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Research Article  |  Open Access  |  30 May 2024

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

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Complex Eng Syst 2024;4:9.
10.20517/ces.2024.15 |  © The Author(s) 2024.
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Abstract

Most existing risk prediction methods focus on constructing risk element sets and analyzing their uncertainties but do not deeply explore the correlation types and intensities of factors, resulting in large errors in the comprehensive risk prediction results. In this paper, a new integrated risk prediction method is proposed based on the correlation types of tasks in defense task planning and execution. The approach mainly includes the following steps: First, based on the difference of the sequence and mode of action of link tasks, three correlation types (hierarchical, synergistic, and independent) are defined among them, and various correlation measurement techniques are proposed to model these abstract correlation relations and provide data basis for constructing risk decision graphs. Secondly, the rotation extraction strategy is introduced to excavate the internal correlation law between link tasks and generate their hierarchical topology to ensure the rational distribution of their hierarchy positions in defense missions. Then, the intra-layer risk weight is determined based on the centrality of each node in the topology structure, and then the comprehensive risk prediction weighting graph is constructed. Finally, the path analysis is used to assess the rationality of the hierarchical topology structure of the link tasks, and the validity of the proposed method is verified using the test sample set. The results show that compared with other approaches, the predicted results of the proposed method more closely approximate the actual outcomes.

Keywords

Defense mission planning, risk prediction, associative relationships, hierarchical topology, decision mapping

1. INTRODUCTION

Defense task planning[13] involves the comprehensive layout and detailed strategizing of specific defense measures by commanders before executing a specific task. This process mainly includes task understanding, analysis and judgment, formulation of concepts, program development, program selection, plan development, and other links. Each stage aims to achieve the overall objective with a clear purpose and specific function. The reasonableness of mission planning and the effectiveness of its execution directly determine the outcome of the entire defense mission. Therefore, risk decision-making for the specific defense mission is a necessary precondition to ensure its successful implementation.

Scholars have achieved certain research results on risk prediction in defense mission planning[4], medical system research and development[5], commercial project risk management[6], and risk assessment of construction projects[7], mainly in the methods of multi-attribute decision-making[8], decision tree analysis[9], and so on. Wu et al. analyzed various factors influencing the slope stability, including hydrological conditions of open-pit mines, the geologic formations of slopes, the internal structure[10], etc. They identified the attribute sets affecting slope stability and proposed a large-scale multi-attribute group decision-making model based on intuitionistic fuzzy sets. This kind of decision model requires relative independence among the attributes in the set. Decision tree analysis is an analytical method for systematic decision-making after a unified combination arrangement on the premise of classifying the attributes affecting the event; e.g., Ikwan et al. constructed a decision tree from the attributes of environmental impacts, management constraints, human/equipment failure factors, and risk factors of the tanks themselves and put forth a fishbone diagram-decision tree-risk matrix analysis strategy, which realized an effective prediction of the effective prediction of tank leakage risk[11]. Zheng et al. explored the composition of the mission conception risk assessment system from the levels of thoroughness, mobility, and protection by combining the conception elements and research and judgment data and solved the problem of quantifying the risk of defense conception by establishing the quantitative analysis model of each assessment index[12]. Song et al. proposed a program based on the task derivation by establishing a program index system under the background of the joint task risk analysis model[13]. Ryczyński et al. proposed a risk analysis technique integrating the Kaplan method, the Garrick method, and fuzzy theory to realize the risk management decision-making in the liquid fuel material supply program during military operations[14]. Kim et al. introduced the Risk Management Framework (RMF) and developed an assessment model of the weapon system, which offers a theoretical foundation for the mission plan formulation[15]. Most of the above research results focus on constructing risk element sets and analyzing their uncertainty; however, comprehensive prediction is needed on the premise that risk factors are independent.

Subsequently, researchers have made significant efforts in comprehensive risk prediction based on factor associations; e.g., Zhang et al.[16] proposed the project portfolio risk (PPR) evolution and response model to solve the problem of project risk interaction, which effectively reflected the real-time interaction in the evolution process of PPRs and helped decision makers quickly identify key strategic intrusion nodes[16]. Zhang et al.[16] proposed a category-based association measurement technology (the Measuring Attractiveness by a Categorical-Based Evaluation Technique (MACBETH)) to measure the associative relationships between risks and construct a risk response model for the selection of risk relationships in the medical system research and development project (Risk Response Actions (RRA)), so as to maximize the expectations of the management of medical system research and development project under budget constraints[5]. Abedzadeh et al. used fuzzy decision tree analysis to determine the possible combinatorial relationships among social, economic, environmental, and water damage index attributes to achieve an integrated risk management decision for developing water resources[17]. These methods provide a quantitative assessment of the similarity of risk associations and, to some extent, reveal the linkages and impacts between various risks. In fact, in defense mission planning, not only do the tasks adhere to a strict time sequence, but they also follow layer-by-layer refinement. Directly applying the correlation method mentioned to the defense mission planning for risk prediction will lead to large errors in the prediction results. Hierarchical directed topology lays out the nodes of a complex system according to a hierarchical structure. Inspired by this, hierarchical topology is used in the integrated risk prediction of defense tasks. This technique not only effectively highlights the inter-association relationship among link tasks but also reveals the roles of each task on the defense missions, thereby enhancing the accuracy of the integrated risk prediction.

