Safety Assessment of Fuze Based on T-S Fuzzy Fault Tree and Interval Triangular Fuzzy Multi-State Bayesian Network

Abstract

In response to the relevant provisions of safety design criteria for fuze, and considering that Traditional Fault Tree Analysis (TFTA) struggles to describe system failure behavior, such as in its multi-state system faults and probabilistic logic linkages among components, this paper proposed a method for analyzing fuze system failure based on the integration of T-S Fuzzy Fault Tree (T-SFFT) and Bayesian Network (BN), introducing an interval triangular fuzzy subset method for describing failure rates in the safety assessment of the fuze system. Taking the fault tree of the fuze function prior to the initiation of the ordained arming and safety-interruption sequence as an example, using this approach, the analysis and calculation results indicated that the fuzzy subsets of failure probability for the top event under the complete failure state of the fuze system were of the same order of magnitude as those obtained using the TFTA method. This therefore validated the feasibility and effectiveness of this method in fuze system safety assessment. Furthermore, using BN to obtain the posterior probabilities of nodes, this approach provided a data foundation for fuze system fault diagnosis, holding significant engineering significance for fuze system safety assessment.

1. Introduction

Based on the description of a ‘fuze (fuze system)’ in MIL-STD-1316 [1] regarding fuze safety design standards, it can be seen that the fuze [2] serves as the core control unit for both the munition and the entire weapon system, whose safety and reliability directly impact the munition’s operational capability and accuracy of target engagement. The safety assessment of fuze is not only critical to the performance of munitions in combat, but also a pivotal subject in ammunition engineering. Its core lies in accurately predicting fuze system failure risks and identifying vulnerabilities. Correctly understanding the safety of fuze and scientifically conducting fuze safety analysis and assessment work [3] and of significance to the national defense construction of all nations.

Both domestic and international military fuze safety standards [1,4,5,6] stipulate that the safety failure rate of fuze system should not exceed 1 in 1,000,000. This requirement poses significant challenges for fuze system safety engineering [7]. Consequently, employing scientific methods to conduct safety assessments for fuze is an essential task to ensure that it meets safety requirements. Traditional safety assessment methods heavily rely on probability statistics based on historical data and fault tree analysis using Boolean logic. Sharp et al. [8] used probabilistic design and physical failure analysis to study the safety of cluster munition fuze, ensuring the safe failure rates of unexploded ordnance stayed below one percent. However, as probabilistic design relies heavily on extensive historical data, and physical analysis cannot fully simulate the relationships between internal mechanisms within the system, the universality of this approach is limited. Li et al. [9] proposed a modelling approach that integrated structure and function. This method employed structural modelling to describe the physical connections of fuze components and functional models to reflect mission processes. It established a reliability block diagram for the fuze system and assessed its ballistic reliability using fuze ballistic simulation test data. Further research is required into the interactions and interrelationships between various components within different fuze systems. However, in the context of new model development and high-tech adversarial environments, ‘small sample sizes, multi-state conditions, and cognitive ambiguity’ have become common challenges in fuze safety evaluation. Consequently, developing high-precision assessment theories capable of effectively addressing uncertainty, dynamic behaviors, and complex correlations has emerged as an urgent and practically significant task in the field of fuze technology.

Since its invention at Bell Laboratories by H. A. Watson [10] in 1961, FTA [11] has become a highly popular and powerful analytical technique. It is an inverted tree diagram used to illustrate the logical relationships between system failures and component failures, and it is also one of the key analytical methods in system safety engineering [12]. Dixon et al. [13] applied FTA to the fault tree of a pressure tank rupture, employing a bottom-up approach to determine the minimal cut sets and calculating the failure probability of the entire system under boundary conditions. Although fault tree methods have been applied in this field, they exhibit inherent limitations. Firstly, TFTA rely on rigid Boolean logic and binary-state assumptions for components, failing to characterize gradual performance degradation, multi-state failures, or dynamic system failures. Secondly, traditional probabilistic approaches require exact failure probability data as inputs, which sharply conflicts with the scarcity of historical data caused by the high reliability of fuze. When relying on expert judgment, the intrinsic epistemic ambiguity and cognitive uncertainty cannot be effectively represented by singular fuzzy numbers or precise values. These limitations often render traditional assessment models overly conservative and less credible when applied to complex fuze systems.

To address uncertainty, researchers have imported fuzzy set theory [14,15] into FTA to represent the failure probabilities of basic events, thereby better aligning with engineering practice. Liu et al. [16] applied fuzzy set theory to FTA, quantifying uncertainty in subjective information such as experts’ assessments or engineers’ experience to estimate the failure probability of lubricating oil warning systems; Lavasani et al. [17] employed fuzzy theory combined with FTA to analyze leakage through PA oil and natural-gas wells based on expert judgement. Traditional fuzzy numbers can indeed handle ambiguity; however, they neglect interpretations where experts disagree. Forcibly merging differing expert opinions into a single deterministic membership function leads to information loss, rendering the model insufficiently robust.

The T-SFFT is a fuzzy logic extension of traditional fault trees, which quantifies uncertain fault logic through membership functions and integrates fuzzy rules for handling system fault information. The T-SFFT has been employed to achieve a polymorphic description of system fault states. Bai et al. [18] employed T-SFFT fuzzy rules to quantify uncertainty in fault logic, and analyzed the diagnostic method for failures in the hydraulic power systems of coal mine tunnel drilling rigs. Considering the substantial uncertainties inherent in industrial robotic systems, Bai et al. [19] employed fuzzy T-SFFT method rules in place of Boolean logic gates to handle the uncertainty in component failure probabilities, thereby identifying the weak links within the system. Tan et al. [20] employed a T-SFFT to conduct a reliability analysis of the propulsion system in a multi-rotor UAV, overcoming the limitations inherent to TFTA. Lei et al. [21] employed fuzzy possibility to describe the degree of component faults and utilized the T-SFFT method to analyze automatic transmission slippage faults; diagnostic results for system faults were obtained. For the INS/GPS system, Song et al. [22] employed fuzzy logic and fuzzy fault methods to calculate and analyze the reliability of the integrated navigation system. However, the T-SFFT remains fundamentally a forward-looking static logical model; after obtaining system data, it lacks the capability for reverse diagnosis of fault sources.

Many systems lack precise fault data, such as insufficient historical records and variable environmental conditions. Solving T-SFFT using BN can effectively reflect the stochastic uncertainty among variables and demonstrates significant advantages in bidirectional reasoning, processing fuzzy information, and model construction. Bi et al.’s [23] proposed methodology integrates T-SFFT and BN, employing triangular fuzzy numbers to describe event probabilities and constructing conditional probability tables (CPTs) to achieve bidirectional reasoning for pumping station system failures. Wang et al. [24] conducted a safety risk assessment for the entire lifecycle of a construction robot and established a risk assessment model by combining the T-SFFT with the BNs. Although BNs can perfectly address this shortcoming, existing research generally lacks systematic treatment of interval-based cognitive uncertainty in input parameters when converting T-SFFT into BNs. This results in uncertainty information being distorted or lost during the model conversion process.

Therefore, within the field of fuze safety assessment, there remains a lack of comprehensive evaluation frameworks capable of uniformly addressing information ambiguity, expert cognitive differences, and system polymorphic failures while achieving traceability of uncertainty. To fill this research gap, this paper proposes a novel integrated evaluation framework for fuze systems based on the standard ‘Handbook for Fuze Typical Fault Tree’ (GJB/Z 29A-2021) [25]. This framework integrates interval triangular fuzzy subsets, T-SFFT, and fuzzy BN. The core innovations of this research are manifested in the following three aspects:

(1)

Uncertainty Input: Interval triangular fuzzy subsets are introduced to characterize the failure probabilities of basic events in fuze systems. This approach not only captures the intrinsic fuzziness of failure events but also accommodates and quantifies epistemic discrepancies among expert judgments through interval-valued membership functions, thereby providing more robust and engineering-practical uncertainty inputs for the model.

(2)

System Behavior Modeling: Constructing a T-SFFT for the fuze system and employing ‘if-then’ rules to meticulously describe the intricate fault propagation mechanisms within the fuze system, the limitations of traditional fault tree analysis for fuze systems are overcome.

(3)

Reasoning and Assessment: The T-SFFT of the fuze system is mapped to a fuzzy BN, with CPTs and probabilistic reasoning models constructed. This model not only enables forward safety prediction from fuze components to the fuze system but also facilitates reverse diagnosis from fuze system failures to underlying component causes.

