A Two-Layer Factor and Cloud Model-Based Approach to Reliability Allocation

Abstract

Traditional reliability allocation methods face three inherent limitations: (1) the insufficient consideration of influencing factors, (2) the equal treatment of impact factors, and (3) the presence of fuzzy information. To overcome these issues, we propose an integrated reliability allocation method that combines a two-layer integrated factors framework with a cloud model. First, a two-layer framework of influencing factors was constructed, and a mathematical model for global evaluation based on these factors was developed, along with an indicator allocation model derived from the evaluation model. The cloud model was employed to quantitatively express experts’ linguistic evaluations, eliminating the influence of randomness and fuzziness in the assessment process. Both theoretical analysis and practical examples show that the proposed method outperforms three other reliability allocation methods.

1. Introduction

Reliability allocation is a crucial process in the development of new systems, ensuring that reliability targets, such as failure rates or reliability assigned to various components, meet overall system requirements.

Several conventional reliability allocation methods are available, including the equalization allocation method [1], ARINC [2], The Advisory Group on Reliability of Electronic Equipment (AGREE) [3], and the FOO technique [1]. To achieve more balanced allocation results, Kuo [4] proposed the AWM method, which includes two reliability models that utilize average scores from multiple experts. De Felice [5] developed the integrated factors method (IFM), which considers a large number of influencing factors. Subsequently, Bona [6] introduced the IFM target 2.0 method, which enhances the original IFM method by incorporating new maintenance-related factors. In addition, to reduce the subjective influence of experts, researchers have applied results from other reliability analysis techniques, such as FMEA [7,8,9] and reliability prediction [10] to conduct reliability allocation, typically in combination with conventional reliability allocation methods.

In the above approaches, allocation factors are generally treated as equally weighted. However, in practice, some factors have a greater impact on reliability than others. For instance, Bracha [11] considered that technical level is more important than three other factors but did not provide a specific method for quantifying this importance. Chang [12] addressed two fundamental problems—measurement scale and unequal weighting of different factors—and proposed a reliability allocation method using the maximal entropy ordered weighted average method (ME-OWA) [13]. Liaw and Chang [10] proposed a ME-OWA-based DEMATEL reliability apportionment method to address five fundamental challenges. Bona [14] proposed the Analytic IFM method (A-IFM), which combines the IFM method with the Analytic Hierarchy Process (AHP), and Dai [15] applied the AHP-IFM approach to agricultural machinery, demonstrating its feasibility.

In addition to the individual methods mentioned above, integrated approaches to combining multiple allocation methods for reliability allocation have been explored. For instance, Boyd [16] combined the equal allocation method with the ARINC method, while Wang [17] proposed a method based on probabilistic safety analysis method to determine the reliability index of safety-related components.

From the above literature, it is evident that expert scoring forms the foundation of most reliability allocation methods. However, the impact of randomness and fuzziness in expert decision-making has not been considered. Additionally, these methods generally assume that influencing factors are independent. However, in practice, interdependencies often exist among various influencing factors. For example, the more complex a subsystem is, the more difficult its maintenance tends to be. Subsystems with longer operation times are more crucial for the entire machine to complete specified tasks, and their failures cause greater fault losses.

To address these limitations, we propose a new reliability allocation method that integrates a two-layer factors method with a cloud model and grey relational analysis. Our method systematically incorporates granular influencing factors to establish a comprehensive indicator allocation framework. Furthermore, it introduces a cloud model to address both the fuzziness and randomness inherent in expert scoring processes, while grey relational analysis is employed to quantify interdependencies among influencing factors, yielding more accurate indicator vectors.

2. Materials and Methods

2.1. Two-Layer IFM with Cloud Model

This paper proposes a reliability allocation method based on a two-layer integrated factors method, cloud model, and grey relational analysis. The framework of the proposed method is illustrated in Figure 1.

A two-layer framework of influencing factors was constructed, and a mathematical model for global evaluation based on these factors was developed, along with an indicator allocation model derived from it.