The main contributions of this paper are as follows: (1) oriented to the task execution process in defense mission planning, the association relationship between tasks in different planning links is studied, and the normalized description of hierarchical, synergistic, and independent relationships and the measurement calculation method are defined; (2) the hierarchical topology between link tasks is generated using the rotational extraction method; (3) on the premise of clarifying the role of each link task on the whole defense mission, a new integrated risk prediction approach is proposed by constructing a weighted map of integrated risk prediction; and (4) combined with specific defense tasks, the proposed method is analyzed qualitatively and quantitatively, and its effectiveness and feasibility are verified.

The structure of this paper is outlined as follows: Section 2 gives the research idea and schematic diagram for constructing a model of the integrated risk prediction problem for defense mission planning; Section 3 defines the three association types and their quantification techniques; Section 4 describes the process of generating the hierarchical topology for link missions; Section 5 proposes a new integrated risk prediction method based on weighted mapping; Section 6 analyzes and validates the proposed approach; Section 7 provides the conclusion.

2. PROBLEM MODELING

To improve the reliability of risk prediction of defense mission planning, this paper, based on the "OODA" (Observe, Orient, Decide, and Act) ring, decomposes the tasks involved into six links and further divides them into four aspects: acquiring information about the defense mission, analyzing the composition of the execution force, evaluating the capability of the attacking entity, and assessing the execution program and plan. The paper specifically examines 15 links of task risk, such as clarifying the defense information, identifying the force of our unit, determining the scale of the attacking entity, and evaluating the resilience of the program. Aimed at the strong temporal sequence of link tasks and their complexity and intertwined relationships, this paper mainly examined the following three aspects: determining their relationships, constructing their hierarchical topology, and developing a comprehensive risk prediction model. The schematic diagram of this prediction model is shown in Figure 1.

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

Figure 1. Schematic diagram of the integrated risk prediction model.

The functions of each part are as follows:

(ⅰ) As each link follows a strict timing sequence and contains various subtasks, many complex associations may exist between different link tasks. Rationally distinguishing and describing these relationships is the basis for improving the reliability of comprehensive risk prediction. In this paper, we determine the associations by mining the relationship between their time sequences and hierarchies. We specify the association types, give a normalized description of these relationships, and outline the calculation method of the association measures to provide numerical inputs for establishing the probability association matrix between link tasks.

(ⅱ) The relationship between link tasks directly determines the mode and intensity of the role of each link to the total task, and the correlation type varies among various link tasks, which directly affects the comprehensive risk prediction. The correct construction of link task hierarchical topology is the key to improving the reliability of comprehensive risk prediction. Therefore, this paper clarifies the role and intensity of link tasks on the whole defense task based on the association measurement size of each task in (ⅰ), establishes the possibility association matrix, and introduces the rotation extraction method to determine the hierarchical position of the link tasks, which provides a theoretical basis for constructing the comprehensive risk decision-making mapping.

(ⅲ) The risk of each link task acts on the total risk of the defense task based on the topology between link tasks. However, the correlation strength between link tasks is independent, so determining the link task risk weights based on the correlation distribution is a guarantee to enhance the reliability of the comprehensive risk prediction. In this paper, a three-scale hierarchical analysis is conducted based on the centrality of each link task to determine its hierarchical weights, and then a weighted map is constructed to ensure the reliability of the comprehensive risk prediction results.

3. PERCEPTION OF THE CORRELATION BETWEEN LINK TASKS

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 $$ {{R}_{8}} $$.

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

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[2022], which measures the HR correlation between the link tasks, expressed as

$$ \zeta _{i\to j}^{\left( \text{HR} \right)}=\frac{2\left\{ SI({{R}_{i}})\cap SI({{R}_{j}}) \right\}}{\left\{ SI({{R}_{i}}) \right\}+\left\{ SI({{R}_{j}}) \right\}} $$

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[2325] 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

$$ \zeta _{i\to k}^{\text{SR}}=\frac{\text{Im}\left( {{R}_{i}}\to {{R}_{k}} \right)}{\mathop{\sum }_{i=1}^{n}\text{Im}\left( {{R}_{i}}\to {{R}_{k}} \right)} $$

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:

$$ \Gamma ({{R}_{i}}, {{R}_{j}})\text{=}\langle {{R}_{i}}, \zeta _{i\to j}^{\left( * \right)}, {{R}_{j}} \rangle $$

