A comprehensive risk prediction method for defense mission planning based on probabilistic reasoning and hierarchical analysis

3.1. Analysis of the correlation between the link tasks

Defense task planning needs to strictly follow the chronological order of the specific measures taken by each link to gradually refine; when a subsequent link task depends on a preceding task, there is a hierarchical relationship between them^[18]; when multiple tasks converge into a single task, they share a synergistic relationship; when there is no intrinsic link between the link tasks, they have an independent relationship^[19]. The specific description is as follows:

(1) Hierarchical relationship: the correlation between link tasks $$ {{R}_{i}} $$ and $$ {{R}_{j}} $$ is submissive transfer, and $$ {{R}_{j}} $$ needs to depend on the information passed in one direction by $$ {{R}_{i}} $$ to execute, noted as HR.

(2) Synergy relationship: link tasks $$ {{R}_{i}} $$ and $$ {{R}_{j}} $$ act synergistically on link task $$ {{R}_{k}} $$; then, the correlation between $$ {{R}_{i}} $$, $$ {{R}_{j}} $$ and $$ {{R}_{k}} $$ is called synergy relationship, and $$ {{R}_{k}} $$ needs to depend on the output information of both $$ {{R}_{i}} $$ and $$ {{R}_{j}} $$ to execute, denoted as SR.

(3) Independent relationship: there is no interaction of information between link tasks $$ {{R}_{i}} $$ and $$ {{R}_{j}} $$; there is no dependency between them, and each can execute independently, which is recorded as IR.

In principle, the three types of relationships mentioned are unique: only one type can exist between tasks of the same link. Meanwhile, given the sequential execution of the link tasks and the gradual elaboration of the planning content, all these relationships are unidirectional, from previous to subsequent link tasks.

The schematic diagram of the correlation between the link tasks is depicted in Figure 2. The colors represent the specific link in which each task is located, such as the hierarchical relationship between $$ {{R}_{1}} $$ and $$ {{R}_{2}} $$, $$ {{R}_{3}} $$, the synergistic relationship between $$ {{R}_{4}} $$, $$ {{R}_{5}} $$ and $$ {{R}_{6}} $$, and no obvious relationship between $$ {{R}_{3}} $$, $$ {{R}_{4}} $$ and $$ {{R}_{8}} $$.

Figure 2. Schematic diagram of the correlation between the link tasks

3.2. Measurement of the degree of correlation between link tasks

This study uses the degree of association to measure the relationship between link tasks. Subsequently, we determine this degree from the nature of various association relationships.

(1) Calculation of HR relevance

Since the information between the tasks of each link in HR has one-way transferability and the correlation between tasks increases with the similarity of the situational information affecting them, the number of basic situational information inputs to any two link tasks is analyzed using the ensemble similarity measure function^[20–22], which measures the HR correlation between the link tasks, expressed as

where $$ SI(g) $$ is the basic situation information input when performing the g-th session task, $$ \left\{ SI(g) \right\} $$ is the number of situation posture information input when performing the g-th session task, and $$ \left\{ SI({{R}_{i}})\cap SI({{R}_{j}}) \right\} $$ denotes the number of intersections of $$ SI({{R}_{i}}) $$ and $$ SI({{R}_{j}}) $$.

(2) SR relevance calculation

Unlike HR, SR is more inclined to the joint impact of the execution effect of two or more link tasks on another. Fuzzy hierarchical analysis^[23–25] is considered to measure the effect factor of each link task on its acted task, that is, to calculate the SR correlation between link tasks from a functional perspective.

Assuming that a total of $$ n $$ session tasks have a common effect on $$ {{R}_{k}} $$ when a defense task is planned, the SR correlation between the i-th session task $$ {{R}_{i}} $$ and the k-th session task $$ {{R}_{k}} $$ ($$ i\in [1, n] $$, $$ k\in [1, n]\left( k\ne i \right) $$) is

where $$ \operatorname{Im}({{R}_{i}}\to {{R}_{k}}) $$ denotes the action factor of the $$ i $$-th link task $$ {{R}_{\text{i}}} $$ on $$ {{R}_{k}} $$, which needs to be inferred using fuzzy hierarchical analysis to derive the extent to which each of the $$ n $$ link tasks acts on $$ {{R}_{k}} $$.