2. Construction of Fuzzy BNs for Fuze Systems

2.1. Methods of Constructing BNs

2.1.1. BN Flowchart Construction

T-SFFT is a fault tree analysis method developed on the basis of TFTA by incorporating the Takagi-Sugeno (T-S) model. T-SFFT, by integrating the T-S fuzzy model, fuzzy numbers, IF-THEN rules, fuzzy gates, and the fuzzy possibility measure of system failure, constructs an advanced FTA method. This approach enables more accurate assessment of component importance and system reliability within engineering systems. T-S fuzzy model is a model based on fuzzy logic that can handle uncertainty and ambiguity. Fuzzy numbers are used to describe the multiple potential states of components within a system, more accurately reflecting the complexity of component states in real-word systems. The IF-THEN rule is the core component of the T-S model, describing the logical relationship between component status (input) and system status (output). Combining IF-THEN rules to construct fuzzy gates for representing logical relationships between component states addresses ambiguity and uncertainty in TFTA. The fuzzy probability of system failure is derived from the fuzzy failure rate of basic events. This approach overcomes the reliance of TFTA methods on extensive failure data and in-depth research into failure mechanisms, thereby enriching the methods for assessing system reliability.

BNs are graphical models used to represent probabilistic connections between variables, consisting of nodes, directed edges, and conditional probability distributions that form a Directed Acyclic Graph (DAG). The DAG structure of traditional BN is identical to that of fuzzy BN.

The BNs constructed and used in this methodology are illustrated in Figure 1.

2.1.2. Algorithmic Transformation Method for Constructing BNs

Based on the structural characteristics of TFTA, the TFTA is first converted into T-SFFT; the basic events (BEs), intermediate events (IEs), and top events (TEs) in the T-SFFT correspond one-to-one with the BEs, IEs, and TEs in the TFT. The logic gates in the TFT are replaced with T-S gates according to the description rules to construct the T-SFFT.

Subsequently, the T-SFFT is transformed into BNs.

In the mapping relationship between T-SFFT and BNs, BEs can be mapped to root nodes. Within the T-SFFT framework, BEs are the most granular event units, incapable of further decomposition. Similarly, in Bayesian networks, root nodes are the sole input variables lacking parent nodes, which are likewise indivisible, thereby embodying structural compatibility. The occurrence probability of BEs is primarily derived from historical data or experimental determination. The probability distribution at the root nodes must be predefined, with its data source consistent with the T-SFFT to ensure data source coherence. It can thus be seen that the root nodes are entirely consistent with the nature of the BEs. Similarly, IEs are mapped as intermediate nodes in Bayesian networks, as, in T-SFFT, these events both result from root events and cause higher-level events. Within BNs, intermediate nodes serve both as outputs from parent nodes and as inputs to child nodes, thereby establishing a correspondence between the two. TEs are mapped to leaf nodes in Bayesian networks, as, in T-SFFT, the TEs represent the ultimate output events situated at the highest level of the tree. Similarly, leaf nodes in BNs denote output variables with no child nodes, situated at the apex of the network. The two correspond to one another. When constructing BN models, the structure of T-SFFT is directly mapped as DAGs. For each T-S gate, all input events passing through the T-S gate will result in the occurrence of an output event. Mapped to a BN, a directed edge is established from all input event nodes to output event node, with the direction pointing from the parent node to the child node. Finally, we construct a Conditional Probability Table (CPT) for each non-root node based on the T-S gate, and output a BN with a complete structure and defined parameters.

To elaborate further, here is an example:

Suppose a simple system’s T-SFFT comprises two basic events BE₁ and BE₂, which output a top event TE via a T-S gate. Both BE₁ and BE₂ states are {Low, Medium, High}. The T-S gate rule library within this T-SFFT comprises the following rules:

Rule 1: IF ‘BE₁ is Low’ and ‘BE₂ is Low’, THEN ‘TE is Low’ with weight = 1.0

Rule 2: IF ‘BE₁ is Low’ and ‘BE₂ is Medium’, THEN ‘TE is Low’ with weight = 0.7; ‘TE is Medium’ with weight = 0.3

Rule 3: IF ‘BE₁ is Low’ and ‘BE₂ is High’, THEN ‘TE is Low’ with weight = 0.4; ‘TE is Medium’ with weight = 0.4; ‘TE is High’ with weight = 0.2

Rule 4: IF ‘BE₁ is Medium’ and ‘BE₂ is Low’, THEN ‘TE is Low’ with weight = 0.8; ‘TE is Medium’ with weight = 0.2

Rule 5: IF ‘BE₁ is Medium’ and ‘BE₂ is Medium’, THEN ‘TE is Medium’ with weight = 1

Rule 6: IF ‘BE₁ is Medium’ and ‘BE₂ is High’, THEN ‘TE is Medium’ with weight = 0.7; ‘TE is High’ with weight = 0.3

Rule 7: IF ‘BE₁ is High’ and ‘BE₂ is Low’, THEN ‘TE is Low’ with weight = 0.4; ‘TE is Medium’ with weight = 0.4; ‘TE is High’ with weight = 0.2

Rule 8: IF ‘BE₁ is High’ and ‘BE₂ is Medium’, THEN ‘TE is Medium’ with weight = 0.7; ‘TE is High’ with weight = 0.3

Rule 9: IF ‘BE₁ is High’ and ‘BE₂ is High’, THEN ‘TE is High’ with weight = 1

Fill in the conditional probability table for leaf node TE according to the above rules. For example:

When BE₁ = Low and BE₂ = Medium, in the TE’s CPT, P(TE = Medium) = 0.3, and P(TE = High) = 0.

When BE₁ = High and BE₂ = Low, in the TE’s CPT, P(TE = Low) = 0.4, P (TE = Medium) = 0.4. P(TE = High) = 0.2

As can be seen from the example, each ‘IF-THEN’ rule in the T-S gate rule library is converted into a conditional probability entry in the CPT. IF ‘BE₁ is $A_1^K$’ and ‘BE₂ is${ A}_2^K$’ …and ‘BE_n is $A_N^K$’, THEN ‘TE is${ B}^K$’ with w^K, meaning that in BN, when the parent nodes are (A₁^K, A₂^K, …, A_n^K), the probability of the child node TE being B^K is w^K. For all possible combinations of the parent node’s state, the conditional probability table is populated according to this rule, ensuring that the uncertainty reasoning capability of the T-SFFT is fully transferred to the BN.

2.2. Node Description Method for BN

Fuzzy BNs constitute a probabilistic model integrating fuzzy mathematics with conventional BNs. Their nodes’ description in a fuzzy Bayesian network comprises two components: the fault state description and the fault rate description. These components are employed to model the representation of uncertain information concerning the internal elements of the system.

2.2.1. The Fault State Description of Nodes

TFTA assumes systems exist only in two states: ‘normal’ and ‘failure’. However, in practice, the behavior of systems cannot be accurately described due to information fuzziness, such as uncertain probabilities and fault severity. Linguistic variables are essential tools for handling uncertain information in fuzzy logic. They enable the representation of vague phenomena that cannot be precisely defined using exact numerical values, such as “tall,” “short,” “fast,” or “slow.” Linguistic variables are particularly effective for managing the fuzzy and uncertainty prevalent in the real world. For system faults, a linguistic value set is employed to describe the fault state of a node: {‘Fully operational’, ‘Minor fault’, …, ‘Moderate fault’, …, ‘Severe fault’, …, ‘Complete failure’}. The corresponding linguistic values may be described using fuzzy numbers: {‘0’, …, ‘0.3’, …, ‘0.5’, …, ‘0.8’, …, ‘1’}.

Suppose that the fuzzy fault state of BE $x_i\left(i = 1,2,\dots ,n\right)$ is $S_i^{{(a}_i)}\left(a_i=1, 2, \dots ,{ k}_i\right)$, specifically described as the fault state vector $S_i^{{(a}_i)}=\left(S_i^{\left(1\right)},{ S}_i^{\left(2\right)},\dots , S_i^{\left(k_i\right)}\right)$, satisfying $0\le S_i^{\left(a_i\right)}\le 1$. The corresponding fuzzy fault probability is $\stackrel{\mathrm{~}}{P}\left({x_i=S}_i^{{(a}_i)}\right)$, typically derived from existing data or prior knowledge, also termed the prior probability.

When constructing a set of fault state linguistic values, the sum of the membership degrees for each fault state must equal 1. The numerical expression is as follows:

$${\mathsf{\mu }}_{\widetilde{{\mathrm{S}}_{\mathrm{i}}^{\left(1\right)}}}({\mathrm{x}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)})+{\mathsf{\mu }}_{\widetilde{{\mathrm{S}}_{\mathrm{i}}^{\left(2\right)}}}({\mathrm{x}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)})+\dots +{\mathsf{\mu }}_{\stackrel{~}{{\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{k}}_{\mathrm{i}}\right)}}}({\mathrm{x}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)})=1$$

(1)

Considering the general nature of system failures, construct a trapezoidal membership function, as shown in Figure 2. Generally speaking, $S_i^{\left(1\right)}=0,{ S}_i^{\left(k_i\right)}=1.$ The center of gravity of the fuzzy number support set is $S_i^{{(a}_i)}$; the left support radii is $s_L$; the right support radii is $s_R$; the left fuzzy area is $m_L$; and the right fuzzy area is $m_R$.