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. LINK TASK HIERARCHY TOPOLOGY GENERATION

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

$$ {{\zeta }_{i\to j}}\text{=}\begin{matrix} \zeta _{i\to j}^{\left( \text{HR} \right)} & or & \zeta _{i\to j}^{\left( \text{SR} \right)} & or & \zeta _{i\to j}^{\left( \text{IR} \right)} \\ \end{matrix} $$

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.

$$ {{\Gamma }_{n\times n}}\text{=}{{\left( {{\zeta }_{i\to j}} \right)}_{n\times n}}=\left[ \begin{matrix} 0 & ... & ... & {{\zeta }_{1\to j}} & ... & {{\zeta }_{1\to n}} \\ 0 & 0 & ... & ... & ... & ... \\ 0 & 0 & 0 & {{\zeta }_{i\to j}} & ... & {{\zeta }_{i\to n}} \\ 0 & 0 & 0 & 0 & ... & ... \\ 0 & 0 & 0 & 0 & 0 & ... \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{matrix} \right] $$

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 $$ M_{n\times n}^{i\to j} $$:

$$ M_{n\times n}^{i\to j}={{\left( {{\left( {{\zeta }_{i\to j}} \right)}_{n\times n}}+{{\left( I \right)}_{n\times n}} \right)}^{k+1}}={{\left( {{\left( {{\zeta }_{i\to j}} \right)}_{n\times n}}+{{\left( I \right)}_{n\times n}} \right)}^{k}}\ne {{\left( {{\left( {{\zeta }_{i\to j}} \right)}_{n\times n}}+{{\left( I \right)}_{n\times n}} \right)}^{k-1}} $$

(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.

$$ \begin{array}{c} \left\{\begin{array}{l} \text { Divide the extracted session tasks to the innermost layer when } C\left(R_{t}\right) = A\left(R_{t}\right) \bigcap B\left(R_{t}\right) = A\left(R_{t}\right) \\ \text { Divide the extracted session tasks to the outermost layer when } C\left(R_{t}\right) = A\left(R_{t}\right) \bigcap B\left(R_{t}\right) = B\left(R_{t}\right) \end{array}\right. \end{array} $$

(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 $$ {{E}_{n\times n}} $$:

$$ {{E}_{n\times n}}\text{=}{{\left( {{e}_{ij}} \right)}_{n\times n}}=M_{n\times n}^{i\to j}-{{\left( M_{n\times n}^{i\to j}-{{\left( I \right)}_{n\times n}} \right)}^{2}}-{{\left( I \right)}_{n\times n}} $$

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

5. INTEGRATED RISK PREDICTION MODELING

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:

$$ C({{F}_{t}})=\sum\limits_{j=1}^{n}{{{e}_{tj}}}+\sum\limits_{i=1}^{n}{{{e}_{it}}} $$

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:

$$ {{b}_{lk}}=\left\{ \begin{matrix} 2 & C({{F}_{l}})>C({{F}_{k}}) \\ 1 & C({{F}_{l}})=C({{F}_{k}}) \\ 0 & C({{F}_{l}})<C({{F}_{k}}) \\ \end{matrix}\text{ } \right., l, k=1, 2, ..., {{g}_{hv}} $$

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

$$ {{B}_{h}}=\left[ \begin{matrix} {{b}_{11}} & \cdots & {{b}_{1j}} & \cdots & {{b}_{1{{g}_{hv}}}} \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ {{b}_{i1}} & \cdots & {{b}_{ij}} & \cdots & {{b}_{i{{g}_{hv}}}} \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ {{b}_{{{g}_{hv}}1}} & \cdots & {{b}_{{{g}_{hv}}j}} & \cdots & {{b}_{{{g}_{hv}}{{g}_{hv}}}} \\ \end{matrix} \right] $$

(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

$$ {{u}_{lk}}=\exp \left( \frac{1}{{{g}_{hv}}}\sum\limits_{q=1}^{{{g}_{hv}}}{({{b}_{lq}}-{{b}_{kq}})} \right) $$

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:

$$ {{\xi }_{h}}\prime \text{=(}{{\xi }_{h{{g}_{h1}}}}\prime , \cdots {{\xi }_{h{{g}_{h2}}}}\prime , ..., {{\xi }_{h{{g}_{hv}}}}\prime \text{)} $$

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

$$ {{\omega }_{{{g}_{hr}}}}=\frac{{{\xi }_{h{{g}_{hr}}}}\prime }{\sum\limits_{h=1}^{K}{\sum\limits_{r=1}^{{{g}_{hv}}}{{{\xi }_{h{{g}_{h1}}}}\prime \text{+}{{\xi }_{h{{g}_{h2}}}}\prime \text{+}...\text{+}{{\xi }_{h{{g}_{hr}}}}\prime }}} $$

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

$$ F=\sum\nolimits_{h=1}^{K}{{{\sum\nolimits_{r=1}^{{{g}_{hv}}}{{{F}_{h{{g}_{hr}}}}\times \omega }}_{{{g}_{hr}}}}} $$

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.