(3) IR relevance calculation

When the link task $$ {{R}_{i}} $$ is related to $$ {{R}_{j}} $$ as IR, there is no interdependence between these two tasks. Therefore, for any link task, when the relationship between them is IR, then there is $$ \zeta _{i\to j}^{\left( \text{IR} \right)}=0 $$.

In summary, for a particular defense task, the correlation $$ \Gamma ({{R}_{i}}, {{R}_{j}}) $$ between the link tasks in its planning can be expressed in a triple as follows:

where $$ {{R}_{i}} $$ and $$ {{R}_{j}} $$ are the $$ i $$-th and $$ j $$-th link tasks, respectively; $$ \zeta _{i\to j}^{\left( * \right)} $$ denotes the degree of association, when the association between $$ {{R}_{i}} $$ and $$ {{R}_{j}} $$ is a $$ * $$-relationship, and $$ *\in \left\{ \text{HR}, \text{SR}, \text{IR} \right\} $$, $$ \zeta _{i\to j}^{\left( * \right)}\in [0, 1] $$.

4.1. Adjacency probability correlation matrix

Assuming that a total of $$ n $$ link tasks are involved in planning a specific defense task, for two link tasks $$ {{R}_{i}} $$ and $$ {{R}_{j}} $$, since HR, SR, and IR are unique, the correlation between them can be expressed as

In addition, due to the temporal nature of the adjacent links in planning, for the convenience of the subsequent study, the link tasks are sorted in strict time order of execution. Thus, the adjacency correlation matrix between the link tasks is established based on the values of the degree of correlation between them. Considering that the degree of correlation takes values between 0 and 1, instead of only 0 and 1, we distinguish it from the traditional adjacency correlation matrix, which is called the adjacency possibility correlation matrix $$ {{\Gamma }_{n\times n}} $$ here.

Since the adjacency possibility association matrix $$ {{\Gamma }_{n\times n}} $$ emphasizes the unidirectional association between different link tasks and does not consider the own association of the link task, Equation (5) is a triangular array with diagonal elements of zero.

4.2. Calculation of the possibility hierarchy position for the link tasks

To ensure that the link tasks directly related to the total task are classified in the innermost layer, while the fundamental link tasks that affect the total task are classified in the outermost layer, the concept of rotational extraction^[26] is used to calculate the possibility hierarchy position of each link task in the total task^[27,28]; that is, the innermost and outermost layers are determined first. Subsequently, the next inner and outer layers are followed by the next-sub-inner and -outer layers until all link tasks are assigned. The implementation process is as follows:

(ⅰ) Combining the unit matrix $$ I $$, the adjacency possibility correlation matrix $$ {{\Gamma }_{n\times n}} $$ is concatenated until the matrix does not change to obtain the reachable correlation matrix $$ M_{n\times n}^{i\to j} $$:

(ⅱ) The link tasks in each column and row corresponding to all elements in row $$ t $$ and column $$ t $$ of $$ M_{n\times n}^{i\to j} $$ that are not zero are noted as the sets $$ A({{R}_{t}}) $$ and $$ B({{R}_{t}}) $$, respectively. Let the intersection of the two sets be $$ C({{R}_{t}}) $$. Subsequently, the link tasks in the innermost and outermost layers are determined according to the following rules.

(ⅲ) The link tasks set as the innermost and outermost layers in Step (2) are removed from the reachable association matrix $$ M_{n\times n}^{i\to j} $$. If $$ m $$ link task items are removed at this point, a reachable association matrix $$ M_{\left( n-m \right)\times \left( n-m \right)}^{i\to j\left( 1 \right)} $$ of order $$ n-m $$ is obtained, and Step (2) is repeated to determine the link tasks located in the next inner and outer layers.

