Multi-Environmental Reliability Evaluation for Complex Equipment: A Strict Intuitionistic Fuzzy Distance Measure-Based Multi-Attribute Group Decision-Making Framework

Abstract

The theoretical reliability of complex equipment often significantly deviates from real-world performance due to the inherent influence of diverse environmental and operational factors, making scientific reliability evaluation particularly challenging. This study proposes a multi-attribute group decision-making (MAGDM) evaluation framework based on a strict intuitionistic fuzzy distance and an improved TOPSIS approach. First, an improved strict intuitionistic fuzzy distance measure (ISIFDisM) is rigorously developed to overcome the limitations of existing methods, exhibiting high robustness, monotonicity, and discriminability. Second, building upon ISIFDisM, a systematic MAGDM evaluation model is constructed, comprising three key steps: (1) data acquisition through structured questionnaire surveys; (2) attribute weights determined using the entropy weight method; and (3) alternative ranking through normalized priority coefficients derived from intuitionistic fuzzy distance calculations. Third, the proposed framework is applied to a practical case study focused on reliability assessment of ship equipment, enabling effective ranking of various marine engines. Finally, through static comparative analyses and dynamic scenario simulations, the feasibility, robustness, and methodological superiority of the proposed framework are thoroughly validated.

1. Introduction

Contemporary industrial technology is advancing at an unprecedented pace. This rapid progress has driven the complex equipment, such as aerospace systems, high-end manufacturing machinery, and intelligent weapons systems, to develop towards higher complexity and sophistication. The reliability of such equipment is intrinsically linked to operational safety, economic efficiency, and mission success. Complex equipment is typically characterized by multi-component coupling, dynamic operating environments, and multiple failure modes. These features give rise to multi-state behaviors, interdependencies, and environmental sensitivity [1]. A paradigmatic example is marine vessels. Identical equipment deployed in oceanic, coastal, and inland environments demonstrates significant variations in failure rates and degradation patterns of critical components. Traditional reliability assessment frameworks, based on single-environment assumptions, fail to capture real-world complexities, thereby limiting their effectiveness in predictive maintenance and design optimization. Furthermore, complex equipment advancement is characterized by component coupling, functional integration, and strong interdependencies. This progression has intensified aging-related challenges, thereby amplifying system uncertainties. Implementing multi-environmental reliability assessment can provide dynamic and differentiated decision support for equipment design optimization, environmental adaptability enhancement, and maintenance strategy formulation. This comprehensive approach is crucial for improving both the robustness and cost-effectiveness of equipment lifecycle management.

Current reliability assessment methodologies for complex equipment systems primarily utilize three approaches: simulation-based, data-driven, and fuzzy comprehensive evaluation methods. These established techniques form the methodological foundation of the field, each offering distinct advantages while addressing the unique challenges presented by complex equipment systems.

Simulation-based methods commonly utilize non-exponential distributions to derive component lifetimes, effectively capturing time-dependent failure rate behaviors. Furthermore, these methods integrate failure mode and effects analysis (FMEA) with fault tree modeling, enabling comprehensive system reliability assessment through Monte Carlo simulations [2]. While these methodologies enable mechanistic simulation modeling with high precision, they require extensive sampling of diverse failure scenarios. This requirement significantly increases the computational time and complexity, as both difficulty and cost grow exponentially with increasing precision. Moreover, the boundary conditions of simulation models often fail to accurately capture the impacts of multifactor coupling under actual operational conditions.

Concurrently, data-driven methods have attracted considerable research interest, leveraging both historical failure data and simulation outputs to enhance reliability assessment [3]. Failure data encompasses information collected during the development, testing, and operational phases, encompassing various elements such as failure modes, types, causes, product impacts, and occurrence timing. These data capture the fundamental characteristics of failure events. Reliability evaluation metrics can be systematically constructed using multivariate datasets from complex equipment systems. These metrics incorporate key indicators such as reliability coefficient, failure rate function, mean time between failures (MTBF), and mean time to repair (MTTR). Empirical research indicates that these metrics are influenced by specific probability distributions, particularly exponential [4], Weibull [3,5], and Gamma processes [6]. Consequently, probabilistic data-driven approaches have proven effective for conducting reliability assessments. For example, BahooToroody et al. [7] introduced a probabilistic framework to evaluate the reliability of ship mechanical systems operating at different autonomy levels.

However, for complex equipment, the uncertainty of evaluation indices escalates owing to the inherent and external uncertainties of the equipment. Its reliability is susceptible to interference from complex environmental conditions and variable operational conditions. Conventional approaches relying solely on indicators like failure rate to estimate the reliability level may overestimate the impact of minor failures on evaluation results. To address these limitations, Xu et al. [3] developed a statistical evaluation technique to measure equipment reliability more accurately by combining multi-state characteristics with fuzzy failure data. Wang et al. [8] proposed a dynamic Bayesian forward reasoning and reverse diagnosis for sub-sea mud lifting system reliability assessment. Research indicates that multiple factors, such as environmental variability, failure severity, and data reliability, must be considered when developing robust evaluation models for complex systems.

Fuzzy comprehensive evaluation methods have become increasingly prevalent in reliability assessments, effectively addressing the complexities and environmental uncertainties. Duan et al. [9] introduced an integrated approach combining interval-valued triangular fuzzy weighted mean and evidence networks to handle uncertainty in component failure rates. Wang et al. [10] proposed a similarity-based cloud model to assess the operating conditions of complex mechatronic systems. By integrating objective and subjective factors, this model evaluates reliability by calculating the similarity between actual and standardized cloud models, offering a novel approach to address data deficiency. Yue [11] presented a VIKOR-based software reliability framework utilizing picture fuzzy sets for questionnaire information. He et al. [12] proposed a multi-level fuzzy evaluation model with combined authorization to assess the reliability of integrated energy systems. Zhang et al. [13] proposed a reliability-based group decision-making approach using two-dimensional linguistic information and quantum probability theory to assess delivery risks. Jia and Jia [14] developed an innovative multi-attribute group decision-making evaluation model for ship equipment reliability assessment in complex and diverse standard environments, based on symbolic information and an improved TOPSIS framework. Recent studies demonstrate that, in multi-environment scenarios, the reliable assessment of complex equipment demands balancing environmental parameter fluctuations, multi-attribute coupling, and expert cognitive fuzziness. Fuzzy multi-attribute group decision-making (MAGDM) methods provide effective tools for this purpose. These methods can accurately capture the gray features of the environment-reliability link using fuzzy expression tools, such as fuzzy linguistic sets, symbolic information, and cloud models. Additionally, they integrate dynamic performance indicators, such as failure rates under environmental stress, repair-response times, and degradation rates from different environments. Through group decision-making, they reduce uncertainties from single-environment assumptions and data deficiencies. Furthermore, they support the dynamic optimization of multi-environment weights, attribute correlations, and expert consensus, enhancing the adaptability and engineering guidance of the assessment results.

While fuzzy MAGDM methods demonstrate promising capabilities for multiple environments, their foundational theories, particularly the measurement models of intuitionistic fuzzy sets (IFSs), present notable deficiencies. Intuitionistic fuzzy distance measures (IFDisM) and intuitionistic fuzzy similarity measures (IFSimM), which are fundamental to fuzzy MAGDM, have primarily focused on two aspects: two-dimensional (2-D) and three-dimensional (3-D) representations of IFSs [15]. The 2-D representation considers only membership and non-membership, while the 3-D representation includes membership, non-membership, and hesitancy. However, because hesitancy is uniquely determined by membership and non-membership, the intuitionistic fuzzy value space is essentially a two-dimensional topological structure. This indicates that many existing distance and similarity measures, such as those in [16,17,18,19], do not comply with the axiomatic definition of IFDisM and IFSimM [20].

In addition, many newly defined IFDisM and IFSimM have several significant shortcomings: (1) for any $\alpha =〈\mu ,v〉$, where $\mu \ne 0$ and $v\ne 0$, the value of IFDisM between $\alpha$ and $〈\mathrm{0,0}〉$ is equal to the maximum value 1; (2) for any $\mu ,v\in (0, 1]$, the value of IFDisM between $〈\mu ,0〉$ and $〈0,v〉$ is equal to the maximum value 1; and (3) for $\alpha$ = $〈{\mu }_{\alpha },v_{\alpha }〉$ and $\beta =〈{\mu }_{\beta },v_{\beta }〉$, when any one of ${\mu }_{\alpha },$ $v_{\alpha }$, ${\mu }_{\beta },$ and $v_{\beta }$ is equal to 0, the value of IFDisM cannot be calculated (see Section 2.3). However, these results are unreasonable and cannot be explained. To address these deficiencies, we propose a new IFDisM with parameters and show that it is an improved strict IFDisM.

The rest of this paper is organized as follows: Section 2 systematically reviews the existing distance measures for intuitionistic fuzzy sets (IFSs), introduces an improved strict intuitionistic fuzzy distance measure (ISIFDisM), and rigorously derives its mathematical properties with formal proofs. Section 3 develops a novel fuzzy MAGDM method based on ISIFDisM. Section 4 demonstrates the practical applicability of the new method through a case study on the reliability assessment of ship equipment in a shipbuilding enterprise. Section 5 presents static and dynamic comparative analyses of the results. Finally, Section 6 concludes the study and suggests directions for future research.

2. Intuitionistic Fuzzy Set (IFS) and Its Measures

2.1. Intuitionistic Fuzzy Set (IFS)

Definition 1 

([21]). Let $X$ be a universal set, then an intuitionistic fuzzy set (IFS) $A$ in $X$ can be denoted as $A=\{〈x,{\mu }_A\left(x\right),{\nu }_A\left(x\right)〉\left|x\in X\right.\rangle \}$, where ${\mu }_A\left(x\right): X\mathit{\to }[0, 1]$ and ${\nu }_A\left(x\right): X\mathit{\to }[0, 1]$ are membership and non-membership functions, respectively, and $\forall x\in X$, $0\le {\mu }_A\left(x\right)+{\nu }_A\left(x\right)\le 1$. The degree of hesitancy is given by ${\pi }_A\left(x\right)=1-{\mu }_A\left(x\right) -{\nu }_A\left(x\right)$, and $\forall x\in X$, $0\le {\pi }_A\left(x\right)\le 1$.

Let $\mathrm{I}\mathrm{F}\mathrm{S}\left(X\right)$ denote the set of all IFSs in the universal set X. An element $\alpha =〈{\mu }_{\alpha },{\nu }_{\alpha },{\pi }_{\alpha }〉$ is called an intuitionistic fuzzy number (IFN), where ${\mu }_{\alpha }$,${\nu }_{\alpha }$,${\pi }_{\alpha }$ $\in [0, 1]$, $0\le {\mu }_{\alpha }+{\nu }_{\alpha }\le 1$, and ${\pi }_{\alpha }=1-{\mu }_{\alpha }-{\nu }_{\alpha }$. For simplicity, $\alpha$ can be characterized as $〈{\mu }_{\alpha },{\nu }_{\alpha }〉$. Additionally, denote ${\alpha }^C=〈{\nu }_{\alpha },{\mu }_{\alpha }〉$.

For two IFNs $\alpha =〈{\mu }_{\alpha },{\nu }_{\alpha }〉$ and $\beta =〈{\mu }_{\beta },{\nu }_{\beta }〉$, Atanassov [22] introduced a partial order relation that $\alpha \subset \beta$ if and only if ${\mu }_{\alpha }\le {\mu }_{\beta }$ and ${\nu }_{\alpha }\ge {\nu }_{\beta }$. Furthermore, $\alpha ⫋\beta$ if and only if $\alpha \subset \beta$ and $\alpha \ne \beta$.

Definition 2 

([23]). Let $\alpha =〈{\mu }_{\alpha },{\nu }_{\alpha }〉$ and $\beta =〈{\mu }_{\beta },{\nu }_{\beta }〉$ be two IFNs, then the arithmetic of IFNs is as follows:

• (1) $\alpha ⨁\beta =〈{\mu }_{\alpha }+{\mu }_{\beta }-{\mu }_{\alpha }{\mu }_{\beta },{\nu }_{\alpha }{\nu }_{\beta }〉$;
• (2) $\alpha ⨂\beta =〈{\mu }_{\alpha }{\mu }_{\beta },{\nu }_{\alpha }+{\nu }_{\beta }-{\nu }_{\alpha }{\nu }_{\beta }〉$;
• (3) $\lambda \left(\alpha \right)=〈{1-{(1-\mu }_{\alpha })}^{\lambda },{\nu }_{\alpha }^{\lambda }〉, \lambda >0$;
• (4) ${\alpha }^{ \lambda }=〈{\mu }_{\alpha }^{\lambda },1-{(1-{\nu }_{\alpha })}^{\lambda }〉, \lambda >0$.

Definition 3 

([23]). Let ${\alpha }_i$=$〈{\mu }_{{\alpha }_i},{\nu }_{{\alpha }_i}〉$ and $i=\mathrm{1,2},\cdots ,n$ be IFNs, then the intuitionistic fuzzy weighted averaging (IFWA) operator is defined as follows:

$${IFWA}_w\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right)=\langle 1-\prod _{i=1}^n{(1-{\mu }_{{\alpha }_i})}^{w_i},\prod _{i=1}^n{{\nu }_{{\alpha }_i}}^{w_i}\rangle$$

(1)

where $w=\left(w_1,w_2,\cdots ,w_n\right)$ is the weight vector of ${\alpha }_i(i=\mathrm{1,2},\cdots ,n)$, with $w_i\in \left[\mathrm{0,1}\right]$ and ${\sum }_{i=1}^n{w}_i=1$.

2.2. Distance/Similarity Measures for IFNs

Definition 4 

([24]). Let $X$ be a universal set; $\forall \alpha ,\beta ,\gamma \in IFS\left(X\right)$; a mapping $d: IFS\left(X\right)\times IFS\left(X\right)\to \left[\mathrm{0,1}\right]$ is called a strict intuitionistic fuzzy distance measure (SIFDisM) on $IFS\left(X\right)$ if it satisfies the following conditions:

• (C1) $0\le d\left(\alpha ,\beta \right)\le 1$;
• (C2) $d\left(\alpha ,\beta \right)=0$ if and only if $\alpha =\beta$;
• (C3) $d\left(\alpha ,\beta \right)=d\left(\beta ,\alpha \right)$;
• (C4) If $\alpha ⫋\beta ⫋\gamma$, then $d\left(\alpha ,\gamma \right)>d\left(\alpha ,\beta \right)$ and $d\left(\alpha ,\gamma \right)>d\left(\beta ,\gamma \right)$;
• (C5) $d\left(\alpha ,\beta \right)=1$ if and only if $\alpha =\langle 1,0\rangle$ and $\beta =\langle 0,1\rangle$ or $\alpha =\langle 0,1\rangle$ and $\beta =\langle 1,0\rangle$.

Definition 5. 

Let $X$ be a universal set; $\forall \alpha$,$\beta$,$\gamma \in IFS\left(X\right)$; a mapping $d: IFS\left(X\right)\times IFS\left(X\right)\to \left[\mathrm{0,1}\right]$ is called an improved strict intuitionistic fuzzy distance measure (ISIFDisM) on $IFS\left(X\right)$ if it satisfies (C1) to (C5) in Definition 3 and (C6) described by the following:

• (C6) $d\left(\alpha ,\gamma \right)\le d\left(\alpha ,\beta \right)+d\left(\beta ,\gamma \right)$.

Dually, a mapping $s:$ $\mathrm{I}\mathrm{F}\mathrm{S}\left(X\right)\times \mathrm{I}\mathrm{F}\mathrm{S}\left(X\right)\to \left[\mathrm{0,1}\right]$ is called an improved intuitionistic fuzzy similarity measure (ISIFSimM) on IFS(X) if the mapping $d\left(\alpha ,\beta \right)=1-s\left(\alpha ,\beta \right)$ is an ISIFDisM on $\mathrm{I}\mathrm{F}\mathrm{S}\left(X\right)$.

2.3. Some Existing IFDisM and Drawbacks

Definition 6 

([25]). Let $X$ be a universal set and $\alpha$ = $〈{\mu }_{\alpha },{\nu }_{\alpha }〉$ and $\beta =〈{\mu }_{\beta },{\nu }_{\beta }〉$ be two IFNs in $IFS\left(X\right)$; the IFDisM $d_M$ is defined as follows:

$$d_M\left(\alpha ,\beta \right)={\displaystyle \frac{\left|{\mu }_{\alpha }-{\mu }_{\beta }\right|+\left|{\nu }_{\alpha }-{\nu }_{\beta }\right|}{{\mu }_{\alpha }+{\mu }_{\beta }+{\nu }_{\alpha }+{\nu }_{\beta }}}$$

(2)

However, $d_M$ has the following drawbacks: (1) for any $〈\mu ,\nu 〉$, where $\mu \ne 0$ and $\nu \ne 0$, $d_M\left(〈\mu ,\nu 〉,〈\mathrm{0,0}〉\right)=1$; (2) for any $\mu , \nu \in \left(\mathrm{0,1}\right]$, $d_M\left(〈\mu ,0〉,〈0,\nu 〉\right)=1$. However, these results are unreasonable and cannot be explained.

Definition 7 

([26]). Let $X$ be a universal set and $\alpha$ = $〈{\mu }_{\alpha },{\nu }_{\alpha }〉$ and $\beta =〈{\mu }_{\beta },{\nu }_{\beta }〉$ be two IFNs in $IFS\left(X\right)$; the IFDisM $d_V$ is defined as follows:

$$d_V\left(\alpha ,\beta \right)={\mu }_{\alpha }\mathrm{l}\mathrm{n}{\displaystyle \frac{{2\mu }_{\alpha }}{{\mu }_{\alpha }+{\mu }_{\beta }}}+{\mu }_{\beta }\mathrm{l}\mathrm{n}{\displaystyle \frac{{2\mu }_{\beta }}{{\mu }_{\alpha }+{\mu }_{\beta }}}+v_{\alpha }\mathrm{l}\mathrm{n}{\displaystyle \frac{2{\nu }_{\alpha }}{{\nu }_{\alpha }+{\nu }_{\beta }}}+{\nu }_{\beta }\mathrm{l}\mathrm{n}{\displaystyle \frac{{2\nu }_{\beta }}{{\nu }_{\alpha }+{\nu }_{\beta }}}$$

(3)

Also, $d_V$ has certain drawbacks: (1) When any one of ${\mu }_{\alpha }$, ${\nu }_{\alpha }$, ${\mu }_{\beta },$ and ${\nu }_{\beta }$ is equal to 0, the value of $d_V$ cannot be calculated. (2) Let $\alpha$ = $〈\mathrm{0.33,0.36}〉$, $\beta$ = $〈1/3,1/3〉,$ and $\gamma$ = $〈\mathrm{0.334,0.3333}〉$; according to Atanassov [22], we know that $\alpha ⫋\beta ⫋\gamma$. Calculations show that $d_V\left(\alpha ,\beta \right)=0.2282$ and $d_V\left(\alpha ,\gamma \right)=0.0005$, resulting in $d_V\left(\alpha ,\beta \right)>d_V\left(\alpha ,\gamma \right)$. This contradicts condition (C5) of Definition 4.

Definition 8 

([27]). Let $X$ be a universal set and $\alpha$ = $〈{\mu }_{\alpha },{\nu }_{\alpha }〉$ and $\beta =〈{\mu }_{\beta },{\nu }_{\beta }〉$ be two IFNs in $IFS\left(X\right)$; the IFDisM $d_Y$ is defined as follows:

$$d_Y\left(\alpha ,\beta \right)=1-{\displaystyle \frac{{\mu }_{\alpha }{\mu }_{\beta }+{\nu }_{\alpha }{\nu }_{\beta }}{\sqrt{{\mu }_{\alpha }^2+{\nu }_{\alpha }^2}\sqrt{{\mu }_{\beta }^2+{\nu }_{\beta }^2}}}$$

(4)

Similarly, we can verify that $d_Y$ has the following drawbacks: (1) When $\alpha$ = $〈\mathrm{0,0}〉$ or $\beta = 〈\mathrm{0,0}〉$, $d_Y$ cannot be calculated. (2) Let $\alpha$ = $〈\mathrm{0.33,0.36}〉$, $\beta$ = $〈1/3,1/3〉$, and $\gamma$ = $〈\mathrm{0.334,0.333333}〉$; meanwhile, $d_Y\left(\alpha ,\beta \right)=0.00094$, $d_Y\left(\beta ,\gamma \right)=0$, and $d_Y\left(\alpha ,\gamma \right)=0.00099$, where $d_Y\left(\alpha ,\beta \right)+d_Y\left(\beta ,\gamma \right)<d_Y\left(\alpha ,\gamma \right)$. This result contradicts condition (C6) of Definition 5.

