# Quantitative Risk Assessment of a Large-Scale Central Spherical Detector System by an Improved Fuzzy Fault Tree Analysis

Authors: Runze Tan; Xiaobin Li; Jingxia Yue; Zhipeng Du
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Central spherical detector is one of the most essential and complicated support systems for the large-scale underground neutrino observatory. If a central spherical detector accident happens, the consequences may be disastrous with huge economic loss. However, there is very few published research works which focus on its quantitative risk analysis. In this paper, an improved fuzzy fault tree analysis (FFTA) method incorporated with the weakest t-norm (Tω) algorithm, the confidence level concept, and the analytic hierarchy process (AHP) approach is proposed to perform its risk assessment. By carrying out the identification of failure modes and failure reasons, fault tree (FT) model of central detector is constructed based on logical relationship among subcomponents. Fuzzy set theory is applied to obtain failure data, and Tω algorithm is exploited to eliminate fuzzy accumulation in the aggregation process. In addition, a confidence level coefficient and AHP approach are employed to enhance the reliability of the evaluation. Both importance and sensitivity analysis have been conducted to identify the critical basic events and provide improvement measures. Finally, the comparison of the occurrence possibility of detector failure is used to verify the applicability and the feasibility of proposed method. The calculated results indicate that the improved approach is more consistent with real situation and can provide a more effective engineering reference for the risk decision of central spherical detector.


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## Body

## 1. Introduction

Precise determination of neutrino mass hierarchy and oscillation parameters has been one of the most crucial focuses in neutrino physics research, which promotes the development of cosmology and increases the knowledge of particle physics principles [1]. In order to improve the efficiency and accuracy of detection, the construction scale of neutrino detectors has been expanding in the past decades, and they are located in deeper land surface or water surface areas, such as Kamioka Liquid-scintillator Anti-Neutrino Detector (KamLAND) [2], Hyper-Kamiokande experiment [3], Baikal Gigaton Volume Detector (Baikal-GVD) [4], IceCube Neutrino Observatory [5], and Jiangmen Underground Neutrino Observatory (JUNO) [6]. Moreover, the unique characteristics of neutrinos, which have very little mass, no electrical charge, and extreme penetrating property [7, 8], lead to the central detector needing thousands of large format photomultiplier tubes (PMTs) to overcome the challenge of capturing them.As a crucial component of the detector, PMTs are composed of ellipsoid glass shells with thin thickness and internal electrical devices in the vacuum atmosphere, which works in an environment filled with ultrapure water [9]. Even if PMT is manufactured with strict inspection, microscopic cracks in PMT can cause glass breakage under huge hydrostatic pressure loads, which leads to a chain reaction explosion [10]. In addition, PMT implosion may be caused by several risks such as liquid corrosion, temperature or load change, initial defects, material creep, and operator errors, etc. For example, in 2001, a catastrophic accident of PMTs sympathetic implosion, which occurred in the Super-Kamiokande detector in Japan, with extremely serious consequences has been attracting the widespread concern of the international society. As a consequence, the cascade of implosions destroyed a total of 7877 PMTs, which directly resulted in an economic loss of about $30 million and set back progress in the neutrino experiment by 4 years [11].In the context of such implosion failures of the Super-Kamiokande detector, extensive investigations have been made to evaluate the reliability of PMT performance [10, 12, 13]. Although such experimental and numerical research can help disclose the mechanisms of the shock wave propagation across PMTs, it is still hard to determine all the relevant risks caused by various factors [14, 15]. One way to perform the risk assessment is employing probabilistic methods for finding root causes of the accidents, including analyzing the internal and external factors [16–18]. As a newly advanced detecting facility, there is scarcely any comprehensive and quantified research for risk assessment of neutrino detector. Therefore, carrying out an appropriate risk assessment approach to perform hazard analysis of central neutrino detector is essential for designers and managers so as to establish and improve the rational preparative projects of risk prevention.There are a variety of methods for comprehensive probabilistic risk assessment, such as the fault tree analysis (FTA) [19–21], functional resonance analysis method [22, 23], event-based diagnosis model [24], and Bayesian network [25, 26], etc. Among the approaches mentioned above, fault tree analysis is the most widely applied method for identifying the system risks and carrying out reliability estimation [27]. Further, a fault tree (FT) can describe the logical interactions between the system failure event and the fundamental reasons in a graphic format, which can effectively analyze complex systems in a concretization way compared with other common approaches [28, 29]. Mottahedi et al. [30] conducted both qualitative and quantitative analysis based on the fault tree method and investigated the risk evolution mechanism of the coal burst. Hu et al. [31] proposed a fault tree model, which is primarily based on the hierarchical relationships of subsystem combined with logical conditions, to diagnose the component failure faults of flowback water storage. Ikwan et al. [32] adopted a data-based risk evaluation method combining the risk matrix and FTA to obtain the leakage causes of a storage tank. Song et al. [33] spread out FTA in Colored Petri nets to analyze the railway system reliability. Matuzas [34] and Deng [35] developed an upgraded Binary Decision Diagram (BDD) algorithm for improving FTA efficiency. Other examples include the use of dynamic FT [36, 37] and Pandora temporal FT [38] for performing dependability analysis. Most of the above research on FTA methods is mainly to ameliorate the risk identification of complex systems in different engineering fields; however, there is a leading limitation requiring precise probability values to quantify the occurrence of failures. The fact is that the probability data of complex engineering accidents is usually rare, which result in a significant challenge to pick up statistical data. Furthermore, due to the uniqueness of each failure event, existing failure rate databases may not be suitable for providing an objective reference for it.Therefore, in addition to the FTA methods described above, complementary data acquisition techniques are required for providing failure probabilities for basic events. A very effective and reasonable tool is the subject expert evaluation based on the fuzzy set theory [21, 31, 39]. The superiority of the method is that it not only describes the fault level of events derived from fuzzy membership functions but also converts professional experiences and knowledge of each expert to failure possibility. To decrease nonconsensus generated from the procedure of expert evaluation, a great number of researchers adopted the aggregation method with AHP, which could express the contribution to total judgement results by quantifying the otherness of various experts [40–43]. More importantly, other studies introduced the weight of confidence level to assist experts accurately in representing their views against the deficiency of certainty and self-assurance [44, 45].Furthermore, the arithmetic operations of fuzzy numbers are rarely discussed whether it is suitable for the aggregate calculation or not in FTA. The algorithms of diverse types of fuzzy numbers, generally including sup-min convolution [46] and α-cut level [47, 48], were derived from the extension principle proposed by Zadeh [49]. However, the traditional fuzzy arithmetic accumulates the fuzziness in reiterative computation where the effect of left/right fuzzy probability interval excessively spreading is ignored [50]. The weakest t-norm arithmetic (Tω) is carried out for decreasing the fuzzy accumulation, which could both maintain the shape of initial fuzzy numbers and cut down dispensable fuzzy extension relatively [51, 52]. However, the application of the Tω to the fuzzy reliabilities analysis of practical systems is currently restricted to a few projects such as paper plant washing systems [53], healthcare institutions [54], and standby redundant system [55].Therefore, an improved FFTA approach to assess the failure probability of the neutrino detector is proposed in this paper. Meanwhile, six subject experts from the Institute of High Energy Physics, Chinese Academy of Sciences (IHEP-CAS), were also invited to conduct a qualitative assessment of each basic event. This paper focuses on performing a complete risk quantification of the central neutrino detector by building a FT model. In this risk analysis of the neutrino detector, the effects of internal faults and external shocks are fully considered. Moreover, the approach combines AHP technology, the concept of the confidence level, andTω operators to improve the reliability of obtaining probability.The rest of the paper is structured as follows. Section2 focuses on the introduction of this improved FFTA approach for quantitative risk evaluation. In Section 3, the proposed approach is used to evaluate the failure probability of the neutrino detector and the critical events are identified by conducting importance measures. Section 4 discusses the difference of probability results between the proposed FFTA approach and traditional approach, and Section 5 concludes the paper.


## 2. Improved FFTA Approach

This section is mainly about the proposed approach applied to the quantitative risk analysis of neutrino detector. As shown in Figure1, four stages of this approach are described in detail.Figure 1
Flowchart for the proposed approach.


### 2.1. Fault Tree Analysis (FTA) Method

FTA is used to diagnose the different types of faults and access the reliability based on a tree-like diagram. In this method, a FT is constructed between the top event and subevents, which are connected by various logic gates, e.g., “AND” and “OR.” Usually, the fault likelihood of basic events (BEs) plays an essential role of determining the occurrence probability of the top event (TE) based on fuzzy arithmetic operations [56].Building up a FT of the neutrino detector relies on a systematic and adequate investigation of its system functions and possible failure processes. Based on the failure modes and effects analysis (FMEA), all the failure modes, failure causes, and failure events are identified and analyzed. The identification process consists of four steps:(i)
Collect data about the structure and function of the detector and divide it into several components.(ii)
Identify the failure modes of the detector, which consist of potential and functional failures.(iii)
According to failure modes, analyze all the possible failure causes of the detector, which include direct failures of its component and indirect failures of human operator error and environmental factors.(iv)
Based on the function and structure hierarchy of the detector, combine with the failure modes to determine the failure effects.


### 2.2. Evaluation and Fuzzification Process

#### 2.2.1. Subject Experts’ Evaluation

To ensure the objectivity of the evaluation results, the subject experts should be from different disciplines as much as possible, and they should have different work experience [44]. In this study, six experts have been invited, and four items are used to define their diversified professional background, including position, service time, education level, and age. More details are summarized in Table 1, where Ei means the ith expert. All the experts have been required to complete the same questionnaire.Table 1
Experts information.
ParametersClassificationE1E2E3E4E5E6Professional positionSenior academic●Junior academic●●Engineer●Technician●Worker●Service time≥3020–29●10–19●●●●6–9●≤5Education levelPhD●Master●●●Bachelor●Higher national diploma●School levelAge≥50●40–49●●30–39●●<30●


#### 2.2.2. Fuzzification of Probability Scales and Confidence Scales

To consider the uncertainty of BEs’ fault and the absence of statistical data, the trapezoidal fuzzy number is employed in this paper, and it can be expressed byA˜ (a1, a2, a3, a4) with its membership function denoted by (1)μA˜x=0,x<a1,x−a1a2−a1,a1≤x≤a2,1,a2≤x≤a3,a4−xa4−a3,a3≤x≤a4,0,x>a4,where ai (i = 1, 2, 3, 4) is a characteristic number of A˜.For some special engineering projects, such as neutrino detectors, due to the lack of industry standards, the definition of the classification of the failure degree is still not clear enough. The frequency of neutrino detector failures has been low over the past few decades, with only one explosion accident at the Super-Kamiokande detector [11]. Since nuclear power plant accidents also have an extremely low frequency of occurrence, the fuzzification of probability scales for the detector is determined by referring to it in this research. Based on the available failure data from previous projects, seven linguistic terms have been adopted to classify the failure possibilities of nuclear power plants, as shown in Table 2 [56].Table 2
Failure possibilities and failure likelihood values.
Failure possibilityFailure likelihood valuesExtremely Low (EL)<10−8Low (L)10−8–10−7Relatively Low (RL)10−7–10−6Medium (M)10−6–10−5Relatively High (RH)10−5–10−4High (H)10−4–10−3Extremely High (EH)>10−3Meanwhile, the classic evaluation of human brain is seven plus-minus two chunks, which signifies that the suitable scale of linguistic terms for making a more appropriate judgement is defined between 5 and 9 [57]. Therefore, 7-level fuzzy linguistic scales are used to transform linguistic description of fuzzy numbers for the neutrino detector failure, as shown in Figure 2(a), by referring to [31]. In addition, based on Miller’s principle, the minimum range (5-level) is also selected as the scale of confidence grade to avoid increasing the immoderate complexity of judgement process, as shown in Figure 2(b) [58].Figure 2
Fuzzy membership functions. (a) 7-level fuzzy linguistic scales. (b) 5-level fuzzy linguistic scales.
(a)(b)


### 2.3. Similarity Aggregation and Defuzzification

#### 2.3.1. Weighting Factors of Experts Based on AHP Approach

According to the hierarchy structure of expert competence shown in Figure3, four criteria are determined to construct the judgement matrix, including the professional position (PP), service time (ST), educational level (EL), and age (AG). Additionally, based on the individual information of each expert described in Table 1, there are five comparison matrices employed for grading experts’ weightiness in the aggregated result; five comparison matrices are proposed to obtain the weight coefficients of each expert for the aggregated results, as shown from equations (2) to (6). First of all, the matrix DEu means the priority relationship among the four criteria. DPP is a comparison matrix between different experts based on the professional position as the judgement criteria, and DST, DEL, and DAG, respectively, correspond to the other three criteria.(2)DEu=19/59/89/25/915/85/28/98/518/22/92/52/81,(3)DPP=17/57/47/57/27/35/715/45/55/25/34/74/514/54/24/35/75/55/415/25/32/75/22/42/512/33/73/53/43/53/21,(4)DST=16/46/46/56/36/44/614/44/54/34/44/64/414/54/34/45/65/45/415/35/43/63/43/43/513/44/64/44/44/54/31(5)DEL=17/57/57/77/37/45/715/55/75/35/45/75/515/75/35/47/77/57/517/37/43/73/53/53/713/44/74/54/54/74/31,(6)DAG=16/56/36/36/26/55/615/35/35/25/53/63/513/33/23/53/63/53/313/23/52/62/52/32/312/55/65/55/35/35/21.Figure 3
Expert capabilities hierarchy structure of AHP.In equations (2)–(6), dij is the relationship between factor i and factor j; the value of dij is from 1/9 to 9 [54]. Particularly, the D is positive reciprocal matrix which means that dij = 1/dji.Based on the optimal matrix transformation theory [40, 41], the process of calculating weighting factors with AHP is as follows.Step 1.
Generate the approximate optimal transfer matrixD∗.(7)Hvariable=hijk×k,variable=Euhijm×m,variable=PP,ST,EL,AG,hij=lgdij,G=gijn×n,gij=1n∑z=1nhiz−hjz,Dvariable∗=dij∗k×k,variable=Eudij∗m×m,variable=PP,ST,EL,AG,dij∗=10gij,where H, G, and D∗ represent antisymmetric matrix, optimal transfer matrix, and approximate optimal transfer matrix, respectively. k is the total number of criteria, and m represents the total number of the experts.Step 2.
Check the consistency ratio.
Applying equation (8) to check the consistency ratio (CR) of the approximate optimal transfer matrix D∗, the results illustrate that five transfer matrices are all consistent.(8)CR=λmax−nn−1RI,where RI is the random consistency index, λmax is the maximum eigenvalue, and n is the order of a matrix.Step 3.
Calculate the weight of each expert.
The weight vectorW is then obtained with the geometric mean method.(9)Wvariable=∏j=1kdij∗k∑i=1k∏j=1kdij∗,variable=Eu,∏j=1mdij∗m∑i=1m∏j=1mdij∗,variable=PP,ST,EL,AG.
Eventually, the weight factor of expertEu (u = 1, 2,…, m) is calculated by(10)WEu=WEu,PP×WPP,Eu+WEu,ST×WST,Eu+WEu,EL×WEL,Eu+WEu,AG×WAG,Eu,where W (Eu) represents the weight of expert Eu. WEu, PP is the weighting factor of professional position; WPP, Eu means the weighting factor of expert Eu that characterizes the professional position as a judging criterion. The others are corresponding to service time, educational level, and age.