Table 1

Integrated risk prediction model pseudo-code

Integrated risk prediction model pseudo-code
Perceived linkages between session tasksStep 1Determine the types of interlinked relationships between link tasks based on the concepts of hierarchical, synergistic and independent relationships.
Step 2Using Equations (1) and (2) to calculate the link inter-task correlation, respectively.
Hierarchical topology generationStep 3Using Equations (5) and (6) to obtain the adjacency likelihood correlation matrix and the reachability correlation matrix, respectively.
Step 4Use Equation (7) to determine the hierarchical position of the tasks in each session.
Step 5The general skeleton matrix is computed using Equation (8) and the hierarchical topology is generated.
Consolidated risk projectionsStep 6Using Equations (9-11) to construct the weight judgment matrix of task risk for each layer of the link $$ {{B}_{h}} $$.
Step 7Calculate the weighting coefficients of each segment's task risk in the composite risk using Equations (12-14) to construct a composite risk decision weighting map.
Step 8The combined risk is predicted using Equation (15).

6. CASE VERIFICATION AND COMPARATIVE ANALYSIS

To verify the feasibility and superiority of the proposed method, this section presents an example of defense mission planning.

Assuming that an attacking entity is expected to strike our T-area at a certain time, the commanders plan defense tasks with the acquired basic situational information. A total of 15 tasks within six planning segments are involved in this planning process. The values of risk incurred by the link tasks are shown in Table 2.

Table 2

Information about the session tasks and their corresponding risk items in a given scenario

Session taskCorresponding risk itemsRisk value
Defensive message clarity $$ {{R}_{1}} $$Unclear defensive information $$ {{F}_{1}} $$0.658
Comprehensive information on the attacking entity $$ {{R}_{2}} $$Incomplete information about the attacking entity $$ {{F}_{2}} $$0.790
Our defense unit strength determination $$ {{R}_{3}} $$Our defense unit strength was incorrectly determined $$ {{F}_{3}} $$0.566
Judgment of the direction of movement of the attacking entity $$ {{R}_{4}} $$Inaccurate judgment of the direction of movement of the attacking entity $$ {{F}_{4}} $$0.778
Attack entity size determination $$ {{R}_{5}} $$The size of the attacking entity was incorrectly identified $$ {{F}_{5}} $$0.721
Unit platform response capability $$ {{R}_{6}} $$Inadequate response capability of the unitary platform $$ {{F}_{6}} $$0.618
Reasonable setting of task indicators $$ {{R}_{7}} $$Unreasonable setting of task indicators $$ {{F}_{7}} $$0.692
Reasonableness of the selection of the target of fire collection $$ {{R}_{8}} $$Unreasonable selection of fire target $$ {{F}_{8}} $$0.764
Rationalization of resource integration $$ {{R}_{9}} $$Unreasonable resource integration $$ {{F}_{9}} $$0.625
Protective capability in conception $$ {{R}_{10}} $$Insufficient conceptual protection $$ {{F}_{10}} $$0.750
Reasonableness of programming standards $$ {{R}_{11}} $$Unreasonable standards for program development $$ {{F}_{11}} $$0.615
Program Resilience $$ {{R}_{12}} $$Inadequate program resilience $$ {{F}_{12}} $$0.632
Reasonableness of the index system of program preference $$ {{R}_{13}} $$The index system of program preference is not reasonable $$ {{F}_{13}} $$0.625
Execution of branch plans $$ {{R}_{14}} $$Inadequate execution of branch plans $$ {{F}_{14}} $$0.724
Monitoring capabilities of the program $$ {{R}_{15}} $$Insufficient monitoring power of the plan $$ {{F}_{15}} $$0.625

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).

Table 3

Basic situation information related to $$ {{R}_{7}} $$ and $$ {{R}_{9}} $$

Serial number$$ SI({{R}_{7}}) $$$$ SI({{R}_{9}}) $$$$ SI({{R}_{7}})\cap SI({{R}_{9}}) $$
1Expected means of mission defenseExpected means of mission defense1
2Expected Achievement GoalsExpected Achievement Goals1
3Destruction parameter requirementsDestruction parameter requirements1
4CoverageCoverage1
5Duration of missionDuration of mission1
6The capability of reconnaissance and early warning unit capabilityThe capability of reconnaissance and early warning unit capability1
7The capability of firepower unit capabilityThe capability of firepower unit capability1
8The capability of electronic countermeasures unit capabilityThe capability of electronic countermeasures unit capability1
9The capability of integrated assurance unitThe capability of integrated assurance unit1
10The capability of interceptor strike unit capabilityThe capability of interceptor strike unit capability1
11Attack entity equipment type0
12Number of attacking entity equipment0
13Attack entity incoming main direction0
14Attack entity incoming sub-direction0
15Attack entity incoming proximity direction0
16Attack entity incoming intent0