The fault state of the top event T is denoted as $T_q(q=1, 2, \dots ,k_q )$, satisfying $0\le T_{k_q}\le 1$. It is specifically described as the fault state vector $T_q= \left(T_1,,,T_2,,\dots ,,T_{\left(k_q\right)}\right)$. The occurrence fuzzy probability of this state is $\stackrel{\mathrm{~}}{P}(T=T_{\left(k_q\right)})$.

The fault state of an IE $y_j (j=1,2,\dots ,N)$ is $S_{y_j}^{{(b}_j)} (b_j=1,2,\dots ,e_{j })$, specifically described as the fault state vector $S_{y_j}^{{(b}_j)}=\left(S_{y_j}^{\left(1\right)},S_{y_j}^{\left(2\right)},\dots ,S_{y_j}^{\left(e_j\right)}\right)$, satisfying $0 \le { S}_{y_j}^{{(b}_j)}\le 1$. The occurrence fuzzy probability is $\stackrel{\mathrm{~}}{P}(y_j=S_{y_j}^{{(b}_j)})$.

2.2.2. The Fault Rate Description of Nodes

The failure state of a unit exhibits uncertainty, and an interval triangular fuzzy subset can be employed to characterize the fuzzy occurrence possibilities of basic events. The fuzzy possibility of root node $x_i\left(i=1, 2, \dots , n\right)$ is in a failure state ${\mathrm{S}}_{\mathrm{i}}^{{(\mathrm{a}}_{\mathrm{i}})},\left({\mathrm{a}}_{\mathrm{i}}=1, 2, \dots , {\mathrm{k}}_{\mathrm{i}}\right)$. Its fuzzy possibility $\stackrel{\mathrm{~}}{P}\left({x_i=S}_i^{{(a}_i)}\right)$ is expressed as follows:

$$\stackrel{\mathrm{~}}{\mathrm{P}}\left({{\mathrm{x}}_{\mathrm{i}}=\mathrm{S}}_{\mathrm{i}}^{{(\mathrm{a}}_{\mathrm{i}})}\right)=\left[\left({\mathrm{g}}^1,{\mathrm{g}}_{\mathrm{L}}^1\right);\left({\mathrm{g}}^{\mathrm{m}}\right);\left({\mathrm{g}}_{\mathrm{L}}^{\mathrm{u}},{\mathrm{g}}^{\mathrm{u}}\right)\right]=\left\{\begin{array}{c}\left\{{\mathrm{g}}^1,{\mathrm{g}}^{\mathrm{m}},{\mathrm{g}}^{\mathrm{u}}\right\}\\ \left\{{\mathrm{g}}_{\mathrm{L}}^1{,\mathrm{g}}^{\mathrm{m}},{\mathrm{g}}_{\mathrm{L}}^{\mathrm{u}}\right\} \end{array}\right.$$

(2)

In the formula, $g^m$ is the center of the fuzzy subset, representing the approximate system failure rate; $\left(g^1,g_L^1\right)$ denotes the interval containing the left endpoint of the fuzzy number’s support set; and $\left(g_L^u,g^u\right)$ A denotes the interval containing the right endpoint of the fuzzy number’s support set. According to this definition, a triangular fuzzy subset can be decomposed into a lower fuzzy number and an upper fuzzy number, where the lower fuzzy number corresponds to the minimum membership degree, with a left endpoint of $g_L^1$, a right endpoint of $g_L^u$, and a core value of $g^m$. The upper fuzzy number corresponds to the maximum membership degree, with a left endpoint of $g^l$, a right endpoint of $g^u$, and a core value of $g^m$.

${\mu }_{\stackrel{\mathrm{~}}{P}\left(S_i^{{(a}_i)}\right)}\left(g\right)$ is the membership function of $\stackrel{\mathrm{~}}{P}\left(S_i^{{(a}_i)}\right)$, where g represents the failure probability.

$${\mu }_{\stackrel{\mathrm{~}}{P}\left(S_i^{{(a}_i)}\right)}\left(g\right)=\left\{\begin{array}{ll}0;& 0\le \mathrm{g}\le {\mathrm{g}}^1\\ \left[{\mathrm{g}}^1,\mathrm{g}\right]; & { \mathrm{g}}^1\le \mathrm{g}\le {\mathrm{g}}_{\mathrm{L}}^1\\ \left[1-\frac{{\mathrm{g}}^{\mathrm{m}}-\mathrm{g}}{{\mathrm{g}}^{\mathrm{m}}-{\mathrm{g}}_{\mathrm{L}}^1},1-\frac{{\mathrm{g}}^{\mathrm{m}}-\mathrm{g}}{{\mathrm{g}}^{\mathrm{m}}-{\mathrm{g}}^1}\right]; & {\mathrm{g}}_{\mathrm{L}}^1\le \mathrm{g}\le {\mathrm{g}}^{\mathrm{m}}\\ \left[1-\frac{{\mathrm{g}-\mathrm{g}}^{\mathrm{m}}}{{\mathrm{g}}_{\mathrm{L}}^{\mathrm{u}}-{\mathrm{g}}^{\mathrm{m}}},1-\frac{{\mathrm{g}-\mathrm{g}}^{\mathrm{m}}}{{\mathrm{g}}^{\mathrm{u}}-{\mathrm{g}}^{\mathrm{m}}}\right]; & { \mathrm{g}}^{\mathrm{m}}\le \mathrm{g}\le {\mathrm{g}}_{\mathrm{L}}^{\mathrm{u}}\\ {[\mathrm{g}}_{\mathrm{L}}^{\mathrm{u}},\mathrm{g}]; & {\mathrm{g}}_{\mathrm{L}}^{\mathrm{u}}\le \mathrm{g}\le {\mathrm{g}}^{\mathrm{u}}\\ 0; & { \mathrm{g}}^{\mathrm{u}}\le \mathrm{g}\le 1\end{array}\right.$$

(3)

2.2.3. CPT Construction

Convert T-S description rules into conditional probabilities in BN. Given the operation rules $l\left(l=1, 2, \dots , r\right)$, where $r={\mathsf{\prod }}_{i=1}^n{k}_i=k_1{k}_2\dots k_n$, we assume the failure states of the basic events $x_1,x_2,\dots ,x_n$ are, respectively, $S_1^{\left(a_1\right)}$; then, the fuzzy conditional probabilities of the higher-level event y having failure states $S_y^{\left(1\right)}$,${ S}_y^{\left(2\right)}$,…, $S_y^{\left(e_j\right)}$ are ${\stackrel{\mathrm{~}}{P}}_{\left(l\right)}\left(y=S_y^{\left(1\right)}/x_1,x_2,\dots ,x_n\right)$,${ \stackrel{\mathrm{~}}{P}}_{\left(l\right)}\left(y=S_y^{\left(2\right)}/x_1,x_2,\dots ,x_n\right)$,…, ${ \stackrel{\mathrm{~}}{P}}_{\left(l\right)}\left(y=S_y^{\left(e_j\right)}/x_1,x_2,\dots ,x_n\right)$. This enables the expression of uncertain logical relationships between components, as shown in

Table 1. CPT of interval fuzzy multi-state BN.

