Reliability Analysis of Aerospace Blade Manufacturing Equipment: A Multi-Source Uncertainty FMECA Method for Five-Axis CNC Machine Tool Spindle Systems

Abstract

Five-axis Computerized Numerical Control (CNC) machine tools play a pivotal role in the precision manufacturing of aeroengine turbine blades, where ultra-high reliability and accuracy are essential. Failure Mode, Effects and Criticality Analysis (FMECA) has been widely applied in the reliability assessment of such advanced machining systems due to its systematic evaluation of potential failure modes. However, traditional FMECA approaches often overlook the ambiguity of human cognition and the interdependence among expert evaluations, limiting their effectiveness in complex aerospace manufacturing environments. To address these issues, this paper proposes a novel FMECA framework based on generalized intuitionistic linguistic theory. A new Generalized Intuitionistic Linguistic Weighted Geometric Average (GILWGA) operator is introduced to couple multi-source expert information and quantify the fuzziness inherent in subjective assessments. Additionally, an intuitionistic linguistic entropy-based weighting scheme is developed to dynamically evaluate key risk factors, including severity, occurrence, detectability, and controllability. The proposed framework is applied to a case study involving the spindle system of a five-axis CNC machine tool used in aeroengine blade production. The results demonstrate that the proposed method offers more robust and consistent failure mode prioritization, providing effective decision support for reliability-centered maintenance in aerospace equipment manufacturing.

1. Introduction

With the rapid advancement of aerospace engineering, ensuring the reliability of intelligent manufacturing systems has become essential. Yang et al. [1] improved CNC reliability by modeling spatiotemporal fault propagation. Li et al. [2] developed a Bayesian network-based framework to enhance reliability while reducing operational costs in heavy-duty CNC systems. Wang et al. [3] proposed a fault rate allocation method to support reliable system design. These studies highlight the critical role of reliability in intelligent manufacturing systems.

In particular, five-axis CNC machine tools are widely employed in the machining of aerospace components, such as turbine blades, impellers, and structural parts, where geometric complexity, surface integrity, and dimensional accuracy are paramount [4,5]. The failure of these systems may lead to production interruptions, costly scrap, or safety-critical defects in flight hardware. Therefore, performing a rigorous and systematic reliability assessment of five-axis CNC machine tools—especially their spindle systems—is essential to guarantee the stable operation of aerospace manufacturing processes [6,7].

Failure Mode, Effects and Criticality Analysis (FMECA) has long been recognized as an effective methodology for identifying and prioritizing potential failure modes in complex electromechanical systems. It is extensively applied in the reliability evaluation of CNC machine tools due to its structured approach in assessing the severity, occurrence, and detectability of faults [8,9,10]. However, traditional FMECA models still face limitations in handling ambiguous human judgment, incomplete information, and the mutual influence among multiple experts. These deficiencies become particularly critical in aerospace manufacturing, where decisions must be both data-informed and resilient to uncertainty.

In recent years, system reliability analysis has garnered significant attention in intelligent manufacturing, particularly in high-risk, high-value sectors such as aerospace. In this context, FMECA has been widely applied across various industrial scenarios due to its practical effectiveness. Dhalmahapatra et al. [11] applied FMECA to improve safety in steel manufacturing systems, while Qiu et al. [12] utilized FMECA to enhance the reliability of heavy machine tools. These studies demonstrate the wide applicability of FMECA in complex manufacturing systems.

In the context of five-axis CNC machine tools, FMECA has been used to systematically score failure modes based on severity (S), occurrence (O), and detectability (D) and to calculate a risk priority number (RPN = S × O × D) to support maintenance decisions [13,14].

Researchers from various industries have used traditional FMECA to perform reliability analyses of complex systems. However, despite its widespread adoption, traditional FMECA presents three notable limitations that restrict its applicability to aerospace manufacturing systems involving five-axis CNC machine tools. Figure 1 shows the traditional FMECA process and its existing problems.

1. Incomplete risk factor modeling. Conventional FMECA relies on severity (S), occurrence (O), and detectability (D), often overlooking controllability, which is particularly important in high-precision aerospace machining, where recovery measures are complex and time-sensitive. This limitation may lead to ambiguous RPN results and hinder accurate risk assessment for complex equipment.

2. Neglect of cognitive ambiguity. Expert evaluations are inherently fuzzy, especially in scenarios where data is scarce or judgment is experience-driven, as often encountered in aerospace production lines. However, traditional methods assume deterministic reasoning, ignoring the ambiguity and hesitation in human cognition, which compromises the accuracy and robustness of decision outcomes.

3. Ignorance of expert information coupling. Multi-expert assessments are common in reliability evaluations, but most FMECA models treat expert opinions as independent. In reality, interpersonal influence and shared cognition among experts are inevitable, especially in tightly coordinated engineering teams. Failure to account for such coupling can distort the aggregation process and reduce consistency in failure mode prioritization.

In summary, existing FMECA approaches fall short in fully modeling risk factors, accounting for cognitive ambiguity, and capturing the coupling effects in multi-expert evaluations—factors that are particularly critical in aerospace manufacturing contexts [15,16]. To address these limitations, this paper proposes a novel generalized intuitionistic linguistic FMECA framework. Building upon the traditional S-O-D structure, the method introduces controllability (C) as an additional risk dimension and redefines the risk priority number accordingly. Furthermore, a Generalized Intuitionistic Linguistic Weighted Geometric Average (GILWGA) operator is developed to aggregate expert opinions while preserving interdependencies and cognitive fuzziness. By combining linguistic entropy-based weighting with intuitive scoring and exact functions, the proposed framework enhances both the objectivity and expressiveness of failure mode evaluations under multi-source uncertainty. This method provides a practical and scalable decision-support tool for improving the reliability of high-end equipment in aerospace manufacturing, offering both methodological advancement and strong applicability in engineering practice. In addition, the incorporation of extended linguistic terms further improves the granularity and consistency of the evaluation process.

The remainder of this paper is organized as follows. Section 2 reviews the latest developments in FMECA and system reliability analysis for aerospace manufacturing systems. Section 3 presents the theoretical basis of the proposed method, including key definitions and the formulation of the Generalized Intuitionistic Linguistic Weighted Geometric Average (GILWGA) operator. Section 4 details the implementation of the framework based on linguistic entropy and expert information aggregation. Section 5 provides a case study on the spindle system of a five-axis CNC machine tool to validate the method in an aerospace-relevant context. Section 6 concludes the paper and discusses future work.

2. Literature Review

2.1. Applications of FMECA in Aerospace Manufacturing and CNC Machine Tools

As the aerospace industry continues to demand higher levels of precision, stability, and processing flexibility, five-axis CNC machine tools have become indispensable in the manufacturing of complex components such as turbine blades, impellers, and structural parts. Ensuring their operational reliability is critical to maintaining the quality and consistency of aerospace production. Accordingly, reliability analysis methods for such equipment have evolved significantly, progressing from early approaches like fault tree analysis (FTA) and event tree analysis (ETA) to more advanced models that consider multiple failure modes and system-level interactions. Among them, traditional FMECA has been widely applied to assess the impact of individual failure modes. However, as aerospace manufacturing systems grow more complex and interconnected, conventional single-mode analysis often fails to capture the full spectrum of potential risks.

Tian et al. [17] refined the evaluation process by decomposing severity, occurrence, and detectability into seven sub-indicators, thereby improving the precision of fault assessment for CNC machine tools operating in complex environments. Yu et al. [18] further enhanced risk differentiation by subdividing severity and occurrence into three metrics and incorporating buyer satisfaction (B) to better rank the importance of failure modes. These approaches offer valuable insights for high-end manufacturing scenarios, where system reliability and fault traceability are critical. However, both studies overlook the ambiguity inherent in expert judgment, such as variation in subjective fault identification. This issue is particularly prominent in aerospace domains, where limited fault data and high system complexity exacerbate cognitive uncertainty.

To improve the accuracy of FMECA in complex manufacturing systems, researchers have integrated various computational techniques to better capture interdependencies and multidimensional failure modes. Yu et al. [19] combined fuzzy mathematics with data envelopment analysis (DEA) to optimize risk prioritization during the manufacturing stage of CNC machine tools. Alkabaa et al. [20] developed an FMECA framework incorporating reliability-centered maintenance (RCM) and the analytic network process (ANP), enhancing decision reliability. These methods significantly improve the reliability assessment under dynamic and multi-factor conditions of high-end equipment, such as five-axis CNC machine tools used in the aerospace manufacturing industry. However, most of these approaches still neglect the subjective cognitive criteria involved in expert fault evaluations. Incorporating standardized expert evaluation frameworks could further improve consistency and accuracy.

Recent studies have explored structured aggregation methods to improve consistency in expert evaluations for complex engineering systems. Wang et al. [21] applied group decision-making theory to unify expert opinions in the FMECA of CNC machine tools, offering insights into enhancing assessment objectivity in aerospace manufacturing environments, where expert involvement is indispensable. Yang et al. [22] employed intuitionistic fuzzy sets (IFSs) and the VIKOR method to incorporate diverse expert preferences, thereby improving the rationality of failure mode prioritization. While these approaches mitigate the impact of cognitive ambiguity, they do not fully consider multi-source heterogeneity or the information coupling among experts. In high-stakes domains such as aerospace, where system failures can result in mission-critical consequences, addressing both the fuzziness of human reasoning and the interdependence of expert knowledge is essential for reliable risk assessment.

2.2. Generalized Intuitionistic Language Operator

In multi-criteria decision-making problems, particularly in complex engineering systems such as aerospace manufacturing, expert evaluations often vary due to differences in disciplinary background, experience, and cognitive perspectives. Such discrepancies are especially pronounced in the reliability assessment of high-precision equipment, where expert judgments must integrate uncertain, fragmented, or qualitative engineering information. To address this issue, previous studies have incorporated fuzzy numbers into the FMECA framework and developed fuzzy FMECA aggregation operators to integrate expert opinions, thereby improving the rationality and stability of the evaluation results.

Benbachir et al. [23] applied fuzzy FMECA to wastewater treatment networks, improving the evaluation of repair strategies and priority ranking within wastewater systems. Zuniga et al. [24] employed fuzzy FMECA to represent experts’ perceptions of risk, enabling a more accurate prioritization of failure modes in power grid reliability analysis. Giardina et al. [25] compared fuzzy FMECA with the conventional FMECA approach and demonstrated the advantages of fuzzy FMECA in the context of fault radiotherapy systems. Although these studies enhanced the FMECA framework by introducing fuzzy numbers, the coupling among multiple expert opinions was not explicitly considered. As a result, the evaluation outcomes may still exhibit a relatively high degree of subjectivity.