Cloud model theory [18,19], grounded in fuzzy mathematics and probability theory, provides a robust approach for handling uncertainty in transitions between qualitative concepts and quantitative data. Unlike conventional methods that rely on precise numerical inputs, the cloud model represents knowledge using stochastic cloud droplets, which denote the possible membership degrees of a specific value within a linguistic term [20]. This allows it to effectively capture the fuzziness and randomness inherent in expert judgments, making it particularly well-suited for expressing experts’ preferences in reliability allocation.

The cloud model is employed to quantitatively represent the linguistic evaluations of decision-making experts, reducing the influence of randomness and fuzziness in the assessment process. Grey relational analysis is then introduced to process disordered correlation data among influencing factors, identify inter-factor relationships, and reveal primary contradictions—i.e., factor pairs with high relational grades but conflicting effects (e.g., cost vs. reliability). Based on their degrees of influence, all relevant factors are ranked according to their overall impact.

2.2. Influencing Factor Selection

Five influencing factors have been identified to improve the rationality and effectiveness of reliability allocation: technology level, importance, complexity, operating environment, and cost-effectiveness. This factor set can be adapted by designers to meet specific product requirements. During the early design phase, a minimal factor set (≥4) is recommended to maintain computational efficiency while ensuring adequate subsystem characterization. As the design matures, additional factors (≤8) may be incorporated to enhance model precision. This balanced approach prevents both oversimplification (insufficient subsystem representation) and excessive computational overhead.

From a lifecycle perspective, the first-layer technology level factor is jointly affected by the design, manufacturing, and assembly phases, and is modeled as a multiplicative effect. The importance factor reflects how critical a component is to system function and safety; it is determined by two second-layer factors: failure consequence and probability importance. Failure consequence measures the impact severity, while probability importance reflects the likelihood that the component contributes to system failure. The first-layer complexity factor represents both the structural intricacy and operational challenges. It is determined by two second-layer factors: the number of constituent parts and the difficulty of disassembly and assembly, which reflect design complexity and maintenance accessibility during operation, respectively. The first-layer environmental condition factor captures external physical and operational stresses on the component’s reliability. It is influenced by two second-layer factors: operation conditions (e.g., temperature, humidity, vibration, or radiation) and the duty cycle, which indicates how frequently and intensively the component is used during its service life. Together, these factors characterize the environmental stress acting on the component. The first-layer cost-effectiveness factor represents the balance between reliability improvement and resource investment over the product’s lifecycle. It is determined by three second-layer factors: cost-effectiveness at the design level, manufacturing level, and assembly level, each representing the relative benefit of reliability enhancement versus cost input at the corresponding stage. Together, they evaluate the economic efficiency of reliability measures throughout the product development process. These first-layer factors and their corresponding subfactors are summarized in

Table 1. Factors and subfactors.

Factors	Subfactors
Technology level (K1)	Design level (k11)
	Manufacturing level (k12)
	Assembly level (k13)
Importance (K2)	Failure consequence (k21)
	Probability importance (k22)
Complexity (K3)	Number of the parts (k31)
	Disassembly/assembly difficulty (k32)
Environmental condition (K4)	Operation conditions (k41)
	Duty cycle of the equipment (k42)
Cost-effectiveness (K5)	Cost-effectiveness of design (k51)
	Cost-effectiveness of manufacturing (k52)
	Cost-effectiveness of assembly (k53)

.

As an illustrative example, consider the Electric Motor Control Center. Its technology level (K₁) is relatively high, supported by advanced control logic in the design phase (k₁₁), precision manufacturing (k₁₂), and tight assembly tolerances (k₁₃). In terms of importance (K₂), a failure in this subsystem could lead to core damage (k₂₁), and its probability importance (k₂₂) indicates a strong impact on overall system reliability. The subsystem also exhibits high complexity, characterized by a large number of components (k₃₁) and considerable difficulty in disassembly and assembly (k₃₂). Regarding environmental conditions (K₄), it operates under high temperatures (k₄₁) and typically experiences a low duty cycle (k₄₂). For cost-effectiveness (K₅), improving reliability varies at the design stage (k₅₁) is relatively cost-effective, whereas enhancements at the manufacturing (k₅₂) and assembly (k₅₃) stages tend to be more resource-intensive due to stricter process and quality requirements.