Rules	x1	x2	xn	${\displaystyle \stackrel{\mathbf{~}}{\mathbf{P}}\mathbf{(}\mathbf{y}\mathbf{=}{\mathbf{S}}_{\mathbf{y}}^{\left(\mathbf{1}\right)}\mathbf{|}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{,}\dots \mathbf{,}{\mathbf{x}}_{\mathbf{n}}\mathbf{)}}$	$\stackrel{\mathbf{~}}{\mathbf{P}}\mathbf{(}\mathbf{y}\mathbf{=}{\mathbf{S}}_{\mathbf{y}}^{\left(\mathbf{2}\right)}\mathbf{|}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{,}\dots \mathbf{,}{\mathbf{x}}_{\mathbf{n}}\mathbf{)}$	$\stackrel{\mathbf{~}}{\mathbf{P}}\mathbf{(}\mathbf{y}\mathbf{=}{\mathbf{S}}_{\mathbf{y}}^{\left({\mathbf{e}}_{\mathbf{j}}\right)}\mathbf{|}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{,}\dots \mathbf{,}{\mathbf{x}}_{\mathbf{n}}\mathbf{)}$
L	${\mathrm{S}}_1^{{(\mathrm{a}}_1)}$	${\mathrm{S}}_2^{{(\mathrm{a}}_2)}$	${\mathrm{S}}_{\mathrm{n}}^{{(\mathrm{a}}_{\mathrm{n}})}$	${\stackrel{\mathrm{~}}{\mathrm{P}}}_{\left(\mathrm{L}\right)}\left(\mathrm{y}={\mathrm{S}}_{\mathrm{y}}^{\left(1\right)}\right)$	${\stackrel{\mathrm{~}}{\mathrm{P}}}_{\left(\mathrm{L}\right)}\left(\mathrm{y}={\mathrm{S}}_{\mathrm{y}}^{\left(2\right)}\right)$	${\stackrel{\mathrm{~}}{\mathrm{P}}}_{\left(\mathrm{L}\right)}\left(\mathrm{y}={\mathrm{S}}_{\mathrm{y}}^{\left({\mathrm{e}}_{\mathrm{j}}\right)}\right)$

.

2.2.4. Leaf Node Interval Fuzzy Subset

Based on the fault probability of the triangular interval fuzzy subset $\stackrel{\mathrm{~}}{P}\left(x_i=S_i^{\left(a_i\right)}\right)$ at the root node $x_i$ under various fault states, the interval fuzzy possibility of the leaf node T in the fault state T_q can be determined.

$$\begin{gathered} \stackrel{\mathrm{~}}{\mathrm{P}}\left(\mathrm{T}={\mathrm{T}}_{\mathrm{q}}\right)=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{{\mathrm{x}}_1,\dots ,{\mathrm{x}}_{\mathrm{i}},\dots ,{\mathrm{x}}_{\mathrm{n}},}{{\mathrm{y}}_1,\dots ,{\mathrm{y}}_{\mathrm{j}},\dots ,{\mathrm{y}}_{\mathrm{N}}}}}\stackrel{\mathrm{~}}{\mathrm{P}}({\mathrm{x}}_1,\dots ,{\mathrm{x}}_{\mathrm{i}},\dots ,{\mathrm{x}}_{\mathrm{n}},{\mathrm{y}}_1,\dots ,{\mathrm{y}}_{\mathrm{j}},\dots ,{\mathrm{y}}_{\mathrm{N}},\mathrm{T}={\mathrm{T}}_{\mathrm{q}}) \\ =\sum _{\mathsf{\pi }\left(\mathrm{T}\right)}\stackrel{\mathrm{~}}{\mathrm{P}}\left(\mathrm{T}={\mathrm{T}}_{\mathrm{q}}|\mathsf{\pi }\left(\mathrm{T}\right)\right)\sum _{\mathsf{\pi }\left({\mathrm{y}}_1\right)} \stackrel{\mathrm{~}}{\mathrm{P}}\left({\mathrm{y}}_1|\mathsf{\pi }({\mathrm{y}}_1)\right)\dots \sum _{\mathsf{\pi }({\mathrm{y}}_{\mathrm{j}})}\stackrel{\mathrm{~}}{\mathrm{P}}\left({\mathrm{y}}_{\mathrm{j}}|\mathsf{\pi }({\mathrm{y}}_{\mathrm{j}})\right)\dots \sum _{\mathsf{\pi }\left({\mathrm{y}}_{\mathrm{N}}\right)}\stackrel{\mathrm{~}}{\mathrm{P}}\left({\mathrm{y}}_{\mathrm{N}}|\mathsf{\pi }({\mathrm{y}}_{\mathrm{N}})\right)\dots \stackrel{\mathrm{~}}{\mathrm{P}}\left({{\mathrm{x}}_1=\mathrm{S}}_1^{{(\mathrm{a}}_1)}\right)\stackrel{\mathrm{~}}{\mathrm{P}}\left({{\mathrm{x}}_2=\mathrm{S}}_2^{{(\mathrm{a}}_2)}\right)\dots \stackrel{\mathrm{~}}{\mathrm{P}}\left({{\mathrm{x}}_{\mathrm{n}}=\mathrm{S}}_{\mathrm{n}}^{{(\mathrm{a}}_{\mathrm{n}})}\right) \end{gathered}$$

(4)

In this formula, ${ \stackrel{\mathrm{~}}{P}(x}_1,\dots ,x_i,\dots ,x_n,y_1,\dots ,y_j,\dots ,y_N,T=T_q)$ represents the joint distribution probability of all root nodes $x_i$ and intermediate nodes $y_j$; $\mathsf{\pi }\left(T\right)$ represents the subordinate events of leaf node T; and $\mathsf{\pi }\left(y_j\right)$ denotes the subordinate events of the intermediate node $y_j$. $\stackrel{\mathrm{~}}{P}\left(T=T_q|\mathsf{\pi }\left(T\right)\right)$ is the interval-valued fuzzy possibility of the leaf node $T$ being in failure state $Tq$ given the occurrence of its subordinate events $\mathsf{\pi }\left(T\right)$.

Based on the current fault states of the root nodes being $x_1^{\prime },\dots ,x_i^{\prime },\dots ,x_n^{\prime }$, the fuzzy possibility of leaf node T in the fault state $T_q$ can be calculated as shown in Formula (5):

$$\begin{gathered} \tilde{P}\left(\mathrm{T}={\mathrm{T}}_{\mathrm{q}}\right)=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{{\mathrm{x}}_1,\dots ,{\mathrm{x}}_{\mathrm{i}},\dots ,{\mathrm{x}}_{\mathrm{n}},}{{\mathrm{y}}_1,\dots ,{\mathrm{y}}_{\mathrm{j}},\dots ,{\mathrm{y}}_{\mathrm{N}}}}}\tilde{P}\left({\mathrm{x}}_1^{\prime },\dots ,{\mathrm{x}}_{\mathrm{i}}^{\prime },\dots ,{\mathrm{x}}_{\mathrm{n}}^{\prime },{\mathrm{y}}_1^{\prime },\dots ,{\mathrm{y}}_{\mathrm{i}}^{\prime },\dots ,{\mathrm{y}}_{\mathrm{n}}^{\prime },\mathrm{T}={\mathrm{T}}_{\mathrm{q}}\right) \\ =\sum _{\mathsf{\pi }\left(\mathrm{T}\right)} \stackrel{\mathrm{~}}{\mathrm{P}}\left(\mathrm{T}={\mathrm{T}}_{\mathrm{q}}|\mathsf{\pi }\left(\mathrm{T}\right)\right)\sum _{\mathsf{\pi }\left({\mathrm{y}}_1\right)} \stackrel{\mathrm{~}}{\mathrm{P}}\left({\mathrm{y}}_1^{\prime }|\mathsf{\pi }({\mathrm{y}}_1^{\prime })\right)\dots \sum _{\mathsf{\pi }\left({\mathrm{y}}_{\mathrm{j}}\right)} \stackrel{\mathrm{~}}{\mathrm{P}}\left({\mathrm{y}}_{\mathrm{j}}^{\prime }|\mathsf{\pi }({\mathrm{y}}_{\mathrm{j}}^{\prime })\right)\dots \sum _{\mathsf{\pi }\left({\mathrm{y}}_{\mathrm{N}}\right)} \stackrel{\mathrm{~}}{\mathrm{P}}\left({\mathrm{y}}_{\mathrm{N}}^{\prime }|\mathsf{\pi }({\mathrm{y}}_{\mathrm{N}}^{\prime })\right)\dots {\mathsf{\mu }}_{\stackrel{~}{{\mathrm{x}}_1^{\left({\mathrm{a}}_1\right)}}}\dots {\mathsf{\mu }}_{\stackrel{~}{{\mathrm{x}}_1^{\left({\mathrm{a}}_{\mathrm{n}}\right)}}} \end{gathered}$$

(5)

2.2.5. Posteriori Probability

Given known system failures, calculating the posteriori probability of component failures provides further insight into system reliability. Through backward inference in BNs, the interval fuzzy possibility of any root node can be derived.

$$\stackrel{\mathrm{~}}{\mathrm{P}}\left({\mathrm{x}}_{\mathrm{i}}={\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}|\mathrm{T}={\mathrm{T}}_{\mathrm{q}}\right)=\mathrm{E}[{\displaystyle \frac{\stackrel{\mathrm{~}}{\mathrm{P}}\left({\mathrm{x}}_{\mathrm{i}}={\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)},\mathrm{T}={\mathrm{T}}_{\mathrm{q}}\right)}{\stackrel{\mathrm{~}}{\mathrm{P}}\left(\mathrm{T}={\mathrm{T}}_{\mathrm{q}}\right)}}={\displaystyle \frac{\sum _{{\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{\mathrm{n}}}\stackrel{\mathrm{~}}{\mathrm{P}}\left({\mathrm{x}}_1,\dots ,{\mathrm{x}}_{\mathrm{i}}={\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)},\dots ,{\mathrm{x}}_{\mathrm{n}},\mathrm{T}={\mathrm{T}}_{\mathrm{q}}\right)}{\stackrel{\mathrm{~}}{\mathrm{P}}\left(\mathrm{T}={\mathrm{T}}_{\mathrm{q}}\right)}}]$$

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In the formula, $\stackrel{\mathrm{~}}{P}\left(x_i=S_i^{\left(a_i\right)},T=T_q\right)$ denotes the interval fuzzy subset where the root node $x_i$ is in fault state $S_i^{\left(a_i\right)}$ and the leaf node $T$ is in fault state $T=T_q$.

