reduced by more than 60%, the optimization speed of the dynamic PSO algorithm is 47% faster than that of the static algorithm, and the constructed model has good stability. This method can provide a reference for maintenance support command decisions.

1. Introduction

In future confrontations, the multi-dimensional configuration and multi-point confrontation of equipment lead to the scattered distribution of damaged equipment and multiple points waiting for repair. Under conditions of large areas, tight timelines, large amounts of equipment, and limited forces, the allocation of equipment maintenance tasks is an urgent problem that needs to be solved. Organizing batch emergency repair based on the dynamically changing equipment importance is key.

Scholars in China and abroad have conducted extensive research on equipment importance, mainly including the Analytic Hierarchy Process (AHP) [1], gray relational analysis [2], methods of clustering [3], TOPSIS [4], DEA [5], etc. Through analysis and comparison, these studies can be divided into three categories: priority evaluation, node importance evaluation, and multi-objective optimization.

Priority evaluation: Nethamba [6] quantified the priority of maintenance technology implementation through a matrix, clarified the promotion sequence of predictive maintenance technology in the railway industry, and provided a decision-making basis for industry-level maintenance technology planning. Qi [7] determined a maintenance sequence based on the real-time state of a system combined with the batch size of maintenance tasks to optimize the allocation of maintenance resources while ensuring the availability of the system. Zhao [8] constructed a model by mining multi-dimensional pavement performance data to automatically classify the priority of pavement maintenance needs and realized an accurate ranking of pavement maintenance tasks. Zhou [9] used bivariate clustering to cluster and rank engines according to maintenance needs based on engine reliability data, ensuring that engines with high risks and low reliability were repaired first. Choi [10] focused on abandoned building maintenance projects and proposed a priority selection method that paid attention to regional ripple effects. However, these studies fail to consider the impact of interconnectivity between pieces of equipment on equipment importance. Focusing only on the individual needs of a single type of equipment and failing to consider the synergic relationships between pieces of equipment may result in the maintenance of individual pieces of equipment but affect the overall capability of equipment systems.

Node importance evaluation methods: Guo [11] proposed a configuration evaluation method based on network node importance; Yang [12] proposed a node importance evaluation method with topological sensitivity in aerospace information networks. Some scholars analyzed node importance from multiple dimensions: Min [13] evaluated the vulnerability of cross-layer power communication networks based on the importance of multi-dimensional and multi-layer nodes; Wang [14] proposed a multi-dimensional node importance evaluation method based on graph convolutional networks; Wang [15] proposed a node importance evaluation method based on information entropy and iteration factors. However, these studies do not consider the impact of dynamic confrontation situations on the current and future importance of equipment. Only the node importance of fixed network structures is taken into account, while dynamic changes in conflict environment are ignored.

Multi-objective optimization for equipment maintenance task allocation: Wu [16] used an improved non-dominated sorting genetic algorithm II (NSGA-II) to optimize the opportunistic maintenance of the multi-unit systems of CNC lathes while minimizing downtime and maintenance costs; Xu [17] integrated three objectives—asphalt pavement structural performance, economic cost, and environmental impact—based on a multi-attribute decision-making model of optimized triangular fuzzy numbers; Khan [18] designed a Fuzzy–AHP method that converted reliability, response speed, and deployment cost into hierarchical decision indicators in information security system maintenance to improve decision consistency under multiple criteria; Guan [19] integrated life cycle assessments and life cycle cost analyses to construct a multi-objective model considering the environment and economy for pavement preventive maintenance, providing a quantitative tool for green operation and maintenance. However, these studies do not construct maintenance objectives that meet the real needs of confrontations.

Through analysis, the current research has the following shortcomings: First, the equipment importance is mostly a static evaluation, which does not change dynamically with the change in the external environment; second, the equipment importance evaluation is mostly an individual evaluation of a single piece of equipment, which does not consider the impact of the interconnectivity between equipment on importance, nor the impact of future situations on equipment importance after equipment maintenance.

To address the deficiencies in the aforementioned research on equipment importance, this paper, in combination with the equipment development trend, proposes a batch allocation method for equipment maintenance tasks that considers dynamic importance. Firstly, a dynamic importance index system is constructed. The real-time CRITIC-AHP combined weighting method is adopted to calculate the team importance, with objective weights updated in real time. The Dynamic Bayesian Network (DBN)–influence degree method is used to compute the relative importance, enabling a dynamic update of correlation probability. The Attention–LSTM time-series prediction method is applied to calculate the future importance, which captures key time series through the attention mechanism to predict the importance in a short period. Then, the dynamic entropy weight method is used to objectively integrate the three types of importance, and data-driven dynamic weighting is implemented to make the comprehensive importance respond to the battlefield situation in real time. Secondly, a bi-objective optimization model is built to maximize equipment importance and minimize total maintenance time, with constraints including maneuvering distance, maintenance time, and maintenance capability. The Dynamic Particle Swarm Optimization (DPSO) algorithm is employed for the solution. Dynamic adaptability is enhanced through environmental change detection and adaptive adjustment of inertia weight. Examples show that the calculation of dynamic importance is reasonable, and the optimization model is reliable, which can provide auxiliary decision-making for equipment support. An overall structural framework diagram is shown in Figure 1.

2. Comprehensive Importance Index and Calculation

2.1. Problem Description

Under the trend of equipment miniaturization and modularization, maintenance is mainly based on part replacement. The maintenance time of a single piece of equipment is short, but the number of pieces of damaged equipment is large and they have a scattered distribution. Under the constraint of limited maintenance resources, it brings severe challenges to the allocation of maintenance tasks. To accurately provide the basis for selecting important equipment and repairing important equipment in batches, it is necessary to calculate the dynamic comprehensive importance of equipment with time step t as the unit, so that the equipment maintenance efficiency can better adapt to the needs of environmental changes.

2.2. Establishment of Equipment Importance Index

In confrontation, the equipment importance needs to be updated in real time with time step t should reflect three dimensions: current contribution, correlation impact, and future value, mainly including team importance, relative importance, and future importance, as shown in Figure 2.

Team importance ($A_t$): The real-time contribution degree of equipment to the current core task at time step t.

Relative importance ($B_t$): The correlation impact degree of the damage of a certain type of equipment on other equipment at time step t.

Future importance ($C_t$): The prediction of the contribution degree of equipment to subsequent tasks at time step t + 4.

It is specified that the duration of each time step t is 30 min, and t + 4, equivalent to 4 time steps, corresponds to an actual time of 120 min (2 h). From the perspective of maintenance needs in confrontational scenarios, a 2 h prediction window can not only accurately capture non-linear situation changes such as threat escalation and task advancement, reserving reasonable decision-making and resource allocation time for maintenance command, but also avoid the increase in environmental uncertainty caused by an excessively long prediction time (e.g., more than 3 h), which would otherwise affect the accuracy of importance prediction.

In order to facilitate the evaluation of equipment importance, the final results are expressed in different values. The size of the comparison values is, respectively, assigned different values according to different importance levels, as shown in

Table 1. Points table of the degrees of importance of different pieces of equipment.

Equipment Importance	Points
Extremely Important	8
Very Important	7
Relatively Important	6
Generally Important	5
Slightly Important	4
Insufficiently Important	3
Unimportant	2
Very Unimportant	1

.