#### 2.3.2. Adding a Confidence Coefficient to the Experts’ Judgement

To obtain more accurate and reliable results through subject expert judgement, Yazdi proposed a method based on the concept ofZ-numbers [44]. According to this method, a confidence coefficient can be added to the original results to reflect the reliability of various expert opinions, and it can be calculated by centre of area (COA) method. In addition, the confidence coefficient of fuzzy number E˜zu can be denoted as follows:(11)Z∗=∫e1ze2zx−e1z/e2z−e1zxdx+∫e2ze3zxdx+∫e3ze4ze4z−x/e4z−e3zxdx∫e1ze2zx−e1z/e2z−e1zdx+∫e2ze3zdx+∫e3ze4ze4z−x/e4z−e3zdx=13e4z+e3z2−e4ze3z−e1z+e2z2+e1ze2ze4z+e3z−e2z−e1z,where Z∗ is the confidence coefficient.Then, the final fuzzy judgement from an expertEu is obtained by equation(12)E˜u′=Z∗E˜u,where ∼Eu means the expert Eu initial fuzzy opinion on the failure probability of BE, and E˜u′ is the transformed result.


#### 2.3.3. Aggregation of Various Expert Opinions and Defuzzification

To fuse different expert opinions and reach a consensus, Similarity Aggregation Method (SAM) is developed to integrate each expert’s opinion on a certain BE [47]; the process is as follows:Step 1: Calculate the similarity degree.It is assumed that the linguistic judgements of the expertEu and Ev can be symbolized by E˜u′ and E˜v′ if considering the coincidence level. Thus, the similarity degree between E˜u′ (eu1', eu2', eu3', eu4') and E˜v′ (ev1', ev2', ev3', ev4') is calculated as follows:(13)SE˜u',E˜v'=∫xminμE˜u'x,μE˜v'xdx∫xmaxμE˜u'x,μE˜v'xdx=1−14∑i=14eui'−evi',whereS (E˜u′, E˜v′) is the similarity degree between two experts.Step 2: Calculate the average agreement degree (AA), the relative agreement degree (RA), and the consensus coefficient degree (CC).(14)SAAEu=1m−1∑u≠vv=1mSuv,SRAEu=SAAEu∑i=1mSAAEu,SCCEu=β⋅WEu+1−β⋅SRAEu,whereβ is the relaxation factor, β ∈ [0, 1], SAA (Eu) is the average agreement degree, SRA (Eu) is the relative agreement degree, and SCC (Eu) is the consensus coefficient degree.Step 3: Aggregate results of a single BE byTω algorithm.In this paper, the arithmetic operation based onTω uses the Tω algorithm to replace the traditional “min” operator and the operator of two trapezoidal fuzzy numbers can be stated as follows:(15)A˜1∘A˜2=supTωA˜1a1i∘A˜2a2i.According to theTω arithmetic operations [51, 54], the aggregating results are denoted by(16)X˜i=SCCE1×E˜1′⊕TωSCCE2×E˜2′⊕Tω⋯⊕TωSCCEu×E˜u′,where∼Xi is the final aggregating fuzzy numbers of the ith BE, and the “⊕Tω” symbol symbolizes the fuzzy plus operation of Tω.Step 4: Defuzzification of fuzzy numbers.The COA method is applied to perform the defuzzification of results to convert the aggregated fuzzy number into the fuzzy possibility score (FPS) as follows:(17)FPS=13xi4+xi32−xi4xi3−xi1+xi22+xi1xi2xi4+xi3−xi2−xi1,wherexij (j = 1, 2, 3, 4) is a characteristic number of ∼Xi.In addition, the FPS is required to be transformed into the fuzzy failure probability (FFP).(18)FFP=1101−FPS/FPS1/3×2.301,FPS≠0,0,FPS=0.


### 2.4. Quantitative Analysis

#### 2.4.1. Probability of TE

According to the probability results of all the MCSs and the structure function of the detector, the occurrence probability of TE could be calculated by theTω operator as follows:(19)PTE=PMCS1∪MCS2∪⋯∪MCSn=∑i=1nPMCSi−∑i<j=2nPMCSi∩MCSj+∑i<j<g=3nPMCSi∩MCSj∩MCSg+⋯+−1n−1PMCS1∩MCS2∩⋯∩MCSn,where P (TE) is the probability of the top event, and P (MCSi) is the probability of the ith minimal cut set.


#### 2.4.2. Importance Measures and Sensitive Analysis

By employing the importance measures, it can assist the decision makers to identify faults and enhance the reliability of weak components correctly. In this study, the Birnbaum (QBI) and criticality importance (QCI) of each BE are calculated, and the Fussell–Vesely importance of minimal cut sets (QMCSI) [19] is adopted as well. Besides, in order to verify the established model, the risk reduction worth (RRW) method is used to perform the sensitivity analysis [29, 58].(20)QBIBEi=∂PTE∂PBEi,QCIBEi=PBEiPTE∂PTE∂PBEi=PBEiPTEQBIBEi,QMCSIMCSi=PMCSiPTE,QRRWBEi=PTEPTE|PBEi=0,where P (BEi) is the probability of the ith BE, QRRW (BEi) is the risk reduction worth of the ith BE, P (TE)|P (BEi) = 0 is the TE probability with the BE probability set to zero.


## 2.1. Fault Tree Analysis (FTA) Method

FTA is used to diagnose the different types of faults and access the reliability based on a tree-like diagram. In this method, a FT is constructed between the top event and subevents, which are connected by various logic gates, e.g., “AND” and “OR.” Usually, the fault likelihood of basic events (BEs) plays an essential role of determining the occurrence probability of the top event (TE) based on fuzzy arithmetic operations [56].Building up a FT of the neutrino detector relies on a systematic and adequate investigation of its system functions and possible failure processes. Based on the failure modes and effects analysis (FMEA), all the failure modes, failure causes, and failure events are identified and analyzed. The identification process consists of four steps:(i)
Collect data about the structure and function of the detector and divide it into several components.(ii)
Identify the failure modes of the detector, which consist of potential and functional failures.(iii)
According to failure modes, analyze all the possible failure causes of the detector, which include direct failures of its component and indirect failures of human operator error and environmental factors.(iv)
Based on the function and structure hierarchy of the detector, combine with the failure modes to determine the failure effects.


## 2.2. Evaluation and Fuzzification Process

### 2.2.1. Subject Experts’ Evaluation

To ensure the objectivity of the evaluation results, the subject experts should be from different disciplines as much as possible, and they should have different work experience [44]. In this study, six experts have been invited, and four items are used to define their diversified professional background, including position, service time, education level, and age. More details are summarized in Table 1, where Ei means the ith expert. All the experts have been required to complete the same questionnaire.Table 1
Experts information.
ParametersClassificationE1E2E3E4E5E6Professional positionSenior academic●Junior academic●●Engineer●Technician●Worker●Service time≥3020–29●10–19●●●●6–9●≤5Education levelPhD●Master●●●Bachelor●Higher national diploma●School levelAge≥50●40–49●●30–39●●<30●


### 2.2.2. Fuzzification of Probability Scales and Confidence Scales

To consider the uncertainty of BEs’ fault and the absence of statistical data, the trapezoidal fuzzy number is employed in this paper, and it can be expressed byA˜ (a1, a2, a3, a4) with its membership function denoted by (1)μA˜x=0,x<a1,x−a1a2−a1,a1≤x≤a2,1,a2≤x≤a3,a4−xa4−a3,a3≤x≤a4,0,x>a4,where ai (i = 1, 2, 3, 4) is a characteristic number of A˜.For some special engineering projects, such as neutrino detectors, due to the lack of industry standards, the definition of the classification of the failure degree is still not clear enough. The frequency of neutrino detector failures has been low over the past few decades, with only one explosion accident at the Super-Kamiokande detector [11]. Since nuclear power plant accidents also have an extremely low frequency of occurrence, the fuzzification of probability scales for the detector is determined by referring to it in this research. Based on the available failure data from previous projects, seven linguistic terms have been adopted to classify the failure possibilities of nuclear power plants, as shown in Table 2 [56].Table 2
Failure possibilities and failure likelihood values.
Failure possibilityFailure likelihood valuesExtremely Low (EL)<10−8Low (L)10−8–10−7Relatively Low (RL)10−7–10−6Medium (M)10−6–10−5Relatively High (RH)10−5–10−4High (H)10−4–10−3Extremely High (EH)>10−3Meanwhile, the classic evaluation of human brain is seven plus-minus two chunks, which signifies that the suitable scale of linguistic terms for making a more appropriate judgement is defined between 5 and 9 [57]. Therefore, 7-level fuzzy linguistic scales are used to transform linguistic description of fuzzy numbers for the neutrino detector failure, as shown in Figure 2(a), by referring to [31]. In addition, based on Miller’s principle, the minimum range (5-level) is also selected as the scale of confidence grade to avoid increasing the immoderate complexity of judgement process, as shown in Figure 2(b) [58].Figure 2
Fuzzy membership functions. (a) 7-level fuzzy linguistic scales. (b) 5-level fuzzy linguistic scales.
(a)(b)


## 2.2.1. Subject Experts’ Evaluation

To ensure the objectivity of the evaluation results, the subject experts should be from different disciplines as much as possible, and they should have different work experience [44]. In this study, six experts have been invited, and four items are used to define their diversified professional background, including position, service time, education level, and age. More details are summarized in Table 1, where Ei means the ith expert. All the experts have been required to complete the same questionnaire.Table 1
Experts information.
ParametersClassificationE1E2E3E4E5E6Professional positionSenior academic●Junior academic●●Engineer●Technician●Worker●Service time≥3020–29●10–19●●●●6–9●≤5Education levelPhD●Master●●●Bachelor●Higher national diploma●School levelAge≥50●40–49●●30–39●●<30●


## 2.2.2. Fuzzification of Probability Scales and Confidence Scales

To consider the uncertainty of BEs’ fault and the absence of statistical data, the trapezoidal fuzzy number is employed in this paper, and it can be expressed byA˜ (a1, a2, a3, a4) with its membership function denoted by (1)μA˜x=0,x<a1,x−a1a2−a1,a1≤x≤a2,1,a2≤x≤a3,a4−xa4−a3,a3≤x≤a4,0,x>a4,where ai (i = 1, 2, 3, 4) is a characteristic number of A˜.For some special engineering projects, such as neutrino detectors, due to the lack of industry standards, the definition of the classification of the failure degree is still not clear enough. The frequency of neutrino detector failures has been low over the past few decades, with only one explosion accident at the Super-Kamiokande detector [11]. Since nuclear power plant accidents also have an extremely low frequency of occurrence, the fuzzification of probability scales for the detector is determined by referring to it in this research. Based on the available failure data from previous projects, seven linguistic terms have been adopted to classify the failure possibilities of nuclear power plants, as shown in Table 2 [56].Table 2
Failure possibilities and failure likelihood values.
Failure possibilityFailure likelihood valuesExtremely Low (EL)<10−8Low (L)10−8–10−7Relatively Low (RL)10−7–10−6Medium (M)10−6–10−5Relatively High (RH)10−5–10−4High (H)10−4–10−3Extremely High (EH)>10−3Meanwhile, the classic evaluation of human brain is seven plus-minus two chunks, which signifies that the suitable scale of linguistic terms for making a more appropriate judgement is defined between 5 and 9 [57]. Therefore, 7-level fuzzy linguistic scales are used to transform linguistic description of fuzzy numbers for the neutrino detector failure, as shown in Figure 2(a), by referring to [31]. In addition, based on Miller’s principle, the minimum range (5-level) is also selected as the scale of confidence grade to avoid increasing the immoderate complexity of judgement process, as shown in Figure 2(b) [58].Figure 2
Fuzzy membership functions. (a) 7-level fuzzy linguistic scales. (b) 5-level fuzzy linguistic scales.
(a)(b)