Finally, the centroid values of the posteriori probabilities are calculated using the interval midpoint method for defuzzification, facilitating the processing and analysis of fuzzy information.

3. Case Study Analysis

An accidental fuze explosion under non-intended conditions is by no means a mere “malfunction”; rather, it constitutes a systemic safety failure that can trigger a catastrophic cascade of events. Its risk spectrum spans devastating consequences: from the instantaneous annihilation of personnel and complete destruction of equipment, to mission failure and strategic disadvantage, and further extending to the erosion of trust, societal panic, and environmental devastation. Consequently, the safety of the fuze constitutes one of the paramount priorities in weapon system design. To prevent fuze safety failures, the probability of safety failure must be calculated for all phases of handling and operation, in accordance with relevant standards. This paper takes the fuze functioned prior to initiation of the ordained arming and safety-interruption sequence as an example to conduct a safety assessment of the fuze system.

3.1. Constructing the DAG of a BN

Based on GJB/Z 29A-2021 <Handbook for Fuze typical fault tree>, the primary causes of failure for the fuze functioned prior to initiation of the ordained arming and safety-interruption sequence $T_{12}$ are safety failure before the initiation of the ordained arming and safety-interruption sequence ${(B}_{12})$ and detonator initiation occurring prematurely ${(A}_{12})$. The main reasons for $A_{12}$ are electric detonator ignition ($C_{12})$ or electric detonator self-detonation ${(x}_1)$. The detonation of $\left(C_{12}\right)$ can only occur through the combined action of the three factors: removal of the short-circuit pin $\left(x_2\right)$, premature power activation ${(x}_3)$, and abnormal switch tube conduction $\left(x_4\right)$.

The primary cause of $B_{12}$ is the combined effect of the fuze-achieved interruption clearance prior to initiation of the ordained arming and safety-interruption sequence ${(T}_{11})$ and inherent failure of the fuze safety system ${(D}_{12})$. Manufacturing defects in blast-isolation material ($x_5$), inadequate strength of blast-isolation material ($x_6$), or insufficient blast-isolation distance ($x_7$) can all induce inherent failure in the safety system.

Based on the logical relationships in the conventional fault tree for the fuze at this stage, we constructed a T-SFT and then mapped it to BN, as shown in Figure 3 and Figure 4. $T_{12}$ is a leaf node; $C_{12}$, $D_{12}$, $T_{11}$, $A_{12}$, and ${\mathrm{B}}_{12}$ are intermediate nodes; and $x_1,$ $x_2,$ $x_3,$ $x_4,$ $x_5, { x}_6$, and $x_7$ are root nodes.

3.2. CPTs with Fuze Function Initiation Prior to Scheduled Arming and Enabling Sequence

3.2.1. Fuzzy Fault State Description of Fuze Nodes

As the fuze is a multi-state system, certain nodes exist in half-failure states and could be represented by fuzzy numbers. This paper investigated the failure states of a fuze system through three fault conditions (corresponding to the linguistic value set {0, 0.5, 1}):

0 = No failure state;

1 = Complete failure state;

0.5 = Partial failure state.

Considering the fuzziness of the fuze fault state, a trapezoidal membership function is employed to construct the membership function for the fuze fault state. Set $s_R$=$s_L=0.1$; $m_R$=$m_L=0.3$.

The three-state fault membership function for the system is constructed based on Figure 2, with the formula as follows:

$${\mathsf{\mu }}_{\stackrel{\mathrm{~}}{0}}\left({\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}\right)=\left\{\begin{array}{c} 1 0\le {\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}\le 0.1\\ {\displaystyle \frac{0.4-{\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}}{0.3}} 0.1\le {\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}\le 0.4\\ 0 \mathrm{O}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{s}\end{array}\right.$$

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$${\mathsf{\mu }}_{\stackrel{\mathrm{~}}{0.5}}\left({\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}\right)=\left\{\begin{array}{c}{\displaystyle \frac{{\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}-0.1}{0.3}} 0.1\le {\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}\le 0.4\\ 1 0.4\le {\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}\le 0.6 \\ {\displaystyle \frac{0.9-{\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}}{0.3} 0.6\le {\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}\le 0.9 }\\ 0 \mathrm{Others}\end{array}\right.$$

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$${\mathsf{\mu }}_{\stackrel{\mathrm{~}}{1}}\left({\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}\right)=\left\{\begin{array}{c} 1 0.9\le {\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}\le 1\\ {\displaystyle \frac{{\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}-0.6}{0.3}} 0.6\le {\mathrm{S}}_{\mathrm{i}}^{\left({\mathrm{a}}_{\mathrm{i}}\right)}\le 0.9\\ 0 \mathrm{O}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{s}\end{array}\right.$$

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Assuming the node’s current fault state is 0.45, according to Formulas (7)–(9), the membership degree for the no failure state is 0, the membership degree for the partial failure state is 2/3, and the membership degree for the complete failure state is 1/3. The sum of these membership degrees equals 1.

3.2.2. Fuzzy Fault Probability Description of the Fuze Nodes

The fuze system is a critical component within weapon systems, and achieving high reliability is one of its core design objectives. To meet this requirement, field data distinguishing between ‘partial failure’ and ‘complete failure’ is exceedingly scarce, with independent statistical data for the two categories virtually non-existent. Forcibly assigning highly uncertain fuzzy sets would introduce greater risks of subjectivity due to the lack of substantive data support. In fuze systems, the fault state manifests as a continuous gradient process from “partial fault (performance degradation)” to “complete fault (functional failure)”. Fuzzy subsets can quantify this gradual process through membership degree measurement, thereby fuzzing the boundaries between “partial fault” and “complete fault”. When the membership degrees of partial failure and complete failure are identical, this effectively discretizes continuous states into finite categories such as ‘partial failure’ or ‘complete failure’. This approach aligns with the membership function characteristics of fuzzy mathematics and is consistent with the engineering principle of ‘functional integrity first’ in assessment.

On the other hand, the safety and reliability analysis of fuze systems necessitates handling numerous conditional probability events to evaluate safety probabilities under various failure modes. Employing fuzzy subsets with identical numerical values reduces the dimensionality of the state space, thereby avoiding the need to traverse all possible combinations of damage severity. This approach disregards subtle distinctions between intermediate states, significantly enhancing computational efficiency while effectively mitigating the risk of parameter explosion at the root node level.

In this paper, the interval-valued fuzzy subsets for root nodes under partial failure state are identical in value to those under complete failure state. Based on GJB 373B-2019 <Fuze design, safety criteria for> [26], historical data, and expert experience, the interval-valued fuzzy subsets for root nodes under complete failure state were obtained, as shown in

Table 2. Interval triangular fuzzy subsets of the fuze root nodes, where the fault state is 1.

Root Nodes	Event Name	Interval Triangular Fuzzy Subsets
${\mathit{T}}_{11}$	the fuze achieved interruption clearance prior to initiation of the ordained arming and safety-interruption sequence	[(6.2,6.3); 6.4; (6.5,6.6)] × 10−9
${\mathit{x}}_1$	electric detonator self-detonation	[(2,2.5); 3; (3.5,4)] × 10−5
${\mathit{x}}_2$	removal of the short-circuit pin	[(1,1.5); 2; (2.5,3)] × 10−5
${\mathit{x}}_3$	premature power activation	[(2,3);4;(5,6)] × 10−5
${\mathit{x}}_4$	abnormal switch tube conduction	[(4,4.5); 5; (5.5,6)] × 10−5
${\mathit{x}}_5$	manufacturing defect in blast-isolation material	[(2,2.5); 3; (3.5,4)] × 10−5
${\mathit{x}}_6$	inadequate strength of blast-isolation material	[(0.5,0.7); 1; (1.5,1.7)] × 10−5
${\mathit{x}}_7$	insufficient blast-isolation distance	[(6.5,7.5); 8; (8.5,9.5)] × 10−5

.