Definition 9 

([28]). Let $X$ be a universal set and $\alpha$ = $〈{\mu }_{\alpha },{\nu }_{\alpha }〉$ and $\beta =〈{\mu }_{\beta },{\nu }_{\beta }〉$ be two IFNs in $IFS\left(X\right)$; the parametric IFDisM $d_W$ is defined as follows:

$$d_{W,\epsilon }\left(\alpha ,\beta \right)={\displaystyle \frac{2+\epsilon }{2}}\left({\displaystyle \frac{\left|{\mu }_{\alpha }-{\mu }_{\beta }\right|+\left|{\nu }_{\alpha }-{\nu }_{\beta }\right|}{{\mu }_{\alpha }+v_{\alpha }+{\mu }_{\beta }+{\nu }_{\beta }+\epsilon }}\right)$$

(5)

where $\epsilon \ge 0$ is the parameter to adjust the difference.

Obviously, $d_{W,\epsilon }$ is a generalization of $d_M$. Although $d_W$ can yield diverse measurement results by adjusting parameter values, it also has drawbacks. Let $\alpha$ = $〈\mathrm{0.2166,0.6656}〉, \beta = 〈\mathrm{0.2992,0.6828}〉,\mathrm{ and }\gamma =〈\mathrm{0.6128,0.2569}〉, \mathrm{taking} \epsilon =0.1$; meanwhile, $d_{W,\epsilon }\left(\alpha ,\beta \right)=0.0534$, $d_{W,\epsilon }\left(\beta ,\gamma \right)=0.3977$, and $d_{W,\epsilon }\left(\alpha ,\gamma \right)=0.4563$, where $d_{W,\epsilon }\left(\alpha ,\beta \right)+d_{W,\epsilon }\left(\beta ,\gamma \right)<d_{W,\epsilon }\left(\alpha ,\gamma \right)$. This result contradicts condition (C6) of Definition 5.

2.4. A Novel ISIFDisM/ISIFSimM with Parametric

Definition 10. 

Let $X$ be a universal set and $\alpha$= $〈{\mu }_{\alpha },{\nu }_{\alpha }〉$ and $\beta =〈{\mu }_{\beta },{\nu }_{\beta }〉$ be two IFNs in $IFS\left(X\right)$; the mapping $d_{\epsilon }(\alpha ,\beta ):$$IFS\left(X\right)\times IFS\left(X\right)\to \left[\mathrm{0,1}\right]$ is defined by the following:

$$d_{\epsilon }\left(\alpha ,\beta \right)={\displaystyle \frac{1+\epsilon }{2}}\left({\displaystyle \frac{\left|{\mu }_{\alpha }-{\mu }_{\beta }\right|}{{\mu }_{\alpha }+{\mu }_{\beta }+\epsilon }}+{\displaystyle \frac{\left|{\nu }_{\alpha }-{\nu }_{\beta }\right|}{{\nu }_{\alpha }+{\nu }_{\beta }+\epsilon }}\right)$$

(6)

where $0<\epsilon \le 1$ is the parameter to adjust the difference.

In Equation (6), $\epsilon$ functions as a “smoothing weight” that modulates the trade-off between membership or non-membership degrees and uncertainty in IFSs. Specifically, if $\epsilon \le 0$, the denominator of Equation (6) will be zero under certain circumstances, thereby rendering the distance measure mathematically ill-defined. Conversely, when $\epsilon >1$, the proposed measure violates condition (C1) as stipulated in Definition 4. Geometrically, $\epsilon$ corresponds to the “uncertainty radius” in the unit simplex where membership degrees and non-membership degrees are plotted. A radius greater than one would extend beyond the boundaries of this simplex, resulting in a physically meaningless interpretation of uncertainty in the context of intuitionistic fuzzy sets.

Theorem 1. 

$d_{\epsilon }(\alpha ,\beta )$ is an ISIFDisM on $IFS\left(X\right).$

Proof of Theorem 1. 

(C1) Note that $0\le \left|{\mu }_{\alpha }-{\mu }_{\beta }\right|\le {\mu }_{\alpha }+{\mu }_{\beta }$ and $0\le \left|{\mu }_{\alpha }-{\mu }_{\beta }\right|\le 1$, thus $0\le \left(1+\epsilon \right)\left|{\mu }_{\alpha }-{\mu }_{\beta }\right|\le {\mu }_{\alpha }+{\mu }_{\beta }+\epsilon$ and $0\le \left|{\mu }_{\alpha }-{\mu }_{\beta }\right|/{(\mu }_{\alpha }+{\mu }_{\beta }+\epsilon )\le 1/(1+\epsilon )$. Similarly, we have $0\le \left|{\nu }_{\alpha }-{\nu }_{\beta }\right|/({\nu }_{\alpha }+{\nu }_{\beta }+\epsilon )\le 1/(1+\epsilon )$. It follows that ${0\le d}_{\epsilon }\left(\alpha ,\beta \right)\le 1$.

(C2) The necessity is obvious. Given that $0\le \epsilon \le 1$, according to Equation (6), $d_{\epsilon }\left(\alpha ,\beta \right)=0$ if and only if $\left|{\mu }_{\alpha }-{\mu }_{\beta }\right|/{(\mu }_{\alpha }+{\mu }_{\beta }+\epsilon )=0$ and $\left|{\nu }_{\alpha }-{\nu }_{\beta }\right|/({\nu }_{\alpha }+{\nu }_{\beta }+\epsilon )=0$. This condition holds if and only if ${\mu }_{\alpha }={\mu }_{\beta }$ and ${\nu }_{\alpha }={\nu }_{\beta }$, thereby proving the sufficiency.

(C3) From Equation (6), this property is obvious.

(C4) For any $\alpha$ = $〈{\mu }_{\alpha },{\nu }_{\alpha }〉$, $\beta =〈{\mu }_{\beta },{\nu }_{\beta }〉$, and $\gamma =〈{\mu }_{\gamma },{\nu }_{\gamma }〉$; since $\alpha ⫋\beta ⫋\gamma$, we have ${\mu }_{\alpha }<{\mu }_{\beta }<{\mu }_{\gamma }$ and ${\nu }_{\alpha }>{\nu }_{\beta }>{\nu }_{\gamma }$. It is clear that

$$\begin{array}{cc}\hfill d_{\epsilon }\left(\alpha ,\gamma \right)-d_{\epsilon }\left(\alpha ,\beta \right)& {\displaystyle =\frac{1+\epsilon }{2}}\left({\displaystyle \frac{{\mu }_{\gamma }-{\mu }_{\alpha }}{{\mu }_{\alpha }+{\mu }_{\gamma }+\epsilon }}+{\displaystyle \frac{{{\mu }_{\alpha }-\mu }_{\beta }}{{\mu }_{\alpha }+{\mu }_{\beta }+\epsilon }}+{\displaystyle \frac{{\nu }_{\alpha }-{\nu }_{\gamma }}{{\nu }_{\alpha }+{\nu }_{\gamma }+\epsilon }}+{\displaystyle \frac{{\nu }_{\beta }-{\nu }_{\alpha }}{{\nu }_{\alpha }+{\nu }_{\beta }+\epsilon }}\right)\hfill \\ & \ge {\displaystyle \frac{1+\epsilon }{2}}\left({\displaystyle \frac{{\mu }_{\gamma }-{\mu }_{\alpha }}{{\mu }_{\alpha }+{\mu }_{\gamma }+\epsilon }}+{\displaystyle \frac{{{\mu }_{\alpha }-\mu }_{\beta }}{{\mu }_{\alpha }+{\mu }_{\gamma }+\epsilon }}+{\displaystyle \frac{{\nu }_{\alpha }-{\nu }_{\gamma }}{{\nu }_{\alpha }+{\nu }_{\beta }+\epsilon }}+{\displaystyle \frac{{\nu }_{\beta }-{\nu }_{\alpha }}{{\nu }_{\alpha }+{\nu }_{\beta }+\epsilon }}\right)\hfill \\ & ={\displaystyle \frac{1+\epsilon }{2}}\left({\displaystyle \frac{{\mu }_{\gamma }{-\mu }_{\beta }}{{\mu }_{\alpha }+{\mu }_{\gamma }+\epsilon }}+{\displaystyle \frac{{\nu }_{\beta }-{\nu }_{\gamma }}{{\nu }_{\alpha }+{\nu }_{\beta }+\epsilon }}\right)\ge 0\hfill \end{array}$$

(7)

Thus, $d_{\epsilon }\left(\alpha ,\gamma \right)>d_{\epsilon }\left(\alpha ,\beta \right)$. Similarly, we have $d_{\epsilon }\left(\alpha ,\gamma \right)>d_{\epsilon }\left(\beta ,\gamma \right)$.

(C5) First, we prove the sufficiency. According to Equation (6), it is clear that $d_{\epsilon }\left(\langle 1,0\rangle ,\langle 0,1\rangle \right)=d_{\epsilon }\left(\langle 0,1\rangle ,\langle 1,0\rangle \right)=1$. Thus, the necessity is proven. According to Equation (6), $d_{\epsilon }\left(\alpha ,\beta \right)=1$ if and only if $\left|{\mu }_{\alpha }-{\mu }_{\beta }\right|/{(\mu }_{\alpha }+{\mu }_{\beta }+\epsilon )+\left|{\nu }_{\alpha }-{\nu }_{\beta }\right|/({\nu }_{\alpha }+{\nu }_{\beta }+\epsilon )=2/(1+\epsilon )$, which is equivalent to $\left[\left|{\mu }_{\alpha }-{\mu }_{\beta }\right|/{(\mu }_{\alpha }+{\mu }_{\beta }+\epsilon )-1/\left(1+\epsilon \right)\right]+\left[\left|{\nu }_{\alpha }-{\nu }_{\beta }\right|/\left({\nu }_{\alpha }+{\nu }_{\beta }+\epsilon \right)-1/\left(1+\epsilon \right)\right]=0$. According to the proof procedure of (C1), we know that $\left|{\mu }_{\alpha }-{\mu }_{\beta }\right|/{(\mu }_{\alpha }+{\mu }_{\beta }+\epsilon )-1/\left(1+\epsilon \right)\le 0$ and $\left|{\nu }_{\alpha }-{\nu }_{\beta }\right|/\left({\nu }_{\alpha }+{\nu }_{\beta }+\epsilon \right)-1/\left(1+\epsilon \right)\le 0$. Therefore, $d_{\epsilon }\left(\alpha ,\beta \right)=1$ if and only if $\left|{\mu }_{\alpha }-{\mu }_{\beta }\right|/{(\mu }_{\alpha }+{\mu }_{\beta }+\epsilon )=1/\left(1+\epsilon \right)$ and $\left|{\nu }_{\alpha }-{\nu }_{\beta }\right|/\left({\nu }_{\alpha }+{\nu }_{\beta }+\epsilon \right)=1/\left(1+\epsilon \right)$, which is equivalent to $\left|{\mu }_{\alpha }-{\mu }_{\beta }\right|=1$, ${\mu }_{\alpha }+{\mu }_{\beta }=1$, and $\left|{\nu }_{\alpha }-{\nu }_{\beta }\right|=1$, ${\nu }_{\alpha }+{\nu }_{\beta }=1$. Consequently, ${\mu }_{\alpha }=1$, ${\mu }_{\beta }=0$, ${\nu }_{\alpha }=0$, and ${\nu }_{\beta }=1$, or ${\mu }_{\alpha }=0$, ${\mu }_{\beta }=1$, ${\nu }_{\alpha }=1$, and ${\nu }_{\beta }=0$. In conclusion, $\alpha =\langle 1,0\rangle$ and $\beta =\langle 0,1\rangle$ or $\alpha =\langle 0,1\rangle$ and $\beta =\langle 1,0\rangle$.

(C6) For any $\alpha$ = $〈{\mu }_{\alpha },{\nu }_{\alpha }〉$, $\beta =〈{\mu }_{\beta },{\nu }_{\beta }〉$, and $\gamma =〈{\mu }_{\gamma },{\nu }_{\gamma }〉$, we first prove that the following formula holds:

$${\displaystyle \frac{\left|{\mu }_{\alpha }-{\mu }_{\beta }\right|}{{\mu }_{\alpha }+{\mu }_{\beta }+\epsilon }}+{\displaystyle \frac{\left|{\mu }_{\beta }-{\mu }_{\gamma }\right|}{{\mu }_{\beta }+{\mu }_{\gamma }+\epsilon }}\ge {\displaystyle \frac{\left|{\mu }_{\alpha }-{\mu }_{\gamma }\right|}{{\mu }_{\alpha }+{\mu }_{\gamma }+\epsilon }}$$

(8)

The above inequality is valid in four cases: (a) ${\mu }_{\alpha }\ge {\mu }_{\beta }\ge {\mu }_{\gamma }$; (b) ${\mu }_{\alpha }\le {\mu }_{\beta }\le {\mu }_{\gamma }$; (c) ${\mu }_{\beta }\ge \mathrm{m}\mathrm{a}\mathrm{x}({\mu }_{\alpha },{\mu }_{\gamma })$; and (d) ${\mu }_{\beta }\le \mathrm{m}\mathrm{i}\mathrm{n}({\mu }_{\alpha },{\mu }_{\gamma })$.

Under the case (a), the inequality (8) is equivalent to

$${\displaystyle \frac{{\mu }_{\alpha }-{\mu }_{\beta }}{{\mu }_{\alpha }+{\mu }_{\beta }+\epsilon }}+{\displaystyle \frac{{\mu }_{\beta }-{\mu }_{\gamma }}{{\mu }_{\beta }+{\mu }_{\gamma }+\epsilon }}\ge {\displaystyle \frac{{\mu }_{\alpha }-{\mu }_{\gamma }}{{\mu }_{\alpha }+{\mu }_{\gamma }+\epsilon }}$$

(9)

By direct calculation, we have

$$\begin{array}{c}{\displaystyle \frac{{\mu }_{\alpha }-{\mu }_{\beta }}{{\mu }_{\alpha }+{\mu }_{\beta }+\epsilon }}+{\displaystyle \frac{{\mu }_{\beta }-{\mu }_{\gamma }}{{\mu }_{\beta }+{\mu }_{\gamma }+\epsilon }}-{\displaystyle \frac{{\mu }_{\alpha }-{\mu }_{\gamma }}{{\mu }_{\alpha }+{\mu }_{\gamma }+\epsilon }}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{\left({\mu }_{\alpha }-{\mu }_{\beta }\right)\left({\mu }_{\beta }-{\mu }_{\gamma }\right)\left({\mu }_{\alpha }-{\mu }_{\gamma }\right)}{\left({\mu }_{\alpha }+{\mu }_{\beta }+\epsilon \right)\left({\mu }_{\beta }+{\mu }_{\gamma }+\epsilon \right)\left({\mu }_{\alpha }+{\mu }_{\gamma }+\epsilon \right)}}\ge 0\end{array}$$

(10)

Consequently, under situation (a), inequality (8) is satisfied. By employing an analogous approach, the validity of inequality (8) can also be demonstrated for case (b).

Case (c) encompasses two distinct situations, which are described as follows:

• If ${\mu }_{\beta }\ge {\mu }_{\alpha }\ge {\mu }_{\gamma }$, we have

$$\begin{array}{c}{\displaystyle \frac{\left|{\mu }_{\alpha }-{\mu }_{\beta }\right|}{{\mu }_{\alpha }+{\mu }_{\beta }+\epsilon }}+{\displaystyle \frac{\left|{\mu }_{\beta }-{\mu }_{\gamma }\right|}{{\mu }_{\beta }+{\mu }_{\gamma }+\epsilon }}-{\displaystyle \frac{\left|{\mu }_{\alpha }-{\mu }_{\gamma }\right|}{{\mu }_{\alpha }+{\mu }_{\gamma }+\epsilon }}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{{\mu }_{\beta }{-\mu }_{\alpha }}{{\mu }_{\alpha }+{\mu }_{\beta }+\epsilon }}+{\displaystyle \frac{{\mu }_{\beta }-{\mu }_{\gamma }}{{\mu }_{\beta }+{\mu }_{\gamma }+\epsilon }}-{\displaystyle \frac{{\mu }_{\alpha }-{\mu }_{\gamma }}{{\mu }_{\alpha }+{\mu }_{\gamma }+\epsilon }}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge {\displaystyle \frac{\left({\mu }_{\beta }{-\mu }_{\alpha }\right)+\left({\mu }_{\beta }-{\mu }_{\gamma }\right)-\left({\mu }_{\alpha }-{\mu }_{\gamma }\right)}{{\mu }_{\alpha }+{\mu }_{\beta }+\epsilon }}={\displaystyle \frac{2({\mu }_{\beta }-{\mu }_{\alpha })}{{\mu }_{\alpha }+{\mu }_{\beta }+\epsilon }}\ge 0\hfill \end{array}$$

(11)

If ${\mu }_{\beta }\ge {\mu }_{\gamma }\ge {\mu }_{\alpha }$, we have

$$\begin{array}{c}{\displaystyle \frac{\left|{\mu }_{\alpha }-{\mu }_{\beta }\right|}{{\mu }_{\alpha }+{\mu }_{\beta }+\epsilon }}+{\displaystyle \frac{\left|{\mu }_{\beta }-{\mu }_{\gamma }\right|}{{\mu }_{\beta }+{\mu }_{\gamma }+\epsilon }}-{\displaystyle \frac{\left|{\mu }_{\alpha }-{\mu }_{\gamma }\right|}{{\mu }_{\alpha }+{\mu }_{\gamma }+\epsilon }}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{{\mu }_{\beta }{-\mu }_{\alpha }}{{\mu }_{\alpha }+{\mu }_{\beta }+\epsilon }}+{\displaystyle \frac{{\mu }_{\beta }-{\mu }_{\gamma }}{{\mu }_{\beta }+{\mu }_{\gamma }+\epsilon }}-{\displaystyle \frac{{\mu }_{\gamma }-{\mu }_{\alpha }}{{\mu }_{\alpha }+{\mu }_{\gamma }+\epsilon }}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge {\displaystyle \frac{\left({\mu }_{\beta }{-\mu }_{\alpha }\right)+\left({\mu }_{\beta }-{\mu }_{\gamma }\right)-\left({\mu }_{\gamma }-{\mu }_{\alpha }\right)}{{\mu }_{\alpha }+{\mu }_{\gamma }+\epsilon }}={\displaystyle \frac{2({\mu }_{\beta }-{\mu }_{\gamma })}{{\mu }_{\alpha }+{\mu }_{\beta }+\epsilon }}\ge 0\hfill \end{array}$$

(12)

Therefore, inequality (8) is established under the condition of case (c).

Case (d) is symmetric to case (c), and it also encompasses two distinct situations: ${\mu }_{\beta }\le {\mu }_{\alpha }\le {\mu }_{\gamma }$ and ${\mu }_{\beta }\le {\mu }_{\gamma }\le {\mu }_{\alpha }$. The proof is similar. Therefore, inequality (8) still holds.