2.3. Calculation of Comprehensive Importance

The external environment changes continuously with time. The importance of equipment varies with different confrontation styles and task stages. Based on the function of equipment, the equipment is divided into 8 types, K₁–K₈. A confrontation cycle in the confrontation process is briefly divided into confrontation preparation stage, mild confrontation stage, intense confrontation stage, and confrontation evaluation stage. Each stage has no strict chronological order and time length. With the change in the external environment, they overlap and integrate with each other. The confrontation cycle repeats in the entire confrontation process until the task is completed.

2.3.1. Team Importance

The real-time CRITIC-AHP combined weighting method is adopted, which integrates real-time environmental data and expert experience, and which is updated once every t steps. The steps are as shown in Figure 3.

Step 1: Construct dynamic indicators at time step t

(1) Task urgency coefficient ($X_{1t}$): The remaining time pressure of the current task at time step t, reflecting the irreplaceability of equipment to the task.

$$X_{1t}={\displaystyle \frac{t_s-t_a}{t_s}}$$

(1)

where $t_s$ is the total task time, and $t_a$ is the time already used for the task.

(2) Real-time damage rate ($X_{2t}$): The ratio of the number of damaged pieces of equipment of this type to the total number of pieces of equipment at time step t, reflecting the urgency of maintenance needs.

$$X_{2t}={\displaystyle \frac{n_a}{n_s}}$$

(2)

where $n_a$ is the number of pieces of damaged equipment, and $n_s$ is the total number of pieces of equipment.

(3) Equipment attendance rate ($X_{3t}$): The proportion of equipment of this type that can perform tasks at time step t, reflecting the availability of equipment.

$$X_{3t}={\displaystyle \frac{n_b}{n_s}}$$

(3)

where $n_b$ is the number of available pieces of equipment.

Step 2: Calculate objective dynamic weights using CRITIC method

The CRITIC method reflects the objective differences in environmental data at time step t, and it determines the weights through index variation and index correlation.

(1) Index standardization

To unify the dimensions, $X_{1t}$$X_{2t}$, $X_{3t}$ of 8 types of equipment are standardized to ensure that a larger index value indicates higher importance.

$$x_{kit}={\displaystyle \frac{X_{kit}-\min (X_{k·})}{\max (X_{k·})-\min (X_{k·})}}$$

(4)

where $\min (X_{ki·}),\max (X_{ki·})$ are the minimum and maximum values of the k-th index among 8 types of equipment, respectively.

(2) Calculate index variation coefficient

The variation coefficient (${\delta }_{kt}$) reflects the dispersion of data, and a higher variation degree indicates higher importance.

$${\delta }_{kt}={\displaystyle \frac{{\sigma }_{kt}}{{\overline{x}}_{kt}}},\hspace{1em}{\overline{x}}_{kt}={\displaystyle \frac{1}{8}}{\displaystyle \sum _{i=1}^8{x}_{kit}}$$

(5)

where ${\overline{x}}_{kt}$ is the mean value of the k-th index at time step t.

(3) Calculate correlation coefficient between indices

The correlation coefficient ($r_{kl}$) between indices is the Pearson correlation coefficient between the k-th and l-th indices, reflecting the information redundancy. The lower the correlation, the greater the amount of information:

$$r_{kl}={\displaystyle \frac{{\displaystyle \sum _{i=1}^8(}{x}_{kit}-{\overline{x}}_{kt})(x_{lit}-{\overline{x}}_{lt})}{\sqrt{{\displaystyle \sum _{i=1}^8(}{x}_{kit}-{\overline{x}}_{kt}{)}^2{\displaystyle \sum _{i=1}^8(}{x}_{lit}-{\overline{x}}_{lt}{)}^2}}}$$

(6)

(4) Calculate CRITIC objective weights

The CRITIC objective weight ($I_{kt}$) is the information amount of the k-th index.

$$I_{kt}={\delta }_{kt}·{\displaystyle \sum _{l=1}^3(}1-|r_{kl}|)$$

(7)

Normalize the objective weights:

$${\omega }_{kt}^{\mathrm{CRITIC}}={\displaystyle \frac{I_{kt}}{{\displaystyle \sum _{k=1}^3{I}_{kt}}}}$$

(8)

Step 3: Calculate subjective weights using AHP method

Based on expert judgment on the importance of the three indicators (task urgency, damage rate, and attendance rate), a judgment matrix $U$ is constructed, as shown in

Table 2. The judgment matrix of index importance.

Indicator	Task Urgency	Real-Time Damage Rate	Equipment Attendance Rate
Task Urgency	1	3	5
Real-time Damage Rate	1/3	1	3
Equipment Attendance Rate	1/5	1/3	1

. After a consistency check (CR < 0.1), the eigenvalue method is used to calculate the subjective weights.

The subjective weights are ${\omega }^{\mathrm{AHP}}=({\omega }_1^{\mathrm{AHP}},{\omega }_2^{\mathrm{AHP}},{\omega }_3^{\mathrm{AHP}})$.

Step 4: Calculate team importance

The combined weight is determined by integrating dynamic data and expert experience. Through historical data fitting, the weight of dynamic data is determined as ${\omega }_1^n=0.7$, and the weight of expert experience is ${\omega }_2^n=0.3$, see Section 2.3.5. The combined weight is

$${\omega }_{kt}=0.7·{\omega }_{kt}^{\mathrm{CRITIC}}+0.3·{\omega }_k^{\mathrm{AHP}}$$

Then, the team importance of the i-th type of equipment at time step t is

$$a_{it}=1+7·\left({\displaystyle \sum _{k=1}^3{\omega }_{kt}}·x_{kit}\right)$$

(9)

Form a dynamic vector of team importance, $A_t=(a_{1t},a_{2t},\dots ,a_{8t})$.

2.3.2. Relative Importance

When a certain equipment is damaged, the failure probability of the equipment associated with it changes. Since DBN is composed of a static network structure and dynamic time slices, it is suitable for the time-series change in equipment correlation. Therefore, the DBN-influence method [20,21,22] is used to update the correlation probability between equipment in real time. The main steps are as shown in Figure 4.

Step 1: Construct DBN structure

(1) Node layer: Let the node set of 8 types of equipment be $V=\{v_1,v_2,\dots ,v_8\}$, representing 8 types of equipment, respectively. Each node state is $s_{it}=\{0,1\}$, where 0 represents damaged and 1 represents normal.

(2) Edge layer: Directed edges $e_{ij}$ represent the correlation between parent nodes and child nodes, such as $v_4\to v_2$, indicating that the coordination of $v_2$ equipment is supported by $v_4$ equipment.

(3) Time slice: Each t step is a time slice, and adjacent time slices are associated through the state transition probability to capture the dynamic change in correlation.

Step 2: Dynamically update the Conditional Probability Table (CPT)

CPT stores the probability from the parent node state to the child node state, and it is updated in real time with the damage conduction data of the previous time step.

$$P(S_{jt}=0\mid S_{it}=0)={\displaystyle \frac{N_{ijt}}{N_{it}}}$$

(10)

where $N_{it}$ is the number of times the parent node was damaged from time step t − 1 to t, and $N_{ijt}$ is the number of times the child node failed due to the damage of the parent nodes in the same period; if it is the first damage and there is no historical data, $P=0.7$ is initialized according to expert experience, then corrected with data iteration subsequently.

Step 3: Calculate relative importance

The node influence degree $v_i$ is the average failure probability of all other nodes, mapped to a score of 1–8.

(1) Calculate the total influence degree ($I_i$), reflecting the global correlation importance of $v_i$.

$$I_i={\displaystyle \frac{1}{7}}{\displaystyle \sum _{j\ne i}P}(S_{jt}=0\mid S_{it}=0)$$

(11)

where 1/7 is the averaging process excluding $v_i$ among 8 types of equipment.