3.2.3. Fuzzy CPT Construction of the Fuze Nodes

During the safety analysis of the fuze system, the multi-state modelling approach must be fully considered. Through the Conditional Probability Table (CPT) derived from rigorously defined T-S gate rules, this method clearly elucidates how a single (or multiple) root node can lead to fundamental differences at the system level, namely ‘partial failure’ or ‘total failure’.

Root node ${\mathrm{x}}_2$ has two failure states, represented by the fuzzy numbers (0, 1). Root nodes $x_3$, $x_4$ and child node $C_{12}$ have three failure states, represented by the fuzzy numbers (0, 0.5, 1), respectively. Considering the inherent logical uncertainty in inter-node relationships within practical conditions, the CPT for node $C_{12}$ is established as shown in

Table 3. CPT at node $C_{12}$.

Rules	${\mathbf{x}}_{\mathbf{2}}$	${\mathbf{x}}_{\mathbf{3}}$	${\mathbf{x}}_{\mathbf{4}}$	$\mathbf{P}\mathbf{(}{\mathbf{C}}_{\mathbf{12}}\mathbf{=}\mathbf{0}\mathbf{|}{\mathbf{x}}_{\mathbf{2}}\mathbf{,}{\mathbf{x}}_{\mathbf{3}}\mathbf{,}{\mathbf{x}}_{\mathbf{4}}\mathbf{)}$	$\mathbf{P}\mathbf{(}{\mathbf{C}}_{\mathbf{12}}\mathbf{=}\mathbf{0.5}\mathbf{|}{\mathbf{x}}_{\mathbf{2}}\mathbf{,}{\mathbf{x}}_{\mathbf{3}}\mathbf{,}{\mathbf{x}}_{\mathbf{4}}\mathbf{)}$	$\mathbf{P}\mathbf{(}{\mathbf{C}}_{\mathbf{12}}\mathbf{=}\mathbf{1}\mathbf{|}{\mathbf{x}}_{\mathbf{2}}\mathbf{,}{\mathbf{x}}_{\mathbf{3}}\mathbf{,}{\mathbf{x}}_{\mathbf{4}}\mathbf{)}$
1	0	0	0	1	0	0
2	0	0	0.5	1	0	0
3	0	0	1	1	0	0
4	0	0.5	0	1	0	0
5	0	0.5	0.5	1	0	0
6	0	0.5	1	1	0	0
7	0	1	0	1	0	0
8	0	1	0.5	1	0	0
9	0	1	1	1	0	0
10	1	0	0	1	0	0
11	1	0	0.5	1	0	0
12	1	0	1	1	0	0
13	1	0.5	0	1	0	0
14	1	0.5	0.5	0	0.7	0.3
15	1	0.5	1	0	0.3	0.7
16	1	1	0	1	0	0
17	1	1	0.5	0	0.3	0.7
18	1	1	1	0	0	1

.

Root nodes $x_5$, $x_6{, x}_7$ and child node $D_{12}$ have three failure states, represented by the fuzzy numbers (0, 0.5, 1), respectively. Considering the inherent logical uncertainty in inter-node relationships within practical conditions, the CPT for node $D_{12}$ is established as shown in

Table 4. CPT at node $D_{12}$.

Rules	${\mathbf{x}}_{\mathbf{5}}$	${\mathbf{x}}_{\mathbf{6}}$	${\mathbf{x}}_{\mathbf{7}}$	$\mathbf{P}\mathbf{(}{\mathbf{D}}_{\mathbf{12}}\mathbf{=}\mathbf{0}\mathbf{|}{\mathbf{x}}_{\mathbf{5}}\mathbf{,}{\mathbf{x}}_{\mathbf{6}}\mathbf{,}{\mathbf{x}}_{\mathbf{7}}\mathbf{)}$	$\mathbf{P}\mathbf{(}{\mathbf{D}}_{\mathbf{12}}\mathbf{=}\mathbf{0.5}\mathbf{|}{\mathbf{x}}_{\mathbf{5}}\mathbf{,}{\mathbf{x}}_{\mathbf{6}}\mathbf{,}{\mathbf{x}}_{\mathbf{7}}\mathbf{)}$	$\mathbf{P}\mathbf{(}{\mathbf{D}}_{\mathbf{12}}\mathbf{=}\mathbf{1}\mathbf{|}{\mathbf{x}}_{\mathbf{5}}\mathbf{,}{\mathbf{x}}_{\mathbf{6}}\mathbf{,}{\mathbf{x}}_{\mathbf{7}}\mathbf{)}$
1	0	0	0	1	0	0
2	0	0	0.5	0	0.7	0.3
3	0	0	1	0	0	1
4	0	0.5	0	0	0.7	0.3
5	0	0.5	0.5	0	0.8	0.2
6	0	0.5	1	0	0	1
7	0	1	0	0	0	1
8	0	1	0.5	0	0	1
9	0	1	1	0	0	1
10	0.5	0	0	0	0.7	0.3
11	0.5	0	0.5	0	0.8	0.2
12	0.5	0	1	0	0	1
13	0.5	0.5	0	0	0.8	0.2
14	0.5	0.5	0.5	0	0.9	0.1
15	0.5	0.5	1	0	0	1
16	0.5	1	0	0	0	1
17	0.5	1	0.5	0	0	1
18	0.5	1	1	0	0	1
19	1	0	0	0	0	1
20	1	0	0.5	0	0	1
21	1	0	1	0	0	1
22	1	0.5	0	0	0	1
23	1	0.5	0.5	0	0	1
24	1	0.5	1	0	0	1
25	1	1	0	0	0	1
26	1	1	0.5	0	0	1
27	1	1	1	0	0	1

.

Root node $x_1$ has two failure states, represented by the fuzzy numbers (0, 1). Parent nodes $C_{12}$ and child node $A_{12}$ have three failure states, represented by the fuzzy numbers (0, 0.5, 1), respectively. Considering the inherent logical uncertainty in inter-node relationships within practical conditions, the CPT for node $A_{12}$ is established as shown in

Table 5. CPT at node $A_{12}$.

Rules	${\mathbf{x}}_{\mathbf{1}}$	${\mathbf{C}}_{\mathbf{12}}$	$\mathbf{P}\mathbf{(}{\mathbf{A}}_{\mathbf{12}}\mathbf{=}\mathbf{0}\mathbf{|}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{C}}_{\mathbf{12}}\mathbf{)}$	$\mathbf{P}\mathbf{(}{\mathbf{A}}_{\mathbf{12}}\mathbf{=}\mathbf{0}\mathbf{|}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{C}}_{\mathbf{12}}\mathbf{)}$	$\mathbf{P}\mathbf{(}{\mathbf{A}}_{\mathbf{12}}\mathbf{=}\mathbf{0}\mathbf{|}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{C}}_{\mathbf{12}}\mathbf{)}$
1	0	0	1	0	0
2	0	0.5	0	0.7	0.3
3	0	1	0	0	1
4	1	0	0	0	1
5	1	0.5	0	0	1
6	1	1	0	0	1

.

Parent nodes $D_{12}$, $T_{11}$ and child node $B_{12}$ have three failure states, represented by the fuzzy numbers (0, 0.5, 1), respectively. Considering the inherent logical uncertainty in inter-node relationships within practical conditions, the CPT for node $B_{12}$ is established as shown in

Table 6. CPT at node $B_{12}$.

Rules	${\mathbf{D}}_{\mathbf{12}}$	${\mathbf{T}}_{\mathbf{11}}$	$\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{12}}\mathbf{=}\mathbf{0}\mathbf{|}{\mathbf{D}}_{\mathbf{12}}\mathbf{,}{\mathbf{T}}_{\mathbf{11}}\mathbf{)}$	$\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{12}}\mathbf{=}\mathbf{0}\mathbf{|}{\mathbf{D}}_{\mathbf{12}}\mathbf{,}{\mathbf{T}}_{\mathbf{11}}\mathbf{)}$	$\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{12}}\mathbf{=}\mathbf{0}\mathbf{|}{\mathbf{D}}_{\mathbf{12}}\mathbf{,}{\mathbf{T}}_{\mathbf{11}}\mathbf{)}$
1	0	0	1	0	0
2	0	0.5	0	0.7	0.3
3	0	1	0	0	1
4	0.5	0	0	0.7	0.3
5	0.5	0.5	0	0.5	0.5
6	0.5	1	0	0	1
7	1	0	0	0	1
8	1	0.5	0	0	1
9	1	1	0	0	1

.