To better integrate expert information and further mitigate the influence of cognitive ambiguity in human judgments, Generalized Intuitionistic Linguistic Operators (GILOs) have been proposed as an effective tool for representing and aggregating fuzzy expert opinions. These operators have been widely applied in areas such as investment evaluation and industrial risk analysis and are particularly suitable for fuzzy multi-attribute decision-making problems under uncertainty. Generalized Intuitionistic Linguistic Operators (GILOs) have emerged as an effective tool for addressing the inherent uncertainty and subjectivity in expert evaluations. Built upon intuitionistic fuzzy set theory and incorporating linguistic variables, this approach is particularly suitable for multi-attribute decision-making (MADM) problems under vague or imprecise conditions.

In general, a generalized intuitionistic linguistic term can be represented by a triplet ⟨s, μ, ν⟩, where s denotes the linguistic label, μ represents the degree of support for the evaluation result, and ν denotes the degree of opposition. The residual term π = 1 − μ − ν quantifies the hesitation or uncertainty of the evaluator. This representation not only extends the expressive capability of conventional linguistic terms but also captures the inherent vagueness involved in human reasoning. The operational mechanism of the GILO framework is illustrated in Figure 2.

Over the past decade, Generalized Intuitionistic Linguistic Operators (GILOs) have been widely applied in various decision-making contexts, including investment evaluation, risk prioritization, supplier selection, and environmental impact assessment. Building on this framework, several extended aggregation models have been developed, such as hybrid GILO models based on the Choquet integral and entropy-weighted variants. These models enable flexible integration of linguistic evaluations from multiple sources while preserving the semantic information contained in the original inputs.

Brafman and De Giacomo [26] developed a model based on GILOs. The model represents complex dependency relationships through intuitive mathematical formulations. It supports decision-making under cognitive constraints. Dyczkowski et al. [27] combined GILOs with interval-valued fuzzy sets. This approach enhanced the reasoning capability of the Simpful library and improved operational efficiency under uncertain conditions. Frischauf et al. [28] applied generalized linguistic decision functions to study the inverse problem of neural network hypothesis functions. Their work provided a more intuitive perspective for analyzing convergence properties. Although GILO methods have been applied in many domains, most studies focus on improving algorithmic performance. Their limitations in complex decision environments are less discussed. As a result, their application in highly complex systems remains limited. This issue becomes particularly evident in aerospace manufacturing systems. In such environments, GILO-based fuzzy reasoning still faces two major challenges.

First, traditional approaches often assume that expert evaluations are independent. They ignore the information coupling and cognitive interdependence that naturally arise in multi-expert assessments. This assumption may weaken the validity of aggregated results in collaborative decision settings. Second, human decision-making often involves vague preferences and hesitation. Traditional GILO models cannot fully capture these characteristics.

In modern intelligent manufacturing environments, data-driven monitoring techniques have been widely adopted for equipment condition monitoring and fault diagnosis. Advanced sensing systems, real-time data acquisition, and predictive maintenance approaches enable early detection of abnormal operating states, thereby significantly reducing the risk of undetected failures.

Yu et al. [29] addressed equipment condition monitoring under limited and imbalanced data by integrating multi-source information. Jiang et al. [30] proposed an image-based deep learning method for monitoring and predictive maintenance of large-scale industrial turbines. These studies demonstrate the increasing adoption of data-driven approaches in practical industrial scenarios.

Building upon these developments, modern intelligent manufacturing systems increasingly rely on data-driven monitoring and diagnostic mechanisms rather than traditional manual inspection or experience. Advanced sensing systems, real-time data acquisition, and predictive maintenance techniques enable early identification of abnormal operating conditions, thereby reducing the risk of undetected faults. In this context, accurately evaluating detectability (D) becomes crucial for providing reliable risk assessment and decision-making.

Based on the existing literature, although the application of FMECA in CNC machine tool reliability analysis has made notable progress, several key limitations remain. These limitations restrict its effectiveness in aerospace manufacturing and other high-reliability domains. First, most existing models consider only a limited set of risk dimensions. Important attributes such as controllability are often overlooked, and due to the lack of methodological specificity, the level of detectability (D) is difficult to evaluate accurately. Second, these models do not fully capture the inherent fuzziness in human cognition. This issue may affect the consistency of expert evaluations. Third, many approaches ignore the heterogeneity and interdependence among experts from different domains. As a result, decision information cannot be fully integrated. These issues reduce the accuracy of failure mode prioritization under uncertainty and weaken the credibility of reliability assessment results.

To address these challenges, this study proposes a new Generalized Intuitionistic Linguistic Weighted Geometric Averaging (GILWGA) operator and integrates it into an enhanced FMECA framework. The proposed method combines judgments from multiple experts while preserving cognitive fuzziness and evaluation coupling. This mechanism supports more interpretable, adaptive, and robust prioritization of failure modes in complex engineering systems. The approach is particularly suitable for reliability assessment of aerospace-grade five-axis CNC machine tools.

3. Methodology

3.1. Engineering Advantages of Controllability (C)

In the conventional FMECA framework, the three classical indicators—severity (S), occurrence (O), and detectability (D)—are generally sufficient for prioritizing most failure modes. However, recent studies on the reliability assessment of complex engineering systems have shown that relying solely on these three indicators may not fully capture the overall risk characteristics of a system. This limitation becomes particularly evident in equipment equipped with active protection mechanisms and intelligent control systems, where the ability to control or mitigate failures after they occur can significantly influence their ultimate consequences. To better reflect the risk characteristics of engineering systems, several studies have proposed extending the traditional RPN framework by incorporating additional evaluation dimensions, such as machine hazard (M), personal hazard (P), or buyer satisfaction (B) [18,19].

However, most of these extended indicators mainly describe risk consequences or stakeholder impacts. They rarely reflect the engineering capability of a system to intervene after a failure occurs. In highly automated equipment, multiple protection and control mechanisms are usually integrated into the system. With these mechanisms, some failures can still be contained through timely control or mitigation, even after they occur. Therefore, relying only on severity (S), occurrence (O), and detectability (D) makes it difficult to distinguish failure modes that produce similar consequences but differ significantly in controllability. As a result, the resulting risk ranking may deviate from actual engineering conditions.

This issue is particularly evident in five-axis CNC machine tools. Modern machine tool systems operate under strict requirements in terms of structural complexity, machining precision, and operational safety. Key components are usually equipped with multiple layers of protection and control strategies. When a potential failure is detected, the system can often reduce its impact through automatic control, parameter compensation, or maintenance intervention.

In such systems, the risk of a failure mode is therefore not determined only by its occurrence probability and severity. It is also closely related to the system’s ability to control the failure after it occurs. Based on this engineering characteristic, this study introduces controllability (C) into the traditional FMECA indicator framework. The controllability factor describes the ability of a system to suppress or mitigate the consequences of a failure through control measures after the fault occurs or is detected. This extension allows the risk prioritization results to better reflect the actual operating characteristics of CNC machine tool systems.

Controllability describes the extent to which a potential failure mode can be contained, mitigated, or compensated once it occurs or is detected. This indicator reflects the availability and effectiveness of built-in protection measures, adaptive control strategies, and maintenance interventions. A higher level of controllability indicates that the system has stronger safety barriers and countermeasures, which can prevent a failure from developing into more severe consequences.

When guiding domain experts to evaluate controllability, four main aspects are considered. These include the design of protective measures, adaptive control and compensation capability, the convenience of isolation and repair, and the coordination between monitoring and control mechanisms. Based on these considerations, controllability is classified according to the following levels.

Very low $s_4$: Effective measures are almost absent. Once a failure occurs, the consequences are generally unacceptable.

Low $s_3$: Some control measures exist, but the response is slow or the effect is limited.

Medium $s_2$: The system has certain control or compensation mechanisms that can reduce the impact to a moderate extent.

High $s_1$: Multiple control measures are available and can effectively limit failure consequences in most situations.

Very high $s_{1/2}$: Comprehensive control measures are in place, and the failure consequences can be effectively controlled in almost all situations.

3.2. Explanation and Usage of Intuitionistic Linguistic Numbers for Experts

To reduce ambiguity in expert scoring and improve evaluation accuracy, several representative examples are provided. Common failure modes are associated with their corresponding control measures so that experts can conduct the assessment in a more intuitive engineering context. Typical protection and control mechanisms of five-axis CNC machine tools are also listed. This enables experts to make judgments based on actual machine configurations rather than abstract descriptions.

During the evaluation process, experts assess each failure mode from four dimensions: severity (S), occurrence (O), detectability (D), and controllability (C). Severity reflects the potential impact of a failure mode on system safety, machining accuracy, and operational stability. Occurrence represents the likelihood that the failure mode will occur during machine operation. Detectability indicates whether the fault can be identified in time by existing monitoring or diagnostic mechanisms. In modern intelligent manufacturing environments, a large number of data-driven monitoring technologies have been widely applied to equipment condition monitoring and fault diagnosis. Advanced sensing systems, real-time data acquisition, and predictive maintenance techniques enable abnormal operating conditions to be identified at an early stage, significantly reducing the risk of undetected faults. Under such conditions, accurately evaluating detectability (D) becomes particularly important for reliable risk assessment. Controllability describes the ability of the system to limit or mitigate the consequences of the fault through protection or control measures after it occurs or is detected.

This study explicitly distinguishes detectability (D) from controllability (C). Detectability concerns whether a fault can be detected in time, whereas controllability focuses on whether the system can effectively limit the impact of the fault through control actions after it occurs or is detected. The objective is to reduce potential losses as much as possible.

The following example illustrates the controllability assessment for CNC machine tools based on several failure modes considered in this study.

FM2 Oil leakage

Severity (S) mainly depends on the potential impact of oil leakage on machine operation stability, machining accuracy, and environmental contamination risk. If the leakage may lead to lubrication failure or contamination of critical components, the severity is usually considered high. If the leakage only causes minor contamination and does not significantly affect machine operation, the severity may be evaluated as medium.