Moreover, the following equations are satisfied:

$${\displaystyle \frac{\left|{\nu }_{\alpha }-{\nu }_{\beta }\right|}{{\nu }_{\alpha }+{\nu }_{\beta }+\epsilon }}+{\displaystyle \frac{\left|{\nu }_{\beta }-{\nu }_{\gamma }\right|}{{\nu }_{\beta }+{\nu }_{\gamma }+\epsilon }}\ge {\displaystyle \frac{\left|{\nu }_{\alpha }-{\nu }_{\gamma }\right|}{{\nu }_{\alpha }+{\nu }_{\gamma }+\epsilon }}$$

(13)

Based on inequalities (8) and (13), we can prove the following proposition:

$$\begin{array}{c}{d}_{\epsilon }\left(\alpha ,\beta \right)+d_{\epsilon }\left(\beta ,\gamma \right)={\displaystyle \frac{1+\epsilon }{2}}\left({\displaystyle \frac{\left|{\mu }_{\alpha }-{\mu }_{\beta }\right|}{{\mu }_{\alpha }+{\mu }_{\beta }+\epsilon }}+{\displaystyle \frac{\left|{\nu }_{\alpha }-{\nu }_{\beta }\right|}{{\nu }_{\alpha }+{\nu }_{\beta }+\epsilon }}+{\displaystyle \frac{\left|{\mu }_{\beta }-{\mu }_{\gamma }\right|}{{\mu }_{\beta }+{\mu }_{\gamma }+\epsilon }}+{\displaystyle \frac{\left|{\nu }_{\beta }-{\nu }_{\gamma }\right|}{{\nu }_{\beta }+{\nu }_{\gamma }+\epsilon }}\right)\\ \ge {\displaystyle \frac{1+\epsilon }{2}}\left({\displaystyle \frac{\left|{\mu }_{\alpha }-{\mu }_{\gamma }\right|}{{\mu }_{\alpha }+{\mu }_{\gamma }+\epsilon }}+{\displaystyle \frac{\left|{\nu }_{\alpha }-{\nu }_{\gamma }\right|}{{\nu }_{\alpha }+{\nu }_{\gamma }+\epsilon }}\right)=d_{\epsilon }\left(\alpha ,\gamma \right)\end{array}$$

(14)

From the above discussion, $s(\alpha ,\beta )$ is an ISIFDisM on $IFS\left(X\right)$.□

Theorem 2. 

The function $1-d_{\epsilon }(\alpha ,\beta )$ is an ISIFSimM on $IFS\left(X\right).$

Definition 11. 

Let $X=\{x_1,x_2,\cdots ,x_n\}$ be a universal set and $A=\{\langle x_i,{\mu }_{\alpha }\left(x_i\right),{\nu }_{\alpha }\left(x_i\right)\rangle |x_i\in X\}$ and $B=\{\langle x_i,{\mu }_{\beta }\left(x_i\right),{\nu }_{\beta }\left(x_i\right)\rangle |x_i\in X\}$ be two IFSs; the distance between $A$ and $B$ is expressed as the following:

$$d_{\epsilon }\left(A,B\right)={\displaystyle \frac{1+\epsilon }{2n}}\sum _{i=1}^n\left({\displaystyle \frac{\left|{\mu }_{\alpha }\left(x_i\right)-{\mu }_{\beta }\left(x_i\right)\right|}{{\mu }_{\alpha }\left(x_i\right)+{\mu }_{\beta }\left(x_i\right)+\epsilon }}+{\displaystyle \frac{\left|{\nu }_{\alpha }\left(x_i\right)-{\nu }_{\beta }\left(x_i\right)\right|}{{\nu }_{\alpha }\left(x_i\right)+{\nu }_{\beta }\left(x_i\right)+\epsilon }}\right)$$

(15)

The weighted distance between $A$ and $B$ is expressed as the following:

$$d_{\epsilon }\left(A,B\right)={\displaystyle \frac{1+\epsilon }{2n}}\sum _{i=1}^n{w}_i\left({\displaystyle \frac{\left|{\mu }_{\alpha }\left(x_i\right)-{\mu }_{\beta }\left(x_i\right)\right|}{{\mu }_{\alpha }\left(x_i\right)+{\mu }_{\beta }\left(x_i\right)+\epsilon }}+{\displaystyle \frac{\left|{\nu }_{\alpha }\left(x_i\right)-{\nu }_{\beta }\left(x_i\right)\right|}{{\nu }_{\alpha }\left(x_i\right)+{\nu }_{\beta }\left(x_i\right)+\epsilon }}\right)$$

(16)

where $0<\epsilon \le 1$ is the parameter to adjust the difference, and $\mathit{w}{ =(w}_1,w_2,\dots ,w_n)$ is the weight vector of $x_i$ with ${0\le w}_i\le 1$ and ${\sum }_{i=1}^n{w}_i=1$.

2.5. Advantages of the Proposed Distance

The subsequent examples demonstrate that the proposed distance metric exhibits identical superiority when applied to the same numerical examples in the context of pattern recognition.

Example 1. 

Assume that the IFSs on $X=\left\{x\right\}$ are given by $\alpha$ = $〈\mu ,\nu 〉$ and $\beta =〈\nu ,\mu 〉$. Then the ISIFDisM $d_{\epsilon }\left(\alpha ,\beta \right)$ is equal to $\left(\right(1+\epsilon \left)\right|\mu -\nu \left|\right)/(\mu +\nu +\epsilon )$. To show the changing trend of the distances between $\alpha$ and $\beta$, the parametric values of $\epsilon =0.01, 0.1, 0.3, 0.5, 0.7, 0.9$ are selected to plot the relationship between $\mu$, $\nu ,$ and $d_{\epsilon }$, as depicted in Figure 1. It is shown that $d_{\epsilon }\left(\alpha ,\beta \right)=0$ if and only if $\mu =\nu$, and $d_{\epsilon }\left(\alpha ,\beta \right)=1$ if only if $\mu =1$, $\nu =0$ or $\mu =0$, $\nu =1$. In decision-making, $\alpha$ and $\beta$ can represent two complementary assessments of a single alternative (e.g., the “reliability” and “unreliability” of a product). The distance $d_{\epsilon }\left(\alpha ,\beta \right)$ quantifies the discrepancy between these two assessments. When $\mu =\nu$, $d_{\epsilon }\left(\alpha ,\beta \right)$ indicates perfect symmetry between the two assessments—this implies that the alternative is evaluated as neutral or ambiguous, potentially signaling hesitation in decision-making (e.g., a product perceived to have equal levels of reliability and unreliability may necessitate further investigation prior to selection). Conversely, $d_{\epsilon }\left(\alpha ,\beta \right)=1$ reflects extreme asymmetry, indicating that one evaluation dominates entirely over the other.

Example 2. 

Assume that the IFSs on $X=\left\{x\right\}$ are given by $\alpha$ = $〈\mu ,\nu 〉$ and $\beta =〈0,0〉$. Then the ISIFDisM $d_{\epsilon }\left(\alpha ,\beta \right)$ is equal to $(1+\epsilon )/2 (\mu /(\mu +\epsilon )+v/(\nu +\epsilon \left)\right)$. To illustrate the changing trend of the distances between $\alpha$ and $\beta$, the parameter values $\epsilon =0.01, 0.1, 0.3, 0.5, 0.7, 0.9$ are also chosen to plot the relationship between $\mu$, $\nu$, and $d_{\epsilon }$, as depicted in Figure 2. It is revealed that $d_{\epsilon }\left(\alpha ,\beta \right)<1$, and it changes with $\alpha$. In decision-making, $\beta$ can be seen as a null reference representing a state of complete uncertainty or lack of prior information about an alternative. Conversely, $\alpha$ reflects the accumulated information or expert judgments about the alternative. The distance $d_{\epsilon }\left(\alpha ,\beta \right)$ quantifies how much the available information about the alternative deviates from this state of complete uncertainty. A higher $d_{\epsilon }$ (while still less than 1) indicates that the alternative possesses distinct characteristics that differentiate it from the unknown—this enables decision-makers to prioritize alternatives with more clearly defined profiles.

Example 3. 

Assume that the IFSs on $X=\left\{x\right\}$ are given by $\alpha$=$〈\mu ,0〉$ and $\beta =〈0,\nu 〉$. Then the ISIFDisM $d_{\epsilon }\left(\alpha ,\beta \right)$ is equal to $(1+\epsilon )/2 (\mu /(\mu +\epsilon )+v/(\nu +\epsilon \left)\right)$. To illustrate the changing trend of the distances between $\alpha$ and $\beta$, the parameter values $\epsilon =0.01, 0.1, 0.3, 0.5, 0.7, 0.9$ are also chosen to plot the relationship between $\mu$, $\nu ,$ and $d_{\epsilon }$, as depicted in Figure 3. It is revealed that $d_{\epsilon }\left(\alpha ,\beta \right)<1$ except for the case when $\mu =\nu =1$. In decision-making, $\alpha$ can represent an assessment that focuses exclusively on a “positive attribute”: μ quantifies the degree of belief in the strength of an alternative (e.g., “energy efficiency” of a device). Conversely, $\beta$ captures an assessment centered on a “negative attribute”: ν measures the degree of belief in a weakness (e.g., “energy consumption” of the device). The distance $d_{\epsilon }\left(\alpha ,\beta \right)$ quantifies the tension or discrepancy between these opposing attributes. A higher $d_{\epsilon }$ indicates a greater imbalance. When $\mu =\nu =1$, $d_{\epsilon }\left(\alpha ,\beta \right)=1$. This edge case represents a scenario where an alternative simultaneously exhibits maximum strength in the positive attribute and maximum weakness in the negative attribute (e.g., a product that is both “perfectly efficient” and “perfectly costly”). Such a situation is often paradoxical in real-world decisions, signifying an extreme conflict that may necessitate re-evaluation (e.g., revising cost estimates or efficiency claims) prior to making a choice.

Example 4. 

Consider $\alpha$ = $〈\mathrm{0,0}〉$, $\beta =〈0.2,0.8〉$, $\gamma =〈\mathrm{0.4,0.6}〉$, and $\delta =〈\mathrm{0.5,0.5}〉,$ which are defined on the universal set $X=\left\{x\right\}$. To assess the similarity among the IFSs ($\alpha ,\beta$), ($\alpha ,\gamma$), and ($\alpha ,\delta$), and determine which of the IFSs $\beta$, $\gamma$, and $\delta$ is closer to IFS $\alpha$. For the pairs ($\alpha ,\beta$), ($\alpha ,\gamma$), and ($\alpha ,\delta$), those with higher similarity are regarded as more similar. The similarity measures of these pairs are calculated using the existing distance formula in Section 2.3, Definition 8, and some existing distances in the literature, such as, $d_K$ [29], $d_X$ [30], $d_W$ [24], $d_G$ [15], and $d_{qROFS}$ with $q=1$ [31,32]. The results are presented in

Table 1. Comparative analysis results of different similarity measures in Example 4.

Distance	Similarity Measure			Comment
	$1-\mathit{d}(\mathit{\alpha },\mathit{\beta })$	$1-\mathit{d}(\mathit{\alpha },\mathit{\gamma })$	$1-\mathit{d}(\mathit{\alpha },\mathit{\delta })$
$d_M$ [25]	0	0	0	Ineffective
$d_V$ [26]	0.3069	0.3069	0.3069	Ineffective
$d_Y$ [27]	NaN	NaN	NaN	Ineffective
$d_{W,\epsilon =0.1}$ [28]	0.0454	0.0454	0.0454	Ineffective
$d_{W,\epsilon =0.5}$ [28]	0.1667	0.1667	0.1667	Ineffective
$d_{W,\epsilon =0.9}$ [28]	0.2368	0.2368	0.2368	Ineffective
$d_K$ [29]	0.5556	0.6250	0.6667	Justified, $\delta$
$d_X$ [30]	0.2593	0.2262	0.2222	Justified, $\beta$
$d_W$ [24]	NaN	NaN	NaN	Ineffective
$d_G$ [15]	0.4375	0.4375	0.4375	Ineffective
$d_{qROFS1,q=1}$ [31]	0.2500	0.2500	0.2500	Ineffective
$d_{qROFS2,q=1}$ [32]	0.9239	0.9239	0.9239	Ineffective
Proposed $d_{\epsilon =0.1}$	0.1444	0.0886	0.0833	Justified, $\beta$
Proposed $d_{\epsilon =0.5}$	0.3242	0.2575	0.2500	Justified, $\beta$
Proposed $d_{\epsilon =0.9}$	0.3802	0.3277	0.3214	Justified, $\beta$

, which show that the proposed distance measure $d_{\epsilon }$ and $d_X$ justifies that $\alpha$ is more similar to $\beta$, while $d_K$ justifies that $\alpha$ is more similar to $\delta$, and the other distance measures are ineffective.

Example 5. 

Consider a pattern classification problem with three classes $P=\{P_1, P_2, P_3\}$ and three attributes $A=\{a_1, a_2, a_3\}$. The IFSs are as follows:

$$P_1=\{〈\mathrm{0.2,0.3}〉, 〈\mathrm{0.1,0.4}〉, 〈\mathrm{0.2,0.6}〉\},$$

$$P_2=\left\{〈\mathrm{0.3,0.2}〉, 〈\mathrm{0.4,0.1}〉, 〈\mathrm{0.5,0.3}〉\right\},$$

$$P_3=\left\{〈\mathrm{0.2,0.3}〉, 〈\mathrm{0.4,0.1}〉, 〈\mathrm{0.5,0.3}〉\right\}.$$

Our goal is to classify the unknown test sample $S_1=\{〈\mathrm{0.1,0.2}〉,〈\mathrm{0.4,0.5}〉,〈\mathrm{0,0}〉\}$ into one of the patterns $P_1$, $P_2$, and $P_3$. If we take the weight vector $\omega$ of the three attributes as $(1/3, 1/3,1/3)$, then, according to the principle of the maximum of IFSimMs, the pattern classification results obtained using different distances are listed in

Table 2. Pattern recognition results, using different similarity measures in Example 5.

Distance	Similarity Measure			Classification
	$1-\mathit{d}({\mathit{P}}_1,{\mathit{S}}_1)$	$1-\mathit{d}({\mathit{P}}_2,{\mathit{S}}_1)$	$1-\mathit{d}({\mathit{P}}_3,{\mathit{S}}_1)$
$d_M$ [25]	0.4881	0.4881	0.4881	Not classify
$d_V$ [26]	0.7722	0.7492	0.7576	Classified, $P_1$
$d_Y$ [27]	NaN	NaN	NaN	Not classify
$d_{W,\epsilon =0.1}$ [28]	0.5178	0.5178	0.5178	Not classify
$d_{W,\epsilon =0.5}$ [28]	0.5918	0.5918	0.5918	Not classify
$d_{W,\epsilon =0.9}$ [28]	0.6316	0.6316	0.6316	Not classify
$d_K$ [29]	0.7678	0.7381	0.7633	Classified, $P_1$
$d_X$ [30]	0.6693	0.6510	0.6483	Classified, $P_1$
$d_W$ [24]	NaN	NaN	NaN	Not classify
$d_G$ [15]	0.7129	0.7129	0.7129	Not classify
$d_{qROFS1,q=1}$ [31]	0.8100	0.8100	0.8100	Not classify
$d_{qROFS2,q=1}$ [32]	0.9641	0.9656	0.9656	Not classify
Proposed $d_{\epsilon =0.1}$	0.5342	0.5316	0.5286	Classified, $P_1$
Proposed $d_{\epsilon =0.5}$	0.6431	0.6348	0.6341	Classified, $P_1$
Proposed $d_{\epsilon =0.9}$	0.6813	0.6746	0.6743	Classified, $P_1$

. The results show that the proposed distances $d_{\epsilon }$, $d_V$, $d_K$, and $d_X$ recognize the unknown test sample $S_1$ as $P_1$, while the other distances are ineffective because they have the same similarity measures.

As illustrated in Examples 4 and 5, the parameter $\epsilon$ exhibits no discernible impact on the outcomes of similarity recognition and pattern classification when set to 0.1, 0.5, or 0.9. To rigorously assess the potential influence of $\epsilon$ on the results, comprehensive sensitivity analyses are conducted. Within the range of 0.1 to 1.0, $\epsilon$ is systematically varied at intervals of 0.1, and the corresponding changes in results are quantified and evaluated. The analyses revealed that for similarity recognition and pattern classification, all evaluation results remained stable, and the optimal range of $\epsilon$ was between 0.2 and 0.3.

2.6. Intuitionistic Fuzzy Entropy Measure Based on the Proposed Distance

IFSs effectively capture subjective evaluations made by decision-makers. However, the information embedded in IFSs may inherently contain ambiguity or incompleteness. As a fundamental information measure, entropy plays a critical role in quantifying uncertainty and imprecision within IFSs. Patel et al. [33] successfully exploited this relationship to derive attribute weights, thereby validating the efficacy of entropy-based approaches. For an IFS $A=\{〈x_i,{\mu }_{\alpha }\left(x_i\right),{\nu }_{\alpha }\left(x_i\right)〉\left|x_i\in X\right.\rangle \}$, the entropy measure calculated using the Intuitionistic Fuzzy Similarity Measure (IFSimM) is defined as $E\left(A\right)=IFSimM(A,A^C)$. Therefore, we formally define the intuitionistic fuzzy entropy measure (IFEnM) as follows:

Definition 12. 

Let $X$ be a universal set; the IFEnM of IFS $A=\left\{〈x_i,{\mu }_{\alpha }\left(x_i\right),{\nu }_{\alpha }\left(x_i\right)〉\left|x_i\in X\right.\right\}$ is given by the following equation:

$$E\left(A\right)=1-{\displaystyle \frac{1+\epsilon }{n}}\sum _{i=1}^n\left({\displaystyle \frac{\left|{\mu }_{\alpha }\left(x_i\right)-{\nu }_{\alpha }\left(x_i\right)\right|}{{\mu }_{\alpha }\left(x_i\right)+{\nu }_{\alpha }\left(x_i\right)+\epsilon }}\right)$$

(17)

where $0<\epsilon \le 1$ is the parameter to adjust the difference.

Obviously, Equation (17) satisfies the properties of the entropy measure defined in [34]. The following rules and examples are presented to demonstrate the feasibility and effectiveness of the proposed entropy measure.

For any positive real number $r$ and IFS $A=\{〈x,\mu \left(x\right),\nu \left(x\right)〉\left|x\in X\right.$}, the operation $A^r$ is defined as $A^r=\{〈x,{\left[\mu \left(x\right)\right]}^r,1-{(1-\nu \left(x\right))}^r〉\left|x_i\in X\right.$} [35].

If the IFS $A$ is regarded as “Large”, then the corresponding IFSs $A^{1/2}$, $A^2$, $A^3$, and $A^4$ convey the following meanings: “More or less large”, “Very large”, “Quite very large”, and “Very very large”, respectively. The entropy values of these IFSs follow the order $A^{1/2}\succ A\succ A^2\succ A^3\succ A^4$. In other words, $E\left(A^{1/2}\right)>E\left(A\right)>E\left(A^2\right)>E\left(A^3\right)>E\left(A^4\right)$, where $E(\cdot )$ represents the entropy function [36].

Example 6. 

Let $A=\{〈0.2,0.8〉,〈0.3,0.5〉,〈0.7,0.2〉,〈0.9,0〉,〈1,0〉\}$. The entropy of $A^{1/2}$, $A$, $A^2$, $A^3$, and $A^4$ are calculated by Equation (17) with $\epsilon =0.14$. Specifically, $E\left(A^{1/2}\right)=0.873>E\left(A\right)=0.865>E\left(A^2\right)=0.848>E\left(A^3\right)=0.838>E\left(A^4\right)=0.827$, which proves the effectiveness.