For reliability allocation methods based on weighted factors, a larger weight results in lower allocated reliability, and vice versa. The weight of the vth subsystem can be calculated as follows:

$$w^v={\displaystyle \frac{{GI}^v}{\sum _{v=1}^n{GI}^v}}$$

(1)

where ${GI}^v$ denotes the global index obtained from the influencing factors;

• n denotes the number of subsystems or components in the system.

The above equation shows that the global index is inversely related to system reliability. The following analysis explores how different influencing factors affect the global index. According to the principles of reliability allocation, higher values for second-level subfactors such as k₃₁, k₃₂, k₄₁, k₄₂, k₅₁, k₅₂, and k₅₃ correspond to lower allocated reliability. Since the global indicator is inversely proportional to reliability, these second-level subfactors—k₃₁, k₃₂, k₄₁, k₄₂, k₅₁, k₅₂, and k₅₃—are directly proportional to the global indicator. Similarly, the higher the values for second-level subfactors k₁₁, k₁₂, k₁₃, k₂₁, k₂₂, the higher the allocation reliability. Therefore, these subfactors (k₁₁, k₁₂, k₁₃, k₂₁, k₂₂) are inversely proportional to the global indicator. We used the same notation to represent expert ratings. The weight coefficients of the first-layer factors were used to determine the influence levels of their respective second-layer subfactors. Based on this, the mathematical model for calculating the global indicator was defined as follows:

$${\mathrm{GI}}^v={\left[{\displaystyle \frac{1}{k_{11}^v{k}_{12}^v{k}_{13}^v}}\right]}^{K_1^v}{\left[{\displaystyle \frac{1}{k_{21}^v{k}_{22}^v}}\right]}^{K_2^v}{\left[k_{31}^v{k}_{32}^v\right]}^{K_3^v}{\left[k_{41}^v{k}_{42}^v\right]}^{K_4^v}{\left[k_{51}^v{k}_{52}^v{k}_{53}^v\right]}^{K_5^v}$$

(2)

where $K_i^v$ denotes the weight coefficient of the first-layer factors influencing the vth subsystem;

• $k_{ij}^v$ represents expert scores based on second-layer subfactors.

2.3. Coefficients of Second-Layer Factors Based on the Cloud Model

Cloud model theory, which integrates fuzziness and probabilistic theory, addresses real-world imprecision and uncertainty. It can credibly describe fuzzy concepts and random variables using a series of discrete points instead of fixed numerical values. The most common normal cloud is written as C = (Ex, En, He). Ex (Expectation) describes a constituent’s formal concept, En (entropy) describes the random and fuzzy nature of qualitative concepts, and He (hyper-entropy) indicates uncertainty in members.

The computational rules for the two cloud models are as follows:

$$C_1+C_2=\left(E_{x1}+E_{x2},\sqrt{E_{n1}^2+E_{n2}^2},\sqrt{H_{e1}^2+H_{e2}^2}\right)$$

$$C_1-C_2=\left(E_{x1}-E_{x2},\sqrt{E_{n1}^2+E_{n2}^2},\sqrt{H_{e1}^2+H_{e2}^2}\right)$$

$$C_1\times C_2=\left(E_{x1}·E_{x2},|E_{x1}·E_{x2}|\sqrt{{({\displaystyle \frac{E_{n1}}{E_{x1}}})}^2+{({\displaystyle \frac{E_{n2}}{E_{x2}}})}^2},|E_{x1}·E_{x2}|\sqrt{{({\displaystyle \frac{E_{e1}}{E_{x1}}})}^2+{({\displaystyle \frac{E_{e2}}{E_{x2}}})}^2}\right)$$

$$C_1∕C_2=\left({\displaystyle \frac{E_{x1}}{E_{x2}}},|{\displaystyle \frac{E_{x1}}{E_{x2}}}|\sqrt{{({\displaystyle \frac{E_{n1}}{E_{x1}}})}^2+{({\displaystyle \frac{E_{n2}}{E_{x2}}})}^2},|{\displaystyle \frac{E_{x1}}{E_{x2}}}|\sqrt{{({\displaystyle \frac{E_{e1}}{E_{x1}}})}^2+{({\displaystyle \frac{E_{e2}}{E_{x2}}})}^2}\right)$$

In reliability allocation research, it is necessary to employ expert decision-making method to provide the relevant evaluation data upon which the reliability allocation coefficients of each subsystem are calculated. However, due to factors such as the accumulated work experience of decision-making experts, psychological influences during the decision-making process, and cognitive fuzziness or hesitancy in judgment, the resulting evaluations often exhibit randomness and ambiguity, ultimately affecting the accuracy of the allocation results.