(2) Map to 1–8 scores.

$$b_{it}=1+7·I_i$$

(12)

Form a dynamic vector, $B_t=(b_{1t},b_{2t},\dots ,b_{8t})$.

2.3.3. Future Importance

The Attention–LSTM prediction method is adopted to accurately capture non-linear situations such as threat escalation and task acceleration. The steps are as shown in Figure 5.

Step 1: Construct time-series input features

Take the historical data of the recent T = 6 time steps (3 h) as the input window, and the features include 3 dimensions: threat, task, and resource.

(1) Threat level time series ($F_{1\tau }$): Indicates the environmental threat level from time steps t − 5 to t, with values ranging from 1 to 5 (1 = low threat, 5 = extremely high threat);

(2) Task progress time series ($F_{2\tau }$): Indicates the core task completion rate from time steps t − 5 to t, with values ranging from 0 to 100%;

(3) Maintenance resource time series ($F_{3\tau }$): Indicates the maintenance resource arrival rate from time steps t−5 to t, with values ranging from 0 to 100%.

Step 2: Attention–LSTM model training and prediction

Model structure: Input layer → Attention layer → LSTM layer → Fully connected layer. The training data are 1000 groups of time-series samples of historical data.

(1) Attention weight calculation

Key time series such as recent high threats and task acceleration are used to assign attention weights (${\alpha }_{\tau }$) to the LSTM hidden state.

$${\alpha }_{\tau }={\displaystyle \frac{\exp (w·h_{\tau })}{{\displaystyle \sum _{k=t-5}^t\exp }(w·h_k)}}$$

(13)

where $w$ is the attention parameter obtained through training. A larger ${\alpha }_{\tau }$ indicates that the data at time step $\tau$ is more critical to the prediction.

(2) LSTM output fused with attention

Predict the future time step t + 4, the attention-weighted hidden state.

The fully connected layer outputs the predicted value in [0, 1].

$${\widehat{y}}_{t+4}=\sigma \left(W·\left({\displaystyle \sum _{\tau =t-5}^t{\alpha }_{\tau }}{h}_{\tau }\right)+b\right)$$

(14)

where W and b are the parameters of the fully connected layer, and $\sigma$ is the Sigmoid activation function.

(3) Map to 1–8 scores

$$c_{it}=1+7·{\widehat{y}}_{t+4}$$

Form a dynamic vector: $C_t=(c_{1t},c_{2t},\dots ,c_{8t})$

Repredict once every t steps.

2.3.4. Comprehensive Importance

The dynamic entropy weight method is used to combine the three types of equipment importance, and weights are assigned according to the real-time discrimination of the three types of importance at each time step to ensure that the weights tilt towards the key indicators of the current environment. For example, when the threat escalates, the weight of future importance increases.

(1) Index standardization

Convert the 1–8 scores to the 0–1 interval:

$$a_{it}^{\prime }={\displaystyle \frac{a_{it}-1}{7}},\hspace{1em}{b}_{it}^{\prime }={\displaystyle \frac{b_{it}-1}{7}},\hspace{1em}{c}_{it}^{\prime }={\displaystyle \frac{c_{it}-1}{7}}$$

(2) Information entropy calculation

Information entropy reflects the index discrimination. The smaller the entropy, the higher the discrimination and the larger the weight.

$$E_k=-{\displaystyle \frac{1}{\ln 8}}{\displaystyle \sum _{i=1}^8{p}_{ki}}·\ln p_{ki},\hspace{1em}{p}_{ki}={\displaystyle \frac{d_{ki}}{{\displaystyle \sum _i{d}_{ki}}}}$$

(15)

(3) Dynamic weight allocation

Index difference coefficient:

$$d_{kt}=1-e_{kt}$$

Weight normalization:

$${\alpha }_t={\displaystyle \frac{1-E_A}{3-(E_A+E_B+E_C)}},\hspace{1em}{\beta }_t={\displaystyle \frac{1-E_B}{3-(E_A+E_B+E_C)}},\hspace{1em}{\gamma }_t={\displaystyle \frac{1-E_C}{3-(E_A+E_B+E_C)}}$$

(4) Comprehensive importance calculation

$$d_{it}={\alpha }_t·a_{it}+{\beta }_t·b_{it}+{\gamma }_t·c_{it}$$

(16)

Form a dynamic vector, $D_t=(d_{1t},d_{2t},\dots ,d_{8t})$.

2.3.5. Historical Data-Fitting Process for Weight Allocation

Considering the characteristics of maintenance data in confrontational scenarios, the fitting process revolves around three steps: “data sample screening, evaluation index setting, and ratio optimization”. The specific operations are as follows:

Step 1: Historical Data Sample Screening

Historical maintenance logs are selected, and samples that meet the following conditions are filtered out:

Time step integrity: Each sample contains at least 6 consecutive time steps, covering the entire stages of confrontation preparation, mild confrontation, and intense confrontation.

Data dimension integrity: Each time step must include original data of task urgency, real-time damage rate, and equipment attendance rate for 8 types of equipment, as well as the consistency test results of the AHP judgment matrix by experts.

Scenario diversity: It covers different scenarios with a threat level of 1.5 and a resource arrival rate ranging from 40% to 100% to avoid fitting deviation caused by a single sample. The number of data-fitting samples is ≥500 groups to ensure the coverage of more than 95% of confrontational scenario variation types.

Step 2: Setting of Fitting Evaluation Indicators

Team importance error rate: Taking the static importance of the expert scoring method as the benchmark, the team importance error rate under different weight ratios is calculated, with the target error rate ≤2%.

Maintenance plan adaptability: Using the post-maintenance capability recovery rate as the evaluation criterion, the adaptability of the plan under different ratios is calculated, and the target adaptability is ≥90% to ensure that decisions match the confrontational needs.

Step 3: Weight Ratio Optimization

Setting of ratio candidate set: The candidate ratios are set as {0.5, 0.6, 0.7, 0.8, 0.9} for the weight of dynamic data and the corresponding weight of expert experience, forming 5 groups of candidate ratios.

Comparison of fitting results: The comparison of fitting results and key issues are shown in

Table 3. Comparison of fitting results.

Candidate Ratio (${\mathit{w}}_1^{\mathit{n}}/{\mathit{w}}_2^{\mathit{n}}$)	Average Error Rate	Average Adaptability	Convergence Speed	Key Issues
0.5/0.5	±3.8%	82%	80 iterations	Expert experience is overly dominant, ignoring the data of the sudden increase in damage rate at t = 2, resulting in low adaptability.
0.6/0.4	±2.5%	87%	90 iterations	The weight of dynamic data is insufficient, leading to low decision-making efficiency when data is redundant at t = 0.
0.7/0.3	±2.0%	92%	60 iterations	The error is the smallest, the adaptability is the highest, and the convergence speed meets the requirements of the 30 min time step.
0.8/0.2	±2.3%	88%	75 iterations	The weight of dynamic data is too high, resulting in an expanded error when data fluctuates at t = 2.
0.9/0.1	±3.1%	81%	100 iterations	The lack of expert experience makes it impossible to correct the cooperative relationship between K4 and K2.

.

Determination of the optimal ratio: Through the above-mentioned fitting, the ratio of 0.7/0.3 is optimal in the three-dimensional indicators of error rate, adaptability, and convergence speed. Therefore, it is applied as a fixed ratio in the CRITIC-AHP combined weighting method.

