Introduction

As land-based oil and gas resources are increasingly depleted, global attention is shifting towards the exploration and development of offshore fields1. Consequently, the exploitation of marine resources has been advancing rapidly. Subsea production systems, the primary mode of oil and gas development, consist of equipment installed on the seabed for oil recovery operations2. A key component of these systems is the subsea pipeline, which connects production facilities and transports oil, gas, and water. Often referred to as the “lifeline” of subsea oil and gas production, these pipelines are exposed to harsh conditions, including high pressure, temperature fluctuations, and corrosion, as well as potential damage from falling objects, fishing nets, and other hazards3,4. Given these challenges, researching the reliability of subsea pipeline systems is critical to ensuring their safe and reliable operation. However, these systems face significant challenges due to their complexity, exposure to harsh environmental conditions, and the inherent uncertainties surrounding their operation. As a result, assessing the reliability of subsea pipeline systems is an inherently difficult task, often complicated by uncertainties such as imprecise failure data, environmental variability, and the multifaceted nature of component failures. A reliable method for evaluating these uncertainties and improving risk management strategies is therefore essential5,6,7.

The reliability of subsea pipeline systems is influenced by a range of factors, including material degradation, corrosion, mechanical failures, and external events such as seismic activity or vessel impacts. A key challenge in reliability analysis is the uncertainty surrounding failure rates for individual components of the pipeline system. These failure rates are typically based on historical data, which may be scarce or unreliable, especially for subsea operations where the lack of direct monitoring and long-term operational history complicates the determination of failure rates8,9. Moreover, subsea systems often exhibit polymorphism in failure modes, meaning that a single component can fail in multiple ways depending on the conditions it faces. This polymorphism further complicates reliability analysis, as traditional methods struggle to account for the variety of potential failure states of a component. Additionally, environmental conditions, such as temperature, pressure, and marine corrosion, add layers of uncertainty that traditional models may not adequately capture10,11,12,13,14.

Traditional reliability analysis techniques, such as Fault Free Analysis (FTA), Event Tree Analysis (ETA), and Monte Carlo simulations, have limitations when it comes to incorporating these uncertainties5,15,16,17,18. For instance, fault trees typically assume that component failure rates are constant and do not account for the dynamic nature of system failure modes under varying environmental conditions. Furthermore, these methods often struggle with imprecision in failure data and ambiguous causal relationships, leading to overly simplistic or inaccurate results in complex systems like subsea pipelines.

To address these challenges, there is a growing need for advanced reliability analysis techniques that can incorporate the inherent uncertainties and polymorphism in failure modes. One promising approach is the use of Bayesian Network (BN), which have proven effective in modeling complex systems with uncertain or incomplete information. BN is a probabilistic graphical model that represents the dependencies among various components and allows for the propagation of uncertainty through the system. BN is particularly useful in reliability analysis as they can model both direct and indirect failure dependencies, update beliefs based on observed data, and provide a probabilistic interpretation of system performance. Zheng et al.19 proposed to model the high-voltage drive motor system with a multistate BN and make quantitative judgement on the system to accurately portray the various fault states of the high-voltage drive motor system, which provides a theoretical reference for improving the high-voltage drive motor system of pure electric commercial vehicles. Bai et al.20 proposed a polymorphic fuzzy reliability-based emergency management strategy for construction safety, integrating system reliability and emergency management methods. They transformed a fault tree analysis model into a reliability-based polymorphic Bayesian Network (BN), addressing issues of event polymorphism and uncertainty in logical relationships. Chan et al.21 introduced a subset simulation sampling technique in BN inference, showed how to estimate component importance measures using subset simulated samples, and demonstrated it by applying it to two road networks affected by earthquakes. Animah et al.22 provided a comprehensive review of the application of BN in the maritime industry over the past 20 years. Khakzad et al.23 took offshore drilling operations as the research object and applied BN method to conduct quantitative assessment of system failure risk. Li et al.24 further investigated the reliability analysis of floating offshore wind turbines using BN, which yielded results that were more in line with the statistical data as compared to fault tree analysis. Xiao et al.25 used BN to improve fault diagnosis of traction transformers by correlating various types of tests and faults. Sun et al.26 proposed a combination of fault physics and BN for evaluating complex electronic systems in response to the limitations of traditional methods. Seghier et al.27 proposed the importance sampling method for estimating reliability analysis results in terms of probability of failure and reliability indices, which was applied to a real case study of subsea pipeline. Yang et al.28 presented a methodology for analyzing observed anomalous events to assess the condition of submarine pipelines.