The cloud model addresses these issues by uniformly representing the randomness and fuzziness inherent in expert linguistic assessments using three numerical characteristics: Expected Value (Ex), Entropy (En), and Hyper-Entropy (He). This representation prevents the loss of linguistic information during the decision-making process.

Therefore, in this study, we introduced the cloud model to quantify the linguistic information from expert evaluations during comprehensive reliability analysis, thereby enabling more accurate reliability assessment. The cloud model terminology sets used in this study are shown in

Table 2. Cloud model terminology sets used in this study.

Linguistic Terms	Label	Cloud Model
Very low/easy/few	VL	(0, 16.7, 0.424)
Low/easy/few	L	(19.1, 10.31, 0.262)
Rather low/easy/few	RL	(30.9, 6.37, 0.162)
Medium	M	(50, 3.93, 0.1)
Rather high/hard/much	RH	(69.1, 6.37, 0.162)
High/hard/much	H	(80.9, 10.31, 0.262)
Very high/hard/much	VH	(100, 16.7, 0.424)

.

The cloud model of the dth expert for the second-level influencing factor k_ij in the vth subsystem is as follows:

$$C_{d,ij}^v=(E_{x,d,ij}^v{,E}_{n,d,ij}^v,H_{e,d,ij}^v)$$

(3)

The integrated cloud model can be determined by the following formula:

$$C_{ij}^v=\sum _{d=1}^k{\gamma }_d{C}_{d,ij}^v=(\sum _{d=1}^k{\gamma }_d{E}_{x,ij},\sqrt{\sum _{d=1}^k{\gamma }_d{E_{n,ij}}^2},\sqrt{\sum _{d=1}^k{\gamma }_d{H_{e,ij}}^2})$$

(4)

${\gamma }_1$ represents the weight of the ith experts. The coefficient of the second-layer factors can be determined by the integrated cloud model.

2.4. Coefficients of First-Layer Factors Based on Grey Relational Analysis

Traditional comprehensive factor methods do not account for the interrelationships among different types of influencing factors. To address this limitation, the grey relational analysis method was introduced to process disordered correlation data within the influencing factor sequences, identify the correlations and principal contradictions among all analyzed factors, and rank the degree of influence each factor exerts on the overall system. The method for determining the weight coefficients of class factors using grey relational analysis is as follows.

For the first-layer factors K = {K₁, K₂, K₃, K₄, K₅}, the score provided by the eth expert for the ith class factor of the vth subsystem is denoted as $u_{ei}^v$, the scores of the first-layer factors of vth subsystem are shown in

Table 3. The scores of first-layer factors by m experts.

Experts	First-Layer Factors
	K1	K2	K3	K4	K5
1	$u_{11}^v$	$u_{12}^v$	$u_{13}^v$	$u_{14}^v$	$u_{15}^v$
2	$u_{21}^v$	$u_{22}^v$	$u_{23}^v$	$u_{24}^v$	$u_{25}^v$
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
m	$u_{m1}^v$	$u_{m2}^v$	$u_{m3}^v$	$u_{m4}^v$	$u_{m5}^v$

.

The method for determining weights based on grey relational theory treats the group influencing factors as the characteristic sequence of the system, and the individual influencing factors as the behavioral sequence. By comparing the behavioral sequence with the characteristic sequence, the relational degree values of the individual factors are calculated. These values are then normalized to determine the relational weights of the class-level influencing factors on the subsystem. The higher the relational weight of an individual factor, the greater its influence on the system.