3. A New Fuzzy MAGDM Method Based on the Proposed ISIFDisM

For clarity and convenience, the sets and symbols employed in the subsequent evaluation process are rigorously defined as follows:

• Set of alternatives (complex equipment): Denoted as $E=\left\{E_i|i\in M\right\}$, where $M=\{\mathrm{1,2},\dots ,m\}$. Here, each $E_i$ represents the complex equipment of a different model, and set $M$ encompasses all indices that uniquely identify the complex equipment.
• Set of attributes (critical components): Represented as $C=\left\{c_j|j\in N\right\}$, where $N=\{\mathrm{1,2},\dots ,n\}$. Each $c_j$ corresponds to a specific core component, and set $N$ contains all indices for these components.
• Attribute weight (importance of components) vector: Designated as $\mathit{w}=(w_1,w_2,\dots ,w_n)$, where $0\le w_j\le 1$ for $\left(j\in N\right)$ and ${\sum }_{j=1}^n{w}_j=1$. The elements of vector $\mathit{w}$ quantify the relative significance of each core component within the evaluation framework.
• Set of decision-makers (operational environment): Denoted as $D=\left\{d_k|k\in T\right\}$, where $T=\{\mathrm{1,2},\dots ,t\}$. Each $d_k$ represents a decision-maker or a specific aspect of the operational environment, and $T$ contains all the relevant indices.

3.1. Accessing and Converting Decision Data from Questionnaires

The decision-making dataset employed in this study is collected through questionnaires administered to complex equipment operators across various environmental categories and operational scenarios. Guided by the equipment’s operational mechanics and design specifications, a specialized decision-making committee composed of experts in mechanical engineering, materials science, and systems biology is required to collaboratively identify critical components and establish a scientific basis for state evaluation. In this study, the operational states of critical components are classified into three distinct levels as follows:

• Normal operation status: The equipment functions as intended, with all parameters remaining within the normal range, and no faults or abnormalities are present.
• Abnormal operation status: The equipment exhibits minor defects or parameter deviations from the normal operating range but can still operate. Continuous monitoring and adjustments are required.
• Failure shutdown status: Severe faults render the equipment inoperable, necessitating shutdown for component repair or replacement.

For complex equipment integrated with a health management system, the real-time states of critical components and the entire system can be accurately evaluated. However, for low-intelligence equipment lacking built-in monitoring capabilities, a questionnaire-based approach is indispensable for evaluating the states of components within the same operational cycle. Thus, the questionnaire presented in

Table 3. Questionnaire format.

Equipment	Environment	Use	Component 1			Component 2				Component n
			Normal	Abnormal	Failure	Normal	Abnormal	Failure	…	Normal	Abnormal	Failure
E1
E2
…

is developed to systematically capture state parameters.

Statistical data regarding the operational conditions of critical components across diverse environments and usage scenarios are collected. This information is represented as a three-dimensional interval vector $(\left[{\alpha }_{kj}^{il},{\alpha }_{kj}^{iu}\right],\left[{\beta }_{kj}^{il},{\beta }_{kj}^{iu}\right],\left[{\gamma }_{kj}^{il},{\gamma }_{kj}^{iu}\right])$, where ${\alpha }_{kj}^{il}$ denotes the minimum percentage of the normal operational status for the $j$th component of the $i$th equipment across all usages within the $k$th operational environment, while ${\alpha }_{kj}^{iu}$ represents the corresponding maximum percentage. Similarly, ${\beta }_{kj}^{il}$ and ${\beta }_{kj}^{iu}$ signify the minimum and maximum percentages of abnormal operation, and ${\gamma }_{kj}^{il}$ and ${\gamma }_{kj}^{iu}$ denote those of failure shutdown status, respectively. According to [23], the interval $\left[{\alpha }_{kj}^{il},{\alpha }_{kj}^{iu}\right]$ can be converted into an IFN $〈{\mu }_{kj}^{\alpha i},v_{kj}^{\alpha i}〉$, where ${\mu }_{kj}^{\alpha i}={\alpha }_{kj}^{il}$ and ${\nu }_{kj}^{\alpha i}=1-{\alpha }_{kj}^{iu}$. The original fuzzy MAGDM matrices ${\mathit{X}}_i={\left(\left(〈{\mu }_{kj}^{\alpha i},{\nu }_{kj}^{\alpha i}〉,〈{\mu }_{kj}^{\beta i},{\nu }_{kj}^{\beta i}〉,〈{\mu }_{kj}^{\gamma i},{\nu }_{kj}^{\gamma i}〉\right)\right)}_{t\times n}$ are constructed as the following:

$${\mathit{X}}_i={\left(\left(〈{\mu }_{kj}^{\alpha i},{\nu }_{kj}^{\alpha i}〉,〈{\mu }_{kj}^{\beta i},{\nu }_{kj}^{\beta i}〉,〈{\mu }_{kj}^{\gamma i},{\nu }_{kj}^{\gamma i}〉\right)\right)}_{t\times n}$$

(18)

where the IFN vector $(〈{\mu }_{kj}^{\alpha i},{\nu }_{kj}^{\alpha i}〉,〈{\mu }_{kj}^{\beta i},{\nu }_{kj}^{\beta i}〉,〈{\mu }_{kj}^{\gamma i},{\nu }_{kj}^{\gamma i}〉)$ is the evaluation result of the $j$th critical component of the $i$th equipment device in the $k$th operational environment.

3.2. Determining the Weights of Decision Attributes

The determination of decision-makers and attribute weights plays a pivotal role in addressing the MAGDM problem. In this study, all distinct environmental conditions and vessel categories are assumed to hold equal importance, hence being assigned uniform weights. Conversely, the entropy weight method is employed to determine the attribute weight or component importance, which considers the distributional characteristics embedded within the data itself.

The entropy weight method recognizes that if all decision-makers provide identical values for a specific attribute across all available alternatives in a decision-making process, it implies that the attribute does not influence the final decision. In such instances, allocating a weight to the attribute is redundant. Conversely, attributes with more dispersed data distributions should be endowed with higher weights to enhance the discriminative accuracy of the results. In this study, if the condition data of a component remains largely consistent across different environmental and usage scenarios, it is inferred that the component has negligible influence on equipment reliability, justifying a lower weight assignment. Conversely, if the condition data of a component exhibits significant variability across diverse environmental and usage scenarios, it is considered to have substantial influence on equipment reliability, justifying a higher weight allocation. This ensures that the weights are dynamically calibrated to reflect the informational entropy of attribute variations, thereby optimizing the assessment framework’s sensitivity to critical components.

Subsequently, the attribute weights or component importance indices can be expressed as the following:

$$w_j={\displaystyle \frac{1-E_j}{n-{\sum }_{j=1}^n{E}_j}},j\in N$$

(19)

where $E_j$ is given by the following equation:

$$E_j=-{\displaystyle \frac{1}{3m}}\sum _{i=1}^m\left(E\left({\alpha }_j^i\right)+E\left({\beta }_j^i\right)+E\left({\gamma }_j^i\right)\right),j\in N$$

(20)

$$\begin{gathered} E\left({\alpha }_j^i\right)=E\left({\left(〈{\mu }_{1j}^{\alpha i},{\nu }_{1j}^{\alpha i}〉,〈{\mu }_{2j}^{\alpha i},{\nu }_{2j}^{\alpha i}〉,\dots ,〈{\mu }_{tj}^{\alpha i},{\nu }_{tj}^{\alpha i}〉\right)}^T\right) \\ E\left({\beta }_j^i\right)=E\left({\left(〈{\mu }_{1j}^{\beta i},{\nu }_{1j}^{\beta i}〉,〈{\mu }_{2j}^{\beta i},{\nu }_{2j}^{\beta i}〉,\dots ,〈{\mu }_{tj}^{\beta i},{\nu }_{tj}^{\beta i}〉\right)}^T\right) \\ E\left({\gamma }_j^i\right)=E\left({\left(〈{\mu }_{1j}^{\gamma i},{\nu }_{1j}^{\gamma i}〉,〈{\mu }_{2j}^{\gamma i},{\nu }_{2j}^{\gamma i}〉,\dots ,〈{\mu }_{tj}^{\gamma i},{\nu }_{tj}^{\gamma i}〉\right)}^T\right) \end{gathered}$$

(21)

3.3. Ranking Preference Order of Alternatives

The attribute weight vector is obtained from Equations (19)–(21), and the attribute weight matrix $\mathit{W}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(w_1,w_2,\dots ,w_n)$ can be determined. Additionally, the weighted fuzzy MAGDM matrix for equipment $E_i$ can be established as follows:

$${\mathit{Y}}_i={\mathit{X}}_i\mathit{W}={\left(\left(〈{\mu }_{kj}^{\rho i},{\nu }_{kj}^{\rho i}〉, 〈{\mu }_{kj}^{\phi i},{\nu }_{kj}^{\phi i}〉, 〈{\mu }_{kj}^{\varphi i},{\nu }_{kj}^{\varphi i}〉\right)\right)}_{t\times n}$$

(22)

where $〈{\mu }_{kj}^{\rho i},{\nu }_{kj}^{\rho i}〉=〈{1-(1-{\mu }_{kj}^{\alpha i})}^{w_j},{\left({\nu }_{kj}^{\alpha i}\right)}^{w_j}〉$, $〈{\mu }_{kj}^{\phi i},{\nu }_{kj}^{\phi i}〉=〈{1-(1-{\mu }_{kj}^{\beta i})}^{w_j},{\left({\nu }_{kj}^{\beta i}\right)}^{w_j}〉$, and $〈{\mu }_{kj}^{\varphi i},{\nu }_{kj}^{\varphi i}〉=〈{1-(1-{\mu }_{kj}^{\gamma i})}^{w_j},{\left({\nu }_{kj}^{\gamma i}\right)}^{w_j}〉$.

Since $〈{\mu }_{kj}^{\rho i},{\nu }_{kj}^{\rho i}〉, 〈{\mu }_{kj}^{\phi i},{\nu }_{kj}^{\phi i}〉,$ and $〈{\mu }_{kj}^{\varphi i},{\nu }_{kj}^{\varphi i}〉$ are the evaluation values of normal, abnormal, and failure states, respectively, to distinguish the differences among different states, $〈{\mu }_{kj}^{\rho i},{\nu }_{kj}^{\rho i}〉, 〈{\mu }_{kj}^{\phi i},{\nu }_{kj}^{\phi i}〉,$ and $〈{\mu }_{kj}^{\varphi i},{\nu }_{kj}^{\varphi i}〉$ are multiplied by the corresponding state parameters. The three states can be expressed by three-level fuzzy linguistic values $S_1$, $S_2$, and $S_3$. The corresponding three-level fuzzy language can also be converted into IFNs $\eta =〈{\mu }_{\eta },{\nu }_{\eta }〉, \theta =〈{\mu }_{\theta },{\nu }_{\theta }〉,$ and $\vartheta =〈{\mu }_{\vartheta },{\nu }_{\vartheta }〉$. Thus, the final fuzzy MAGDM matrix is graded as follows:

$${\mathit{Y}}_i^{\prime }={\left(\left(〈{\mu }_{kj}^{\eta i},{\nu }_{kj}^{\eta i}〉, 〈{\mu }_{kj}^{\theta i},{\nu }_{kj}^{\theta i}〉, 〈{\mu }_{kj}^{\vartheta i},{\nu }_{kj}^{\vartheta i}〉\right)\right)}_{t\times n}$$

(23)

where $〈{\mu }_{kj}^{\eta i},{\nu }_{kj}^{\eta i}〉=\eta ⨂〈{\mu }_{kj}^{\rho i},{\nu }_{kj}^{\rho i}〉=〈{\mu }_{\eta }{\mu }_{kj}^{\rho i},{\nu }_{\eta }+{\nu }_{kj}^{\rho i}-{{\nu }_{\eta }\nu }_{kj}^{\rho i}〉$, $〈{\mu }_{kj}^{\theta i},{\nu }_{kj}^{\theta i}〉= \theta ⨂〈{\mu }_{kj}^{\phi i},{\nu }_{kj}^{\phi i}〉=〈{\mu }_{\theta }{\mu }_{kj}^{\phi i},{\nu }_{\theta }+{\nu }_{kj}^{\phi i}-{\nu }_{\theta }{v}_{kj}^{\phi i}〉$, and $〈{\mu }_{kj}^{\vartheta i},{\nu }_{kj}^{\vartheta i}〉=\vartheta ⨂〈{\mu }_{kj}^{\varphi i},{\nu }_{kj}^{\varphi i}〉=〈{\mu }_{\vartheta }{\mu }_{kj}^{\varphi i},{\nu }_{\vartheta }+{\nu }_{kj}^{\varphi i}-{\nu }_{\vartheta }{\nu }_{kj}^{\varphi i}〉$.

In order to achieve a consensus on group decision-making information in different environments, the IFWA operator is used for integration. For weighted fuzzy MAGDM matrices ${\mathit{Y}}_i(i\in M)$, the integrated information of the equipment $E_i$ can be calculated using the IFWA operator:

$${\mathit{Z}}_i={\left(\left(〈{\mu }_j^{Ai},{\nu }_j^{Ai}〉, 〈{\mu }_j^{Bi},{\nu }_j^{Bi}〉, 〈{\mu }_j^{\Gamma i},{\nu }_j^{\Gamma i}〉\right)\right)}_{1\times n}$$

(24)

where $\left(〈{\mu }_j^{Ai},{\nu }_j^{Ai}〉, 〈{\mu }_j^{Bi},{\nu }_j^{Bi}〉, 〈{\mu }_j^{\Gamma i},{\nu }_j^{\Gamma i}〉\right)$ represents the consensus IFN for the $j$th critical component of the $i$th equipment, computed as $〈{\mu }_j^{Ai},{\nu }_j^{Ai}〉=〈1-\prod _{k=1}^t{\left(1-{\mu }_{kj}^{\eta i}\right)}^{1/t},\prod _{k=1}^n{\left({\nu }_{kj}^{\eta i}\right)}^{1/t}〉$, $〈{\mu }_j^{Bi},{\nu }_j^{Bi}〉=〈1-\prod _{k=1}^t{\left(1-{\mu }_{kj}^{\theta i}\right)}^{1/t},\prod _{k=1}^n{\left({\nu }_{kj}^{\theta i}\right)}^{1/t}〉$, and $〈{\mu }_j^{\mathsf{\Gamma }i},{\nu }_j^{\mathsf{\Gamma }i}〉=〈1-\prod _{k=1}^t{\left(1-{\mu }_{kj}^{\vartheta i}\right)}^{1/t},\prod _{k=1}^n{\left({\nu }_{kj}^{\vartheta i}\right)}^{1/t}〉$.

The exponent $1/t$ ensures equal weighting across all $t$ operational environments. This aligns with our foundational assumption that all distinct environmental conditions and vessel categories are accorded equal significance.

Therefore, the positive ideal decision (PID) and negative ideal decision (NID) are determined as follows:

$$\begin{gathered} {\mathit{Z}}_{+}={\left(\left(〈{\mu }_{j+}^{Ai},{\nu }_{j+}^{Ai}〉, 〈{\mu }_{j+}^{{\rm B}i},{\nu }_{j+}^{{\rm B}i}〉, 〈{\mu }_{j+}^{\Gamma i},{\nu }_{j+}^{\Gamma i}〉\right)\right)}_{1\times n} \\ {\mathit{Z}}_{-}={\left(\left(〈{\mu }_{j-}^{Ai},{\nu }_{j-}^{Ai}〉, 〈{\mu }_{j-}^{{\rm B}i},{\nu }_{j-}^{{\rm B}i}〉, 〈{\mu }_{j-}^{\Gamma i},{\nu }_{j-}^{\Gamma i}〉\right)\right)}_{1\times n} \end{gathered}$$

(25)

where $〈{\mu }_{j+}^{Ai},{\nu }_{j+}^{Ai}〉=〈\mathrm{m}\mathrm{a}\mathrm{x}\left({\mu }_j^{Ai}\right),\mathrm{m}\mathrm{i}\mathrm{n}\left({\nu }_j^{Ai}\right)〉$, $〈{\mu }_{j+}^{{\rm B}i},{\nu }_{j+}^{{\rm B}i}〉=〈\mathrm{m}\mathrm{a}\mathrm{x}\left({\mu }_j^{Bi}\right),\mathrm{m}\mathrm{i}\mathrm{n}\left({\nu }_j^{Bi}\right)〉$, and $〈{\mu }_{j+}^{\Gamma i},{\nu }_{j+}^{\Gamma i}〉=〈\mathrm{m}\mathrm{a}\mathrm{x}\left({\mu }_j^{\Gamma i}\right),\mathrm{m}\mathrm{i}\mathrm{n}\left({\nu }_j^{\Gamma i}\right)〉$; $〈{\mu }_{j-}^{Ai},{\nu }_{j-}^{Ai}〉=〈\mathrm{m}\mathrm{i}\mathrm{n}\left({\mu }_j^{Ai}\right),\mathrm{m}\mathrm{a}\mathrm{x}\left({\nu }_j^{Ai}\right)〉$, $〈{\mu }_{j-}^{{\rm B}i},{\nu }_{j-}^{{\rm B}i}〉=〈\mathrm{m}\mathrm{i}\mathrm{n}\left({\mu }_j^{Bi}\right),\mathrm{m}\mathrm{a}\mathrm{x}\left({\nu }_j^{Bi}\right)〉$ and $〈{\mu }_{j-}^{\mathsf{\Gamma }i},v_{j-}^{\mathsf{\Gamma }i}〉=〈\mathrm{m}\mathrm{i}\mathrm{n}\left({\mu }_j^{\mathsf{\Gamma }i}\right),\mathrm{m}\mathrm{a}\mathrm{x}\left({\nu }_j^{\mathsf{\Gamma }i}\right)〉$.

Subsequently, according to Equation (15), the distance between each weighted MAGDM matrix ${\mathit{Z}}_i$ and the PID or NID can be calculated as follows:

$$\begin{gathered} d_{\epsilon }\left({\mathit{Z}}_i,{\mathit{Z}}_{+}\right)={\displaystyle \frac{1}{n}}\sum _{j=1}^n{d}_{\epsilon }\left(\left(〈{\mu }_j^{Ai},{\nu }_j^{Ai}〉, 〈{\mu }_j^{Bi},{\nu }_j^{Bi}〉, 〈{\mu }_j^{\Gamma i},{\nu }_j^{\Gamma i}〉\right),\left(〈{\mu }_{j+}^{Ai},{\nu }_{j+}^{Ai}〉, 〈{\mu }_{j+}^{{\rm B}i},{\nu }_{j+}^{{\rm B}i}〉, 〈{\mu }_{j+}^{\Gamma i},{\nu }_{j+}^{\Gamma i}〉\right)\right) \\ {d}_{\epsilon }\left({\mathit{Z}}_i,{\mathit{Z}}_{-}\right)={\displaystyle \frac{1}{n}}\sum _{j=1}^n{d}_{\epsilon }\left(\left(〈{\mu }_j^{Ai},{\nu }_j^{Ai}〉, 〈{\mu }_j^{Bi},{\nu }_j^{Bi}〉, 〈{\mu }_j^{\Gamma i},{\nu }_j^{\Gamma i}〉\right),\left(〈{\mu }_{j-}^{Ai},{\nu }_{j-}^{Ai}〉, 〈{\mu }_{j-}^{{\rm B}i},{\nu }_{j-}^{{\rm B}i}〉, 〈{\mu }_{j-}^{\Gamma i},{\nu }_{j-}^{\Gamma i}〉\right)\right) \end{gathered}$$

(26)

According to Definition 11, the closer the $d_{\epsilon } \left({\mathit{Z}}_i,{\mathit{Z}}_{+}\right)$ is to 0, and the closer the $d_{\epsilon }\left({\mathit{Z}}_i,{\mathit{Z}}_{-}\right)$ is to 1, the better the performance of the complex equipment $E_i$. To provide a comprehensive reliability evaluation result that considers the proximity of ${\mathit{Z}}_i$ to both the PID and NID simultaneously, we define the normalized priority coefficient for complex equipment $E_i$ as follows:

$${NPC}_i={\displaystyle \frac{d_{\epsilon }\left({\mathit{Z}}_i,{\mathit{Z}}_{-}\right)}{d_{\epsilon }\left({\mathit{Z}}_i,{\mathit{Z}}_{-}\right)+d_{\epsilon }\left({\mathit{Z}}_i,{\mathit{Z}}_{+}\right)}},i\in M$$

(27)

Hence, the closer the value of ${NPC}_i$ is to 1, the better the performance of the complex equipment $E_i$.