Though scholars have conducted reliability analysis on subsea pipeline system, traditional BN still face limitations when dealing with fuzziness and polymorphism in failure modes. In real-world applications, failure data is often imprecise, and there is often uncertainty regarding the exact nature of failure modes. These shortcomings have led to the exploration of more sophisticated extensions of BN, such as Fuzzy BN, which integrate fuzzy logic with BN to handle imprecise data and accommodate vague relationships. While Fuzzy BN have shown promise in capturing the fuzziness in failure rates and system behavior, they still struggle with polymorphic failure modes, where components can fail in multiple distinct ways under varying conditions. This is particularly problematic in subsea pipeline systems, where a component may fail due to corrosion, mechanical damage, or operational stress, among other factors. Consequently, traditional Fuzzy BN may not adequately address the complexity of subsea pipeline system failure modes, leading to incomplete or misleading reliability assessments. With the growth of artificial intelligence, the integration of algorithms with reliability analysis methods has also developed29,30,31,32,33.

To overcome these challenges, this study proposes a novel reliability analysis method based on a Fuzzy Polymorphic Bayesian Network (FPBN). The FPBN combines the flexibility of Bayesian networks with the capability of fuzzy logic to handle imprecise data, while also incorporating polymorphic failure modes into the model. By introducing a multi-state fault tree structure, the FPBN can model different failure states for each component and account for the fuzziness in failure rates. This integration of fuzziness and polymorphism provides a more accurate representation of the complex and uncertain nature of subsea pipeline systems. The FPBN has unique advantages compared with the traditional reliability analysis methods for complex systems.

  1. (1)

    Multi-state fault handling: FPBN allows for the representation and analysis of multiple fault states, going beyond simple binary fault models, thereby providing a more nuanced and detailed reliability assessment.

  2. (2)

    Handling uncertainty: FPBN effectively manages various uncertainties in the system, including fuzziness and randomness, making the analysis results more comprehensive and accurate.

  3. (3)

    Flexibility: FPBN offers flexibility in representing and managing complex system structures and dependencies. Its network-based model facilitates the visualization of various factors and failure modes, enhancing ease of understanding and analysis.

  4. (4)

    Modeling complex systems: FPBN possesses robust modeling capabilities, enabling it to handle systems with intricate structures and multiple dependencies. It can represent complex interactions and effects among components, thereby offering a more precise overall reliability assessment.

Compared to traditional reliability analysis methods, the key innovation of the FPBN lies in its ability to model multiple failure modes for each component, along with the uncertainty surrounding the failure rates. In traditional models, each component is typically treated as having a single failure mode with a fixed failure rate, which may not reflect the reality of systems exposed to dynamic and uncertain environmental conditions. By allowing for multiple failure states and incorporating fuzzy logic into the BN framework, the FPBN can model more realistic failure behavior, leading to a more robust and reliable analysis of system performance. Furthermore, the FPBN allows for a more nuanced understanding of system risks through reverse fault diagnosis. By calculating the posterior probabilities of each root node in the Bayesian network, the FPBN can identify weak points in the system, which is crucial for prioritizing maintenance activities and improving risk management strategies. This diagnostic capability is particularly useful in subsea pipeline systems, where early detection of potential failures can help prevent catastrophic events and reduce operational costs.

In summary, FPBN provide strong support for reliability analysis, fault diagnosis and prediction of complex systems. In this study, the Bayesian network model is constructed through the “system level - equipment level - component level”, and the triangular fuzzy set theory is introduced to upgrade the traditional node variables to fuzzy node variables, and the fault polymorphic factors are considered to solve the problems of inaccurate fault rate and unclear fault logic, so that the reliability analysis of the subsea pipeline system is closer to the actual needs of the project.

The remaining sections of this study are organized as follows. Section Subsea pipeline system describes the components and roles of subsea pipeline system. Section Fuzzy Polymorphic Bayesian Network introduces the FPBN approach. Section FPBN analysis for subsea pipeline system presents the reliability analysis of subsea pipeline system. Section Conclusion introduces the conclusion.