## 2.3. Similarity Aggregation and Defuzzification

### 2.3.1. Weighting Factors of Experts Based on AHP Approach

According to the hierarchy structure of expert competence shown in Figure3, four criteria are determined to construct the judgement matrix, including the professional position (PP), service time (ST), educational level (EL), and age (AG). Additionally, based on the individual information of each expert described in Table 1, there are five comparison matrices employed for grading experts’ weightiness in the aggregated result; five comparison matrices are proposed to obtain the weight coefficients of each expert for the aggregated results, as shown from equations (2) to (6). First of all, the matrix DEu means the priority relationship among the four criteria. DPP is a comparison matrix between different experts based on the professional position as the judgement criteria, and DST, DEL, and DAG, respectively, correspond to the other three criteria.(2)DEu=19/59/89/25/915/85/28/98/518/22/92/52/81,(3)DPP=17/57/47/57/27/35/715/45/55/25/34/74/514/54/24/35/75/55/415/25/32/75/22/42/512/33/73/53/43/53/21,(4)DST=16/46/46/56/36/44/614/44/54/34/44/64/414/54/34/45/65/45/415/35/43/63/43/43/513/44/64/44/44/54/31(5)DEL=17/57/57/77/37/45/715/55/75/35/45/75/515/75/35/47/77/57/517/37/43/73/53/53/713/44/74/54/54/74/31,(6)DAG=16/56/36/36/26/55/615/35/35/25/53/63/513/33/23/53/63/53/313/23/52/62/52/32/312/55/65/55/35/35/21.Figure 3
Expert capabilities hierarchy structure of AHP.In equations (2)–(6), dij is the relationship between factor i and factor j; the value of dij is from 1/9 to 9 [54]. Particularly, the D is positive reciprocal matrix which means that dij = 1/dji.Based on the optimal matrix transformation theory [40, 41], the process of calculating weighting factors with AHP is as follows.Step 1.
Generate the approximate optimal transfer matrixD∗.(7)Hvariable=hijk×k,variable=Euhijm×m,variable=PP,ST,EL,AG,hij=lgdij,G=gijn×n,gij=1n∑z=1nhiz−hjz,Dvariable∗=dij∗k×k,variable=Eudij∗m×m,variable=PP,ST,EL,AG,dij∗=10gij,where H, G, and D∗ represent antisymmetric matrix, optimal transfer matrix, and approximate optimal transfer matrix, respectively. k is the total number of criteria, and m represents the total number of the experts.Step 2.
Check the consistency ratio.
Applying equation (8) to check the consistency ratio (CR) of the approximate optimal transfer matrix D∗, the results illustrate that five transfer matrices are all consistent.(8)CR=λmax−nn−1RI,where RI is the random consistency index, λmax is the maximum eigenvalue, and n is the order of a matrix.Step 3.
Calculate the weight of each expert.
The weight vectorW is then obtained with the geometric mean method.(9)Wvariable=∏j=1kdij∗k∑i=1k∏j=1kdij∗,variable=Eu,∏j=1mdij∗m∑i=1m∏j=1mdij∗,variable=PP,ST,EL,AG.
Eventually, the weight factor of expertEu (u = 1, 2,…, m) is calculated by(10)WEu=WEu,PP×WPP,Eu+WEu,ST×WST,Eu+WEu,EL×WEL,Eu+WEu,AG×WAG,Eu,where W (Eu) represents the weight of expert Eu. WEu, PP is the weighting factor of professional position; WPP, Eu means the weighting factor of expert Eu that characterizes the professional position as a judging criterion. The others are corresponding to service time, educational level, and age.


### 2.3.2. Adding a Confidence Coefficient to the Experts’ Judgement

To obtain more accurate and reliable results through subject expert judgement, Yazdi proposed a method based on the concept ofZ-numbers [44]. According to this method, a confidence coefficient can be added to the original results to reflect the reliability of various expert opinions, and it can be calculated by centre of area (COA) method. In addition, the confidence coefficient of fuzzy number E˜zu can be denoted as follows:(11)Z∗=∫e1ze2zx−e1z/e2z−e1zxdx+∫e2ze3zxdx+∫e3ze4ze4z−x/e4z−e3zxdx∫e1ze2zx−e1z/e2z−e1zdx+∫e2ze3zdx+∫e3ze4ze4z−x/e4z−e3zdx=13e4z+e3z2−e4ze3z−e1z+e2z2+e1ze2ze4z+e3z−e2z−e1z,where Z∗ is the confidence coefficient.Then, the final fuzzy judgement from an expertEu is obtained by equation(12)E˜u′=Z∗E˜u,where ∼Eu means the expert Eu initial fuzzy opinion on the failure probability of BE, and E˜u′ is the transformed result.


### 2.3.3. Aggregation of Various Expert Opinions and Defuzzification

To fuse different expert opinions and reach a consensus, Similarity Aggregation Method (SAM) is developed to integrate each expert’s opinion on a certain BE [47]; the process is as follows:Step 1: Calculate the similarity degree.It is assumed that the linguistic judgements of the expertEu and Ev can be symbolized by E˜u′ and E˜v′ if considering the coincidence level. Thus, the similarity degree between E˜u′ (eu1', eu2', eu3', eu4') and E˜v′ (ev1', ev2', ev3', ev4') is calculated as follows:(13)SE˜u',E˜v'=∫xminμE˜u'x,μE˜v'xdx∫xmaxμE˜u'x,μE˜v'xdx=1−14∑i=14eui'−evi',whereS (E˜u′, E˜v′) is the similarity degree between two experts.Step 2: Calculate the average agreement degree (AA), the relative agreement degree (RA), and the consensus coefficient degree (CC).(14)SAAEu=1m−1∑u≠vv=1mSuv,SRAEu=SAAEu∑i=1mSAAEu,SCCEu=β⋅WEu+1−β⋅SRAEu,whereβ is the relaxation factor, β ∈ [0, 1], SAA (Eu) is the average agreement degree, SRA (Eu) is the relative agreement degree, and SCC (Eu) is the consensus coefficient degree.Step 3: Aggregate results of a single BE byTω algorithm.In this paper, the arithmetic operation based onTω uses the Tω algorithm to replace the traditional “min” operator and the operator of two trapezoidal fuzzy numbers can be stated as follows:(15)A˜1∘A˜2=supTωA˜1a1i∘A˜2a2i.According to theTω arithmetic operations [51, 54], the aggregating results are denoted by(16)X˜i=SCCE1×E˜1′⊕TωSCCE2×E˜2′⊕Tω⋯⊕TωSCCEu×E˜u′,where∼Xi is the final aggregating fuzzy numbers of the ith BE, and the “⊕Tω” symbol symbolizes the fuzzy plus operation of Tω.Step 4: Defuzzification of fuzzy numbers.The COA method is applied to perform the defuzzification of results to convert the aggregated fuzzy number into the fuzzy possibility score (FPS) as follows:(17)FPS=13xi4+xi32−xi4xi3−xi1+xi22+xi1xi2xi4+xi3−xi2−xi1,wherexij (j = 1, 2, 3, 4) is a characteristic number of ∼Xi.In addition, the FPS is required to be transformed into the fuzzy failure probability (FFP).(18)FFP=1101−FPS/FPS1/3×2.301,FPS≠0,0,FPS=0.


## 2.3.1. Weighting Factors of Experts Based on AHP Approach

According to the hierarchy structure of expert competence shown in Figure3, four criteria are determined to construct the judgement matrix, including the professional position (PP), service time (ST), educational level (EL), and age (AG). Additionally, based on the individual information of each expert described in Table 1, there are five comparison matrices employed for grading experts’ weightiness in the aggregated result; five comparison matrices are proposed to obtain the weight coefficients of each expert for the aggregated results, as shown from equations (2) to (6). First of all, the matrix DEu means the priority relationship among the four criteria. DPP is a comparison matrix between different experts based on the professional position as the judgement criteria, and DST, DEL, and DAG, respectively, correspond to the other three criteria.(2)DEu=19/59/89/25/915/85/28/98/518/22/92/52/81,(3)DPP=17/57/47/57/27/35/715/45/55/25/34/74/514/54/24/35/75/55/415/25/32/75/22/42/512/33/73/53/43/53/21,(4)DST=16/46/46/56/36/44/614/44/54/34/44/64/414/54/34/45/65/45/415/35/43/63/43/43/513/44/64/44/44/54/31(5)DEL=17/57/57/77/37/45/715/55/75/35/45/75/515/75/35/47/77/57/517/37/43/73/53/53/713/44/74/54/54/74/31,(6)DAG=16/56/36/36/26/55/615/35/35/25/53/63/513/33/23/53/63/53/313/23/52/62/52/32/312/55/65/55/35/35/21.Figure 3
Expert capabilities hierarchy structure of AHP.In equations (2)–(6), dij is the relationship between factor i and factor j; the value of dij is from 1/9 to 9 [54]. Particularly, the D is positive reciprocal matrix which means that dij = 1/dji.Based on the optimal matrix transformation theory [40, 41], the process of calculating weighting factors with AHP is as follows.Step 1.
Generate the approximate optimal transfer matrixD∗.(7)Hvariable=hijk×k,variable=Euhijm×m,variable=PP,ST,EL,AG,hij=lgdij,G=gijn×n,gij=1n∑z=1nhiz−hjz,Dvariable∗=dij∗k×k,variable=Eudij∗m×m,variable=PP,ST,EL,AG,dij∗=10gij,where H, G, and D∗ represent antisymmetric matrix, optimal transfer matrix, and approximate optimal transfer matrix, respectively. k is the total number of criteria, and m represents the total number of the experts.Step 2.
Check the consistency ratio.
Applying equation (8) to check the consistency ratio (CR) of the approximate optimal transfer matrix D∗, the results illustrate that five transfer matrices are all consistent.(8)CR=λmax−nn−1RI,where RI is the random consistency index, λmax is the maximum eigenvalue, and n is the order of a matrix.Step 3.
Calculate the weight of each expert.
The weight vectorW is then obtained with the geometric mean method.(9)Wvariable=∏j=1kdij∗k∑i=1k∏j=1kdij∗,variable=Eu,∏j=1mdij∗m∑i=1m∏j=1mdij∗,variable=PP,ST,EL,AG.
Eventually, the weight factor of expertEu (u = 1, 2,…, m) is calculated by(10)WEu=WEu,PP×WPP,Eu+WEu,ST×WST,Eu+WEu,EL×WEL,Eu+WEu,AG×WAG,Eu,where W (Eu) represents the weight of expert Eu. WEu, PP is the weighting factor of professional position; WPP, Eu means the weighting factor of expert Eu that characterizes the professional position as a judging criterion. The others are corresponding to service time, educational level, and age.


## 2.3.2. Adding a Confidence Coefficient to the Experts’ Judgement

To obtain more accurate and reliable results through subject expert judgement, Yazdi proposed a method based on the concept ofZ-numbers [44]. According to this method, a confidence coefficient can be added to the original results to reflect the reliability of various expert opinions, and it can be calculated by centre of area (COA) method. In addition, the confidence coefficient of fuzzy number E˜zu can be denoted as follows:(11)Z∗=∫e1ze2zx−e1z/e2z−e1zxdx+∫e2ze3zxdx+∫e3ze4ze4z−x/e4z−e3zxdx∫e1ze2zx−e1z/e2z−e1zdx+∫e2ze3zdx+∫e3ze4ze4z−x/e4z−e3zdx=13e4z+e3z2−e4ze3z−e1z+e2z2+e1ze2ze4z+e3z−e2z−e1z,where Z∗ is the confidence coefficient.Then, the final fuzzy judgement from an expertEu is obtained by equation(12)E˜u′=Z∗E˜u,where ∼Eu means the expert Eu initial fuzzy opinion on the failure probability of BE, and E˜u′ is the transformed result.


## 2.3.3. Aggregation of Various Expert Opinions and Defuzzification

To fuse different expert opinions and reach a consensus, Similarity Aggregation Method (SAM) is developed to integrate each expert’s opinion on a certain BE [47]; the process is as follows:Step 1: Calculate the similarity degree.It is assumed that the linguistic judgements of the expertEu and Ev can be symbolized by E˜u′ and E˜v′ if considering the coincidence level. Thus, the similarity degree between E˜u′ (eu1', eu2', eu3', eu4') and E˜v′ (ev1', ev2', ev3', ev4') is calculated as follows:(13)SE˜u',E˜v'=∫xminμE˜u'x,μE˜v'xdx∫xmaxμE˜u'x,μE˜v'xdx=1−14∑i=14eui'−evi',whereS (E˜u′, E˜v′) is the similarity degree between two experts.Step 2: Calculate the average agreement degree (AA), the relative agreement degree (RA), and the consensus coefficient degree (CC).(14)SAAEu=1m−1∑u≠vv=1mSuv,SRAEu=SAAEu∑i=1mSAAEu,SCCEu=β⋅WEu+1−β⋅SRAEu,whereβ is the relaxation factor, β ∈ [0, 1], SAA (Eu) is the average agreement degree, SRA (Eu) is the relative agreement degree, and SCC (Eu) is the consensus coefficient degree.Step 3: Aggregate results of a single BE byTω algorithm.In this paper, the arithmetic operation based onTω uses the Tω algorithm to replace the traditional “min” operator and the operator of two trapezoidal fuzzy numbers can be stated as follows:(15)A˜1∘A˜2=supTωA˜1a1i∘A˜2a2i.According to theTω arithmetic operations [51, 54], the aggregating results are denoted by(16)X˜i=SCCE1×E˜1′⊕TωSCCE2×E˜2′⊕Tω⋯⊕TωSCCEu×E˜u′,where∼Xi is the final aggregating fuzzy numbers of the ith BE, and the “⊕Tω” symbol symbolizes the fuzzy plus operation of Tω.Step 4: Defuzzification of fuzzy numbers.The COA method is applied to perform the defuzzification of results to convert the aggregated fuzzy number into the fuzzy possibility score (FPS) as follows:(17)FPS=13xi4+xi32−xi4xi3−xi1+xi22+xi1xi2xi4+xi3−xi2−xi1,wherexij (j = 1, 2, 3, 4) is a characteristic number of ∼Xi.In addition, the FPS is required to be transformed into the fuzzy failure probability (FFP).(18)FFP=1101−FPS/FPS1/3×2.301,FPS≠0,0,FPS=0.