To determine the characteristic and behavioral sequences of the influencing factors for the subsystem, the characteristic sequence of the vth subsystem is denoted as

$$U_o^v=\{u_{o1}^v,u_{o2}^v,\cdots ,u_{oe}^v,\cdots ,u_{om}^v\}$$

$$u_{oe}^v=max\{u_{e1}^v,u_{e2}^v,\cdots ,u_{en}^v\}$$

The behavioral sequence of the influencing factors is presented in the columns in Table 3.

$$U_{ei}^v=\{u_{1i}^v,u_{2i}^v,\cdots ,u_{mi}^v\}$$

To calculate the grey correlation coefficient and correlation degree of the class factors, let the absolute residual sequence between the characteristic sequence and the behavioral sequence of each influencing factor, as obtained by the eth expert, be denoted as ${∆}_e^v$. Each column of this sequence is then treated as an individual influencing factor and substituted into the following equation to determine the corresponding correlation coefficient.

$${\xi }_i^v(e)={\displaystyle \frac{\underset{i}{\mathit{min}} \underset{e}{\mathit{min}}{∆}_i^v\left(e\right)+\rho \underset{i}{\mathit{max}} \underset{e}{\mathit{max}}{∆}_i^v(e)}{{∆}_i^v(e)+\rho \underset{i}{\mathit{max}} \underset{e}{\mathit{max}}{∆}_i^v\left(e\right)}}$$

(5)

• $\underset{i}{\mathit{min}}\underset{e}{\mathit{min}}{∆}_i^v\left(e\right)$ represents the minimum absolute residual value;
• $\underset{i}{\mathit{max}}\underset{e}{\mathit{max}}{∆}_i^v\left(e\right)$ represents the maximum absolute residual value.

The distinguishing coefficient was set to 0.5 in this study. The correlation coefficient can be calculated by the following formula:

$$r_i^v={\displaystyle \frac{1}{m}}\sum _{e=1}^m{\xi }_i^v\left(e\right)$$

(6)

By normalizing the correlation degrees, the weight vector of class factors for the vth subsystem can be obtained as

$$K^v=(K_1^v,K_2^v,K_3^v,K_4^v,K_5^v)$$

$$K_i^v={\displaystyle \frac{r_i^v}{\sum _{i=1}^n{r}_i^v}}$$

(7)

3. Results

3.1. System Description

The automatic depressurization system (ADS), a key part of the reactor coolant system, regulates system depressurization through the sequential actuation of valves. This process maintains safety injection flow within design limits, ensuring emergency core cooling. Figure 2 illustrates the structural configuration, and Figure 3 presents the corresponding reliability block diagram.

3.2. Reliability Allocation Process with Our Method

Based on these systems, we selected five influencing factors: technology level, importance, complexity, environmental conditions, and cost-effectiveness. The fuzzy language used for scoring, along with the defined terminology set for the cloud model, is presented in Table 2. The weights assigned to each expert are listed in

Table 4. The weights of the experts.

Experts	Weight
No. 1	0.2
No. 2	0.3
No. 3	0.2
No. 4	0.1
No. 5	0.1
No. 6	0.1

.

The evaluation of second-level subfactors was conducted according to the criteria established in Table 2. The resultant scores are systematically presented in

Table 5. The scores of second-layer factors for valve #1.

Factors	No. 1	No. 2	No. 3	No. 4	No. 5	No. 6
k11	RL	RH	M	M	RL	RH
k12	RL	M	M	M	RH	M
k13	RL	L	L	M	M	RL
k21	H	VH	VH	VH	H	VH
k22	L	L	L	VL	L	L
k31	L	VL	VL	VL	L	VL
k32	M	M	RL	RL	M	RL
k41	RL	RL	H	M	L	M
k42	RL	RL	RL	M	M	M
k51	RL	M	H	RL	M	L
k52	H	RH	H	M	M	RH
k53	M	RH	M	RL	M	M

.

The integrated cloud model is obtained by Equation (4) and presented in

Table 6. Integrated cloud model of the second-layer factors for valve #1.