Subsea pipeline system

Subsea pipeline system refers to all or part of the subsea steel pipelines suspended on the seabed or buried in the soil underground, which are used for connecting and transporting oil and gas water resources in subsea production facilities, and are regarded as the “lifeline” of the subsea oil and gas production system34. The major roles and functions of the pipeline system include:

  1. (1)

    Used for transporting mixtures of oil and gas resources, water, etc.

  2. (2)

    Water injection and gas lift tasks within offshore oil and gas fields.

According to the basic working principle and functional role of the subsea pipeline system, construct the functional structure block diagram, as shown in Fig. 1.

Fig. 1
figure 1

Functional block diagram of subsea pipeline system.

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Table 1 details the subsea pipeline system subdivision into subunits and components. The boundary applies to all subsea pipelines connecting a subsea production facility to a receiving terminal, such as another subsea facility or a topside production facility (floating or fixed). Flexible or rigid pipes from the seabed to the hang-off at the receiving installation are classified as riser equipment.

Table 1 Subsea pipeline system subdivision into subunits and components.
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Fuzzy polymorphic bayesian network

Bayesian network

Bayesian Network (BN) is a mathematical model of probabilistic networks that combines graph theory, probability theory and decision analysis and is suitable for analyzing uncertainty and probabilistic problems35. BN is mainly composed of two parts: a directed acyclic graph (DAG) and a number of conditional probability tables (CPT), which can be denoted as \({\text{BN=}}\left. {\left\langle {{\text{G,P}}} \right.} \right\rangle\), with G denoting the nodes and the DAG composed of directed arcs, to describe the network topology of the BN, as the qualitative analysis part of the BN36, and P denoting the conditional probability table of the nodes, to describe the strength of the relationship between the child nodes and the parent nodes, as the quantitative analysis part of the BN. Figure 2 gives a sketch of the BN, A and B represent the root nodes also known as the parent nodes, C is the child nodes also known as the leaf nodes, and the nodes are connected to each other by directed arcs L1 and L2.

Fig. 2
figure 2

A simple example of Bayesian network.

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The DAG not only describes the association between nodes, but also determines the conditional independence between nodes. Suppose that under the condition of the parent node of a given node \({x_i}\) and the node is conditionally independent from nodes other than the parent node, as shown in Eq. (1).

$$P\left( {{x_i}\left| {Pa\left( {{x_i}} \right),A\left( {{x_i}} \right)} \right.} \right)=P\left( {{x_i}\left| {Pa\left( {{x_i}} \right)} \right.} \right)$$
(1)

in which, \(Pa\left( {{x_i}} \right)\) is the set of parent nodes of node \({x_i}\) and \(A\left( {{x_i}} \right)\) is the set of nodes other than the parent nodes of node \({x_i}\).

Therefore, the joint probability distribution \(P\left( X \right)\) of the BN correlation variable \(x=\{ {x_1},{x_2} \cdot \cdot \cdot ,{x_n}\}\) is obtained, which can be expressed as shown in Eq. (2).

$$P\left( {{X_1}, \cdot \cdot \cdot ,{X_N}} \right)=\prod\limits_{{{X_i} \in \left( {{X_1}, \cdot \cdot \cdot ,{X_N}} \right)}} {P\left( {{x_i}\left| {Pa\left( {{x_i}} \right)} \right.} \right)}$$
(2)

In Eq. (2), \(P\left( {{X_1}, \cdot \cdot \cdot ,{X_N}} \right)\) reflects the properties of the BN, and the causal relationship between the nodes depends on the conditional probability function and is in accordance with Bayes’ Theorem. Assuming that given a variable Y, then the conditional probability of X is shown in Eq. (3).

$$P\left( {\left. X \right|Y} \right)=E\left[ {\frac{{P\left( X \right)P\left( {\left. Y \right|X} \right)}}{{P\left( Y \right)}}} \right]$$
(3)

The traditional steps of BN analysis are as follows: firstly, analyze the structural composition of the target system and the common failure modes of the components and investigate the relevant reliability data; then take the system failure as the leaf node, and progressively upward deductive reasoning on the intermediate events until the bottom event, which is the root node, to establish the BN model; quantitatively analyze the system, and analyze the weak links of the system. The basic process of BN analysis is shown in Fig. 3.