## 2.4. Quantitative Analysis

### 2.4.1. Probability of TE

According to the probability results of all the MCSs and the structure function of the detector, the occurrence probability of TE could be calculated by theTω operator as follows:(19)PTE=PMCS1∪MCS2∪⋯∪MCSn=∑i=1nPMCSi−∑i<j=2nPMCSi∩MCSj+∑i<j<g=3nPMCSi∩MCSj∩MCSg+⋯+−1n−1PMCS1∩MCS2∩⋯∩MCSn,where P (TE) is the probability of the top event, and P (MCSi) is the probability of the ith minimal cut set.


### 2.4.2. Importance Measures and Sensitive Analysis

By employing the importance measures, it can assist the decision makers to identify faults and enhance the reliability of weak components correctly. In this study, the Birnbaum (QBI) and criticality importance (QCI) of each BE are calculated, and the Fussell–Vesely importance of minimal cut sets (QMCSI) [19] is adopted as well. Besides, in order to verify the established model, the risk reduction worth (RRW) method is used to perform the sensitivity analysis [29, 58].(20)QBIBEi=∂PTE∂PBEi,QCIBEi=PBEiPTE∂PTE∂PBEi=PBEiPTEQBIBEi,QMCSIMCSi=PMCSiPTE,QRRWBEi=PTEPTE|PBEi=0,where P (BEi) is the probability of the ith BE, QRRW (BEi) is the risk reduction worth of the ith BE, P (TE)|P (BEi) = 0 is the TE probability with the BE probability set to zero.


## 2.4.1. Probability of TE

According to the probability results of all the MCSs and the structure function of the detector, the occurrence probability of TE could be calculated by theTω operator as follows:(19)PTE=PMCS1∪MCS2∪⋯∪MCSn=∑i=1nPMCSi−∑i<j=2nPMCSi∩MCSj+∑i<j<g=3nPMCSi∩MCSj∩MCSg+⋯+−1n−1PMCS1∩MCS2∩⋯∩MCSn,where P (TE) is the probability of the top event, and P (MCSi) is the probability of the ith minimal cut set.


## 2.4.2. Importance Measures and Sensitive Analysis

By employing the importance measures, it can assist the decision makers to identify faults and enhance the reliability of weak components correctly. In this study, the Birnbaum (QBI) and criticality importance (QCI) of each BE are calculated, and the Fussell–Vesely importance of minimal cut sets (QMCSI) [19] is adopted as well. Besides, in order to verify the established model, the risk reduction worth (RRW) method is used to perform the sensitivity analysis [29, 58].(20)QBIBEi=∂PTE∂PBEi,QCIBEi=PBEiPTE∂PTE∂PBEi=PBEiPTEQBIBEi,QMCSIMCSi=PMCSiPTE,QRRWBEi=PTEPTE|PBEi=0,where P (BEi) is the probability of the ith BE, QRRW (BEi) is the risk reduction worth of the ith BE, P (TE)|P (BEi) = 0 is the TE probability with the BE probability set to zero.


## 3. Case Study

### 3.1. Central Neutrino Detector

The underground neutrino observatory is 53 kilometers away from both the Yangjiang and Taishan nuclear reactors approximately. In addition, it is built under 700 meters of rock and primarily consists of a central detector and a cylindrical pool, as shown in Figure4. It mainly determines the neutrino mass level of the reactor by accurately detecting the energy spectrum and oscillation signal of the neutrino.Figure 4
A schematic view of central neutrino detector. (i) PMTs, (ii) ultrapure water, (iii) support structure, (iv) liquid scintillator, (v) bracing structure, and (vi) acrylic sphere.The central detector will be filled with 20 kilotons of liquid scintillator and immersed in a pool of ultrapure water to shield the surrounding rock from radioactivity. Besides, it is composed of 18000 PMT modules and an acrylic sphere supported with a structurally stainless-steel frame. The PMT module is assembled by a 20-inch PMT and a “closed-housing” protective device, which mainly detects the Cerenkov light generated by neutrinos. In addition, the acrylic sphere refers to a receptacle and the stainless-steel structures contribute mechanical support to the first two components.


### 3.2. Fault Identification and Fault Tree Construction

In this work, the neutrino detector failure is defined as a TE. Regarding the neutrino detector as a multicomponent system, where it consists of PMT modules, acrylic sphere, and stainless-steel structures, each subsystem failure occurring could result in the malfunction of the whole facility. By analyzing the failure modes and failure causes of all subsystems, the corresponding FT model can be constructed, as shown in Figure5. The qualitative analysis indicates that 34 BEs are the essential causes, and detailed information is listed in Table 3. TE consists of three conditional intermediate events; no matter which intermediate event occurs, the detector will experience severe structural failure. Thus, an “OR” gate is employed to link three intermediate events. Finally, by performing qualitative analysis of the fault tree, all the MCSs of detector failure can be found out and the summarized results of MCSs are shown in Table 4.Figure 5
The fault tree for detector failure.
(a)(b)Table 3
Description of basic events.
EventsBasic events descriptionEventsBasic events descriptionX1Temperature variation induced by ambient conditionX18Bending defect during girders machiningX2Pressure variation caused by liquid supply/drainageX19Unreasonable connection in neighbor PMTs protectionX3PMT failure due to ultrapure water corrosionX20Bad welding of neighbor PMTs protectionX4Damage during repairX21Unreasonable strength of bottom cover in neighbor PMTs protectionX5Initial crack of one PMTX22Unsuitable bottom cover opening area in neighbor PMTs protectionX6Thin thickness of one PMTX23Initial crack of top cover in neighbor PMTs protectionX7Unreasonable connectionX24Thin thickness of top cover in neighbor PMTs protectionX8Bad weldingX25Strains given at the bottom work in neighbor PMTs protectionX9Unreasonable strength of bottom coverX26Design defect of sphere materialX10Unsuitable bottom cover opening areaX27Unsuitable thickness distributionX11Initial crack of top coverX28High out-of-roundness caused by improper assemblyX12Thin thickness of top coverX29Sphere failure due to water and liquid scintillator corrosionX13Strains given at the bottom workX30Service load far outweighs design loadX14Failure of detector due to devastating earthquakeX31Excessive stress due to external load on structureX15Failure of detector due to serious rock subsidenceX32Loss of external anticorrosive coatingX16Expected earthquake occurs during service timeX33Structure rupture due to using unreasonable materialX17Loose connection between acrylic sphere and bracing structureX34Construction defect of weld-seamTable 4
Results of MCSs.
MCS orderCountEvents12X14, X1527X16 × X28, X16 × X17, X16 × X30, X16 × X27, X16 × X18, X16 × X29, X16 × X263296X4 × X10 × X22, X2 × X13 × X22, X6 × X10 × X25, X1 × X10 × X20, X1 × X13 × X24, X3 × X11 × X23, X16 × X31 × X34, X5 × X9 × X23, X4 × X12 × X25, X6 × X8 × X22, X6 × X10 × X23…41X3 × X16 × X31 × X32The first failure mode of detector is “PMTs chain reaction implosion,” which means the implosion of one or a few PMTs causing thousands of PMTs to explode and then resulting in the final failure. A PMT module consists of a PMT upper acrylic protective cover, lower stainless-steel protective cover, and bolted connection, as shown in Figure6(a). For the PMT implosion failure, the most common cause may be glass cracks due to crushing during initial manufacturing or repair. Changes in operating conditions (temperature or pressure) are a nonnegligible factor in evaluating implosion occurrence. In terms of the upper protective cover failure, in the inspection operation of the PMT at the bottom of the detector, it is easy to put it in a state of compression, which is likely to cause residual deformation defects, resulting in collapse. For lower protective cover failure, the open area is the primary key to optimizing shock wave pressure by controlling flow or velocity. Improper connections can lead to undue stress concentration at the joint with increasing the risk of protective cover’s rupture failure. It should be noted that it is assumed that two-layer PMT modules are considered in the FT of the detector, and the performance of all second-layer PMTs is consistent. Furthermore, assume that the chain reaction explosion is caused by only one PMT implosion.Figure 6
The sketch of substructure. (a) The PMT with protection with (i) PMT, (ii) upper protective cover, (iii) bolted connection, and (iv) lower protective cover. (b) The joint on the acrylic sphere with (i) fastener, (ii) support bar, (iii) sphere glass, and (iv) patched glass. (c) Central acrylic sphere.
(a)(b)(c)The second failure mode is the overall functional failure caused by “severe geological hazard.” Since the detector is built in a specific underground environment, the cause of its failure should include earthquakes and severe rock subsidence.Other failure modes can be acrylic sphere failure and stainless-steel structural failure. In this paper, it is assumed that “expected earthquake occurs during service time” is a necessary precondition for these two failure modes. Concerning the stainless-steel structure, it is constructed by a double-layer truss with bottom bracing pillars. The weakest part of the structure may be the joints between inner layer frame and sphere glass, as shown in Figure6(b). In order to ensure the integrity and reliability of the sphere, an additional piece of glass is bonded to a part of the main glass through secondary polymeric casting. Besides, a fastener preburied in the patched glass is used to link the sphere with the support bar. The other reasons for structure failure may be welding defects, loss of external anticorrosive coating, and excess stress. As to the central sphere shown in Figure 6(c), it is subject to various external pressure along the height direction due to the different densities of the inner and outer liquids. Therefore, inappropriate thickness distribution may lead to buckling more easily. Another reason for sphere failure is high out-of-roundness caused by improper assembly. The corrosion caused by the liquid medium is a critical hazard issue for the sphere as well.


### 3.3. Probability Occurrence of BEs

#### 3.3.1. Evaluation Stage

Each expert is specialized in a specific relevant subfield of neutrinos and participates in the detector project. In accordance with the linguistic scales, judgements of BEs’ occurrence likelihood and their confidence degree are described in Table5. In order to generate more reasonable judgements, an additional step is to combine the confidence coefficient with the initial fuzzy opinion of the experts. For example, with reference to the basic event X1, the transforming process is expressed in Table 6.Table 5
Summary of for all basic events.
EventsProbability levelsConfidence levelsE1E2E3E4E5E6E1E2E3E4E5E6X1LLRLMRLRLHHMHMMMHX2MHRHMMMMMHMHMLX3LRLELMELELMMLHMHMHMX4HRHMHMHHHHMHHHX5MRLRLLRLLMMHMHLMLMHX6LRLRLRLLLMHMMMLMMHX7HMRLRHMRHMMMHHMLMX8LRLLMRLLHHMHMHMMX9RLLLLMLMHHMMHMLMX10RHHRHMRHHHMHHHMHHX11LRLRLLRLLMLMHHHMLHX12RLLMLRLELMHMLMMMHHX13MRHHRHHRHMMMLMHHHX14ELELELELELELMMHMHHMLMX15ELELELELELELMHHMHMMMLX16ELLELELLLMMHMHMMLMX17RHMRHRHMRLHMHMMHHHX18ELELLRLLLMHHMHMMHHX19MMRLLLRLMMMHHMLMX20LRLLMRLLHHMHMHMMX21RLLLLMLMHHMMHMLMX22RHHRHMRHHHMHHHMHHX23LRLRLLRLLMLMHHHMLHX24RLLMLRLELMHMLMMMHHX25MRHHRHHRHMMMLMHHHX26ELELELELLELHMHHMHMHHX27ELRLELLMLMHMHMHMMX28RHMRHMRHMHHMHMHMX29ELLELELLMMHMMMMMHX30ELLLRLRLLHHMHHMMHX31RLELELELMMMHMMHMMHMX32ELELELELRLRLHHMHMHMMX33ELELELELLELHHHMHMHHX34MMLLELELMHMHHHMHMHTable 6
Calculating the fuzzy judgement corresponding toX1.
CodesConfidence linguisticsTrapezoidal numbersZ∗Probability linguisticsTrapezoidal numbersTransformed fuzzy numbersE1H(0.8, 0.9, 1, 1)0.96L(0.1, 0.2, 0.2, 0.3)(0.096, 0.192, 0.192, 0.288)E2H(0.8, 0.9, 1, 1)0.96L(0.1, 0.2, 0.2, 0.3)(0.096, 0.192, 0.192, 0.288)E3MH(0.6, 0.7, 0.7, 0.9)0.86RL(0.2, 0.3, 0.4, 0.5)(0.173, 0.260, 0.346, 0.433)E4M(0.3, 0.5, 0.5, 0.7)0.70M(0.4, 0.5, 0.5, 0.6)(0.283, 0.354, 0.354, 0.424)E5M(0.3, 0.5, 0.5, 0.7)0.70RL(0.2, 0.3, 0.4, 0.5)(0.141, 0.212, 0.283, 0.354)E6MH(0.6, 0.7, 0.7, 0.9)0.86RL(0.2, 0.3, 0.4, 0.5)(0.173, 0.260, 0.346, 0.433)