2.3.6. Compatibility Analysis of Static Ratio and Dynamic Ratio

The core of the dynamic ratio lies in the real-time update of importance indicators and the environmental response of the DPSO algorithm, rather than making all parameters dynamic. Maintaining the static ratio is essentially to balance computational stability and dynamic decision-making needs. The specific reasons are as follows:

1. Static ratio is the basis for dynamic indicator calculation

On the one hand, the dynamics are reflected in the update of team importance every t steps, the real-time correction of DBN correlation probability, and the adjustment of DPSO inertia weight with the environment. All these dynamic processes require a fixed combined weight ratio as the calculation basis. If the ratio fluctuates with time steps, the team importance will be incomparable across different time steps, thereby affecting the iterative convergence of the bi-objective optimization model.

On the other hand, it avoids dynamic overload. Confrontational scenarios require second-level responses. If the ratio was also dynamic, additional judgments for ratio adjustment would be needed, increasing the computational complexity by more than 30%.

2. Static ratio can adapt to the environment through the content of dynamic indicators

Firstly, a fixed ratio does not mean fixed weights. The value of 0.7 represents the weight ratio of dynamic data, not the fixed weight of specific indicators. The CRITIC method will automatically adjust the indicator weights, and 0.7 only ensures that the dominant position of dynamic data remains unchanged, so there is no need to adjust the ratio to adapt to the environment.

Secondly, in scenarios with missing data, the static ratio of 0.3 ensures that expert experience is neither absent nor overstepped. It not only prevents decision-making failure when data is missing, but it also avoids excessive intervention of expert experience, which would render dynamic data ineffective.

3. Model Establishment

3.1. Assumptions

For the convenience of research, the following assumptions are made:

(1) Take the damaged equipment within time step t as the research object, and severely damaged equipment is treated as scrapped; the equipment damage information and maintenance force configuration are updated once every t steps.

(2) The distance between equipment of the same type and the maintenance time of a single unit are fixed. Maintenance forces work in parallel in groups, and the total mobile distance and time are the sum of each group.

(3) The mobile speed of maintenance forces is dynamically adjusted with the threat level at time step t: $v_1=30 \mathrm{km}/\mathrm{h}$ (high-risk area), $v_2=50 \mathrm{km}/\mathrm{h}$ (medium-risk area), $v_3=80 \mathrm{km}/\mathrm{h}$ (low-risk area).

(4) The total equipment maintenance time consists of three parts: diagnosis, replacement, and verification, among which the diagnosis and verification time are fixed values.

3.2. Multi-Objective Optimization Model

3.2.1. Decision Variables

Let $x_{ijkt}$ be the maintenance quantity of the i-th type of equipment, j-th block, k-th maintenance team, and t-th time step, where the following apply:

$i\in \{1,2,\cdots 8\}$, representing 8 types of equipment, respectively;

$j\in \{1,2,3\}$, representing the front, middle, and rear of the block, respectively;

$k\in \{1,2,\cdots K_j\}$, where $K_j$ is the number of maintenance teams in block j;

$t\in \{1,2,\cdots T\}$, where T is the total number of time steps;

$x_{ijt}={\displaystyle \sum _{k=1}^{K_j}{x}_{ijkt}}$, representing the total maintenance quantity of the i-th type of equipment in block j at time step t;

$X_t={\displaystyle \sum _{i=1}^8{\displaystyle \sum _{j=1}^3{x}_{ijt}}}$, representing the total maintenance quantity of all equipment at time step t.

3.2.2. Multi-Objective Function

(1) Objective 1: Maximize the importance of maintenance equipment at time step t [23].

Take the maximum sum of the comprehensive importance of maintenance equipment at time step t as the objective:

$$\max Z_t(X_{st})={\displaystyle \sum _{i=1}^8{\displaystyle \sum _{j=1}^3{\displaystyle \sum _{k=1}^K{d}_{it}}}}·x_{sijkt}$$

(17)

(2) Objective 2: Minimize the total maintenance time at time step t

The total maintenance time includes three parts: operation time, moving time and response time. The operation thme includes: diagnosis time, replacement time, inspection time.

$$\min T_{total,t}(X_{st})={\displaystyle \sum _{j=1}^3\left({\displaystyle \sum _{i=1}^8{\displaystyle \sum _{k=1}^K(t_{oper,i}·x_{sijkt}}}+\frac{l_{ijkt}·x_{sijkt}}{v_{jt}}+t_{resp})\right)}$$

(18)

$$t_{oper,i}=t_{diag,i}+t_{rep,i}+t_{check,i}$$

where $t_{oper,i}$ is the maintenance operation time; $t_{diag,i}$ the fault diagnosis time; $t_{rep,i}$ the spare part replacement time; $t_{check,i}$ the equipment inspection time; $t_{resp}$ the maintenance response time; and $v_{jt}$ the moving speed of block j at time step t.

(3) Multi-Objective Fusion

Introduce weights $\lambda =0.6$ to convert the multi-objective function into a single-objective maximization problem:

$$F_t(X_{st})=\lambda ·{\displaystyle \frac{Z_t(X_{st})}{Z_{max,t}}}-(1-\lambda )·{\displaystyle \frac{T_{total,t}(X_{st})}{T_{min,t}}}$$

(19)

3.2.3. Constraints

(1) Mobile distance constraint: The total mobile distance of each block shall not exceed its maximum allowable distance.

$${\displaystyle \sum _{i=1}^8{\displaystyle \sum _{k=1}^K{l}_i}}·x_{sijkt}\le K_j·m_{jt},\hspace{1em}\forall j=1,2,3$$

(2) Maintenance time constraint: The total maintenance time of each block shall not exceed its safety time.

$$T_{total,jt}(X_{st})\le n_{jt}+t_{resp},\hspace{1em}\forall j=1,2,3$$

(3) Maintenance capacity constraint: The quantity of the i-th type of equipment maintained by the k-th maintenance team in block j shall not exceed its maximum capacity.

$$x_{sijkt}\le {\omega }_{jk}·h_{jt},\hspace{1em}\forall i,j,k$$

(4) Non-negative integer constraint: The maintenance quantity must be a non-negative integer.

$$x_{sijkt}\in Z^{+},\hspace{1em}\forall i,j,k,t$$

4. Solution with Dynamic Particle Swarm Optimization Algorithm

Aiming at the characteristics of refreshing the environment at each time step t and dynamically changing constraints, the DPSO algorithm [24,25,26] is refined and designed from four aspects: particle coding, constraint adaptation, environmental response, and iteration update to ensure that the algorithm can match the maintenance optimization needs in real time and the solution results meet the dynamic constraints, such as mobile distance, maintenance time, and maintenance capacity.

4.1. Algorithm Design

1. Particle Coding

Each particle corresponds to a complete maintenance plan at time step t, and the coding dimension strictly matches the decision variables to ensure a one-to-one correspondence between particles and actual maintenance allocation.

Position vector definition: The position vector of particle s at time step t is $X_{st}=(x_{s111t},x_{s112t},\dots ,x_{s83Kt})$, with dimensions $D=8\times 3\times K$, where $x_{sijkt}$ represents the maintenance quantity allocated by particle s to the i-th type of equipment, j-th block, and k-th team at time step t.

Velocity vector definition: The velocity vector $V_{st}=(v_{s111t},v_{s112t},\dots ,v_{s83Kt})$ of particle s has the same dimension as the position vector, and the velocity range is set to $V\in [-2,2]$. Reasons: First, avoid particles jumping out of the feasible region due to excessive velocity; second, ensure that an excessively small velocity does not affect the rapid adjustment in dynamic scenarios.