Fig. 3
figure 3

Flow chart of BN analysis.

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Construction of FPBN

Compared with the traditional BN analysis, the FPBN is an extension of the traditional BN by blurring the variables of each node, which improves the ability of the BN to deal with uncertainty.

Node failure rate description

To address the uncertainty in node failure rates within BN, this study replaces exact values with fuzzy subsets. The triangular membership function, widely used for its simplicity and ease of implementation, is adopted to model the node failure rate37,38. It is known that the fuzzy likelihood of a node \({x_i}\left( {1,2, \cdot \cdot \cdot ,n} \right)\) when the fault state is case \(x_{i}^{{{a_i}}}\left( {{a_i}=1,2, \cdot \cdot \cdot ,{k_i}} \right)\) is denoted by\(P\left( {x_{i}^{{{a_i}}}} \right)\), i.e., as a fuzzy subset \(\{ g_{{i{a_i}}}^{l},g_{{i{a_i}}}^{m},g_{{i{a_i}}}^{r}\}\), where \(g_{{i{a_i}}}^{m}\) is the center of the fuzzy subset, \(g_{{i{a_i}}}^{m} - g_{{i{a_i}}}^{l}\)and \(g_{{i{a_i}}}^{r} - g_{{i{a_i}}}^{m}\)are the left and right fuzzy zones, and \({\mu _{\tilde {P}\left( {x_{i}^{{{a_i}}}} \right)}}\left( g \right)\)is the affiliation function of \(P\left( {x_{i}^{{{a_i}}}} \right)\)as shown in Fig. 4 and Eq. (4).

$${\mu _{\tilde {P}\left( {x_{i}^{{{a_i}}}} \right)}}\left( g \right)=\left\{ {\begin{array}{*{20}{c}} {\hbox{max} \{ 0,1 - \frac{{g_{{i{a_i}}}^{m} - g}}{{g_{{i{a_i}}}^{m} - g_{{i{a_i}}}^{l}}}\} \begin{array}{*{20}{c}} {}&{}&{0 \leqslant g \leqslant g_{{i{a_i}}}^{m}} \end{array}} \\ {\hbox{max} \{ 0,1 - \frac{{g - g_{{i{a_i}}}^{m}}}{{g_{{i{a_i}}}^{r} - g_{{i{a_i}}}^{m}}}\} \begin{array}{*{20}{c}} {}&{}&{g_{{i{a_i}}}^{m} \leqslant g \leqslant 1} \end{array}} \end{array}} \right.$$
(4)

in which, g is the failure probability of the node.

Fig. 4
figure 4

The membership function of \({P_{i{a_i}}}\left( t \right)\)

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Node failure state description

Traditional system reliability analysis describes the failure state of a system as two states, however, due to the changes in the operating environment of the system and different characteristics of the system itself, most of the system failure states present polymorphism and uncertainty. In this study, the system failure state is assumed to be three states (no failure, minor failure, and complete failure), and the set of linguistic values denoted as\(\{ 0,0.5,1\}\) is used to describe the failure state of nodes. BN is known to a node \({x_i}\left( {1,2, \cdot \cdot \cdot ,n} \right)\) fault state \(x_{i}^{{{a_i}}}\left( {{a_i}=1,2, \cdot \cdot \cdot ,{k_i}} \right)\), establish the affiliation function as in Eq. (5), and satisfy the sum of the three fault state affiliations of the corresponding node is 1.

$${\mu _{\tilde {0}}}\left( {x_{i}^{{{a_i}}}} \right)+{\mu _{\tilde {0}0.5}}\left( {x_{i}^{{{a_i}}}} \right)+{\mu _{\tilde {1}}}\left( {x_{i}^{{{a_i}}}} \right)=1$$
(5)

In this study, the fuzzy radius is selected as 0.1 trapezoidal affiliation function, combined with the above Eq. (5) to construct the trapezoidal affiliation function as shown in Fig. 5.

Fig. 5
figure 5

Membership function graph of language values.