Occurrence (O) is mainly related to seal wear, the condition of oil circuit connections, and the operating time of the equipment. If the seals have been in service for a long period and the maintenance interval is long, the occurrence probability is typically high. If the machine is well maintained and seals are replaced periodically, the occurrence probability is usually low.

Detectability (D) depends on whether the system is equipped with leakage detection mechanisms or regular inspection procedures. If leakage sensors or pressure monitoring systems are available to identify abnormalities in time, the detectability score is generally low, indicating that the fault can be detected relatively easily. If detection relies mainly on manual inspection, the detectability score is usually higher, reflecting greater difficulty in identifying the fault in time.

Possible mitigation measures include isolation valves, leakage detection systems, scheduled seal replacement, and contamination control. If effective isolation is absent and seal replacement requires a long maintenance time, controllability (C) is usually low or very low. If isolation mechanisms are available and seals can be replaced quickly under established maintenance procedures, controllability can increase to a medium or high level.

3.3. Basic Theory of GILWGA Operator

The GILWGA operator is a fuzzy aggregation method that integrates both membership and non-membership functions within a weighted geometric averaging framework [31,32]. Unlike conventional averaging methods, the GILWGA operator incorporates expert subjectivity by dynamically assigning expert weights based on individual scoring patterns, and then coupling these weighted evaluations to generate a more objective and representative fuzzy value [33,34]. This mechanism enhances the credibility of the decision-making results of reliability assessment of aerospace manufacturing equipment in complex and uncertain environments. The GILWGA operator offers several advantages:

1. Effective integration of multi-expert evaluations: Traditional approaches often apply equal-weight averaging across expert inputs, which fails to account for varying levels of expertise or evaluation consistency. In contrast, the GILWGA operator derives expert weights from their score distributions and integrates them accordingly, producing more balanced and informative results.

2. Improved handling of fuzziness and ambiguity: Human assessments often carry inherent uncertainty, particularly in complex technical domains. While conventional weighted methods mitigate some of this ambiguity, they still rely on single-point evaluations. The GILWGA operator captures both support and opposition through the use of membership and non-membership functions, allowing for more nuanced modeling of expert reasoning. This reduces subjectivity and enhances both the robustness and accuracy of the final decision.

In aerospace system evaluation, where reliability and decision accuracy are critical, robust aggregation methods are essential. The Generalized Intuitionistic Linguistic Weighted Geometric Average (GILWGA) operator addresses fuzzy multi-attribute decision-making by integrating membership and non-membership degrees. Its structure supports stable and interpretable expert evaluation under uncertainty. Now we give the basic theory of the GILWGA operator:

Definition 1. 

Suppose the language term set is $S=\left\{s_{\theta }\right| \theta =1/\tau ,\cdots ,1/2,1,2,\cdots ,\tau \}$, where $s_{\theta }$ is the language term, $s_{1/\tau }$ and $s_{\tau }$ are the lower limit and upper limit of the language term respectively, $\tau$ is a positive integer, and $S$ satisfies the following conditions:

(1). If $a>b$, then $s_a>s_b$.

(2). There exists a negative operator $\mathrm{neg}(s_a)=s_b$ such that $ab=1$; in particular, $\mathrm{neg}(s_1)=s_1$.

For example, when $\tau =4$, the set of language terms is $S=\{s_{1/4}=\mathrm{Very} \mathrm{bad},s_{1/3}=\mathrm{Poor},s_{1/2}=\mathrm{Bad},s_1=\mathrm{Average},s_2=\mathrm{Good},s_3=\mathrm{Better},s_4=\mathrm{Excellent}\}$. The scoring criteria can be based on those given in

Table 1. Evaluation score criteria for language terminology $s_{\theta }$.

Serial	Evaluation Index	Standard	Scores
1	Occurrence (O)	85~100%	4
		70~85%	3
		55~70%	2
		40~55%	1
		≤40%	1/2
2	Severity (S)	>80%	4
		65~80%	3
		50~65%	2
		35~50%	1
		≤35%	1/2
3	Detectability (D)	>80%	4
		65~80%	3
		50~65%	2
		35~50%	1
		≤35%	1/2
4	Controllability (C)	>90%	4
		75~90%	3
		60~75%	2
		45~60%	1
		≤45%	1/2

.

It is worth noting that linguistic terms such as 1/3 and 1/4 are not explicitly listed in Table 1, as they are generated as extended linguistic terms during the computation process to improve granularity. The evaluation criteria in Table 1 correspond to the basic linguistic term set.

In order to make the calculation more convenient and prevent the loss of decision information, an extended language term set $\tilde{S}=\{s_{\theta }| \theta \in [1/q,q]\}$ is defined based on the original language term set $S$, where $q(q>1)$ is a sufficiently large natural number. If $s_{\theta }\in S$, then $s_{\theta }$ is called a primitive term; otherwise it is called an extended term. The original terminology is generally used to evaluate decision options, while the extended terminology is used in the process of language calculation and ranking of decision options.

Definition 2. 

Assume $X$ is a non-empty domain $X=(x_1,x_2,\cdots ,x_n)$, and an intuitionistic fuzzy set on $X$ can be represented as $A=\{(x,\langle {\mu }_A(x),{\nu }_A(x)\rangle )| x\in X\}$, where ${\mu }_A:X\to [0,1]$ and ${\nu }_A:X\to [0,1]$ are both membership functions of $X$, and $0⩽{\mu }_A(x)+{\nu }_A(x)⩽1,{\mu }_A(x),{\nu }_A(x)$, the membership and non-membership of element $x$ to $A$ . For each intuitionistic fuzzy set on $X$, ${\pi }_A(x)=1-{\mu }_A(x)-{\nu }_A(x)$ is called the intuition index of element $x$ in intuitionistic fuzzy set $A$, which indicates the hesitation of element $x$ to belong to $A$.

Definition 3. 

Let $X$ be a domain, $s_{\theta (x)}\in \tilde{S}$. Then an intuitive language set $A$ is defined as follows:

$$A=\{(x,\langle s_{\theta (x)},{\mu }_A(x),{\nu }_A(x)\rangle )| x\in X\}.$$

Here, $s_{\theta (x)}$ is a language term; ${\mu }_A(x)$ is a membership function, which indicates the degree to which $x$ belongs to the language evaluation value $s_{\theta (x)}$, ${\mu }_A:X\to \tilde{S}\to [0,1]$; $x\mapsto s_{\theta (x)}\mapsto {\mu }_A(x),{\nu }_A(x)$ is a non-membership function, which indicates the degree to which $x$ does not belong to the language evaluation value $s_{\theta (x)}$, ${\nu }_A:X\to \tilde{S}\to [0,1],x\mapsto s_{\theta (x)}\mapsto {\nu }_A(x)$; and $x\in X,{\pi }_A(x)=1-{\mu }_A(x)-{\nu }_A(x)$ indicates the hesitation degree of $x$ belonging to $s_{\theta (x)}$ if the condition is met.

Definition 4. 

Let ${\alpha }_1=\langle s_{\theta ({\alpha }_1)},\mu ({\alpha }_1),\nu ({\alpha }_1)\rangle$ and ${\alpha }_2=\langle s_{\theta ({\alpha }_2)},\mu ({\alpha }_2),\nu ({\alpha }_2)\rangle$ be two intuitive language numbers, $\lambda ⩾0$ . Then,

(1). ${\alpha }_1+{\alpha }_2=\langle s_{\theta ({\alpha }_1)+\theta ({\alpha }_2)},1-(1-\mu ({\alpha }_1))(1-\mu ({\alpha }_2)),\nu ({\alpha }_1)\nu ({\alpha }_2)\rangle$

(2). ${\alpha }_1\otimes {\alpha }_2=\langle s_{\theta ({\alpha }_1)\times \theta ({\alpha }_2)},\mu ({\alpha }_1)\mu ({\alpha }_2),\nu ({\alpha }_1)+\nu ({\alpha }_2)-\nu ({\alpha }_1)\nu ({\alpha }_2)\rangle$;

(3). $\lambda {\alpha }_1=\langle s_{\lambda \times \theta ({\alpha }_1)},1-{(1-\mu ({\alpha }_1))}^{\lambda },{(\nu ({\alpha }_1))}^{\lambda }\rangle$

(4). ${\alpha }_1^{\lambda }=\langle s_{{(\theta ({\alpha }_1))}^{\lambda }},{(\mu ({\alpha }_1))}^{\lambda },1-{(1-\nu ({\alpha }_1))}^{\lambda }\rangle$

Further, we can deduce that

(1). ${\alpha }_1+{\alpha }_2={\alpha }_2+{\alpha }_1,{\alpha }_1\otimes {\alpha }_2={\alpha }_2\otimes {\alpha }_1$

(2). $\lambda ({\alpha }_1+{\alpha }_2)=\lambda {\alpha }_2+\lambda {\alpha }_1$

(3). ${\lambda }_1{\alpha }_1+{\lambda }_2{\alpha }_1=({\lambda }_1+{\lambda }_2){\alpha }_1$

(4). ${\alpha }_1^{{\lambda }_1}\otimes {\alpha }_1^{{\lambda }_2}={({\alpha }_1)}^{{\lambda }_1+{\lambda }_2}$

(5). ${\alpha }_1^{{\lambda }_1}\otimes {\alpha }_2^{{\lambda }_1}={({\alpha }_1\otimes {\alpha }_2)}^{{\lambda }_1}$

Definition 5. 

$\alpha =\langle s_{\theta (\alpha )},\mu (\alpha ),\nu (\alpha )\rangle$ is an intuitive language number. Then the score function $h(\alpha )$ and the exact function $H(\alpha )$ of $\alpha =\langle s_{\theta (\alpha )},\mu (\alpha ),\nu (\alpha )\rangle$ are

$$h(\alpha )=\theta (\alpha )(\mu (\alpha )-\nu (\alpha ))$$

$$H(\alpha )=\theta (\alpha )(\mu (\alpha )+\nu (\alpha ))$$

Definition 6. 

Assume ${\alpha }_1=\langle s_{\theta ({\alpha }_1)},\mu ({\alpha }_1),\nu ({\alpha }_1)\rangle$ and ${\alpha }_2=\langle s_{\theta ({\alpha }_2)},\mu ({\alpha }_2),\nu ({\alpha }_2)\rangle$ are two intuitive language numbers. Then the size relationship and order are as follows:

(1) If $h({\alpha }_1)>h({\alpha }_2)$, then ${\alpha }_1$ is greater than ${\alpha }_2$; that is, ${\alpha }_1>{\alpha }_2$.