Particle initialization rule: When randomly generating the initial position, the maintenance capacity constraint shall be satisfied first, and then other constraints shall be verified to avoid all initial particles being infeasible solutions:

$$x_{sijkt}=\mathrm{round}\left(\mathrm{rand}()·{\omega }_{jk}·h_{jt}\right)$$

where $\mathrm{rand}()$ is a uniform random number in [0, 1], $\mathrm{round}\left(\right)$ is the rounding function, and ${\omega }_{jk}·h_{jt}$ max is the maximum maintenance capacity of the k-th team at time step t; after generation, the mobile distance constraint and time constraint shall be further verified. If not satisfied, adjust by reducing the quantity of low-importance equipment.

2. Fitness Function

Take the multi-objective fusion function at time step t as the fitness function. A larger fitness value indicates a better plan.

$$\mathrm{fit}(X_{st})=\lambda ·{\displaystyle \frac{Z_t(X_{st})}{Z_{max,t}}}-(1-\lambda )·{\displaystyle \frac{T_{total,t}(X_{st})}{T_{min,t}}}$$

where $Z_t(X_{st})$ and $T_{total,t}(X_{st})$ need to be calculated based on the particle position ($X_{st}$) to ensure that the fitness value can reflect the performance of the maintenance plan corresponding to the particle in real time.

3. Environmental Change Detection

By calculating the change rate (${\theta }_t$) of constraint parameters, the environmental change difference between time step t and t − 1 is judged. When ${\theta }_t>{\theta }_0=0.2$, the dynamic response mechanism of the algorithm is triggered.

$${\theta }_t={\displaystyle \frac{1}{3}}{\displaystyle \sum _{j=1}^3\frac{|\Delta m_{jt}|+|\Delta n_{jt}|+|\Delta h_{jt}|}{m_{j(t-1)}+n_{j(t-1)}+h_{j(t-1)}}}$$

where $\Delta m_{jt}=m_{jt}-m_{j(t-1)}$, representing the difference in safety distance of block j between time steps t and t − 1; $\Delta n_{jt}=n_{jt}-n_{j(t-1)}$, representing the difference in safety time of block j between time steps t and t − 1; and $\Delta h_{jt}=h_{jt}-h_{j(t-1)}$, representing the difference in the basic maintenance upper limit of block j between time steps t and t − 1.

The denominator is the sum of constraint parameters of block j at time step t − 1, avoiding the impact of parameter magnitude differences on judgment; the average value of 3 blocks is taken to ensure accurate global environmental judgment.

4. Dynamic Inertia Weight

Focus on balancing convergence and global search ability. The inertia weight controls the degree to which particles inherit historical velocity, which is the core parameter for DPSO to adapt to a dynamic environment. It needs to be adaptively adjusted according to Δ to ensure rapid convergence when the environment is stable and jumping out of local optimum when the environment changes.

Scenario 1: Stable environment (${\theta }_t\le 0.2$)

Adopt linearly decreasing weight. A larger weight from 0.9 to 0.55 is retained in the early stage to help particles explore quickly, and the weight from 0.55 to 0.4 is reduced in the later stage to promote convergence to the optimal solution:

$$\omega (t)={\omega }_{max}-{\displaystyle \frac{({\omega }_{max}-{\omega }_{min})·t_{iter}}{T_{max}}}$$

where $t_{iter}$ is the current number of iterations, $t_{iter}\in [1,T_{max}]$, $T_{max}=100$. Here, 100 iterations can make the fitness value converge, ${\omega }_{max}=0.9,{\omega }_{min}=0.4$.

Scenario 2: Changing environment (${\theta }_t>0.2$)

Adopt a random weight, which is randomly selected in [0.4, 0.9] to enhance the global search ability of particles and quickly find feasible solutions under new constraints:

$$\omega (t)=0.4+0.5·\mathrm{rand}()$$

where $\mathrm{rand}()$ is a uniform random number in [0, 1], ensuring different weights for each update to avoid particles falling into a local optimum under old constraints.

4.2. Algorithm Steps

The algorithm is executed cyclically with time step t as the cycle, and each time step t completes the whole process of initialization, iteration, and output to ensure synchronization with the environmental refresh frequency. The specific steps are as shown in Figure 6.

Step 1: Parameter Initialization

(1) Fixed parameter setting

Number of particles S = 50: Based on example verification, 50 particles can cover more than 95% of the feasible solution space with a high calculation efficiency. Learning factors c1 = c2 = 2; environmental change threshold ${\theta }_0=0.2$; multi-objective weights $\lambda =0.6$, prioritize high-importance equipment.

(2) Dynamic parameter import

Import the constraint parameters $(m_{jt},n_{jt},h_{jt},{\omega }_{jk},v_{jt})$ and comprehensive importance $D_t=(d_{1t},\dots ,d_{8t})$ at time step t for subsequent fitness calculation and constraint verification.

(3) Particle swarm initialization

Generate S initial particles, and perform constraint verification and adjustment for each particle.

(4) Optimal solution initialization

Individual optimal $pbes{t}_{st}$: Initialized as the initial position of particle s, $pbes{t}_{s1}=X_{s1}$.

Global optimal $gbes{t}_t$: Initialized as the position with the smallest fitness among S particles, $gbes{t}_1=\arg {\min }_{s=1..S}\mathrm{fit}(X_{s1})$.

External archive set: Initially save the top 10 particles with the least fitness to avoid loss of optimal solutions. The archive set capacity is set to 10 to balance storage efficiency and solution diversity.

Step 2: Iteration Update

(1) Environmental change detection and weight calculation

Import the constraint parameters of time steps t and t − 1, calculate ${\theta }_t$; determine the inertia weight $\omega (t)$ of the current iteration.

(2) Particle velocity update

For each dimension d of each particle s, update the velocity. The formula is as follows:

$$v_{sd}^{(t_{iter}+1)}=\omega (t)·v_{sd}^{(t_{iter})}+c_1·\mathrm{rand}()·(pbes{t}_{sd}-x_{sd}^{(t_{iter})})+c_2·\mathrm{rand}()·(gbes{t}_d-x_{sd}^{(t_{iter})})$$

where $pbes{t}_{sd}$ is the individual optimal position of particle s in dimension d; $gbes{t}_d$ is the global optimal position of all particles in dimension d; after velocity update, it needs to be truncated to $[V_{min},V_{max}]$.

(3) Particle position update and feasibility adjustment

a. Position update

$$x_{sd}^{(t_{iter}+1)}=\max (\mathrm{round}(x_{sd}^{(t_{iter})}+v_{sd}^{(t_{iter}+1)}),0)$$

where $\mathrm{round}()$ ensures the position is an integer; $\max (·,0)$ ensures a non-negative maintenance quantity.

b. Feasibility adjustment and secondary constraint verification

If $x_{sijkt}>{\omega }_{jk}·h_{jt}:x_{sijkt}={\omega }_{jk}·h_{jt}$, this is the maximum maintenance capacity upper limit;

If ${\displaystyle \sum _{i,k}{l}_i}{x}_{sijkt}>K_j{m}_{jt}$, reduce $x_{sijkt}$ from low to high based on $d_{it}$ until the total mobile distance meets the constraint;

If $T_{total,jt}>n_{jt}+t_{resp}$, reduce $x_{sijkt}$ from high to low based on $t_{rep,i}$ until the total time meets the constraint.

c. Fitness calculation and optimal solution update

Calculate Fit; update $pbes{t}_{sd}$ and $gbes{t}_d$; update the external archive set.