#### 3.3.2. Aggregation Stage and Defuzzification Stage

To obtain the consensus coefficient, the expert’s weight factor is calculated by normalizing the comparison matrix. As a result, the comprehensive weight factors of every expert are calculated, as provided in the second part of Table7. Afterwards, any expert opinion should be considered in the aggregate calculation of the fuzzy likelihood of BE. The aggregated results of BEs are listed in Table 8 (the second column) by employing the Tω arithmetic operations, where the value of the relaxation factor β is regarded as 0.5 [19, 58].Table 7
The results for weight factors of different experts.
Weighting scores of four criteriaCriteriaNormalized scoresProfessional position0.3751Service time0.2083Educational level0.3334Age0.0833Weighting scores of expertEuCodesProfessional position (WPP,Eu)Service time (WST,Eu)Educational level (WEL,Eu)Age (WAG,Eu)W (Eu)E10.23320.20810.20600.22160.2179E20.18630.15880.16460.19630.1742E30.16060.15880.16460.13960.1598E40.18630.18430.20600.13960.1886E50.10110.13110.11710.10660.1131E60.13250.15880.14180.19630.1464Table 8
Evaluation results.
EventsAggregated fuzzy numbersFPSFFPRankaResults of FFP by traditional FFTARankbX1(0.226, 0.245, 0.282, 0.301)0.2645.771E – 04111.088E – 0312X2(0.390, 0.406, 0.422, 0.437)0.4142.613E – 0398.211E – 038X3(0.113, 0.128, 0.176, 0.191)0.1528.378E – 05263.094E – 0425X4(0.616, 0.634, 0.650, 0.669)0.6421.280E – 0211.724E – 021X5(0.198, 0.212, 0.248, 0.263)0.2333.659E – 04151.219E – 0311X6(0.161, 0.177, 0.210, 0.226)0.1931.994E – 04236.831E – 0417X7(0.397, 0.414, 0.456, 0.473)0.4353.096E – 0388.954E – 036X8(0.223, 0.242, 0.268, 0.286)0.2555.165E – 04138.451E – 0413X9(0.177, 0.193, 0.210, 0.226)0.2012.300E – 04215.714E – 0421X10(0.583, 0.601, 0.647, 0.666)0.6241.144E – 0221.563E – 022X11(0.178, 0.195, 0.233, 0.250)0.2142.842E – 04176.190E – 0419X12(0.169, 0.186, 0.230, 0.246)0.2082.567E – 04196.924E – 0415X13(0.480, 0.495, 0.538, 0.553)0.5175.636E – 0341.458E – 024X14(0.000, 0.000, 0.077, 0.094)0.0433.413E – 07335.662E – 0633X15(0.000, 0.000, 0.077, 0.093)0.0423.362E – 07345.662E – 0634X16(0.051, 0.065, 0.106, 0.120)0.0858.698E – 06305.275E – 0530X17(0.439, 0.458, 0.518, 0.536)0.4884.595E – 0377.545E – 039X18(0.105, 0.120, 0.165, 0.182)0.1436.689E – 05271.533E – 0427X19(0.234, 0.251, 0.277, 0.293)0.2645.806E – 04101.661E – 0310X20(0.223, 0.242, 0.268, 0.286)0.2555.165E – 04128.451E – 0414X21(0.177, 0.193, 0.210, 0.226)0.2012.300E – 04205.714E – 0422X22(0.583, 0.601, 0.647, 0.666)0.6241.144E – 0231.563E – 023X23(0.178, 0.195, 0.233, 0.250)0.2142.842E – 04166.190E – 0420X24(0.169, 0.186, 0.230, 0.246)0.2082.567E – 04186.924E – 0416X25(0.480, 0.495, 0.538, 0.553)0.5175.636E – 0351.458E – 025X26(0.012, 0.024, 0.103, 0.122)0.0652.661E – 06321.178E – 0531X27(0.123, 0.138, 0.183, 0.199)0.1611.029E – 04252.883E – 0426X28(0.458, 0.476, 0.522, 0.541)0.4994.985E – 0368.228E – 037X29(0.098, 0.112, 0.152, 0.169)0.1335.089E – 05281.361E – 0428X30(0.152, 0.168, 0.214, 0.232)0.1911.923E – 04243.135E – 0424X31(0.150, 0.166, 0.222, 0.238)0.1942.012E – 04224.734E – 0423X32(0.052, 0.063, 0.148, 0.167)0.1082.229E – 05297.986E – 0529X33(0.012, 0.024, 0.105, 0.123)0.0662.813E – 06311.178E – 0532X34(0.204, 0.221, 0.247, 0.264)0.2343.868E – 04146.748E – 0418a the ranking of FFP by improved FFTA approach. b the ranking of FFP by traditional FFTA approach.To represent the probability occurrence of TE in an intuitionistic way, the aggregated fuzzy numbers of BEs are required to be converted into a precise value; FPS and FFP of all BEs are listed in the third column and fourth column of Table8, respectively. Moreover, in order to highlight the reasonability and applicability of the improved FFTA approach, results of FFP are further calculated by the traditional FFTA approach, as shown in the fifth column of Table 8. For the traditional approach, it is mainly adopting a simple way (likewise using in [30]) to deal with weight factors and not considering the confidence level and the Tω arithmetic.


### 3.4. Further Quantitative Analysis

#### 3.4.1. Occurrence Probability of TE

In this study, the calculated failure probability of the detector is 7.4316E – 06 based on the improved FFTA approach, which corresponds to the possible interval of “Medium” (10−6∼10−5) level according to Table 2. However, if using the traditional approach, the failure probability is 3.6171E – 05 which is more than twice the results from the improved FFTA approach and at the level of “RH” (10−5∼10−4). Since a series of protection measures have been taken in the design to reduce the failure probability, the results from the improved FFTA approach are more reasonable and more consistent with the actual situation, while the traditional approach is excessively conservative. It should be addressed that only two layers of PMT are considered in the calculations based on both approaches.However, if FT involves more layers of PMT modules, the failure probability will be further reduced, as clearly shown in Figure7. The horizontal axis is the number of layers of PMTs with protection, in which “0” means that each PMT is baring with no cover and “1” means that only the exploding PMT fit out with the defensive cover; the vertical axis represents the logarithmic probability of failure. Furthermore, two groups of failure probabilities corresponding to different number of layers of PMT have been calculated, including TE probability and the intermediate event “PMTs chain reaction implosion” probability, and they have been plotted in Figure 7 and correspond to black color and green color. For the TE failure probability, the results of the two approaches drop sharply with the number of layers from 0 to 3, while when the number of layers is greater than 4, the curve fluctuates smoothly and the results stabilize at 7.6436E – 07 and 1.1785E – 05, respectively. It can be clearly observed that the level of the improved approach drops to “RL” according to Table 2, while the traditional approach is still at the “RH” level. For “PMTs chain reaction implosion” failure probability, the results of both methods keep decreasing as the number of layers increases, suggesting that the more the PMTs installing protection, the safer the detector. Eventually, even though the evaluated probability of the improved approach is relatively low, it should be attached with sufficient concern because of resulting in postponing the development of neutrino research severely and suffering great losses of economy in case of an accident.Figure 7
Failure probability of TE and the intermediate event “PMTs chain reaction implosion.”


#### 3.4.2. Importance Measures and Sensitive Analysis

The results of QBI and QCI for 34 BEs are presented in Table9, and results for the top 17 critical MSCs are listed in Table 10. Subsequently, according to the sorting results of the fuzzy probability, CI, BI, and MCSI, the top 17 BEs and MCS are plotted in Figure 8, which are represented with blue, pink, orange and black histograms, respectively, and the height of the histogram corresponds to the results value.Table 9
Importance measures and RRW of BEs.
EventsQBIQCIQRRWX13.9533E – 043.0703E – 021.0317X23.9613E – 041.3930E – 011.1618X33.9513E – 044.4549E – 031.0045X44.0023E – 046.8977E – 013.2234X53.9524E – 041.9462E – 021.0198X63.9518E – 041.0605E – 021.0107X73.0705E – 041.2793E – 011.1467X83.0626E – 042.1288E – 021.0218X93.0617E – 049.4768E – 031.0096X103.0964E – 044.7665E – 011.9108X113.0619E – 041.1711E – 021.0119X123.0618E – 041.0578E – 021.0107X133.0783E – 042.3348E – 011.3046X149.9999E – 014.5935E – 021.0481X159.9999E – 014.5248E – 021.0474X169.9694E – 031.1669E – 021.0118X172.6285E – 055.3499E – 031.0054X182.6166E – 057.7528E – 051.0001X193.4729E – 042.7133E – 021.0279X203.4726E – 042.4139E – 021.0247X213.4716E – 041.0746E – 021.0109X223.5110E – 045.4047E – 012.1762X233.4718E – 041.3279E – 021.0135X243.4717E – 041.1994E – 021.0121X253.4905E – 042.6474E – 011.3601X262.6164E – 053.0846E – 061.0000X272.6167E – 051.1927E – 041.0001X282.6295E – 055.8058E – 031.0058X292.6166E – 055.8983E – 051.0001X302.6169E – 052.2291E – 041.0002X311.0196E – 089.0853E – 081.0000X324.4091E – 134.3539E – 131.0000X335.2622E – 096.5565E – 101.0000X345.2642E – 099.0197E – 081.0000Table 10
Information of the top seventeen important MSCs.
CodesEvent setsOrderP (MCSi)QMCSIMCS1X4 × X10 × X2231.6762E – 062.2252E – 01MCS2X4 × X13 × X2238.2589E – 071.0964E – 01MCS3X4 × X10 × X2538.2589E – 071.0964E – 01MCS4X4 × X7 × X2234.5370E – 076.0228E – 02MCS5X4 × X13 × X2534.0693E – 075.4019E – 02MCS6X2 × X10 × X2233.4201E – 074.5402E – 02MCS7X1413.4137E – 074.5317E – 02MCS8X1513.3627E – 074.4640E – 02MCS9X4 × X7 × X2532.2354E – 072.9675E – 02MCS10X2 × X13 × X2231.6851E – 072.2370E – 02MCS11X2 × X10 × X2531.6851E – 072.2370E – 02MCS12X2 × X7 × X2239.2570E – 081.2289E – 02MCS13X4 × X10 × X1938.5074E – 081.1294E – 02MCS14X2 × X13 × X2538.3028E – 081.1022E – 02MCS15X4 × X8 × X2237.5691E – 081.0048E – 02MCS16X4 × X10 × X2037.5691E – 081.0048E – 02MCS17X1 × X10 × X2237.5536E – 081.0027E – 02Figure 8
Top17 basic events based on different ranking results. (a) Fuzzy failure probability. (b) Critical importance. (c) Birnbaum importance. (d) Cut set importance.
(a)(b)(c)(d)According to the ranking results of failure probability, Figure8 shows that the BEs X4, X10, X22, X13, X25, X28, and X17 are ranked higher, which means that these events are more likely to cause detector accidents. Even though X28 and X17 belong to the top seven BEs depending on the probability value, these two BEs have little contribution to the occurrence of the detector accident because they are both linked to TE with BE X16 through an “AND” gate. In the FT model, their abovementioned logical relationship weakens the influence of any one of them (e.g., X28, X17, or X16) on TE. Moreover, in terms of sorting the Birnbaum importance, X14, X15, X16, X4, X2, X1, and X5 are the first seven BEs, whereas the forefront of them (e.g., X14 and X15) have little influence on the detector failure due to their extremely low occurrence likelihood. By the comprehensive consideration of the ranking result of QBI, QCI, and QMCSI, it is supposed to observe that the intermediate event “PMTs chain reaction implosion” accounts for a substantial part of the top seventeen BEs. Therefore, the PMT module should receive more attention in the installation and operation of detectors. Finally, a total of six BEs (e.g., X4, X10, X21, X13, X25, and X7) were identified as the most critical BEs.In addition, through the comparison of the ranking results of CI and RRW, it is concluded thatX4 is the most critical basic event, and its RRW value is also the largest among other BEs. Therefore, both ranking results based on the FT model have good consistency, which further verifies the rationality of the proposed approach.Referring to the concluded results, several improvement measures corresponding to the most critical events are summarized as follows:(i)
In terms ofX4: “damage during repair,” strict monitoring technology should be used to identify the impact damage of the photomultiplier tube, and the damaged photomultiplier tube can be quickly replaced when a defect occurs. During the transportation of photomultiplier tubes, flexible soft materials with better adaptability and higher impact resistance should be considered to cover the photomultiplier tubes.(ii)
As forX10 (likewise, the X21): “unsuitable bottom cover opening area,” the total inlet area corresponding to the shockwave threshold that caused the adjacent PMTs to explode needs to be calculated cautiously by a great many numerical and experimental tests. Moreover, it also should improve the bottom cover construction technology and carry out strict sampling inspection of the inflow area.(iii)
As toX13 (likewise, the X25): “strains given at the bottom work,” when the operators work at the bottom of the detector, the thick impact energy-absorption material or special scaffold should be used to prevent residual stress on the top protection cover.(iv)
With regard toX7: “unreasonable connection,” to alleviate the stress concentration that happened near the bolted hole of the top cover, it is advised to adopt six vertical bolt fastenings for clamping the acrylic protection. In addition, a rubber sleeve is sheathed at the end of each bolt and the connection between bolts and the top cover is relatively loose to keep a little margin acting as cushioning.


## 3.1. Central Neutrino Detector

The underground neutrino observatory is 53 kilometers away from both the Yangjiang and Taishan nuclear reactors approximately. In addition, it is built under 700 meters of rock and primarily consists of a central detector and a cylindrical pool, as shown in Figure4. It mainly determines the neutrino mass level of the reactor by accurately detecting the energy spectrum and oscillation signal of the neutrino.Figure 4
A schematic view of central neutrino detector. (i) PMTs, (ii) ultrapure water, (iii) support structure, (iv) liquid scintillator, (v) bracing structure, and (vi) acrylic sphere.The central detector will be filled with 20 kilotons of liquid scintillator and immersed in a pool of ultrapure water to shield the surrounding rock from radioactivity. Besides, it is composed of 18000 PMT modules and an acrylic sphere supported with a structurally stainless-steel frame. The PMT module is assembled by a 20-inch PMT and a “closed-housing” protective device, which mainly detects the Cerenkov light generated by neutrinos. In addition, the acrylic sphere refers to a receptacle and the stainless-steel structures contribute mechanical support to the first two components.