(2) If $h({\alpha }_1)<h({\alpha }_2)$, then ${\alpha }_1$ is less than ${\alpha }_2$; that is, ${\alpha }_1<{\alpha }_2$.

(3) If $h({\alpha }_1)=h({\alpha }_2)$, then:

① If $H({\alpha }_1)=H({\alpha }_2)$, then ${\alpha }_1$ is equal to ${\alpha }_2$; that is, ${\alpha }_1={\alpha }_2$.

② If $H({\alpha }_1)<H({\alpha }_2)$, then ${\alpha }_1$ is less than ${\alpha }_2$; that is, ${\alpha }_1<{\alpha }_2$.

③ If $H({\alpha }_1)>H({\alpha }_2)$, then ${\alpha }_1$ is greater than ${\alpha }_2$; that is, ${\alpha }_1>{\alpha }_2$.

3.4. Intuitive Language Entropy Determines Weights

Aiming at the decision-making problem with unknown indicator weights, this paper proposes intuitive linguistic entropy based on the intuitionistic fuzzy entropy proposed in reference [35]. The GILWGA operator is determined by the intuitive linguistic entropy. The specific formula is as follows:

Definition 7. 

Assume $X=(x_1,x_2,\cdots ,x_n)$ and intuitive language set $A=\{(x,\langle s_{\theta (x)},{\mu }_A(x),{\nu }_A(x)\rangle )| x\in X\}$. Then the intuitive language entropy of $A$ is

$$E(A)={\displaystyle \frac{1}{n}}{\displaystyle {\displaystyle \sum }_{i=1}^n}{\displaystyle \frac{(1-| {\mu }_A(x_i)-{\nu }_A(x_i)| +{\pi }_A(x_i))}{(1+| {\mu }_A(x_i)-{\nu }_A(x_i)| +{\pi }_A(x_i))}}{\displaystyle \frac{\theta (x_i)}{\tau }}.$$

(1)

In the intuitive language decision matrix $F=(c_{ij})$, for any intuitive language number $c_{ij}=\langle s_{\theta (x)},{\mu }_A(x),{\nu }_A(x)\rangle$, its intuitive language entropy can be calculated by the above formula, which is simply recorded as $e_{ij}$. Here, $e_{ij}$ represents the uncertainty of the attribute value $c_{ij}$. The larger $e_{ij}$ is, the greater the uncertainty of $c_{ij}$ is. For indicator $c_{ij}$, its entropy can be expressed as $E_j=l_1{e}_{1j}+l_2{e}_{2j}+\cdots +l_m{e}_{mj}$, where $l_i$ is the weight of the $i$-th solution, because the importance of each solution is the same, so

$$E_j={\displaystyle \frac{1}{m}}{\displaystyle {\displaystyle \sum }_{i=1}^m}{e}_{ij}.$$

(2)

Therefore, the weight of the $j$-th indicator can be calculated as follows:

$${\omega }_j={\displaystyle \frac{1-E_j}{{{\displaystyle \sum }}_j^n\left(1-E_j\right)}}.$$

(3)

Then, the generalized intuitive language operator is used to obtain the comprehensive evaluation value of each scheme, in which the attribute weight is calculated by the intuitive language entropy.

3.5. Properties and Calculation Methods of GILWGA

The calculation method of GILWGA is as follows:

Definition 8. 

Let  ${\alpha }_j=\langle s_{\theta ({\alpha }_j)},\mu ({\alpha }_j),\nu ({\alpha }_j)\rangle (j=1,2,\cdots ,n)$ be a set of intuitive language numbers, and let GILWGA be ${\mathsf{\Omega }}^n\to \mathsf{\Omega }$ . If

$$\mathrm{GILWGA}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)={\displaystyle \frac{1}{\lambda }}\left({\displaystyle {\displaystyle \prod }_{j=1}^n}{(\lambda {\alpha }_j)}^{{\omega }_j}\right)$$

then the function GILWGA is called the generalized intuitionistic language weighted geometric average (GILWGA) operator, where $\omega =({\omega }_1,{\omega }_2,\cdots ,{\omega }_n)$ is the attribute weight, satisfying ${\omega }_j\in [0,1]$, and ${\displaystyle {\displaystyle \sum }_{j=1}^n}{\omega }_j=1,\lambda >0$ is an arbitrary real number. According to the intuitive language number operation rules, Formula (3) can be further deduced to obtain

$$\begin{array}{l}\mathrm{GILWGA}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)=\langle s_{{\displaystyle \frac{1}{\lambda }}({\displaystyle {\displaystyle \prod }_{j=1}^n}{\left(\lambda \theta ({\alpha }_j)\right)}^{{\omega }_j})},1-{\left(1-{\displaystyle {\displaystyle \prod }_{j=1}^n}{(1-{(1-\mu ({\alpha }_j))}^{\lambda })}^{{\omega }_j}\right)}^{1/\lambda }\\ ,{\left(1-{\displaystyle {\displaystyle \prod }_{j=1}^n}{(1-\nu {({\alpha }_j)}^{\lambda })}^{{\omega }_j}\right)}^{1/\lambda }\rangle \end{array}$$

(4)

For the sake of calculation simplicity, we take $\lambda =1$; then

$$\mathrm{GILWGA}({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n)=\langle s_{{\displaystyle {\displaystyle \prod }_{j=1}^n}{\left(\theta ({\alpha }_j)\right)}^{\omega }j},{\displaystyle {\displaystyle \prod }_{j=1}^n}{\left(\mu ({\alpha }_j)\right)}^{{\omega }_j},1-{\displaystyle {\displaystyle \prod }_{j=1}^n}{\left(1-\nu ({\alpha }_j)\right)}^{{\omega }_j}\rangle$$

(5)

The GILWGA operator synthesizes the intuitionistic linguistic values of multiple experts based on their corresponding attribute weights. The detailed mechanism of expert opinion aggregation through the GILWGA operator is illustrated in Figure 3.

The properties of the GILWGA operator are presented in Appendix A.

Therefore, from Equations (A6) and (A7), we can see that the theorem holds. Because of the idempotence, monotonicity and boundary properties of the GILWGA operator, it is very suitable for multi-criteria decision-making (MCDM) in complex systems. The mechanism of the GILWGA operator operation in this paper is shown in Figure 4.

The idempotence of the GILWGA operator ensures consistency and stability across iterative computations, which is essential for reliability modeling in aerospace systems. Monotonicity guarantees logical coherence among evaluation criteria, enhancing decision credibility in multi-attribute assessments. The boundary property effectively suppresses the influence of outliers, reducing computational volatility and preventing erroneous judgments in high-stakes environments such as aerospace fault diagnosis and risk analysis.

4. FMECA Based on Intuitive Language Entropy and GILWGA Operator

Suppose that there are $m$ solutions $x_i(i=1,2,\cdots ,m)$ constituting solution set $X=\{x_1,x_2,\cdots ,x_m\}$, $n$ attributes $c_j(j=1,2,\cdots ,n)$ constituting attribute set $C=\{c_1,c_2,\cdots ,c_n\}$, and $l$ experts $d_k\left(k=1,2,\cdots ,l\right)$ constituting decision expert set $D=\{d_1,d_2,\cdots ,d_l\}$, and that $\omega =({\omega }_1,{\omega }_2,\cdots ,{\omega }_n)$ is the attribute weight. Then ${\omega }_j\in [0,1],{\displaystyle {\displaystyle \sum }_{j=1}^n}{\omega }_j=1$, $e=(e_1,e_2,\cdots ,e_l)$ are expert weights, and $e_k\in [0,1],{\displaystyle {\displaystyle \sum }_{k=1}^l}{e}_k=1$. The evaluation value of expert $e_k$ on solution $x_i$ under attribute $c_j$ is expressed as ${\alpha }_{ij}^k=\langle s_{\theta ({\alpha }_{ij}^k)},\mu ({\alpha }_{ij}^k),\nu ({\alpha }_{\mathrm{ij}}^k)\rangle$ in intuitive language, and the decision matrix given by decision-maker $e_k$ is $(k=1,2,\cdots ,l)$.

The solution process of the reliability multi-attribute group analysis model based on intuitive language entropy and GILWGA operator is as follows:

Step 1: The target object is systematically evaluated to establish a complete set of failure modes. The specific process is: integrate cross-enterprise operation data of similar equipment, identify potential failure types, and screen out core failure modes that have a significant impact on enterprise operations and have unclear priorities.

Step 2: An FMEA table is prepared, and relevant experts and scholars are invited to evaluate it. Relevant information sets on key failure modes are collected to understand the sources of risks and the impact and losses that the risks may cause. Then, an expert system consisting of $k$ relevant experts, scholars and risk managers is invited to score all failure mode sets under different risk factor indicators in the form of intuitive language numbers and develop an FMEA table.

Step 3: The data is normalized, and then the attribute weights are calculated using the data in the FMECA table using intuitive language entropy. Different attribute weights ${\omega }^k=({\omega }_1^k,{\omega }_2^k,\cdots ,{\omega }_n^k)$ are obtained from the decision matrices of different experts, and then the average value is used to obtain the attribute weight ${\omega }_j={\displaystyle \frac{{\displaystyle {\displaystyle \sum }_{k=1}^l}{\omega }_j^k}{l}}$.

Step 4: The GILWGA operator is used to aggregate attribute values, couple the scores of multiple experts, calculate the comprehensive evaluation value ${\alpha }_i^k$ of each decision expert under different schemes and construct the matrix, where ${\alpha }_i^k=\mathrm{GILWGA}({\alpha }_{i1}^k,{\alpha }_{i2}^k,\cdots ,{\alpha }_{in}^k)={\displaystyle \frac{1}{\lambda }}\left({\displaystyle {\displaystyle \prod }_{j=1}^n}{(\lambda {\alpha }_j)}^{{\omega }_j}\right)$.

Step 5: The score function and the exact function are used to calculate each intuitive language number, and each score function value in the matrix is introduced into the RPN formula, where $RPN=h({\alpha }_s)\times h({\alpha }_o)\times h({\alpha }_D)\times h({\alpha }_c)$. $h({\alpha }_s),h({\alpha }_o),h({\alpha }_D),h({\alpha }_c)$ are the scoring functions of severity (S), occurrence (O), detectability (D), and controllability (C), respectively, which are then used to calculate the criticality ranking of each fault mode.