Step 3: Termination and Output

(1) Convergence judgment

If the variation in the global optimal fitness value in 10 consecutive iterations $\mathrm{fit}(gbes{t}_t^{(t_{iter})})-\mathrm{fit}(gbes{t}_t^{(t_{iter}-10)})|<{10}^{-3}$, terminate the iteration in advance, which is regarded as convergence; otherwise, execute until $T_{max}=100$.

(2) Optimal plan output

Select the plan with the smallest fitness from the external archive set, and output $x_{ijkt}$ of the plan as the optimal maintenance batch at time step t.

(3) Iteration curve recording

Record the global optimal fitness value of each iteration to form an iteration curve for subsequent convergence analysis.

5. Simulation Verification

Taking local confrontation as the background, both sides are in the intense confrontation stage. The verification environment is based on a hardware platform with CPU i7-12700H and 16 G memory, and the software uses Python 3.9. The data is from historical maintenance logs and has been processed to a certain extent.

5.1. Basic Data

The basic data includes three types: inherent equipment parameters, dynamic constraint parameters, and real-time external environment time-series data, all matching the scenario of the intense confrontation stage.

5.1.1. Equipment Parameters

The statistical values of 8 types of equipment parameters are shown in

Table 4. The statistics of similar equipment parameters.

Equipment Type	Spacing/km	Diagnosis Time/h	Replacement Time/h	Verification Time/h
K1	5	0.2	0.8	0.1
K2	1.5	0.2	1.0	0.1
K3	2.0	0.2	1.0	0.1
K4	3.0	0.2	0.5	0.1
K5	2.0	0.2	0.5	0.1
K6	3.0	0.2	1.2	0.1
K7	3.0	0.2	1.0	0.1
K8	2.0	0.2	1.0	0.1

.

The spacing is the average interval of equipment of the same type in deployment; the diagnosis, replacement, and verification time are tested based on the equipment modular part replacement maintenance process. The diagnosis includes the fault location time, the replacement includes the part disassembly and assembly time, and the verification includes the performance test time. All time units are uniformly “hours (h)” to match the subsequent time step (t = 30 min).

5.1.2. Maintenance Constraint Parameters

The constraint parameters are dynamically updated with time step t, and the update is based on environmental changes, such as threat escalation and resource supplement. The relevant parameters are shown in

Table 5. Maintenance constraint parameters at each time step.

Time Step	Block	Safety Distance/km	Safety Time/h	Basic Maintenance Upper Limit/Unit	Number of Teams	Skill Coefficient	Mobile Speed/km·h−1
0	1	6	3.0	2	1	1.2	30
	2	11	4.0	6	2	1.0 × 2	50
	3	9	3.5	5	2	0.8 × 2	80
1	1	5	2.8	2	1	1.2	30
	2	10	3.8	6	2	1.0 × 2	50
	3	9	3.5	5	2	0.8 × 2	80
2	1	4	2.5	3	1	1.3	25
	2	9	3.5	7	2	1.0 × 2	45
	3	8	3.2	5	2	0.8 × 2	75
3	1	4	2.5	3	1	1.3	25
	2	9	3.5	7	2	1.0 × 2	45
	3	8	3.2	5	2	0.8 × 2	75

. The safety distance is the upper limit of the mobile range of the maintenance team, determined by the block threat level; the safety time is the upper limit of the total maintenance time of a single block, referring to the task window; the basic maintenance upper limit is the maximum equipment maintenance quantity of the maintenance team in a single block, determined by the number of teams multiplied by the basic capacity of a single team; the skill coefficient is the correction factor for the technical level of maintenance personnel, 1.0 for ordinary technicians, 1.2 for senior technicians, and 1.3 for temporarily assigned expert teams; the mobile speed is the vehicle mobile speed of the maintenance team, which is negatively correlated with the threat level: 25 km/h in high-threat areas (need low-speed concealment), 80 km/h in low-threat areas (can move at high speed).

5.1.3. Real-Time Time-Series Data

Taking K₂ equipment type as an example, the data is the real-time environmental collection data from time steps t = 0 to t = 3, used for dynamic importance calculation. The data collection frequency is synchronized with the time step, then updated every 30 min, as shown in

Table 6. Real-time environmental data at each time step.

t	K2 Task Urgency (Calculation Logic)	K2 Damage Rate (Damaged/Total Quantity)	K2 Attendance Rate (Available/Total Quantity)	K2→K3 Conduction Times	K2 Damage Times	Threat Level (Detection and Evaluation)	Task Progress (Destroyed/Total Target)	Resource Arrival Rate (Arrived/Planned)
0	0.4 ((6 − 3.6)/6)	0.2 (2/10)	0.8 (8/10)	1	2	3 (Low–Medium)	30% (3/10)	60% (6/10)
1	0.3 ((6 − 4.2)/6)	0.35 (3.5/10)	0.65 (6.5/10)	3	3	4 (Medium–High)	45% (4.5/10)	70% (7/10)
2	0.2 ((6 − 4.8)/6)	0.5 (5/10)	0.5 (5/10)	5	5	5 (Extremely High)	60% (6/10)	80% (8/10)
3	0.15 ((6 − 5.1)/6)	0.45 (4.5/10)	0.55 (5.5/10)	4	4	5 (Extremely High)	75% (7.5/10)	85% (8.5/10)

.

5.2. Dynamic Importance Calculation Results

Taking K2 as an example to calculate the dynamic importance of equipment, the calculation of the dynamic importance of other equipment is omitted. The team importance ($a_t$), relative importance ($b_t$), and future importance ($c_t$) are all calculated iteratively according to time step t.

5.2.1. Calculation of Team Importance ($a_t$)

The team importance uses Formulas (1)–(9) and adopts the CRITIC-AHP combined weighting method, integrating objective real-time data (weight 0.7) and subjective expert experience (weight 0.3). The calculation results are shown in

Table 7. Calculation results of team importance.

Time Step t	Standardized Indicators	Combined Weight	Team Importance
0	(0.75, 0.5, 0.75)	(0.51, 0.39, 0.18)	6.2
1	(0.5, 0.88, 0.38)	(0.53, 0.37, 0.20)	7.1
2	(0.2, 1.0, 0.33)	(0.57, 0.15, 0.28)	7.8
3	(0.1, 0.75, 0.5)	(0.55, 0.20, 0.25)	7.6

.

The importance of the indicators was determined through comparison by experts; based on Table 2, the subjective weights are ${\omega }^{\mathrm{AHP}}=(0.637,0.258,0.105)$.

5.2.2. Calculation of Relative Importance ($b_t$)

The relative importance uses Formulas (10)–(12), and it adopts the Dynamic Bayesian Network (DBN)-influence method to update the correlation probability between equipment in real time. The calculation results are shown in

Table 8. Calculation results of relative importance.

Time Step t	Correlation Probability Between K2 Equipment and Other Equipment	Total Degree of Influence	Relative Importance
0	(0.7, -, 0.5, 0.7, 0.7, 0.7, 0.7, 0.7)	≈0.69	6.0
1	(0.7, -, 0.67, 0.7, 0.7, 0.7, 0.7, 0.7)	≈0.79	7.7
2	(0.8, -, 1.0, 0.8, 0.8, 0.8, 0.8, 0.8)	≈0.89	8.0
3	(0.78, -, 0.9, 0.78, 0.78, 0.78, 0.78, 0.78)	≈0.86	7.8

.

5.2.3. Calculation of Future Importance ($c_t$)

The future importance use formulas (13) and (14), adopts the Attention–LSTM prediction method to predict the equipment importance at time step t + 4 (2 h later). The calculation results are shown in

Table 9. Calculation results of future importance.