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:

$$ \left( \begin{matrix} 0.5 & 0.55 & 0.15 \\ 0.45 & 0.5 & 0.1 \\ 0.85 & 0.9 & 0.5 \\ \end{matrix} \right) \left( \begin{matrix} 0.5 & 0.5 & 0.2 \\ 0.5 & 0.5 & 0.1 \\ 0.8 & 0.9 & 0.5 \\ \end{matrix} \right) \left( \begin{matrix} 0.5 & 0.6 & 0.15 \\ 0.4 & 0.5 & 0.05 \\ 0.85 & 0.95 & 0.5 \\ \end{matrix} \right) \left( \begin{matrix} 0.5 & 0.5 & 0.2 \\ 0.5 & 0.5 & 0.2 \\ 0.8 & 0.8 & 0.5 \\ \end{matrix} \right) \left( \begin{matrix} 0.5 & 0.6 & 0.1 \\ 0.4 & 0.5 & 0.15 \\ 0.9 & 0.85 & 0.5 \\ \end{matrix} \right) $$

$$ \left( \begin{matrix} 0.5 & 0.55 & 0.15 \\ 0.45 & 0.5 & 0.2 \\ 0.85 & 0.8 & 0.5 \\ \end{matrix} \right) \left( \begin{matrix} 0.5 & 0.5 & 0.1 \\ 0.5 & 0.5 & 0.05 \\ 0.9 & 0.95 & 0.5 \\ \end{matrix} \right) \left( \begin{matrix} 0.5 & 0.55 & 0.2 \\ 0.45 & 0.5 & 0.1 \\ 0.8 & 0.9 & 0.5 \\ \end{matrix} \right) \left( \begin{matrix} 0.5 & 0.6 & 0.15 \\ 0.4 & 0.5 & 0.2 \\ 0.85 & 0.8 & 0.5 \\ \end{matrix} \right) \left( \begin{matrix} 0.5 & 0.55 & 0.2 \\ 0.45 & 0.5 & 0.05 \\ 0.8 & 0.95 & 0.5 \\ \end{matrix} \right) $$

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 $$ \zeta _{5\to 9}^{(S\text{R})}=0.459 $$.

Table 4

The action factors of the link tasks $$ {{R}_{1}} $$, $$ {{R}_{3}} $$ and $$ {{R}_{5}} $$ on $$ {{R}_{9}} $$ obtained from FAHP

$$ \operatorname{Im}({{R}_{1}}\to {{R}_{9}}) $$$$ \operatorname{Im}({{R}_{3}}\to {{R}_{9}}) $$$$ \operatorname{Im}({{R}_{5}}\to {{R}_{9}}) $$
Action factor0.2830.2580.459

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:

Table 5

The adjacency probability correlation matrix

$$ {{\Gamma }_{15\times 15}}={{\left( {{\zeta }_{i\to j}} \right)}_{15\times 15}} $$$$ {{R}_{1}} $$$$ {{R}_{2}} $$$$ {{R}_{3}} $$$$ {{R}_{4}} $$$$ {{R}_{5}} $$$$ {{R}_{6}} $$$$ {{R}_{7}} $$$$ {{R}_{8}} $$$$ {{R}_{9}} $$$$ {{R}_{10}} $$$$ {{R}_{11}} $$$$ {{R}_{12}} $$$$ {{R}_{13}} $$$$ {{R}_{14}} $$$$ {{R}_{15}} $$
$$ {{R}_{1}} $$0000000.275/00.283/0000.4*00
$$ {{R}_{2}} $$000.75*0.462*000.6/00000000
$$ {{R}_{3}} $$000000.913*0.125/00.258/000000
$$ {{R}_{4}} $$00000000.7/0000000
$$ {{R}_{5}} $$00000000.3/0.459/000000
$$ {{R}_{6}} $$0000000000.606*0000.92*0.467*
$$ {{R}_{7}} $$000000000.769*0000.324*00
$$ {{R}_{8}} $$000000000000000
$$ {{R}_{9}} $$00000000000.64*0000
$$ {{R}_{10}} $$00000000000000.541*0
$$ {{R}_{11}} $$000000000000000
$$ {{R}_{12}} $$000000000000000
$$ {{R}_{13}} $$000000000000000
$$ {{R}_{14}} $$000000000000000
$$ {{R}_{15}} $$000000000000000

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 $$ M_{15\times 15}^{i\to j} $$[3234] and the general skeleton matrix $$ {{\left( E \right)}_{15\times 15}} $$ can be determined, as provided in Tables 6 and 7, respectively:

Table 6

The reachable correlation matrix

$$ M_{15\times 15}^{i\to j} $$$$ {{R}_{1}} $$$$ {{R}_{2}} $$$$ {{R}_{3}} $$$$ {{R}_{4}} $$$$ {{R}_{5}} $$$$ {{R}_{6}} $$$$ {{R}_{7}} $$$$ {{R}_{8}} $$$$ {{R}_{9}} $$$$ {{R}_{10}} $$$$ {{R}_{11}} $$$$ {{R}_{12}} $$$$ {{R}_{13}} $$$$ {{R}_{14}} $$$$ {{R}_{15}} $$
$$ {{R}_{1}} $$1000000.27500.28300.28300.400
$$ {{R}_{2}} $$0100.750.46200.60.70.600.600.32400
$$ {{R}_{3}} $$001000.9130.12500.2580.6060.25800.1250.9130.467
$$ {{R}_{4}} $$00010000.70000000
$$ {{R}_{5}} $$00001000.30.45900.4590000
$$ {{R}_{6}} $$0000010000.6060000.920.467
$$ {{R}_{7}} $$000000100.76900.6400.32400
$$ {{R}_{8}} $$000000010000000
$$ {{R}_{9}} $$00000000100.640000
$$ {{R}_{10}} $$00000000010000.5410
$$ {{R}_{11}} $$000000000010000
$$ {{R}_{12}} $$000000000001000
$$ {{R}_{13}} $$000000000000100
$$ {{R}_{14}} $$000000000000010
$$ {{R}_{15}} $$000000000000001
Table 7

The general skeleton matrix

$$ {{\left( E \right)}_{15\times 15}} $$$$ {{R}_{1}} $$$$ {{R}_{2}} $$$$ {{R}_{3}} $$$$ {{R}_{4}} $$$$ {{R}_{5}} $$$$ {{R}_{6}} $$$$ {{R}_{7}} $$$$ {{R}_{8}} $$$$ {{R}_{9}} $$$$ {{R}_{10}} $$$$ {{R}_{11}} $$$$ {{R}_{12}} $$$$ {{R}_{13}} $$$$ {{R}_{14}} $$$$ {{R}_{15}} $$
$$ {{R}_{1}} $$0000000.27500.0080000.12500
$$ {{R}_{2}} $$0000.750.46200.600000000
$$ {{R}_{3}} $$000000.9130.12500.133000000
$$ {{R}_{4}} $$00000000.70000000
$$ {{R}_{5}} $$00000000.30.459000000
$$ {{R}_{6}} $$0000000000.6060000.3790.467
$$ {{R}_{7}} $$000000000.7690000.32400
$$ {{R}_{8}} $$000000000000000
$$ {{R}_{9}} $$00000000000.640000
$$ {{R}_{10}} $$00000000000000.5410
$$ {{R}_{11}} $$000000000000000
$$ {{R}_{12}} $$000000000000000
$$ {{R}_{13}} $$000000000000000
$$ {{R}_{14}} $$000000000000000
$$ {{R}_{15}} $$000000000000000

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

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

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[3537], 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.

Table 8

Regression coefficients for structural modeling of link task associations

PathStandardized coefficientPPathStandardized coefficientP
$$ {{R}_{\text{2}}}\to {{R}_{\text{4}}} $$0.961$$< $$1% (0.000***)$$ {{R}_{\text{9}}}\to {{R}_{\text{11}}} $$0.983$$< $$1% (0.000***)
$$ {{R}_{\text{2}}}\to {{R}_{\text{5}}} $$0.953$$< $$1% (0.000***)$$ {{R}_{\text{1}}}\to {{R}_{\text{13}}} $$0.147$$< $$5% (0.042**)
$$ {{R}_{\text{3}}}\to {{R}_{\text{6}}} $$0.963$$< $$1% (0.000***)$$ {{R}_{\text{7}}}\to {{R}_{\text{13}}} $$0.843$$< $$1% (0.000***)
$$ {{R}_{\text{1}}}\to {{R}_{\text{7}}} $$0.447$$< $$1% (0.000***)$$ {{R}_{\text{6}}}\to {{R}_{\text{14}}} $$0.499$$< $$1% (0.001***)
$$ {{R}_{\text{3}}}\to {{R}_{\text{7}}} $$0.547$$< $$1% (0.000***)$$ {{R}_{\text{10}}}\to {{R}_{\text{14}}} $$0.495$$< $$1% (0.01***)
$$ {{R}_{\text{4}}}\to {{R}_{\text{8}}} $$0.515$$< $$1% (0.000***)$$ {{R}_{\text{6}}}\to {{R}_{\text{15}}} $$0.974$$< $$1% (0.000***)
$$ {{R}_{\text{5}}}\to {{R}_{\text{8}}} $$0.479$$< $$1% (0.000***)$$ {{R}_{\text{8}}}\to R $$0.154$$< $$1% (0.000***)
$$ {{R}_{\text{1}}}\to {{R}_{\text{9}}} $$0.201$$< $$5% (0.029**)$$ {{R}_{\text{11}}}\to R $$0.269$$< $$1% (0.000***)
$$ {{R}_{\text{3}}}\to {{R}_{\text{9}}} $$0.229$$< $$5% (0.023**)$$ {{R}_{\text{12}}}\to R $$0.081$$< $$5% (0.025**)
$$ {{R}_{\text{5}}}\to {{R}_{\text{9}}} $$0.181$$< $$1% (0.07***)$$ {{R}_{\text{13}}}\to R $$0.127$$< $$1% (0.07***)
$$ {{R}_{\text{7}}}\to {{R}_{\text{9}}} $$0.430$$< $$1% (0.01***)$$ {{R}_{\text{14}}}\to R $$0.251$$< $$1% (0.000***)
$$ {{R}_{\text{6}}}\to {{R}_{\text{10}}} $$0.976$$< $$1% (0.000***)$$ {{R}_{\text{15}}}\to R $$0.190$$< $$1% (0.01***)