Factors	Integrated Cloud Model
k11	(51.91, 5.747, 0.146)
k12	(48.09, 4.794, 0.121)
k13	(28.82, 8.271, 0.210)
k21	(94.27, 15.070, 0.382)
k22	(17.19, 11.115, 0.282)
k31	(5.73, 15.070, 0.382)
k32	(42.36, 5.049, 0.128)
k41	(43.54, 7.434, 0.188)
k42	(36.63, 5.747, 0.146)
k51	(47.36, 7.088, 0.180)
k52	(70, 7.863, 0.1998)
k53	(53.82, 5.049, 0.128)

. The results represent a synthesis of multiple experts’ knowledge, reflecting their collective understanding of the key factors affecting reliability. For example, factor k₂₁ (failure consequence) has a high expected value (94.27), indicating its high importance. However, its entropy (15.070) is also relatively high, suggesting that expert opinions or data regarding this factor are widely dispersed, implying lower confidence in its assessment. Through this process, the global indicators of the second-layer influencing factors for valve #1 were determined.

The expert scoring results for the first-layer factors of valve #1 are presented in

Table 7. The scores of first-layer factors for valve #1.

Factors	No. 1	No. 2	No. 3	No. 4	No. 5	No. 6
K1	H	M	M	RH	RH	M
K2	RH	M	RH	RH	H	M
K3	M	RH	M	RH	M	M
K4	L	L	RL	L	M	RL
K5	RH	M	RH	M	RH	M

. These evaluations are then quantified using the cloud model. Scores for the other subsystems can be obtained in the same manner.

The characteristic sequence is $U_0=\{H,RH,RH,RH,H,M\}$. Each row is treated as a behavior sequence $U_i$. The absolute residual sequence can be obtained by comparing U₀ and U_i, and the results are presented in

Table 8. Absolute residual sequence.

Factors	No. 1	No. 2	No. 3	No. 4	No. 5	No. 6
$∆K1$	0	19.1	19.1	11.8	11.8	0
$∆K2$	11.8	30.9	11.8	11.8	0	0
$∆K3$	19.1	0	19.1	0	19.1	0
$∆K4$	30.9	30.9	19.1	30.9	0	19.1
$∆K5$	0	19.1	0	19.1	0	0

.

$$\underset{i}{\mathit{min}} \underset{e}{\mathit{min}}∆K_i min=0,\underset{i}{\mathit{max}} \underset{e}{\mathit{max}}∆K_i=30.9$$

Let $\rho =0.5$; the coefficients are derived using the specific formula, and the results are presented in

Table 9. Coefficient factors.

Factors	No. 1	No. 2	No. 3	No. 4	No. 5	No. 6
$∆{\xi }_{K1}$	1	0.5	0.5	0.567	0.567	0.5
$∆{\xi }_{K2}$	0.724	0.5	0.724	0.724	1	0.5
$∆{\xi }_{K3}$	0.618	1	0.618	1	0.618	0.618
$∆{\xi }_{K4}$	0.5	0.5	0.618	0.5	1	0.618
$∆{\xi }_{K5}$	1	0.618	1	0.618	1	0.618

.

$$r_{Ki}={\displaystyle \frac{1}{6}}\sum _{e=1}^6{\xi }_{Ki}\left(e\right)=(0.826,0.890,0.769,0.443,0.780)$$

After normalization, the grey relational weight vector for valve reliability allocation factors is obtained as

$$K=(0.222,0.239,0.207,0.119,0.210)$$

Substituting the grey relational weight vector K and the aforementioned data into the equation yields the global index of the valve as

$$\begin{gathered} {GI}^{valve\#1}=({\displaystyle \frac{1}{51.91\times 48.09\times 28.82}}{)}^{0.222}({\displaystyle \frac{1}{94.27\times 17.19}}{)}^{0.239}(5.73\times 42.36{)}^{0.207} \\ \left({43.54\times 36.63)}^{0.119}\right(47.36\times 70\times 53.82{)}^{0.210} \end{gathered}$$

(8)

$${GI}^{valve\#1}=1.356$$

Similarly, the global index values of the other subsystems can be calculated as follows:

$${GI}^{valve\#2}=1.356$$

$${GI}^{EMCC}=1.441$$

$${GI}^{DCDP}=1.644$$

$${GI}^{SPS}=0.898$$

$${GI}^{MSS}=0.797$$

Substituting the global index values of each subsystem into Equation (2), the global index weights of each subsystem are obtained sequentially as

$$w=(0.181,0.181,0.192,0.220,0.120,0.106)$$

3.3. Comparison Between Our Method and Three Other Methods

The weights obtained using our method were compared with those from three other methods: IFM, EWM-OWA, and the multiplication normalization method, as shown in Figure 4.