3.4. The Decision-Making Procedure with the Proposed Method

The reliability evaluation of complex equipment is contingent upon the employment of the extended TOPSIS approach and ISIFDisM. This paper presents a novel fuzzy MAGDM method through the following sequential procedures:

Step 1: Identify critical components and design the questionnaire.

First, identify a comprehensive list of complex equipment from different brands and their critical components, which serve as the foundation for questionnaire design. The decision-making committee should work collaboratively to determine the critical components and establish criteria for assessing their status. Subsequently, based on Table 3, a questionnaire should be meticulously prepared.

Step 2: Collect statistical data and construct interval-valued matrices.

Systematically collect multivariate statistical data describing the operational profiles of critical components across diverse environmental gradients and real-world usage scenarios. The collected data is used to construct the interval-valued MAGDM matrices ${\tilde{\mathit{X}}}_i={\left(\left(\left[{\alpha }_{kj}^{il},{\alpha }_{kj}^{iu}\right],\left[{\beta }_{kj}^{il},{\beta }_{kj}^{iu}\right],\left[{\gamma }_{kj}^{il},{\gamma }_{kj}^{iu}\right]\right)\right)}_{t\times n}$.

Step 3: Create the original fuzzy MAGDM matrices.

Transform the interval-valued matrices into IFN matrices. Specifically, the interval value $\left[{\alpha }_{kj}^{il},{\alpha }_{kj}^{iu}\right]$ is converted into an IFN $〈{\mu }_{kj}^{\alpha i},{\nu }_{kj}^{\alpha i}〉$, where ${\mu }_{kj}^{\alpha i}={\alpha }_{kj}^{il}$, ${\nu }_{kj}^{\alpha i}=1-{\alpha }_{kj}^{iu}$. Therefore, the original fuzzy MAGDM matrices ${\mathit{X}}_i={\left(\left(〈{\mu }_{kj}^{\alpha i},{\nu }_{kj}^{\alpha i}〉,〈{\mu }_{kj}^{\beta i},{\nu }_{kj}^{\beta i}〉,〈{\mu }_{kj}^{\gamma i},{\nu }_{kj}^{\gamma i}〉\right)\right)}_{t\times n}$ are constructed according to Equation (18).

Step 4: Determine the weights of the critical components.

Determine the weights of the critical components using the entropy weight method. The importance vector of the components $\mathit{w}=(w_1,w_2,\cdots ,w_n)$ is calculated through Equations (19)–(21).

Step 5: Construct the weighted fuzzy MAGDM matrices.

Given the weight matrix $\mathit{W}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(w_1,w_2,\cdots ,w_n)$, the weighted fuzzy MAGDM matrices ${\mathit{Y}}_i={\left(\left(〈{\mu }_{kj}^{\rho i},{\nu }_{kj}^{\rho i}〉, 〈{\mu }_{kj}^{\phi i},{\nu }_{kj}^{\phi i}〉, 〈{\mu }_{kj}^{\varphi i},{\nu }_{kj}^{\varphi i}〉\right)\right)}_{t\times n}$ for the reliability of complex equipment are computed by Equation (22).

Step 6: Construct the final fuzzy MAGDM matrices.

The three-level fuzzy linguistic values $S_1$, $S_2$, and $S_3$ are converted into IFNs $\eta ,$ $\theta$, and $\vartheta$. Subsequently, these IFNs are multiplied by ${\mathit{Y}}_i$ to obtain the final fuzzy MAGDM matrices ${\mathit{Y}}_i^{\prime }={\left(\left(〈{\mu }_{kj}^{\mathsf{\eta }i},{\nu }_{kj}^{\mathsf{\eta }i}〉, 〈{\mu }_{kj}^{\mathsf{\theta }i},{\nu }_{kj}^{\mathsf{\theta }i}〉, 〈{\mu }_{kj}^{\mathsf{\vartheta }i},{\nu }_{kj}^{\mathsf{\vartheta }i}〉\right)\right)}_{t\times n}$ using Equation (23).

Step 7: Integrate information from different operational environments.

Employing the IFWA operator, the final fuzzy MAGDM matrices across different operational environments are aggregated to calculate the integration information matrices ${\mathit{Z}}_i={\left(\left(〈{\mu }_j^{Ai},{\nu }_j^{Ai}〉, 〈{\mu }_j^{Bi},{\nu }_j^{Bi}〉, 〈{\mu }_j^{\Gamma i},{\nu }_j^{\Gamma i}〉\right)\right)}_{1\times n}$ of equipment $E_i$ by Equation (24).

Step 8: Determine ideal decisions.

According to the framework of the TOPSIS approach, PID ${\mathit{Z}}_{+}$ and NID ${\mathit{Z}}_{+}$ are determined by Equation (25).

Step 9: Calculate the distance between the complex equipment and the ideal decisions.

Calculate the closeness of each weighted fuzzy MAGDM matrix ${\mathit{Z}}_i$ to the ideal decisions. The distances between ${\mathit{Z}}_i$ and both the PID as well as the NID are determined by Equation (26).

Step 10: Calculate the priority coefficient.

The normalized priority coefficient ${NPC}_i$ for complex equipment $E_i$ is determined by Equation (27).

Step 11: Rank the reliability of complex equipment.

The reliability of complex equipment is ranked from best to worst based on ${NPC}_i$, in descending order.

For clarity, a schematic of the proposed method is shown in Figure 4.

4. Application

This section presents a practical reliability assessment of ship equipment manufactured by a shipbuilding enterprise in Jiangsu, China. The methodology proposed in Section 3 is systematically applied to comprehensively validate its feasibility and effectiveness, demonstrating the approach’s utility in real-world industrial contexts.

4.1. Describe the Evaluation Problem

According to Step 1, five different types of marine engines are evaluated, denoted as $E=\{E_1,E_2,E_3,E_4,E_5\}$. Each engine comprises six critical components, represented as $C=\left\{c_1,c_2,c_3,c_4,c_5,c_6\right\}=${moving parts, gas distribution mechanism, fuel system, engine body, cooling system, lubrication system}. The operational environments of the investigated engines, which correspond to different ships, are classified into four types: ocean-going ships, near-ocean ships, coastal ships, and inland ships, denoted as $D=\{d_1,d_2,d_3,d_4\}$, respectively. Subsequently, these environments can be further subdivided into three vessel categories: transport vessels (TVs), engineering vessels (EVs), and special working vessels (SWVs). The key distinctions among these environmental classifications and vessel types are as follows.

Vessels are primarily categorized according to their navigational reach and geographical constraints:

• Ocean-going ships are purpose-built for prolonged intercontinental voyages, integrating cutting-edge technologies essential for deep-sea operations.
• Near-ocean ships serve as workhorses for maritime operations in transboundary waters, typically executing short- to medium-range missions between adjacent nations.
• Coastal ships are legally and technically confined to operating within a nation’s territorial waters, prioritizing shallow-water adaptability and regional logistics.
• Inland ships are custom designed for navigation on inland water bodies, including rivers, lakes, and canal networks.

Functional specialization further delineates vessel types into distinct operational categories:

• Transport vessels (TVs) are mainly employed for transporting goods or passengers.
• Engineering vessels (EVs) are predominantly utilized in activities such as dredging, sub-sea operations, and other technically demanding tasks.
• Specialized working vessels (SWVs) are specifically designed for the transportation of large objects and engineered to handle oversized cargo that cannot be accommodated by standard transport vessels.

The proposed classification approach presents a robust and meticulously designed framework. It can precisely gauge the reliability of marine diesel engines across diverse environmental conditions and multifaceted applications. This guarantees the comprehensiveness and thoroughness of the evaluation process, leaving no critical aspect overlooked.

To operationalize this methodology, domain experts are tasked with completing a rigorously structured questionnaire. The questionnaire completion process is fundamentally anchored in a comprehensive analysis of the engine’s real-world operational conditions during its first maintenance cycle, detailed daily operational records, and comprehensive maintenance logs. Within this framework, the notations “N”, “A”, and “F” are systematically employed to denote normal operational status, abnormal operational status, and fault shutdown status, respectively. Notably, the operational performance of a critical component can be effectively assessed using a parsimonious yet informative metric: a higher proportion of “N” ratings, coupled with lower proportions of “A” and “F” ratings, directly correlates with a superior and more stable operational state of the component.

4.2. Processing Evaluation Data

According to Step 2, the statistical results of the original questionnaire data are presented in

Table 4. Statistical results of the original questionnaire data.

$\mathit{E}$	$\mathit{D}$	Use	Total Number	${\mathit{c}}_1$			${\mathit{c}}_2$			${\mathit{c}}_3$			${\mathit{c}}_4$			${\mathit{c}}_5$			${\mathit{c}}_6$
				N	A	F	N	A	F	N	A	F	N	A	F	N	A	F	N	A	F
$E_1$	$d_1$	TV	45	32	9	4	30	11	4	28	13	4	36	6	3	31	10	4	29	12	4
		EV	42	28	10	4	25	12	5	24	14	4	31	8	3	27	11	4	26	13	3
		SWV	38	26	8	4	23	10	5	22	12	4	29	6	3	25	9	4	24	10	4
	$d_2$	TV	35	28	5	2	26	6	3	27	6	2	30	4	1	25	7	3	26	7	2
		EV	32	24	6	2	22	7	3	20	9	3	25	5	2	22	8	2	21	9	2
		SWV	28	20	5	3	18	7	3	17	8	3	22	4	2	19	6	3	18	7	3
	$d_3$	TV	40	31	7	2	29	8	3	30	7	3	34	5	1	28	9	3	29	8	3
		EV	37	27	7	3	24	9	4	23	10	4	29	6	2	25	9	3	24	10	3
		SWV	30	24	4	2	21	6	3	20	7	3	25	3	2	21	6	3	20	7	3
	$d_4$	TV	25	19	4	2	16	6	3	17	5	3	20	3	2	17	5	3	16	6	3
		EV	22	17	3	2	14	5	3	13	6	3	17	3	2	14	5	3	13	6	3
		SWV	18	14	2	2	12	4	2	11	5	2	14	2	2	12	4	2	11	5	2
$E_2$	$d_1$	TV	42	30	8	4	28	10	4	26	12	4	33	6	3	29	9	4	27	11	4
		EV	38	26	8	4	23	10	5	22	12	4	29	6	3	25	9	4	23	11	4
		SWV	35	25	7	3	22	9	4	21	10	4	27	5	3	23	8	4	22	9	4
	$d_2$	TV	32	26	4	2	24	5	3	25	5	2	28	3	1	23	6	3	24	6	2
		EV	28	22	4	2	19	6	3	18	7	3	23	3	2	20	5	3	19	6	3
		SWV	25	20	3	2	17	5	3	16	6	3	20	3	2	17	5	3	16	6	3
	$d_3$	TV	39	32	5	2	30	6	3	31	6	2	34	4	1	29	7	3	30	7	2
		EV	33	26	5	2	23	7	3	22	8	3	26	5	2	23	7	3	22	8	3
		SWV	27	22	3	2	19	5	3	18	6	3	21	4	2	19	5	3	18	6	3
	$d_4$	TV	22	18	2	2	15	4	3	16	4	2	19	2	1	16	4	2	15	5	2
		EV	18	14	2	2	12	4	2	11	5	2	14	2	2	12	4	2	11	5	2
		SWV	15	12	1	2	10	3	2	9	4	2	12	1	2	10	3	2	9	4	2
$E_3$	$d_1$	TV	46	34	8	4	32	10	4	30	12	4	37	6	3	33	9	4	31	11	4
		EV	41	29	8	4	26	10	5	25	12	4	32	6	3	28	9	4	26	11	4
		SWV	37	27	7	3	24	9	4	23	10	4	29	5	3	25	8	4	24	9	4
	$d_2$	TV	34	28	4	2	26	5	3	27	5	2	29	3	2	25	6	3	26	6	2
		EV	29	23	4	2	20	6	3	19	7	3	24	3	2	21	5	3	20	6	3
		SWV	25	20	3	2	17	5	3	16	6	3	20	3	2	17	5	3	16	6	3
	$d_3$	TV	42	35	5	2	33	6	3	34	6	2	37	4	1	32	7	3	33	7	2
		EV	36	29	5	2	26	7	3	25	8	3	30	4	2	27	7	2	26	8	2
		SWV	31	25	4	2	22	6	3	21	7	3	25	4	2	22	6	3	21	7	3
	$d_4$	TV	24	20	2	2	17	4	3	18	4	2	21	2	1	18	4	2	17	5	2
		EV	20	16	2	2	13	4	3	12	5	3	15	3	2	13	4	3	12	5	3
		SWV	16	13	1	2	11	3	2	10	4	2	13	1	2	11	3	2	10	4	2
$E_4$	$d_1$	TV	38	26	8	4	23	10	5	22	11	5	28	7	3	24	10	4	23	11	4
		EV	34	24	7	3	20	9	5	18	11	5	23	8	3	20	10	4	19	11	4
		SWV	31	22	6	3	18	8	5	16	10	5	21	7	3	17	9	5	16	10	5
	$d_2$	TV	24	18	4	2	15	6	3	14	7	3	18	4	2	15	6	3	14	7	3
		EV	20	14	4	2	11	6	3	9	8	3	12	5	3	10	7	3	9	8	3
		SWV	17	12	3	2	9	5	3	7	7	3	10	5	2	8	6	3	7	7	3
	$d_3$	TV	30	22	6	2	19	8	3	18	9	3	22	6	2	19	8	3	18	9	3
		EV	26	19	5	2	15	8	3	13	10	3	17	7	2	15	8	3	13	10	3
		SWV	22	16	4	2	12	7	3	10	9	3	14	6	2	12	7	3	10	9	3
	$d_4$	TV	16	12	2	2	9	5	2	8	6	2	11	3	2	9	5	2	8	6	2
		EV	13	9	2	2	6	5	2	5	6	2	7	4	2	6	5	2	5	6	2
		SWV	10	7	2	1	4	4	2	3	5	2	5	3	2	4	4	2	3	5	2
$E_5$	$d_1$	TV	40	28	8	4	25	10	5	24	11	5	30	7	3	26	10	4	25	11	4
		EV	36	26	7	3	23	9	4	21	11	4	27	7	2	23	9	4	22	10	4
		SWV	32	24	6	2	20	8	4	18	10	4	23	7	2	20	8	4	18	10	4
	$d_2$	TV	27	20	5	2	16	8	3	15	9	3	19	6	2	16	8	3	15	9	3
		EV	23	17	4	2	13	7	3	11	9	3	15	6	2	13	7	3	11	9	3
		SWV	19	14	3	2	11	6	2	9	8	2	12	5	2	10	7	2	9	8	2
	$d_3$	TV	35	26	6	3	22	9	4	21	10	4	26	7	2	22	9	4	21	10	4
		EV	30	22	5	3	18	8	4	16	10	4	21	7	2	18	8	4	16	10	4
		SWV	26	19	4	3	14	8	4	12	10	4	17	7	2	14	8	4	12	10	4
	$d_4$	TV	21	16	3	2	12	6	3	11	7	3	14	5	2	12	6	3	11	7	3
		EV	17	12	3	2	9	6	2	7	7	3	10	5	2	8	7	2	7	7	3
		SWV	14	10	3	1	7	5	2	6	6	2	8	4	2	7	5	2	6	6	2

. Subsequently, the original interval value vectors can be derived. For example, ${\alpha }_{11}^{1l}=\min \left\{32/45, 28/\mathrm{42,26}/38\right\}$, ${\alpha }_{11}^{1u}=\max \left\{32/45, 28/42, 26/38\right\}$, ${\beta }_{11}^{1l}=\min \left\{9/45, 10/42, 8/38\right\}$, ${\beta }_{11}^{1u}=\max \left\{9/45, 10/42, 8/38\right\}$, ${\gamma }_{11}^{1l}=\min \left\{4/45, 4/42, 4/38\right\}$, and ${\gamma }_{11}^{1u}=\min \left\{4/45, 4/42, 4/38\right\}$. Thus, the interval vector $\left(\left[{\alpha }_{11}^{1l},{\alpha }_{11}^{1u}\right], \left[{\beta }_{11}^{1l},{\beta }_{11}^{1u}\right], \left[{\gamma }_{11}^{1l},{\gamma }_{11}^{1u}\right]\right)=(\left[0.6667, 0.7111\right], \left[0.2000, 0.2381\right], [0.0889, 0.1053\left]\right)$. Consequently, the original interval MAGDM matrices are summarized in

Table 5. The original interval MAGDM matrices.