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.

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

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:

$$ \left\{ \begin{matrix} \sum\nolimits_{i=1}^{15}{{{e}_{ij}}}\text{+}\left( N-1 \right) & \text{The impact value of the other tasks on the }i\text{th link task} \\ \sum\nolimits_{j=1}^{15}{{{e}_{ij}}}\text{+}\left( N-1 \right) & \text{The impact value of the }i\text{th link task on other tasks} \\ \end{matrix} \right. $$

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.

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

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.

Table 9

The centrality of the link task risk

$$ {{F}_{i}} $$$$ C({{F}_{i}}) $$$$ {{F}_{i}} $$$$ C({{F}_{i}}) $$$$ {{F}_{i}} $$$$ C({{F}_{i}}) $$
$$ {{F}_{1}} $$0.4080$$ {{F}_{6}} $$2.3650$$ {{F}_{11}} $$0.6400
$$ {{F}_{2}} $$1.8120$$ {{F}_{7}} $$2.0930$$ {{F}_{12}} $$0.0000
$$ {{F}_{3}} $$1.1710$$ {{F}_{8}} $$1.0000$$ {{F}_{13}} $$0.4490
$$ {{F}_{4}} $$1.4500$$ {{F}_{9}} $$2.0090$$ {{F}_{14}} $$0.9200
$$ {{F}_{5}} $$1.2210$$ {{F}_{10}} $$1.1470$$ {{F}_{15}} $$0.4670

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.

Table 10

Table of intra-tier weights for the link tasks

Weight judgment matrix$$ {{B}_{h}}={{\left( {{b}_{ij}} \right)}_{{{g}_{h}}\times {{g}_{h}}}} $$Methodology of this studyBottom-up extraction methodEntropy method
LevelsIntra-layer weightsComposite weightsLevelsIntra-layer weightsComposite weights
$${{F}_{8}}$$1222221st Floor0.32780.08222nd Floor0.17130.04430.0480
$${{F}_{11}}$$0122020.16840.04221st Floor0.73110.18870.0631
$${{F}_{12}}$$0010000.06190.01564th Floor0.10150.02620.0725
$${{F}_{13}}$$0021000.08640.02172nd Floor0.07700.01990.0674
$${{F}_{14}}$$0222120.23490.05891st Floor0.26890.06940.0343
$${{F}_{15}}$$0022010.12060.03032nd Floor0.11480.02960.0451
$${{F}_{4}}$$1022nd Floor0.28890.08223rd Floor0.16740.00740.0842
$${{F}_{9}}$$2120.56280.04222nd Floor0.38130.18870.0654
$${{F}_{10}}$$0010.14830.05892nd Floor0.25560.06940.0535
$${{F}_{5}}$$1003rd Floor0.14830.04923rd Floor0.10150.02360.0631
$${{F}_{6}}$$2120.56280.14810.45510.09900.0720
$${{F}_{7}}$$2010.28890.03390.27600.07190.0474
$${{F}_{1}}$$1004th Floor0.56270.13154th Floor0.45510.03100.1244
$${{F}_{2}}$$2120.14840.03300.16740.01210.0642
$${{F}_{3}}$$2010.28890.17010.27600.11880.0954

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[4244], 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.