## 3.2. Fault Identification and Fault Tree Construction

In this work, the neutrino detector failure is defined as a TE. Regarding the neutrino detector as a multicomponent system, where it consists of PMT modules, acrylic sphere, and stainless-steel structures, each subsystem failure occurring could result in the malfunction of the whole facility. By analyzing the failure modes and failure causes of all subsystems, the corresponding FT model can be constructed, as shown in Figure5. The qualitative analysis indicates that 34 BEs are the essential causes, and detailed information is listed in Table 3. TE consists of three conditional intermediate events; no matter which intermediate event occurs, the detector will experience severe structural failure. Thus, an “OR” gate is employed to link three intermediate events. Finally, by performing qualitative analysis of the fault tree, all the MCSs of detector failure can be found out and the summarized results of MCSs are shown in Table 4.Figure 5
The fault tree for detector failure.
(a)(b)Table 3
Description of basic events.
EventsBasic events descriptionEventsBasic events descriptionX1Temperature variation induced by ambient conditionX18Bending defect during girders machiningX2Pressure variation caused by liquid supply/drainageX19Unreasonable connection in neighbor PMTs protectionX3PMT failure due to ultrapure water corrosionX20Bad welding of neighbor PMTs protectionX4Damage during repairX21Unreasonable strength of bottom cover in neighbor PMTs protectionX5Initial crack of one PMTX22Unsuitable bottom cover opening area in neighbor PMTs protectionX6Thin thickness of one PMTX23Initial crack of top cover in neighbor PMTs protectionX7Unreasonable connectionX24Thin thickness of top cover in neighbor PMTs protectionX8Bad weldingX25Strains given at the bottom work in neighbor PMTs protectionX9Unreasonable strength of bottom coverX26Design defect of sphere materialX10Unsuitable bottom cover opening areaX27Unsuitable thickness distributionX11Initial crack of top coverX28High out-of-roundness caused by improper assemblyX12Thin thickness of top coverX29Sphere failure due to water and liquid scintillator corrosionX13Strains given at the bottom workX30Service load far outweighs design loadX14Failure of detector due to devastating earthquakeX31Excessive stress due to external load on structureX15Failure of detector due to serious rock subsidenceX32Loss of external anticorrosive coatingX16Expected earthquake occurs during service timeX33Structure rupture due to using unreasonable materialX17Loose connection between acrylic sphere and bracing structureX34Construction defect of weld-seamTable 4
Results of MCSs.
MCS orderCountEvents12X14, X1527X16 × X28, X16 × X17, X16 × X30, X16 × X27, X16 × X18, X16 × X29, X16 × X263296X4 × X10 × X22, X2 × X13 × X22, X6 × X10 × X25, X1 × X10 × X20, X1 × X13 × X24, X3 × X11 × X23, X16 × X31 × X34, X5 × X9 × X23, X4 × X12 × X25, X6 × X8 × X22, X6 × X10 × X23…41X3 × X16 × X31 × X32The first failure mode of detector is “PMTs chain reaction implosion,” which means the implosion of one or a few PMTs causing thousands of PMTs to explode and then resulting in the final failure. A PMT module consists of a PMT upper acrylic protective cover, lower stainless-steel protective cover, and bolted connection, as shown in Figure6(a). For the PMT implosion failure, the most common cause may be glass cracks due to crushing during initial manufacturing or repair. Changes in operating conditions (temperature or pressure) are a nonnegligible factor in evaluating implosion occurrence. In terms of the upper protective cover failure, in the inspection operation of the PMT at the bottom of the detector, it is easy to put it in a state of compression, which is likely to cause residual deformation defects, resulting in collapse. For lower protective cover failure, the open area is the primary key to optimizing shock wave pressure by controlling flow or velocity. Improper connections can lead to undue stress concentration at the joint with increasing the risk of protective cover’s rupture failure. It should be noted that it is assumed that two-layer PMT modules are considered in the FT of the detector, and the performance of all second-layer PMTs is consistent. Furthermore, assume that the chain reaction explosion is caused by only one PMT implosion.Figure 6
The sketch of substructure. (a) The PMT with protection with (i) PMT, (ii) upper protective cover, (iii) bolted connection, and (iv) lower protective cover. (b) The joint on the acrylic sphere with (i) fastener, (ii) support bar, (iii) sphere glass, and (iv) patched glass. (c) Central acrylic sphere.
(a)(b)(c)The second failure mode is the overall functional failure caused by “severe geological hazard.” Since the detector is built in a specific underground environment, the cause of its failure should include earthquakes and severe rock subsidence.Other failure modes can be acrylic sphere failure and stainless-steel structural failure. In this paper, it is assumed that “expected earthquake occurs during service time” is a necessary precondition for these two failure modes. Concerning the stainless-steel structure, it is constructed by a double-layer truss with bottom bracing pillars. The weakest part of the structure may be the joints between inner layer frame and sphere glass, as shown in Figure6(b). In order to ensure the integrity and reliability of the sphere, an additional piece of glass is bonded to a part of the main glass through secondary polymeric casting. Besides, a fastener preburied in the patched glass is used to link the sphere with the support bar. The other reasons for structure failure may be welding defects, loss of external anticorrosive coating, and excess stress. As to the central sphere shown in Figure 6(c), it is subject to various external pressure along the height direction due to the different densities of the inner and outer liquids. Therefore, inappropriate thickness distribution may lead to buckling more easily. Another reason for sphere failure is high out-of-roundness caused by improper assembly. The corrosion caused by the liquid medium is a critical hazard issue for the sphere as well.


## 3.3. Probability Occurrence of BEs

### 3.3.1. Evaluation Stage

Each expert is specialized in a specific relevant subfield of neutrinos and participates in the detector project. In accordance with the linguistic scales, judgements of BEs’ occurrence likelihood and their confidence degree are described in Table5. In order to generate more reasonable judgements, an additional step is to combine the confidence coefficient with the initial fuzzy opinion of the experts. For example, with reference to the basic event X1, the transforming process is expressed in Table 6.Table 5
Summary of for all basic events.
EventsProbability levelsConfidence levelsE1E2E3E4E5E6E1E2E3E4E5E6X1LLRLMRLRLHHMHMMMHX2MHRHMMMMMHMHMLX3LRLELMELELMMLHMHMHMX4HRHMHMHHHHMHHHX5MRLRLLRLLMMHMHLMLMHX6LRLRLRLLLMHMMMLMMHX7HMRLRHMRHMMMHHMLMX8LRLLMRLLHHMHMHMMX9RLLLLMLMHHMMHMLMX10RHHRHMRHHHMHHHMHHX11LRLRLLRLLMLMHHHMLHX12RLLMLRLELMHMLMMMHHX13MRHHRHHRHMMMLMHHHX14ELELELELELELMMHMHHMLMX15ELELELELELELMHHMHMMMLX16ELLELELLLMMHMHMMLMX17RHMRHRHMRLHMHMMHHHX18ELELLRLLLMHHMHMMHHX19MMRLLLRLMMMHHMLMX20LRLLMRLLHHMHMHMMX21RLLLLMLMHHMMHMLMX22RHHRHMRHHHMHHHMHHX23LRLRLLRLLMLMHHHMLHX24RLLMLRLELMHMLMMMHHX25MRHHRHHRHMMMLMHHHX26ELELELELLELHMHHMHMHHX27ELRLELLMLMHMHMHMMX28RHMRHMRHMHHMHMHMX29ELLELELLMMHMMMMMHX30ELLLRLRLLHHMHHMMHX31RLELELELMMMHMMHMMHMX32ELELELELRLRLHHMHMHMMX33ELELELELLELHHHMHMHHX34MMLLELELMHMHHHMHMHTable 6
Calculating the fuzzy judgement corresponding toX1.
CodesConfidence linguisticsTrapezoidal numbersZ∗Probability linguisticsTrapezoidal numbersTransformed fuzzy numbersE1H(0.8, 0.9, 1, 1)0.96L(0.1, 0.2, 0.2, 0.3)(0.096, 0.192, 0.192, 0.288)E2H(0.8, 0.9, 1, 1)0.96L(0.1, 0.2, 0.2, 0.3)(0.096, 0.192, 0.192, 0.288)E3MH(0.6, 0.7, 0.7, 0.9)0.86RL(0.2, 0.3, 0.4, 0.5)(0.173, 0.260, 0.346, 0.433)E4M(0.3, 0.5, 0.5, 0.7)0.70M(0.4, 0.5, 0.5, 0.6)(0.283, 0.354, 0.354, 0.424)E5M(0.3, 0.5, 0.5, 0.7)0.70RL(0.2, 0.3, 0.4, 0.5)(0.141, 0.212, 0.283, 0.354)E6MH(0.6, 0.7, 0.7, 0.9)0.86RL(0.2, 0.3, 0.4, 0.5)(0.173, 0.260, 0.346, 0.433)


### 3.3.2. Aggregation Stage and Defuzzification Stage

To obtain the consensus coefficient, the expert’s weight factor is calculated by normalizing the comparison matrix. As a result, the comprehensive weight factors of every expert are calculated, as provided in the second part of Table7. Afterwards, any expert opinion should be considered in the aggregate calculation of the fuzzy likelihood of BE. The aggregated results of BEs are listed in Table 8 (the second column) by employing the Tω arithmetic operations, where the value of the relaxation factor β is regarded as 0.5 [19, 58].Table 7
The results for weight factors of different experts.
Weighting scores of four criteriaCriteriaNormalized scoresProfessional position0.3751Service time0.2083Educational level0.3334Age0.0833Weighting scores of expertEuCodesProfessional position (WPP,Eu)Service time (WST,Eu)Educational level (WEL,Eu)Age (WAG,Eu)W (Eu)E10.23320.20810.20600.22160.2179E20.18630.15880.16460.19630.1742E30.16060.15880.16460.13960.1598E40.18630.18430.20600.13960.1886E50.10110.13110.11710.10660.1131E60.13250.15880.14180.19630.1464Table 8
Evaluation results.
EventsAggregated fuzzy numbersFPSFFPRankaResults of FFP by traditional FFTARankbX1(0.226, 0.245, 0.282, 0.301)0.2645.771E – 04111.088E – 0312X2(0.390, 0.406, 0.422, 0.437)0.4142.613E – 0398.211E – 038X3(0.113, 0.128, 0.176, 0.191)0.1528.378E – 05263.094E – 0425X4(0.616, 0.634, 0.650, 0.669)0.6421.280E – 0211.724E – 021X5(0.198, 0.212, 0.248, 0.263)0.2333.659E – 04151.219E – 0311X6(0.161, 0.177, 0.210, 0.226)0.1931.994E – 04236.831E – 0417X7(0.397, 0.414, 0.456, 0.473)0.4353.096E – 0388.954E – 036X8(0.223, 0.242, 0.268, 0.286)0.2555.165E – 04138.451E – 0413X9(0.177, 0.193, 0.210, 0.226)0.2012.300E – 04215.714E – 0421X10(0.583, 0.601, 0.647, 0.666)0.6241.144E – 0221.563E – 022X11(0.178, 0.195, 0.233, 0.250)0.2142.842E – 04176.190E – 0419X12(0.169, 0.186, 0.230, 0.246)0.2082.567E – 04196.924E – 0415X13(0.480, 0.495, 0.538, 0.553)0.5175.636E – 0341.458E – 024X14(0.000, 0.000, 0.077, 0.094)0.0433.413E – 07335.662E – 0633X15(0.000, 0.000, 0.077, 0.093)0.0423.362E – 07345.662E – 0634X16(0.051, 0.065, 0.106, 0.120)0.0858.698E – 06305.275E – 0530X17(0.439, 0.458, 0.518, 0.536)0.4884.595E – 0377.545E – 039X18(0.105, 0.120, 0.165, 0.182)0.1436.689E – 05271.533E – 0427X19(0.234, 0.251, 0.277, 0.293)0.2645.806E – 04101.661E – 0310X20(0.223, 0.242, 0.268, 0.286)0.2555.165E – 04128.451E – 0414X21(0.177, 0.193, 0.210, 0.226)0.2012.300E – 04205.714E – 0422X22(0.583, 0.601, 0.647, 0.666)0.6241.144E – 0231.563E – 023X23(0.178, 0.195, 0.233, 0.250)0.2142.842E – 04166.190E – 0420X24(0.169, 0.186, 0.230, 0.246)0.2082.567E – 04186.924E – 0416X25(0.480, 0.495, 0.538, 0.553)0.5175.636E – 0351.458E – 025X26(0.012, 0.024, 0.103, 0.122)0.0652.661E – 06321.178E – 0531X27(0.123, 0.138, 0.183, 0.199)0.1611.029E – 04252.883E – 0426X28(0.458, 0.476, 0.522, 0.541)0.4994.985E – 0368.228E – 037X29(0.098, 0.112, 0.152, 0.169)0.1335.089E – 05281.361E – 0428X30(0.152, 0.168, 0.214, 0.232)0.1911.923E – 04243.135E – 0424X31(0.150, 0.166, 0.222, 0.238)0.1942.012E – 04224.734E – 0423X32(0.052, 0.063, 0.148, 0.167)0.1082.229E – 05297.986E – 0529X33(0.012, 0.024, 0.105, 0.123)0.0662.813E – 06311.178E – 0532X34(0.204, 0.221, 0.247, 0.264)0.2343.868E – 04146.748E – 0418a the ranking of FFP by improved FFTA approach. b the ranking of FFP by traditional FFTA approach.To represent the probability occurrence of TE in an intuitionistic way, the aggregated fuzzy numbers of BEs are required to be converted into a precise value; FPS and FFP of all BEs are listed in the third column and fourth column of Table8, respectively. Moreover, in order to highlight the reasonability and applicability of the improved FFTA approach, results of FFP are further calculated by the traditional FFTA approach, as shown in the fifth column of Table 8. For the traditional approach, it is mainly adopting a simple way (likewise using in [30]) to deal with weight factors and not considering the confidence level and the Tω arithmetic.