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$${\mu _{\tilde {0}\left( {x_{i}^{{{a_i}}}} \right)}}=\;\;\;\left\{ {\begin{array}{*{20}{c}} {\;\begin{array}{*{20}{c}} 1&{\;\;\;\;\;\;\;\;\;}&{0<x_{i}^{{{a_i}}} \leqslant 0.1} \end{array}} \\ \begin{gathered} \begin{array}{*{20}{c}} {\frac{{0.4 - x_{i}^{{{a_i}}}}}{{0.3}}}&{}&{0.1<x_{i}^{{{a_i}}} \leqslant 0.4} \end{array} \hfill \\ 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Other \hfill \\ \end{gathered} \end{array}} \right.$$
(6)
$${\mu _{0.\tilde {5}\left( {x_{i}^{{{a_i}}}} \right)}}=\;\;\;\left\{ {\begin{array}{*{20}{c}} \begin{gathered} \frac{{x_{i}^{{{a_i}}} - 0.1}}{{0.3}}\;\;\;\;\;\;\;0.4<x_{i}^{{{a_i}}} \leqslant 0.6 \hfill \\ \begin{array}{*{20}{c}} {\;\;\;1}&{\;\;\;\;\;\;\;\;\;}&{0.4<x_{i}^{{{a_i}}} \leqslant 0.6} \end{array} \hfill \\ \end{gathered} \\ \begin{gathered} \begin{array}{*{20}{c}} {\frac{{0.9 - x_{i}^{{{a_i}}}}}{{0.3}}}&{}&{0.6<x_{i}^{{{a_i}}} \leqslant 0.9} \end{array} \hfill \\ 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Other \hfill \\ \end{gathered} \end{array}} \right.$$
(7)
$${\mu _{\tilde {1}\left( {x_{i}^{{{a_i}}}} \right)}}=\;\;\;\left\{ {\begin{array}{*{20}{c}} {\;\begin{array}{*{20}{c}} 1&{\;\;\;\;\;\;\;\;\;}&{0.9<x_{i}^{{{a_i}}} \leqslant 1} \end{array}} \\ \begin{gathered} \begin{array}{*{20}{c}} {\frac{{x_{i}^{{{a_i}}} - 0.6}}{{0.3}}}&{}&{0.6<x_{i}^{{{a_i}}} \leqslant 0.9} \end{array} \hfill \\ 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Other \hfill \\ \end{gathered} \end{array}} \right.$$
(8)

Assuming that the current \({x_i}\) fault state is 0.7, according to Eq. (6) - Eq. (8) it can be calculated that: the affiliation belonging to the mild fault state is 2/3, that belonging to the complete fault state is 1/3, and that belonging to the no-fault state is 0, \({\mu _{\tilde {0}}}\left( {0.7} \right)+{\mu _{\tilde {0}0.5}}\left( {0.7} \right)+{\mu _{\tilde {1}}}\left( {0.7} \right)=1\).

DAG for FPBN

The DAG of fuzzy Bayes and the DAG of conventional BN are consistent in the topological map, and the mapping relationship between the two corresponds to each other using the Fault Free (FT) to BN transformation method. The logic gates in FT correspond to the connection strengths of BN, and the input and output relationships between them are also consistent, as shown in Fig. 6, where a leaf node Y corresponds to multiple root nodes X.

Fig. 6
figure 6

DAG of FPBN.

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CPT for FPBN

In practical engineering applications, the fault logics between system components are often uncertain relationships, and a simple description of the fault logics between components using the CPT of traditional BN is bound to be distorted. Therefore, the CPT of FPBN is used and combined with expert knowledge and practical experience, and describes the multiple fault states of the nodes by adopting different values, and the CPT of FPBN corresponding to Fig. 6 DAG is given in Table 2.

Table 2 CPT of FPBN.
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in which, n denotes the number of root nodes, \({Y^{{b_j}}}\)denotes the fault state taken by the leaf node, \({b_j}\)denotes the number of fault states, and \(P(Y={Y^{({b_j})}}|X1=1, \cdot \cdot \cdot ,Xn=1)\) denotes the conditional probability value of the leaf node corresponding to the fault state of \({Y^{{b_j}}}\)when the root node fails.

Leaf node fuzzy possibility analysis

The root node \({X_i}\left( {i=1,2, \cdot \cdot \cdot ,n} \right)\), the intermediate node \({Y_j}\left( {j=1,2, \cdot \cdot \cdot ,m} \right)\), and the leaf node T in the BN are used to express the basic, intermediate, and top events in the system, respectively. In FNBN, fuzzy numbers \(X_{i}^{{{a_i}}}\left( {{a_i}=1,2, \cdot \cdot \cdot ,{l_i}} \right)\), \(Y_{j}^{{{b_j}}}\left( {{b_j}=1,2, \cdot \cdot \cdot ,{l_j}} \right)\) and \({T_c}\left( {c=1,2, \cdot \cdot \cdot ,l} \right)\) are used to describe the fault states of each node, where \({a_i},{b_j},{T_c}\) expresses the number of fault states.