Table 11

TOPSIS decision-making matrix

Segment task risk itemsMethodology of this articleBottom-up extraction methodEntropy methodSegment task risk itemsMethodology of this articleBottom-up extraction methodEntropy method
$$ {{F}_{1}} $$0.00150.0990.081$$ {{F}_{9}} $$0.05780.08870.0348
$$ {{F}_{2}} $$0.0030.01790.0404$$ {{F}_{10}} $$0.00890.01940.0135
$$ {{F}_{3}} $$0.00020.05150.0995$$ {{F}_{11}} $$0.05780.08870.0288
$$ {{F}_{4}} $$0.00220.07260.0219$$ {{F}_{12}} $$0.00260.01320.0701
$$ {{F}_{5}} $$0.02920.00360.0336$$ {{F}_{13}} $$0.00440.00620.0605
$$ {{F}_{6}} $$0.01710.0320.0762$$ {{F}_{14}} $$0.03690.04740.0387
$$ {{F}_{7}} $$0.00390.04190.0472$$ {{F}_{15}} $$0.00870.0080.0442
$$ {{F}_{8}} $$0.00620.03170.006
Table 12

TOPSIS evaluation results

MethodPositive ideal solution distanceNegative ideal solution distanceComposite score indexArrange in order
Methodology of this article1.064223493.522068850.767955591
Bottom-up extraction method2.856819122.184872420.433360992
Entropy method2.980843382.117875220.415374023

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

Comparison of integrated risk forecast results

Actual combined risk valueBottom-up extraction methodEntropy methodMethodology of this article
Comprehensive Risk0.66210.64850.67640.6676

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.

Table 14

Deviation between the predicted value of each integrated risk and the actual integrated risk value

Bottom-up extraction methodEntropy methodMethodology of this article
Deviation13.82%15.83%6.08%
A comprehensive risk prediction method for defense mission planning based on probabilistic reasoning and hierarchical analysis

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.

7. CONCLUSION

In this paper, the relation between links and tasks is systematically expounded. On the premise of considering different association types, the hierarchy determination method of the link task in the whole defense task is explored. The decision graph of link task risk level for efficient defense task planning is generated. Then, a comprehensive risk prediction model is constructed. The problem of inaccurate risk prediction results caused by non-independence between tasks is solved. Through the simulation of test samples, the feasibility and rationality of the proposed method are compared and analyzed. The work and innovation of this paper are mainly reflected in:

(ⅰ) Defining the types of link inter-task associations as hierarchical, synergistic and independent relationships, further deepening the connotation of associative relationships, and proposing a calculation method for link inter-task association measurement, which provides theoretical guidance for establishing associative relationships.

(ⅱ) Introducing the idea of hierarchical topology, a method for determining the hierarchical positions of link tasks based on rotational extraction is proposed, and the reasonableness of the generated hierarchical topology paths is analyzed based on the level of significance in the path analysis method.

(ⅲ) A hierarchical decision weighted mapping of integrated risk is constructed using the link task centrality degree. Compared with the entropy weight method, this paper considers the correlation type and intensity of tasks in weight calculation. The result is more consistent with the actual situation, and the relative deviation of risk is reduced by 61.6%. Compared with the bottom-up extraction method, the nodes in the graph constructed in this paper are more centralized to the root node, and the relative risk deviation is reduced by 56%. To some extent, the proposed method solves the problem that the existing decision-making methods are difficult to reflect the correlation strength of link tasks, which leads to unreasonable prediction results.

In summary, the integrated risk prediction method for defense mission planning with hierarchical weighted mapping proposed in this paper shows significant advantages in dealing with the complex risk environment of planning missions in the field of national defense and can provide scientific decision support for strategic deployment. At the same time, the method can also be applied to risk decision-making in financial investment, emergency management, urban planning and construction. However, the construction of risk terms proposed in this paper is not comprehensive enough, and we will further improve the accuracy of comprehensive risk prediction by refining and expanding the risk terms in the future.

DECLARATIONS

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

Availability of data and materials

Not applicable.

Financial support and sponsorship

None.

Conflicts of interest

Both authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Copyright

© The Author(s) 2024.

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OAE Style

Du WW, Chen XW. A comprehensive risk prediction method for defense mission planning based on probabilistic reasoning and hierarchical analysis. Complex Eng Syst 2024;4:9. http://dx.doi.org/10.20517/ces.2024.15

AMA Style

Du WW, Chen XW. A comprehensive risk prediction method for defense mission planning based on probabilistic reasoning and hierarchical analysis. Complex Engineering Systems. 2024; 4(2): 9. http://dx.doi.org/10.20517/ces.2024.15

Chicago/Turabian Style

WeiWei Du, XiaoWei Chen. 2024. "A comprehensive risk prediction method for defense mission planning based on probabilistic reasoning and hierarchical analysis" Complex Engineering Systems. 4, no.2: 9. http://dx.doi.org/10.20517/ces.2024.15

ACS Style

Du, WW.; Chen XW. A comprehensive risk prediction method for defense mission planning based on probabilistic reasoning and hierarchical analysis. Complex. Eng. Syst. 2024, 4, 9. http://dx.doi.org/10.20517/ces.2024.15

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© The Author(s) 2024. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, sharing, adaptation, distribution and reproduction in any medium or format, for any purpose, even commercially, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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