Parent nodes $A_{12}$, $B_{12}$ and child node $T_{12}$ have three failure states, represented by the fuzzy numbers (0, 0.5, 1), respectively. Considering the inherent logical uncertainty in inter-node relationships within practical conditions, the CPT for node $T_{12}$ is established as shown in

Table 7. CPT at node $T_{12}$.

Rules	${\mathbf{A}}_{\mathbf{12}}$	${\mathbf{B}}_{\mathbf{12}}$	$\mathbf{P}\mathbf{(}{\mathbf{T}}_{\mathbf{12}}\mathbf{=}\mathbf{0}\mathbf{|}{\mathbf{A}}_{\mathbf{12}}\mathbf{,}{\mathbf{B}}_{\mathbf{12}}\mathbf{)}$	$\mathbf{P}\mathbf{(}{\mathbf{T}}_{\mathbf{12}}\mathbf{=}\mathbf{0}\mathbf{|}{\mathbf{A}}_{\mathbf{12}}\mathbf{,}{\mathbf{B}}_{\mathbf{12}}\mathbf{)}$	$\mathbf{P}\mathbf{(}{\mathbf{T}}_{\mathbf{12}}\mathbf{=}\mathbf{0}\mathbf{|}{\mathbf{A}}_{\mathbf{12}}\mathbf{,}{\mathbf{B}}_{\mathbf{12}}\mathbf{)}$
1	0	0	1	0	0
2	0	0.5	0.8	0.2	0
3	0	1	1	0	0
4	0.5	0	1	0	0
5	0.5	0.5	0.5	0.5	0
6	0.5	1	0.2	0.8	0
7	1	0	1	0	0
8	1	0.5	1	0	0
9	1	1	0	0	1

.

4. Numerical Analysis

4.1. Interval-Valued Fuzzy Subsets of Nodes Under Various Fault States

By using Formula (4), interval-valued fuzzy subsets for each fault state of $T_{12}$ can be determined. The calculation results are presented in

Table 8. Fault state is 0 for each node interval triangular fuzzy subset.

Nodes	Fault State is 0, Interval Triangular Fuzzy Subsets × 10−1
T12	[(9.999851, 9.999868); 9.999882; (9.999895, 9.999912)]
A12	[(9.999600, 9.999650); 9.999700; (9.999750, 9.999800)]
B12	[(9.996960, 9.997300); 9.997600; (9.997860, 9.998200)]
C12	[(9.999999, 9.999999); 9.999999; (9.999999, 9.999999)]
D12	[(9.998200, 9.997860);9.997600; (9.9973002, 9.996960)]

,

Table 9. Fault state is 0.5 for each node interval triangular fuzzy subset.

Nodes	Fault State is 0.5, Interval Triangular Fuzzy Subsets
T12	[(0.882021, 1.048590); 1.175956; (1.322911, 1.489462)] × 10−5
A12	[(0.727985, 1.842703); 3.639890; (6.256031, 9.827607)] × 10−14
B12	[(4.410192, 5.243080); 5.879958; (6.614786, 7.447609)] × 10−5
C12	[(1.04, 2.6325); 5.2; (8.9375, 14.04)] × 10−14
D12	[(6.299655, 7.489485); 8.3993000; (9.449045, 10.6387810)] × 10−5

and

Table 10. Fault state is 1 or each node interval triangular fuzzy subset.

Nodes	Fault State is 1, Interval Triangular Fuzzy Subsets
T12	[(2.718071, 4.039287); 5.435977; (7.134611, 9.180509)] × 10−9
A12	[(2.000000, 2.500000); 3.000000; (3.500000, 4.0000000)] × 10−5
B12	[(1.359036, 1.615715); 1.811992; (2.038460, 2.2951273)] × 10−4
C12	[(2.16, 5.4675); 10.8; (18.5625, 29.16)] × 10−14
D12	[(1.1699665, 1.39095); 1.559930; (1.754905,1.97587810)] × 10−4

below. For detailed calculations regarding the relevant nodes, please refer to Appendix A.

$$\begin{gathered} \stackrel{\mathrm{~}}{\mathrm{P}}\left({\mathrm{T}}_{12}={\mathrm{T}}_{\mathrm{q}}\right)=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{{\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_7}{{\mathrm{A}}_{12,}{\mathrm{B}}_{12},{\mathrm{C}}_{12},{\mathrm{D}}_{12,}{\mathrm{T}}_{11}}}}\stackrel{\mathrm{~}}{\mathrm{P}}\left({\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_7,{\mathrm{A}}_{12,}{\mathrm{B}}_{12},{\mathrm{C}}_{12},{\mathrm{D}}_{12,}{\mathrm{T}}_{11},{\mathrm{T}}_{12}=1\right)=\sum _{{\mathrm{A}}_{12},{\mathrm{B}}_{12}}\stackrel{\mathrm{~}}{\mathrm{P}}({\mathrm{T}}_{12}={\mathrm{T}}_{\mathrm{q}}|{\mathrm{A}}_{12},{\mathrm{B}}_{12}) \\ \sum _{{\mathrm{C}}_{12},{\mathrm{x}}_1}\stackrel{\mathrm{~}}{\mathrm{P}}({\mathrm{A}}_{12}|{\mathrm{C}}_{12},{\mathrm{x}}_1)·\sum _{{\mathrm{D}}_{12},{\mathrm{T}}_{11}}\stackrel{\mathrm{~}}{\mathrm{P}}({\mathrm{B}}_{12}|{\mathrm{D}}_{12},{\mathrm{T}}_{11})·\sum _{{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4}\stackrel{\mathrm{~}}{\mathrm{P}}({\mathrm{C}}_{12}|{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4)·\sum _{{\mathrm{x}}_5,{\mathrm{x}}_6,{\mathrm{x}}_7}\stackrel{\mathrm{~}}{\mathrm{P}}({\mathrm{D}}_{12}|{\mathrm{x}}_5,{\mathrm{x}}_6,{\mathrm{x}}_7)·\stackrel{\mathrm{~}}{\mathrm{P}}({\mathrm{T}}_{11})·\prod _{\mathrm{i}=1}^7\stackrel{\mathrm{~}}{\mathrm{P}}\left({\mathrm{x}}_{\mathrm{i}}\right) \end{gathered}$$

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4.2. Probability of Top Event Occurring Under the Current Failure States

We assume the root nodes are in current failure states as follows: ${\mathrm{x}}_1^{\prime }=0$; ${\mathrm{x}}_2^{\prime }=1$; ${\mathrm{x}}_3^{\prime }=0.1$; ${\mathrm{x}}_4^{\prime }=0.8$; ${\mathrm{x}}_5^{\prime }=0.2$; ${\mathrm{x}}_6^{\prime }=0.3$; ${\mathrm{x}}_7^{\prime }=0.7$; ${\mathrm{T}}_{11}^{\prime }=0.3$. Firstly, use Formulas (7)–(9) to calculate the membership degrees of the root nodes’ current fault states. Secondly, apply Formula (5) to compute the fuzzy probabilities of the system under the root nodes’ current fault states. The calculation results are presented in

Table 11. Membership degree of failure states for each root node.

Root Nodes	Failure States	Membership Degree
		0	0.5	1
${\mathrm{x}}_1^{\prime }=0$	0	1	0	0
${\mathrm{x}}_2^{\prime }=1$	1	0	0	1
${\mathrm{x}}_3^{\prime }=0.1$	0.1	1	0	0
${\mathrm{x}}_4^{\prime }=0.8$	0.8	0	0.33333	0.66667
${\mathrm{x}}_5^{\prime }=0.2$	0.2	0.66667	0.33333	0
${\mathrm{x}}_6^{\prime }=0.3$	0.3	0.33333	0.66667	0
${\mathrm{x}}_7^{\prime }=0.7$	0.7	0	0.66667	0.33333
${\mathrm{T}}_{11}^{\prime }=0.3$	0.3	0.33333	0.66667	0

and

Table 12. Fuzzy probabilities of the fuze system.