## 3.3.1. Evaluation Stage

Each expert is specialized in a specific relevant subfield of neutrinos and participates in the detector project. In accordance with the linguistic scales, judgements of BEs’ occurrence likelihood and their confidence degree are described in Table5. In order to generate more reasonable judgements, an additional step is to combine the confidence coefficient with the initial fuzzy opinion of the experts. For example, with reference to the basic event X1, the transforming process is expressed in Table 6.Table 5
Summary of for all basic events.
EventsProbability levelsConfidence levelsE1E2E3E4E5E6E1E2E3E4E5E6X1LLRLMRLRLHHMHMMMHX2MHRHMMMMMHMHMLX3LRLELMELELMMLHMHMHMX4HRHMHMHHHHMHHHX5MRLRLLRLLMMHMHLMLMHX6LRLRLRLLLMHMMMLMMHX7HMRLRHMRHMMMHHMLMX8LRLLMRLLHHMHMHMMX9RLLLLMLMHHMMHMLMX10RHHRHMRHHHMHHHMHHX11LRLRLLRLLMLMHHHMLHX12RLLMLRLELMHMLMMMHHX13MRHHRHHRHMMMLMHHHX14ELELELELELELMMHMHHMLMX15ELELELELELELMHHMHMMMLX16ELLELELLLMMHMHMMLMX17RHMRHRHMRLHMHMMHHHX18ELELLRLLLMHHMHMMHHX19MMRLLLRLMMMHHMLMX20LRLLMRLLHHMHMHMMX21RLLLLMLMHHMMHMLMX22RHHRHMRHHHMHHHMHHX23LRLRLLRLLMLMHHHMLHX24RLLMLRLELMHMLMMMHHX25MRHHRHHRHMMMLMHHHX26ELELELELLELHMHHMHMHHX27ELRLELLMLMHMHMHMMX28RHMRHMRHMHHMHMHMX29ELLELELLMMHMMMMMHX30ELLLRLRLLHHMHHMMHX31RLELELELMMMHMMHMMHMX32ELELELELRLRLHHMHMHMMX33ELELELELLELHHHMHMHHX34MMLLELELMHMHHHMHMHTable 6
Calculating the fuzzy judgement corresponding toX1.
CodesConfidence linguisticsTrapezoidal numbersZ∗Probability linguisticsTrapezoidal numbersTransformed fuzzy numbersE1H(0.8, 0.9, 1, 1)0.96L(0.1, 0.2, 0.2, 0.3)(0.096, 0.192, 0.192, 0.288)E2H(0.8, 0.9, 1, 1)0.96L(0.1, 0.2, 0.2, 0.3)(0.096, 0.192, 0.192, 0.288)E3MH(0.6, 0.7, 0.7, 0.9)0.86RL(0.2, 0.3, 0.4, 0.5)(0.173, 0.260, 0.346, 0.433)E4M(0.3, 0.5, 0.5, 0.7)0.70M(0.4, 0.5, 0.5, 0.6)(0.283, 0.354, 0.354, 0.424)E5M(0.3, 0.5, 0.5, 0.7)0.70RL(0.2, 0.3, 0.4, 0.5)(0.141, 0.212, 0.283, 0.354)E6MH(0.6, 0.7, 0.7, 0.9)0.86RL(0.2, 0.3, 0.4, 0.5)(0.173, 0.260, 0.346, 0.433)


## 3.3.2. Aggregation Stage and Defuzzification Stage

To obtain the consensus coefficient, the expert’s weight factor is calculated by normalizing the comparison matrix. As a result, the comprehensive weight factors of every expert are calculated, as provided in the second part of Table7. Afterwards, any expert opinion should be considered in the aggregate calculation of the fuzzy likelihood of BE. The aggregated results of BEs are listed in Table 8 (the second column) by employing the Tω arithmetic operations, where the value of the relaxation factor β is regarded as 0.5 [19, 58].Table 7
The results for weight factors of different experts.
Weighting scores of four criteriaCriteriaNormalized scoresProfessional position0.3751Service time0.2083Educational level0.3334Age0.0833Weighting scores of expertEuCodesProfessional position (WPP,Eu)Service time (WST,Eu)Educational level (WEL,Eu)Age (WAG,Eu)W (Eu)E10.23320.20810.20600.22160.2179E20.18630.15880.16460.19630.1742E30.16060.15880.16460.13960.1598E40.18630.18430.20600.13960.1886E50.10110.13110.11710.10660.1131E60.13250.15880.14180.19630.1464Table 8
Evaluation results.
EventsAggregated fuzzy numbersFPSFFPRankaResults of FFP by traditional FFTARankbX1(0.226, 0.245, 0.282, 0.301)0.2645.771E – 04111.088E – 0312X2(0.390, 0.406, 0.422, 0.437)0.4142.613E – 0398.211E – 038X3(0.113, 0.128, 0.176, 0.191)0.1528.378E – 05263.094E – 0425X4(0.616, 0.634, 0.650, 0.669)0.6421.280E – 0211.724E – 021X5(0.198, 0.212, 0.248, 0.263)0.2333.659E – 04151.219E – 0311X6(0.161, 0.177, 0.210, 0.226)0.1931.994E – 04236.831E – 0417X7(0.397, 0.414, 0.456, 0.473)0.4353.096E – 0388.954E – 036X8(0.223, 0.242, 0.268, 0.286)0.2555.165E – 04138.451E – 0413X9(0.177, 0.193, 0.210, 0.226)0.2012.300E – 04215.714E – 0421X10(0.583, 0.601, 0.647, 0.666)0.6241.144E – 0221.563E – 022X11(0.178, 0.195, 0.233, 0.250)0.2142.842E – 04176.190E – 0419X12(0.169, 0.186, 0.230, 0.246)0.2082.567E – 04196.924E – 0415X13(0.480, 0.495, 0.538, 0.553)0.5175.636E – 0341.458E – 024X14(0.000, 0.000, 0.077, 0.094)0.0433.413E – 07335.662E – 0633X15(0.000, 0.000, 0.077, 0.093)0.0423.362E – 07345.662E – 0634X16(0.051, 0.065, 0.106, 0.120)0.0858.698E – 06305.275E – 0530X17(0.439, 0.458, 0.518, 0.536)0.4884.595E – 0377.545E – 039X18(0.105, 0.120, 0.165, 0.182)0.1436.689E – 05271.533E – 0427X19(0.234, 0.251, 0.277, 0.293)0.2645.806E – 04101.661E – 0310X20(0.223, 0.242, 0.268, 0.286)0.2555.165E – 04128.451E – 0414X21(0.177, 0.193, 0.210, 0.226)0.2012.300E – 04205.714E – 0422X22(0.583, 0.601, 0.647, 0.666)0.6241.144E – 0231.563E – 023X23(0.178, 0.195, 0.233, 0.250)0.2142.842E – 04166.190E – 0420X24(0.169, 0.186, 0.230, 0.246)0.2082.567E – 04186.924E – 0416X25(0.480, 0.495, 0.538, 0.553)0.5175.636E – 0351.458E – 025X26(0.012, 0.024, 0.103, 0.122)0.0652.661E – 06321.178E – 0531X27(0.123, 0.138, 0.183, 0.199)0.1611.029E – 04252.883E – 0426X28(0.458, 0.476, 0.522, 0.541)0.4994.985E – 0368.228E – 037X29(0.098, 0.112, 0.152, 0.169)0.1335.089E – 05281.361E – 0428X30(0.152, 0.168, 0.214, 0.232)0.1911.923E – 04243.135E – 0424X31(0.150, 0.166, 0.222, 0.238)0.1942.012E – 04224.734E – 0423X32(0.052, 0.063, 0.148, 0.167)0.1082.229E – 05297.986E – 0529X33(0.012, 0.024, 0.105, 0.123)0.0662.813E – 06311.178E – 0532X34(0.204, 0.221, 0.247, 0.264)0.2343.868E – 04146.748E – 0418a the ranking of FFP by improved FFTA approach. b the ranking of FFP by traditional FFTA approach.To represent the probability occurrence of TE in an intuitionistic way, the aggregated fuzzy numbers of BEs are required to be converted into a precise value; FPS and FFP of all BEs are listed in the third column and fourth column of Table8, respectively. Moreover, in order to highlight the reasonability and applicability of the improved FFTA approach, results of FFP are further calculated by the traditional FFTA approach, as shown in the fifth column of Table 8. For the traditional approach, it is mainly adopting a simple way (likewise using in [30]) to deal with weight factors and not considering the confidence level and the Tω arithmetic.


## 3.4. Further Quantitative Analysis

### 3.4.1. Occurrence Probability of TE

In this study, the calculated failure probability of the detector is 7.4316E – 06 based on the improved FFTA approach, which corresponds to the possible interval of “Medium” (10−6∼10−5) level according to Table 2. However, if using the traditional approach, the failure probability is 3.6171E – 05 which is more than twice the results from the improved FFTA approach and at the level of “RH” (10−5∼10−4). Since a series of protection measures have been taken in the design to reduce the failure probability, the results from the improved FFTA approach are more reasonable and more consistent with the actual situation, while the traditional approach is excessively conservative. It should be addressed that only two layers of PMT are considered in the calculations based on both approaches.However, if FT involves more layers of PMT modules, the failure probability will be further reduced, as clearly shown in Figure7. The horizontal axis is the number of layers of PMTs with protection, in which “0” means that each PMT is baring with no cover and “1” means that only the exploding PMT fit out with the defensive cover; the vertical axis represents the logarithmic probability of failure. Furthermore, two groups of failure probabilities corresponding to different number of layers of PMT have been calculated, including TE probability and the intermediate event “PMTs chain reaction implosion” probability, and they have been plotted in Figure 7 and correspond to black color and green color. For the TE failure probability, the results of the two approaches drop sharply with the number of layers from 0 to 3, while when the number of layers is greater than 4, the curve fluctuates smoothly and the results stabilize at 7.6436E – 07 and 1.1785E – 05, respectively. It can be clearly observed that the level of the improved approach drops to “RL” according to Table 2, while the traditional approach is still at the “RH” level. For “PMTs chain reaction implosion” failure probability, the results of both methods keep decreasing as the number of layers increases, suggesting that the more the PMTs installing protection, the safer the detector. Eventually, even though the evaluated probability of the improved approach is relatively low, it should be attached with sufficient concern because of resulting in postponing the development of neutrino research severely and suffering great losses of economy in case of an accident.Figure 7
Failure probability of TE and the intermediate event “PMTs chain reaction implosion.”


### 3.4.2. Importance Measures and Sensitive Analysis

The results of QBI and QCI for 34 BEs are presented in Table9, and results for the top 17 critical MSCs are listed in Table 10. Subsequently, according to the sorting results of the fuzzy probability, CI, BI, and MCSI, the top 17 BEs and MCS are plotted in Figure 8, which are represented with blue, pink, orange and black histograms, respectively, and the height of the histogram corresponds to the results value.Table 9
Importance measures and RRW of BEs.
EventsQBIQCIQRRWX13.9533E – 043.0703E – 021.0317X23.9613E – 041.3930E – 011.1618X33.9513E – 044.4549E – 031.0045X44.0023E – 046.8977E – 013.2234X53.9524E – 041.9462E – 021.0198X63.9518E – 041.0605E – 021.0107X73.0705E – 041.2793E – 011.1467X83.0626E – 042.1288E – 021.0218X93.0617E – 049.4768E – 031.0096X103.0964E – 044.7665E – 011.9108X113.0619E – 041.1711E – 021.0119X123.0618E – 041.0578E – 021.0107X133.0783E – 042.3348E – 011.3046X149.9999E – 014.5935E – 021.0481X159.9999E – 014.5248E – 021.0474X169.9694E – 031.1669E – 021.0118X172.6285E – 055.3499E – 031.0054X182.6166E – 057.7528E – 051.0001X193.4729E – 042.7133E – 021.0279X203.4726E – 042.4139E – 021.0247X213.4716E – 041.0746E – 021.0109X223.5110E – 045.4047E – 012.1762X233.4718E – 041.3279E – 021.0135X243.4717E – 041.1994E – 021.0121X253.4905E – 042.6474E – 011.3601X262.6164E – 053.0846E – 061.0000X272.6167E – 051.1927E – 041.0001X282.6295E – 055.8058E – 031.0058X292.6166E – 055.8983E – 051.0001X302.6169E – 052.2291E – 041.0002X311.0196E – 089.0853E – 081.0000X324.4091E – 134.3539E – 131.0000X335.2622E – 096.5565E – 101.0000X345.2642E – 099.0197E – 081.0000Table 10
Information of the top seventeen important MSCs.
CodesEvent setsOrderP (MCSi)QMCSIMCS1X4 × X10 × X2231.6762E – 062.2252E – 01MCS2X4 × X13 × X2238.2589E – 071.0964E – 01MCS3X4 × X10 × X2538.2589E – 071.0964E – 01MCS4X4 × X7 × X2234.5370E – 076.0228E – 02MCS5X4 × X13 × X2534.0693E – 075.4019E – 02MCS6X2 × X10 × X2233.4201E – 074.5402E – 02MCS7X1413.4137E – 074.5317E – 02MCS8X1513.3627E – 074.4640E – 02MCS9X4 × X7 × X2532.2354E – 072.9675E – 02MCS10X2 × X13 × X2231.6851E – 072.2370E – 02MCS11X2 × X10 × X2531.6851E – 072.2370E – 02MCS12X2 × X7 × X2239.2570E – 081.2289E – 02MCS13X4 × X10 × X1938.5074E – 081.1294E – 02MCS14X2 × X13 × X2538.3028E – 081.1022E – 02MCS15X4 × X8 × X2237.5691E – 081.0048E – 02MCS16X4 × X10 × X2037.5691E – 081.0048E – 02MCS17X1 × X10 × X2237.5536E – 081.0027E – 02Figure 8
Top17 basic events based on different ranking results. (a) Fuzzy failure probability. (b) Critical importance. (c) Birnbaum importance. (d) Cut set importance.
(a)(b)(c)(d)According to the ranking results of failure probability, Figure8 shows that the BEs X4, X10, X22, X13, X25, X28, and X17 are ranked higher, which means that these events are more likely to cause detector accidents. Even though X28 and X17 belong to the top seven BEs depending on the probability value, these two BEs have little contribution to the occurrence of the detector accident because they are both linked to TE with BE X16 through an “AND” gate. In the FT model, their abovementioned logical relationship weakens the influence of any one of them (e.g., X28, X17, or X16) on TE. Moreover, in terms of sorting the Birnbaum importance, X14, X15, X16, X4, X2, X1, and X5 are the first seven BEs, whereas the forefront of them (e.g., X14 and X15) have little influence on the detector failure due to their extremely low occurrence likelihood. By the comprehensive consideration of the ranking result of QBI, QCI, and QMCSI, it is supposed to observe that the intermediate event “PMTs chain reaction implosion” accounts for a substantial part of the top seventeen BEs. Therefore, the PMT module should receive more attention in the installation and operation of detectors. Finally, a total of six BEs (e.g., X4, X10, X21, X13, X25, and X7) were identified as the most critical BEs.In addition, through the comparison of the ranking results of CI and RRW, it is concluded thatX4 is the most critical basic event, and its RRW value is also the largest among other BEs. Therefore, both ranking results based on the FT model have good consistency, which further verifies the rationality of the proposed approach.Referring to the concluded results, several improvement measures corresponding to the most critical events are summarized as follows:(i)
In terms ofX4: “damage during repair,” strict monitoring technology should be used to identify the impact damage of the photomultiplier tube, and the damaged photomultiplier tube can be quickly replaced when a defect occurs. During the transportation of photomultiplier tubes, flexible soft materials with better adaptability and higher impact resistance should be considered to cover the photomultiplier tubes.(ii)
As forX10 (likewise, the X21): “unsuitable bottom cover opening area,” the total inlet area corresponding to the shockwave threshold that caused the adjacent PMTs to explode needs to be calculated cautiously by a great many numerical and experimental tests. Moreover, it also should improve the bottom cover construction technology and carry out strict sampling inspection of the inflow area.(iii)
As toX13 (likewise, the X25): “strains given at the bottom work,” when the operators work at the bottom of the detector, the thick impact energy-absorption material or special scaffold should be used to prevent residual stress on the top protection cover.(iv)
With regard toX7: “unreasonable connection,” to alleviate the stress concentration that happened near the bolted hole of the top cover, it is advised to adopt six vertical bolt fastenings for clamping the acrylic protection. In addition, a rubber sleeve is sheathed at the end of each bolt and the connection between bolts and the top cover is relatively loose to keep a little margin acting as cushioning.