$\mathit{E}$	$\mathit{D}$	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$	${\mathit{c}}_4$	${\mathit{c}}_5$	${\mathit{c}}_6$
$E_1$	$d_1$	([0.6667,0.7111], [0.2000,0.2381], [0.0889,0.1053])	([0.5952,0.6667], [0.2444,0.2857], [0.0889,0.1316])	([0.5714,0.6222], [0.2889,0.3333], [0.0889,0.1053])	([0.7381,0.8000], [0.1333,0.1905], [0.0667,0.0789])	([0.6429,0.6889], [0.2222,0.2619], [0.0889,0.1053])	([0.6190,0.6444], [0.2632,0.3095], [0.0714,0.1053])
	$d_2$	([0.7143,0.8000], [0.1429,0.1875], [0.0571,0.1071])	([0.6429,0.7429], [0.1714,0.2500], [0.0857,0.1071])	([0.6071,0.7714], [0.1714,0.2857], [0.0571,0.1071])	([0.7813,0.8571], [0.1143,0.1563], [0.0286,0.0714])	([0.6786,0.7143], [0.2000,0.2500], [0.0625,0.1071])	([0.6429,0.7429], [0.2000,0.2813], [0.0571,0.1071])
	$d_3$	([0.7297,0.8000], [0.1333,0.1892], [0.0500,0.0811])	([0.6486,0.7250], [0.2000,0.2432], [0.0750,0.1081])	([0.6216,0.7500], [0.1750,0.2703], [0.0750,0.1081])	([0.7838,0.8500], [0.1000,0.1622], [0.0250,0.0667])	([0.6757,0.7000], [0.2000,0.2432], [0.0750,0.1000])	([0.6486,0.7250], [0.2000,0.2703], [0.0750,0.1000])
	$d_4$	([0.7600,0.7778], [0.1111,0.1600], [0.0800,0.1111])	([0.6364,0.6667], [0.2222,0.2400], [0.1111,0.1364])	([0.5909,0.6800], [0.2000,0.2778], [0.1111,0.1364])	([0.7727,0.8000], [0.1111,0.1364], [0.0800,0.1111])	([0.6364,0.6800], [0.2000,0.2273], [0.1111,0.1364])	([0.5909,0.6400], [0.2400,0.2778], [0.1111,0.1364])
$E_2$	$d_1$	([0.6842,0.7143], [0.1905,0.2105], [0.0857,0.1053])	([0.6053,0.6667], [0.2381,0.2632], [0.0952,0.1316])	([0.5789,0.6190], [0.2857,0.3158], [0.0952,0.1143])	([0.7632,0.7857], [0.1429,0.1579], [0.0714,0.0857])	([0.6571,0.6905], [0.2143,0.2368], [0.0952,0.1143])	([0.6053,0.6429], [0.2571,0.2895], [0.0952,0.1143])
	$d_2$	([0.7857,0.8125], [0.1200,0.1429], [0.0625,0.0800])	([0.6786,0.7500], [0.1563,0.2143], [0.0938,0.1200])	([0.6400,0.7813], [0.1563,0.2500], [0.0625,0.1200])	([0.8000,0.8750], [0.0938,0.1200], [0.0313,0.0800])	([0.6800,0.7188], [0.1786,0.2000], [0.0938,0.1200])	([0.6400,0.7500], [0.1875,0.2400], [0.0625,0.1200])
	$d_3$	([0.7879,0.8205], [0.1111,0.1515], [0.0513,0.0741])	([0.697,0.7692], [0.1538,0.2121], [0.0769,0.1111])	([0.6667,0.7949], [0.1538,0.2424], [0.0513,0.1111])	([0.7778,0.8718], [0.1026,0.1515], [0.0256,0.0741])	([0.697,0.7436], [0.1795,0.2121], [0.0769,0.1111])	([0.6667,0.7692], [0.1795,0.2424], [0.0513,0.1111])
	$d_4$	([0.7778,0.8182], [0.0667,0.1111], [0.0909,0.1333])	([0.6667,0.6818], [0.1818,0.2222], [0.1111,0.1364])	([0.6000,0.7273], [0.1818,0.2778], [0.0909,0.1333])	([0.7778,0.8636], [0.0667,0.1111], [0.0455,0.1333])	([0.6667,0.7273], [0.1818,0.2222], [0.0909,0.1333])	([0.6000,0.6818], [0.2273,0.2778], [0.0909,0.1333])
$E_3$	$d_1$	([0.7073,0.7391], [0.1739,0.1951], [0.0811,0.0976])	([0.6341,0.6957], [0.2174,0.2439], [0.0870,0.1220])	([0.6098,0.6522], [0.2609,0.2927], [0.0870,0.1081])	([0.7805,0.8043], [0.1304,0.1463], [0.0652,0.0811])	([0.6757,0.7174], [0.1957,0.2195], [0.0807,0.1081])	([0.6341,0.6739], [0.2391,0.2683], [0.0870,0.1081])
	$d_2$	([0.7931,0.8235], [0.1176,0.1379], [0.0588,0.0800])	([0.6800,0.7647], [0.1471,0.2069], [0.0882,0.1200])	([0.6400,0.7941], [0.1471,0.2414], [0.0588,0.1200])	([0.800,0.8529], [0.0882,0.1200], [0.0588,0.0800])	([0.6800,0.7353], [0.1724,0.2000], [0.0882,0.1200])	([0.6400,0.7647], [0.1765,0.2400], [0.0588,0.1200])
	$d_3$	([0.8056,0.8333], [0.1190,0.1389], [0.0476,0.0645])	([0.7097,0.7857], [0.1429,0.1944], [0.0714,0.0968])	([0.6774,0.8095], [0.1429,0.2258], [0.0476,0.0968])	([0.8065,0.8810], [0.0952,0.1290], [0.0238,0.0645])	([0.7097,0.7619], [0.1667,0.1944], [0.0556,0.0968])	([0.6774,0.7857], [0.1667,0.2258], [0.0476,0.0968])
	$d_4$	([0.8000,0.8333], [0.0625,0.1000], [0.0833,0.1250])	([0.6500,0.7083], [0.1667,0.2000], [0.1250,0.1500])	([0.600,0.7500], [0.1667,0.2500], [0.0833,0.1500])	([0.7500,0.8750], [0.0625,0.1500], [0.0417,0.1250])	([0.6500,0.7500], [0.1667,0.2000], [0.0833,0.1500])	([0.600,0.7083], [0.2083,0.2500], [0.0833,0.1500])
$E_4$	$d_1$	([0.6842,0.7097], [0.1935,0.2105], [0.0882,0.1053])	([0.5806,0.6053], [0.2581,0.2647], [0.1316,0.1613])	([0.5161,0.5789], [0.2895,0.3235], [0.1316,0.1613])	([0.6765,0.7368], [0.1842,0.2353], [0.0789,0.0968])	([0.5484,0.6316], [0.2632,0.2941], [0.1053,0.1613])	([0.5161,0.6053], [0.2895,0.3235], [0.1053,0.1613])
	$d_2$	([0.7000,0.7500], [0.1667,0.2000], [0.0833,0.1176])	([0.5294,0.6250], [0.2500,0.3000], [0.1250,0.1765])	([0.4118,0.5833], [0.2917,0.4118], [0.1250,0.1765])	([0.5882,0.7500], [0.1667,0.2941], [0.0833,0.1500])	([0.4706,0.6250], [0.2500,0.3529], [0.1250,0.1765])	([0.4118,0.5833], [0.2917,0.4118], [0.1250,0.1765])
	$d_3$	([0.7273,0.7333], [0.1818,0.2000], [0.0667,0.0909])	([0.5455,0.6333], [0.2667,0.3182], [0.1000,0.1364])	([0.4545,0.6000], [0.3000,0.4091], [0.1000,0.1364])	([0.6364,0.7333], [0.2000,0.2727], [0.0667,0.0909])	([0.5455,0.6333], [0.2667,0.3182], [0.1000,0.1364])	([0.4545,0.6000], [0.3000,0.4091], [0.1000,0.1364])
	$d_4$	([0.6923,0.7500], [0.1250,0.2000], [0.1000,0.1538])	([0.4000,0.5625], [0.3125,0.4000], [0.1250,0.2000])	([0.3000,0.5000], [0.3750,0.5000], [0.1250,0.2000])	([0.5000,0.6875], [0.1875,0.3077], [0.1250,0.2000])	([0.4000,0.5625], [0.3125,0.4000], [0.1250,0.2000])	([0.3000,0.5000], [0.3750,0.5000], [0.1250,0.2000])
$E_5$	$d_1$	([0.7000,0.7500], [0.1875,0.2000], [0.0625,0.1000])	([0.6250,0.6389], [0.2500,0.2500], [0.1111,0.1250])	([0.5625,0.6000], [0.2750,0.3125], [0.1111,0.1250])	([0.7188,0.7500], [0.1750,0.2188], [0.0556,0.0750])	([0.6250,0.6500], [0.2500,0.2500], [0.1000,0.1250])	([0.5625,0.6250], [0.2750,0.3125], [0.1000,0.1250])
	$d_2$	([0.7368,0.7407], [0.1579,0.1852], [0.0741,0.1053])	([0.5652,0.5926], [0.2963,0.3158], [0.1053,0.1304])	([0.4737,0.5556], [0.3333,0.4211], [0.1053,0.1304])	([0.6316,0.7037], [0.2222,0.2632], [0.0741,0.1053])	([0.5263,0.5926], [0.2963,0.3684], [0.1053,0.1304])	([0.4737,0.5556], [0.3333,0.4211], [0.1053,0.1304])
	$d_3$	([0.7308,0.7429], [0.1538,0.1714], [0.0857,0.1154])	([0.5385,0.6286], [0.2571,0.3077], [0.1143,0.1538])	([0.4615,0.6000], [0.2857,0.3846], [0.1143,0.1538])	([0.6538,0.7429], [0.2000,0.2692], [0.0571,0.0769])	([0.5385,0.6286], [0.2571,0.3077], [0.1143,0.1538])	([0.4615,0.6000], [0.2857,0.3846], [0.1143,0.1538])
	$d_4$	([0.7059,0.7619], [0.1429,0.2143], [0.0714,0.1176])	([0.5000,0.5714], [0.2857,0.3571], [0.1176,0.1429])	([0.4118,0.5238], [0.3333,0.4286], [0.1429,0.1765])	([0.5714,0.6667], [0.2381,0.2941], [0.0952,0.1429])	([0.4706,0.5714], [0.2857,0.4118], [0.1176,0.1429])	([0.4118,0.5238], [0.3333,0.4286], [0.1429,0.1765])

.

By Step 3, the intuitionistic fuzzy MAGDM matrices are shown in

Table 6. The original fuzzy MAGDM matrices.

$\mathit{E}$	$\mathit{D}$	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$	${\mathit{c}}_4$	${\mathit{c}}_5$	${\mathit{c}}_6$
$E_1$	$d_1$	(⟨0.6667,0.2889⟩, ⟨0.2000,0.7619⟩, ⟨0.0889,0.8947⟩)	(⟨0.5952,0.3333⟩, ⟨0.2444,0.7143⟩, ⟨0.0889,0.8684⟩)	(⟨0.5714,0.3778⟩, ⟨0.2889,0.6667⟩, ⟨0.0889,0.8947⟩)	(⟨0.7381,0.2000⟩, ⟨0.1333,0.8095⟩, ⟨0.0667,0.9211⟩)	(⟨0.6429,0.3111⟩, ⟨0.2222,0.7381⟩, ⟨0.0889,0.8947⟩)	(⟨0.6190,0.3556⟩, ⟨0.2632,0.6905⟩, ⟨0.0714,0.8947⟩)
	$d_2$	(⟨0.7143,0.2000⟩, ⟨0.1429,0.8125⟩, ⟨0.0571,0.8929⟩)	(⟨0.6429,0.2571⟩, ⟨0.1714,0.7500⟩, ⟨0.0857,0.8929⟩)	(⟨0.6071,0.2286⟩, ⟨0.1714,0.7143⟩, ⟨0.0571,0.8929⟩)	(⟨0.7813,0.1429⟩, ⟨0.1143,0.8438⟩, ⟨0.0286,0.9286⟩)	(⟨0.6786,0.2857⟩, ⟨0.2000,0.7500⟩, ⟨0.0625,0.8929⟩)	(⟨0.6429,0.2571⟩, ⟨0.2000,0.7188⟩, ⟨0.0571,0.8929⟩)
	$d_3$	(⟨0.7297,0.2000⟩, ⟨0.1333,0.8108⟩, ⟨0.0500,0.9189⟩)	(⟨0.6486,0.2750⟩, ⟨0.2000,0.7568⟩, ⟨0.0750,0.8919⟩)	(⟨0.6216,0.2500⟩, ⟨0.1750,0.7297⟩, ⟨0.0750,0.8919⟩)	(⟨0.7838,0.1500⟩, ⟨0.1000,0.8378⟩, ⟨0.0250,0.9333⟩)	(⟨0.6757,0.3000⟩, ⟨0.2000,0.7568⟩, ⟨0.0750,0.9000⟩)	(⟨0.6486,0.2750⟩, ⟨0.2000,0.7297⟩, ⟨0.0750,0.9000⟩)
	$d_4$	(⟨0.7600,0.2222⟩, ⟨0.1111,0.8400⟩, ⟨0.0800,0.8889⟩)	(⟨0.6364,0.3333⟩, ⟨0.2222,0.7600⟩, ⟨0.1111,0.8636⟩)	(⟨0.5909,0.3200⟩, ⟨0.2000,0.7222⟩, ⟨0.1111,0.8636⟩)	(⟨0.7727,0.2000⟩, ⟨0.1111,0.8636⟩, ⟨0.0800,0.8889⟩)	(⟨0.6364,0.3200⟩, ⟨0.2000,0.7727⟩, ⟨0.1111,0.8636⟩)	(⟨0.5909,0.3600⟩, ⟨0.2400,0.7222⟩, ⟨0.1111,0.8636⟩)
$E_2$	$d_1$	(⟨0.6842,0.2857⟩, ⟨0.1905,0.7895⟩, ⟨0.0857,0.8947⟩)	(⟨0.6053,0.3333⟩, ⟨0.2381,0.7368⟩, ⟨0.0952,0.8684⟩)	(⟨0.5789,0.381⟩, ⟨0.2857,0.6842⟩, ⟨0.0952,0.8857⟩)	(⟨0.7632,0.2143⟩, ⟨0.1429,0.8421⟩, ⟨0.0714,0.9143⟩)	(⟨0.6571,0.3095⟩, ⟨0.2143,0.7632⟩, ⟨0.0952,0.8857⟩)	(⟨0.6053,0.3571⟩, ⟨0.2571,0.7105⟩, ⟨0.0952,0.8857⟩)
	$d_2$	(⟨0.7857,0.1875⟩, ⟨0.1200,0.8571⟩, ⟨0.0625,0.9200⟩)	(⟨0.6786,0.2500⟩, ⟨0.1563,0.7857⟩, ⟨0.0938,0.8800⟩)	(⟨0.6400,0.2188⟩, ⟨0.1563,0.7500⟩, ⟨0.0625,0.8800⟩)	(⟨0.800,0.1250⟩, ⟨0.0938,0.8800⟩, ⟨0.0313,0.9200⟩)	(⟨0.6800,0.2813⟩, ⟨0.1786,0.8000⟩, ⟨0.0938,0.8800⟩)	(⟨0.6400,0.2500⟩, ⟨0.1875,0.7600⟩, ⟨0.0625,0.8800⟩)
	$d_3$	(⟨0.7879,0.1795⟩, ⟨0.1111,0.8485⟩, ⟨0.0513,0.9259⟩)	(⟨0.697,0.2308⟩, ⟨0.1538,0.7879⟩, ⟨0.0769,0.8889⟩)	(⟨0.6667,0.2051⟩, ⟨0.1538,0.7576⟩, ⟨0.0513,0.8889⟩)	(⟨0.7778,0.1282⟩, ⟨0.1026,0.8485⟩, ⟨0.0256,0.9259⟩)	(⟨0.6970,0.2564⟩, ⟨0.1795,0.7879⟩, ⟨0.0769,0.8889⟩)	(⟨0.6667,0.2308⟩, ⟨0.1795,0.7576⟩, ⟨0.0513,0.8889⟩)
	$d_4$	(⟨0.7778,0.1818⟩, ⟨0.0667,0.8889⟩, ⟨0.0909,0.8667⟩)	(⟨0.6667,0.3182⟩, ⟨0.1818,0.7778⟩, ⟨0.1111,0.8636⟩)	(⟨0.6000,0.2727⟩, ⟨0.1818,0.7222⟩, ⟨0.0909,0.8667⟩)	(⟨0.7778,0.1364⟩, ⟨0.0667,0.8889⟩, ⟨0.0455,0.8667⟩)	(⟨0.6667,0.2727⟩, ⟨0.1818,0.7778⟩, ⟨0.0909,0.8667⟩)	(⟨0.6000,0.3182⟩, ⟨0.2273,0.7222⟩, ⟨0.0909,0.8667⟩)
$E_3$	$d_1$	(⟨0.7073,0.2609⟩, ⟨0.1739,0.8049⟩, ⟨0.0811,0.9024⟩)	(⟨0.6341,0.3043⟩, ⟨0.2174,0.7561⟩, ⟨0.0870,0.8780⟩)	(⟨0.6098,0.3478⟩, ⟨0.2609,0.7073⟩, ⟨0.0870,0.8919⟩)	(⟨0.7805,0.1957⟩, ⟨0.1304,0.8537⟩, ⟨0.0652,0.9189⟩)	(⟨0.6757,0.2826⟩, ⟨0.1957,0.7805⟩, ⟨0.0870,0.8919⟩)	(⟨0.6341,0.3261⟩, ⟨0.2391,0.7317⟩, ⟨0.0870,0.8919⟩)
	$d_2$	(⟨0.7931,0.1765⟩, ⟨0.1176,0.8621⟩, ⟨0.0588,0.9200⟩)	(⟨0.6800,0.2353⟩, ⟨0.1471,0.7931⟩, ⟨0.0882,0.8800⟩)	(⟨0.6400,0.2059⟩, ⟨0.1471,0.7586⟩, ⟨0.0588,0.8800⟩)	(⟨0.800,0.1471⟩, ⟨0.0882,0.8800⟩, ⟨0.0588,0.9200⟩)	(⟨0.6800,0.2647⟩, ⟨0.1724,0.8000⟩, ⟨0.0882,0.8800⟩)	(⟨0.6400,0.2353⟩, ⟨0.1765,0.7600⟩, ⟨0.0588,0.8800⟩)
	$d_3$	(⟨0.8056,0.1667⟩, ⟨0.1190,0.8611⟩, ⟨0.0476,0.9355⟩)	(⟨0.7097,0.2143⟩, ⟨0.1429,0.8056⟩, ⟨0.0714,0.9032⟩)	(⟨0.6774,0.1905⟩, ⟨0.1429,0.7742⟩, ⟨0.0476,0.9032⟩)	(⟨0.8065,0.119⟩, ⟨0.0952,0.8710⟩, ⟨0.0238,0.9355⟩)	(⟨0.7097,0.2381⟩, ⟨0.1667,0.8056⟩, ⟨0.0556,0.9032⟩)	(⟨0.6774,0.2143⟩, ⟨0.1667,0.7742⟩, ⟨0.0476,0.9032⟩)
	$d_4$	(⟨0.8000,0.1667⟩, ⟨0.0625,0.9000⟩, ⟨0.0833,0.8750⟩)	(⟨0.6500,0.2917⟩, ⟨0.1667,0.8000⟩, ⟨0.1250,0.8500⟩)	(⟨0.6000,0.2500⟩, ⟨0.1667,0.7500⟩, ⟨0.0833,0.8500⟩)	(⟨0.7500,0.1250⟩, ⟨0.0625,0.8500⟩, ⟨0.0417,0.8750⟩)	(⟨0.6500,0.2500⟩, ⟨0.1667,0.8000⟩, ⟨0.0833,0.8500⟩)	(⟨0.6000,0.2917⟩, ⟨0.2083,0.7500⟩, ⟨0.0833,0.8500⟩)
$E_4$	$d_1$	(⟨0.6842,0.2903⟩, ⟨0.1935,0.7895⟩, ⟨0.0882,0.8947⟩)	(⟨0.5806,0.3947⟩, ⟨0.2581,0.7353⟩, ⟨0.1316,0.8387⟩)	(⟨0.5161,0.4211⟩, ⟨0.2895,0.6765⟩, ⟨0.1316,0.8387⟩)	(⟨0.6765,0.2632⟩, ⟨0.1842,0.7647⟩, ⟨0.0789,0.9032⟩)	(⟨0.5484,0.3684⟩, ⟨0.2632,0.7059⟩, ⟨0.1053,0.8387⟩)	(⟨0.5161,0.3947⟩, ⟨0.2895,0.6765⟩, ⟨0.1053,0.8387⟩)
	$d_2$	(⟨0.7000,0.2500⟩, ⟨0.1667,0.8000⟩, ⟨0.0833,0.8824⟩)	(⟨0.5294,0.3750⟩, ⟨0.2500,0.7000⟩, ⟨0.1250,0.8235⟩)	(⟨0.4118,0.4167⟩, ⟨0.2917,0.5882⟩, ⟨0.1250,0.8235⟩)	(⟨0.5882,0.2500⟩, ⟨0.1667,0.7059⟩, ⟨0.0833,0.8500⟩)	(⟨0.4706,0.3750⟩, ⟨0.2500,0.6471⟩, ⟨0.1250,0.8235⟩)	(⟨0.4118,0.4167⟩, ⟨0.2917,0.5882⟩, ⟨0.1250,0.8235⟩)
	$d_3$	(⟨0.7273,0.2667⟩, ⟨0.1818,0.8000⟩, ⟨0.0667,0.9091⟩)	(⟨0.5455,0.3667⟩, ⟨0.2667,0.6818⟩, ⟨0.1000,0.8636⟩)	(⟨0.4545,0.4000⟩, ⟨0.3000,0.5909⟩, ⟨0.1000,0.8636⟩)	(⟨0.6364,0.2667⟩, ⟨0.2000,0.7273⟩, ⟨0.0667,0.9091⟩)	(⟨0.5455,0.3667⟩, ⟨0.2667,0.6818⟩, ⟨0.1000,0.8636⟩)	(⟨0.4545,0.4000⟩, ⟨0.3000,0.5909⟩, ⟨0.1000,0.8636⟩)
	$d_4$	(⟨0.6923,0.2500⟩, ⟨0.1250,0.8000⟩, ⟨0.1000,0.8462⟩)	(⟨0.4000,0.4375⟩, ⟨0.3125,0.6000⟩, ⟨0.1250,0.8000⟩)	(⟨0.3000,0.5000⟩, ⟨0.3750,0.5000⟩, ⟨0.1250,0.8000⟩)	(⟨0.5000,0.3125⟩, ⟨0.1875,0.6923⟩, ⟨0.1250,0.8000⟩)	(⟨0.4000,0.4375⟩, ⟨0.3125,0.6000⟩, ⟨0.1250,0.8000⟩)	(⟨0.3000,0.5000⟩, ⟨0.3750,0.5000⟩, ⟨0.1250,0.8000⟩)
$E_5$	$d_1$	(⟨0.7000,0.2500⟩, ⟨0.1875,0.8000⟩, ⟨0.0625,0.9000⟩)	(⟨0.6250,0.3611⟩, ⟨0.2500,0.7500⟩, ⟨0.1111,0.8750⟩)	(⟨0.5625,0.4000⟩, ⟨0.2750,0.6875⟩, ⟨0.1111,0.875⟩)	(⟨0.7188,0.2500⟩, ⟨0.1750,0.7813⟩, ⟨0.0556,0.9250⟩)	(⟨0.6250,0.3500⟩, ⟨0.2500,0.7500⟩, ⟨0.1000,0.8750⟩)	(⟨0.5625,0.3750⟩, ⟨0.2750,0.6875⟩, ⟨0.1000,0.8750⟩)
	$d_2$	(⟨0.7368,0.2593⟩, ⟨0.1579,0.8148⟩, ⟨0.0741,0.8947⟩)	(⟨0.5652,0.4074⟩, ⟨0.2963,0.6842⟩, ⟨0.1053,0.8696⟩)	(⟨0.4737,0.4444⟩, ⟨0.3333,0.5789⟩, ⟨0.1053,0.8696⟩)	(⟨0.6316,0.2963⟩, ⟨0.2222,0.7368⟩, ⟨0.0741,0.8947⟩)	(⟨0.5263,0.4074⟩, ⟨0.2963,0.6316⟩, ⟨0.1053,0.8696⟩)	(⟨0.4737,0.4444⟩, ⟨0.3333,0.5789⟩, ⟨0.1053,0.8696⟩)
	$d_3$	(⟨0.7308,0.2571⟩, ⟨0.1538,0.8286⟩, ⟨0.0857,0.8846⟩)	(⟨0.5385,0.3714⟩, ⟨0.2571,0.6923⟩, ⟨0.1143,0.8462⟩)	(⟨0.4615,0.4000⟩, ⟨0.2857,0.6154⟩, ⟨0.1143,0.8462⟩)	(⟨0.6538,0.2571⟩, ⟨0.2000,0.7308⟩, ⟨0.0571,0.9231⟩)	(⟨0.5385,0.3714⟩, ⟨0.2571,0.6923⟩, ⟨0.1143,0.8462⟩)	(⟨0.4615,0.4000⟩, ⟨0.2857,0.6154⟩, ⟨0.1143,0.8462⟩)
	$d_4$	(⟨0.7059,0.2381⟩, ⟨0.1429,0.7857⟩, ⟨0.0714,0.8824⟩)	(⟨0.5000,0.4286⟩, ⟨0.2857,0.6429⟩, ⟨0.1176,0.8571⟩)	(⟨0.4118,0.4762⟩, ⟨0.3333,0.5714⟩, ⟨0.1429,0.8235⟩)	(⟨0.5714,0.3333⟩, ⟨0.2381,0.7059⟩, ⟨0.0952,0.8571⟩)	(⟨0.4706,0.4286⟩, ⟨0.2857,0.5882⟩, ⟨0.1176,0.8571⟩)	(⟨0.4118,0.4762⟩, ⟨0.3333,0.5714⟩, ⟨0.1429,0.8235⟩)