Assuming that the interval fuzzy likelihood of each root node under different fault state conditions,\(P^{\prime } \left( {X_{i}^{{a_{i} }} } \right)\), the interval fuzzy likelihood of the leaf node T being in the fault state \({T_C}\) can be known denoted as \(P^{\prime } \left( {T={T_C}} \right)\). The solution process by applying the bucket elimination method is shown in Eq. (9).

$$\begin{gathered} {P^{\prime }}\left( {T={T_C}} \right)=\sum\limits_{{{X_1}, \cdot \cdot \cdot ,{X_n},{Y_1}, \cdot \cdot \cdot ,{Y_m}}} {{P^{\prime }}\left( {{X_1}, \cdot \cdot \cdot ,{X_n},{Y_1}, \cdot \cdot \cdot ,{Y_m},{T_C}} \right)} \hfill \\ =\sum\limits_{{Pa\left( T \right)}} {{P^{\prime }}\left( {T={T_C}\left| {Pa\left( T \right)} \right.} \right)} \prod\limits_{{j=1}}^{m} {{P^{\prime }}\left( {{Y_j}\left| {Pa\left( {{Y_j}} \right)} \right.} \right)} \prod\limits_{{i=1}}^{n} {{P^{\prime }}\left( {{X_i},{a_i}} \right)} \hfill \\ \end{gathered}$$
(9)

in which, \(Pa\left( T \right)\) is the set of parent nodes of leaf node T and \(Pa\left( {{Y_j}} \right)\)is the set of parent nodes of intermediate node\({Y_j}\).

If the state of each root node is known to be \(X_{1}^{\prime } ,X_{2}^{\prime } , \cdots ,X_{n}^{\prime }\) under the current fault state condition, the interval fuzzy probability that the leaf node T is in the fault state \({T_C}\) can be known to be denoted as \(\tilde{P}^{\prime } \left( {T = T_{C} } \right)\). Applying the bucket elimination method, the solution process is shown in Eq. (10).

$$\begin{gathered} \tilde{P}^{\prime } \left( {T = T_{C} } \right) = \sum\limits_{{X_{1} , \cdot \cdot \cdot ,X_{n} ,Y_{1} , \cdot \cdot \cdot ,Y_{m} }} {\tilde{P}^{\prime } \left( {X_{1}^{\prime } ,X_{2}^{\prime } ,....,X_{n}^{\prime } ,Y_{1} , \cdot \cdot \cdot ,Y_{m} ,T_{C} } \right)} \hfill \\ = \sum\limits_{{Pa\left( T \right)}} {\tilde{P}^{\prime } \left( {T = T_{C} \left| {Pa\left( T \right)} \right.} \right)} \prod\limits_{{j = 1}}^{m} {\tilde{P}^{\prime } \left( {Y_{j} \left| {Pa\left( {Y_{j} } \right)} \right.} \right)} \prod\limits_{{i = 1}}^{n} {\mu _{{\tilde{x}_{i}^{{a_{i} }} }} \left( {X_{i}^{\prime } } \right)} \hfill \\ \end{gathered}$$
(10)

in which, \(\mu _{{\tilde{x}_{i}^{{a_{i} }} }} \left( {X_{i}^{\prime } } \right)\left( {i = 1,2,...,n} \right)\)denotes the affiliation degree of the current fault state \(X_{i}^{\prime }\) belonging to the fuzzy state subset \(\tilde {x}_{i}^{{{a_i}}}\) of nodes in the CPT of BN.