From Figure 4, the allocation weights of each method exhibit similar overall trends, with relatively critical subsystems receiving higher weights. However, the multiplication normalization method may produce an “amplification effect”, wherein larger weights are disproportionately larger while smaller weights become even smaller.

The multiplication normalization method integrates weights obtained from both the IFM and EWM-OWA methods, followed by normalization. The calculation formula is as follows:

$$w^v={\displaystyle \frac{w_{IFM}^v\times w_{EWM-OWA}^v}{\sum _{v=1}^n{w}_{IFM}^v\times w_{EWM-OWA}^v}}$$

(9)

This demonstrates that the amplification effect arises due to the multiplicative nature of the formulation.

4. Discussion

In this study, the proposed two-layer evaluation factor framework serves as an extension of the traditional IFM method by decomposing influencing factors into a more refined structure. This enables a more comprehensive and structured assessment of the factors affecting system reliability. The framework is particularly suited for early-stage application, as it allows for a minimal yet representative factor set, thereby maintaining a balance between computational efficiency and subsystem characterization—especially important in scenarios where operational data is limited. Furthermore, by employing the cloud model to transform linguistic evaluations into quantifiable representations of uncertainty, the method effectively captures the richness and subjectivity of expert knowledge, which is critical during the conceptual design phase when empirical data is often unavailable.

Due to space limitations, this study demonstrates the application of the proposed method using only the Automatic Depressurization System (ADS) from the nuclear domain as an illustrative case. However, the underlying framework is generalizable and can be extended to other domains, such as aerospace, transportation, and AI-integrated systems. To ensure domain relevance, the reliability evaluation factors should be tailored to the specific characteristics of each field. For example, in aerospace systems, key considerations include redundancy, fault tolerance, and operation under harsh environmental conditions. In transportation systems—such as railways, automotive, and maritime—emphasis is placed on operational profiles, human factors, and lifecycle maintenance. For AI-integrated systems, reliability may encompass both hardware/software components and the probabilistic behavior of machine learning algorithms.

While the proposed method demonstrates strong performance in ADS applications, its computational efficiency may degrade when applied to large-scale systems. To ensure scalability for complex scenarios (e.g., aerospace or transportation systems), parallel computing architectures can be introduced for the efficiency issue. Two distinct parallelization approaches can be implemented:

(1)

Parallel evaluation of independent subsystems: In our framework, reliability allocation for each subsystem requires independent calculations of global indicators based on their respective factor scores and weights. These subsystems are structurally and computationally independent, as their factor evaluations (e.g., technology level, complexity) and grey relational weights are derived from subsystem-specific data. This independence allows task distribution across multiple computing nodes, where each node processes a subset of subsystems simultaneously, significantly reducing overall computation time for large-scale systems with numerous subsystems.

(2)

Parallel computation of distinct level factors: The two-layer factor structure (first-layer factors and second-layer subfactors enables parallelization within factor calculations. For example, the cloud model-based integration of second-layer subfactors for a subsystem involves independent processing of each subfactor’s expert evaluations; these can be computed in parallel across different nodes.

5. Conclusions

Compared to traditional methods, the approach proposed in this paper offers several advantages:

• It comprehensively considers and refines reliability-influencing factors, resulting in a more thorough analysis;
• The proposed method effectively addresses both the fuzziness and stochasticity inherent in expert evaluations;
• By incorporating grey relational analysis, the proposed method explicitly accounts for interdependencies among influencing factors, leading to more rational allocation results.

This study focused on non-repairable systems as analytical objects. However, most complex systems are repairable. Future research could further relax this assumption and explore reliability allocation methods for repairable systems. Additionally, data supplementation could be achieved through various means such as simulation or neural networks. While the present study adopts the cloud model to characterize uncertainty, future research may explore comparative analyses with alternative uncertainty modeling frameworks—such as Bayesian networks, fuzzy Analytic Hierarchy Process (AHP), and Dempster–Shafer theory—within analogous reliability allocation scenarios. Such comparisons could offer deeper insights into the suitability and performance of each method under varying system conditions and data availability constraints.