Step 6: Based on the calculation results, the failure modes are ranked according to their severity, and preventive measures are taken for failure modes with higher severity to minimize the damage and losses. Figure 5 shows the calculation process of FMECA combined with intuitive language entropy and the GILWGA operator.

5. Case Analysis

In aerospace manufacturing, the spindle system of a five-axis CNC machine tool plays a critical role in ensuring the precision and surface integrity of complex components such as turbine blades and structural parts. Given the system’s complexity and its direct impact on machining accuracy, ensuring its reliability is essential. This section applies the proposed FMECA method to analyze the failure modes of a representative spindle system, aiming to validate the method’s practicality and robustness. The case demonstrates the use of generalized intuitionistic linguistic numbers to evaluate failure indicators and highlights the effectiveness of entropy-based dynamic weighting in handling multi-criteria decision-making under uncertainty. A set of representative failure modes is selected to illustrate the full computational process.

5.1. Calculation Process

In the aerospace manufacturing field, the traditional FMECA method for CNC machine tools usually lists all potential failure modes, evaluates their severity, and ranks them to identify key risks. However, given the complexity of the five-axis CNC spindle system, the number of possible failure modes may be very large, so a comprehensive listing is not practical. To ensure representativeness and feasibility, this study follows the case method in reference [36] to select failure modes based on the statistical intersection of failure records, maintenance logs, and user feedback collected in the enterprise. Focus is placed on those modes that have a significant impact on operation but whose severity has not yet been determined.

Step 1: A comprehensive fault set was established by combining empirical data and expert judgment. Nine failure modes with notable enterprise-level risk were identified, though their relative criticality remained uncertain: FM1, gear wear; FM2, oil leakage; FM3, disk spring failure; FM4, gear shift failure; FM5, excessive noise; FM6, bearing temperature is too high; FM7, low spindle positioning accuracy; FM8, positioning key wear; and FM9, bearing burnout.

Given the stringent precision and reliability requirements in aerospace applications, failure containment becomes as critical as failure prediction. To address this, the proposed method extends the conventional risk structure by incorporating controllability (C) as a fourth criterion. The four risk dimensions form the basis of the refined risk assessment. The corresponding evaluation levels and criteria are summarized in

Table 2. Risk attribute indicator assessment level and basis.

Evaluation Index	Standard	Scores
Occurrence (O)	85~100%	The ratio of the number of times that a fault occurs to the total number of times that all faults occur in a unit time is used for reference.
	70~85%
	55~70%
	40~55%
	≤40%
Severity (S)	>80%	Based on historical data or expert experience, the impact of failure on the operation of the enterprise and the proportion of income loss generated are used as references.
	65~80%
	50~65%
	35~50%
	≤35%
Detectability (D)	>80%	Based on the existing fault detection data, the fault detection ability is evaluated from the point of difficulty and cost of detection.
	65~80%
	50~65%
	35~50%
	≤35%
Controllability (C)	>90%	According to the previous data and expert experience, the risk shows a weakening trend after taking emergency measures.
	75~90%
	60~75%
	45~60%
	≤45%

.

Step 2: An FMEA table was prepared, and relevant experts and scholars were invited to evaluate it. After investigating factories in different regions and interviewing experts in the factories, we collected and summarized the relevant information sets of key failure modes. The causes of failures and the local and final impacts of the failures are shown in

Table 3. System failure mode FMEA table.

Serial	Failure Mode	Failure Cause	Failure Impact
			Local Impact	Final Impact
FM1	gear wear	Poor lubrication, excessive load	Poor gear meshing	Reduced transmission efficiency
FM2	oil leakage	Aging and wear of seals	Oil-contaminated parts	System performance degradation
FM3	disk spring failure	Long-term high load	Spring force decreases	Unstable spindle operation
FM4	gear shift failure	Worn-out mechanical parts, abnormal control signals	Gear shifting failure	Reduced processing accuracy
FM5	excessive noise	Aging equipment and improper maintenance	Performance degradation is obvious	Interference with peripheral equipment, causing unstable system operation
FM6	bearing temperature is too high	Cooling system failure, improper lubrication	Reduced bearing life	Spindle not working
FM7	low spindle positioning accuracy	Bearing wear or large clearance	Workpiece position deviation	Decreased workpiece quality and effect on efficiency
FM8	positioning key wear	Excessive friction due to long-term stress	Positioning accuracy decreases	Unstable spindle operation
FM9	bearing burnout	Poor lubrication or excessive grease, overheating, severe overload	Bearing damage	Loss of function, machine tool alarm

.

The expert system was invited to conduct scoring. According to the actual situation of the industry, only three experts from different fields were invited to conduct the evaluation. A researcher from a mechanical product design institute, an engineer from a machine tool manufacturing company, and a workshop management director from a mechanical processing company were invited to form a scoring expert system.

According to the evaluation levels and basis in Table 2, the FMEA method was applied to analyze the failure mode set in this article. The three experts scored the fault mode set in the form of intuitive language numbers from four perspectives: probability of occurrence (O), severity (S), detectability (D) and controllability (C). The specific data are shown in

Table 4. Expert 1’s scores.

Serial	Failure Mode	Severity (S)	Occurrence (O)	Detectability (D)	Controllability (C)
FM1	gear wear	<s3, 0.9, 0.1>	<s3, 0.7, 0.3>	<s2, 0.8, 0.1>	<s1/2, 0.7, 0.2>
FM2	oil leakage	<s3, 0.7, 0.1>	<s1, 0.7, 0.2>	<s3, 0.7, 0.2>	<s3, 0.7, 0.3>
FM3	disk spring failure	<s2, 0.8, 0.1>	<s4, 0.7, 0.2>	<s3, 0.7, 0.1>	<s3, 0.7, 0.3>
FM4	gear shift failure	<s2, 0.8, 0.2>	<s2, 0.7, 0.2>	<s2, 0.9, 0.1>	<s1, 0.7, 0.1>
FM5	excessive noise	<s3, 0.7, 0.2>	<s2, 0.8, 0.2>	<s2, 0.7, 0.3>	<s3, 0.7, 0.2>
FM6	bearing temperature is too high	<s2, 0.8, 0.1>	<s3, 0.8, 0.1>	<s2, 0.7, 0.2>	<s1/2, 0.7, 0.1>
FM7	low spindle positioning accuracy	<s3, 0.7, 0.3>	<s2, 0.8, 0.1>	<s2, 0.7, 0.3>	<s1, 0.8, 0.1>
FM8	positioning key wear	<s2, 0.7, 0.2>	<s1, 0.7, 0.2>	<s1, 0.7, 0.3>	<s2, 0.7, 0.3>
FM9	bearing burnout	<s3, 0.9, 0.1>	<s1, 0.7, 0.2>	<s1, 0.7, 0.2>	<s1, 0.7, 0.1>

,

Table 5. Expert 2’s scores.

Serial	Failure Mode	Severity (S)	Occurrence (O)	Detectability (D)	Controllability (C)
FM1	gear wear	<s1/2, 0.7, 0.1>	<s2, 0.8, 0.2>	<s2, 0.7, 0.1>	<s3, 0.8, 0.2>
FM2	oil leakage	<s2, 0.7, 0.1>	<s2, 0.8, 0.1>	<s2, 0.7, 0.3>	<s2, 0.7, 0.2>
FM3	disk spring failure	<s3, 0.7, 0.2>	<s3, 0.8, 0.2>	<s2, 0.8, 0.2>	<s2, 0.9, 0.1>
FM4	gear shift failure	<s2, 0.7, 0.1>	<s1, 0.7, 0.1>	<s2, 0.8, 0.2>	<s1, 0.7, 0.3>
FM5	excessive noise	<s3, 0.9, 0.1>	<s2, 0.7, 0.3>	<s2, 0.9, 0.1>	<s2, 0.8, 0.2>
FM6	bearing temperature is too high	<s3, 0.9, 0.1>	<s1/2, 0.7, 0.2>	<s1, 0.7, 0.3>	<s2, 0.7, 0.3>
FM7	low spindle positioning accuracy	<s2, 0.9, 0.1>	<s2, 0.7, 0.3>	<s2, 0.7, 0.2>	<s1, 0.7, 0.2>
FM8	positioning key wear	<s1, 0.8, 0.1>	<s2, 0.8, 0.2>	<s1, 0.8, 0.2>	<s1, 0.7, 0.1>
FM9	bearing burnout	<s2, 0.7, 0.2>	<s1/2, 0.7, 0.3>	<s1, 0.7, 0.3>	<s2, 0.8, 0.2>

and

Table 6. Expert 3’s scores.

Serial	Failure Mode	Severity (S)	Occurrence (O)	Detectability (D)	Controllability (C)
FM1	gear wear	<s3, 0.8, 0.1>	<s2, 0.7, 0.2>	<s1/2, 0.7, 0.3>	<s2, 0.7, 0.3>
FM2	oil leakage	<s2, 0.7, 0.1>	<s2, 0.8, 0.1>	<s2, 0.8, 0.1>	<s2, 0.8, 0.2>
FM3	disk spring failure	<s3, 0.9, 0.1>	<s3, 0.7, 0.3>	<s2, 0.7, 0.2>	<s2, 0.7, 0.2>
FM4	gear shift failure	<s2, 0.7, 0.1>	<s1, 0.8, 0.1>	<s1, 0.7, 0.2>	<s2, 0.7, 0.1>
FM5	excessive noise	<s3, 0.9, 0.1>	<s2, 0.7, 0.1>	<s2, 0.7, 0.2>	<s2, 0.8, 0.1>
FM6	bearing temperature is too high	<s2, 0.8, 0.2>	<s1/2, 0.7, 0.3>	<s2, 0.8, 0.1>	<s2, 0.8, 0.1>
FM7	low spindle positioning accuracy	<s2, 0.7, 0.1>	<s2, 0.8, 0.2>	<s2, 0.7, 0.2>	<s1, 0.7, 0.3>
FM8	positioning key wear	<s2, 0.9, 0.1>	<s1, 0.8, 0.1>	<s1, 0.7, 0.1>	<s1, 0.8, 0.2>
FM9	bearing burnout	<s2, 0.8, 0.1>	<s2, 0.9, 0.1>	<s1, 0.7, 0.3>	<s1/2, 0.7, 0.2>

.