Root node posterior probability analysis

The posteriori probability value of any root node variable \({X_i}\left( {1,2, \cdot \cdot \cdot ,n} \right)\) is thus introduced using the BN backward inference capability and node variable de-blurring38. The posteriori probability of the root node variable \({X_i}\left( {1,2, \cdot \cdot \cdot ,n} \right)\) failing to state \(X_{i}^{{{a_i}}}\left( {{a_i}=1,2, \cdot \cdot \cdot ,{l_i}} \right)\) is deduced when the leaf node T fails to state \({T_C}\) as shown in Eq. (11).

$$P\left( {{X_i}=X_{i}^{{{a_i}}}\left| {T=T{}_{C}} \right.} \right)=E\left[ {\frac{{{P^{\prime }}\left( {{X_i}=X_{i}^{{{a_i}}},T=T{}_{C}} \right)}}{{{P^{\prime }}\left( {T=T{}_{C}} \right)}}} \right]$$
(11)

in which, \({P^{\prime }}\left( {{X_i}=X_{i}^{{{a_i}}},T=T{}_{C}} \right)\) is the joint probability value of root node \({X_i}\left( {1,2, \cdot \cdot \cdot ,n} \right)\) in fault state \(X_{i}^{{{a_i}}}\left( {{a_i}=1,2, \cdot \cdot \cdot ,{l_i}} \right)\) and leaf node T in fault state \({T_C}\). \(E\left[ {\frac{{{P^{\prime }}\left( {{X_i}=X_{i}^{{{a_i}}},T=T{}_{C}} \right)}}{{{P^{\prime }}\left( {T=T{}_{C}} \right)}}} \right]\) is the center of gravity value of interval fuzzy likelihood.

FPBN analysis for subsea pipeline system

Construction of BN model for subsea pipeline system

In this study, taking the failure of subsea pipeline system as an example, firstly, a fault tree model is established, as shown in Fig. 7, Then, the model is transformed into a network model according to the FT to BN transformation method, the conversion rule is shown in Fig. 8, and the BN model is shown in Fig. 9, which is then quantitatively analyzed to complete the instance calculation of the BN model. Among them, the root node X series, the middle node Y series, and the leaf node T, correspond to the events of the fault tree in Fig. 7. The name of each event is shown in Table 3.

Fig. 7
figure 7

Fault tree model of subsea pipeline system failures.

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Fig. 8
figure 8

Fault tree conversion Bayesian network model rule.

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Fig. 9
figure 9

Bayesian network model of subsea pipeline system failures.

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Table 3 Node event coding table.
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Assuming that the root node \({X_1} - {X_{{\text{13}}}}\), intermediate node \({Y_1} - {Y_7}\) and leaf node T fault states of the system are described by the fuzzy numbers 0, 0.5, and 1, and based on the historical data of OREDA (2015)39 and expert experience the interval fuzzy likelihood can be obtained for the root node with a fault state of 0.5, and it is specified that the fault state has the same interval fuzzy likelihood of 0.5 and 1, as shown in Table 4.

Table 4 Fuzzy possibility of failure rate interval of each component.
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Due to the problem of uncertainty in the logical relationship between components, for this reason, this study comprehensively considers the OREDA (2015) historical data and expert experience from which the CPT of each sub-node is obtained, as shown in Tables 5, 6 and 7.

Table 5 CPT of node T.
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Table 6 CPT of node Y1.
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Table 7 CPT of node Y2.
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Similarly, the CPT for the remaining nodes E1-E5 can be obtained (due to space issues involving none of the remaining CPT are given).

Fuzzy subsets of each fault state in leaf nodes

According to the model algorithm given in the previous section, the fuzzy likelihood of the leaf node T being in different fault state conditions is thus obtained, the fuzzy likelihood of the leaf node is calculated using Eq. (9) and Eq. (10), as shown in Eq. (12).

