An effective hybrid method for analysis the large-scale reliability block diagram model

Highlights Abstract ▪ This paper proposes extended diagrams (e.g., plus and multi-functional structures). ▪ A structure identification method is proposed for large-scale RBD. ▪ An analysis method based on BDD is proposed to enhance the efficiency of RBD. The reliability block diagram (RBD) is a graphical tool used for reliability modeling and analysis in various industries, including shipbuilding, aviation, and aerospace. Typically, RBDs are transformed into Bayesian networks for quantitative analysis of systems. Bayesian networks are probabilistic graphical models that can capture the uncertainties and causal relationships in complex systems. They can provide various reliability metrics such as failure probability, mean time to failure, availability, etc. However, these techniques have several drawbacks, especially for large-scale models, such as being extremely time and memory-consuming. To address these issues, we propose a hybrid method for quantitative analysis of large-scale RBDs based on the structure identification approach and binary decision diagrams. Theoretical analysis and case verification demonstrate that the proposed method is significantly more efficient than the current one.


Introduction
The reliability block diagram (RBD) is a widely used graphical modeling tool for analyzing the reliability of a system. It expresses a system as a connection of several components by their logical relation of reliability 6, and is used in various industries including aerospace 7 ships [9,15], supply chains 8 and more.
RBDs consist of series, parallel, voting, and bridging compositions, which can also be replaced by parallel-series or series-parallel combinations. Rauzy expanded k-out-of-n gates by applying the decomposition rather than expanding as a sum of products 15. Rodrigues  Traditionally, paper-and-pencil proof methods have been used for RBD-based analysis; but these methods are constrained and cannot guarantee 100% correctness. Laura 1 developed an efficient library for RBD in C programming language, and they demonstrated that their library outperformed SHARPE.
However, the tool still relies on mathematical formulas to evaluate the probability of series, parallel, k-out-of-n, and bridging blocks. There are two main drawbacks to this approach: 1) the path sets cannot be directly collected, and 2) It is impossible to determine the reliability of RBD system 5 with repeated events.
In addition to these methods, another approach is to transform the original RBD model into a secondary structure and analyze it there. For instance, an RBD can be transformed into a fault tree, which can subsequently be handled using algorithms designed to solve fault trees. Methods for calculating the reliability of fault tree with repeated events (duplicate nodes) are as follows: 1) Factorizing algorithm 20.

5) Bayesian networks 13.
All of these methods can be used directly to solve the fault tree problem, and the fault relationship between the system and its components can also be represented by RBD. Therefore, fault trees are logically equivalent to RBD 2, and these methods can be applied to RBD problems as well. The procedure for converting an RBD to a BN was described by Tchangani

Reliability block diagram
The Reliability Block Diagram (RBD) is a graphical modeling approach used to describe the logical relationships of reliability among system components. The RBD incorporates several fundamental logical structures, including series, parallel, voting, and plus structures, among others. These structures provide a means of representing the dependencies and interrelationships between system components and enable engineers to evaluate the overall reliability of complex systems.
(1) Series structure The reliability of the series structure can be calculated using eq.1: where R is the reliability of the series system; i is the index of components in the series system, and Ri represents the reliability of the i-th component.
(2) Parallel structure The reliability of the parallel structure can be calculated using eq.2: where R is the reliability of the parallel system; i is the index of components in the parallel system, and Ri represents the reliability of the i-th component. (

3) Vote structure
The reliability of the vote structure can be calculated using eq.3: where R is the reliability of the system; k is the vote value; j is the number of components in functioning state in the vote system, n is the number of components consisting of system, and r represents the reliability of the components which are independent and identically distributed.
Plus structure S S1 S2 U Sn 1 2 1 The reliability of the plus structure shown in Fig. 1 can be calculated using eq.4: where R is the reliability of the plus system; i is the index of components in the plus system; ωi is the weight of the component, and Ri represents the reliability of the i-th component.
Moreover, the RBD can be extended to represent multifunctional systems, as described in various literatures such as GJB813-1900. This extension of RBD can be evaluated using our proposed method.

Binary decision diagram
A binary decision diagram (BDD) is a tree-like structure consisting of nodes, including root nodes, non-terminal nodes, and terminal nodes, connected by directed edges. It is a data structure used to express Boolean functions. The BDD method was first proposed by Lee 10 in 1959. In 1993, Rauzy 15 introduced the BDD method for analyzing fault trees. It has been shown that BDD-based methods typically require less memory and computational time than other methods.
There are three main elements of binary decision diagrams, each of which has the following meaning: Circle: non-leaf node (non-terminal node), representing an event; Box: leaf node (terminal node) with two node values ("1" for normal, "0" for failure);

The process of the proposed method
The analysis method of the reliability block diagram model we proposed has the following key features: (1) Enhanced modeling capability for large-scale complex systems; (2) Optimized the memory usage for efficient analysis; (3) Improved the calculation efficiency for large-scale complex systems; (4) Support the solution of RBD models with plus structures; And our method consists of the following three steps: (1) Identify the basic structure of the reliability block diagram, which includes separate structures such as series, parallel, voting, plus, bridging, etc., and create a simple reliability block diagram model with a hierarchical structure.
(2) Transform the hierarchical reliability block diagram model into an equivalent binary decision diagram model.
(3) Assess the reliability level of the complex system using the BDD model that has been created.