(ⅳ) The sub-inner and sub-outer link tasks identified in Step (3) are removed from the reachable association matrix $$ M_{\left( n-m \right)\times \left( n-m \right)}^{i\to j\left( 1 \right)} $$. If $$ c $$ link task items are removed at this point, a reachable association matrix $$ M_{\left( n-m-c \right)\times \left( n-m-c \right)}^{i\to j\left( 2 \right)} $$ of order $$ n-m-c $$ is obtained, and Step (2) is repeated to identify the link tasks located in the sub-inner and sub-outer layers until all link task items have been set.

(ⅴ) The following operations are performed on the reachable correlation matrix $$ M_{n\times n}^{i\to j} $$ to create a general skeleton matrix $$ {{E}_{n\times n}} $$:

The position of the possibility hierarchy for each link task is determined based on Step (4).

5.1. Calculation of task risk weights and integrated risk

The process of realization is as follows:

(ⅰ) With the link task risk as the sub-node and the total task risk as the root node, the link task level position determined in Section 4 is taken as the position of each sub-node relative to the root node, and the hierarchical decision graph of the comprehensive risk is preliminarily determined. Determine the total number of layers $$ K $$. If the $$ h\in [1, K] $$ layer contains $$ {{g}_{hv}} $$ child nodes, calculate the centrality of each child node according to the general skeleton matrix $$ {{E}_{n\times n}} $$ as follows:

where $$ {{F}_{t}} $$ is the $$ t $$th child node, $$ {{e}_{tj}} $$ is the degree of association between the $$ t $$th child node and the $$ j\left( j=1, 2, \cdots , n \right) $$th child node, and $$ {{e}_{it}} $$ is the degree of association between the $$ i $$th child node and the $$ t $$th child node.

The child node centrality at layer $$ h $$ is compared pairwise and assigned on a three-scale degree:

Then, the weight judgment matrix of the $$ h $$th layer sub-node is built.

(ⅱ) According to the weight judgment matrix $$ {{B}_{h}} $$, the proposed optimal transfer matrix^[29]$$ {{U}_{h}}\text{=}{{\left( {{u}_{lk}} \right)}_{{{g}_{hv}}\times {{g}_{hv}}}} $$ is established, and the element $$ {{u}_{lk}} $$ is determined using

The maximum eigenvalue $$ {{\lambda }_{\max }} $$ of $$ {{U}_{h}} $$ is calculated and its corresponding eigenvector $$ {{\xi }_{h}}\text{=(}{{\xi }_{h1}}, {{\xi }_{h2}}, ..., {{\xi }_{h{{g}_{hv}}}}\text{)} $$ is obtained and normalized to obtain the intra-layer weight vector of the $$ {{g}_{h}} $$sub-nodes of the $$ h $$th layer, as expressed below:

where $$ {{\xi }_{h{{g}_{hr}}}}\prime $$ denotes the intra-layer weight coefficient of the $$ r\in [1, v] $$th child node in the $$ h $$-th layer. The intra-layer risk weight of each link is used as the connection edge strength between the child nodes to represent the degree of influence of the previous on the subsequent child nodes. In this way, the weighted graph of comprehensive risk prediction is generated.

(ⅲ) Combining the weight vectors of the child nodes within all the layers, the weight of the impact of each link's risk on the total risk is determined from the top level downwards, and the combined weight of the $$ r $$-th child node in the $$ h $$-th layer in the root node is calculated by

(ⅳ) The weighted fusion of task risks for each segment yields a combined risk prediction of

where $$ {{F}_{h{{g}_{hr}}}} $$ represents the value at risk corresponding to the tasks in each session.

5.2. Comprehensive risk prediction based on weighted mapping

The pseudo-code of the integrated risk prediction model based on weighted mapping is shown in Table 1.