Figure 6, Figure 7 and Figure 8 show the visualization results of the intuitive language numerical evaluation values of nine failure modes under four risk factors (severity, occurrence, detectability, and controllability) by three different experts in the expert system. The data visualization results intuitively reflect the differences in perception and judgment between different experts.

Step 3: Experts’ scoring of faults is based on effectiveness and does not need to be standardized. Substituting the data in the table into formulas (1) and (2) can obtain the intuitive language entropy of each indicator, and then substituting the intuitive language entropy into formula (3) obtains the indicator weights of each expert decision matrix, which are respectively: ${\omega }^1=(0.2489,0.2485,0.2502,0.2524)$, ${\omega }^2=(0.2510,0.2497,0.2501,0.2492)$, and ${\omega }^3=(0.2501,0.2499,0.2502,0.2499)$. Then the average value is found to get the attribute weight $\omega =(0.2500,0.2494,0.2502,0.2505)$.

Step 4: Combined with the attribute weights calculated in step 3, the data in Table 4, Table 5 and Table 6 are substituted into formula (5) to calculate the comprehensive evaluation value of each expert’s solution, and the language term matrix $S={\left({s_{\theta }}_{ij}\right)}_{9\times 4}$, membership matrix $U={\left({\mu }_{ij}\right)}_{9\times 4}$, and non-membership matrix $V={\left({\nu }_{ij}\right)}_{9\times 4}$ are obtained.

$$S={\left({s_{\theta }}_{ij}\right)}_{9\times 4}=\left[\begin{array}{cccc}1.4565& 1.8584& 1.1894& 1.3168\\ 1.8612& 1.4130& 2.0610& 1.8635\\ 2.0598& 2.4442& 2.0610& 1.8635\\ 1.6818& 1.1887& 1.4146& 1.1896\\ 2.2795& 1.6797& 1.6825& 1.8635\\ 1.8612& 0.9308& 1.6825& 1.1896\\ 1.8612& 1.6797& 1.6825& 1.0000\\ 1.4142& 1.1887& 1.0000& 1.1896\\ 1.8612& 1.0000& 1.0000& 1.0000\end{array}\right]$$

$$U={\left({\mu }_{ij}\right)}_{9\times 4}=\left[\begin{array}{cccc}0.8426& 0.7917& 0.8180& 0.7909\\ 0.7653& 0.8185& 0.7911& 0.7909\\ 0.8426& 0.7917& 0.7651& 0.8146\\ 0.7913& 0.7917& 0.8677& 0.7649\\ 0.8678& 0.7917& 0.7651& 0.8178\\ 0.8712& 0.7917& 0.7911& 0.7909\\ 0.8149& 0.8185& 0.7651& 0.7909\\ 0.8426& 0.8185& 0.7651& 0.7909\\ 0.8426& 0.8153& 0.7651& 0.7909\end{array}\right]$$

$$V={\left({\nu }_{ij}\right)}_{9\times 4}=\left[\begin{array}{cccc}0.9240& 0.8185& 0.8677& 0.8178\\ 0.9240& 0.8974& 0.8711& 0.8178\\ 0.8972& 0.8185& 0.8971& 0.8423\\ 0.8972& 0.8974& 0.8971& 0.8675\\ 0.8972& 0.8429& 0.7911& 0.8709\\ 0.8972& 0.8429& 0.8711& 0.8675\\ 0.8678& 0.8429& 0.7911& 0.8423\\ 0.8972& 0.8715& 0.8148& 0.8423\\ 0.8972& 0.8429& 0.7911& 0.8709\end{array}\right]$$

Figure 9 standardizes the evaluation values of different experts in the expert system for nine failure modes under four major risk factors (severity, occurrence, detectability and controllability). It can be seen that the overall trend of the evaluation data among different experts is consistent, but there are still cognitive differences for some individuals.

Step 5: According to Definitions 5 and 6, the scheme is calculated to obtain the score function $h(\alpha )$ of each indicator, as shown in

Table 7. The score function value of each indicator.

Serial	Failure Mode	Severity (S)	Occurrence (O)	Detectability (D)	Controllability (C)
FM1	gear wear	0.1186	0.0498	0.0591	0.0354
FM2	oil leakage	0.2954	0.1115	0.1649	0.0501
FM3	disk spring failure	0.1125	0.0655	0.2721	0.0516
FM4	gear shift failure	0.1781	0.1256	0.0416	0.1221
FM5	excessive noise	0.067	0.086	0.0437	0.099
FM6	bearing temperature is too high	0.0484	0.0477	0.1346	0.0911
FM7	low spindle positioning accuracy	0.0985	0.041	0.0437	0.0514
FM8	positioning key wear	0.0772	0.063	0.0497	0.0611
FM9	bearing burnout	0.1016	0.0276	0.026	0.08

.

The normalized evaluation data were integrated into the revised RPN formulation, and the resulting values were calculated and ranked as shown in

Table 8. Failure mode evaluation ranking.

Serial	Failure Mode	RPN (10−5)	Sorting
FM1	gear wear	1.2357	7
FM2	oil leakage	27.2110	1
FM3	disk spring failure	10.3460	3
FM4	gear shift failure	11.3622	2
FM5	excessive noise	2.4928	5
FM6	bearing temperature is too high	2.8309	4
FM7	low spindle positioning accuracy	0.9071	8
FM8	positioning key wear	1.4769	6
FM9	bearing burnout	0.5833	9

. Figure 10 visualizes the relative criticality of the nine failure modes, reflecting the distribution of high-risk points within the spindle system of the five-axis CNC machine tool.

Step 6: Failure mode ranking and reliability decision support. Based on the scoring function and the proposed RPN calculation method, the risk priority number (RPN) of all nine failure modes is calculated. The ranking in Table 8 reflects the comprehensive dangerousness of the failure mode under the multi-attribute evaluation integrating expert cognitive fuzziness and coupling scores.

5.2. Effectiveness Analysis and Comparison

To evaluate the stability of the proposed algorithm, a sensitivity analysis was conducted with respect to the number of experts and the granularity parameter. The case with seven experts and a granularity parameter of six was selected as the baseline. These two parameters were then varied and the rankings were recalculated. Kendall’s τ coefficient was used to measure the consistency of the results under different conditions. The corresponding results are presented in

Table 9. Sensitivity analysis of changes in the number of experts and the granularity parameter.

Number of Experts	Granularity Parameter	RPN									Rank	Kendall τ
		FM1	FM2	FM3	FM4	FM5	FM6	FM7	FM8	FM9
3	5	1.2357	27.211	10.346	11.362	2.4928	2.8309	0.9071	1.4769	0.5833	FM2 > FM4 > FM3 > FM6 > FM5 > FM8 > FM1 > FM7 > FM9	0.333
5	5	76.285	460.1	182.44	229.18	128.94	154.95	60.824	25.611	11.935	FM2 > FM4 > FM3 > FM6 > FM5 > FM1 > FM7 > FM8 > FM9	1
7	5	1059.8	5295.8	2793.5	3341.5	1460.9	1741.4	755.59	199.23	19.294	FM2 > FM4 > FM3 > FM6 > FM5 > FM1 > FM7 > FM8 > FM9	1
3	6	1.4078	16.999	6.2889	11.088	2.9537	5.371	1.5586	0.3346	0.4938	FM2 > FM4 > FM3 > FM6 > FM5 > FM7 > FM1 > FM9 > FM8	0.556
5	6	53.503	494.87	202.84	229.62	142.61	189.93	47.863	7.1507	6.4426	FM2 > FM4 > FM3 > FM6 > FM5 > FM1 > FM7 > FM8 > FM9	1
7	6	532.85	5698.3	3092.2	3345.5	1168	2000.7	498.53	55.568	10.409	FM2 > FM4 > FM3 > FM6 > FM5 > FM1 > FM7 > FM8 > FM9	1
3	7	1.1846	20.777	6.2611	12.263	2.9482	5.3837	1.7176	0.3107	0.6477	FM2 > FM4 > FM3 > FM6 > FM5 > FM7 > FM1 > FM9 > FM8	0.556
5	7	53.56	606.64	202.44	253.91	142.53	189.7	47.722	8.7678	5.9935	FM2 > FM4 > FM3 > FM6 > FM5 > FM1 > FM7 > FM8 > FM9	1
7	7	533.22	6985.1	3085.2	3698.5	1169.5	2001	498.02	57.054	9.7069	FM2 > FM4 > FM3 > FM6 > FM5 > FM1 > FM7 > FM8 > FM9	1

.

Based on the data in the table, the average value of Kendall’s τ is calculated as 0.827. This indicates that the ranking results remain highly consistent under different numbers of experts and granularity parameters. The result suggests that the proposed algorithm has good stability. Figure 11 shows the variation in RPN values under different expert numbers and granularity parameters.

As shown in the figure, the algorithm remains generally stable when the number of experts and the granularity parameter change, and only minor variations in the ranking results can be observed. The table also indicates that some fluctuations appear when the number of experts is relatively small. As the number of experts increases and the granularity parameter becomes larger, the ranking gradually stabilizes. At the same time, failure modes with higher hazard levels become more prominent in the ranking. These results suggest that the proposed method can effectively reduce the influence of fuzziness in multi-expert evaluations and can identify high-risk failure modes in a more objective and reliable manner.

The sensitivity analysis indicates that when the number of experts is three and the granularity parameter is five, the algorithm remains relatively stable. However, some deviations in the ranking results can still be observed. Therefore, this study adopts the case with seven experts and a granularity parameter of seven for subsequent analysis. This combination shows higher stability in the sensitivity analysis, and the influence of cognitive fuzziness is reduced when more experts are involved. Based on this relatively stable dataset, reliability improvement decisions are further analyzed and compared with other results in the following section.

To clarify the effect of the parameter λ on the decision results, we further examine the changes in the score function values and ranking results of the alternatives under different values of λ. The corresponding results are presented in

Table 10. Sensitivity analysis of the parameter λ on the score function values and ranking results.