.

By Step 4, the weights $\mathit{w}=\left(\mathrm{0.1541,0.1715,0.1749,0.1527,0.1712,0.1755}\right)$, indicating that the fuel system and lubrication system are more susceptible to failure compared with other systems. In Steps 5 and 6, when the IFNs of the three-level fuzzy linguistic values $S_1$, $S_2$, and $S_3$ are selected as ⟨0.8,0.2⟩, ⟨0.5,0.5⟩, and ⟨0.2,0.8⟩, respectively, the weighted three-level fuzzy MAGDM matrices are presented in

Table 7. The weighted three-level fuzzy MAGDM matrices.

$\mathit{E}$	$\mathit{D}$	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$	${\mathit{c}}_4$	${\mathit{c}}_5$	${\mathit{c}}_6$
$E_1$	$d_1$	(⟨0.1246,0.8607⟩, ⟨0.0169,0.9795⟩, ⟨0.0028,0.9966⟩)	(⟨0.1150,0.8626⟩, ⟨0.0235,0.9720⟩, ⟨0.0032,0.9952⟩)	(⟨0.1102,0.8747⟩, ⟨0.0289,0.9658⟩, ⟨0.0032,0.9961⟩)	(⟨0.1480,0.8256⟩, ⟨0.0108,0.9841⟩, ⟨0.0021,0.9975⟩)	(⟨0.1293,0.8550⟩, ⟨0.0211,0.9747⟩, ⟨0.0032,0.9962⟩)	(⟨0.1247,0.8672⟩, ⟨0.0261,0.9685⟩, ⟨0.0026,0.9961⟩)
	$d_2$	(⟨0.1404,0.8243⟩, ⟨0.0117,0.9843⟩, ⟨0.0018,0.9965⟩)	(⟨0.1295,0.8338⟩, ⟨0.0159,0.9759⟩, ⟨0.0031,0.9962⟩)	(⟨0.1206,0.8180⟩, ⟨0.0162,0.9714⟩, ⟨0.0020,0.9961⟩)	(⟨0.1657,0.7943⟩, ⟨0.0092,0.9872⟩, ⟨0.0009,0.9977⟩)	(⟨0.1413,0.8456⟩, ⟨0.0187,0.9760⟩, ⟨0.0022,0.9962⟩)	(⟨0.1323,0.8303⟩, ⟨0.0192,0.9718⟩, ⟨0.0021,0.9961⟩)
	$d_3$	(⟨0.1460,0.8243⟩, ⟨0.0109,0.9841⟩, ⟨0.0016,0.9974⟩)	(⟨0.1314,0.8411⟩, ⟨0.0188,0.9767⟩, ⟨0.0027,0.9961⟩)	(⟨0.1251,0.8277⟩, ⟨0.0165,0.9732⟩, ⟨0.0027,0.9960⟩)	(⟨0.1669,0.7987⟩, ⟨0.0080,0.9867⟩, ⟨0.0008,0.9979⟩)	(⟨0.1403,0.8510⟩, ⟨0.0187,0.9767⟩, ⟨0.0027,0.9964⟩)	(⟨0.1342,0.8378⟩, ⟨0.0192,0.9731⟩, ⟨0.0027,0.9963⟩)
	$d_4$	(⟨0.1579,0.8345⟩, ⟨0.0090,0.9867⟩, ⟨0.0026,0.9964⟩)	(⟨0.1274,0.8626⟩, ⟨0.0211,0.9770⟩, ⟨0.0040,0.9950⟩)	(⟨0.1158,0.8554⟩, ⟨0.0191,0.9723⟩, ⟨0.0041,0.9949⟩)	(⟨0.1620,0.8256⟩, ⟨0.0089,0.9889⟩, ⟨0.0025,0.9964⟩)	(⟨0.1272,0.8582⟩, ⟨0.0187,0.9784⟩, ⟨0.0040,0.9950⟩)	(⟨0.1162,0.8687⟩, ⟨0.0235,0.9722⟩, ⟨0.0041,0.9949⟩)
$E_2$	$d_1$	(⟨0.1302,0.8596⟩, ⟨0.0160,0.9821⟩, ⟨0.0027,0.9966⟩)	(⟨0.1179,0.8626⟩, ⟨0.0228,0.9745⟩, ⟨0.0034,0.9952⟩)	(⟨0.1123,0.8757⟩, ⟨0.0286,0.9679⟩, ⟨0.0035,0.9958⟩)	(⟨0.1580,0.8323⟩, ⟨0.0116,0.9870⟩, ⟨0.0023,0.9973⟩)	(⟨0.1340,0.8545⟩, ⟨0.0202,0.9774⟩, ⟨0.0034,0.9959⟩)	(⟨0.1204,0.8677⟩, ⟨0.0254,0.9709⟩, ⟨0.0035,0.9958⟩)
	$d_2$	(⟨0.1690,0.8181⟩, ⟨0.0098,0.9883⟩, ⟨0.0020,0.9974⟩)	(⟨0.1415,0.8307⟩, ⟨0.0144,0.9797⟩, ⟨0.0033,0.9957⟩)	(⟨0.1309,0.8132⟩, ⟨0.0146,0.9755⟩, ⟨0.0022,0.9956⟩)	(⟨0.1744,0.7823⟩, ⟨0.0075,0.9903⟩, ⟨0.0010,0.9975⟩)	(⟨0.1418,0.8438⟩, ⟨0.0166,0.9813⟩, ⟨0.0033,0.9957⟩)	(⟨0.1313,0.8272⟩, ⟨0.0179,0.9765⟩, ⟨0.0023,0.9956⟩)
	$d_3$	(⟨0.1700,0.8140⟩, ⟨0.0090,0.9875⟩, ⟨0.0016,0.9976⟩)	(⟨0.1481,0.8221⟩, ⟨0.0141,0.9800⟩, ⟨0.0027,0.9960⟩)	(⟨0.1399,0.8064⟩, ⟨0.0144,0.9763⟩, ⟨0.0018,0.9959⟩)	(⟨0.1642,0.7846⟩, ⟨0.0082,0.9876⟩, ⟨0.0008,0.9977⟩)	(⟨0.1479,0.8337⟩, ⟨0.0167,0.9800⟩, ⟨0.0027,0.9960⟩)	(⟨0.1403,0.8185⟩, ⟨0.0171,0.9762⟩, ⟨0.0018,0.9959⟩)
	$d_4$	(⟨0.1655,0.8152⟩, ⟨0.0053,0.9910⟩, ⟨0.0029,0.9956⟩)	(⟨0.1374,0.8573⟩, ⟨0.0169,0.9789⟩, ⟨0.0040,0.9950⟩)	(⟨0.1185,0.8373⟩, ⟨0.0172,0.9723⟩, ⟨0.0033,0.9951⟩)	(⟨0.1642,0.7901⟩, ⟨0.0052,0.9911⟩, ⟨0.0014,0.9957⟩)	(⟨0.1372,0.8404⟩, ⟨0.0169,0.9789⟩, ⟨0.0032,0.9952⟩)	(⟨0.1188,0.8543⟩, ⟨0.0221,0.9722⟩, ⟨0.0033,0.9950⟩)
$E_3$	$d_1$	(⟨0.1380,0.8504⟩, ⟨0.0145,0.9836⟩, ⟨0.0026,0.9969⟩)	(⟨0.1267,0.8524⟩, ⟨0.0206,0.9766⟩, ⟨0.0031,0.9956⟩)	(⟨0.1214,0.865⟩, ⟨0.0258,0.9706⟩, ⟨0.0032,0.9960⟩)	(⟨0.1654,0.8235⟩, ⟨0.0106,0.9881⟩, ⟨0.0020,0.9974⟩)	(⟨0.1403,0.8444⟩, ⟨0.0183,0.9792⟩, ⟨0.0031,0.9961⟩)	(⟨0.1294,0.8572⟩, ⟨0.0234,0.9733⟩, ⟨0.0032,0.9960⟩)
	$d_2$	(⟨0.1724,0.8124⟩, ⟨0.0095,0.9887⟩, ⟨0.0019,0.9974⟩)	(⟨0.1420,0.8242⟩, ⟨0.0135,0.9805⟩, ⟨0.0031,0.9957⟩)	(⟨0.1309,0.8068⟩, ⟨0.0137,0.9764⟩, ⟨0.0021,0.9956⟩)	(⟨0.1744,0.7969⟩, ⟨0.0070,0.9903⟩, ⟨0.0018,0.9975⟩)	(⟨0.1418,0.8372⟩, ⟨0.0159,0.9813⟩, ⟨0.0031,0.9957⟩)	(⟨0.1313,0.8206⟩, ⟨0.0168,0.9765⟩, ⟨0.0021,0.9956⟩)
	$d_3$	(⟨0.1784,0.8070⟩, ⟨0.0097,0.9886⟩, ⟨0.0015,0.9980⟩)	(⟨0.1529,0.8143⟩, ⟨0.0130,0.9818⟩, ⟨0.0025,0.9965⟩)	(⟨0.1437,0.7986⟩, ⟨0.0133,0.9781⟩, ⟨0.0017,0.9965⟩)	(⟨0.1775,0.7780⟩, ⟨0.0076,0.9896⟩, ⟨0.0007,0.9980⟩)	(⟨0.1527,0.8257⟩, ⟨0.0154,0.9818⟩, ⟨0.0019,0.9965⟩)	(⟨0.1441,0.8105⟩, ⟨0.0157,0.9780⟩, ⟨0.0017,0.9965⟩)
	$d_4$	(⟨0.1757,0.8070⟩, ⟨0.0049,0.9919⟩, ⟨0.0027,0.9959⟩)	(⟨0.1318,0.8476⟩, ⟨0.0154,0.9812⟩, ⟨0.0045,0.9945⟩)	(⟨0.1185,0.8277⟩, ⟨0.0157,0.9755⟩, ⟨0.0030,0.9944⟩)	(⟨0.1527,0.7823⟩, ⟨0.0049,0.9877⟩, ⟨0.0013,0.9960⟩)	(⟨0.1316,0.8310⟩, ⟨0.0154,0.9813⟩, ⟨0.0030,0.9945⟩)	(⟨0.1188,0.8444⟩, ⟨0.0201,0.9754⟩, ⟨0.0030,0.9944⟩)
$E_4$	$d_1$	(⟨0.1302,0.8612⟩, ⟨0.0163,0.9821⟩, ⟨0.0028,0.9966⟩)	(⟨0.1108,0.8821⟩, ⟨0.0250,0.9743⟩, ⟨0.0048,0.9941⟩)	(⟨0.0954,0.8877⟩, ⟨0.0290,0.9670⟩, ⟨0.0049,0.9939⟩)	(⟨0.1267,0.8524⟩, ⟨0.0153,0.9799⟩, ⟨0.0025,0.9969⟩)	(⟨0.1018,0.8743⟩, ⟨0.0255,0.9711⟩, ⟨0.0038,0.9941⟩)	(⟨0.0957,0.8796⟩, ⟨0.0291,0.9668⟩, ⟨0.0039,0.9939⟩)
	$d_2$	(⟨0.1354,0.8462⟩, ⟨0.0138,0.9831⟩, ⟨0.0027,0.9962⟩)	(⟨0.0970,0.8761⟩, ⟨0.0241,0.9703⟩, ⟨0.0045,0.9934⟩)	(⟨0.0709,0.8864⟩, ⟨0.0293,0.9557⟩, ⟨0.0046,0.9933⟩)	(⟨0.1014,0.8473⟩, ⟨0.0137,0.9741⟩, ⟨0.0026,0.9951⟩)	(⟨0.0825,0.8763⟩, ⟨0.0240,0.9641⟩, ⟨0.0045,0.9935⟩)	(⟨0.0711,0.886⟩, ⟨0.0294,0.9555⟩, ⟨0.0046,0.9933⟩)
	$d_3$	(⟨0.1451,0.8526⟩, ⟨0.0152,0.9831⟩, ⟨0.0021,0.9971⟩)	(⟨0.1012,0.8735⟩, ⟨0.0259,0.9682⟩, ⟨0.0036,0.9950⟩)	(⟨0.0805,0.8815⟩, ⟨0.0302,0.9560⟩, ⟨0.0037,0.9949⟩)	(⟨0.1145,0.8537⟩, ⟨0.0168,0.9763⟩, ⟨0.0021,0.9971⟩)	(⟨0.1010,0.8737⟩, ⟨0.0259,0.9683⟩, ⟨0.0036,0.9950⟩)	(⟨0.0807,0.8812⟩, ⟨0.0303,0.9559⟩, ⟨0.0037,0.9949⟩)
	$d_4$	(⟨0.1328,0.8462⟩, ⟨0.0102,0.9831⟩, ⟨0.0032,0.9949⟩)	(⟨0.0671,0.8942⟩, ⟨0.0311,0.9581⟩, ⟨0.0045,0.9925⟩)	(⟨0.0484,0.9086⟩, ⟨0.0395,0.9429⟩, ⟨0.0046,0.9923⟩)	(⟨0.0804,0.8698⟩, ⟨0.0156,0.9727⟩, ⟨0.0040,0.9933⟩)	(⟨0.067,0.8944⟩, ⟨0.0311,0.9581⟩, ⟨0.0045,0.9925⟩)	(⟨0.0485,0.9084⟩, ⟨0.0396,0.9427⟩, ⟨0.0046,0.9923⟩)
$E_5$	$d_1$	(⟨0.1354,0.8462⟩, ⟨0.0157,0.9831⟩, ⟨0.0020,0.9968⟩)	(⟨0.1239,0.8718⟩, ⟨0.0241,0.9759⟩, ⟨0.0040,0.9955⟩)	(⟨0.1077,0.8815⟩, ⟨0.0274,0.9683⟩, ⟨0.0041,0.9954⟩)	(⟨0.1409,0.8473⟩, ⟨0.0145,0.9815⟩, ⟨0.0017,0.9976⟩)	(⟨0.1237,0.8684⟩, ⟨0.0240,0.9760⟩, ⟨0.0036,0.9955⟩)	(⟨0.1081,0.8735⟩, ⟨0.0274,0.9682⟩, ⟨0.0037,0.9954⟩)
	$d_2$	(⟨0.1487,0.8498⟩, ⟨0.0131,0.9845⟩, ⟨0.0024,0.9966⟩)	(⟨0.1065,0.8858⟩, ⟨0.0292,0.9685⟩, ⟨0.0038,0.9953⟩)	(⟨0.085,0.8942⟩, ⟨0.0342,0.9544⟩, ⟨0.0039,0.9952⟩)	(⟨0.1132,0.8644⟩, ⟨0.0188,0.9772⟩, ⟨0.0023,0.9966⟩)	(⟨0.0961,0.8860⟩, ⟨0.0292,0.9622⟩, ⟨0.0038,0.9953⟩)	(⟨0.0852,0.8939⟩, ⟨0.0343,0.9543⟩, ⟨0.0039,0.9952⟩)
	$d_3$	(⟨0.1464,0.849⟩, ⟨0.0127,0.9857⟩, ⟨0.0027,0.9963⟩)	(⟨0.0994,0.8750⟩, ⟨0.0249,0.9694⟩, ⟨0.0041,0.9944⟩)	(⟨0.0821,0.8815⟩, ⟨0.0286,0.9593⟩, ⟨0.0042,0.9942⟩)	(⟨0.1197,0.8501⟩, ⟨0.0168,0.9766⟩, ⟨0.0018,0.9976⟩)	(⟨0.0992,0.8752⟩, ⟨0.0248,0.9695⟩, ⟨0.0041,0.9944⟩)	(⟨0.0824,0.8812⟩, ⟨0.0287,0.9592⟩, ⟨0.0042,0.9942⟩)
	$d_4$	(⟨0.1375,0.8413⟩, ⟨0.0117,0.9818⟩, ⟨0.0023,0.9962⟩)	(⟨0.0897,0.8918⟩, ⟨0.0280,0.9635⟩, ⟨0.0042,0.9948⟩)	(⟨0.0709,0.9026⟩, ⟨0.0342,0.9534⟩, ⟨0.0053,0.9933⟩)	(⟨0.0971,0.8764⟩, ⟨0.0203,0.9741⟩, ⟨0.0030,0.9953⟩)	(⟨0.0825,0.8920⟩, ⟨0.0280,0.9566⟩, ⟨0.0042,0.9948⟩)	(⟨0.0711,0.9023⟩, ⟨0.0343,0.9532⟩, ⟨0.0053,0.9933⟩)

.

Through Steps 7 and 8, the integration information matrices, together with the PID and NID, are systematically presented in

Table 8. The integration information matrices, PID, and NID.