## 3.4.1. Occurrence Probability of TE

In this study, the calculated failure probability of the detector is 7.4316E – 06 based on the improved FFTA approach, which corresponds to the possible interval of “Medium” (10−6∼10−5) level according to Table 2. However, if using the traditional approach, the failure probability is 3.6171E – 05 which is more than twice the results from the improved FFTA approach and at the level of “RH” (10−5∼10−4). Since a series of protection measures have been taken in the design to reduce the failure probability, the results from the improved FFTA approach are more reasonable and more consistent with the actual situation, while the traditional approach is excessively conservative. It should be addressed that only two layers of PMT are considered in the calculations based on both approaches.However, if FT involves more layers of PMT modules, the failure probability will be further reduced, as clearly shown in Figure7. The horizontal axis is the number of layers of PMTs with protection, in which “0” means that each PMT is baring with no cover and “1” means that only the exploding PMT fit out with the defensive cover; the vertical axis represents the logarithmic probability of failure. Furthermore, two groups of failure probabilities corresponding to different number of layers of PMT have been calculated, including TE probability and the intermediate event “PMTs chain reaction implosion” probability, and they have been plotted in Figure 7 and correspond to black color and green color. For the TE failure probability, the results of the two approaches drop sharply with the number of layers from 0 to 3, while when the number of layers is greater than 4, the curve fluctuates smoothly and the results stabilize at 7.6436E – 07 and 1.1785E – 05, respectively. It can be clearly observed that the level of the improved approach drops to “RL” according to Table 2, while the traditional approach is still at the “RH” level. For “PMTs chain reaction implosion” failure probability, the results of both methods keep decreasing as the number of layers increases, suggesting that the more the PMTs installing protection, the safer the detector. Eventually, even though the evaluated probability of the improved approach is relatively low, it should be attached with sufficient concern because of resulting in postponing the development of neutrino research severely and suffering great losses of economy in case of an accident.Figure 7
Failure probability of TE and the intermediate event “PMTs chain reaction implosion.”


## 3.4.2. Importance Measures and Sensitive Analysis

The results of QBI and QCI for 34 BEs are presented in Table9, and results for the top 17 critical MSCs are listed in Table 10. Subsequently, according to the sorting results of the fuzzy probability, CI, BI, and MCSI, the top 17 BEs and MCS are plotted in Figure 8, which are represented with blue, pink, orange and black histograms, respectively, and the height of the histogram corresponds to the results value.Table 9
Importance measures and RRW of BEs.
EventsQBIQCIQRRWX13.9533E – 043.0703E – 021.0317X23.9613E – 041.3930E – 011.1618X33.9513E – 044.4549E – 031.0045X44.0023E – 046.8977E – 013.2234X53.9524E – 041.9462E – 021.0198X63.9518E – 041.0605E – 021.0107X73.0705E – 041.2793E – 011.1467X83.0626E – 042.1288E – 021.0218X93.0617E – 049.4768E – 031.0096X103.0964E – 044.7665E – 011.9108X113.0619E – 041.1711E – 021.0119X123.0618E – 041.0578E – 021.0107X133.0783E – 042.3348E – 011.3046X149.9999E – 014.5935E – 021.0481X159.9999E – 014.5248E – 021.0474X169.9694E – 031.1669E – 021.0118X172.6285E – 055.3499E – 031.0054X182.6166E – 057.7528E – 051.0001X193.4729E – 042.7133E – 021.0279X203.4726E – 042.4139E – 021.0247X213.4716E – 041.0746E – 021.0109X223.5110E – 045.4047E – 012.1762X233.4718E – 041.3279E – 021.0135X243.4717E – 041.1994E – 021.0121X253.4905E – 042.6474E – 011.3601X262.6164E – 053.0846E – 061.0000X272.6167E – 051.1927E – 041.0001X282.6295E – 055.8058E – 031.0058X292.6166E – 055.8983E – 051.0001X302.6169E – 052.2291E – 041.0002X311.0196E – 089.0853E – 081.0000X324.4091E – 134.3539E – 131.0000X335.2622E – 096.5565E – 101.0000X345.2642E – 099.0197E – 081.0000Table 10
Information of the top seventeen important MSCs.
CodesEvent setsOrderP (MCSi)QMCSIMCS1X4 × X10 × X2231.6762E – 062.2252E – 01MCS2X4 × X13 × X2238.2589E – 071.0964E – 01MCS3X4 × X10 × X2538.2589E – 071.0964E – 01MCS4X4 × X7 × X2234.5370E – 076.0228E – 02MCS5X4 × X13 × X2534.0693E – 075.4019E – 02MCS6X2 × X10 × X2233.4201E – 074.5402E – 02MCS7X1413.4137E – 074.5317E – 02MCS8X1513.3627E – 074.4640E – 02MCS9X4 × X7 × X2532.2354E – 072.9675E – 02MCS10X2 × X13 × X2231.6851E – 072.2370E – 02MCS11X2 × X10 × X2531.6851E – 072.2370E – 02MCS12X2 × X7 × X2239.2570E – 081.2289E – 02MCS13X4 × X10 × X1938.5074E – 081.1294E – 02MCS14X2 × X13 × X2538.3028E – 081.1022E – 02MCS15X4 × X8 × X2237.5691E – 081.0048E – 02MCS16X4 × X10 × X2037.5691E – 081.0048E – 02MCS17X1 × X10 × X2237.5536E – 081.0027E – 02Figure 8
Top17 basic events based on different ranking results. (a) Fuzzy failure probability. (b) Critical importance. (c) Birnbaum importance. (d) Cut set importance.
(a)(b)(c)(d)According to the ranking results of failure probability, Figure8 shows that the BEs X4, X10, X22, X13, X25, X28, and X17 are ranked higher, which means that these events are more likely to cause detector accidents. Even though X28 and X17 belong to the top seven BEs depending on the probability value, these two BEs have little contribution to the occurrence of the detector accident because they are both linked to TE with BE X16 through an “AND” gate. In the FT model, their abovementioned logical relationship weakens the influence of any one of them (e.g., X28, X17, or X16) on TE. Moreover, in terms of sorting the Birnbaum importance, X14, X15, X16, X4, X2, X1, and X5 are the first seven BEs, whereas the forefront of them (e.g., X14 and X15) have little influence on the detector failure due to their extremely low occurrence likelihood. By the comprehensive consideration of the ranking result of QBI, QCI, and QMCSI, it is supposed to observe that the intermediate event “PMTs chain reaction implosion” accounts for a substantial part of the top seventeen BEs. Therefore, the PMT module should receive more attention in the installation and operation of detectors. Finally, a total of six BEs (e.g., X4, X10, X21, X13, X25, and X7) were identified as the most critical BEs.In addition, through the comparison of the ranking results of CI and RRW, it is concluded thatX4 is the most critical basic event, and its RRW value is also the largest among other BEs. Therefore, both ranking results based on the FT model have good consistency, which further verifies the rationality of the proposed approach.Referring to the concluded results, several improvement measures corresponding to the most critical events are summarized as follows:(i)
In terms ofX4: “damage during repair,” strict monitoring technology should be used to identify the impact damage of the photomultiplier tube, and the damaged photomultiplier tube can be quickly replaced when a defect occurs. During the transportation of photomultiplier tubes, flexible soft materials with better adaptability and higher impact resistance should be considered to cover the photomultiplier tubes.(ii)
As forX10 (likewise, the X21): “unsuitable bottom cover opening area,” the total inlet area corresponding to the shockwave threshold that caused the adjacent PMTs to explode needs to be calculated cautiously by a great many numerical and experimental tests. Moreover, it also should improve the bottom cover construction technology and carry out strict sampling inspection of the inflow area.(iii)
As toX13 (likewise, the X25): “strains given at the bottom work,” when the operators work at the bottom of the detector, the thick impact energy-absorption material or special scaffold should be used to prevent residual stress on the top protection cover.(iv)
With regard toX7: “unreasonable connection,” to alleviate the stress concentration that happened near the bolted hole of the top cover, it is advised to adopt six vertical bolt fastenings for clamping the acrylic protection. In addition, a rubber sleeve is sheathed at the end of each bolt and the connection between bolts and the top cover is relatively loose to keep a little margin acting as cushioning.


## 4. Further Discussions

The FFP can be calculated by both FFTA and traditional approach, respectively, and their results are represented by 34 circle and 34 square dots accordingly in Figure9. Graphically, all the circle dots are below the square dots, and this indicates that the FFP values obtained by the improved FFTA approach are all smaller than those results from traditional approach. In addition, from another perspective, the square dots mean relatively larger FFP calculated by the traditional method. This should be the essential reason that results in conservative TE probability, as is consistent with the discussion in Section 3.4.1. Although the variation trends of the ranking sequences given by the two approaches are basically consistent, the difference in FFP values between them can reach more than 75% for some BEs, such as X14, X15, X26, and X33. Furthermore, for those events with higher FFP values, the results are closer to those obtained from traditional approaches. The influences of the Tω algorithm, AHP, and confidence coefficient on the difference between the two methods are discussed below.Figure 9
Comparison between the proposed method and traditional method.One of the reasons why there are differences between both approaches may be due to theTω arithmetic operator in the improved approach. Specifically, this operator helps alleviate the continuous accumulation of ambiguity and significantly enhances the authenticity of the estimation results by improving the algorithm in the aggregation calculation. Take BE X1 as an example and check its membership function μ(x) based on both approaches, as shown in Figure 10. Obviously, the blue area obtained from Tω method is surrounded by the yellow area obtained from the traditional method, and this means a smaller fuzzy interval. Therefore, the improved approach could generate relatively reliable failure probability of TE.Figure 10
The aggregated results between the improved FFTA and traditional approach.In addition, the main reason causing such differences shown in Figure9 should be the application of AHP and confidence level. To count the quantitative judgement more accurately on the failure probability of BEs, the former approach fully considers the experience and knowledge of different experts, and the latter enhances the credibility of expert opinions. In addition, Figure 11 shows a blue curve based on the Onisawa transformed data and 34 circle dots which represent various BEs. Furthermore, referring to the fuzzy mathematic theory of α-cut set, elements would belong to a fuzzy interval with a greater membership degree under the condition of α ∈ [0.5, 1]. Therefore, when the membership value is over 0.5, the interval [0, 0.15] is determined as the “EL” level of FPS. To describe the event distribution more intuitively, it also shows the range of 7 linguistic terms on the FPS interval [0,1] according to the 7th linguistic scale of judgements. It is obviously observed that if the FPS value is in the “EL” level or “EH” level, the slope of the curve increases significantly. For these intervals, changing expert weights and adding confidence coefficients may have a relatively significant impact on the probability values of BE (e.g., X14, X15, X26, and X33) in the aggregation of expert fuzzy opinions. As a result, experts should pay more attention to their opinions close to the level of “EL” and “EH” because small changes of judgements may lead to huge fluctuation of FFP.Figure 11
Schematics of Onisawa converting method and events distribution (EL–Extremely Low; L–Low; RL–Relatively Low; M–Medium; RH–Relatively High; H–High; EH–Extremely High).


## 5. Conclusions

In this paper, the FFTA approach has been integrated with AHP technology, the concept of the confidence level, andTω operators to perform risk assessment of the neutrino detector failure. The improved FFTA approach provides an option to perform the effective quantity risk evaluation of neutrino detector with very little measured data. Compared with the results from the traditional approach, the occurrence probability of detector failure is relative smaller. This is more reasonable and more consistent with the reality that several important protective measures have been applied in the project. The Tω aggregated algorithms are adopted to reduce the vagueness accumulation. In addition, the comparison of aggregated fuzzy numbers shows that the Tω algorithm can result in a less range of left/right interval, which means an improved accuracy. PMT module is the most critical device of the detector and a series of measures have been proposed to improve the performance of vulnerable components. Due to the relatively small number of neutrino detectors built worldwide and the lack of sufficient failure data for detector evaluation, it is very necessary to build such a large-scale neutrino detector and continuously develop its related database in the future.



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*Source: 5104612-2022-04-22.xml*