Failure States	0	0.5	1
${\mathrm{T}}_{12}$	0.939556	0.060444	0

below.

$$\begin{gathered} \stackrel{\mathrm{~}}{\mathrm{P}}\left({\mathrm{T}}_{12}=0.5\right)=\sum _{{\displaystyle \genfrac{}{}{0pt}{}{{\mathrm{x}}_1,\dots ,{\mathrm{x}}_{\mathrm{i}},\dots ,{\mathrm{x}}_{\mathrm{n}},}{{\mathrm{y}}_1,\dots ,{\mathrm{y}}_{\mathrm{j}},\dots ,{\mathrm{y}}_{\mathrm{N}}}}}\stackrel{\mathrm{~}}{\mathrm{P}}({\mathrm{x}}_1^{\prime },\dots ,{\mathrm{x}}_{\mathrm{i}}^{\prime },\dots ,{\mathrm{x}}_7^{\prime },{\mathrm{A}}_{12}^{\prime },{\mathrm{B}}_{12}^{\prime },{\mathrm{C}}_{12}^{\prime },{\mathrm{D}}_{12}^{\prime },{\mathrm{T}}_{11}^{\prime },\mathrm{T}={\mathrm{T}}_{\mathrm{q}})=\sum _{{\mathrm{A}}_{12},{\mathrm{B}}_{12}}\stackrel{\mathrm{~}}{\mathrm{P}}({\mathrm{T}}_{12}={\mathrm{T}}_{\mathrm{q}}|{\mathrm{A}}_{12}^{\prime },{\mathrm{B}}_{12}^{\prime }) · \\ \sum _{{\mathrm{C}}_{12},{\mathrm{x}}_1}\stackrel{\mathrm{~}}{\mathrm{P}}({\mathrm{A}}_{12}^{\prime }|{\mathrm{C}}_{12}^{\prime },{\mathrm{x}}_1^{\prime })·\sum _{{\mathrm{D}}_{12},{\mathrm{T}}_{11}}\stackrel{\mathrm{~}}{\mathrm{P}}({\mathrm{B}}_{12}^{\prime }|{\mathrm{D}}_{12}^{\prime },{\mathrm{T}}_{11}^{\prime })·\sum _{{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4}\stackrel{\mathrm{~}}{\mathrm{P}}({\mathrm{C}}_{12}^{\prime }|{\mathrm{x}}_2^{\prime },{\mathrm{x}}_3^{\prime },{\mathrm{x}}_4^{\prime })·\sum _{{\mathrm{x}}_5,{\mathrm{x}}_6,{\mathrm{x}}_7}\stackrel{\mathrm{~}}{\mathrm{P}}({\mathrm{D}}_{12}^{\prime }|{\mathrm{x}}_5^{\prime },{\mathrm{x}}_6^{\prime },{\mathrm{x}}_7^{\prime })·{\mathsf{\mu }}_{{\tilde{\mathrm{x}}}_1^{\left(0.5\right)}}\dots {\mathsf{\mu }}_{{\tilde{\mathrm{x}}}_7^{\prime \left(0.5\right)}}·{\mathsf{\mu }}_{{\tilde{\mathrm{T}}}_{11}^{\prime \left(0.5\right)}} \end{gathered}$$

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4.3. Posteriori Probability

Employing Formula (6) and the midpoint method, calculate the posteriori probabilities of half-failure and complete failure at the root nodes under the condition of total failure of the fuze system.The calculation results are presented in

Table 13. Posteriori probabilities of each root node.

Root Nodes ${\mathit{x}}_{\mathit{i}}$	Posteriori Probability
	$\mathbf{P}\mathbf{(}{\mathbf{x}}_{\mathbf{i}}\mathbf{=}\mathbf{0.5}\mathbf{|}{\mathbf{T}}_{\mathbf{12}}\mathbf{=}\mathbf{1}\mathbf{)}$	$\mathbf{P}\mathbf{(}{\mathbf{x}}_{\mathbf{i}}\mathbf{=}\mathbf{1}\mathbf{|}{\mathbf{T}}_{\mathbf{12}}\mathbf{=}\mathbf{1}\mathbf{)}$
$x_1$	--	2.999999988 × 10−5
$x_2$	--	4.000823959 × 10−10
$x_3$	1.600069 × 10−9	1.600095457 × 10−9
$x_4$	2.500086 × 10−9	2.50011931 × 10−9
$x_5$	2.533311 × 10−6	4.966908836 × 10−6
$x_6$	2.814439 × 10−7	1.103774237 × 10−6
$x_7$	1.801391 × 10−5	3.532024061 × 10−5
$T_{11}$	6.784334 × 10−14	2.260495399 × 10−13

.

4.4. Result Analysis

Based on the above results, the following can be concluded:

(1)

The fault probability at leaf node $T_{12}$ is in the order of 10⁻⁵ for a fault state of 0.5 and in the order of 10⁻⁹ for a fault state of 1. This indicates that both half-failure and complete failure states occurring in this fuze system prior to initiation of the ordained arming and safety-interruption sequence are low-probability events, with the complete fault state being smaller by four orders of magnitude compared to the half-failure fault state;

(2)

Based on calculations, the sum of the fault probabilities for leaf node $T_{12}$ across its three fault states (state:0, 0.5, 1) equals 1, validating the feasibility of the computational methodology. Furthermore, the fault probability magnitude for the complete fault state (state 1) complies with the requirements stipulated in China National Military Standard;

(3)

When the fault state of leaf node $T_{12}$ is 0, the probability value reaches 0.9999…, demonstrating that the system exhibits an extremely high likelihood of normal operation during this phase. This observation aligns with TFTA, thereby validating the feasibility and accuracy of employing the interval-triangular fuzzy multi-state Bayesian method for fuze safety assessment;

(4)

According to the posteriori probabilities of each node, under the condition of fuze complete failure before initiation of the ordained arming and safety-interruption sequence, the descending order of priority for detecting half-failure nodes is as follows: $x_7$, $x_5$, $x_6$, $x_4$, $x_3$, $T_{11}$ (notably, $x_1$ and $x_2$ exhibit no half-failure states). Conversely, the descending priority order for detecting complete-fault nodes is as follows: $x_7$, $x_1$, $x_5$, $x_6$, $x_4$, $x_3$, $x_2$, $T_{11}$.

5. Conclusions and Outlook

In the safety assessment of fuze systems, addressing the limitations of traditional evaluation methods in handling multi-source uncertainty and fault attribution, this paper systematically explores and constructs an integrated assessment framework combining T-SFFT with BNs. The core contribution lies in the integration and innovation of fuze system evaluation methodologies, providing a novel safety analysis and assessment tool for fuze systems characterized by sparse data and complex mechanisms. The following conclusions are drawn from this research:

(1)

This study proposes a safety assessment framework for fuze systems, effectively addressing multi-source uncertainties within the systems by integrating interval-based triangular fuzzy subsets, T-SFFT, and BNs. This framework captures both the inherent information ambiguity in system failures (uncertainty in fault boundaries) and the intrinsic vagueness in expert judgements (interval-based fuzzy quantification). Consequently, it overcomes the limitations of traditional Boolean logic when modelling multi-state failure systems;

(2)

This study defines conversion rules and algorithms for transforming T-S fault trees into Bayesian networks within fuze systems. It ensures that the complex and uncertain information contained within the T-S gate rule library is fully and faithfully mapped onto BNs, thereby establishing Bayesian structures and parameters. This methodology lays the foundation for bidirectional reasoning in fuze system fault diagnosis, encompassing both forward prediction and backward diagnosis.

(3)

Case studies indicate that the probability of the fuze functioning before the initiation of the ordained arming and safety-interruption sequence is 5.435977 × 10⁻⁹. Comparison with GJB standards and the relevant literature revealed that the calculated results were of the same order of magnitude and closely aligned in value.

(4)

This framework enables the scientific quantification of safety risks within fuze systems and facilitates fault root cause analysis. Compared to conventional methods, the analytical outcomes provided by this framework offer greater engineering guidance. It not only determines the failure probability of fuze systems but also identifies, through reverse analysis, the sources of underlying uncertainty and their propagation pathways.

Although the safety assessment framework for fuze systems developed in this study has demonstrated considerable potential, further exploration is required in the following four areas:

(1)

Optimization of Fault State Modelling: The current model employs simplified processing, classifying certain faults and complete failure states into identical fuzzy subsets. Future work will focus on exploring more uncertainty modelling approaches, establishing logically related yet mutually exclusive fuzzy subsets for different fault states to further enhance the precision and accuracy of the evaluation model.

(2)

Data Source Optimization: Within the existing framework, fault data for fuze systems primarily originates from expert assessments. Future work will incorporate physical models as prior information, thereby reducing reliance on expert assessments and enhancing the persuasiveness of evaluation outcomes.

(3)

Dynamic Analysis Extension: This research currently focuses on static safety analysis of fuze systems. Given the dynamic nature of fuze operational environments, extending this framework to dynamic BNs represents a key research direction.

(4)

Computational Efficiency Enhancement: To address the combinatorial explosion of conditional probability tables caused by multiple state nodes and parent nodes within the framework, alongside complex operations arising from fuzzy number propagation, subsequent work may incorporate technical strategies such as parallel computing or cloud computing to meet large-scale computational demands.

Looking ahead, through persistent exploration of the aforementioned issues, we believe the theoretical framework of this fuze safety assessment system will become increasingly refined, with its engineering applicability and practical value further enhanced.