$$\begin{aligned} P^{\prime } \left( {T = 1} \right) = & \sum\limits_{{X_{1} , \cdot \cdot \cdot X_{{13}} ,Y_{{\text{1}}} , \cdot \cdot \cdot ,Y_{7} }} {P^{\prime } \left( {X_{1} , \cdot \cdot \cdot X_{{13}} ,Y_{{\text{1}}} , \cdot \cdot \cdot ,Y_{7} ,T = 1} \right)} \\ = \sum\limits_{{Y_{{\text{1}}} ,Y_{{\text{2}}} }} {P^{\prime } \left( {T = 1\left| {Y_{{\text{1}}} ,Y_{2} } \right.} \right)} \times \sum\limits_{{Y_{3} ,Y_{4} ,Y_{5} ,X_{9} }} {P^{\prime } \left( {Y_{{\text{1}}} \left| {Y_{3} ,Y_{4} ,Y_{5} ,X_{9} } \right.} \right)} \times \sum\limits_{{Y_{6} ,Y_{7} }} {P^{\prime } \left( {Y_{2} \left| {Y_{6} ,Y_{7} } \right.} \right)} \\ = \sum\limits_{{X_{1} ,X_{2} ,X_{3} }} {P^{\prime } \left( {X_{1} ,X_{2} ,X_{3} } \right)} \times \cdot \cdot \cdot \times \sum\limits_{{X_{{12}} ,X_{{13}} }} {P^{\prime } \left( {Y_{7} \left| {X_{{12}} ,X_{{13}} } \right.} \right)} \times \prod\limits_{{i = 1}}^{{13}} {P^{\prime } \left( {X_{i} } \right)} \\ {\text{ = }}[{\text{5}}{\text{.626}},{\text{7}}{\text{.171}},{\text{9}}{\text{.324}}]\left( {10^{{ - 5}} /h} \right) \\ P^{\prime } \left( {T = 0.5} \right) = [1.329{\text{1}},2.2378,2.7145]\left( {10^{{ - 5}} /h} \right) \\ \end{aligned}$$
(12)

Root node posterior probability analysis

In this study, node X1 is taken as an example and Eq. (11) is used to obtain the posteriori probability that the root node X1 has a fault state of 1 in case the leaf node T has a fault state of 1 (the calculation retains four valid digits), as shown in Eq. (13).

$$P^{{\prime \prime }} \left( {X_{1} = 1\left| {T = 1} \right.} \right) = E\left[ {\frac{{P^{\prime } \left( {X_{1} = 1,T = 1} \right)}}{{P^{\prime } \left( {T = 1} \right)}}} \right] = 0.0211$$
(13)

Similarly, the posteriori probability values for each of the remaining root nodes can be obtained and are shown in Table 8.

Table 8 Posterior probability of root node.
Full size table

To view the posterior probability distribution of each root node more intuitively, the paper makes a line graph comparing the posterior probability in each state, as shown in Fig. 10.

Fig. 10
figure 10

Root node posteriori probability comparison line graph.

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According to the posteriori probability of each root node in Table 8, further suggestions and guidance can be provided for the fault diagnosis analysis of the subsea pipeline system, the corresponding root nodes are detected according to the order of the root nodes in terms of the size of the posteriori probability of the root nodes. As can be seen intuitively in Fig. 10, in the case of the system in the state of minor failure (T = 0.5) and complete failure (T = 1), the posteriori probability values of root nodes X11 and X12 are relatively large, priority should be given to detecting the subsea pipeline support structure and the subsea pipeline production isolation valve, to improve the efficiency of system fault diagnosis.

Conclusion

To solve the problem of reliability analysis of uncertain systems with multiple failure states, a method based on Fuzzy Multistate Bayesian Network (FPBN) is proposed in this study. The method establishes a multi-state system reliability analysis model, incorporates fuzzy set theory, and applies it to the risk analysis of subsea pipeline systems. By introducing interval variables in the root node failure probability, the method enhances the ability of Bayesian networks to deal with uncertainty and extends the traditional Bayesian networks. The failure probabilities of the subsea pipeline system when it is in a fault state are analyzed through quantitative calculations, which include the fuzzy failure probabilities of minor and complete failures, and the inverse failure probabilities of the system are deduced. After calculation, the posteriori probabilities of nodes X11 and X12 are the largest among all root nodes, which are relatively weak links in the system, and should be prioritized for detection or preventive maintenance when the system fails. In this way, the reliability prediction of the system is completed, which provides some guidance for the fault diagnosis of the system.

Despite its effectiveness, the proposed FPBN method has some limitations, such as its reliance on the availability of fuzzy failure data, which may be difficult to obtain in real-world scenarios. Additionally, the complexity of model may hinder its implementation in large-scale systems with numerous components. Future work could focus on improving data acquisition methods and simplifying the model for broader applications in subsea pipeline risk analysis and other engineering domains. In addition, future work can further optimize the structure and inference algorithms of FPBN models to improve computational efficiency. Combine with deep learning technology to identify potential failure risks in advance. Further develop scientific preventive maintenance strategies to reduce the probability of failure, thus improving the reliability and stability of the system.