6.1. Calculation of the correlation between the link tasks

Here, the reasonableness of resource coordination $$ {{R}_{9}} $$ is used as an example to illustrate the calculation of its correlation with the others of the tasks. To formulate resource coordination rationality, it is necessary to combine the indicators expected to be achieved by the overall defense task, such as the coverage of the defense area and destruction requirements. The rationality of task indicator setting $$ {{R}_{7}} $$ is a specific description of the indicators expected to be achieved by the task. Thus, the relationship between $$ {{R}_{9}} $$ and $$ {{R}_{7}} $$ belongs to a hierarchy. When combining the basic situational information involved in $$ {{R}_{7}} $$ and $$ {{R}_{9}} $$, and their intersection Table 3, the hierarchical correlation between $$ {{R}_{7}} $$ and $$ {{R}_{9}} $$ can be calculated as $$ \zeta _{7\to 9}^{(\text{HR})}=0.769 $$ using Equation (1).

In addition to the hierarchical relationship between $$ {{R}_{9}} $$ and $$ {{R}_{7}} $$, $$ {{R}_{9}} $$ also has a synergistic relationship with defense information clarity $$ {{R}_{1}} $$, our defense unit strength determination $$ {{R}_{3}} $$, and attack entity size determination $$ {{R}_{5}} $$. For example, when planning a mission, if we ignore the determination of the strength of the specific unit defense measures available to us and the overall determination of the size of the attacking entity of the opponent and directly conceptualize the unit strength, the allocation of unit strength resources will likely be unreasonable and even cause the deployment to fail. In other words, the clarity of defense information, the determination of our defense unit forces, and the determination of the size of the attacking entity all affect the rationality of resource coordination in the conceptualization process, which, together, determine the rationality of our resource coordination. Therefore, a synergistic relationship exists between them and the reasonableness of our resource integration.

To measure the degree of synergy between link tasks, ten experts in related fields determined the role factors of link tasks $$ {{R}_{1}} $$, $$ {{R}_{3}} $$ and $$ {{R}_{5}} $$ on $$ {{R}_{9}} $$ using fuzzy hierarchical analysis, the fuzzy complementary judgment matrix is as follows:

Calculating the action factors of the link tasks $$ {{R}_{1}} $$, $$ {{R}_{3}} $$ and $$ {{R}_{5}} $$ on $$ {{R}_{9}} $$ based on the above matrix, as shown in Table 4. Based on Table 4, the degrees of correlation between $$ {{R}_{1}} $$, $$ {{R}_{3}} $$, $$ {{R}_{5}} $$ and $$ {{R}_{9}} $$ were calculated using Equation (2) to obtain $$ \zeta _{1\to 9}^{(S\text{R})}=0.283 $$, $$ \zeta _{3\to 9}^{(S\text{R})}=0.258 $$, and $$ \zeta _{5\to 9}^{(S\text{R})}=0.459 $$.

Since program resilience is not associated with other link tasks, it is considered an independent relationship and acts directly on the total task. The degrees of synergy between all tasks can be calculated based on the analysis of the remaining tasks, which will not be repeated here owing to length constraints. Thus, the adjacency probability correlation matrix $$ {{\Gamma }_{15\times 15}} $$^[30,31] between the 15 link tasks can be generated, as shown in Table 5:

Where data followed by "*" indicates that the type of relationship between tasks in the session is hierarchical; data followed by "/" denotes that the relationship type between tasks in the session is synergistic.

In turn, the reachable correlation matrix $$ M_{15\times 15}^{i\to j} $$^[32–34] and the general skeleton matrix $$ {{\left( E \right)}_{15\times 15}} $$ can be determined, as provided in Tables 6 and 7, respectively:

Therefore, according to the results of link task relevance calculation, the link task relevance structure model can be obtained, as given in Figure 3.