$\mathit{E}$	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$	${\mathit{c}}_4$	${\mathit{c}}_5$	${\mathit{c}}_6$
$E_1$	(⟨0.1423,0.8358⟩, ⟨0.0121,0.9836⟩, ⟨0.0022,0.9967⟩)	(⟨0.1258,0.8499⟩, ⟨0.0198,0.9754⟩, ⟨0.0032,0.9956⟩)	(⟨0.1180,0.8437⟩, ⟨0.0202,0.9707⟩, ⟨0.0030,0.9958⟩)	(⟨0.1607,0.8110⟩, ⟨0.0092,0.9867⟩, ⟨0.0016,0.9974⟩)	(⟨0.1345,0.8524⟩, ⟨0.0193,0.9764⟩, ⟨0.0030,0.9960⟩)	(⟨0.1268,0.8508⟩, ⟨0.0220,0.9714⟩, ⟨0.0029,0.9959⟩)
$E_2$	(⟨0.1588,0.8265⟩, ⟨0.0100,0.9872⟩, ⟨0.0023,0.9968⟩)	(⟨0.1363,0.843⟩, ⟨0.0171,0.9783⟩, ⟨0.0034,0.9955⟩)	(⟨0.1255,0.8327⟩, ⟨0.0187,0.9730⟩, ⟨0.0027,0.9956⟩)	(⟨0.1652,0.7971⟩, ⟨0.0081,0.9890⟩, ⟨0.0014,0.9970⟩)	(⟨0.1402,0.8431⟩, ⟨0.0176,0.9794⟩, ⟨0.0032,0.9957⟩)	(⟨0.1278,0.8417⟩, ⟨0.0206,0.9740⟩, ⟨0.0027,0.9956⟩)
$E_3$	(⟨0.1663,0.8190⟩, ⟨0.0097,0.9882⟩, ⟨0.0022,0.9970⟩)	(⟨0.1384,0.8344⟩, ⟨0.0156,0.9800⟩, ⟨0.0033,0.9956⟩)	(⟨0.1287,0.8241⟩, ⟨0.0171,0.9751⟩, ⟨0.0025,0.9956⟩)	(⟨0.1675,0.7950⟩, ⟨0.0075,0.9889⟩, ⟨0.0015,0.9972⟩)	(⟨0.1416,0.8345⟩, ⟨0.0162,0.9809⟩, ⟨0.0028,0.9957⟩)	(⟨0.1310,0.8329⟩, ⟨0.0190,0.9758⟩, ⟨0.0025,0.9956⟩)
$E_4$	(⟨0.1359,0.8515⟩, ⟨0.0139,0.9829⟩, ⟨0.0027,0.9962⟩)	(⟨0.0942,0.8815⟩, ⟨0.0265,0.9677⟩, ⟨0.0044,0.9938⟩)	(⟨0.0740,0.8910⟩, ⟨0.0320,0.9554⟩, ⟨0.0044,0.9936⟩)	(⟨0.1059,0.8558⟩, ⟨0.0154,0.9757⟩, ⟨0.0028,0.9956⟩)	(⟨0.0882,0.8797⟩, ⟨0.0266,0.9654⟩, ⟨0.0041,0.9938⟩)	(⟨0.0742,0.8887⟩, ⟨0.0321,0.9552⟩, ⟨0.0042,0.9936⟩)
$E_5$	(⟨0.1420,0.8465⟩, ⟨0.0133,0.9838⟩, ⟨0.0023,0.9965⟩)	(⟨0.1049,0.8811⟩, ⟨0.0266,0.9693⟩, ⟨0.0040,0.9950⟩)	(⟨0.0865,0.8899⟩, ⟨0.0311,0.9588⟩, ⟨0.0044,0.9945⟩)	(⟨0.1179,0.8595⟩, ⟨0.0176,0.9774⟩, ⟨0.0022,0.9968⟩)	(⟨0.1005,0.8803⟩, ⟨0.0265,0.9660⟩, ⟨0.0039,0.9950⟩)	(⟨0.0868,0.8876⟩, ⟨0.0312,0.9587⟩, ⟨0.0043,0.9945⟩)
PID	(⟨0.1663,0.8190⟩, ⟨0.0139,0.9829⟩, ⟨0.0027,0.9962⟩)	(⟨0.1384,0.8344⟩, ⟨0.0266,0.9677⟩, ⟨0.0044,0.9938⟩)	(⟨0.1287,0.8241⟩, ⟨0.0320,0.9554⟩, ⟨0.0044,0.9936⟩)	(⟨0.1675,0.7950⟩, ⟨0.0176,0.9757⟩, ⟨0.0028,0.9956⟩)	(⟨0.1416,0.8345⟩, ⟨0.0266,0.9654⟩, ⟨0.0041,0.9938⟩)	(⟨0.1310,0.8329⟩, ⟨0.0321,0.9552⟩, ⟨0.0043,0.9936⟩)
NID	(⟨0.1359,0.8515⟩, ⟨0.0097,0.9882⟩, ⟨0.0022,0.9970⟩)	(⟨0.0942,0.8815⟩, ⟨0.0156,0.9800⟩, ⟨0.0032,0.9956⟩)	(⟨0.0740,0.8910⟩, ⟨0.0171,0.9751⟩, ⟨0.0025,0.9958⟩)	(⟨0.1059,0.8595⟩, ⟨0.0075,0.9890⟩, ⟨0.0014,0.9974⟩)	(⟨0.0882,0.8803⟩, ⟨0.0162,0.9809⟩, ⟨0.0028,0.9960⟩)	(⟨0.0742,0.8887⟩, ⟨0.0190,0.9758⟩, ⟨0.0025,0.9959⟩)

.

4.3. Providing Evaluation Results

By Steps 9, 10, and 11, the closeness of each weighted fuzzy MAGDM matrix ${\mathit{Z}}_i$ to the ideal decisions are determined. Subsequently, the normalized priority coefficient ${NPC}_i$ for engine $E_i$ is computed. Finally, the ranking order of the engines’ reliability is calculated and presented in

Table 9. The reliability evaluation results of marine engines.

$\mathit{E}$	${\mathit{d}}_{\mathit{\epsilon }}\left({\mathit{Z}}_{\mathit{i}},{\mathit{Z}}_{+}\right)$		${\mathit{d}}_{\mathit{\epsilon }}\left({\mathit{Z}}_{\mathit{i}},{\mathit{Z}}_{-}\right)$		${\mathit{N}\mathit{P}\mathit{C}}_{\mathit{i}}$
	Value	Ranking	Value	Ranking	Value	Ranking
$E_1$	0.0108	3	0.0194	3	0.6431	3
$E_2$	0.0083	2	0.0219	2	0.7250	2
$E_3$	0.0072	1	0.0230	1	0.7619	1
$E_4$	0.0231	5	0.0071	5	0.2359	5
$E_5$	0.0195	4	0.0108	4	0.3556	4

.

Table 9 presents the reliability evaluation results of the five engines, demonstrating a ranking of $E_3\succ E_2\succ E_1\succ E_5\succ E_4$. This indicates that engine $E_3$ exhibits the highest reliability, followed by engine $E_2$, engine $E_1$, engine $E_5$, and engine $E_4,$ in descending order. Moreover, $d_{\epsilon }\left({\mathit{Z}}_i,{\mathit{Z}}_{+}\right)$ and $d_{\epsilon }\left({\mathit{Z}}_i,{\mathit{Z}}_{-}\right)$ are two critical metrics employed for the reliability assessment. Specifically, a smaller value of $d_{\epsilon }\left({\mathit{Z}}_i,{\mathit{Z}}_{+}\right)$ or a larger value of $d_{\epsilon }\left({\mathit{Z}}_i,{\mathit{Z}}_{-}\right)$ corresponds to a more favorable reliability index. Notably, the ranking derived from these two metrics, $E_3\succ E_2\succ E_1\succ E_5\succ E_4$, aligns precisely with the ranking result of ${NPC}_i$.

5. Findings and Discussion

To comprehensively compare the novel approach with the existing methods, this section presents a comprehensive analysis of reliability evaluation results for marine engines, derived from different processing techniques and various distance measures of the IFNs.

First, the superiority of the new method in weighted processing and the integration of state information is validated. The reliability evaluation results are listed in

Table 10. The reliability evaluation results of marine engines based on different methods.

Method	Ranking	Best Reliability	Worst Reliability	Difference
Weighted without state information	$E_3\succ E_2\succ E_1\succ E_5\succ E_4$	$E_3$	$E_4$	0.2650
Weighted after integration	$E_2\succ E_3\succ E_1\succ E_5\succ E_4$	$E_2$	$E_4$	0.1947
Weighted with state information	$E_3\succ E_2\succ E_1\succ E_5\succ E_4$	$E_3$	$E_4$	0.5259

.

The results demonstrated that the reliability ranking results derived from diverse methods are approximately similar. Specifically, engine $E_2$ and engine $E_3$ are identified as having high reliability, while engine $E_4$ has the worst reliability. However, when weighting is conducted after integration, the results diverge from those of the other two methods. When comparing weighting methods with and without state information, identical ranking results are derived. However, for weighting methods without state information, the differences in the ${NPC}_i$ values among the different engines are minimal, which is not favorable for identification.

The reliability of marine engines is meticulously evaluated by employing different distance measures for IFNs, as introduced in Section 2. The comprehensive evaluation results are presented in

Table 11. The reliability evaluation results of marine engines based on different measures.

Measure	Ranking	Best Reliability
$d_M$	$E_3\succ E_2\succ E_1\succ E_5\succ E_4$	$E_3$
$d_V$	$E_2\succ E_3\succ E_1\succ E_5\succ E_4$	$E_2$
$d_Y$	$E_3\succ E_2\succ E_1\succ E_5\succ E_4$	$E_3$
$d_{W,\epsilon =0.5}$	$E_3\succ E_2\succ E_1\succ E_5\succ E_4$	$E_3$
$d_{\epsilon =0.5}$	$E_3\succ E_2\succ E_1\succ E_5\succ E_4$	$E_3$

.

Table 11 demonstrates that the ranking of $d_M$, $d_Y$, and $d_{W,\epsilon }$ with $\epsilon =0.5$ are entirely consistent with ISIFDisM under the same $\epsilon$. This consistency highlights the strong compatibility between the proposed parametric ISIFDisM in this study and the existing distance measures for IFNs.

To validate the reliability and robustness of the proposed method, a comprehensive comparative analysis was performed against established distance measures and multi-attribute decision-making (MADM) methods reported in the literature, including representative methods such as VIKOR [11] and TOPSIS [14,20]. Specifically, to further validate the practical applicability of the proposed method, the interval-valued q-rung picture fuzzy set (I q-RPFS) combined with the TOPSIS method, as described in [14], is employed. Four evaluation attributes are identified: good reputation, low cost, quick repair, and easy recovery, and experts from marine engineering, marine electrical and electronic, and other related professions are to participate in the assessment process. The comparison results of different methods are shown in

Table 12. The comparison of reliability evaluation results with existing MADM methods.

Methods	Ranking	Best Reliability
VIKOR in [11]	$E_3\succ E_2\succ E_1\succ E_4\succ E_5$	$E_3$
TOPSIS in [20]	$E_3\succ E_2\succ E_1\succ E_5\succ E_4$	$E_3$
TOPSIS in [14]	$E_3\succ E_2\succ E_4\succ E_1\succ E_5$	$E_3$
I q-RPFS+TOPSIS in [14]	$E_3\succ E_2\succ E_5\succ E_1\succ E_4$	$E_3$
Proposed $d_{\epsilon }$+ TOPSIS	$E_3\succ E_2\succ E_1\succ E_5\succ E_4$	$E_3$

below.

Observing from Table 12, while there exist discrepancies in the overall rankings of the alternatives, the consistent identification of $E_3$ as the optimal reliability alternative across all evaluation methods. This comparative outcome offers robust evidence substantiating the practical applicability and reliability of our proposed method within the context of this assessment framework.

Furthermore, dynamic effect comparisons of the evaluation results are conducted by analyzing parameter changes and variations in IFNs with three-level state information.

As the parameter $\epsilon$ in Equation (6) fluctuates, the corresponding ${NPC}_i$ values are illustrated in Figure 5.

Figure 5 demonstrates that as the parameter $\epsilon$ increases, the reliability ranking order of $E_3\succ E_2\succ E_1\succ E_5\succ E_4$ remains consistent and distinguishable. Since $\epsilon$ serves as a discriminative parameter for highlighting the disparities among IFNs, a larger $\epsilon$ value significantly accentuates these differences. Consequently, with the increase in $\epsilon$, the differences among the engines become more pronounced, which facilitates the identification of the engine with the highest reliability. These results align with the theoretical expectations, validating the rationality of the proposed analytical framework.

For IFNs $\eta ,$ $\theta ,$ and $\vartheta$ (corresponding to the three-level fuzzy language set $S_1$, $S_2$ and $S_3$ in step 6), four distinct combinations are selected, where $\vartheta ={\eta }^C$ and $\theta =⟨\mathrm{0.5,0.5}⟩$. The corresponding results are shown in Figure 6.

As demonstrated in Figure 6, with the variation of $\eta$, the reliability ranking results remain unchanged, exhibiting remarkable stability. Since $\eta ,$ $\theta$, and $\vartheta$ are employed to reflect the deterministic situation of the evaluation results, a larger membership degree of $\eta$ leads to a more significant disparity among the evaluation results. Consequently, a higher membership degree of $\eta$ is more conducive to the reliability assessment of marine engines. This finding is in perfect agreement with logical expectations, further validating the rationality and effectiveness of the proposed method.

To facilitate shipbuilding enterprises in making well-informed decisions on selecting suitable engine types for ships operating in different environments, this study meticulously selected and comprehensively analyzed data from various environments using the proposed methodology. The results are presented in

Table 13. The reliability evaluation results of marine engines based on different environments.

Operational Environment	Ranking	Best Reliability
Ocean-going ships ($d_1$)	$E_3\succ E_2\succ E_1\succ E_5\succ E_4$	$E_3$
Near-ocean ships $\left(d_2\right)$	$E_1\succ E_2\succ E_3\succ E_4\succ E_5$	$E_1$
Coastal ships ($d_3$)	$E_1\succ E_2\succ E_3\succ E_4\succ E_5$	$E_1$
Inland ships ($d_4$)	$E_2\succ E_1\succ E_3\succ E_4\succ E_5$	$E_2$

.

As clearly shown in Table 13, for ocean-going vessels, engine $E_3$ is the most optimal choice. For near-ocean or coastal vessels, engine $E_1$ is recommended, while engine $E_2$ is the preferred option for an inland vessel.

Additionally, the reliability scores of critical components for five distinct engine types are separately calculated using the proposed method, and the results are compared, as illustrated in Figure 7. Specifically, the reliability score of each component ${RS}_j$ can be computed by the following formula:

$$S_j={\displaystyle \frac{d_{\epsilon }\left({\mathit{Z}}_{ij},{\mathit{Z}}_{+j}\right)}{d_{\epsilon }\left({\mathit{Z}}_{ij},{\mathit{Z}}_{-j}\right)+d_{\epsilon }\left({\mathit{Z}}_{ij},{\mathit{Z}}_{+j}\right)}},j\in N$$

(28)

where $d_{\epsilon }\left({\mathit{Z}}_{ij},{\mathit{Z}}_{+j}\right)$ and $d_{\epsilon }\left({\mathit{Z}}_{ij},{\mathit{Z}}_{-j}\right)$ are given by the following equations:

$$\begin{gathered} d_{\epsilon }\left({\mathit{Z}}_{ij},{\mathit{Z}}_{+j}\right)=d_{\epsilon }\left(\left(〈{\mu }_j^{Ai},{\nu }_j^{Ai}〉, 〈{\mu }_j^{Bi},{\nu }_j^{Bi}〉, 〈{\mu }_j^{\Gamma i},{\nu }_j^{\Gamma i}〉\right),\left(〈{\mu }_{j+}^{Ai},{\nu }_{j+}^{Ai}〉, 〈{\mu }_{j+}^{{\rm B}i},{\nu }_{j+}^{{\rm B}i}〉, 〈{\mu }_{j+}^{\Gamma i},{\nu }_{j+}^{\Gamma i}〉\right)\right) \\ {d}_{\epsilon }\left({\mathit{Z}}_{ij},{\mathit{Z}}_{-j}\right)=d_{\epsilon }\left(\left(〈{\mu }_j^{Ai},{\nu }_j^{Ai}〉, 〈{\mu }_j^{Bi},{\nu }_j^{Bi}〉, 〈{\mu }_j^{\Gamma i},{\nu }_j^{\Gamma i}〉\right),\left(〈{\mu }_{j-}^{Ai},{\nu }_{j-}^{Ai}〉, 〈{\mu }_{j-}^{{\rm B}i},{\nu }_{j-}^{{\rm B}i}〉, 〈{\mu }_{j-}^{\Gamma i},{\nu }_{j-}^{\Gamma i}〉\right)\right) \end{gathered}$$

(29)

As depicted in Figure 7, a comparative analysis of the five engines demonstrates that engine $E_3$ exhibits superior reliability across all critical components when compared to its counterparts. Engine $E_2$ and engine $E_1$ follow in descending order of component reliability. Notably, while engine $E_2$ demonstrates relatively high overall reliability, its cooling system ($c_5$) registers a significantly lower reliability score, which warrants focused attention during subsequent optimization efforts. In contrast, both engines $E_4$ and $E_5$ display uniformly low reliability scores across all components, directly contributing to their subpar overall performance metrics. This finding is highly consistent with earlier assessment results, further emphasizing the need for targeted research and development initiatives in this domain. To enhance the dependability of these engines, future improvement strategies should prioritize in-depth analysis and design modifications of engine $E_4$ and engine $E_5$, with a particular emphasis on optimizing their critical components.

6. Conclusions

This study systematically investigates the critical importance of reliability assessments for complex equipment operating in variable environments, which is essential for achieving economic, efficient, and low-carbon operations. It comprehensively elucidates the multifaceted challenges currently encountered in this domain. A novel reliability assessment methodology for complex equipment is proposed, utilizing intuitionistic fuzzy sets and associated distance measures. The proposed approach offers several unique innovations and advantages.

(1)

A new distance metric for intuitionistic fuzzy sets has been constructed. This metric eliminates the limitations of existing distance measures and exhibits enhanced robustness, monotonicity, and discriminability. By adjusting the parameters, the degree of differentiation can be enhanced, facilitating more precise discrimination between different intuitionistic fuzzy sets.

(2)

The assessment model developed integrates both environmental factors and the utilization of complex equipment. This comprehensive integration facilitates a more holistic understanding of the proximity of decision-making entities and offers a more precise and comprehensive evaluation framework.

(3)

Utilizing a questionnaire-based approach, status-level information is transformed into intuitive fuzzy numbers according to the voting results. This method effectively reduces individual biases and optimally harnesses collective intelligence, thereby enhancing the objectivity and reliability of the assessment results.

(4)

During the integration of evaluation information, the proposed method synchronously incorporates the condition levels of critical equipment and other evaluation data. This synchronous integration significantly improved the differentiation of the evaluation results, facilitating more precise and reliable decision-making.

Despite the strengths of this study, it is important to acknowledge its limitations. Firstly, the identification of critical components in complex equipment is predominantly informed by experience and traditional reliability assessment methods. For high-end and newly developed equipment, future research could employ MADM [37] and optimization model [38] to accurately identify and determine critical components and their respective weights. Secondly, a persistent challenge lies in the clear differentiation between normal and abnormal operational states when evaluating the operational status of critical equipment. To address this issue, subsequent research may focus on data-driven prognostics and a health management (PHM) system for precise state identification [3,39,40]. Finally, regarding the decision-making methodologies, this study primarily focused on distance metrics and the TOPSIS method. Future research could explore a wider array of methods, such as picture fuzzy sets (PFSs) [41], I q-RPFN [14], linguistic q-rung orthopair fuzzy sets (Lq-ROFS) [42], evaluation based on the distance from the average solution [43], extended elimination and choice translating reality (ELECTRE) [44], and the ordinal priority approach (OPA) [45], to further enhance the versatility and effectiveness of the reliability assessment framework.