Method assumptions and scope of application
Throughout the paper, we present a method for the large-scale RBD model, based on the following assumptions: (1) The system and affiliated components operate in either of two states: functioning or failed. These states are denoted by the values '1' and '0', respectively; (2) The components within the system are mutually independent, and there is no common cause failure; In this paper, the method proposed is a generic reliability assessment method that can provide an accurate value of the evaluated system's reliability, it can be applied to various systems. However, it is not suitable for assessing the availability of a system with repairable elements. To address the limitation, we would integrate Markov model into our method to enhance the capability of representing the repairable system.  Fig. 3.

Structure identification of RBD
(1) Set the end node of the reliability block diagram as the current node. Then traverse forward from the current node until the hierarchical RBD is fully constructed.
(2) Access the node C, and judge whether the node C has predecessor nodes, if not, the hierarchical RBD is constructed completely, and if so, count the number of predecessor nodes.
(3) If the current node C has only one predecessor P, it means that P and C are connected in series. In this case, P and C can be encapsulated into a virtual node whose predecessor is the same as P's predecessor, and whose successor is the same as C's successor.
(4) If the current node C has multiple predecessors, we need to determine whether they have a common predecessor. If so, the predecessors of C can be encapsulated into a virtual node. If not, we can access the predecessors of C in order and set them as the current node. Once this process is complete, return to node C and continue with step (2).

Transform hierarchical RBD model into BDD
After constructing the hierarchical RBD, we proceed to explain the transformation process of the RBD into binary decision diagrams. The transformation process involves the following steps: (1) Traverse the hierarchical RBD from top to bottom until an unconverted unit is found; (2) Move to the lowest module of the unit; (3) Convert these modules into the corresponding binary decision diagram based on the structure type.
(4) Backtracking upwards and converting layer by layer to form binary decision diagram.
The flowchart for the BDD transformation is depicted in Fig. 4  In the corresponding BDD, there is only one path that from the root node to the terminal node 0 indicates the system failure. The probability of system failure can be calculated by the product of the probability of components failure.

c) Voting structure
The voting structure can be expressed as a combination of series and parallel structures, so that the voting structure can first be converted to a series and parallel structure, and then to a corresponding binary decision diagram model. A simple voting structure converted to a series and parallel structure is shown in Fig. 7. Assuming that there are n units in the whole plus structure, n virtual nodes need to be added. The edges of the virtual nodes represent the weight value of the units. An example of the binary decision diagram of a three-unit is shown in Fig. 8. Fig. 8. Binary decision diagram corresponding to the plus structure.
Eksploatacja i Niezawodność -Maintenance and Reliability Vol. 25, No. 3, 2023 In the same way it can be shown that the calculation of the plus structure is also accurate.

System reliability evaluation based on BDD
After the hierarchical RBD has been converted to BDD, traverse the path where the leaf node is 1 in the BDD, multiply the probabilities on the path, and add the results of the multiplication to get the reliability of a complex system, the calculation process is as follows: = {( , )| , ∈ }， , = 1,2, ⋯ , Where N is nodes' sets of the BDD, ( , ) is the path in the path sets .

Case definition and application
The following case is used to demonstrate that the proposed method takes advantage over the three levels Bayesian networks proposed by Toledano [15] .
This reliability block diagram is obtained after the modelling of a complex system, as seen in Fig. 9. The reliability block diagram is identified and hierarchical according to the methods mentioned above, as illustrated in Fig. 10.  The BDD that corresponds to RBD shown in Fig. 10 is depicted in Fig. 11, while the Bayesian network is illustrated in (3)   In addition to theory analysis, to demonstrate our method has an advantage over the BN-based method, we create some RBD models and obtain the run time of the two methods applied to the same model. The RBD structure for efficiency comparison between our method and BN-based method is shown in Fig. 13. The components are connected in parallel then in series. We assume the number of the components in parallel structure and the number of the modules in series structure are m and n respectively. In the first parameters group, fix the parameter n to 50 and vary the parameter m which is determined from Table 1. In the second parameters group, set parameter m to 10 and the parameter n is variable whose value is as in Table 2.  The comparison results between our method and the BNbased method are illustrated in Fig. 14 and Fig. 15. It is obvious that performance of our method is much better than BN-based method. In our case study, the computer configuration used is 11th Gen Intel(R) Core (TM)i7-1165G7 @ 2.80 GHz, 16 GB DDR4 RAM, Microsoft Windows 10 professional, and the two methods all were implemented in C# language.

Conclusions and future work
To address the lengthy computation time and significant memory consumption associated with large-scale RBDs, we propose a novel approach. Our approach offers several advantages over conventional methods: (1) The analyst can recognize the system's units in the hierarchical RBD model because it differs little from the original model.
(2) The proposed method is suitable for general RBDs. In addition to series, parallel and voting, plus structures and multifunctional RBDs.
(3) The method is more efficient and uses less memory than previous methods.
(4) By identifying the structure and hierarchy of the RBD, we can generate a modular RBD that facilitates parallel processing, further improving computational efficiency.
In the future, we will integrate the Markov models into our method to overcome the limitation that the method is unsuitable for maintainable systems. Besides, with regards to the issue of model conversion accuracy, we intend to carry out further investigation and verification in subsequent research.