Figure 3. Structural modeling of link task associations

This paper adopts the path analysis method to test the accuracy and reliability of the link task association structure model. Several domain experts are invited to assess the risk value of each link task and use it as a basis for judgment, and the risk value of the corresponding risk item of each link task is imported into the path analysis model for analysis to verify the validity of the association path between each task and the total task^[35–37], where the standardized coefficient specifically reflects the degree of influence between the two risks; when the p-value is $$ <5\% $$, it can be considered significant, representing a valid path between the risks. The specific results are shown in Table 8.

From the results of Table 8, it can be seen that the p-value of paths $$ {{R}_{1}}\to {{R}_{9}} $$, $$ {{R}_{3}}\to {{R}_{9}} $$, $$ {{R}_{12}}\to R $$ is less than $$ 5\% $$, and that of the rest of the paths is less than $$ 1\% $$, which indicates that the paths existing between risks in the link task association structure model are effective. And the degree of influence between risks can be reflected by the standardized coefficient value; the larger the standardized coefficient value is, the higher the degree of validity of the existence of paths between risks; according to the results of Table 8, it can be seen that each path distance presents a significant positive correlation (standardized coefficient $$ >0 $$). Thus, the reliability of the link task association structure model paths is verified.

6.2. Risk hierarchy decision weighting mapping construction

Based on the general skeleton matrix $$ {{\left( E \right)}_{15\times 15}} $$, the rotational extraction concept described in Section 4 is used to determine the possibility hierarchy position of the link task in the total task and then generate a risk hierarchy decision weighting mapping $$ G\updownarrow $$ for defense task planning, as shown in Figure 4A. It can be observed from Figure 4A that $$ {{F}_{8}} $$, $$ {{F}_{11}} $$, $$ {{F}_{12}} $$, $$ {{F}_{13}} $$, $$ {{F}_{14}} $$, and $$ {{F}_{15}} $$ act directly on the total risk $$ F $$, while $$ {{F}_{1}}\sim{{F}_{7}} $$ and $$ {{F}_{9}}\sim{{F}_{10}} $$ need to act indirectly on $$ F $$ through other intermediate nodes.

Figure 4. Decision mapping generated by the two methods. (A) Decision mapping of the possibility hierarchy determined by the rotattional extractional method; (B) Decision mapping of the possibility hierarchy determined by bottom-upextractional method.

To illustrate that the generated risk hierarchy decision weighting mapping $$ G\updownarrow $$ is more conducive to the prediction of integrated risks, the rationality of the approach in this study is analyzed by comparing it with the risk hierarchy decision weighting mapping $$ G\uparrow $$^[38,39] (shown in Figure 4B) constructed based on the bottom-up explanatory structure model^[40].

By comparing Figure 4A and B, we can find that in the $$ G\uparrow $$ generated based on the bottom-up explanation structure model, $$ {{F}_{8}} $$, $$ {{F}_{12}} $$, $$ {{F}_{13}} $$, and $$ {{F}_{15}} $$, which play a direct role in the total risk, are not assigned to the first layer, which is closer to the total risk; however, $$ {{F}_{8}} $$, $$ {{F}_{13}} $$, and $$ {{F}_{15}} $$ are assigned to the second and $$ {{F}_{12}} $$ is assigned to the forth layer. Moreover, $$ {{F}_{4}} $$ is influenced by the position of the $$ {{F}_{8}} $$ layer and is assigned to the third layer that is further away from the total risk. To a certain extent, the direct influence of $$ {{F}_{4}} $$, $$ {{F}_{8}} $$, $$ {{F}_{12}} $$, $$ {{F}_{13}} $$, and $$ {{F}_{15}} $$ on the total task is weakened, making the corresponding risks of each link task relatively more diffuse in the decision, thereby making it difficult for the decision weighting mapping to be clustered to the total risk. In contrast, $$ G\updownarrow $$ generated by the proposed method, $$ {{F}_{8}} $$ and $$ {{F}_{11}}\sim{{F}_{15}} $$, which play a direct role in the total task, are both assigned to the first layer nearest to the total risk. $$ {{F}_{4}} $$ is assigned to the second layer which is relatively far from the total risk, because it indirectly acts on $$ F $$ through a node $$ {{F}_{8}} $$. The proposed method is more conducive to clustering each link task to the total task.