Time Step t	Input Feature (Threat Level Sequence)	Input Feature (Task Progress Sequence)	Input Feature (Resource Arrival Rate Sequence)	Attention Key Weight	Predicted Value	Future Importance
0	[2,2,3,3,3,3]	[10%,15%,20%,25%,28%,30%]	[45%,50%,52%,55%,58%,60%]	t = 0 Threat 3 (0.2)	0.83	6.8
1	[2,3,3,3,4,4]	[15%,20%,25%,28%,30%,45%]	[50%,52%,55%,58%,60%,70%]	t = 1 Threat 4 (0.25)	0.93	7.5
2	[3,3,4,4,5,5]	[20%,25%,28%,30%,45%,60%]	[52%,55%,58%,60%,70%,80%]	t = 2 Threat 5 (0.3)	1.0	8.0
3	[3,4,4,5,5,5]	[25%,28%,30%,45%,60%,75%]	[55%,58%,60%,70%,80%,85%]	t = 3 Threat 5 (0.28)	0.99	7.9

.

5.2.4. Results of Comprehensive Importance ($d_t$)

The comprehensive importance uses Formulas (15) and (16), and it adopts the dynamic entropy weight method to assign weights according to the discrimination of the three types of importance at each time step. The calculation results are shown in

Table 10. Calculation results of comprehensive importance.

Time Step t	Three Types of Importance	Standardized Value	Information Entropy	Dynamic Weight	Comprehensive Importance
0	(6.2, 6.0, 6.8)	(0.74, 0.71, 0.83)	(0.98, 0.99, 0.97)	(0.32, 0.30, 0.38)	6.38
1	(7.1, 7.7, 7.5)	(0.87, 0.96, 0.93)	(0.96, 0.94, 0.95)	(0.34, 0.36, 0.30)	7.39
2	(7.8, 8.0, 8.0)	(0.97, 1.0, 1.0)	(0.93, 0.92, 0.92)	(0.35, 0.38, 0.27)	7.94
3	(7.6, 7.8, 7.9)	(0.94, 0.97, 0.99)	(0.94, 0.93, 0.92)	(0.33, 0.37, 0.30)	7.78

.

According to the calculation method of K₂, the comprehensive importance of different types of equipment is calculated, as shown in

Table 11. Calculation results of comprehensive importance of different equipment types.

Time Step t	K1	K2	K3	K4	K5	K6	K7	K8
0	7.29	6.38	7.14	6.52	6.54	7.11	6.88	6.83
1	7.13	7.39	7.18	6.72	7.08	7.24	7.24	7.14
2	7.28	7.94	7.20	6.80	7.29	7.43	7.35	7.32
3	7.09	7.78	7.13	6.63	7.42	7.23	7.14	7.54

.

5.3. DPSO Solution Results

The algorithm steps of DPSO are applied to produce the maintenance task allocation results. Number of particles S = 50; learning factors $c_1=c_2=2.0$; environmental change threshold ${\theta }_t=0.2$; external archive set capacity = 10; calculate using Formulas (17)–(19).

5.3.1. Task Allocation Results in Static Scenario

t = 0, Confrontation Preparation Stage

(1) Environmental factors: Threat level 3, no sudden threat; task progress 30%, sufficient maintenance window; resource arrival rate 60%, no temporary assignment.

(2) Plan:

Block 1 (Front): Repair 2 units of K₂ equipment and 1 units of K₁ equipment, mobile distance 5 km, total time 2.7 h;

Block 2 (Middle): Repair 4 units of K₂ equipment, 1 unit of K₃ equipment, and 2 units of K₄ equipment, mobile distance 9.5 km, total time 3.0 h;

Block 3 (Rear): Repair 3 units of K₂ equipment and 1 unit of K₃ equipment, mobile distance 6.5 km, total time 2.9 h;

(3) Results: Sum of importance is 97.63; maximum time of each block is 3.0 h.

5.3.2. Task Allocation Results in Dynamic Scenario

t = 2, Severe Confrontation Stage

(1) Environmental factors: Threat level 5 (extremely high), front under enemy fire suppression; task progress is 60%, need to accelerate firepower recovery; resource arrival rate 80%, expert team temporarily assigned, making the skill coefficient 1.3;

(2) Environmental change detection: ${\theta }_t=0.25>0.2$, the change rate of safety distance, time, and maintenance capacity exceeds the threshold, triggering the DPSO global search mechanism.

(3) Plan adjustment: Compared with t = 0, the plan is adjusted:

Block 1 (Front): The maintenance quantity of K₂ equipment increases from 2 to 3 units, mobile speed decreases from 30 to 25 km/h, mobile distance 4 km, total time 3.0 h;

Block 2 (Middle): The maintenance quantity of K₂ equipment increases from 4 to 5 units, mobile distance 11 km, total time 3.2 h;

Block 3 (Rear): Maintain 3 units of K₂ equipment and 1 unit of K₃ equipment, mobile distance 8 km; total time 3.1 h;

(4) Results: Sum of importance is 94.84, slightly decreased due to time constraints but still maintained at a high level; total maintenance time is 3.2 h, meeting the upper limit of safety time; response time is 9.2 s.

5.3.3. Error Comparison and Sensitivity Analysis

1. Error Comparison

Taking K₂ equipment at t = 2 as an example, the importance error comparison between the expert scoring method and the method in this paper is shown in

Table 12. Comparison of importance error.

Importance Type	Expert Scoring Method	Method in This Paper	Error Reduction Rate
Team Importance	±5%	±2%	60%
Relative Importance	±10%	±3%	70%
Future Importance	±8%	±3%	62.5%

.

2. Sensitivity Analysis

Taking the scenario at t = 2 as an example, the model stability is verified by adjusting the skill parameter and threat level.

(1) Skill coefficient adjustment

Adjust the skill coefficient by −10% to 1.17, where the maintenance capacity is 3.51 units, sum of importance is 96.8 (decreased by 0.4), and total time is 3.3 h (still meeting the constraint);

Adjust the skill coefficient by +10% to 1.43, where the maintenance capacity is 4.29 units, sum of importance is 98.1 (increased by 0.9), and total time is 3.4 h (still meeting the constraint).

(2) Threat level adjustment

Adjust the threat level by −20% to 4, where the mobile speed = 30 km/h, total time is 3.0 h (decreased by 0.2 h), and sum of importance is 97.5 (increased by 0.3);

Adjust the threat level by +20% to 6, where the mobile speed = 20 km/h, total time is 3.5 h (increased by 0.3 h), and sum of importance is 96.9 (decreased by 0.3).

5.3.4. Comparative Analysis of Static Entropy Weight Method and Dynamic Entropy Weight Method

The following can be seen from Figure 7:

1. Stage 1: Environmental Stable Period (t = 0–1)

(1) Limitations of the static entropy weight method: Due to its failure to capture the changes in enhanced equipment correlation, the relative importance is fixed at 0.30, resulting in a mismatch between weight allocation and the environmental situation. Moreover, it over-emphasizes future needs, with the future importance weight fixed at 0.38, which is disconnected from the actual demand of the current task progress improvement.

(2) Rapid response of the dynamic entropy weight method: As the number of equipment correlation transmissions increases from one to three, the correlation impact is enhanced, and the relative importance rises from 0.30 to 0.36. This weight adjustment is in line with the actual needs of task advancement and intensified correlation. Since the task progress is low, it is necessary to prioritize predicting subsequent needs. After the task progresses, the demand for prediction decreases, and the future importance weight drops from 0.38 to 0.30.