Parameter λ	Scores									Ranking Order
	FM1	FM2	FM3	FM4	FM5	FM6	FM7	FM8	FM9
λ = 1	533.22	6985.1	3085.2	3698.5	1169.5	2001	498.02	57.054	9.7069	FM2 > FM4 > FM3 > FM6 > FM5 > FM1 > FM7 > FM8 > FM9
λ = 2	521.34	6832.7	3010.6	3612.9	1145.2	1956.8	489.36	56.175	9.5504	FM2 > FM4 > FM3 > FM6 > FM5 > FM1 > FM7 > FM8 > FM9
λ = 4	498.67	6521.3	2875.4	3422.7	1098.6	1863.5	470.89	53.769	9.11	FM2 > FM4 > FM3 > FM6 > FM5 > FM1 > FM7 > FM8 > FM9
λ = 10	442.57	6018.2	2648.3	3150.6	1015.3	1711.9	448.69	50.124	8.53	FM2 > FM4 > FM3 > FM6 > FM5 > FM7 > FM1 > FM8 > FM9

.

The results of the sensitivity analysis are shown in Figure 12, which illustrates the score variation and relative changes in failure mode evaluations under different values of λ. When λ is set to 1, 2, 4, and 10, the overall variation trends of the normalized scores remain consistent across all failure modes. As λ increases, the scores decrease slightly, while the magnitude of change remains small and the ranking results remain largely stable. This indicates that the proposed method is robust with respect to variations in the parameter λ.

It is worth noting that λ serves as a regulating parameter in the aggregation process and influences the numerical distribution of evaluation results. Therefore, its selection should balance model interpretability and practical applicability. In this study, λ = 1 is adopted based on engineering practice and expert consultation as a commonly used baseline setting in similar multi-criteria decision-making contexts. This choice preserves stable ranking performance without introducing additional preference bias into the aggregation process. Meanwhile, larger values of λ would increase computational complexity due to repeated exponential operations, without providing significant improvement in ranking discrimination. Consequently, λ = 1 is selected as a parsimonious and practically effective configuration for subsequent calculations.

To translate the ranked failure modes into actionable maintenance strategies, different classes of risk levels are addressed with tailored engineering decisions. Based on the computed RPN values from Table 8 and supported by the proposed intuitionistic linguistic FMECA framework, the following recommendations are made:

(1) High risk (FM2, FM4, FM3)—Immediate intervention required

FM2—Oil leakage (RPN: 6985.1 × 10⁻⁵)

Justification: The high RPN arises from strong expert consensus (low entropy) and a substantial controllability penalty. In aerospace-grade five-axis CNC systems, oil leakage can rapidly contaminate precision components and compromise dimensional accuracy. The incorporation of controllability and expert coupling in the proposed method enhances its ability to identify this critical hazard, which may be overlooked in traditional FMECA.

Reliability improvement decisions: Oil leakage shows the most prominent risk level. This indicates that the reliability of the lubrication system plays a critical role in the overall operation of the spindle system. In engineering practice, priority should be given to the stability of sealing structures and lubrication paths. For example, the aging condition of seals should be inspected, and locations prone to leakage should be closely monitored. Simple online monitoring methods can also be introduced at key lubrication points, such as monitoring oil film conditions or lubrication pressure changes. Once abnormalities are detected, corrective actions can be taken in time. This helps prevent lubrication contamination from affecting spindle accuracy and bearing performance.

FM4—Gear shift failure (RPN: 3698.5 × 10⁻⁵)

Justification: Experts highlighted the potential for control instability to affect system-wide performance, a particularly serious concern in aerospace part manufacturing. The GILWGA operator effectively captures such interdependencies through expert cognitive coupling and fuzzy integration, revealing system-level threats that static models fail to detect.

Reliability improvement decisions: Gear shift failure has a significant impact on system stability. The problem is not limited to local transmission components and may also affect the overall machine operation through the control system. In engineering practice, two aspects should receive particular attention. First, the wear condition of the shift actuator and related mechanical components should be inspected to avoid unstable operation caused by long-term use. Second, the stability of the shift control logic should be monitored. For example, abnormal signals or impact events during operation can be recorded and analyzed. These measures help identify potential issues at an early stage and reduce the influence of gear shift failure on the operational rhythm of the system.

FM3—Disk spring failure (RPN: 3085.2 × 10⁻⁵)

Justification: Although the occurrence rate is moderate, entropy-based dynamic weighting significantly elevated its severity score. In aerospace applications, where fatigue-induced deformation can lead to micro-scale positioning errors, identifying such latent risks is critical. This demonstrates the proposed method’s advantage in prioritizing failure modes that are not apparent in conventional analyses.

Reliability improvement decisions: Failures related to disk springs show relatively high importance in the overall evaluation. This indicates that although such faults may not be the most obvious source of failure, their long-term impact should not be neglected. In engineering practice, improvements can be considered from the perspective of structural reliability. For example, the fatigue life of the springs can be evaluated, and appropriate replacement intervals can be determined according to actual operating conditions. In addition, load distribution can be optimized during the design stage to reduce local stress concentration. These measures can help extend the service life of the springs to a certain extent.

(2) Medium risk (FM6, FM5, FM1, FM7)—Preventive measures recommended

FM6—Bearing overheating (RPN: 2001.0 × 10⁻⁵)

Justification: Severity and controllability received high membership ratings under the GILWGA framework. In aerospace machining systems, even moderate thermal deviations can cause microstructural deformation or geometric inaccuracies. Early mitigation is warranted, despite its moderate detectability.

Reliability improvement decisions: Bearing overheating is identified as a failure type that requires continuous attention in the evaluation results. Instead of waiting until the temperature rises significantly, it is more effective to strengthen monitoring during operation. For example, temperature sensors or operational data analysis can be used to track changes in the thermal condition of the bearing. If intervention is carried out when the temperature rise trend first appears, such as adjusting lubrication conditions or operating parameters, more serious failures can often be avoided.

FM5—Excessive noise (RPN: 1169.5 × 10⁻⁵)

Justification: Although traditionally perceived as a minor nuisance, expert fuzzy evaluations reveal its association with mechanical imbalance and operator fatigue. In high-precision aerospace machining environments, such instability may lead to subtle degradation of performance, highlighting the method’s advantage in capturing non-obvious but consequential risks.

Reliability improvement decisions: Noise-related issues are often overlooked in practice. However, the analysis indicates that noise levels are closely associated with the operating condition of the equipment, such as mechanical imbalance or localized wear. Therefore, variations in noise can be treated as an early warning signal during operation. Simple acoustic or vibration monitoring methods can be used to track the system condition over time. If a noticeable change in noise level is detected, the corresponding components should be inspected promptly to prevent the issue from developing further.

FM1—Gear wear (RPN: 533.22 × 10⁻⁵)

Justification: Attribute entropy analysis shows consistent expert judgment, while the hesitation degree indicates uncertainty regarding long-term reliability. In aerospace-grade transmission systems, prolonged gear wear may accumulate into alignment drift or motion errors, necessitating preventive measures.

Reliability improvement decisions: Gear wear is typically a gradual process that accumulates over time. It may not cause immediate and obvious failures. However, as operating time increases, it can gradually affect transmission accuracy. In engineering practice, its progression is better controlled through periodic inspection and proper lubrication management. Attention should be paid to the wear condition of gear surfaces and the lubrication state. This approach allows maintenance to be carried out before the wear becomes severe.

FM7—Low spindle positioning accuracy (RPN: 498.02 × 10⁻⁵)

Justification: This failure mode received low severity and high controllability scores across experts. The proposed model, through expert opinion coupling, identifies it as a situational issue rather than a systemic fault, particularly in aerospace operations, where active calibration protocols are standard.

Reliability improvement decisions: A reduction in spindle positioning accuracy is usually reflected first in changes in machining quality rather than in direct system failure. In engineering practice, stability can be maintained through periodic precision inspection and calibration maintenance. Adjustments can also be made by applying existing calibration procedures or compensation strategies. With appropriate maintenance measures, such issues can generally be kept under effective control.

(3) Low risk (FM8, FM9)—Monitor and defer action

FM8—Bearing burnout (RPN: 57.094 × 10⁻⁵)

Justification: While traditionally classified as a critical fault, the integrated expert assessments in this study indicate high detectability and controllability. When standard maintenance protocols are observed, the practical risk remains low, even under demanding aerospace machining conditions.

Reliability improvement decisions: Although bearing burnout is traditionally considered a severe failure, the overall evaluation results indicate that under the current system conditions this issue can often be detected at an early stage. Therefore, engineering practice can rely more on existing protection mechanisms, such as temperature monitoring and overload protection. As long as these monitoring measures remain effective, corrective actions can be taken before the failure develops into a critical condition.

FM9—Positioning key wear (RPN: 9.7069 × 10⁻⁵)

Justification: Entropy-based weighting reveals low variance among expert judgments, and overall fuzzy evaluations suggest moderate risk. Given the tolerance of aerospace-grade equipment to minor positional deviations, this mode is deemed non-critical.

Reliability improvement decisions: Wear of the positioning key has a relatively minor impact on the overall operation of the system and usually does not lead to immediate and obvious failures. In engineering practice, it only needs to be checked during routine maintenance. The wear condition and changes in the fitting clearance can be observed. If no significant abnormalities are found, maintenance can be carried out according to the normal service schedule.

A summary of reliability risk levels and corresponding engineering strategies is provided in

Table 11. Reliability risk assessment and engineering decision table.

Failure Mode	RPN (10−5)	Risk Level	Reliability Improvement Decisions	Justification
FM2—oil leakage	6985.1	high	Reliability Improvement Decision	High uncontrollability and expert consensus; risk may be underestimated in traditional FMECA
FM4—gear shift failure	3698.5	high	Inspect seals and monitor leakage	Cross-impact potential revealed through expert coupling and fuzzy assessments
FM3—disk spring failure	3085.2	high	Inspect shift actuator and control logic	Low entropy in severity; traditional RPN may ignore this hidden hazard
FM6—bearing temperature is too high	2001	medium	Evaluate fatigue life and optimize load	Fuzzy modeling uncovers systemic impact of perceived non-critical issue
FM5—excessive noise	1169.5	medium	Monitor temperature and adjust lubrication	High severity and controllability scores suggest early intervention
FM1—gear wear	533.22	medium	Acoustic monitoring for imbalance detection	Stable entropy but moderate hesitation indicates long-term degradation risk
FM7—low spindle positioning accuracy	498.02	medium	Periodic inspection and lubrication	High controllability and low severity; situational rather than structural
FM8—positioning key wear	57.054	low	Routine calibration and compensation	High detectability and controllability reduce practical risk
FM9—bearing burnout	9.7069	low	Temperature monitoring and overload protection	Low expert differentiation and moderate fuzzy evaluations

.