To visualize the aforementioned phenomenon, firstly, according to the general skeleton matrix $$ {{\left( E \right)}_{15\times 15}} $$ obtained in Section 5.1, the elements in each nonzero row and column are summed up to obtain the impact value $$ \sum\nolimits_{i=1}^{15}{{{e}_{ij}}} $$ of other tasks on the $$ i $$th link task and the impact value $$ \sum\nolimits_{j=1}^{15}{{{e}_{ij}}} $$ of the $$ i $$th link task on other tasks. Subsequently, the impact values of each link task are processed in combination with its different levels $$ N\left( N=1, 2, 3, 4 \right) $$ as follows:

Taking $$ \sum\nolimits_{i=1}^{15}{{{e}_{ij}}}\text{+}\left( N-1 \right) $$ and $$ \sum\nolimits_{j=1}^{15}{{{e}_{ij}}}\text{+}\left( N-1 \right) $$ as the horizontal and vertical coordinates of the $$ i $$th link task, respectively, the possibility hierarchy position distribution of each link task of $$ {{F}_{1}}\sim{{F}_{15}} $$ can be calculated, as shown in Figure 5. Among them, "0, 1, 2, 3, 4" on the coordinate axis represent the boundaries of the four layer task hierarchy established in the defense task planning process, and their physical meaning is the number of hierarchical levels.

Figure 5. Distribution of link task likelihood hierarchy positions based on two methods.

It can be intuitively observed from Figure 5 that in the hierarchical decision map $$ G\updownarrow $$ generated by our proposed method, the link tasks are more aggregated to the root node $$ F $$. In contrast, the link tasks in the method in the literature^[40], are more inclined to spread to the branch nodes, which weakens the comprehensive effect on the risk to a certain extent. Therefore, it is more reasonable and feasible to use the decision mapping generated by the proposed method for subsequent integrated risk prediction.

6.3. Forecasting methodology for integrated risk

Based on the general skeleton matrix $$ {{\left( E \right)}_{15\times 15}} $$ and the integrated risk hierarchical decision weighting mapping generated in Section 5.1, the centrality of the link task risk is calculated using Equation (9) (as shown in Table 9), and the numerical comparison is used to establish the link task risk weight judgment matrix $$ {{B}_{h}}={{\left( {{b}_{ij}} \right)}_{{{g}_{h}}\times {{g}_{h}}}} $$ for the hth layer, where $$ h=1, 2, 3, 4 $$. Then, Equations (11-14) are used to determine the intra-layer weight and comprehensive weight of the task risk of each layer in the total task risk.

To illustrate the rationality of the above weight determination method, the weights obtained from the bottom-up extraction method^[40] and entropy method^[41] are compared for analysis, respectively, as shown in Table 10.

Compared to the bottom-up extraction method, the weight values of $$ {{F}_{1}} $$, $$ {{F}_{2}} $$, and $$ {{F}_{3}} $$ in $$ F $$ obtained by the proposed method are higher, which is in line with the needs of defense task planning. The main reason is that the link tasks $$ {{F}_{1}} $$, $$ {{F}_{2}} $$, and $$ {{F}_{3}} $$ are all basic situational information. Only by fully grasping them can the emergence of risks in the execution of the later link tasks be avoided and the completion of the total task be ensured. The risks corresponding to $$ {{F}_{8}} $$, $$ {{F}_{13}} $$, and $$ {{F}_{15}} $$ are raised from the second to the first level in the proposed method, and the weight values are increased because they act directly on $$ F $$. Simultaneously, the risk corresponding to $$ {{F}_{4}} $$ increases from the third to the second level, and the weight value is also improved. The aforementioned adjustments are reasonable when combined with the comprehensive risk prediction results calculated in this section.