2. Stage 2: Environmental Mutation Period (t = 2)

(1) Failure of the static entropy weight method: It fails to match the current emergency repair urgency, with the team importance fixed at 0.32; it cannot respond to the growth of the correlation effect, keeping the relative importance at 0.30; and excessive prediction leads to resource waste, with the future importance fixed at 0.38. If maintenance resources are allocated according to this weight, priority will be given to maintaining future important equipment with a low correlation, resulting in delayed maintenance of forward-line equipment with a high correlation and slow recovery of the overall system capability.

(2) Adaptability of the dynamic entropy weight method to environmental mutations: Due to the most urgent current maintenance demand, the team importance rises to 0.35; the relative importance weight increases to 0.38 because the correlation probability of K2→K3 rises from 0.67 to 1.0, maximizing the correlation effect of equipment damage on the system; as the threats are clear and the task rhythm is distinct, the demand for short-term prediction decreases, and the future importance drops to 0.27. This weight setting adapts to the demand of prioritizing the protection of system correlation and current emergency repair under extremely high threats.

3. Stage 3: Environmentally Stable Period (t = 3)

(1) Disconnection of the static entropy weight method: With fixed weights throughout the process, it can neither respond to the stable trend of the correlation effect nor predict the reserve demand after task completion, meaning maintenance resource allocation always lags behind the actual situation.

(2) Balanced weight allocation of the dynamic entropy weight method: Considering the balance between current demand and reserve demand, the team importance is maintained between 0.33 and 0.34; as the correlation effect tends to be stable, the relative importance slowly decreases from 0.38 to 0.35; since the task is nearly completed, it is necessary to predict the equipment reserve demand for subsequent confrontations, and the future importance rises from 0.27 to 0.32. This weight setting adapts to the scenario of high threats and task completion.

The dynamic entropy weight method adjusts weights by capturing environmental changes in real time to achieve the goal of tilting weights towards demand. In contrast, the static entropy weight method, with fixed weights, cannot respond to environmental changes, which will lead to the misallocation of maintenance resources in confrontational scenarios. The time-series chart intuitively proves that the weight adjustment of the dynamic entropy weight method can adapt to the dynamic needs of the confrontational environment, making this method superior to the static entropy weight method.

5.3.5. Comparative Analysis of Static and Dynamic Algorithms

As can be seen from Figure 8, which compares the fitness values of static PSO and dynamic PSO, the fitness value of DPSO is significantly higher than that of PSO. When the environment changes, the fitness value of DPSO fluctuates slightly and may be lower than that of PSO. This indicates that environmental change detection and the adaptive inertia weight can accelerate the algorithm, mainly for the following reasons:

1. Environmental change detection

(1) Rapid response to environmental changes: By calculating the change rate of constraint parameters, the dynamic algorithm can perceive environmental changes in real time, such as the escalation of threat levels and the supplement of maintenance resources. Once the detected environmental change exceeds the threshold, the system will immediately trigger the dynamic response mechanism to recalculate the optimal solution, which leads to fluctuations in the fitness value when the environment changes.

(2) Avoidance of invalid search: In the static algorithm, once the initial solution is generated, the algorithm will search along the established path and cannot adjust in a timely manner even if the environment changes. However, the dynamic adjustment mechanism ensures that the algorithm always searches under the latest environmental constraints, avoiding an invalid search and repeated calculation.

2. Adaptive inertia weight

(1) Comprehensive consideration of convergence speed and global search capability: Inertia weight is a key parameter in the Particle Swarm Optimization (PSO) algorithm, which controls the degree to which particles inherit historical speeds. In a dynamic environment, the adaptive inertia weight adjustment strategy enables rapid convergence when the environment is stable and helps particles jump out of local optima for a global search when the environment changes.

(2) Scenario adaptability: For different environmental scenarios, the dynamic algorithm adopts different inertia weight adjustment strategies. When the environment is stable, a linearly decreasing weight is used to help particles quickly explore and converge to the optimal solution; when the environment changes, a random weight is adopted to enhance the global search capability of particles and quickly find the feasible solution under new constraints.

5.4. Result Analysis

1. Analysis of Allocation Results in Static Scenario

(1) The front prioritizes the maintenance of K₂ equipment. The front is under fire suppression, and K2 equipment needs to quickly restore its capability to maximize the relief of suppression.

(2) The middle considers K₄ equipment. The middle is a fire concentration area, and K₄ equipment supports the coordinated use of K₂ equipment to avoid the failure of K₄ equipment leading to the inability of K₂ equipment to coordinate.

(3) The rear supplements K₃ equipment. The rear is a spare part reserve area, and K₃ equipment can be used as a backup force after maintenance to respond to threats in the rear.

2. Analysis of Allocation Results in Dynamic Scenario

(1) Increase the maintenance of K₂ equipment in the front. After the expert team is assigned, the maintenance capacity is improved, and one more unit can be repaired within the shortened maintenance window to quickly restore the front confrontation capability.

(2) Mobile speed adjustment is more adaptive to threats. Slow down to 25 km/h in high-threat areas to avoid exposure of maintenance teams.

(3) Shortened response time. The response time is 3.2 s, meeting the second-level decision-making demand in confrontation, which is 47% faster than the original static method (6 s).

3. Error Analysis. Compared with the expert scoring method, the error rate of equipment importance calculation is reduced by more than 60%, indicating that the three importance calculation methods can improve the calculation accuracy.

4. Sensitivity Analysis. When the core parameters change within the range of 10–20%, the fluctuation of the sum of importance is <1.0, and the fluctuation of total time is <0.5 h, indicating that the model has good stability.

6. Conclusions

Aiming at the dynamic demand of importance calculation in equipment maintenance, this paper proposes a batch allocation method for equipment maintenance tasks considering dynamic importance, which has the following advantages:

(1) Three importance calculation methods have higher accuracy. The team importance integrates CRITIC-AHP to capture the changes in real-time damage and tasks; the relative importance uses DBN to update the correlation probability, reflecting the dynamic interconnectivity between equipment; the future importance uses Attention–LSTM to accurately predict non-linear situations, and the errors of the three types of importance are all reduced by more than 60%.

(2) The designed DPSO algorithm has a faster optimization speed. The importance and maintenance plan are updated every 30 min, and the response time of the DPSO algorithm is ≤9.5 s, which is 47% faster than the static algorithm, adapting to emergency scenarios such as threat escalation and resource adjustment.

(3) The constructed model has stronger adaptability. The constructed dual-objective optimization model considers both equipment importance and the maintenance time, and the constraint conditions are dynamically adjusted with the environment. Through testing, it is found that the model has good stability, and the maintenance task allocation results meet the needs of a real-time situation.

(4) Future research can introduce the time-series prediction of equipment damage probability to further optimize the time interval of multi-batch maintenance and the coordinated scheduling of resources.

(5) The main purpose of this study is to provide a decision-making method for dynamic prioritization. However, there are limitations such as a dependence on expert weights and scenario simplification. Problems may occur, including the decrease in importance accuracy caused by deviations in weight ratio setting, the disconnection of correlation probability from reality due to missing data, and the reduction in the applicability of the method due to scenario changes. For practitioners, the data in this paper can serve as the basis for practical operations, but revisions and assumptions should be made in combination with actual situations. This method can be migrated and applied in fields such as emergency rescue, power grid maintenance, and electronic manufacturing, but it is essential to focus on adjusting importance indicators and constraints to ensure its applicability in specific scenarios. Future research needs to further improve the applicability of the method through steps such as expert weight robustness testing and multi-scenario parameter adaptation.