This risk-specific decision strategy highlights the capability of the proposed FMECA framework to support fine-grained, expert-informed, and cognitively aware maintenance planning in aerospace manufacturing systems. By integrating intuitionistic linguistic entropy, dynamic weighting, and multi-source expert coupling, the method overcomes the limitations of traditional FMECA and offers a more realistic, interpretable, and robust foundation for reliability-centered decision-making in high-precision engineering contexts.

To validate the effectiveness of the proposed method and highlight the advantages of the intuitionistic linguistic FMECA approach, the RPN values obtained by the proposed method are compared with those derived from several existing approaches. These include the traditional method (the intuitionistic linguistic values provided by experts are directly transformed into a score function for calculation, expressed as $RPN=h\prime ({\alpha }_s)\times h\prime ({\alpha }_o)\times h\prime ({\alpha }_D)$, the method reported in reference [37], Bayesian network FMECA (BN-FMECA, including the OOBN extension), and the Monte Carlo RPN method. The comparison is conducted across different failure modes. For the OOBN-based Bayesian network FMECA method, three sub-nodes E(Si), E(Oi), and E(Di) are constructed to represent severity S, occurrence O, and detectability D, respectively. To incorporate the controllability factor introduced in this study, controllability C is treated as a conversion factor g(Ci) and integrated into the FMECA calculation. For the Monte Carlo RPN method, experts first provide initial expected values for severity S, occurrence O, detectability D, and controllability C for each failure mode. These expectations are used as the mean values, and small normal perturbations are introduced. Random sampling is then performed based on the normal distribution. The number of samples is set to N = 20,000. The conversion factor g(C) is also incorporated into the original RPN formulation.

The resulting RPN values obtained from different methods for each failure mode are presented in

Table 12. RPN of each risk factor under different methods.

Method	Scores									Ranking Order
	FM1	FM2	FM3	FM4	FM5	FM6	FM7	FM8	FM9
Proposed method	533.22	6985.1	3085.2	3698.5	1169.5	2001	498.02	57.054	9.7069	FM2 > FM4 > FM3 > FM6 > FM5 > FM1 > FM7 > FM8 > FM9
Ref [37] method	1.425	3.4578	2.1926	3.372	1.7376	3.1558	2.1312	2.0538	1.2472	FM2 > FM4 > FM6 > FM3 > FM7 > FM8 > FM5 > FM1 > FM9
Traditional method	2.1447	4.2756	2.9927	3.1736	1.7514	1.5548	0.9318	0.6548	1.9756	FM2 > FM4 > FM3 > FM1 > FM9 > FM5 > FM6 > FM7 > FM8
BN-FMECA (OOBN-based)	241.47	395.9	476.26	254.32	215.28	231.67	159.38	145.82	129.36	FM3 > FM2 > FM4 > FM1 > FM6 > FM5 > FM7 > FM8 > FM9
Monte Carlo RPN	2320.28	3988.66	2418.81	1265.83	1511.85	3134.52	277.85	261.26	555.66	FM2 > FM6 > FM3 > FM1 > FM5 > FM4 > FM9 > FM7 > FM8

.

Figure 13 shows the comparative study results of the proposed method, the traditional method, the method in Ref. [37], the BN-FMECA method, and the Monte Carlo method. Meanwhile, in order to present the ranking trends of the five methods more intuitively, all methods are normalized to obtain Figure 13.

Figure 13 is comparison of different methods for failure mode evaluation. The blue line represents the proposed method, the orange line represents the method of Ref. [37], the yellow line represents the traditional method, the purple line represents the BN-FMECA method, and the green line represents the Monte Carlo method.

Figure 13 presents the RPN values calculated by the proposed method and four alternative approaches. Figure 14 compares the normalized ranking results obtained from the five methods. Overall, although the other approaches show partial consistency with the proposed method in the prioritization of some failure modes, the proposed FMECA framework demonstrates higher sensitivity and better rationality in identifying critical failure modes.

Figure 14 is comparison of normalized evaluation results of failure modes obtained by different methods. The blue line represents the proposed method, the orange line represents the method reported in Ref. [37], the yellow line represents the traditional method, the purple line represents the BN-FMECA method, and the green line represents the Monte Carlo method.

For example, FM2 (oil leakage) and FM4 (gear shift failure) obtain significantly higher RPN values under the proposed method. This result is consistent with the risk mechanisms observed in five-axis CNC machines used for aerospace precision manufacturing. Oil leakage in lubrication or hydraulic systems may reduce the operational stability of the spindle or feed units. This instability can lead to dimensional deviations or surface defects when machining high-precision components such as turbine blades and engine disks. Gear shift failure may cause unstable torque transmission in the spindle and sudden changes in machining parameters. Such changes may seriously affect the machining quality of aerospace components and increase the risk of tool wear and surface quality fluctuation.

In contrast, other methods tend to underestimate the importance of these hidden risks. By integrating intuitionistic linguistic entropy with the GILWGA operator, the proposed method reduces cognitive fuzziness in expert evaluations and captures the interactions among risk factors. As a result, the collective judgment of experts on high-risk failure modes is strengthened. This allows critical faults to stand out more clearly in the ranking and better reflects their actual threat to precision manufacturing processes.

Similarly, FM2 (oil leakage) is more prominently ranked in the proposed model compared to FM3 and FM4. This is due to consistent expert evaluation indicating high risk across all four dimensions (O, S, D, and C). Oil leakage in aerospace-grade machine tools poses severe contamination risks to high-tolerance components, and the GILWGA operator effectively aggregates this risk perception. Thus, the framework not only highlights critical failures with greater clarity but also offers engineering insights that are directly actionable in high-stakes aviation environments.

Although conventional FMECA methods and the model in Ref. [37] provide a basic overview of failure mode priorities, they often fail to reflect the real operational burden in high-reliability aerospace contexts. For example, FM5 requires complex actions such as real-time monitoring and component alignment correction, while FM7 may only involve bolt tightening or clearance adjustments. FM8 often entails minimal mechanical intervention. The proposed method captures these gradations more effectively by dynamically adjusting weights and fusing multi-source expert inputs.

Although the four alternative methods are able to provide a general prioritization of failure modes, they often fail to accurately reflect the practical conditions in high-reliability aerospace manufacturing environments. For example, FM5 receives relatively high scores in some methods because it requires real-time monitoring and complex calibration operations. However, such issues mainly reflect maintenance complexity rather than a higher level of system risk.

In contrast, FM9 has a very high hazard level. However, for five-axis CNC machines used in aerospace precision manufacturing, dynamic monitoring systems are usually implemented and maintenance is performed frequently. As a result, the actual occurrence probability of this failure mode is extremely low. Under such conditions, assigning an excessively high score to FM9 is clearly unreasonable.

Compared to additive RPN models or those relying on single-expert scoring, the FMECA framework proposed in this paper integrates nuanced risk factors—such as structural complexity, control margins, and cognitive uncertainty—that are particularly relevant in aerospace-grade manufacturing equipment. This leads to a more comprehensive, objective, and operationally meaningful risk ranking. As such, the proposed method demonstrates strong potential as a decision-support tool for reliability evaluation and maintenance planning in advanced aerospace production systems.

To verify the effectiveness of the proposed method, this study compares it with four alternative approaches. These include the direct summation of risk factors, a weighted aggregation method based on single-expert scoring, Bayesian network FMECA (BN-FMECA, including the OOBN extension), and the Monte Carlo RPN method. The results show that the RPN values obtained from the proposed FMECA framework more accurately reflect the structural complexity and multidimensional risk characteristics of aerospace manufacturing systems.

By introducing differentiated expert weights and refining risk attributes, the proposed method reduces subjectivity in the evaluation process and improves the precision of risk assessment. Therefore, for complex equipment such as five-axis CNC machines used in the machining of critical aerospace components, this approach provides a more accurate and practically valuable basis for reliability analysis and maintenance planning.

6. Conclusions

In response to the increasing demand for high-reliability manufacturing in aerospace systems, this study proposes a novel FMECA framework tailored to complex, high-precision equipment such as five-axis CNC machine tools used in the fabrication of critical aerospace components. The method integrates a GILWGA operator to overcome three key limitations of traditional approaches: inadequate representation of diverse risk factors, insufficient modeling of cognitive ambiguity in expert judgment, and neglect of interdependencies among expert evaluations. By introducing controllability as an additional risk dimension and employing linguistic entropy to dynamically assign weights to both experts and indicators, the proposed FMECA framework significantly enhances the objectivity and adaptability of failure mode prioritization in aerospace manufacturing environments. In addition, the incorporation of extended linguistic terms further improves the granularity and consistency of the evaluation process.

A case study involving the spindle system of a five-axis CNC machine tool—commonly used in the high-precision manufacturing of aerospace components such as turbine blades—demonstrates the practical value of the proposed method. Compared to conventional approaches, the G-FMECA model provides a more realistic representation of system vulnerabilities by capturing multi-source expert consensus and adjusting for expert coupling effects. It effectively reveals latent failure risks that may be overlooked in single-perspective evaluations, offering decision-makers a more robust and cognitively informed tool for reliability-centered maintenance planning.

This research contributes to the development of intelligent, data-driven reliability analysis techniques for complex aerospace manufacturing systems. Future work will extend the method to multi-stage production processes and integrate real-time condition monitoring data to further enhance predictive reliability and system resilience in mission-critical aerospace applications. At the same time, with the rapid development of data-driven process monitoring technologies in intelligent manufacturing environments, fault identification is gradually shifting from approaches that rely on manual experience or predefined thresholds to monitoring frameworks based on adaptive analysis of process data. Emerging data analysis methods are capable of identifying abnormal patterns directly from process data without requiring predefined thresholds or extensive manual labeling, thereby reducing reliance on subjective expert judgment. Urgo et al. [38] generated labeled synthetic data using digital twins to train CNN-based detection models for anomaly identification in manufacturing processes. Kumar et al. [39] integrated unsupervised generative deep learning with feature-sensitive interpretability to enable scalable and cost-effective acoustic monitoring. These studies demonstrate the effectiveness of data-driven, label-free approaches for intelligent anomaly detection in advanced manufacturing.

In this context, future research can explore the integration of such data-driven anomaly detection methods with the proposed FMECA evaluation framework. By combining expert knowledge with process monitoring data, the objectivity and reliability of failure mode assessment can be further improved, thereby enhancing the practical applicability of reliability analysis in complex aerospace manufacturing systems.