Compared to the entropy method, the weight value of $$ {{F}_{2}} $$ obtained using the proposed method is relatively low. The main reason is that the link task $$ {{F}_{2}} $$ has higher value-at-risk when using the entropy method. If the information of the attacking entity is not fully understood, it will seriously hinder the total task execution and even cause a fatal blow. However, this paper argues that $$ {{F}_{2}} $$ is very important. More importantly, we must allocate our forces and respond to physical attacks based on our knowledge of the enemy's posture when making comprehensive risk predictions. Therefore, it is reasonable that the combined weight of $$ {{F}_{2}} $$ obtained by the proposed method is lower than that obtained using the entropy method.

In addition, compared to the scenario strain shortage capacity risk $$ {{F}_{12}} $$, the impact of the unreasonable indicators system of program preference risk $$ {{F}_{13}} $$ on the overall program performance is relatively large, and the weight of $$ {{F}_{12}} $$ in $$ F $$ should be lower than the weight of $$ {{F}_{13}} $$ in $$ F $$. Compared with the literature^[40], it is more effectively reflected in the proposed method.

According to the risk weight value of each link task given by the defense expert system, the link task weight value obtained by each method in Table 10 is analyzed by using the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) ideal point method^[42–44], and the decision matrix^[45] constructed is shown in Table 11, where the elements indicate the difference between the weights of each link task given by different methods and expert systems. Based on Table 11, the calculated TOPSIS evaluation results are shown in Table 12.

As can be seen from the results in Table 12, compared with the other two methods, the comprehensive score index of the method proposed in this paper is higher and is the optimal program, indicating that the risk weight value of each link of the task obtained by our technique is most consistent with the actual situation.

Combining the link task risk values in Table 2 and the link task weights in Table 10, the risk value of the total task is obtained using Equation (15). The integrated risk value evaluated by the air defense expert system was taken as the actual integrated risk value, and was compared and analyzed with the risk values obtained by each of the aforementioned methods, as shown in Table 13.

Table 13 shows that the relative error rate between the integrated risk value obtained by the proposed method and the actual risk value is 0.82%, while the relative error rates of the other two approaches are 2.1% and 2.11%.

6.4. Comparative analysis of different methods

To further illustrate the overall advantages of the proposed method, the risk values of 20 test samples are predicted; the detection samples are obtained by collecting basic data of the defense side in exercise tasks in different scenarios. The real comprehensive risk values based on the expert system were 0.5327, 0.6745, 0.8871, 0.7125, 0.6265, 0.5292, 0.7715, 0.4858, 0.6142, 0.5383, 0.7627, 0.5475, 0.4965, 0.6057, 0.7322, 0.8447, 0.5737, 0.7124, 0.5325, and 0.6581. The risk values predicted by the proposed method, bottom-up extraction method, and fuzzy hierarchical analysis were compared. The validation outcomes illustrate the feasibility and reasonableness of the proposed method. The actual integrated risk values of the 20 test samples and the predicted integrated risk values obtained by different methods are shown in Figure 6. The deviation of each predicted value from the actual integrated risk value is calculated, as given in Table 14.

Figure 6. Comparative chart of integrated risk forecast results

It can be seen from Table 14 that the deviation between the proposed method and the actual comprehensive risk value is minimal. Compared with bottom-up extraction, the relative deviation is reduced by 56%. Compared with the entropy weight method, the relative deviation is reduced by 61.6%. It shows that this method can predict the comprehensive risk of defense mission planning more accurately and exhibits certain feasibility.

Acknowledgments

The authors would like to thank the reviewers for their thoughtful comments and efforts towards improving this manuscript.

Authors' contributions

Methodology, experiment, data analysis, and manuscript drafting: Du W

Conceptualization, manuscript edition and review, and supervision: Chen X

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Conflicts of interest

Both authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

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Copyright

© The Author(s) 2024.